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+{"text":"\\section{Introduction and background}\n\n\nIn this paper, we seek to apply Stein's method -- a technique for obtaining convergence (often CLT-type) results for random variables -- on a vertex-count in the Neighborhood Attack Voter-type model.\n\nVoter models are interacting-particle-system models on finite graphs. The original Voter model (introduced independently in the 1970s by Clifford and Sudbury in 1973, and by Holley and Liggett in 1975, as mentioned in \\cite{Liggett2005}) can be formulated as follows: Take a connected, $r$-regular (each vertex has $r$ edges) graph of size $n$. Assign $1$'s and $-1$'s to the nodes of the graph. Run a Markov chain on the graph with the following transition procedure: each turn, pick a node at random (under some distribution; usually we take the uniform), pick one of its neighbors at random (usually uniformly), and switch the value of the selected neighbor-node to the value of the originally selected node. Under uniformity of node and neighbor selection, this chain converges to one of two absorbing states, in which all nodes have the same values.\n\nThe ``Anti-voter'' model, introduced in \\cite{Matloff1977}, has the selected neighbour node adopt a value opposite to that of the originally selected node. Under uniformity (again, of node and neighbor selection) the resulting chain has a stationary distribution.\n\nPersi Diaconis and Christos Athanasiadis in \\cite{AthaDiac2010} proposed the following variation of the Voter model: upon selecting a node, instead of picking one of its neighbors, flip a coin (with weight $p$, perhaps taken to be a half), and, according to the result of the cointoss, assign either $1$ or $-1$ to the selected nodes and all its neighbors. The model has been labeled the ``Neighborhood Attack'' model.\n\nStein's method (first introduced in \\cite{Stein1972}) provides an infrastructure for the estimation of the distances between certain classes of random variables and certain (usually classical) distributions, most notably the Gaussian and the Poisson distributions. For practical purposes, we can break Stein's method into three key steps: First, one has to use Stein's identities to establish a bound on the distance between a class of random variables and a specific distribution expected to be close to the given class; second, one has to satisfy the conditions generated in the preceding step; and third, one has to evaluate the acquired bound. The last step typically involves something along the lines of reducing an expression involving a function of the variance of the given random variable.\n\nIn \\cite{RR1997}, Yosef Rinott and Vladimir Rotar show, using a Stein's method argument, that the sum of the values of the nodes in the Anti-voter model at stationarity is asymptotically normally distributed. The problem Rinnott and Rotar tackled was posed by Aldous and Fill in a book that touches on Voter models, \\cite{AldousFillDraftbook}. Our goal in the present article is to show that the sum of the values of the nodes in the Neighborhood Attack model is asymptotically normally distributed, using Stein's method techniques different from the ones employed by Rinott and Rotar.\n\n\nFor an application of the Stein technique in a different context, see the paper \\cite{Fulman2004}, in which Jason Fulman shows that the number of descents or inversions in permutations complies to a central limit theorem. Both the current problem and the one examined in \\cite{Fulman2004} can be viewed as random walks on hyperplanes; and hence there is a structural similarity between the approach adopted here, and the one in \\cite{Fulman2004}.\n\nFor more results on the Neighbourhood Attack model, see \\cite{AthaDiac2010}, \\cite{ChungGra2012}. The former paper introduces the model and presents some results on random walks on hyperplane arrangements. The latter paper studies some properties of the distributions of the implicit Markov chains in models similar to the Neighbourhood Attack model.\n\nFor more on Stein's method, see \\cite{ChenGoldShao2011}, \\cite{Barbour1992}, \\cite{Ross2011}. The first two books provide a comprehensive overview of Stein's method in regard to its applications to Normal and Poisson approximations reflexively. The monograph \\cite{Ross2011} is an up-to-date survey of Stein's method literature and a useful entry-level source on the subject.\n\n\nIn Section \\ref{SectionProblemApproach}, we pose our problem. In Section \\ref{SectionSM}, we conduct a brief overview of our main technique: Stein's method. In Section \\ref{SectionResults}, we introduce a few definitions and assumptions, and then list the main result of the paper. In the Section \\ref{SectionProof}, we provide calculations and proofs for the result. Section \\ref{SectionConsequences} interprets the result with some examples of its applicability. We draw conclusions in Section \\ref{SectionConclusions}.\n\n\\section{Problem and Approach}\\label{SectionProblemApproach}\n \nWe apply the Neighbourhood Attack model (introduced in \\cite{AthaDiac2010}) on a given family of (finite) graphs. Randomly assign either $1$ or $-1$ to each node of the graph. As mentioned above, the model does the following each turn:\n\\begin{itemize}\n\\item Selects a node uniformly at random.\n\\item Turns the node and all its immediate neighbours into $1$'s or $-1$'s according to a Bernoulli($p$) distribution with $0<p<1$; we want $p=1\/2$ for the sake of symmetry.\n\\end{itemize}\nGiven: 1) a connected graph; 2) positive probability of selection for all nodes; and 3) positive probabilities of turning into $1$ or $-1$ for the selected node and its neighbours, the underlying Markov chain, the states of which are the possible permutations of $1$'s and $-1$'s, is irreducible and everywhere recurrent on an essential class of its state space, and therefore possesses a stationary distribution. Assume the considered Markov chain begins at this stationary distribution.\n\nLet $X$ be the number of $1$'s at stationarity. Then $N-X$ equals the number of $-1$'s, where $N$ is the number of nodes.\n\nWe want to use Stein's method to show that\n\\begin{equation*}\n\\frac{X - \\mathbb{E} X}{\\sigma_{X}} \\xrightarrow[N\\rightarrow\\infty]{} \\mathcal{Z}\n\\end{equation*}\nwhere $\\mathcal{Z}$ is of the standard normal distribution, and $\\mathbb{E}X$ and $\\sigma_{X}$ are the expectation and standard deviation of $X$.\n\nWe derive our result under an assumption of $r$-regularity for the underlying graphs.\n\nWe seek to apply Stein's method, and in particular we want to use a result along the lines of Theorem 1.2 in \\cite{RR1997}:\n\\begin{theorem}\\label{Theorem1}\nLet $(W,W')$ be exchangeable with $\\mathbb{E}W = 0$ and $\\mathbb{E}W^{2} = 1$. Define the r.v. $R=R(W)$ by\n\\begin{equation}\\label{SteinPairEq}\n\\mathbb{E}(W'|W) = (1-\\lambda)W + R,\n\\end{equation}\nwhere $0<\\lambda<1$. Then, if there is some $A$ for which $|W'-W|\\leq A$, we have \n\\begin{align*}\n\\delta := & \\sup\\{ |\\mathbb{E}h(W) - \\Phi h|: h\\in \\mathcal{H}\\} \\leq \\\\\n& \\leq \\frac{12}{\\lambda} \\sqrt{\\mathop{\\rm Var}\\{ \\mathbb{E} \\left[ (W'-W)^{2} | W \\right] \\}} + 37 \\frac{\\sqrt{\\mathbb{E}R^{2}}}{\\lambda} + 48\\frac{aA^{3}}{\\lambda} + 8\\frac{aA^{2}}{\\sqrt{\\lambda}},\n\\end{align*}\nwhere $\\mathcal{H}$ is such that all functions in it are uniformly bounded in absolute value by 1, for any real numbers $c$ and $d$ and any $h\\in \\mathcal{H}$, the function $h(cx+d)$ is in $\\mathcal{H}$, and for any $\\epsilon > 0$ and any $h\\in\\mathcal{H}$, the functions $h_{\\epsilon}^{+}, h_{\\epsilon}^{-}$ are also in $\\mathcal{H}$, and \n\\begin{equation*}\n\\int \\tilde{h}(x; \\epsilon) \\Phi(dx) \\leq a\\epsilon\n\\end{equation*}\nfor some constant $a$ which depends only on the class $\\mathcal{H}$.\n\\end{theorem}\n\nOur $W$ would be some normalization of a vertex-count on the Voter-type model graphs we deal with.\n\n\\section{Brief overview of Stein's method}\\label{SectionSM}\n\nStein's technique goes as follows: for a given probability distribution, one can come up with an appropriate operator which implicitly defines the distribution. For example, the operator $A$ in $Af(x)=f'(x)-xf(x)$ implicitly defines the Gaussian distribution, in the sense that 1) $\\mathbb{E} Af(Z)=0$ for all absolutely continuous $f$ with $\\mathbb{E}|f'(Z)|<\\infty$, where $Z$ is a variable with the standard normal distribution; and 2) if for some random variable $W$ we have $\\mathbb{E}Af(W)=0$ for all absolutely continuous functions $f$ with $|f'|<\\infty$, then $W$ has the standard normal distribution.\n\nNext, for an appropriately chosen $A$, one can solve the differential equation given by \n\\begin{equation}\\label{eq:SteinDiffEq}\nAf(x) = 1_{w\\leq x}-\\Phi(x),\n\\end{equation} \nwhere $\\Phi(x)$ is the c.d.f. of the target distribution.\n\nBut now, armed with the solution to equation (\\ref{eq:SteinDiffEq}), and within the context of an appropriate metric (above we used the Kolomogorov metric), we can produce a bound on the distance $|P(W\\leq x)-\\Phi(x)|$ between a given distribution we want to analyze, and the target distribution with c.d.f. $\\Phi(x)$.\n\nFor example,\n\\begin{equation*}\nf_{x}(w) = e^{w^{2}\/2}\\int_{w}^{\\infty}e^{-t^{2}\/2} \\left( \\Phi(x)-1_{t\\leq x} \\right) dt\n\\end{equation*}\nis the unique bounded solution to\n\\begin{equation*}\nf'_{x}(w)-wf_{x}(w)=1_{w\\leq x}-\\Phi(x),\n\\end{equation*}\nwhere $\\Phi(x)$ is the c.d.f. of the standard normal. And next, under the Wasserstein metric given by $\\mathcal{H}=\\{h: \\mathbb{R}\\rightarrow \\mathbb{R}: |h(x)-h(y)|\\leq |x-y|\\}$, one can show that (for example, see \\cite[3.1]{Ross2011}) \n\\begin{equation*}\nd_{W}(W,Z)\\leq (A+B)n^{-1\/2}, \\quad A=\\mathbb{E}|X_{1}|^{3}, \\quad B=\\frac{\\sqrt{2\\mathbb{E}[X_{1}^{4}]}}{\\sqrt{\\pi}},\n\\end{equation*}\nwhere $W$ is a normalized sum of $n$ i.i.d. standard normal variables endowed with a fourth moment, and $d_{W}(W,Z)$ stands for the Wasserstein distance between $W$ and the standard normal distribution.\n\nThe potential utility of Stein's technique in producing powerful bounds and obtaining convergence results is clear; and, indeed, Stein's method has been instrumental in the proofs of a variety of interesting convergence and bounding results. In general, there are two standard avenues of research focusing on Stein's method -- one can try to obtain formulas for bounds on the distances between various target distributions and various random variables (or rather, their distributions) -- examples of recent results in this direction include \\cite{FulmanRoss2012} (Exponential distribution), \\cite{PikeRen2012} (Laplace), and \\cite{GoldIslak2013} (zero-bias couplings and concentration inequalities); and one can use these formulas and techniques to obtain results pertaining to specific problems, including many classic problems such as the Birthday Problem or the Coupon Collector Problem -- for examples, refer to \\cite{ChatDiacMeck2005} (comprehensive survey) and \\cite{GoldZhang2011} (Lightbulb process).\n\n\\section{Initial setup and main result}\\label{SectionResults}\n\n\\subsection{Initial setup} \n\nWe first seek to show that (\\ref{SteinPairEq}) holds. To that end, let $X$ be the number of 1's at stationarity. Let $$Y = 2X - N = \\sum_{i=1}^{N} \\xi_{i}.$$\nHere $N$ is the total number of nodes and $\\xi_{i}$ is the value of node $i$ (under an arbitrary indexing). Examining $Y$ is equivalent to examining $X$. Next, define\n\\begin{equation*}\nW:=\\frac{Y-\\mathbb{E}Y} {\\sigma_{Y}}.\n\\end{equation*}\nNote $\\sigma_{Y}$ is a constant dependent on $N$:\n\\begin{equation*}\n\\sigma^{2}_{Y}=\\mathop{\\rm Var}\\sum_{i=1}^{N}\\xi_{i}=\\sum_{i=1}^{N} \\mathop{\\rm Var}\\xi_{i}+2\\sum_{1\\leq i<j\\leq N} \\mathop{\\rm Cov}(\\xi_{i},\\xi_{j})\n\\end{equation*}\nNow, $W$ is mean-0 variance-1. To get the condition for Theorem \\ref{Theorem1}, we first need to define a $W'$ as the equivalent of $W$ after one further turn of the Neighborhood Attack model. That is to say, if $W$ is the normalized node count of the model at some turn of its evolution in stationarity, then $W'$ is the same normalized node count in the next turn.\n\nNote, once again, that we assume $r$-regularity for the graph (i.e. every node has exactly $r$ neighbours). We want $r$-regularity for the sake of symmetry, because without symmetry, the problem under consideration is far less tractable.\n\n\n\\subsection{Main result}\n\n\\begin{theorem}\\label{MainTheorem}\nUnder the assumptions \n\\begin{equation} \\label{rstarAssumption}\nr^{*}_{i} = r^{*} \\hspace{1cm} \\forall i \\in I,\n\\end{equation} \nwhere $I$ is the index-set of the nodes, $r^{*}_{i}$ is the number of first or second order neighbors node $i$ has, and $r^{*}$ is some constant dependent on the graph; \nand \n\\begin{equation}\\label{CovCondition}\n\\mathop{\\rm Cov}(\\eta,\\theta) \\leq 0,\n\\end{equation} \nwhere $\\eta$ is the count of pairs of neighbors or near-neighbors with values both equal to 1, and $\\theta$ is the count of pairs of $-1$'s,\nwe derive the bound on the distance between (the distributions of) $W$ and the standard normal,\n\\begin{equation}\\label{BoundFINAL}\n\\delta \\leq 48 \\frac{r^{2}}{\\sqrt{r+1}\\sqrt{N}} + (2^{19\/2}+48)\\frac{\\sqrt{r+1}}{\\sqrt{N}}.\n\\end{equation}\n\\end{theorem}\nWe first establish bounds on $\\sigma_{Y}$ in Section \\ref{SectionVarianceY}, and then complete the proof in Section \\ref{SectionBigVar1}.\n\n\n\\section{Details and proof}\\label{SectionProof}\n\n\\subsection{Proving $\\mathbb{E}(W'|W)=(1-\\lambda)W$}\\label{SectionLambda}\n\nGiven the $r$-regularity assumption, the sum $\\left( \\sum_{i=1}^{N} \\xi_{i} \\right)$ changes each turn by between $-2(r+1)$ and $2(r+1)$. A basic example of a graph of this type is the circle (2-regular) graph, in which we have a set of nodes arranged in a circle, each node with two neighbours.\n\nWe also assumed uniformity in choosing nodes and in flipping $1$'s or $-1$'s. Under such conditions $\\mathbb{E} \\xi_{i}=0$, i.e. at stationarity each node is $1$ or $-1$ with equal probability. The sum of the node values will tend toward 0 (under certain conditions; one of which, clearly, has to do with the number of neighbors each node has, since our model takes only the extreme values over the complete graph), since if nodes of a certain value ($+1$ or $-1$) dominate the graph, we are less likely to see an increase in the number of the nodes of that value.\n\nWe show in Section \\ref{SectionVarianceY} that, as desired,\n\\begin{equation*}\n\\mathbb{E}(W'|W) = \\left( 1-\\frac{r+1}{N} \\right) W,\n\\end{equation*}\nwhich complies with the Stein linearity condition (\\ref{SteinPairEq})\n\\begin{equation*}\n\\mathbb{E}(W'|W) = \\left( 1-\\lambda \\right) W + R,\n\\end{equation*}\nwhere $0\\leq\\lambda\\leq 1$ and $R$ is a random variable. In our case conveniently $R=0$. As for lambda,\n\\begin{equation*}\n\\lambda=\\frac{r+1}{N}.\n\\end{equation*}\n\n\\subsection{R\\\"{o}llin's result} \nIn general, the next step is to show that $W$ and $W'$ are exchangeable, i.e. $(W,W')=^{d}(W',W)$, as was done in \\cite{Fulman2004}. Exchangeability clearly holds when the Markov chain underlying $W$ and $W'$ is stationary and reversible. Reversibility is not always available or easily proved. \n\nFor example, our chain is clearly not necessarily reversible. Consider the circle graph. It is easy to see that for $N$ large, $Y$ can take the value $N-2$ -- i.e. there is an attainable at stationarity arrangement of values for the nodes in which all nodes but one have the value of 1. Now, the probability of going from that arrangement to the all 1's arrangement for which $Y=N$ is positive; but the probability of going from $Y=N$ to $Y=N-2$ is zero, and hence our chain fails to satisfy the detailed balance equations $\\pi(x)P(x,y) = \\pi(y)P(y,x)$.\n\n\nHowever, a recent result by Adrian R\\\"{o}llin removes the necessity for exchangeability. R\\\"{o}llin's theorem  (see \\cite[Theorem 2.1]{Rollin2008}) states:\n\\begin{theorem}\\label{Rollin}  Assume $W,W'$ are r.v.s on the same probability space, s.t. $\\mathcal{L}(W')=\\mathcal{L}(W)$ ($\\mathcal{L}$ for 'law'), $\\mathbb{E}W=0$, $\\mathop{\\rm Var}(W)=1$. Given $\\mathbb{E}(W'|W)=(1-\\lambda)W+R$, for\n\\begin{equation*} \n\\delta:=\\sup_{h\\in \\mathcal{H}}|\\mathbb{E}h(W) - \\mathbb{E}h( \\mathcal{Z} )|\n\\end{equation*}\n(here $\\mathcal{Z}$ is the standard normal distribution, and $\\mathcal{H}$ is the family of functions associated with the Wasserstein distance), we have\n\\begin{equation*}\n\\delta\\leq \\frac{6}{\\lambda}\\sqrt{\\mathop{\\rm Var} \\mathbb{E}^{W}(W'-W)^{2}}+\\frac{19\\sqrt{\\mathbb{E}R^{2}}}{\\lambda} + 4\\sqrt{\\frac{a \\mathbb{E}|W'-W|^{3}}{\\lambda}}.\n\\end{equation*}\nIf also there exists a constant $A$ s.t. $|W'-W|\\leq A$ a.s., we have\n\\begin{equation*}\n\\delta\\leq \\frac{12}{\\lambda}\\sqrt{\\mathop{\\rm Var} \\mathbb{E}^{W}(W'-W)^{2}}+\\frac{37\\sqrt{\\mathbb{E} R^{2}}}{\\lambda} + 32\\frac{A^{3}}{\\lambda}+6\\frac{A^{2}}{\\sqrt{\\lambda}}.\n\\end{equation*}\n\\end{theorem}\n\\begin{proof}\nSee \\cite[Theorem 2.1]{Rollin2008}.\n\\end{proof}\nIn our case $R=0$; so the bound is\n\\begin{equation}\\label{RollinBoundA}\n\\delta\\leq \\frac{12}{\\lambda}\\sqrt{\\mathop{\\rm Var} \\mathbb{E}[(W'-W)^{2}|W]} + 32\\frac{A^{3}}{\\lambda} + 6\\frac{A^{2}}{\\sqrt{\\lambda}}.\n\\end{equation}\nThe next step is to bound $|W'-W|$. Note $|Y'-Y|\\leq 2(r+1)$. So $$|W'-W|\\leq\\frac{2(r+1)}{\\sigma_{Y}}=A.$$\nThus, the bound becomes\n\\begin{equation}\\label{BigBound}\n\\delta\\leq \\frac{12N}{(r+1)\\sigma_{Y}^{2}}\\sqrt{\\mathop{\\rm Var} \\mathbb{E}[(Y'-Y)^{2}|Y]} + 32\\frac{8(r+1)^{2}N}{\\sigma_{Y}^{3}}  + 6\\frac{4(r+1)^{3\/2}\\sqrt{N}}{\\sigma_{Y}^{2}}.\n\\end{equation}\n\n\\subsection{Bounding the variance of $Y$}\\label{SectionVarianceY}\nIn effect, the next goal is to bound the two terms $\\mathop{\\rm Var} \\left[ \\mathbb{E}(Y'-Y)^{2} | Y \\right]$ and $\\mathop{\\rm Var}(Y') = \\mathop{\\rm Var}(Y) =  \\sigma_{Y}^{2}$. For an explicit formula for the value of the corresponding $\\sigma_{Y}$ in the Anti-voter model, see \\cite[Chapter 14]{AldousFillDraftbook}. Another relevant paper dealing with the Anti-voter case can be found in \\cite{DonnellyWelsh1984}.\n\nLet us first try to find $\\mathop{\\rm Var}{Y}$. Define $\\Delta Y = Y' - Y$ to obtain $Y' = Y + \\Delta Y$.\n\nBy stationarity, it follows that $0 = \\mathop{\\rm Var}(Y')-\\mathop{\\rm Var}(Y).$ Hence:\n\\begin{align}\n\\nonumber 0 &= \\mathbb{E} (Y')^{2} - \\mathbb{E} Y^{2} = \\\\\n\\nonumber &= \\mathbb{E} \\left[ Y^{2} + 2Y\\Delta Y + (\\Delta Y)^{2} - Y^{2} \\right] = \\\\\n&= 2\\mathbb{E} (Y\\Delta Y) + \\mathbb{E} (\\Delta Y)^{2}. \\label{VarY}\n\\end{align}\n\nTo continue, we need to obtain a firmer grip on the r.v. $\\Delta Y$. It is easy to see that $\\Delta Y$ takes values between $-2(r+1)$ and $2(r+1)$, and that the probability distribution of $\\Delta Y$ is a function of certain edge and vertex counts on the coloured graph, themselves random variables. Specifically:\n\nDefine $q_{i}$ as the number of nodes s.t. the sum of the values at the node and all its neighbors equals $i$. Clearly, $i$ takes integer values (all odd or all even depending on the parity of $r$) between $-(r+1)$ and $(r+1)$. Specifically, if $r$ is odd, $i$ takes the values $-(r+1), -(r-1), ..., -2, 0, 2, ..., (r-1), (r+1)$; and if $r$ is even, $i$ takes the values $-(r+1), -(r-1),..., -1, 1, ..., (r-1), (r+1)$. In each case $i$ takes $(r+2)$ distinct values. Call the set of those values (the possible values $i$ can take) $I$.\n\nSimple counting produces two useful identities involving the $q_{i}$'s: \n\\begin{equation*}\n\\sum_{i\\in I} q_{i} = N  \\text{  and  } \\sum_{i\\in I} iq_{i}= (r+1)Y.\n\\end{equation*}\n\nNow, at each turn of the Neighborhood Attack process we pick a node uniformly at random (i.e. with probability $1\/N$), and turn its value and the value of all its neighbors to either $1$ or $-1$ uniformly at random (i.e. with probability $1\/2$).\n\nIt thus follows that $\\Delta Y$ has the (conditional on $\\{q_{i}\\}$) p.d.f.:\n\\begin{equation*}\n\\Delta Y = \\begin{cases}\n  (r+1)-i & \\text{ with probability } q_{i}\\frac{1}{2N} \\\\\n  -(r+1)-i & \\text{ with probability } q_{i}\\frac{1}{2N} \\\\\n\\end{cases}\n\\end{equation*}\nSo for example, $\\Delta Y$ takes the value of $2(r+1) = (r+1) - (-(r+1))$ with probability $q_{r+1}\/2N$; and the value $-2(r+1) = -(r+1) - (r+1)$ with probability $q_{-(r+1)}\/2N$.\n\nThus, we have \n\\begin{align*}\n\\mathbb{E} \\left[ \\Delta Y | Y \\right] &= \\mathbb{E}\\mathbb{E} \\left[ \\left( \\Delta Y | \\{ q_{i} \\} \\right) | Y \\right] =\\\\\n&= \\mathbb{E} \\left( \\sum_{i\\in I} \\left[ (r+1)-i \\right] \\frac{q_{i}}{2N} + \\sum_{i\\in I} \\left[ -(r+1)-i \\right] \\frac{q_{i}}{2N} | Y \\right) = \\\\\n&= -2\\mathbb{E} \\left( \\sum_{i\\in I}i\\frac{q_{i}}{2N} |Y \\right) = -\\frac{(r+1)Y}{N},\n\\end{align*}\nwhich is what we stated in Section \\ref{SectionLambda}. So:\n\\begin{lemma}\nIn the Neighborhood Attack model on an $r$-regular graph,\n\\begin{equation*}\n\\mathbb{E} \\left[ \\Delta Y | Y \\right] = -\\frac{(r+1)Y}{N}.\n\\end{equation*}\nThat is, the Stein linearity condition is satisfied with $\\lambda = \\frac{(r+1)}{N}$ and $R=0$.\n\\end{lemma}\n\nWe continue from (\\ref{VarY}):\n\\begin{align}\n\\nonumber 0 &= \\mathbb{E} (Y')^{2} - \\mathbb{E} Y^{2} = \\\\\n\\nonumber &= 2\\mathbb{E} (Y \\Delta Y) + \\mathbb{E} (\\Delta Y)^{2} = \\\\\n&= -\\frac{2(r+1)}{N}\\mathbb{E} (Y^{2}) + \\mathbb{E}\\mathbb{E} \\left( \\sum_{i\\in I} \\left[ (r+1-i)^{2}+(-(r+1)-i)^{2} \\right] \\frac{q_{i}}{2N} | \\{q_{i}\\} \\right) \\label{VarY2}\n\\end{align}\nLet us focus on the $\\mathbb{E}\\left( \\sum_{i\\in I} \\left[ (r+1-i)^{2}+(-(r+1)-i)^{2} \\right] \\frac{q_{i}}{2N} | \\{q_{i}\\} \\right)$ term:\n\\begin{align}\n\\nonumber & \\mathbb{E} \\left( \\sum_{i\\in I} \\left[ (r+1-i)^{2}+(-(r+1)-i)^{2} \\right] \\frac{q_{i}}{2N} | \\{q_{i}\\} \\right) = \\\\\n&= (r+1)^{2} + \\frac{1}{N} \\left[ \\sum_{i\\in I}i^{2}q_{i} \\right] \\leq \\label{DeltaSquare} \\\\\n\\nonumber &\\leq (r+1)^{2} + \\frac{1}{N} \\left[ \\sum_{i\\in I}(r+1)^{2}q_{i} \\right] = (r+1)^{2} + (r+1)^{2} = 2(r+1)^{2}\n\\end{align}\nAnd therefore, continuing from (\\ref{VarY2}),\n\\begin{align*}\n0 &= \\mathbb{E} (Y')^{2} - \\mathbb{E} Y^{2} = \\\\\n&= -\\frac{2(r+1)}{N}\\mathbb{E} (Y^{2}) + \\mathbb{E}\\mathbb{E} \\left( \\sum_{i\\in I} \\left[ (r+1-i)^{2}+(-(r+1)-i)^{2} \\right] \\frac{q_{i}}{2N} | \\{q_{i}\\} \\right) \\leq \\\\\n&\\leq -\\frac{2(r+1)}{N}\\mathbb{E} (Y^{2}) + 2(r+1)^{2},\n\\end{align*}\nmeaning\n\\begin{align*}\n\\sigma_{Y}^{2} = \\mathop{\\rm Var}(Y) = \\mathbb{E} (Y^{2}) \\leq (r+1)N.\n\\end{align*}\n\n\nHowever, since the $\\sigma_{Y}$ terms appear in the denominators of the terms in (\\ref{BigBound}), we need either a lower bound of $\\sigma_{Y}$ or the exact variance of $Y$.\n\nObserve that we have:\n\\begin{align*}\n\\mathop{\\rm Var}(Y) &= \\mathbb{E}Y^{2} = \\frac{N}{2(r+1)} \\mathbb{E}\\mathbb{E} \\left( \\sum_{i\\in I} \\left[ (r+1-i)^{2}+(-(r+1)-i)^{2} \\right] \\frac{q_{i}}{2N} | \\{q_{i}\\} \\right) =\\\\\n&= \\frac{N}{2(r+1)} \\left[ (r+1)^{2} + \\frac{1}{N}\\mathbb{E} \\left( \\sum_{i\\in I} i^{2}q_{i} \\right) \\right] \\geq \\frac{(r+1)N}{2}.\n\\end{align*}\nThus\n\\begin{lemma}\nFor the Neighborhood Attack model on an $r$-regular graph, for $Y=\\sum_{i=1}^{N} \\xi_{i}$ the sum of the values of the nodes of the graph,\n\\begin{equation*}\n\\frac{(r+1)N}{2} \\leq \\sigma_{Y}^{2} \\leq (r+1)N.\n\\end{equation*}\n\\end{lemma}\n\n\\subsection{Reducing and bounding $\\mathop{\\rm Var} \\mathbb{E}[(Y'-Y)^{2}|Y]$}\\label{SectionBigVar1}\nNow we have to evaluate or bound $\\mathop{\\rm Var} \\mathbb{E}[(Y'-Y)^{2}|Y] = \\mathop{\\rm Var} \\mathbb{E}^{Y}[(Y'-Y)^{2}]$, as in \\cite{RR1997} and \\cite{Fulman2004}.\n\nLet us consider the following:\n\\begin{align*}\n\\mathop{\\rm Var} &\\left[ \\mathbb{E}(Y'-Y)^{2} | Y \\right] = \\mathop{\\rm Var} \\left[ \\mathbb{E}(\\Delta Y)^{2}| Y \\right] \\leq \\mathop{\\rm Var} \\left[ \\mathbb{E}(\\Delta Y)^{2}| \\{q_{i}\\} \\right] =\\\\\n& = \\mathop{\\rm Var} \\left( (r+1)^{2} + \\frac{1}{N} \\left[ \\sum_{i\\in I}i^{2}q_{i}  \\right] \\right) = \\frac{1}{N^{2}}\\mathop{\\rm Var} \\left( \\sum_{i\\in I}i^{2}q_{i} \\right)\n\\end{align*}\nThe transition between the lines follows from (\\ref{DeltaSquare}) and (\\ref{VarY2}). Also,\n\\begin{equation*}\n\\mathop{\\rm Var} \\left[ \\mathbb{E}(Y'-Y)^{2} | Y \\right] = \\frac{1}{N^{2}}\\mathop{\\rm Var} \\left( \\mathbb{E} \\left( \\sum_{i\\in I}i^{2}q_{i} \\right) | \\{ \\xi_{i} \\} \\right) = \\frac{1}{N^{2}} \\mathop{\\rm Var}\\left( \\sum_{k=1}^{N} \\left( \\sum_{j\\in \\mathcal{N}_{k}}\\xi_{j} \\right)^{2} \\right),\n\\end{equation*}\nwhere $\\mathcal{N}_{k}$ is the set of node $k$ and all its neighbors.\n\nNext,\n\\begin{align*}\n& \\mathop{\\rm Var} \\left( \\sum_{k=1}^{N} \\left( \\sum_{j\\in N_{k}}\\xi_{j} \\right)^{2} \\right) = \\\\\n&= \\mathop{\\rm Var} \\left( N(r+1) + \\sum_{0\\leq i,j \\leq N; i,j: d(i,j)=1,2} \\xi_{i}\\xi_{j} \\right) = \\\\\n&= \\mathop{\\rm Var} \\left( 2(\\alpha - \\beta) \\right) = \\mathop{\\rm Var} \\left( 4(\\eta + \\theta) \\right).\n\\end{align*}\nHere $d(i,j)$ is the distance between nodes $i$ and $j$. For the last line, observe that $N(r+1)$ is invariant, and that the sum $\\sum_{0\\leq i,j \\leq N; i,j: d(i,j)=1,2} \\xi_{i}\\xi_{j}$ can be interpreted as a sort of an edge count over our graph, with each pair of neighbors or near neighbors of the same sign participating as a $+1$, and each pair of opposite values participating as a $-1$. Each such pair gets counted twice.\n\nThus let $r^{*}$ be the number of neighbors or near-neighbors (meaning nodes at distances one or two) each node has; $\\alpha$ be the number of pairs of neighbors or near-neighbors with equal node-values; and $\\beta$ be the number of pairs with opposite node-values. Moreover, $\\eta$ corresponds to the count of pairs of neighbors or near-neighbors with values both equal to 1, and $\\theta$ is the count of pairs of $-1$'s.\n\nNow, suppose, with $I$ the indexed set of nodes on our graph, that\n\\begin{equation*}\nr^{*}_{i} = r^{*} \\hspace{1cm} \\forall i \\in I.\n\\end{equation*}\nThat is, we assume $r^{*}$ is some fixed quantity: i.e. the underlying graph possesses sufficient symmetry so that each node has the same number of neighbors or near-neighbors. For example each node in the circle-graph has 4 neighbors\/ near-neighbors.\n\nThe assumption that $r_{i}^{*}$ is fixed for all $i$ is not particularly gratuitous, since, either way, $r_{i}^{*}\\leq r_{i}^{2}$, and under the present assumptions, $r_{i}=r$ is fixed.\n\nNext, since $2(\\alpha+\\beta)=r^{*}N$, we have $2(\\alpha-\\beta)=2\\alpha-(r^{*}N-2\\alpha)=4\\alpha-r^{*}N$. Therefore, $\\mathop{\\rm Var}(2(\\alpha-\\beta)) = \\mathop{\\rm Var}(4\\alpha) = \\mathop{\\rm Var}(4(\\eta+\\theta))$, where $\\eta$ is the number of pairs of neighbors and near neighbors with node-values 1, and $\\theta$ is the count of pairs with values $-1$.\n\nOn the other hand, we have $2(\\eta-\\theta)=r^{*}Y$, and therefore $\\mathop{\\rm Var}(\\eta-\\theta) = \\frac{(r^{*})^2}{4}\\mathop{\\rm Var}(Y)$.\n\nOne naturally wonders if we can use the bound we established for $\\mathop{\\rm Var}(Y)$ to bound $\\mathop{\\rm Var}(\\eta+\\theta)$. From the definition,\n\\begin{equation*}\n\\mathop{\\rm Var}(\\eta + \\theta) = \\mathop{\\rm Var}(\\eta)+2\\mathop{\\rm Cov}(\\eta , \\theta)+\\mathop{\\rm Var}(\\theta)\n\\end{equation*}\n\\begin{equation*}\n\\mathop{\\rm Var}(\\eta - \\theta) = \\mathop{\\rm Var}(\\eta) - 2\\mathop{\\rm Cov}(\\eta , \\theta) + \\mathop{\\rm Var}(\\theta)\n\\end{equation*}\nIt suffices to show that $\\mathop{\\rm Cov}(\\eta , \\theta)\\leq 0$ to obtain $\\mathop{\\rm Var}(\\eta + \\theta) \\leq \\mathop{\\rm Var}(\\eta - \\theta)$. For $\\eta$ and $\\theta$ to be negatively correlated, an increase in one would have to imply a decrease in the other -- meaning, in our setting, that an increase in the number of edges (i.e. pairs of nodes at distance 1) with ones at both ends would have to imply a decrease in the edges with negative ones at both ends -- and vice versa.\n\nOne is tempted to try to use the FKG inequality to prove $\\mathop{\\rm Cov}(\\eta , \\theta)\\leq 0$. Specifically, we know that the lattice $\\{-1,1\\}^{\\Gamma}$ (where $\\Gamma$ is our graph) is a poset; and that $\\eta= f(\\{\\xi_{i}\\})$ is an increasing function of that lattice, while $\\theta = g(\\{ \\xi_{i}\\})$ is a decreasing function on the same lattice. Moreover, if we take an element $x$ from $\\{-1,1\\}^{\\Gamma}$, and suppose that $\\overline{x}$ is the element we obtain by switching all $-1$'s in $x$ to $+1$'s, and all $+1$'s to $-1$'s, then $f(x)=g(\\overline{x})$, and the stationary probability $p_{x}$ of state $x$ occurring in our Markov Chain equals the corresponding probability for state $\\overline{x}$ -- that is, $p_{x} = p_{\\overline{x}}$.\n\nNow, the FKG theorem (after Fortuin, Kasteleyn, and Ginibre \\cite{FKG71}) states that for $\\textbf{X}$ a finite distributive lattice, and $\\mu$ a non-negative function (really a measure) on it, satisfying the ``log-supermodularity condition'' \n\\begin{equation}\\label{logsupermod}\n\\mu(x\\wedge y)\\mu(x\\vee y) \\geq \\mu(x)\\mu(y) \\hspace{1cm} \\forall x,y\\in \\textbf{X} \n\\end{equation}\nyields\n\\begin{equation*}\n\\left( \\sum_{x\\in \\textbf{X}} f(x)g(x)\\mu(x) \\right) \\left( \\sum_{x \\in \\textbf{X}} \\mu(x) \\right) \\geq \\left( \\sum_{x\\in \\textbf {X}} f(x)\\mu(x) \\right)\\left( \\sum_{x\\in \\textbf {X}} g(x)\\mu(x) \\right)\n\\end{equation*}\nfor any two monotonically increasing (or decreasing) on $\\textbf{X}$ functions $f$ and $g$; with the inequality reversed if one of $f$ and $g$ is monotonically increasing and other one monotonically decreasing. \n\nIn our particular case, for the considered lattice, stationary distribution over the lattice, and functions $f$ and $g$, having\n\\begin{equation*}\n\\left( \\sum_{x\\in \\{-1,1\\}^{\\Gamma}} f(x)g(x)p_{x} \\right) \\left( \\sum_{x\\in \\{-1,1\\}^{\\Gamma}} p_{x} \\right) \\leq \\left( \\sum_{x\\in \\{-1,1\\}^{\\Gamma}} f(x)p_{x} \\right)\\left( \\sum_{x\\in \\{-1,1\\}^{\\Gamma}} g(x)p_{x} \\right)\n\\end{equation*}\nwould do the job, since \n\\begin{equation*}\n\\mathbb{E}\\eta\\theta = \\sum_{x\\in \\{-1,1\\}^{\\Gamma}} f(x)g(x)p_{x},\n\\end{equation*}\nand \n\\begin{equation*}\n\\mathbb{E}\\eta = \\sum_{x\\in \\{-1,1\\}^{\\Gamma}} f(x)p_{x} = \\sum_{x\\in \\{-1,1\\}^{\\Gamma}} g(x)p_{x} = \\mathbb{E}\\theta,\n\\end{equation*}\nand \n\\begin{equation*}\n\\mathbb{E}\\eta\\theta \\leq (\\mathbb{E}\\eta)(\\mathbb{E}\\theta)\n\\end{equation*} \nimplies exactly $\\mathop{\\rm Cov}(\\eta,\\theta)\\leq 0$.\n\nUnfortunately, our example fails to necessarily satisfy the log-supermodularity condition (\\ref{logsupermod}).\\footnote{Welsh and Donnelly found that the stationary distribution of the chain underlying the Anti-voted model also fails to satisfy the log-superlinearity condition -- see \\cite[Section 5]{DonnellyWelsh1984}.} For example, take a circle graph of odd length. The state in which $1$'s and $-1$'s alternate along the entire graph cannot occur at stationarity, and therefore has measure zero in the stationary distribution of our chain. But the state $w_{1}$ in which we have one $1$ at some node $i$ and everything else is $-1$; and the state $w_{2}$ in which node $i$ and its two neighbors are $-1$'s, and the rest of the graph consists of alternating $1$'s and $-1$'s, can both occur. But then $p_{w_{1}}p_{w_{2}} > 0$, while $p_{w_{1}\\wedge w_{2}}=0$, meaning that the log-supermodularity condition fails. One can come up with similar examples for other standard families of graphs. So we fail to have the log-supermodularity condition.\n\nStill, the log-supermodularity condition is only sufficient rather than necessary for our desired result. Hence our results might be obtainable via different means. For now, suppose\n\\begin{equation*}\n\\mathop{\\rm Cov}(\\eta,\\theta) \\leq 0.\n\\end{equation*}\n\nGiven (\\ref{CovCondition}),\n\\begin{align*}\n\\mathop{\\rm Var} \\left( \\sum_{i\\in I}i^{2}q_{i} \\right) &= \\mathop{\\rm Var} \\left( 2(\\alpha - \\beta) \\right) = 4\\mathop{\\rm Var} \\left( 2(\\eta + \\theta) \\right) \\leq \\\\\n&\\leq 4 \\mathop{\\rm Var} \\left( 2(\\eta - \\theta) \\right) = 4(r^{*})^{2}\\mathop{\\rm Var}(Y) \\leq 4(r^{*})^{2} (r+1)N.\n\\end{align*}\nIt follows that\n\\begin{equation*}\n\\mathop{\\rm Var}[\\mathbb{E}(Y'-Y)^{2} | Y] \\leq \\frac{4 (r^{*})^{2}(r+1)}{N}\n\\end{equation*}\n\nWe thus arrive at the overall bound (\\ref{BigBound}):\n\\begin{align*}\n\\delta &\\leq \\frac{12N}{(r+1)\\sigma_{Y}^{2}}\\sqrt{\\mathop{\\rm Var} \\mathbb{E}[(Y'-Y)^{2}|Y]} + 32\\frac{8(r+1)^{2}N}{\\sigma_{Y}^{3}}  + 6\\frac{4(r+1)^{3\/2}\\sqrt{N}}{\\sigma_{Y}^{2}} \\leq \\\\\n& \\leq 48\\frac{r^{*}}{\\sqrt{r+1}\\sqrt{N}} + 2^{19\/2}\\frac{\\sqrt{r+1}}{\\sqrt{N}} + 48\\frac{\\sqrt{r+1}}{\\sqrt{N}}\n\\end{align*}\nThus our overall bound is:\n\\begin{equation}\\label{finalBound}\n\\delta \\leq 48 \\frac{r^{*}}{\\sqrt{r+1}\\sqrt{N}} + (2^{19\/2}+48)\\frac{\\sqrt{r+1}}{\\sqrt{N}},\n\\end{equation}\nwhere $r^{*}$ is a constant dependent on the underlying family of graphs and satisfying $r^{*}\\leq r^{2}$.\n\nThis completes the proof of Theorem \\ref{MainTheorem}, and derives (\\ref{BoundFINAL}). The final bound is of $O \\left( \\frac{r^{*}}{\\sqrt{r}}N^{-\\frac{1}{2}} \\right)$. \n\n\\section{Consequences and explanation of main result}\\label{SectionConsequences}\n\nThe bound in (\\ref{finalBound}) implies that (under stationarity) the normalized sum of values of the nodes of the graph, $W$, goes in law to the standard normal distribution as the size of the graph rises given $\\frac{(r^{*})^{2}}{rN}\\rightarrow 0$. Note that $r\\leq r^{*}\\leq r^{2}$.\n\nLet us consider four specific families of graphs.\n\nFirst, the complete graph, in which $r=N-1$. On the complete graph, $Y = \\sigma_{Y}W$ clearly has the uniform binary distribution taking values $\\pm N$. Thus it is to no surprise that our bound on the distance to the normal distribution rises to infinity with $N$.\n\nFrom the other side of the spectrum of regular graphs, we can take the circuit (or circle or simple cycle) graph, in which we have $N$ ordered nodes, each connected to its predecessor and its successor, with node $N$ connected to nodes $N-1$ and $1$. Here $r=2$, and hence $r^{3\/2}\/N^{1\/2}$ goes to 0 as $N$ increases to infinity.\n\nThe argument can be extended to circulant\\footnote{ A circulant graph is such that we can arbitrarily index its nodes with 0,1,...,$N-1$, in such a way that if the nodes corresponding to two indices $x$ and $y$ are adjacent, then any two nodes indexed by $z$ and $(z-x+y) \\mod N$ are adjacent. Here $N$ is the number of nodes and adjacency of two nodes means they are connected by an undirected edge. } graphs: as long as $r$ stays constant as $N$ rises, $Y$ would converge to the normal in distribution.\n\nFor a slightly more complicated example, consider the hypercube graph. One can index the nodes of the $n$-dimensional hypercube graph with a string of $n$ zeros and ones, with nodes differing in exactly one digit being neighbors.\n\nIt is easy to see that for an $n$-dimensional hypercube, $r=n$, and $N=2^{n}$. Since \n\\begin{equation*}\n\\frac{r^{3\/2}}{N^{1\/2}} = \\frac{n^{3\/2}}{2^{n\/2}} \\xrightarrow[n\\rightarrow\\infty]{} 0,\n\\end{equation*}\nwe can conclude that $W$ goes in law to the standard normal distribution for the hypercube family of graphs.\n\nFinally, consider the (complete) bipartite graph of size $N=2M$, with $M$ a natural number. For this family, $r=M$, and $N=2M$. On such a graph, $Y$ would frequently take values near $-N,0$ and $N$, and hence cannot be expected to go to the normal in distribution. Indeed, we have \n\\begin{equation*}\n\\frac{r^{3\/2}}{N^{1\/2}} = \\frac{M^{3\/2}}{(2M)^{1\/2}} \\xrightarrow[M\\rightarrow\\infty]{} \\infty.\n\\end{equation*}\nThe argument can clearly extend to multipartite graphs of a fixed number of partitions.\n\n\\section{Conclusions}\\label{SectionConclusions}\n\nTo sum up, we have shown that, subject to some symmetry assumptions, the normalized sum of the values of the nodes in the Neighborhood Attack model is at a distance of $O \\left( r^{3\/2} N^{-\\frac{1}{2}} \\right)$ to the standard normal distribution in the Wasserstein metric. Hence the sum of the nodes is asymptotically normally distributed as the sizes of the underlying graphs increase, provided that $r^{3\/2}N^{-1\/2}$ goes to zero as $N$ rises to infinity.\n\nAlong the way to the result, we also showed that the node-sum $Y$ in the Neighborhood Attack model on an $r$-regular graph satisfies Stein's linearity condition with $\\lambda = \\frac{r+1}{N}$ and $R=0$; and that $\\sigma_{Y}$ satisfies $\\frac{(r+1)N}{2} \\leq \\sigma_{Y}^{2} \\leq (r+1)N.$\n\n\n\\ack\nThis work is based on the author's 2008-2013 graduate research at the University of Southern California under the advisorship of Prof.  Jason Fulman. The author would like to express his gratitude to Prof. Fulman and the USC Department of Mathematics.\n\n\\bibliographystyle{apt\n\\input{Radoslav_Marinov-Counting_Vertices.bbl} \n\n\\end{document}\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction}\n\\label{intro}\n\nThe improvement of magnetic properties of thin films associated\nwith perpendicular magnetic anisotropy (PMA) system has been\nemphasized, for many decades, due to the possible industrial\napplications like high-density perpendicular recording media,\nmagnetic tunnel junctions, sensors, etc. The PMA systems like\nCoPt, FePt alloys are crystalline, therefore tuning of PMA can be\nstudied by understanding the magneto-crystalline anisotropy in the\nsystem~\\cite{CoPt,FePt,FePt_cap}. However, the theory behind PMA\nin amorphous systems like rare-earth transition-metal (RE-TM),\nCoFeB, CoCrPt, is still elusive till date\n~\\cite{GdFe_amor,TbFe_amor,CoFeB_amor,CoCrPt_amor}. The PMA in\nthese systems can be developed by simply varying thickness,\nstrain-modulation, and interface inducing roughness in the film,\netc. Several approaches have been considered for further\nimprovement in PMA property, by deposition of a suitable\nunderlayer (UL) or seed-layer, incremental in the repetition of\nmultilayers (ML), inducing more interfacial roughness, etc.\n\n\n\nLiterature suggests various UL like Ti, V, Cr, Ni, Cu, Ta, Pd,\netc. have been tested on various films like FeCo, FePt, CoCrPt,\nCoCr, FePt ML, etc. For example, Ti UL was used to improve the\ndegree of order of FePt films~\\cite{FePt}, to improve the PMA\nproperty of CoCrPt~\\cite{CoCrPt_amor} and CoCr \\cite{CoCr} films.\nInterestingly it was observed that Ti and Ta UL made considerable\nimprovement in magnetic properties of FeCo films compared to other\nUL~\\cite{FeCo}. This may be due to the development of grain size\nand the internal stress of the film. Another important thing is,\nTi UL also minimizes the contact resistance of\nsingle-wall-nano-tube and a metal electrode which can be\nbeneficial for various electrical applications~\\cite{SWDW}.\n\nRE-TM films exhibit an impressive PMA property suitable for\npossible spintronics applications. GdFe~\\cite{GdFe_amor}, TbFe\n~\\cite{TbFe_amor}, GdFeCo~\\cite{GdFeCo} many more systems were\ninvestigated on the light of their thickness, composition, and\nvarious UL to enhance PMA property. This tuning of properties\ndepends strictly on the thickness of the film and interface\nproperties. When the film thickness ~$<$ $5$~nm, interface-driven\nproperties or effects become dominant. However, for $\\geq$ $5$~nm\nvolume thickness shows an effective bulk-like property, whereas\ninterface effects become negligible. These bulk properties are\nmostly observed in the amorphous system irrespective of the sample\npreparation technique.\n\n\nHere, we have chosen an amorphous ferrimagnetic TbFe system with\nvarious thicknesses of Ti UL to observe a full extent of variance\nin structural, magnetic, and microscopic properties. First, we\nobserved TbFe of 40~nm thickness gives a clear PMA sign with nice\nstripe magnetic domains. Then, we varied the Ti layer of various\nthicknesses (10, 20, and 40~nm) to observe the effect of interface\nroughness, pinning sites, and hybridization effects of the films\ntowards structural, magnetic, and microscopic phenomena. Here, we\nobserved higher OOP $H_c$ values along with nearly null OOP stray\nfields in UL of 20~nm and 40~nm sample. This observation is quite\nnew in these kinds of bulk PMA system, as of our knowledge.\n\n\\section{Experimental detail}\n\n\nSi$<1 0 0>$ \/ Ti($t_{UL}$) \/ TbFe(40~nm) \/ Cr(3~nm); with $t_{UL}$\n= 0, 10, 20, and 40~nm thin films were prepared by electron beam\nevaporation technique. Cr (3~nm) was used as a capping layer to\nprotect the film from oxidation, while Ti is used as UL. Films\nwere deposited with a rate of deposition $\\approx 3$ \\AA\/sec.\nTb-Fe films were deposited from its composite alloy, after\ndeposition the composition of the film was estimated by energy\ndispersive spectroscopy (EDS) analysis. For $t_{UL}$ = 0~nm can be\nnoted as a bare TbFe film of 40~nm, which means TbFe film does not\nhave any UL influence. The background pressure was maintained at\naround $2\\times 10^{-6}$ $Torr$ during deposition. To understand\nthe effect of various thicknesses of UL on structural formations\ngrazing incident X-ray diffraction (GI-XRD), and the X-ray\nreflectivity (XRR) technique were used. Atomic force microscopy\n(AFM) was used for identifying the topographic information.\nMagnetic force microscopy (MFM) was used to probe the stray fields\nemanating from the film surface. For hysteresis and domain\nreversal studies, we have used polar magneto-optical Kerr\nmicroscopy (PMOKE) (spot size $\\approx$ 3 $\\mu$m).\n\n\n\n\\section{Results and discussions}\n\n\\begin{figure}[ht]\n\\centering\n\\includegraphics[scale=0.5]{fig1}\n\\caption{GIXRD patterns of Tb-Fe film with various thicknesses of\nTi UL (0, 10, 20, and 40~nm).}\n\\label{fig:1}\n\\end{figure}\n\nFrom EDS analysis (not presented here) composition of the film\nturns out to be $\\approx$ $Tb_{30}Fe_{70}$ for all the samples.\nFigure~\\ref{fig:1} represents GIXRD patterns of the Tb-Fe film\nwith various UL thicknesses. Bare 40~nm  thick TbFe film of shows\nan amorphous signature, which is consistent with amorphous nature\nof RE-TM\n~\\cite{RE_TM,GdFe_amorphous,TbFeCo_amorphous,DyTbFeCo_amorphous,TbFe_oPMA_amorphous}.\nHowever, it is observed in akin GdFe films with the Gd composition\nless than 15 at. \\% are not amorphous, while more than 20 at. \\%\nshows an amorphous behavior~\\cite{GdFe_per_amor}. Comparing with\nthe literature database, in this case, Tb composition was\nestimated as $\\approx$ 30 at. \\% for all the TbFe films, therefore\nit is expected to exhibit an amorphous nature in the as-deposited\nstate. For UL of 10 nm, the film shows the hcp phase of Ti, which\nis at $2\\theta$ $\\approx$ 44$^{\\circ}$. Along with this, the Fe-Ti\nphase is observed in the case of UL = 20, and 40 nm. This means\nthat the possibility of diffusion of Fe and Ti may occur at the\ninterface of Ti-UL and TbFe film. The effect of this diffusion\nduring the growth gets enhanced when UL thicknesses are increased\nwhich is also evident from the peak intensity. To further probe\nthe individual layer thickness and rms roughness we implement a\nnon-destructive XRR technique. Figure~\\ref{fig:2} (a) and\n~\\ref{fig:2} (b) represents XRR spectra and corresponding electron\ndensity profile (EDP) of various films respectively. Despite of\nbeing amorphous in the case of UL: 0 nm (observed by GIXRD), we\nobserve Kiessig fringes in the corresponding reflectivity spectra\nbecause of the difference in electron density of TbFe and Cr\nlayers. The extracted data of individual layer thickness (t), root\nmean square (rms) roughness ($\\sigma$), and density ($\\rho$) of\nthe layer are summarized in Table~\\ref{tab1}. The composition of\n$Tb_{0.3}Fe_{0.7}$ well agreed with the experimental spectra,\nwhich is also evident from EDS analysis earlier. Expected\nthickness of TbFe of $40$ nm and Cr of $3$ nm (UL: 0 nm) is found\nto be as $42.5$ nm and $2.7$ nm respectively from XRR analysis.\nSlight increment in inter-diffusion of the TbFe and Ti layer\n(formed as $Ti_{0.3}Fe_{0.7}$) is observed for increase in UL\nthickness. Similar evidence of UL (Cu) diffusion into the RE-TM\nmagnetic layer (Sm$Co_5$) is observed using the elemental map and\nenergy dispersive x-ray spectroscopy line analysis\n~\\cite{underlayer_Cu}. The increment of UL thickness has a\nsignificant impact on the microstructural, magnetic, and\nmicroscopic properties of the sample.\n\n\n\\begin{figure}[ht]\n\\centering\n\\includegraphics[scale=0.35]{fig2}\n\\caption{(color online) (a) X-ray reflectivity spectra as a\nfunction of 2$\\theta$ of various films, UL: 0, 10, 20, and 40 nm.\n(b) Corresponding electron density profile (EDP) of various films.\nThe region$-$(i), (ii), (iii), and (iv) represent Cr, TbFe, Ti UL,\nTiFe inter-diffusion layers.}\n\\label{fig:2}\n\\end{figure}\n\n\n\\begin{table}[ht]\n\\centering\n\\caption{The extracted parameters thickness ($t$ in\n$nm$), rms interfacial roughness ($\\sigma$ in $nm$), and density\n($\\rho$ in $g\/cm^{3}$) of various layers in sample with UL\nthickness: 0, 10, 20, and 40 $nm$ are summarized here.}\n\n    \\begin{tabular}{|p{0.9cm}|p{1.3cm}|p{0.5cm}|p{1.3cm}|p{1.3cm}|p{0.5cm}|}\n    \\hline\n    Sample & Parameter & Ti    & $Ti_{0.3}Fe_{0.7}$ & $Tb_{0.3}Fe_{0.7}$ & Cr \\\\\n    \\hline\n          & t ($\\pm 0.5$) &     -  &    -   & 42.5  & 2.7 \\\\\n\n    0 nm & $\\sigma$ ($\\pm 0.1$) &    -   &    -   & 1.2   & 1.1 \\\\\n\n          & $\\rho$ ($\\pm 0.01$) &   -    &   -    & 8.42   & 6.04 \\\\\n    \\hline\n          & t ($\\pm 0.5$) & 7.9   &  -     & 42.9  & 2.5 \\\\\n\n    10 nm & $\\sigma$ ($\\pm 0.1$) & 1.1   &   -    & 0.6  & 0.4 \\\\\n\n          & $\\rho$ ($\\pm 0.01$) & 4.75  &   -    & 8.50   & 5.18 \\\\\n    \\hline\n          & t ($\\pm 0.5$) & 16.7  & 4.5   & 39.9  & 3.4 \\\\\n\n    20 nm & $\\sigma$ ($\\pm 0.1$) & 1.5   & 1.4   & 1.7   & 1.3 \\\\\n\n          & $\\rho$ ($\\pm 0.01$) & 4.37  & 6.35  & 8.43  & 2.73 \\\\\n    \\hline\n          & t ($\\pm 0.5$) & 34.2  & 5.9   & 34.2  & 2.4 \\\\\n\n    40 nm & $\\sigma$ ($\\pm 0.1$) & 1.9   & 2.2   & 1.8   & 1.5 \\\\\n\n          & $\\rho$ ($\\pm 0.01$) & 4.21  & 6.12  & 9.22  & 6.04 \\\\\n          \\hline\n    \\end{tabular}\n  \\label{tab1}\n\\end{table}\n\n\n\nTi UL affects the microscopic properties of the sample. Figure\n~\\ref{fig:3} shows AFM and MFM images of TbFe films grown with\nvarious thicknesses of Ti UL. AFM images revealed that the surface\nmorphology slightly changed with increase in UL thickness. MFM\nimages are found to be free from topographic influences. TbFe film\n(UL : 0~nm) shows a labyrinth-like stripe pattern as represented\nin fig.~\\ref{fig:3}-(ii) (a). TbFe system also exhibits a\nhoneycomb-like structure as reported in literature\n~\\cite{TbFe_honey}. The yellow, and red colors represent up-, and\ndown-magnetization respectively. The average domain width and\ncontrast are enhanced for UL = 10 nm (fig.~\\ref{fig:3}$-$(ii)\n(b)). These domain-orientations are similar to the case of UL =\n0~nm film. In a way, by placing a 10~nm Ti UL enhances the domain\ncontrast and average domain size in the film. However, domain\ncontrast becomes weaker for further increment in UL thickness. The\nyield in the electron beam evaporation technique is quite high,\nso, it is suspected that roughness may increase by increasing the\nthickness of the Ti UL. It is observed that the interface\nroughness ($\\sigma$, Table~\\ref{tab1}), and surface roughness\n($\\approx$ 0.89~nm, Table~\\ref{table_example}) of UL = 50~nm are\ncomparatively higher than the lower UL thicknesses. XRR analysis\n(Table~\\ref{tab1}) also confirms increase in $\\sigma$ value due to\nincrease in UL thickness. Further increment in UL thickness to\n25~nm, stray fields (probing by MFM) that are emanating from the\nsamples significantly deteriorates resulting in faint maze-like\ndomains as shown in fig.~\\ref{fig:3}-(ii) (c). For UL = 50~nm, a\nnull out-of-plane contrast, which is observed in MFM image (fig.\n~\\ref{fig:3}-(ii) (d)).\n\n\\begin{figure} [ht]\n\\centering\n\\includegraphics[scale=0.5]{fig3}\n\\caption{(color online) AFM and MFM images of various films. Upper\nimages (i) represent AFM images and lower (ii) represent\ncorresponding MFM images. (a) refers for bare TbFe film (UL = 0\nnm), (b), (c), and (d) represents for UL = 10, 20, and 40~nm\nrespectively. The yellow contrast represents up-magnetization\nwhile red contrast as down-magnetization in the MFM images (up-\nand down-arrow is represented on the MFM image of UL = 20 nm).\nSmall white dashed circular dots are some of the point-like\ndefects highlighted on the surface of the film.} \\label{fig:3}\n\\end{figure}\n\nIt was observed that microscopic properties are enhanced for UL\nthickness up to 10~nm. Further increment in UL thickness leads to\ndeterioration of magnetic contrast probably due to the increase of\ninterface-induced-roughness. The root-mean-squared surface\nroughness ($R_q$), areal percentage (\\%) of up- and down-domains\nare summarized in Table {~\\ref{table_example}}. By increasing the\nthickness of Ti UL, $R_q$ is found to be increasing. Although\n$R_q$ can not significantly describe the details of interfacial\nroughness ($\\sigma$), still it hints overall roughness of the\nsample. For UL = 0~nm, the area of down-domains (60 \\%) are more\nthan up-domains (40 \\%). By increasing the UL thickness, \\% of\ndown-domains are decreased compared to \\% of up-domains.\nMeanwhile, this is observed as UL thickness helped to increase the\naverage-domain-size, and after a certain thickness, stray fields\nthat are emanating from these domains become deteriorated. The\ndomain sizes are extracted by taking a line scan over the MFM\nimages (fig.~\\ref{fig:4}). The sign of up- or down-domains are\nmarked as up- or down-arrow in the line scan of UL = 10~nm. Domain\nsize can be calculated by considering two neighboring domains,\nmentioned as $D_i$, black dotted lines over the line scans.\nKeeping in mind, up to UL = 25~nm, we observed the appearance of\ndomains. In the case of UL = 50~nm, we did not observe any\ndomains.\n\n\n\n\\begin{figure} [ht]\n \\centering\n \\includegraphics[scale=0.5]{fig4}\n\\caption{(color online) Line scans (at the middle position) on the\nMFM images of the various sample leveled as UL = 0, 10, 20, and\n40~nm. Up-, and down-magnetization profile are marked in UL =\n10~nm line scan. Domain size (D) can be calculated as mentioned in\nthe line profile, e.g, $D_1$: 0.31 $\\mu$m, $D_2$: 0.55 $\\mu$m,\n$D_3$: 0.78 $\\mu$m.} \\label{fig:4}\n\\end{figure}\n\n\n\nAs we observe various thicknesses of Ti UL fascinates the\nmicroscopic domain evolution of the system. So, the reversal\nmechanism of these magnetic films would be of fundamental\ninterest. The magnetization reversals along with Kerr hysteresis\nare captured by using the PMOKE technique. MOKE can measure the\nchange in rotation of polarization of incident light after\nreflected from the magnetized surface. Here, the Kerr rotation\n(KR) can be detected by even a small change of $\\pm$ 5 $m^\\circ$.\nThe MOKE effect is a relativistic quantum effect, which arises due\nto the combined effect of two central phenomena. One is\ncorresponding to the inherent magnetization of the host\nsample$-$and$-$other one is the spin-orbit interaction and the\nexchange interaction among host atoms. Generally speaking here, KR\nvalues can be considered as additional evidence of correlation of\nperpendicular moments in the system.\n\n\nFigure~\\ref{fig:5}-(i) represents MOKE measurements of 40~nm thick\nTbFe film. The Kerr hysteresis (Kerr signal vs Field) shows a\nnearly square hysteresis behavior. The $H_c$ and KR were recorded\nas 465~Oe and 185 $m^\\circ$ respectively. This film is saturated\nat a field of $\\approx$ 985~Oe. The magnetization reversal is\nobserved to be initially dominated by nucleation then driven by\ndomain wall motion. Similarly, we capture both Kerr loops and\nreversal domains for both UL = 10, 20~nm (fig.~\\ref{fig:5}-(ii),\n(iii)). The $H_c$, KR, and saturation field ($H_s$) values are\nsummarized in the Table~\\ref{table_example}. Magnetization\nreversals of UL = 10, and 20~nm, are  of similar nature to the\ncase of bare TbFe 40~nm film. The reversal mechanism initially\nstarted from nucleation-dominated to wall-motion dominated one in\nall the cases.\n\n\\begin{figure} [htp]\n \\centering\n    \\includegraphics[scale=0.4]{fig5}\n        \\caption{PMOKE measurements of various films.\n        \\textbf{(i)} Kerr hysteresis of bare TbFe of 100 nm (UL = 0 nm),\n        \\textbf{(ii)} Kerr hysteresis of UL = 10 nm,\n        \\textbf{(iii)} Kerr hysteresis of  UL = 20 nm.}\n    \\label{fig:5}\n\\end{figure}\n\n\nFrom Table~\\ref{table_example}, it is found that the increment of\nUL thickness results in gradual increase in OOP $H_c$. This\nsignifies magnetic moments are still lying perpendicular to the\nplane of the sample with increment in UL thickness. OOP $H_c$ of\nUL = 20~nm is $\\approx$ 60 \\% more than in the case of UL = 10~nm.\nWhereas, strength of OOP stray field of UL = 20 nm is weakened\ncompared to UL = 10~nm (fig.~\\ref{fig:3}, and corresponding line\nscans in fig.~\\ref{fig:4}). So, placing a UL ultimately enhance\n$H_c$ and $H_s$ values. A more understanding of the samples can be\nderived by considering KR values. We observe KR of UL = 10~nm (222\n$m^\\circ$) shows an improvement over UL = 0~nm (185 $m^\\circ$)\n(refer Table~\\ref{table_example}). This shows a possible\nenhancement of correlation, and\/or interaction of local magnetic\nmoments, which is due to 10~nm of Ti UL. While in the case of UL =\n20~nm, KR rapidly decreases to 51 $m^\\circ$. This may be due to\nmagnetic moments are intermixed, and\/or trapped within the\nproximity of interface roughness, which leads to a reduction of KR\nvalues as Kerr signals are only sensitive to magnetic moments. So,\none can correlate these $H_c$ and KR values to the strength of\nperpendicular anisotropy.\n\n\\begin{table}[h]\n\\caption{Effect of various thickness of UL (0, 10, 20, and 40~nm)\non the $R_q$, \\% of up-\/down-domain area ($A_d$), $H_c$, $H_s$,\nand Kerr rotation (KR) are summarized here, which have been\nextracted from fig.~\\ref{fig:3} and fig.~\\ref{fig:5}.}\n\\label{table_example} \\centering\n\n\\begin{tabular}{|c|c|c|c|c|c|}\n\n\\hline\n\n  $t_{UL}$ & $R_q$ & $A_d$ & $H_c$ & $H_s$ & KR\\\\\n (nm) & (nm) & ($\\pm$ 5 \\%) & ($\\pm$ 5 Oe) & ($\\pm$ 5 Oe) & ($m^\\circ$)\\\\\n \\hline\n 0 & 0.50 & 40\/60 & 465 & 985 & 185\\\\\n \\hline\n 10 & 0.60 & 48\/52 & 890 & 1045 & 222\\\\\n \\hline\n 20 & 0.85 & 55\/45 & 2410 & 2550 & 51\\\\\n \\hline\n 40 & 0.89 & -- & -- & -- & --\\\\\n \\hline\n\\end{tabular}\n\\end{table}\n\n\nThe enhancement or deterioration of magnetic and\/or microscopic\nproperties with the increment in thicknesses of Ti UL can be\nunderstood by following conclusive understanding. When a film gets\ndeposited on the substrate, the system experiences lots of thermal\nstress, due to difference in thermal expansion coefficients\nbetween substrate and film, or between films, or mismatch lattice\nparameters between adjacent layers. In our case, also, bare\namorphous Tb-Fe film of 40~nm experiences thermal stress, but the\neffect would be minimal due to given thickness. However, the\norigin of PMA is expected to be more prominent by $Tb$-$Fe$\ncorrelations~\\cite{TbFe_origin} which leads to structural\nanisotropy in these kinds of amorphous system. In such a case, one\ncan say that these films are more dominated by magneto-crystalline\nanisotropy, as PMA increases by increasing the thickness of the\nfilm~\\cite{GdFe_thickness}. When we place Ti UL, Tb-Fe film grows\non UL rather on the Si substrate, microscopic property changes. As\nTi shows crystalline behavior (fig.~\\ref{fig:1}), placing as UL\nwill create a lattice mismatch between Ti and Tb-Fe layers, at the\nsame time thermal stress on the TbFe layer reduces. However, these\neffects are not prominent in our case as the TbFe is inherently\namorphous. In addition to this, there could be another phenomenon,\n$d-d$ hybridization, coexist between interface. It is not that\n3d-magnetism leads to large exchange interactions (here, $Fe$\natoms) whereas 4f-magnetism drives large magneto-crystalline\nanisotropy (here, Tb atoms)~\\cite{3d_4f}. The extended length of\nthe hybridization is more prominent in the ultrathin system, where\ninterfacial roughness is very nominal. It is also observed that Ti\n($d^2$) alloys show a faint-magnetic behavior when it is mixed\nwith Fe, Co, or Ni (any strong magnetic material)~\\cite{Ti_book}.\nUp to UL = 10~nm, we observe a nominal roughness $\\approx$ 0.60~nm\n(Table~\\ref{table_example}). The reason for increment in domain\nsize could be the lateral extent of hybridization of Fe ($d^6$)\nand Ti ($d^2$), $d-d$ hybridization, become more prominent than\nthe extent of interfacial roughness. As the thickness of UL\nincreases, the extent of interfacial roughness gets dominated over\nhybridization effects, as a result, local anisotropy becomes weak\nand local moments may get trap in point-like defects present in\nthe sample. These moments seem to be trapped by the proximity of\ninterfacial roughness. One such observation was also found in an\nultrathin TbCo system, $<$ 3~nm~\\cite{Ti_alloy_PMA}. The reason\nfor the enhancement of PMA was elucidated due to the appearance of\na rugged interface.\n\nThe above understanding of magnetic and microscopic properties of\nTbFe films can be (a) by increasing UL thicknesses perpendicular\nanisotropy gradually increases regardless of interfacial\nroughness, (b) increment in UL thickness beyond a certain\nthickness leads to intermixing of magnetic moments with the\nproximity of interface roughness, while magnetic moments still\nlies perpendicular to the film plane. However strength of OOP\nstray fields are decreased. For instance, in UL = 20~nm case, one\ncan observe with a considerable PMA (as $H_c$ is more) while OOP\nstray field is weakened. This contrast mixing properties due to\nthe UL which is addressed in this study. Magnetometry measurements\n$H_c$ of UL = 40~nm has a similar range of value compared to that\nof UL = 20~nm.\n\n\n\n\n\n\n\n\n\\section{Summary}\n\n\nIn summary, we have investigated structural, microscopic, and\nmagnetic properties of magnetic anisotropy in TbFe thin films by\nplacing a wide range of Ti UL thickness. By increasing the\nthickness of UL which results in an increment in interfacial\nroughness with the formation of Fe-Ti phases. Here, we observed\nthat (i) for UL = 10~nm, the possibility of extended $d$-$d$\nhybridization dominates over the influence of interfacial\nroughness, as a result, the strength of stray fields enhanced,\n(ii) for UL = 20~nm and 40~nm, the extent of interfacial roughness\ndominates over the hybridization effects as a result stray fields\ndeteriorated. However, $H_c$ and $H_s$ are gradually increased\nwith the increment of UL thickness. By placing UL of 20~nm, $H_c$\nincreases nearly 6 times more than the bare TbFe film, whereas\nKerr rotation decreased by more than 4 times compared to UL of\n10~nm. The magnetization reversal studies reveal domain nucleation\nfollowed by domain-wall motion in all the films.\n\n\n\n\n\n\n\n\n\n\n\n\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction}\nThis paper is devoted to the study of a class of Kirchhoff type problems driven by a nonlocal fractional operator and involving a singular term and a critical nonlinearity.\nMore precisely, we consider\n\\begin{equation}\\label{P}\n\\left\\{\\begin{array}{ll}\n\\left(\\displaystyle\\iint_{\\mathbb{R}^{2N}}\\frac{|u(x)-u(y)|^2}{|x-y|^{N+2s}}dxdy\\right)^{\\theta-1} (-\\Delta)^s u = \\displaystyle\\frac{\\lambda}{u^\\gamma}+u^{2^*_s-1}&\\quad\\mbox{in } \\Omega,\\\\\nu>0&\\quad\\mbox{in } \\Omega,\\\\\nu=0&\\quad\\mbox{in } \\mathbb{R}^N\\setminus\\Omega,\n\\end{array}\n\\right.\n\\end{equation}\nwhere $\\Omega$ is an open bounded subset of $\\mathbb R^N$ with continuous boundary, dimension $N>2s$ with parameter $s\\in (0,1)$, $2^*_s=2N\/(N-2s)$ is the fractional critical Sobolev exponent, $\\lambda>0$ is a real parameter, exponent $\\theta\\in(1,2^*_s\/2)$ while $\\gamma\\in(0,1)$.\nHere $(-\\Delta)^s$ is the fractional Laplace operator defined, up to normalization factors, by the Riesz potential as\n$$(-\\Delta)^s \\varphi(x)=\\int_{\\mathbb{R}^N}\\frac{2\\varphi(x)-\\varphi(x+y)-\\varphi(x-y)}{|y|^{N+2s}}dy,\\quad x\\in\\mathbb R^N,$$\nalong any $\\varphi\\in C^\\infty_0(\\Omega)$; we refer to \\cite{DPV} and the recent monograph \\cite{MBRS} for further details on the fractional Laplacian and the fractional Sobolev spaces $H^s(\\mathbb R^N)$ and $H^s_0(\\Omega)$.\n\nAs well explained in \\cite{DPV}, problem \\eqref{P} is the fractional version of the following nonlinear problem\n\\begin{equation}\\label{Pc}\n\\left\\{\\begin{array}{ll}\n-M\\left(\\int_\\Omega|\\nabla u(x)|^2dx\\right)\\Delta u = \\displaystyle\\frac{\\lambda}{u^\\gamma}+u^{2^*-1}&\\quad\\mbox{in } \\Omega,\\\\\nu>0&\\quad\\mbox{in } \\Omega,\\\\\nu=0&\\quad\\mbox{in } \\partial\\Omega,\n\\end{array}\n\\right.\n\\end{equation}\nwhere $\\Delta$ denotes the classical Laplace operator while, just for a general discussion, $M(t)=t^{\\theta-1}$ for any $t\\in\\mathbb R^+_0$. In literature, problems like \\eqref{P} and \\eqref{Pc} are called of Kirchhoff type whenever the function $M:\\mathbb R^+_0\\to\\mathbb R^+_0$ models the Kirchhoff prototype, given by\n\\begin{equation}\\label{prot}\nM(t)=a +bt^{\\theta-1},\\quad a,\\,b\\ge0,\\,\\;a+b>0,\\,\\;\\theta\\ge1.\n\\end{equation}\nIn particular, when $M(t)\\ge \\mbox{\\textit{ constant}}>0$ for any $t\\in\\mathbb R^+_0$, Kirchhoff problems are said to be \\textit{non--degenerate}\nand this happens for example if $a>0$ in the model case \\eqref{prot}.\nWhile, if $M(0)=0$ but $M(t)>0$ for any $t\\in\\mathbb R^+$, Kirchhoff problems are called\n\\textit{degenerate}. Of course, for \\eqref{prot} this occurs when $a=0$.\n\nThis kind of nonlocal problems has been widely studied in recent years. We refer to \\cite{LLT, LKLT, LZLT, LS, LTLW} for different Kirchhoff problems with $M$ like in \\eqref{prot}, driven by the Laplace operator and involving a singular term of type $u^{-\\gamma}$. In \\cite{LS}, the authors study a Kirchhoff problem with a singular term and a Hardy potential, by using the Nehari method. The same approach is used in \\cite{LZLT} for a singular Kirchhoff problem with also a subcritical term. In \\cite{LLT}, strongly assuming $a>0$ in \\eqref{prot}, the authors prove the existence of two solutions for a Kirchhoff problem like \\eqref{Pc}, by combining perturbation and variational methods. While in \\cite{LKLT}, they provide a uniqueness result for a singular Kirchhoff problem involving a negative critical nonlinearity, by a minimization argument. By arguing similarly to \\cite{LLT}, the authors of \\cite{LTLW} give the existence of two solutions for a critical Kirchhoff problem with a singular term of type $|x|^{-\\beta}u^{-\\gamma}$.\n\nProblem \\eqref{P} has been studied in \\cite{BIMP} when $\\theta=1$, namely without a Kirchhoff coefficient. In \\cite{BIMP}, they prove the existence of two solutions by applying the sub\/supersolutions and Sattinger methods. In \\cite{CMSS}, the authors generalize the results of Section 3 of \\cite{BIMP} to the delicate case of the $p$--fractional Laplace operator $(-\\Delta)^s_p$. While in the last section of \\cite{AMPP}, the authors provide the existence of a solution for nonlinear fractional problems with a singularity like $u^{-\\gamma}$ and a fractional Hardy term, by perturbation methods. Concerning fractional Kirchhoff problems involving critical nonlinearities, we refer to \\cite{AFP, CP, FP, FP2, FV, PP} for existence results and to \\cite{BFL, F, PS, PXZ, XZQ} for multiplicity results.\nIn particular, in \\cite{CP, FP, FP2, PP} different singular terms appear, but given by the fractional Hardy potential.\n \nInspired by the above works, we study a multiplicity result for problem \\eqref{P}. As far as we know, a fractional Kirchhoff problem involving a singular term of type $u^{-\\gamma}$ has not been studied yet. Thus, we can state our result as follows.\n\\begin{theorem}\\label{main} Let $s\\in(0,1)$, $N>2s$, $\\theta\\in(1,2^*_s\/2)$, $\\gamma\\in(0,1)$ and let $\\Omega$ be an open bounded subset of $\\mathbb R^N$ with $\\partial\\Omega$ continuous. Then, there exists $\\overline{\\lambda}>0$ such that for any $\\lambda\\in(0,\\overline{\\lambda})$ problem \\eqref{P} has at least two different solutions.\n\\end{theorem}\nThe first solution of problem \\eqref{P} is obtained by a suitable minimization argument, where we must pay attention to the nonlocal nature of the fractional Laplacian. Concerning the second solution, because of the presence of $u^{-\\gamma}$, we can not apply the usual critical point theory to problem \\eqref{P}. For this, we first study a perturbed problem obtained truncating the singular term $u^{-\\gamma}$. Then, by approximation we get our second solution of \\eqref{P}.\n\nFinally, we observe that Theorem \\ref{main} generalizes in several directions the first part of \\cite[Theorem 4.1]{BIMP} and \\cite[Theorem 1.1]{LLT}.\n\nThe paper is organized as follows. In Section~\\ref{sec variational}, we discuss the variational formulation of problem \\eqref{P} and we introduce the perturbed problem. In Section~\\ref{sec existence}, we prove the existence of the first solution of \\eqref{P} and we give a possible generalization of this existence result, at the end of the section.\nIn Section~\\ref{sec mountain}, we prove the existence of a mountain pass solution for the perturbed problem.\nIn Section~\\ref{sec finale}, we prove Theorem~\\ref{main}.\n\n\\section{Variational setting}\\label{sec variational}\n{\\em Throughout the paper we assume that $s\\in(0,1)$, $N>2s$, $\\theta\\in(1,2^*_s\/2)$, $\\gamma\\in(0,1)$ and $\\Omega$ is an open bounded subset of $\\mathbb R^N$ with $\\partial\\Omega$ continuous, without further mentioning. As a matter of notations, we denote with $\\varphi^+=\\max\\{\\varphi,0\\}$ and $\\varphi^-=\\max\\{-\\varphi,0\\}$ respectively the positive and negative part of a function $\\varphi$.}\n\nProblem~\\eqref{P} has a variational structure and the natural space where finding solutions is the homogeneous fractional Sobolev space $H^s_0(\\Omega)$. In order to study \\eqref{P}, it is important to encode the ``boundary condition'' $u=0$ in\n$\\mathbb{R}^N\\setminus\\Omega$ in the weak formulation,\nby considering also that the interaction between $\\Omega$ and its complementary in $\\mathbb{R}^N$ gives a positive contribution\nin the so called {\\em Gagliardo norm}, given as\n\\begin{equation}\\label{norma}\n\\left\\|u\\right\\|_{H^s(\\mathbb R^N)}=\\left\\|u\\right\\|_{L^2(\\mathbb R^N)}+\\Big(\\iint_{\\mathbb R^{2N}} \\frac{|u(x)-u(y)|^2}{\\left|x-y\\right|^{N+2s}}dxdy\\Big)^{1\/2}.\n\\end{equation}\n\nThe functional space that takes into account this boundary condition will be denoted by $X_0$ and it is defined as\n$$\nX_0=\\big\\{u\\in H^s(\\mathbb R^N):\\,\\,u=0\\mbox{ a.e. in } \\mathbb R^N\\setminus \\Omega\\big\\}.\n$$\nWe refer to \\cite{SV} for a general definition of $X_0$ and its properties.\nWe also would like to point out that, when $\\partial\\Omega$ is continuous, by \\cite[Theorem~6]{FSV} the space $C^\\infty_0(\\Omega)$ is dense in $X_0$, with respect to the norm \\eqref{norma}. This last point will be used to overcome the singularity in problem \\eqref{P}.\n\nIn $X_0$ we can consider the following norm\n$$\n\\left\\|u\\right\\|_{X_0}=\\Big(\\iint_{\\mathbb R^{2N}} \\frac{|u(x)-u(y)|^2}{\\left|x-y\\right|^{N+2s}}dxdy\\Big)^{1\/2},\n$$\nwhich is equivalent to the usual one defined in \\eqref{norma} (see \\cite[Lemma~6]{SV}).\nWe also recall that $(X_0,\\left\\|\\,\\cdot\\,\\right\\|_{X_0})$ is a Hilbert space, with the scalar product defined as\n$$\n\\left\\langle u,v\\right\\rangle_{X_0}=\\iint_{\\mathbb R^{2N}} \\frac{(u(x)-u(y))(v(x)-v(y))}{\\left|x-y\\right|^{N+2s}}dxdy.\n$$\n{\\it From now on, in order to simplify the notation, we will denote $\\|\\cdot\\|_{X_0}$ and $\\left\\langle \\cdot,\\cdot\\right\\rangle_{X_0}$ by $\\|\\cdot\\|$ and $\\left\\langle \\cdot,\\cdot\\right\\rangle$ respectively, and $\\|\\cdot\\|_{L^q(\\Omega)}$ by $\\|\\cdot\\|_q$ for any $q\\in[1,\\infty]$.}\n\nIn order to present the weak formulation of \\eqref{P} and taking into account that we are looking for positive solutions, we will consider the following Kirchhoff problem\n\\begin{equation}\\label{P+}\n\\left\\{\\begin{array}{ll}\n\\left(\\displaystyle\\iint_{\\mathbb{R}^{2N}}\\frac{|u(x)-u(y)|^2}{|x-y|^{N+2s}}dxdy\\right)^{\\theta-1} (-\\Delta)^s u = \\displaystyle\\frac{\\lambda}{(u^+)^\\gamma}+(u^+)^{2^*_s-1}&\\quad\\mbox{in } \\Omega,\\\\\nu=0&\\quad\\mbox{in } \\mathbb{R}^N\\setminus\\Omega.\n\\end{array}\n\\right.\n\\end{equation}\nWe say that $u\\in X_0$ is a (weak) solution of problem \\eqref{P+}, if $u$ satisfies\n\\begin{equation}\\label{weak}\n\\|u\\|^{2(\\theta-1)}\\langle u,\\varphi\\rangle\n = \\lambda \\int_\\Omega \\frac{\\varphi}{(u^+)^\\gamma}dx\n +\\int_\\Omega(u^+)^{2^*_s-1}\\varphi,\n\\end{equation}\nfor any $\\varphi\\in X_0$.\nProblem \\eqref{P+} has a variational structure and $J_\\lambda:X_0\\to\\mathbb R$, defined by\n\\begin{align*}\nJ_\\lambda(u)=\\frac{1}{2\\theta}\\|u\\|^{2\\theta}-\\frac{\\lambda}{1-\\gamma}\\int_\\Omega (u^+)^{1-\\gamma}dx -\\frac{1}{2^*_s}\\|u^+\\|^{2^*_s}_{2^*_s},\n\\end{align*}\nis the underlying functional associated to \\eqref{P+}.\nBecause of the presence of a singular term in \\eqref{P+}, functional $J_\\lambda$ is not differentiable on $X_0$. Therefore, we can not apply directly the usual critical point theory to $J_\\lambda$, in order to solve problem \\eqref{P+}. However, it is possible to find a first solution of \\eqref{P+} by using a local minimization argument. In order to get the second solution of \\eqref{P+}, we have to study an associated approximating problem. That is, for any $n\\in\\mathbb N$, we consider the following perturbed problem\n\\begin{equation}\\label{Pn}\n\\left\\{\\begin{array}{ll}\n\\left(\\displaystyle\\iint_{\\mathbb{R}^{2N}}\\frac{|u(x)-u(y)|^2}{|x-y|^{N+2s}}dxdy\\right)^{\\theta-1} (-\\Delta)^s u = \\displaystyle\\frac{\\lambda}{(u^++1\/n)^\\gamma}+(u^+)^{2^*_s-1}\\quad\\mbox{in } \\Omega,\\\\\nu=0\\quad\\mbox{in } \\mathbb{R}^N\\setminus\\Omega.\n\\end{array}\n\\right.\n\\end{equation}\nFor this, we say that $u\\in X_0$ is a (weak) solution of problem \\eqref{Pn}, if $u$ satisfies\n\\begin{equation}\\label{weak2}\n\\|u\\|^{2(\\theta-1)}\\langle u,\\varphi\\rangle\n = \\lambda \\int_\\Omega \\frac{\\varphi}{(u^++1\/n)^\\gamma}dx\n +\\int_\\Omega(u^+)^{2^*_s-1}\\varphi,\n\\end{equation}\nfor any $\\varphi\\in X_0$.\nIn this case, solutions of \\eqref{Pn} correspond to the critical points of functional $J_{n,\\lambda}:X_0\\to\\mathbb R$, set as\n\\begin{equation}\\label{jn}\nJ_{n,\\lambda}(u)=\\frac{1}{2\\theta}\\|u\\|^{2\\theta}-\\frac{\\lambda}{1-\\gamma}\\int_\\Omega[(u^++1\/n)^{1-\\gamma}-(1\/n)^{1-\\gamma}]dx -\\frac{1}{2^*_s}\\|u^+\\|^{2^*_s}_{2^*_s}.\n\\end{equation}\nIt is immediate to see that $J_{n,\\lambda}$ is of class $C^1(X_0)$.\n\nWe conclude this section recalling the best constant of the fractional Sobolev embedding, which will be very useful to study the compactness property of functional $J_{n,\\lambda}$. That is, we consider\n\\begin{equation}\\label{S}\nS=\\inf_{\\substack{v\\in H^s(\\mathbb R^N)\\\\\nv\\not\\equiv0}}\\displaystyle\\frac{\\displaystyle\\iint_{\\mathbb{R}^{2N}}\\frac{|v(x)-v(y)|^2}{|x-y|^{N+2s}}dxdy}{\\left(\\int_{\\mathbb R^N}|v(x)|^{2^*_s}dx\\right)^{2\/2^*_s}},\n\\end{equation}\nwhich is well defined and strictly positive, as shown in \\cite[Theorem 1.1]{CT}. \n\n\\section{A first solution for problem \\eqref{P}}\\label{sec existence} \nIn this section we prove the existence of a solution for problem \\eqref{P} by a local minimization argument. For this, we first study the geometry of functional $J_\\lambda$.\n\n\\begin{lemma}\\label{mp} There exist numbers $\\rho\\in(0,1]$, $\\lambda_0=\\lambda_{0}(\\rho)>0$ and $\\alpha=\\alpha(\\rho)>0$ such that $J_\\lambda(u)\\ge\\alpha$ for any $u\\in X_0$, with $\\|u\\|=\\rho$, and for any $\\lambda\\in(0,\\lambda_0]$.\n\nFurthermore, set \n$$m_\\lambda=\\inf\\left\\{J_\\lambda(u):\\,\\, u\\in\\overline{B}_\\rho\\right\\},$$\nwhere $\\overline{B}_\\rho=\\left\\{u\\in X_0:\\,\\,\\|u\\|\\leq\\rho\\right\\}$. Then, $m_\\lambda<0$ for any $\\lambda\\in(0,\\lambda_0]$.\n\\end{lemma}\n\\begin{proof}\nLet $\\lambda>0$. From the H\\\"older inequality and \\eqref{S}, for any $u\\in X_0$ we have \n\\begin{equation}\\label{holder}\n\\int_\\Omega(u^+)^{1-\\gamma}dx\\leq|\\Omega|^{\\frac{2^*_s-1+\\gamma}{2^*_s}}\\|u\\|^{1-\\gamma}_{2^*_s}\\leq|\\Omega|^{\\frac{2^*_s-1+\\gamma}{2^*_s}}S^{-\\frac{1-\\gamma}{2}}\\|u\\|^{1-\\gamma}.\n\\end{equation}\nHence, by using again \\eqref{S} and \\eqref{holder} we get\n$$\nJ_\\lambda(u)\\ge\\frac{1}{2\\theta}\\|u\\|^{2\\theta}-\\frac{S^{-\\frac{2^*_s}{2}}}{2^*_s}\\|u\\|^{2^*_s}-\\frac{\\lambda}{1-\\gamma}|\\Omega|^{\\frac{2^*_s-1+\\gamma}{2^*_s}}S^{-\\frac{1-\\gamma}{2}}\\|u\\|^{1-\\gamma}.\n$$\nSince $1-\\gamma<1<2\\theta<2^*_s$, the function\n$$\\eta(t)=\\frac{1}{2\\theta}t^{2\\theta-1+\\gamma}-\\frac{S^{-\\frac{2^*_s}{2}}}{2^*_s}t^{2^*_s-1+\\gamma},\\quad t\\in[0,1]$$\nadmits a maximum at some $\\rho\\in(0,1]$ small enough, that is $\\displaystyle\\max_{t\\in[0,1]}\\eta(t)=\\eta(\\rho)>0$.\nThus, let\n$$\\lambda_0=\\frac{(1-\\gamma)S^{\\frac{1-\\gamma}{2}}}{2|\\Omega|^{\\frac{2^*_s-1+\\gamma}{2^*_s}}}\\eta(\\rho),\n$$\nthen for any $u\\in X_0$ with $\\|u\\|=\\rho\\leq1$ and for any $\\lambda\\leq\\lambda_0$, we get $J_\\lambda(u)\\ge\\rho^{1-\\gamma}\\eta(\\rho)\/2=\\alpha>0$. \n\nFurthermore, fixed $v\\in X_0$ with $v^+\\not\\equiv0$, for $t\\in(0,1)$ sufficiently small\n$$\nJ_\\lambda(tv)=\\frac{t^{2\\theta}}{2\\theta}\\|v\\|^{2\\theta}-t^{1-\\gamma}\\frac{\\lambda}{1-\\gamma}\\int_\\Omega (v^+)^{1-\\gamma}dx -\\frac{t^{2^*_s}}{2^*_s}\\|v^+\\|^{2^*_s}_{2^*_s}<0,\n$$\nbeing $1-\\gamma<1<2\\theta<2^*_s$. This concludes the proof.\n\\end{proof}\n\nWe are now ready to prove the existence of the first solution of \\eqref{P}.\n\n\\begin{theorem}\\label{prima} Let $\\lambda_0$ be given as in Lemma \\ref{mp}. Then, for any $\\lambda\\in(0,\\lambda_0]$ problem \\eqref{P} has a solution $u_0\\in X_0$, with $J_\\lambda(u_0)<0$.\n\\end{theorem}\n\\begin{proof}\nFix $\\lambda\\in(0,\\lambda_0]$ and let $\\rho$ be as given in Lemma \\ref{mp}. We first prove there exists $u_0\\in \\overline{B}_\\rho$ such that $J_\\lambda(u_0)=m_\\lambda<0$.\nLet $\\{u_k\\}_k\\subset\\overline{B}_\\rho$ be a minimizing sequence for $m_\\lambda$, that is such that \n\\begin{equation}\\label{minimax}\n\\lim_{k\\to\\infty}J_\\lambda(u_k)=m_\\lambda.\n\\end{equation}\nSince $\\{u_k\\}_k$ is bounded in $X_0$, by applying \\cite[Lemma~8]{SV} and \\cite[Theorem 4.9]{B}, there exist a subsequence, still\ndenoted by $\\{u_k\\}_k$, and a function $u_0\\in\\overline{B}_\\rho$ such that, as $k\\to\\infty$ we have\n\\begin{equation}\\label{convergences}\n\\begin{array}{ll}\nu_k\\rightharpoonup u_0\\text{ in }X_0,\\quad &u_k\\rightharpoonup u_0\\text{ in }L^{2^*_s}(\\Omega), \\\\\nu_k\\to u_0\\text{ in }L^p(\\Omega)\\mbox{ for any }p\\in[1,2^*_s),\\quad &u_k\\to u_0\\text{ a.e.  in }\\Omega.\n\\end{array}\n\\end{equation}\nSince $\\gamma\\in(0,1)$, by the H\\\"older inequality, for any $k\\in\\mathbb N$ we have\n$$\\left|\\int_\\Omega (u^+_k)^{1-\\gamma}dx-\\int_\\Omega (u^+_0)^{1-\\gamma}dx\\right|\\leq\\int_\\Omega\\left|u^+_k-u^+_0\\right|^{1-\\gamma}dx\n\\leq\\|u^+_k-u^+_0\\|^{1-\\gamma}_2|\\Omega|^{\\frac{1+\\gamma}{2}},\n$$\nwhich yields, by \\eqref{convergences}\n\\begin{equation}\\label{gamma}\n\\lim_{k\\to\\infty}\\int_\\Omega (u^+_k)^{1-\\gamma}dx=\\int_\\Omega (u^+_0)^{1-\\gamma}dx.\n\\end{equation}\nLet $w_k=u_k-u_0$, by \\cite[Theorem 2]{BL} it holds true that\n\\begin{equation}\\label{bl}\n\\|u_k\\|^2=\\|w_k\\|^2+\\|u_0\\|^2+o(1),\\quad\\|u_k\\|^{2^*_s}_{2^*_s}=\\|w_k\\|^{2^*_s}_{2^*_s}+\\|u_0\\|^{2^*_s}_{2^*_s}+o(1)\n\\end{equation}\nas $k\\to\\infty$.\nSince $\\{u_k\\}_k\\subset\\overline{B}_\\rho$, by \\eqref{bl} for $k$ sufficiently large, we have $w_k\\in\\overline{B}_\\rho$.\nLemma \\ref{mp} implies that for any $u\\in X_0$, with $\\|u\\|=\\rho$, we get\n$$\\frac{1}{2\\theta}\\|u\\|^{2\\theta}-\\frac{1}{2^*_s}\\|u^+\\|^{2^*_s}_{2^*_s}\\geq\\alpha>0,\n$$\nand from this, being $\\rho\\leq1$, for $k$ sufficiently large we have\n\\begin{equation}\\label{2.10}\n\\frac{1}{2\\theta}\\|w_k\\|^{2\\theta}-\\frac{1}{2^*_s}\\|w^+_k\\|^{2^*_s}_{2^*_s}>0.\n\\end{equation}\nThus, by \\eqref{minimax}, \\eqref{gamma}--\\eqref{2.10} and considering $\\theta\\geq1$, it follows that as $k\\to\\infty$\n\\begin{align*}\nm_\\lambda&=J_\\lambda(u_k)+o(1)\\\\\n&=\\frac{1}{2\\theta}\\left(\\|w_k\\|^2+\\|u_0\\|^2\\right)^{\\theta}-\\frac{\\lambda}{1-\\gamma}\\int_\\Omega (u^+_0)^{1-\\gamma}dx -\\frac{1}{2^*_s}\\left(\\|w^+_k\\|^{2^*_s}_{2^*_s}+\\|u^+_0\\|^{2^*_s}_{2^*_s}\\right)+o(1)\\\\\n&\\geq J_\\lambda(u_0)+\\frac{1}{2\\theta}\\|w_k\\|^{2\\theta}-\\frac{1}{2^*_s}\\|w^+_k\\|^{2^*_s}_{2^*_s}+o(1)\\geq J_\\lambda(u_0)+o(1)\\geq m_\\lambda,\n\\end{align*}\nbeing $u_0\\in\\overline{B}_\\rho$.\nHence, $u_0$ is a local minimizer for $J_\\lambda$, with $J_\\lambda(u_0)=m_\\lambda<0$ which implies that $u_0$ is nontrivial.\n\nNow, we prove that $u_0$ is a positive solution of \\eqref{P+}. For any $\\psi\\in X_0$, with $\\psi\\geq0$ a.e. in $\\mathbb R^N$, let us consider a $t>0$ sufficiently small so that $u_0+t\\psi\\in\\overline{B}_\\rho$. Since $u_0$ is a local minimizer for $J_\\lambda$, we have\n\\begin{align*}\n0&\\leq J_\\lambda(u_0+t\\psi)-J_\\lambda(u_0)\\\\\n&=\\frac{1}{2\\theta}\\left(\\|u_0+t\\psi\\|^{2\\theta}-\\|u_0\\|^{2\\theta}\\right)-\\frac{\\lambda}{1-\\gamma}\\int_\\Omega \\left[((u_0+t\\psi)^+)^{1-\\gamma}-(u^+_0)^{1-\\gamma}\\right]dx\\\\\n&\\quad-\\frac{1}{2^*_s}\\left(\\|u_0+t\\psi\\|^{2^*_s}_{2^*_s}-\\|u^+_0\\|^{2^*_s}_{2^*_s}\\right).\n\\end{align*}\nFrom this, dividing by $t>0$ and passing to the limit as $t\\to0^+$, it follows that\n\\begin{equation}\\label{2.12}\n\\liminf_{t\\to0^+}\\frac{\\lambda}{1-\\gamma}\\int_\\Omega\\frac{((u_0+t\\psi)^+)^{1-\\gamma}-(u^+_0)^{1-\\gamma}}{t}dx\\leq\\|u_0\\|^{2(\\theta-1)}\\langle u_0,\\psi\\rangle-\\int_\\Omega(u^+_0)^{2^*_s-1}\\psi dx.\n\\end{equation}\nWe observe that\n$$\n\\frac{1}{1-\\gamma}\\!\\cdot\\!\\frac{((u_0+t\\psi)^+)^{1-\\gamma}-(u^+_0)^{1-\\gamma}}{t}=((u_0+\\xi t\\psi)^+)^{-\\gamma}\\psi\\quad\\mbox{ a.e. in }\\Omega,\n$$\nwith $\\xi\\in(0,1)$ and $((u_0+\\xi t\\psi)^+)^{-\\gamma}\\to(u^+_0)^{-\\gamma}\\psi$ a.e. in $\\Omega$, as $t\\to0^+$.\nThus, by the Fatou lemma, we obtain\n\\begin{equation}\\label{2.12b}\n\\lambda\\int_\\Omega(u^+_0)^{-\\gamma}\\psi dx\\leq\\liminf_{t\\to0^+}\\frac{\\lambda}{1-\\gamma}\\int_\\Omega\\frac{((u_0+t\\psi)^+)^{1-\\gamma}-(u^+_0)^{1-\\gamma}}{t}dx.\n\\end{equation}\nTherefore, combining \\eqref{2.12} and \\eqref{2.12b} we get\n\\begin{equation}\\label{2.13}\n\\|u_0\\|^{2(\\theta-1)}\\langle u_0,\\psi\\rangle-\\lambda\\int_\\Omega(u^+_0)^{-\\gamma}\\psi dx-\\int_\\Omega(u^+_0)^{2^*_s-1}\\psi dx\\geq0,\n\\end{equation}\nfor any $\\psi\\in X_0$, with $\\psi\\geq0$ a.e. in $\\mathbb R^N$.\n\nSince $J_\\lambda(u_0)<0$ and by Lemma \\ref{mp}, we have $u_0\\in B_\\rho$. Hence, there exists $\\delta\\in(0,1)$ such that $(1+t)u_0\\in\\overline{B}_\\rho$ for any $t\\in[-\\delta,\\delta]$. Let us define $I_\\lambda(t)=J_\\lambda((1+t)u_0)$. Since $u_0$ is a local minimizer for $J_\\lambda$ in $\\overline{B}_\\rho$, functional $I_\\lambda$ has a minimum at $t=0$, that is\n\\begin{equation}\\label{2.14}\nI'_\\lambda(0)=\\|u_0\\|^{2\\theta}-\\lambda\\int_\\Omega(u^+_0)^{1-\\gamma}dx-\\int_\\Omega(u^+_0)^{2^*_s}dx=0.\n\\end{equation}\nFor any $\\varphi\\in X_0$ and any $\\varepsilon>0$, let us define $\\psi_\\varepsilon=u^+_0+\\varepsilon\\varphi$. Then, by \\eqref{2.13} we have\n\\begin{equation}\\label{2.15a}\n\\begin{aligned}\n0&\\leq\\|u_0\\|^{2(\\theta-1)}\\langle u_0,\\psi^+_\\varepsilon\\rangle-\\lambda\\int_\\Omega(u^+_0)^{-\\gamma}\\psi^+_\\varepsilon dx-\\int_\\Omega(u^+_0)^{2^*_s-1}\\psi^+_\\varepsilon dx\\\\\n&=\\|u_0\\|^{2(\\theta-1)}\\langle u_0,\\psi_\\varepsilon+\\psi^-_\\varepsilon\\rangle-\\lambda\\int_\\Omega(u^+_0)^{-\\gamma}(\\psi_\\varepsilon+\\psi^-_\\varepsilon) dx-\\int_\\Omega(u^+_0)^{2^*_s-1}(\\psi_\\varepsilon+\\psi^-_\\varepsilon) dx.\n\\end{aligned}\n\\end{equation}\nWe observe that, for a.e. $x$, $y\\in\\mathbb{R}^N$, we obtain\n\\begin{equation}\\label{eur}\n\\begin{aligned}\n(u_0(x)&-u_0(y))(u^-_0(x)-u^-_0(y))\\\\\n&=-u^+_0(x)u^-_0(y)-u^-_0(x)u^+_0(y)-\\big[u^-_0(x)-u^-_0(y)\\big]^2\\\\\n&\\leq-\\left|u^-_0(x)-u^-_0(y)\\right|^2,\n\\end{aligned}\n\\end{equation}\nfrom which we immediately get\n$$\n(u_0(x)-u_0(y))(u^+_0(x)-u^+_0(y))\\leq\\left|u_0(x)-u_0(y)\\right|^2.\n$$\nFrom the last inequality, it follows that\n\\begin{equation}\\label{2.15b}\n\\begin{aligned}\n\\langle u_0,\\psi_\\varepsilon+\\psi^-_\\varepsilon\\rangle&=\\iint_{\\mathbb R^{2N}}\\frac{(u_0(x)-u_0(y))(\\psi_\\varepsilon(x)+\\psi^-_\\varepsilon(x)-\\psi_\\varepsilon(y)-\\psi^-_\\varepsilon(y))}{|x-y|^{N+2s}}dxdy\\\\\n&\\leq\\iint_{\\mathbb R^{2N}}\\frac{|u_0(x)-u_0(y)|^2}{|x-y|^{N+2s}}dxdy+\\varepsilon\\iint_{\\mathbb R^{2N}}\\frac{(u_0(x)-u_0(y))(\\varphi(x)-\\varphi(y))}{|x-y|^{N+2s}}dxdy\\\\\n&\\quad+\\iint_{\\mathbb R^{2N}}\\frac{(u_0(x)-u_0(y))(\\psi^-_\\varepsilon(x)-\\psi^-_\\varepsilon(y))}{|x-y|^{N+2s}}dxdy.\n\\end{aligned}\n\\end{equation}\nHence, denoting with $\\Omega_\\varepsilon=\\left\\{x\\in\\mathbb R^N:\\,\\,u^+_0(x)+\\varepsilon\\varphi(x)\\leq0\\right\\}$ and by combining \\eqref{2.15a} with \\eqref{2.15b}, we get\n\\begin{equation}\\label{2.15c}\n\\begin{aligned}\n0&\\leq\\|u_0\\|^{2\\theta}+\\varepsilon\\|u_0\\|^{2(\\theta-1)}\\langle u_0,\\varphi\\rangle+\\|u_0\\|^{2(\\theta-1)}\\langle u_0,\\psi^-_\\varepsilon\\rangle\\\\\n&\\quad-\\lambda\\int_\\Omega(u^+_0)^{-\\gamma}(u^+_0+\\varepsilon\\varphi) dx-\\int_\\Omega(u^+_0)^{2^*_s-1}(u^+_0+\\varepsilon\\varphi) dx\\\\\n&\\quad+\\lambda\\int_{\\Omega_\\varepsilon}(u^+_0)^{-\\gamma}(u^+_0+\\varepsilon\\varphi)dx+\\int_{\\Omega_\\varepsilon}(u^+_0)^{2^*_s-1}(u^+_0+\\varepsilon\\varphi) dx\\\\\n&\\leq\\|u_0\\|^{2\\theta}-\\lambda\\int_\\Omega(u^+_0)^{1-\\gamma}dx-\\int_\\Omega(u^+_0)^{2^*_s}dx+\\|u_0\\|^{2(\\theta-1)}\\langle u_0,\\psi^-_\\varepsilon\\rangle\\\\\n&\\quad+\\varepsilon\\left[\\|u_0\\|^{2(\\theta-1)}\\langle u_0,\\varphi\\rangle-\\lambda\\int_\\Omega(u^+_0)^{-\\gamma}\\varphi dx-\\int_\\Omega(u^+_0)^{2^*_s-1}\\varphi dx\\right]\\\\\n&=\\|u_0\\|^{2(\\theta-1)}\\langle u_0,\\psi^-_\\varepsilon\\rangle+\\varepsilon\\left[\\|u_0\\|^{2(\\theta-1)}\\langle u_0,\\varphi\\rangle-\\lambda\\int_\\Omega(u^+_0)^{-\\gamma}\\varphi dx-\\int_\\Omega(u^+_0)^{2^*_s-1}\\varphi dx\\right],\n\\end{aligned}\n\\end{equation}\nwhere last equality follows from \\eqref{2.14}.\nArguing similarly to \\eqref{eur}, for a.e. $x$, $y\\in\\mathbb{R}^N$ we have\n\\begin{equation}\\label{eur2}\n(u_0(x)-u_0(y))(u^+_0(x)-u^+_0(y))\\geq\\left|u^+_0(x)-u^+_0(y)\\right|^2.\n\\end{equation}\nThus, denoting with\n$$\n\\mathcal U_\\varepsilon(x,y)=\\frac{(u_0(x)-u_0(y))(\\psi^-_\\varepsilon(x)-\\psi^-_\\varepsilon(y))}{|x-y|^{N+2s}},\n$$\nby the symmetry of the fractional kernel and \\eqref{eur2}, we get\n\\begin{equation}\\label{2.15d}\n\\begin{aligned}\n\\langle u_0,\\psi^-_\\varepsilon\\rangle&=\\iint_{\\Omega_\\varepsilon\\times\\Omega_\\varepsilon}\\mathcal U_\\varepsilon(x,y)dxdy+2\\iint_{\\Omega_\\varepsilon\\times(\\mathbb R^N\\setminus\\Omega_\\varepsilon)}\\mathcal U_\\varepsilon(x,y)dxdy\\\\\n&\\leq-\\varepsilon\\left(\\iint_{\\Omega_\\varepsilon\\times\\Omega_\\varepsilon}\\mathcal U(x,y)dxdy+2\\iint_{\\Omega_\\varepsilon\\times(\\mathbb R^N\\setminus\\Omega_\\varepsilon)}\\mathcal U(x,y)dxdy\\right)\\\\\n&\\leq2\\varepsilon\\iint_{\\Omega_\\varepsilon\\times\\mathbb R^N}\\left|\\mathcal U(x,y)\\right|dxdy,\n\\end{aligned}\n\\end{equation}\nwhere we set\n$$\n\\mathcal U(x,y)=\\frac{(u_0(x)-u_0(y))(\\varphi(x)-\\varphi(y))}{|x-y|^{N+2s}}.\n$$\nClearly $\\mathcal U\\in L^1(\\mathbb R^{2N})$, so that for any $\\sigma>0$ there exists $R_\\sigma$ sufficiently large such that\n$$\\iint_{(\\mbox{\\small supp }\\varphi)\\times(\\mathbb R^N\\setminus B_{R_\\sigma})}\\left|\\mathcal U(x,y)\\right|dxdy<\\frac{\\sigma}{2}.\n$$\nAlso, by definition of $\\Omega_\\varepsilon$, we have $\\Omega_\\varepsilon\\subset\\mbox{\\small supp }\\varphi$ and $|\\Omega_\\varepsilon\\times B_{R_\\sigma}|\\to0$ as $\\varepsilon\\to0^+$. Thus, since $\\mathcal U\\in L^1(\\mathbb R^{2N})$, there exists $\\delta_\\sigma>0$ and $\\varepsilon_\\sigma>0$ such that for any $\\varepsilon\\in(0,\\varepsilon_\\sigma]$\n$$|\\Omega_\\varepsilon\\times B_{R_\\sigma}|<\\delta_\\sigma\\quad\\mbox{and}\\quad\\iint_{\\Omega_\\varepsilon\\times B_{R_\\sigma}}\\left|\\mathcal U(x,y)\\right|dxdy<\\frac{\\sigma}{2}.\n$$\nTherefore, for any $\\varepsilon\\in(0,\\varepsilon_\\sigma]$\n$$\\iint_{\\Omega_\\varepsilon\\times\\mathbb R^N}\\left|\\mathcal U(x,y)\\right|dxdy<\\sigma,\n$$\nfrom which we get\n\\begin{equation}\\label{2.15e}\n\\lim_{\\varepsilon\\to0^+}\\iint_{\\Omega_\\varepsilon\\times\\mathbb R^N}\\left|\\mathcal U(x,y)\\right|dxdy=0.\n\\end{equation}\nCombining \\eqref{2.15c} with \\eqref{2.15d}, dividing by $\\varepsilon$, letting $\\varepsilon\\to0^+$ and using \\eqref{2.15e}, we obtain\n$$\n\\|u_0\\|^{2(\\theta-1)}\\langle u_0,\\varphi\\rangle-\\lambda\\int_\\Omega(u^+_0)^{-\\gamma}\\varphi dx-\\int_\\Omega(u^+_0)^{2^*_s-1}\\varphi dx\\geq0,\n$$\nfor any $\\varphi\\in X_0$. By the arbitrariness of $\\varphi$, we prove that $u_0$ verifies \\eqref{weak}, that is $u_0$ is a nontrivial solution of \\eqref{P+}.\n\nFinally, by considering $\\varphi=u^-_0$ in \\eqref{weak} and using \\eqref{eur}, we see that $\\|u^-_0\\|=0$, which implies that $u_0$ is nonnegative. Moreover, by the maximum principle in \\cite[Proposition 2.2.8]{S}, we can deduce that $u_0$ is a positive solution of \\eqref{P+} and so also solves problem \\eqref{P}. This concludes the proof.\n\\end{proof}\n\nWe end this section observing that the result in Theorem \\ref{prima} can be extended to more general Kirchhoff problems.\nThat is, we can consider the following problem\n\\begin{equation}\\label{Pm}\n\\left\\{\\begin{array}{ll}\nM\\left(\\displaystyle\\iint_{\\mathbb{R}^{2N}}\\frac{|u(x)-u(y)|^p}{|x-y|^{N+ps}}dxdy\\right)(-\\Delta)^s_p u = \\displaystyle\\frac{\\lambda}{u^\\gamma}+u^{p^*_s-1}&\\quad\\mbox{in } \\Omega,\\\\\nu>0&\\quad\\mbox{in } \\Omega,\\\\\nu=0&\\quad\\mbox{in } \\mathbb{R}^N\\setminus\\Omega,\n\\end{array}\n\\right.\n\\end{equation}\nwhere $p^*_s=pN\/(N-ps)$, with here $N>ps$ and $p>1$, while the Kirchhoff coefficient $M$ satisfies condition\n\\begin{enumerate}\n\\item[$(\\mathcal M)$]\n{\\em $M:\\mathbb R^+_0\\rightarrow\\mathbb R^+_0$ is continuous and nondecreasing. There exist numbers $a>0$ and $\\vartheta$ such that for any $t\\in\\mathbb R^+_0$\n$$\n\\mathscr M(t):=\\int_0^t M(\\tau)d\\tau\\geq a\\,t^\\vartheta,\\quad\\mbox{with }\\vartheta\n\\begin{cases}\n\\in(1,p^*_s\/p) & \\mbox{ if }M(0)=0,\\\\\n=1 & \\mbox{ if }M(0)>0.\n\\end{cases}\n$$}\\end{enumerate}\nThe main operator $(-\\Delta)^s_p$ is the fractional $p$--Laplacian which may be defined, for any function $\\varphi\\in C^\\infty_0(\\Omega)$, as\n\\begin{equation*}\n(-\\Delta)^s_p \\varphi(x)= 2\\lim_{\\varepsilon\\rightarrow 0^+}\\int_{\\mathbb{R}^N\\setminus B_\\varepsilon(x)} \\frac{|\\varphi(x)-\\varphi(y)|^{p-2}\\big(\\varphi(x)-\\varphi(y)\\big)}{|x-y|^{N+ps}}dy,\\quad x\\in\\mathbb R^N,\n\\end{equation*}\nwhere $B_\\varepsilon(x)=\\{y\\in\\mathbb{R}^N:\\,\\,|x-y|<\\varepsilon\\}$.\nThen, arguing as in the proof of Theorem \\ref{prima} and observing that we have not used yet the assumption that $\\partial\\Omega$ is continuous, we can prove the following result.\n\n\\begin{theorem} Let $s\\in(0,1)$, $p>1$, $N>ps$, $\\gamma\\in(0,1)$ and let $\\Omega$ be an open bounded subset of $\\mathbb R^N$. Let $M$ satisfy $(\\mathcal M)$. Then, there exists $\\lambda_0>0$ such that for any $\\lambda\\in(0,\\lambda_0]$ problem \\eqref{Pm} admits a solution.\n\\end{theorem}\n\n\n\\section{A mountain pass solution for problem \\eqref{Pn}}\\label{sec mountain}\n\nIn this section we prove the existence of a positive solution for perturbed problem \\eqref{Pn}, by the mountain pass theorem. {\\em For this, throughout this section we assume $n\\in\\mathbb N$, without further mentioning}. Now, we first prove that the related functional $J_{n,\\lambda}$ satisfies all the geometric features required by the mountain pass theorem.\n\n\\begin{lemma}\\label{mp2} Let $\\rho\\in(0,1]$, $\\lambda_0=\\lambda_{0}(\\rho)>0$ and $\\alpha=\\alpha(\\rho)>0$ be given as in Lemma \\ref{mp}.\nThen, for any $\\lambda\\in(0,\\lambda_0]$ and any $u\\in X_0$, with $\\|u\\|\\leq \\rho$, functional $J_{n,\\lambda}(u)\\ge\\alpha$.\n\nFurthermore, there exists $e\\in X_0$, with $\\|e\\|>\\rho$, such that $J_{n,\\lambda}(e)<0$.\n\\end{lemma}\n\\begin{proof}\nSince $\\gamma\\in(0,1)$, by the subadditivity of $t\\mapsto t^{1-\\gamma}$, we have\n\\begin{equation}\\label{subadd}\n(u^++1\/n)^{1-\\gamma}-(1\/n)^{1-\\gamma}\\leq(u^+)^{1-\\gamma}\\quad\\mbox{a.e. in }\\Omega, \n\\end{equation}\nfor any $u\\in X_0$ and any $n\\in\\mathbb N$. Thus, we have $J_{n,\\lambda}(u)\\ge J_\\lambda(u)$ for any $u\\in X_0$ and the first part of lemma directly follows by Lemma \\ref{mp}.\n\nFor any $v\\in X_0$, with $v^+\\not\\equiv0$, and $t>0$ we have\n$$\nJ_{n,\\lambda}(tv)=\\frac{t^{2\\theta}}{2\\theta}\\|v\\|^{2\\theta}-\\frac{\\lambda}{1-\\gamma}\\int_\\Omega\\left[(tv^++1\/n)^{1-\\gamma}-(1\/n)^{1-\\gamma}\\right]dx -\\frac{t^{2^*_s}}{2^*_s}\\|v^+\\|^{2^*_s}_{2^*_s}\\to-\\infty\\quad\\mbox{as }t\\to\\infty,\n$$\nbeing $1-\\gamma<1<2\\theta<2^*_s$. Hence, we can find $e\\in X_0$, with $\\|e\\|>\\rho$ sufficiently large, such that $J_{n,\\lambda}(e)<0$. This concludes the proof.\n\\end{proof}\n\nWe discuss now the compactness property for the functional $J_{n,\\lambda}$, given by\nthe Palais--Smale condition. We recall that $\\{u_k\\}_k\\subset X_0$\nis a Palais--Smale sequence for $J_{n,\\lambda}$ at level $c\\in\\mathbb R$ if\n\\begin{equation}\\label{e2.1}\nJ_{n,\\lambda}(u_k)\\to c\\quad\\mbox{and}\\quad J'_{n,\\lambda}(u_k)\\to 0\\quad\\mbox{in $(X_0)'$ as }k\\to\\infty.\n\\end{equation}\nWe say that $J_{n,\\lambda}$ satisfies the Palais--Smale condition at level $c$ if\nany Palais--Smale sequence $\\{u_k\\}_k$ at level $c$ admits a convergent subsequence in $X_0$.\n\nBefore proving this compactness condition, we introduce the following positive\nconstants which will help us for a better explanation\n\\begin{equation}\\label{costanti}\nD_1=\\left(\\frac{1}{2\\theta}-\\frac{1}{2^*_s}\\right)S^{\\frac{2^*_s\\theta}{2^*_s-2\\theta}}\\qquad D_2=\\frac{\\displaystyle\\left[\\left(\\frac{1}{1-\\gamma}+\\frac{1}{2^*_s}\\right)|\\Omega|^{\\frac{2^*_s-1+\\gamma}{2^*_s}}S^{-\\frac{1-\\gamma}{2}}\\right]^{\\frac{2\\theta}{2\\theta-1+\\gamma}}}{\\displaystyle\\left(\\frac{1}{2\\theta}-\\frac{1}{2^*_s}\\right)^{\\frac{1-\\gamma}{2\\theta-1+\\gamma}}}.\n\\end{equation}\n\n\\begin{lemma}\\label{palais} Let $\\lambda>0$. Then, the functional $J_{n,\\lambda}$ satisfies the Palais--Smale condition at any level $c\\in\\mathbb R$ verifying\n\\begin{equation}\\label{livello}\nc<D_1-D_2\\lambda^{2\\theta\/(2\\theta-1+\\gamma)},\n\\end{equation}\nwith $D_1$, $D_2>0$ given as in \\eqref{costanti}.\n\\end{lemma}\n\n\\begin{proof}\nLet $\\lambda>0$ and let $\\{u_k\\}_k$ be a Palais--Smale sequence in $X_0$ at level $c\\in\\mathbb R$, with $c$ satisfying \\eqref{livello}. We first prove the boundedness of $\\{u_k\\}_k$. By \\eqref{e2.1} there exists $\\sigma>0$ such that, as $k\\to\\infty$\n$$\n\\begin{aligned}\nc+\\sigma\\|u_k\\|+o(1)&\\geq J_{n,\\lambda}(u_k)-\\frac{1}{2^*_s}\\langle J'_{n,\\lambda}(u_k), u_k\\rangle\\\\\n&=\\left(\\frac{1}{2\\theta}-\\frac{1}{2^*_s}\\right)\\|u_k\\|^{2\\theta}-\\frac{\\lambda}{1-\\gamma}\\int_\\Omega[(u^+_k+1\/n)^{1-\\gamma}-(1\/n)^{1-\\gamma}]dx\\\\\n&\\quad+\\frac{\\lambda}{2^*_s}\\int_\\Omega(u^+_k+1\/n)^{-\\gamma}u_kdx\\\\\n&\\geq\\left(\\frac{1}{2\\theta}-\\frac{1}{2^*_s}\\right)\\|u_k\\|^{2\\theta}-\\lambda\\left(\\frac{1}{1-\\gamma}+\\frac{1}{2^*_s}\\right)\\int_\\Omega|u_k|^{1-\\gamma}dx\\\\\n&\\geq\\left(\\frac{1}{2\\theta}-\\frac{1}{2^*_s}\\right)\\|u_k\\|^{2\\theta}-\\lambda\\left(\\frac{1}{1-\\gamma}+\\frac{1}{2^*_s}\\right)|\\Omega|^{\\frac{2^*_s-1+\\gamma}{2^*_s}}S^{-\\frac{1-\\gamma}{2}}\\|u_k\\|^{1-\\gamma}\n\\end{aligned}\n$$\nwhere the last two inequalities follow by \\eqref{S}, \\eqref{subadd} and the H\\\"older inequality.\nTherefore, $\\{u_k\\}_k$ is bounded in $X_0$, being $1-\\gamma<1<2\\theta$.\nAlso, $\\{u^-_k\\}_k$ is bounded in $X_0$ and by \\eqref{e2.1} we have\n$$\n\\lim_{k\\to\\infty}\\langle J'_{n,\\lambda}(u_k), -u^-_k\\rangle=\\lim_{k\\to\\infty}\\|u_k\\|^{2(\\theta-1)}\\langle u_k,-u^-_k\\rangle=0.\n$$\nThus, by inequality \\eqref{eur} we deduce that $\\|u^-_k\\|\\to0$ as $k\\to\\infty$, which yields\n$$\nJ_{n,\\lambda}(u_k)=J_{n,\\lambda}(u_k^+)+o(1)\\quad\\mbox{and}\\quad J'_{n,\\lambda}(u_k)=J'_{n,\\lambda}(u_k^+)+o(1)\\quad\\mbox{as }k\\to\\infty.\n$$\nHence, we can suppose that $\\{u_k\\}_k$ is a sequence of nonnegative functions. \n\nBy the boundedness of $\\{u_k\\}_k$ and by using \\cite[Lemma 8]{SV} and \\cite[Theorem 4.9]{B}, there exist a subsequence, still\ndenoted by $\\{u_k\\}_k$, and a function $u\\in X_0$\nsuch that\n\\begin{equation}\\label{e2.4}\n\\begin{array}{ll}\nu_k\\rightharpoonup u\\text{ in }X_0,\\quad & \\|u_k\\|\\rightarrow \\mu, \\\\\nu_k\\rightharpoonup u\\mbox{ in } L^{2^*_s}(\\Omega),\\quad &\\left\\|u_k-u\\right\\|_{2^*_s}\\to\\ell,\\\\\nu_k\\rightarrow u\\mbox{ in } L^p(\\Omega)\\text{ for any }p\\in[1,2^*_s),\\quad&u_k\\to u\\text{ a.e.  in }\\Omega,\\quad u_k\\leq h\\text{ a.e.  in }\\Omega,\n\\end{array}\n\\end{equation}\nas $k\\to\\infty$, with $h\\in L^p(\\Omega)$ for a fixed $p\\in[1,2^*_s)$.\nIf $\\mu=0$, then immediately $u_k\\to0$ in $X_0$ as $k\\to\\infty$. Hence, let us assume that $\\mu>0$.\n\nSince $n\\in\\mathbb N$, by \\eqref{e2.4} it follows that\n$$\\left|\\frac{u_k-u}{(u_k+1\/n)^\\gamma}\\right|\\leq n^\\gamma(h+|u|)\\quad\\mbox{a.e. in }\\Omega,\n$$\nso by the dominated convergence theorem and \\eqref{e2.4}, we have\n\\begin{equation}\\label{e2.5}\n\\lim_{k\\to\\infty}\\int_{\\Omega}\\frac{u_k-u}{(u_k+1\/n)^\\gamma}dx=0.\n\\end{equation}\nBy \\eqref{e2.4} and \\cite[Theorem 2]{BL} we have\n\\begin{align}\\label{e2.9}\n\\|u_k\\|^2=\\|u_k-u\\|^2+\\|u\\|^2+o(1),\\quad\n\\|u_k\\|_{2^*_s}^{2^*_s}=\\|u_k-u\\|_{2^*_s}^{2^*_s}+\\|u\\|_{2^*_s}^{2^*_s}+o(1)\n\\end{align}\nas $k\\to\\infty$.\nConsequently, we deduce from\n\\eqref{e2.1}, \\eqref{e2.4}, \\eqref{e2.5} and \\eqref{e2.9} that, as $k\\to\\infty$\n\\begin{align*}\no(1)&=\\langle J_{n,\\lambda}^\\prime(u_k),u_k-u\\rangle\n\\nonumber\\\\\n&=\\|u_k\\|^{2(\\theta-1)}\\langle u_k,u_k-u\\rangle\n-\\lambda\\int_{\\Omega}\\frac{u_k-u}{(u_k+1\/n)^\\gamma}dx\n-\\int_{\\Omega}u_k^{2^*_s-1}(u_k-u)dx\\\\\n&=\\mu^{2(\\theta-1)}(\\mu^2-\\|u_k\\|^2)\n-\\|u_k\\|^{2^*_s}_{2^*_s}+\\|u\\|^{2^*_s}_{2^*_s}\n+o(1)=\\mu^{2(\\theta-1)}\\|u_k-u\\|^2\n-\\|u_k-u\\|^{2^*_s}_{2^*_s} +o(1).\\nonumber\n\\end{align*}\nTherefore, we have proved the crucial formula\n\\begin{equation}\\label{I}\n\\mu^{2(\\theta-1)}\\lim_{k\\to\\infty}\\|u_k-u\\|^2=\\lim_{k\\to\\infty}\\|u_k-u\\|^{2^*_s}_{2^*_s}.\n\\end{equation}\nIf $\\ell=0$, since $\\mu>0$, by \\eqref{e2.4} and \\eqref{I} we have $u_k\\to u$ in $X_0$ as $k\\to\\infty$, concluding the proof.\n\nThus, let us assume by contradiction that $\\ell>0$.\nBy \\eqref{S}, the notation in \\eqref{e2.4} and \\eqref{I}, we get\n\\begin{equation}\\label{ll}\n\\ell^{2^*_s}\\ge S\\, \\mu^{2(\\theta-1)}\\ell^2.\n\\end{equation}\nNoting that \\eqref{I} implies in particular that\n$$\\mu^{2(\\theta-1)}\\big(\\mu^2-\\|u\\|^2\\big)=\\ell^{2^*_s},$$\nusing \\eqref{ll}, it follows that\n$$\n\\big(\\ell^{2^*_s}\\big)^{2s\/N}=(\\mu^{2(\\theta-1)})^{2s\/N}\\big(\\mu^2-\\|u\\|^2\\big)^{2s\/N}\\ge S\\,\\mu^{2(\\theta-1)}.\n$$\nFrom this, we obtain\n$$\\mu^{4s\/N}\\ge\\big(\\mu^2-\\|u\\|^2\\big)^{2s\/N}\\ge S\\, (\\mu^{2(\\theta-1)})^{\\frac{N-2s}{N}},$$\nand considering $N<2s\\theta\/(\\theta-1)=2s\\theta'$, we have\n\\begin{align}\\label{eq31}\n\\mu^2\\geq S^{\\frac{N}{2s\\theta-N(\\theta-1)}}.\n\\end{align}\nIndeed, the restriction $N\/(2\\theta')<s$ follows directly from the fact that $1<\\theta<2^*_s\/2=N\/(N-2s)$.\nBy \\eqref{subadd} and considering that $n\\in\\mathbb N$, for any $k\\in\\mathbb N$ we have\n$$\nJ_{n,\\lambda}(u_k)-\\frac{1}{2^*_s}\\left\\langle J'_{n,\\lambda}(u_k),u_k\\right\\rangle\\geq\\left(\\frac{1}{2\\theta}-\\frac{1}{2^*_s}\\right)\\|u_k\\|^{2\\theta}-\\lambda\\left(\\frac{1}{1-\\gamma}+\\frac{1}{2^*_s}\\right)\\int_\\Omega u_k^{1-\\gamma}dx.\n$$\nFrom this, as $k\\to\\infty$, being $\\theta\\geq1$, by \\eqref{e2.1}, \\eqref{e2.4}, \\eqref{e2.9}, \\eqref{eq31}, the H\\\"older inequality and the Young inequality, we obtain\n$$\n\\begin{alignedat}4\nc&\\geq\\left(\\frac{1}{2\\theta}-\\frac{1}{2^*_s}\\right)\\left(\\mu^{2\\theta}+\\|u\\|^{2\\theta}\\right)-\\lambda\\left(\\frac{1}{1-\\gamma}+\\frac{1}{2^*_s}\\right)|\\Omega|^{\\frac{2^*_s-1+\\gamma}{2^*_s}}S^{-\\frac{1-\\gamma}{2}}\\|u\\|^{1-\\gamma}\\\\\n&\\geq\\left(\\frac{1}{2\\theta}-\\frac{1}{2^*_s}\\right)\\left(\\mu^{2\\theta}+\\|u\\|^{2\\theta}\\right)-\\left(\\frac{1}{2\\theta}-\\frac{1}{2^*_s}\\right)\\|u\\|^{2\\theta}\\\\\n&\\quad-\\left(\\frac{1}{2\\theta}-\\frac{1}{2^*_s}\\right)^{-\\frac{1-\\gamma}{2\\theta-1+\\gamma}}\\left[\\lambda\\left(\\frac{1}{1-\\gamma}+\\frac{1}{2^*_s}\\right)|\\Omega|^{\\frac{2^*_s-1+\\gamma}{2^*_s}}S^{-\\frac{1-\\gamma}{2}}\\right]^{\\frac{2\\theta}{2\\theta-1+\\gamma}}\\\\\n&\\geq\\left(\\frac{1}{2\\theta}-\\frac{1}{2^*_s}\\right)S^{\\frac{2^*_s\\theta}{2^*_s-2\\theta}} -\\left(\\frac{1}{2\\theta}-\\frac{1}{2^*_s}\\right)^{-\\frac{1-\\gamma}{2\\theta-1+\\gamma}}\\left[\\lambda\\left(\\frac{1}{1-\\gamma}+\\frac{1}{2^*_s}\\right)|\\Omega|^{\\frac{2^*_s-1+\\gamma}{2^*_s}}S^{-\\frac{1-\\gamma}{2}}\\right]^{\\frac{2\\theta}{2\\theta-1+\\gamma}}\n\\end{alignedat}\n$$\nwhich contradicts \\eqref{livello}, since \\eqref{costanti}. This concludes the proof.\n\\end{proof}\n\nWe now give a control from above for functional $J_{n,\\lambda}$, under a suitable situation.\nFor this, we assume, without loss of generality, that $0\\in\\Omega$.\nBy \\cite{CT} we know that the infimum in \\eqref{S} is attained at the function\n\\begin{equation}\\label{ueps}\nu_\\varepsilon(x)=\\frac{\\varepsilon^{(N-2s)\/2}}{(\\varepsilon^2+|x|^2)^{(N-2s)\/2}},\\quad\\mbox{with }\\varepsilon>0,\n\\end{equation}\nthat is, it holds true that\n$$\\displaystyle\\iint_{\\mathbb{R}^{2N}}\\frac{|u_\\varepsilon(x)-u_\\varepsilon(y)|^2}{|x-y|^{N+2s}}dxdy=S\\|u_\\varepsilon\\|^2_{L^{2^*_s}(\\mathbb R^N)}.$$\nLet us fix $r>0$ such that $B_{4r}\\subset\\Omega$, where $B_{4r}=\\{x\\in\\mathbb R^N:\\,|x|<4r\\}$, and let us introduce a cut--off function $\\phi\\in C^\\infty(\\mathbb R^N,[0,1])$ such that\n\\begin{equation}\\label{phi}\n\\phi=\\begin{cases}\n1 & \\mbox{ in }B_r,\\\\\n0 & \\mbox{ in }\\mathbb R^N\\setminus B_{2r}.\n\\end{cases}\n\\end{equation}\nFor any $\\varepsilon>0$, we set\n\\begin{equation}\\label{psi}\n\\psi_\\varepsilon=\\frac{\\phi u_\\varepsilon}{\\|\\phi u_\\varepsilon\\|^2_{2^*_s}}\\in X_0.\n\\end{equation}\nThen, we can prove the following result.\n\n\\begin{lemma}\\label{lemma3.3} There exist $\\psi\\in X_0$ and $\\lambda_1>0$ such that for any $\\lambda\\in(0,\\lambda_1)$ \n$$\\sup_{t\\geq0} J_{n,\\lambda}(t\\psi)<D_1-D_2\\lambda^{2\\theta\/(2\\theta-1+\\gamma)},$$\nwith $D_1$, $D_2>0$ given as in \\eqref{costanti}.\n\\end{lemma}\n\\begin{proof}\nLet $\\lambda$, $\\varepsilon>0$.\nLet $u_\\varepsilon$ and $\\psi_\\varepsilon$ be as respectively in \\eqref{ueps} and in \\eqref{psi}.\nBy \\eqref{jn}, we have $J_{n,\\lambda}(t\\psi_\\varepsilon)\\to-\\infty$ as $t\\to\\infty$, so that there exists $t_\\varepsilon>0$ such that $J_{n,\\lambda}(t_\\varepsilon \\psi_\\varepsilon)=\\max_{t\\geq0}J_{n,\\lambda}(t \\psi_\\varepsilon)$. By Lemma \\ref{mp2} we know that $J_{n,\\lambda}(t_\\varepsilon \\psi_\\varepsilon)\\geq\\alpha>0$. Hence, by continuity of $J_{n,\\lambda}$, there exist two numbers $t_0$, $t^*>0$ such that $t_0\\leq t_\\varepsilon\\leq t^*$.\n\nNow, since $\\|u_\\varepsilon\\|_{L^{2^*_s}(\\mathbb R^N)}$ is independent of $\\varepsilon$, by \\cite[Proposition 21]{SV2} we have\n$$\\|\\psi_\\varepsilon\\|^2\\leq\\frac{\\displaystyle\\iint_{\\mathbb{R}^{2N}}\\frac{|u_\\varepsilon(x)-u_\\varepsilon(y)|^2}{|x-y|^{N+2s}}dxdy}{\\|\\phi u_\\varepsilon\\|^2_{2^*_s}}=S+O(\\varepsilon^{N-2s}),$$\nfrom which, by the elementary inequality\n$$\n(a+b)^p\\leq a^p+p(a+1)^{p-1}b,\\quad\\mbox{for any }a>0,\\,b\\in[0,1],\\,p\\geq1,\n$$\nwith $p=2\\theta$, it follows that, as $\\varepsilon\\to0^+$\n$$\n\\|\\psi_\\varepsilon\\|^{2\\theta}\\leq S^\\theta+O(\\varepsilon^{N-2s}).\n$$\nHence, by the last inequality, \\eqref{jn} and being $t_0\\leq t_\\varepsilon\\leq t^*$, for any $\\varepsilon>0$ sufficiently small, we have\n\\begin{equation}\\label{binlin1}\nJ_{n,\\lambda}(t_\\varepsilon\\psi_\\varepsilon)\\leq\\frac{t^{2\\theta}}{2\\theta}S^\\theta+C_1\\varepsilon^{N-2s}-\\frac{\\lambda}{1-\\gamma}\\int_\\Omega[(t_0\\psi_\\varepsilon+1\/n)^{1-\\gamma}-(1\/n)^{1-\\gamma}]dx -\\frac{t^{2^*_s}}{2^*_s},\n\\end{equation}\nwith $C_1$ a suitable positive constant. \nWe observe that\n$$\n\\max_{t\\geq0}\\left(\\frac{t^{2\\theta}}{2\\theta}S^\\theta-\\frac{t^{2^*_s}}{2^*_s}\\right)\n=\\left(\\frac{1}{2\\theta}-\\frac{1}{2^*_s}\\right)S^{\\frac{2^*_s\\theta}{2^*_s-2\\theta}}.\n$$\nThus, by \\eqref{binlin1} it follows that\n\\begin{equation}\\label{binlin2}\nJ_{n,\\lambda}(t_\\varepsilon\\psi_\\varepsilon)\\leq\\left(\\frac{1}{2\\theta}-\\frac{1}{2^*_s}\\right)S^{\\frac{2^*_s\\theta}{2^*_s-2\\theta}}+C_1\\varepsilon^{N-2s}-\\frac{\\lambda}{1-\\gamma}\\int_\\Omega[(t_0\\psi_\\varepsilon+1\/n)^{1-\\gamma}-(1\/n)^{1-\\gamma}]dx.\n\\end{equation}\nNow, let us consider a positive number $q$ satisfying\n\\begin{equation}\\label{q}\n\\frac{(N-2s)(1-\\gamma)-2q(N-2s)(1-\\gamma)+2^*_sqN}{2^*_s}\\cdot\\frac{2\\theta}{2\\theta-1+\\gamma}\\cdot\\frac{1}{N-2s}+1-\\frac{2\\theta}{2\\theta-1+\\gamma}<0,\n\\end{equation}\nthat is, being $2<2\\theta<2^*_s$, $N>2s$ and $\\gamma\\in(0,1)$, such that\n$$\n0<q<\\frac{(N-2s)(2^*_s-2\\theta)(1-\\gamma)}{2\\theta N(2^*_s-2)+4\\theta N\\gamma+8\\theta s(1-\\gamma)}.\n$$\nBy the elementary inequality\n$$\na^{1-\\gamma}-(a+b)^{1-\\gamma}\\leq -(1-\\gamma)b^{\\frac{1-\\gamma}{p}}a^{\\frac{(p-1)(1-\\gamma)}{p}},\\quad\\mbox{for any }a>0,\\,b>0\\mbox{ large enough},\\,p>1,\n$$\nwith $p=2^*_s\/2$, and considering $\\varepsilon<r^{1\/q}$ sufficiently small, with $r$ given by \\eqref{phi}, we have\n\\begin{equation}\\label{binlin3}\n\\begin{alignedat}4\n-\\frac{1}{1-\\gamma}\\int_{\\{x\\in\\Omega:\\,|x|\\leq\\varepsilon^q\\}}&[(t_0\\psi_\\varepsilon+1\/n)^{1-\\gamma}-(1\/n)^{1-\\gamma}]dx\\\\\n&\\leq-\\widetilde{C}\\varepsilon^{\\frac{(N-2s)(1-\\gamma)}{2^*_s}}\\int_{\\{x\\in\\Omega:\\,|x|\\leq\\varepsilon^q\\}}\\left[\n\\frac{1}{(|x|^2+\\varepsilon^2)^{\\frac{N-2s}{2}}}\\right]^{\\frac{2(1-\\gamma)}{2^*_s}}dx\\\\\n&\\leq-C_2\\varepsilon^{\\frac{(N-2s)(1-\\gamma)-2q(N-2s)(1-\\gamma)+2^*_s qN}{2^*_s}},\n\\end{alignedat}\n\\end{equation}\nwith $\\widetilde{C}$ and $C_2$ two positive constant independent by $\\varepsilon$.\nBy combining \\eqref{binlin2} with \\eqref{binlin3}, we get\n\\begin{equation}\\label{binlin4}\nJ_{n,\\lambda}(t_\\varepsilon\\psi_\\varepsilon)\\leq\\left(\\frac{1}{2\\theta}-\\frac{1}{2^*_s}\\right)S^{\\frac{2^*_s\\theta}{2^*_s-2\\theta}}+C_1\\varepsilon^{N-2s}-C_2\\lambda\\varepsilon^{\\frac{(N-2s)(1-\\gamma)-2q(N-2s)(1-\\gamma)+2^*_s qN}{2^*_s}}.\n\\end{equation}\nThus, let us consider $\\lambda^*>0$ such that\n$$\nD_1-D_2\\lambda^{\\frac{2\\theta}{2\\theta-1+\\gamma}}>0\\quad\\mbox{for any }\\lambda\\in(0,\\lambda^*)\n$$\nand let us set \n$$\n\\begin{alignedat}4\n\\nu_1=\\displaystyle\\frac{2q\\theta}{(2\\theta-1+\\gamma)(N-2s)},\\quad &\\nu_2=\\displaystyle\\frac{2\\theta[(N-2s)(1-\\gamma)-2q(N-2s)(1-\\gamma)+2^*_s qN]}{2^*_s(2\\theta-1+\\gamma)(N-2s)}+1,\\\\\n\\nu_3=\\displaystyle\\nu_2-\\frac{2\\theta}{2\\theta-1+\\gamma},\\quad&\\lambda_1=\\displaystyle\\min\\left\\{\\lambda^*,r^{1\/\\nu_1},\n\\left(\\frac{C_2}{C_1+D_2}\\right)^{-1\/\\nu_3}\\right\\},\n\\end{alignedat}\n$$\nwhere $r$ and $q$ are given respectively in \\eqref{phi} and \\eqref{q}, while we still consider $D_1$ and $D_2$ as defined in \\eqref{costanti}.\nThen, by considering $\\varepsilon=\\lambda^{\\nu_1\/q}$ in \\eqref{binlin4}, since \\eqref{q} implies that $\\nu_3<0$, for any $\\lambda\\in(0,\\lambda_1)$ we have\n$$\nJ_{n,\\lambda}(t_\\varepsilon\\psi_\\varepsilon)\\leq D_1+C_1\\lambda^{\\frac{2\\theta}{2\\theta-1+\\gamma}}-C_2\\lambda^{\\nu_2}\n=D_1+\\lambda^{\\frac{2\\theta}{2\\theta-1+\\gamma}}(C_1-C_2\\lambda^{\\nu_3})\n<D_1-D_2\\lambda^{\\frac{2\\theta}{2\\theta-1+\\gamma}},\n$$\nwhich concludes the proof.\n\\end{proof}\n\nWe can now prove the existence result for \\eqref{Pn} by applying the mountain pass theorem.\n\\begin{theorem}\\label{seconda} There exists $\\overline{\\lambda}>0$ such that, for any $\\lambda\\in(0,\\overline{\\lambda})$, problem \\eqref{Pn} has a positive solution $v_n\\in X_0$, with \n\\begin{equation}\\label{necessario}\n\\alpha<J_{n,\\lambda}(v_n)<D_1-D_2\\lambda^{2\\theta\/(2\\theta-1+\\gamma)},\n\\end{equation}\nwhere $\\alpha$, $D_1$ and $D_2$ are given respectively in Lemma \\ref{mp} and \\eqref{costanti}.\n\\end{theorem}\n\\begin{proof}\nLet $\\overline{\\lambda}=\\min\\{\\lambda_0,\\lambda_1\\}$, with $\\lambda_0$ and $\\lambda_1$ given respectively in Lemmas \\ref{mp} and \\ref{lemma3.3}.\nLet us consider $\\lambda\\in(0,\\overline{\\lambda})$.\nBy Lemma \\ref{mp2}, functional $J_{n,\\lambda}$ verifies the mountain pass geometry. For this, we can set the critical mountain pass level as\n\\begin{equation}\\label{mm}\nc_{n,\\lambda}=\\inf_{g\\in\\Gamma}\\max_{t\\in[0,1]}J_{n,\\lambda}(g(t)),\n\\end{equation}\nwhere\n$$\\Gamma=\\left\\{g\\in C([0,1],X_0):\\,\\,g(0)=0,\\; J_{n,\\lambda}(g(1))<0\\right\\}.$$\nBy Lemmas \\ref{mp2} and \\ref{lemma3.3}, we get\n$$\n0<\\alpha<c_{n,\\lambda}\\leq\\sup_{t\\geq0} J_{n,\\lambda}(t\\psi)<D_1-D_2\\lambda^{2\\theta\/(2\\theta-1+\\gamma)}.\n$$\nHence, by Lemma \\ref{palais} functional $J_{n,\\lambda}$ satisfies the Palais--Smale condition at level $c_{n,\\lambda}$. Thus, the mountain pass theorem gives the existence of a critical point $v_n\\in X_0$ for $J_{n,\\lambda}$ at level $c_{n,\\lambda}$. Since $J_{n,\\lambda}(v_n)=c_{n,\\lambda}>\\alpha>0=J_{n,\\lambda}(0)$, $v_n$ is a nontrivial solution of \\eqref{Pn}. Furthermore, by \\eqref{weak2} with test function $\\varphi=v^-_n$ and inequality \\eqref{eur}, we can see that $\\|v^-_n\\|=0$, which implies $v_n$ is nonnegative. By the maximum principle in \\cite[Proposition 2.2.8]{S}, we have that $v_n$ is a positive solution of \\eqref{Pn}, concluding the proof.\n\\end{proof}\n\\section{A second solution for problem \\eqref{P}}\\label{sec finale}\nIn this last section we prove the existence of a second solution for problem \\eqref{P}, as a limit of solutions of the perturbed problem \\eqref{Pn}. For this, here we need the assumption that $\\partial\\Omega$ is continuous, in order to apply a density argument for space $X_0$.\n\\begin{proof}[\\textit Proof of Theorem \\ref{main}]\nLet us consider $\\overline{\\lambda}$ as given in Theorem \\ref{seconda} and let $\\lambda\\in(0,\\overline{\\lambda})$.\nSince $\\overline{\\lambda}\\leq\\lambda_0$, by Theorem \\ref{prima} we know that problem \\eqref{P} admits a solution $u_0$ with $J_\\lambda(u_0)<0$.\n\nIn order to find a second solution for \\eqref{P}, let $\\{v_n\\}_n$ be a family of positive solutions of \\eqref{Pn}.\nBy \\eqref{S}, \\eqref{subadd}, \\eqref{necessario} and the H\\\"older inequality, we have\n$$\n\\begin{aligned}\nD_1-D_2\\lambda^{\\frac{2\\theta}{2\\theta-1+\\gamma}}&> J_{n,\\lambda}(v_n)-\\frac{1}{2^*_s}\\langle J'_{n,\\lambda}(v_n), v_n\\rangle\\\\\n&=\\left(\\frac{1}{2\\theta}-\\frac{1}{2^*_s}\\right)\\|v_n\\|^{2\\theta}-\\frac{\\lambda}{1-\\gamma}\\int_\\Omega[(v_n+1\/n)^{1-\\gamma}-(1\/n)^{1-\\gamma}]dx\\\\\n&\\quad+\\frac{\\lambda}{2^*_s}\\int_\\Omega(v_n+1\/n)^{-\\gamma}v_n dx\\\\\n&\\geq\\left(\\frac{1}{2\\theta}-\\frac{1}{2^*_s}\\right)\\|v_n\\|^{2\\theta}-\\frac{\\lambda}{1-\\gamma}\\int_\\Omega v_n^{1-\\gamma}dx\\\\\n&\\geq\\left(\\frac{1}{2\\theta}-\\frac{1}{2^*_s}\\right)\\|v_n\\|^{2\\theta}-\\frac{\\lambda}{1-\\gamma}|\\Omega|^{\\frac{2^*_s-1+\\gamma}{2^*_s}}S^{-\\frac{1-\\gamma}{2}}\\|v_n\\|^{1-\\gamma},\n\\end{aligned}\n$$\nwhich yields that $\\{v_n\\}_n$ is bounded in $X_0$, being $1-\\gamma<1<2\\theta$.\nHence, by using \\cite[Lemma 8]{SV} and \\cite[Theorem 4.9]{B}, there exist a subsequence, still\ndenoted by $\\{v_n\\}_n$, and a function $v_0\\in X_0$\nsuch that\n\\begin{equation}\\label{2.4}\n\\begin{array}{ll}\nv_n\\rightharpoonup v_0\\text{ in }X_0,\\quad & \\|v_n\\|\\rightarrow \\mu, \\\\\nv_n\\rightharpoonup v_0\\mbox{ in } L^{2^*_s}(\\Omega),\\quad &\\left\\|v_n-v_0\\right\\|_{2^*_s}\\to\\ell,\\\\\nv_n\\rightarrow v_0\\mbox{ in } L^p(\\Omega)\\text{ for any }p\\in[1,2^*_s),\\quad&v_n\\to v_0\\text{ a.e.  in }\\Omega.\n\\end{array}\n\\end{equation}\nWe want to prove that $v_n\\to v_0$ in $X_0$ as $n\\to\\infty$. When $\\mu=0$, by \\eqref{2.4} we have $v_n\\to 0$ in $X_0$ as $n\\to\\infty$. For this, we suppose $\\mu>0$.\nWe observe that\n$$0\\leq\\frac{v_n}{(v_n+1\/n)^\\gamma}\\leq v_n^{1-\\gamma}\\quad\\mbox{a.e. in }\\Omega,\n$$\nso by the Vitali convergence theorem and \\eqref{2.4}, it follows that\n\\begin{equation}\\label{2.5}\n\\lim_{n\\to\\infty}\\int_{\\Omega}\\frac{v_n}{(v_n+1\/n)^\\gamma}dx=\\int_{\\Omega}v_0^{1-\\gamma}dx.\n\\end{equation}\nBy using \\eqref{weak2} for $v_n$ and test function $\\varphi=v_n$, by \\eqref{2.4} and \\eqref{2.5}, as $n\\to\\infty$ we have\n\\begin{equation}\\label{4.2}\n\\mu^{2\\theta}-\\lambda \\int_\\Omega v_0^{1-\\gamma}dx+\\|v_n\\|^{2^*_s}_{2^*_s}=o(1).\n\\end{equation}\nFor any $n\\in\\mathbb N$, by an immediate calculation in \\eqref{Pn} we see that\n$$\\|v_n\\|^{2\\theta}(-\\Delta)^sv_n\\geq\\min\\left\\{1,\\frac{\\lambda}{2^\\gamma}\\right\\}\\quad\\mbox{in }\\Omega.\n$$\nThus, since $\\{v_n\\}_n$ is bounded in $X_0$ and by using a standard comparison argument (see \\cite[Lemma 2.1]{BMP}) and the maximum principle in \\cite[Proposition 2.2.8]{S}, for any $\\widetilde{\\Omega}\\subset\\subset\\Omega$, there exists a constant $c_{\\widetilde{\\Omega}}>0$ such that\n\\begin{equation}\\label{basso}\nv_n\\geq c_{\\widetilde{\\Omega}}>0,\\quad\\mbox{a.e. in }\\widetilde{\\Omega}\\mbox{ and for any }n\\in\\mathbb N.\n\\end{equation}\nNow, let $\\varphi\\in C^\\infty_0(\\Omega)$, with $\\mbox{\\small supp }\\varphi=\\widetilde{\\Omega}\\subset\\subset\\Omega$. By \\eqref{basso} we have\n$$0\\leq\\left|\\frac{\\varphi}{(v_n+1\/n)^\\gamma}\\right|\\leq\\frac{|\\varphi|}{c_{\\widetilde{\\Omega}}^\\gamma}\\quad\\mbox{a.e. in }\\Omega,\n$$\nso that by \\eqref{2.4} and the dominated convergence theorem\n\\begin{equation}\\label{2.6}\n\\lim_{n\\to\\infty}\\int_{\\Omega}\\frac{\\varphi}{(v_n+1\/n)^\\gamma}dx=\\int_{\\Omega}v_0^{-\\gamma}\\varphi dx.\n\\end{equation}\nThus, by considering \\eqref{weak2} for $v_n$, sending $n\\to\\infty$ and using \\eqref{2.4} and \\eqref{2.6}, for any $\\varphi\\in C^\\infty_0(\\Omega)$ it follows that\n\\begin{equation}\\label{4.3}\n\\mu^{2(\\theta-1)}\\langle v_0,\\varphi\\rangle-\\lambda \\int_\\Omega v_0^{-\\gamma}\\varphi dx\n +\\int_\\Omega v_0^{2^*_s-1}\\varphi dx=0.\n\\end{equation}\nHowever, since $\\partial\\Omega$ is continuous, by \\cite[Theorem 6]{FSV} the space $C^\\infty_0(\\Omega)$ is dense in $X_0$. Thus, by a strandard density argument, \\eqref{4.3} holds true for any $\\varphi\\in X_0$.\nBy combining \\eqref{4.2} and \\eqref{4.3} with test function $\\varphi=v_0$, as $n\\to\\infty$ we get\n$$\n\\mu^{2(\\theta-1)}(\\mu^2-\\|v_0\\|^2)=\\|v_n\\|^{2^*_s}_{2^*_s}-\\|v_0\\|^{2^*_s}_{2^*_s}+o(1)\n$$\nand by \\eqref{2.4} and \\cite[Theorem 2]{BL} we have\n\\begin{equation}\\label{crucial}\n\\mu^{2(\\theta-1)}\\lim_{n\\to\\infty}\\|v_n-v_0\\|^2=\\ell^{2^*_s}.\n\\end{equation}\nIf $\\ell=0$, then $v_n\\to v_0$ in $X_0$ as $n\\to\\infty$, since $\\mu>0$.\n\nLet us suppose $\\ell>0$ by contradiction.\nArguing as in Lemma \\ref{palais}, by \\eqref{2.4} and \\eqref{crucial} we get \\eqref{eq31}.\nTherefore, being $\\theta\\geq1$, by \\eqref{subadd}, \\eqref{eq31}, \\eqref{necessario}, \\eqref{2.4}, the H\\\"older inequality and the Young inequality, we have \n$$\n\\begin{alignedat}4\nD_1-D_2\\lambda^{\\frac{2\\theta}{2\\theta-1+\\gamma}}&>J_{n,\\lambda}(v_n)-\\frac{1}{2^*_s}\\left\\langle J'_{n,\\lambda}(v_n),v_n\\right\\rangle\\\\\n&\\geq\\left(\\frac{1}{2\\theta}-\\frac{1}{2^*_s}\\right)\\left(\\mu^{2\\theta}+\\|v_0\\|^{2\\theta}\\right)-\\lambda\\left(\\frac{1}{1-\\gamma}+\\frac{1}{2^*_s}\\right)|\\Omega|^{\\frac{2^*_s-1+\\gamma}{2^*_s}}S^{-\\frac{1-\\gamma}{2}}\\|v_0\\|^{1-\\gamma}\\\\\n&\\geq\\left(\\frac{1}{2\\theta}-\\frac{1}{2^*_s}\\right)S^{\\frac{2^*_s\\theta}{2^*_s-2\\theta}}\\\\\n&\\quad-\\left(\\frac{1}{2\\theta}-\\frac{1}{2^*_s}\\right)^{-\\frac{1-\\gamma}{2\\theta-1+\\gamma}}\\left[\\lambda\\left(\\frac{1}{1-\\gamma}+\\frac{1}{2^*_s}\\right)|\\Omega|^{\\frac{2^*_s-1+\\gamma}{2^*_s}}S^{-\\frac{1-\\gamma}{2}}\\right]^{\\frac{2\\theta}{2\\theta-1+\\gamma}}\n\\end{alignedat}\n$$\nwhich is the desired contradiction, thanks to \\eqref{costanti}.\n\nTherefore, $v_n\\to v_0$ in $X_0$ as $n\\to\\infty$ and by \\eqref{weak} and \\eqref{weak2} we immediately see that $v_0$ is a solution of problem \\eqref{P+}. Furthermore, by \\eqref{necessario} we have $J_\\lambda(v_0)\\geq\\alpha>0$, which also implies that $v_0$ is nontrivial. Reasoning as at the end of the proof of Theorem \\ref{seconda}, we conclude that $v_0$ is a positive solution of \\eqref{P+} and so $v_0$ also solves problem \\eqref{P}.\nFinally, $v_0$ is different from $u_0$, since $J_\\lambda(v_0)>0>J_\\lambda(u_0)$.\n\\end{proof}\n\n\n\\section*{Acknowledgments}\nThe author is supported by {\\em Coordena\\c c\\~ao de Aperfei\\c conamento de pessoal de n\\'ivel superior} through the fellowship PNPD--CAPES 33003017003P5.\nThe author is member of the {\\em Gruppo Nazionale per l'Analisi Ma\\-tema\\-tica, la Probabilit\\`a e\nle loro Applicazioni} (GNAMPA) of the {\\em Istituto Nazionale di Alta Matematica ``G. Severi\"} (INdAM).\n\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction}\n\n\n\nLet $\\mathscr{H}$ be a complex Hilbert space with the inner product $\\langle \\cdot,\\cdot \\rangle $ and the corresponding norm $\\|\\cdot\\|$ induced by the inner product. Let $ \\mathbb{B}(\\mathscr{H})$ denote the $C^*$-algebra of all bounded linear operators on $\\mathscr{H}$ with the identity operator  $I$ and the zero operator $O$. Let $A\\in \\mathbb{B}(\\mathscr{H})$. We denote by $|A|=({A^*A})^{\\frac{1}{2}}$ the positive square root of $A$, and  $\\Re(A)=\\frac{1}{2}(A+A^*)$ and $\\Im(A)=\\frac{1}{2\\rm i}(A-A^*)$, respectively, stand for the real and imaginary part of $A$. The numerical range of $A$, denoted by $W(A)$, is defined as $W(A)=\\left \\{\\langle Ax,x \\rangle: x\\in \\mathscr{H}, \\|x\\|=1 \\right \\}.$\nWe denote by $\\|A\\|$ and $w(A)$ the operator norm and the numerical radius of $A$, respectively, and are defined as  $$\\|A\\|=\\sup  \\left \\{\\| Ax\\|: x\\in \\mathscr{H}, \\|x\\|=1 \\right \\}$$ and $$w(A)=\\sup \\left \\{|\\langle Ax,x \\rangle|: x\\in \\mathscr{H}, \\|x\\|=1 \\right \\}.$$ \nIt is well-known that the numerical radius $ w(\\cdot)$ defines a norm on $\\mathbb{B}(\\mathscr{H})$ and is equivalent to the operator norm $\\|\\cdot\\|$. In fact, the following double inequality holds:\n\\begin{eqnarray}\\label{eqv}\n\\frac{1}{2} \\|A\\|\\leq w({A})\\leq\\|A\\|.\n\\end{eqnarray}\nThe inequalities in (\\ref{eqv}) are sharp. The first inequality becomes equality if $A^2=O$, and the second one turns into equality if $A$ is normal. For various refinements of (\\ref{eqv}), we refer the reader to \\cite{BBP_aofa, BP_adm, P18, PK_rm, PSK, YAM1} and references therein. In particular, Kittaneh \\cite{K05} improved the inequalities in (\\ref{eqv}) by establishing that \n\\begin{eqnarray}\\label{k5}\n\\frac{1}{4}\\|A^*A+A{A}^*\\|\\leq w^2({A})\\leq\\frac{1}{2}\\|A^*A+A{A}^*\\|.\n\\label{d}\\end{eqnarray}\n\\smallskip\n Kittaneh \\cite{K03} also improved the upper bound of $w(T)$ in (\\ref{eqv}) to show that \n\\begin{eqnarray}\\label{k3}\n w({A})\\leq\\frac{1}{2}\\left( \\|A\\|+\\|A^2\\|^{1\/2} \\right ).\n\\end{eqnarray}\nFurther, Abu-Omar and Kittaneh \\cite{AK15} obtained the following inequality which refines both the upper bounds in  (\\ref{k5})  and  (\\ref{k3}):\n\\begin{eqnarray}\\label{k15}\n w^2({A})\\leq\\frac{1}{4}\\|A^*A+A{A}^*\\|+\\frac{1}{2}w(A^2).\n\\end{eqnarray}\nRecently, Bhunia and Paul \\cite{PK} also improved both the upper bounds  in (\\ref{k5})  and  (\\ref{k3}) by developing that\n \\begin{eqnarray}\\label{pk21}\n w^2({A})\\leq\\frac{1}{4}\\|A^*A+A{A}^*\\|+\\frac{1}{2}w(|A||A^*|).\n \\end{eqnarray}\n \n In this paper,  we derive  inequalities for the bounds of  the numerical radius which generalize and improve on both in (\\ref{k15})  and  (\\ref{pk21}). Further, we obtain a lower bound for the numerical radius which generalizes and improves on the existing ones. Applications of the obtained inequalities are also given.\n\n\n\\section{Main Results}\n\nWe begin our work with noting that for $A,B\\in \\mathbb{B}(\\mathscr{H})$, the $2\\times 2$ off-diagonal operator matrix $\\begin{pmatrix}\nO&A\\\\ B&O \n\\end{pmatrix}\\in \\mathbb{B}(\\mathscr{H}\\oplus \\mathscr{H})$  and the numerical radius of the matrix is denoted by $w\\begin{pmatrix}\nO&A\\\\ B&O \n\\end{pmatrix}$. To achieve our aim in this paper we need the following four lemmas. First lemma follows easily.\n\n\\begin{lemma}\\label{lem1}\n\tIf $A,B\\in \\mathbb{B}(\\mathscr{H})$, then\n\t$$w\\begin{pmatrix}\n\tA&O\\\\ O&B \n\t\\end{pmatrix}=\\max \\Big \\{w(A), w(B)  \\Big\\}.$$\n\\end{lemma}\n\nSecond lemma deals with positive operators, and  is known as McCarthy inequality.\n\n\\begin{lemma}$($\\cite[p. 20]{simon}$)$.\\label{lem2}\n\tIf $A\\in \\mathbb{B}(\\mathscr{H})$ is positive, then for $r\\geq 1$ \n\t\\[\\langle Ax,x\\rangle^r\\leq \\langle A^rx,x\\rangle,\\]  \n\tfor all $x \\in \\mathscr{H}$ with $\\|x\\|=1.$\n\\end{lemma}\n\nThird lemma deals with  vectors in $\\mathscr{H}$, and is known as Buzano's inequality.\n\n\\begin{lemma}$($\\cite{BU}$)$\\label{lem3}\n\tIf $x,y,e\\in \\mathscr{H}$ with $\\|e\\|=1,$ then \n\t\\[|\\langle x,e\\rangle \\langle e,y\\rangle|\\leq \\frac{1}{2}\\Big(\\|x\\| \\|y\\|+|\\langle x,y\\rangle|\\Big).\\]\n\\end{lemma}\n\nFourth lemma is known as  mixed Schwarz inequality.\n\n\\begin{lemma}$($\\cite{H}$)$\\label{lem4}\n\tIf $A\\in \\mathbb{B}(\\mathscr{H})$, then\n\t\\[|\\langle Ax,y \\rangle| \\leq \\langle |A|x,x \\rangle ^{1\/2} \\langle |A^*|y,y \\rangle ^{1\/2},\\]\n\tfor all $x,y \\in \\mathscr{H}$.\n\\end{lemma}\n\nWe are now in a position to prove our first desired inequality.\n\n\n\\begin{theorem}\\label{th1}\n\tIf $A,B\\in \\mathbb{B}(\\mathscr{H})$, then \n\t\\begin{eqnarray*}\n&&\tw^{2r}\\begin{pmatrix}\n\t\tO&A\\\\ B&O\n\t\\end{pmatrix} \\\\\n&& \\leq  \\frac{1}{4} \\max \\Big \\{ \\left \\| |B|^{2r}+|A^*|^{2r}  \\right \\|, \\left \\| |A|^{2r}+|B^*|^{2r} \\right \\| \\Big\\}+\\frac{(1-\\alpha)}{2} \\max \\Big \\{w^r(AB), w^r(BA) \\Big \\}\\\\\n&&  + \\frac{\\alpha}{2} \\max \\Big \\{\\left \\|\\Re(|B|^r|A^*|^r) \\right \\|,  \\left \\|\\Re(|A|^r|B^*|^r) \\right \\|\\Big \\},\n\t\\end{eqnarray*}\nfor all $\\alpha\\in [0,1]$ and for all $r\\geq 1$.\n\\end{theorem}\n\n\\begin{proof}\nLet $T=\\begin{pmatrix}\nO&A\\\\ B&O\n\\end{pmatrix}$\tand let $x\\in \\mathscr{H}\\oplus \\mathscr{H} $ with $\\|x\\|=1$. Then we have,\n\\begin{eqnarray*}\n&& |\\langle Tx,x\\rangle|^{2r}\\\\\n&=&\\alpha |\\langle Tx,x\\rangle|^{2r}+(1-\\alpha)|\\langle Tx,x\\rangle|^{2r}\\\\\n&\\leq& \\alpha \\left (\\langle |T|x,x\\rangle^{1\/2} \\,\\langle |T^*|x,x\\rangle^{1\/2} \\right)^{2r}+ (1-\\alpha)|\\langle Tx,x\\rangle|^{2r}\\,\\,\\,\\Big(\\textit{by Lemma \\ref{lem4}}\\Big)\\\\\n&=& \\alpha \\left (\\langle |T|x,x\\rangle^{r\/2} \\,\\langle |T^*|x,x\\rangle^{r\/2} \\right)^{2}+ (1-\\alpha)|\\langle Tx,x\\rangle|^{2r}\\\\\n&\\leq& \\alpha \\left (\\langle |T|^rx,x\\rangle^{1\/2} \\,\\langle |T^*|^rx,x\\rangle^{1\/2} \\right)^{2}+ (1-\\alpha)|\\langle Tx,x\\rangle|^{2r} \\,\\,\\,\\Big(\\textit{by Lemma \\ref{lem2}}\\Big)\\\\\n&\\leq & \\alpha \\left ( \\frac{\\langle |T|^r x,x\\rangle+\\langle |T^*|^r x,x\\rangle}{2} \\right)^2+(1-\\alpha)|\\langle Tx,x\\rangle|^{2r} \\\\\n&= & \\alpha \\left ( \\frac{\\langle ( |T|^r+|T^*|^r) x,x\\rangle}{2} \\right)^2+(1-\\alpha)|\\langle Tx,x\\rangle|^{2r}  \\\\\n&\\leq & \\frac{\\alpha}{4} \\langle ( |T|^r+|T^*|^r)^2 x,x\\rangle  +(1-\\alpha)|\\langle Tx,x\\rangle\\, \\langle x,T^*x\\rangle|^r \\,\\,\\,\\Big(\\textit{by Lemma \\ref{lem2}}\\Big)\\\\\n&\\leq & \\frac{\\alpha}{4} \\langle ( |T|^r+|T^*|^r)^2 x,x\\rangle  +\\frac{(1-\\alpha)}{2^r}\\big(\\|Tx\\| \\, \\|T^*x\\|+ |\\langle Tx,T^*x\\rangle|\\big)^r \\,\\,\\,\\Big(\\textit{by Lemma \\ref{lem3}}\\Big)\\\\\n&\\leq & \\frac{\\alpha}{4} \\langle ( |T|^r+|T^*|^r)^2 x,x\\rangle  +\\frac{(1-\\alpha)}{2}\\big(\\|Tx\\|^r \\, \\|T^*x\\|^r+ |\\langle Tx,T^*x\\rangle|^r\\big)\\\\\n&\\leq & \\frac{\\alpha}{4} \\langle ( |T|^r+|T^*|^r)^2 x,x\\rangle  +\\frac{(1-\\alpha)}{2}\\left(\\frac{\\|Tx\\|^{2r}+ \\|T^*x\\|^{2r}}{2}+ |\\langle T^2x,x\\rangle|^r\\right)\\\\\n&\\leq & \\frac{\\alpha}{4} \\langle ( |T|^r+|T^*|^r)^2 x,x\\rangle  +\\frac{(1-\\alpha)}{2}\\left(\\frac{\\langle (|T|^{2r}+ |T^*|^{2r})x,x \\rangle}{2}+ |\\langle T^2x,x\\rangle|^r\\right)\\\\\n&= & \\frac{\\alpha}{4} \\langle \\left( |T|^{2r}+|T^*|^{2r}+|T|^r\\,|T^*|^r+|T^*|^r\\,|T|^r \\right) x,x\\rangle  \\\\\n&& \\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,+\\frac{(1-\\alpha)}{2}\\left(\\frac{\\langle (|T|^{2r}+ |T^*|^{2r})x,x \\rangle}{2}+ |\\langle T^2x,x\\rangle|^r\\right)\\\\\n&= & \\frac{1}{4} \\langle \\left( |T|^{2r}+|T^*|^{2r} \\right) x,x\\rangle +\\frac{\\alpha}{2} \\langle \\Re(|T|^r\\,|T^*|^r)x,x\\rangle +\\frac{(1-\\alpha)}{2}|\\langle T^2x,x\\rangle|^r\\\\\n&\\leq & \\frac{1}{4} \\langle \\left( |T|^{2r}+|T^*|^{2r} \\right) x,x\\rangle +\\frac{\\alpha}{2} |\\langle \\Re(|T|^r\\,|T^*|^r)x,x\\rangle| +\\frac{(1-\\alpha)}{2}|\\langle T^2x,x\\rangle|^r\\\\\n&\\leq & \\frac{1}{4} \\left \\|  |T|^{2r}+|T^*|^{2r} \\right \\| +\\frac{\\alpha}{2} \\left \\| \\Re(|T|^r\\,|T^*|^r) \\right \\|+\\frac{(1-\\alpha)}{2} w^r( T^2 ).\n\\end{eqnarray*}\nThus, taking supremum over $\\|x \\|=1$, we have\n\\begin{eqnarray}\nw^{2r}(T)\n&\\leq & \\frac{1}{4} \\left \\|  |T|^{2r}+|T^*|^{2r} \\right \\| +\\frac{\\alpha}{2} \\left \\| \\Re(|T|^r\\,|T^*|^r) \\right \\|+\\frac{(1-\\alpha)}{2} w^r( T^2 ).\n\\end{eqnarray}\nThat is,\n\\begin{eqnarray*}\nw^{2r}\\begin{pmatrix}\n\tO&A\\\\ B&O\n\\end{pmatrix}\n&\\leq & \\frac{1}{4} \\left \\|  \\begin{pmatrix}\n\t|B|^{2r}+|A^*|^{2r} &O\\\\ O&|A|^{2r}+|B^*|^{2r}\n\\end{pmatrix} \\right \\| \\\\\n&& \\,+\\frac{\\alpha}{2} \\left \\|   \\begin{pmatrix}\n\\Re(|B|^r\\,|A^*|^r)&O\\\\ O&\\Re(|A|^r\\,|B^*|^r)\n\\end{pmatrix} \\right \\|+\\frac{(1-\\alpha)}{2} w^r\\begin{pmatrix}\nAB&O\\\\ O&BA\n\\end{pmatrix}.\n\\end{eqnarray*}\nTherefore, the required inequality follows from Lemma \\ref{lem1}.\n\n\n\t\n\t\n\t\n\\end{proof}\n\n\\begin{remark}\n\nIn particular, considering $\\alpha=1$ and $r=1$ in the above theorem we get that\n\\begin{eqnarray*}\n\t&& w^2\\begin{pmatrix}\n\t\tO&A\\\\ B&O\n\t\\end{pmatrix} \\leq \\\\\n& & \\frac{1}{4} \\max \\Big \\{ \\left \\| |B|^2+|A^*|^2  \\right \\|, \\left \\| |A|^2+|B^*|^2  \\right \\| \\Big\\}  + \\frac{1}{2} \\max \\Big \\{\\left \\|\\Re(|B||A^*|) \\right \\|,  \\left \\|\\Re(|A||B^*|) \\right \\|\\Big \\},\n\\end{eqnarray*}\nwhich refines the existing one \\cite[Th. 2.10]{PK2}, namely\n\\begin{eqnarray*}\n&&\tw^2\\begin{pmatrix}\n\t\tO&A\\\\ B&O\n\t\\end{pmatrix} \\leq \\\\\n& & \\frac{1}{4} \\max \\Big \\{ \\left \\| |B|^2+|A^*|^2  \\right \\|, \\left \\| |A|^2+|B^*|^2  \\right \\| \\Big\\} + \\frac{1}{2} \\max \\Big \\{w(|B||A^*|),  w(|A||B^*|) \\Big \\}.\n\\end{eqnarray*}\n\n\\end{remark}\n\nTo obtain our next result we need the following lemma, which can be found in  \\cite[Lemma 2.1]{HKS}.\n\n\n\\begin{lemma}\\label{lem5}\n\tIf $A,B\\in \\mathbb{B}(\\mathscr{H})$, then\n\t$$w \\left(\\begin{array}{cc}\n\tA & B\\\\\n\tB& A\n\t\\end{array}\\right)=\\max \\Big \\{w(A+B),w(A-B) \\Big\\}.$$\nIn particular, \\,\\,\\,\\,\\,\\,\\, \\,\\,\\,\\,\\,\\,$w \\left( \\begin{array}{cc}\n\tO & B\\\\\n\tB & O\n\t\\end{array} \\right)=w(B).$\n\\end{lemma}\n\n\\begin{cor}\\label{cor1}\nIf $A\\in \\mathbb{B}(\\mathscr{H})$, then\n\\begin{eqnarray*}\nw^{2r}(A) \\leq \\frac{1}{4} \\left \\| |A|^{2r}+|A^*|^{2r}  \\right \\|+ \\frac{1}{2} \\min \\Big \\{ \\left \\|\\Re(|A|^r|A^*|^r) \\right \\|, \\,\\,  w^r(A^2) \\Big \\},\n\\end{eqnarray*}\nfor all $r\\geq 1$.\t\n\\end{cor}\n\\begin{proof}\n\tConsidering $A=B$ in Theorem \\ref{th1}, and then using Lemma \\ref{lem5} we infer that\n\t\\begin{eqnarray}\\label{bound1}\n\t\tw^{2r}(A) \\leq \\frac{1}{4} \\left \\| |A|^{2r}+|A^*|^{2r} \\right \\|+ \\frac{1}{2} \\Big \\{ \\alpha \\left \\|\\Re(|A|^r|A^*|^r) \\right \\| +(1-\\alpha)  w^r(A^2) \\Big \\},\n\t\\end{eqnarray}\nfor all $\\alpha \\in [0,1]$ and for all $r\\geq 1$. This implies the desired inequality. \n\\end{proof}\n\n\\begin{remark}\n\t(i) Since $\\left \\|\\Re(|A||A^*|) \\right \\|\\leq w(|A||A^*|) $, so we would like to remark that Corollary \\ref{cor1} (for $r=1$) gives stronger inequality than that in both (\\ref{k15}) and (\\ref{pk21}). \\\\\n\t(ii) If for norm one sequences $\\{x_n\\}$ in $\\mathscr {H}$ with $|\\langle \\Re(|T|~|T^*|)  x_n,x_n \\rangle| \\to \\|\\Re(|T|~|T^*|)\\|$ and $|\\langle \\Im(|T|~|T^*|)  x_n,x_n \\rangle| \\to \\lambda ( \\neq 0 )$, then Corollary \\ref{cor1} (for $r=1$) gives strictly stronger inequality than that in (\\ref{pk21}). \n\\end{remark}\n\nWe next prove the following theorem.\n\n\\begin{theorem}\n\tLet $A\\in \\mathbb{B}(\\mathscr{H})$  be such that $\\Re\\big(|A||A^*| \\big)=O.$ Then $\\overline{W(A)}$ is a circular disk with center at the origin and radius $\\frac{1}{2}\\sqrt{\\|A^*A+AA^*\\|}$.  \n\\end{theorem}\n\n\\begin{proof}\n\tFrom Corollary \\ref{cor1}, we get for the case $r=1,$\n\t\\begin{eqnarray}\\label{main}\n\tw^{2}(A) \\leq \\frac{1}{4} \\left \\| |A|^{2}+|A^*|^{2} \\right \\|+ \\frac{1}{2}   \\left \\|\\Re(|A||A^*|) \\right \\|. \n\t\\end{eqnarray}\nThe first inequality in (\\ref{k5}) together with (\\ref{main}) we infer that\n $  w(A) = \\frac{1}{2} \\sqrt{\\left \\| |A|^{2}+|A^*|^{2}  \\right \\|} = \\frac{1}{2} \\sqrt{\\|A^*A+AA^*\\|} $ \tif  $\\Re\\big(|A||A^*| \\big)=O.$  \n\tTherefore, from \\cite[Th. 2.14]{BPaul} it follows that  $\\overline{W(A)}$ is a circular disk with center at the origin and radius $\\frac{1}{2}\\sqrt{\\|A^*A+AA^*\\|}$.  \n\t\n\\end{proof}\n\n\n\nWe observe that the converse of the above result is, in general, not true. There exists  an operator $A\\in \\mathbb{B}(\\mathscr{H})$ for which  $w(A) = \\frac{1}{2}\\sqrt{  \\left \\| |A|^2+|A^*|^2  \\right \\|},$ but $\\Re\\big(|A||A^*| \\big)\\neq O$.\n\tConsider  $A=\\begin{pmatrix}\n\t\t0&1&0\\\\\n\t\t0&0&1\\\\\n\t\t0&0&0\n\t\\end{pmatrix}$. Then one can verify  that $w(A) = \\frac{1}{2}\\sqrt{  \\left \\| |A|^2+|A^*|^2  \\right \\|}  =\\frac{1}{\\sqrt{2}} $, but  $\\Re\\big(|A||A^*| \\big)=\\begin{pmatrix}\n\t\t0&0&0\\\\\n\t\t0&1&0\\\\\n\t\t0&0&0\n\t\\end{pmatrix}$.\n\n\nNext we obtain an inequality for the numerical radius of the sum of $n$ operators which generalizes Theorem \\ref{th1}. For this, we need the following \tBohr's inequality which deals with positive real numbers.\n\n\\begin{lemma}$($\\cite{v}$)$\\label{lem4v}\n\tIf $a_i\\geq 0$ for each $i=1,2,\\ldots,n$, then \n\t\\[\\left( \\sum_{i=1}^na_i\\right)^r \\leq n^{r-1}\\sum_{i=1}^na_i^r,\\] for all $r\\geq 1$.\n\\end{lemma}\n\n\n\\begin{theorem}\\label{th1gen}\n\tIf $A_i \\in \\mathbb{B}(\\mathscr{H})$ for $ \\ i=1,2, \\ldots,n$,  then \n\t\\begin{eqnarray*}\n\t\tw^{2r}\\left( \\sum_{i=1}^{n}A_i \\right) &\\leq&     \\frac{n^{2r-1}}{4} \\left \\| \\sum_{i=1}^{n}  \\left( |A_i|^{2r}+ |A_i^*|^{2r} \\right)      \\right\\| \\\\        && +\\frac{n^{2r-1}}{2} \\left(\\alpha \\left \\| \\sum_{i=1}^{n}\\Re \\left (|A_i|^r|A_i^*|^r \\right)\\right \\|+(1-\\alpha) \\sum_{i=1}^{n}w^r \\left( A_i^{2} \\right)      \\right),  \n\t\\end{eqnarray*}\n\tfor all $\\alpha\\in [0,1]$ and for all $r\\geq 1.$\n\\end{theorem}\n\n\\begin{proof}\n\tLet $x\\in \\mathscr{H}$ with $\\|x\\|=1.$ Then  by using Lemma \\ref{lem4v} we infer that \n\t\\begin{eqnarray*}\n\t\t\\left |\\left \\langle \\left( \\sum_{i=1}^{n}A_i \\right)x,x \\right \\rangle \\right |^{2r} &=& \\left |\\sum_{i=1}^{n} \\left \\langle  A_i x,x \\right \\rangle \\right |^{2r}\\\\\n\t\t&\\leq&  \\left (\\sum_{i=1}^{n} |\\left \\langle  A_i x,x \\right \\rangle| \\right )^{2r}\\\\\n\t\t&\\leq&  n^{2r-1} \\left (\\sum_{i=1}^{n} |\\left \\langle  A_i x,x \\right \\rangle|^{2r} \\right )\\\\\n\t\t&=&  n^{2r-1} \\left (   \\alpha \\sum_{i=1}^{n} |\\left \\langle  A_i x,x \\right \\rangle|^{2r} +(1-\\alpha) \\sum_{i=1}^{n} |\\left \\langle  A_i x,x \\right \\rangle|^{2r} \\right ).\n\t\\end{eqnarray*}\n\tNow proceeding similarly  as in Theorem \\ref{th1} we have the desired inequality.\n\\end{proof}\n\n\n\n\nNext we obtain a lower bound for the numerical radius of the $2 \\times 2$ off-diagonal operator matrix $\\begin{pmatrix} \nO&A\\\\ B&O\n\\end{pmatrix}$. By considering the unitary operator matrix $\\begin{pmatrix} \nO&I\\\\ I&O\n\\end{pmatrix}$, the weak unitary invariance property for the numerical radius gives that $w\\begin{pmatrix} \nO&A\\\\ B&O\n\\end{pmatrix}=w\\begin{pmatrix} \n\tO&B\\\\ A&O\n\\end{pmatrix}$.\n\n\\begin{theorem}\\label{th2}\n If $A,B\\in \\mathbb{B}(\\mathscr {H})$, then\n \\begin{eqnarray}\\label{lower1}\n  w\\begin{pmatrix} \n  \tO&A\\\\ B&O\n  \\end{pmatrix}\\geq \\frac{1}{2} \\Big \\| \\Re(A)+{\\rm i}\\, \\Im(B) \\Big \\|+\\frac{1}{4} \\Big | \\, \\|A+B^*\\|-\\|A-B^*\\|\\,  \\Big |,\n\\end{eqnarray}\nand \n \\begin{eqnarray}\\label{lower2}\nw\\begin{pmatrix} \nO&A\\\\ B&O\n\\end{pmatrix}\\geq \\frac{1}{2} \\Big \\| \\Re(B)+{\\rm i}\\, \\Im(A) \\Big \\|+\\frac{1}{4} \\Big | \\, \\|A+B^*\\|-\\|A-B^*\\|\\,  \\Big |.\n\\end{eqnarray}\n \\end{theorem}\n\\begin{proof}\nIt is well-known that \t$$w\\begin{pmatrix} \nO&A\\\\ B&O\n\\end{pmatrix}\\geq \\left\\| \\Re\\begin{pmatrix} \nO&A\\\\ B&O\n\\end{pmatrix} \\right\\|=\\frac{1}{2}\\| A+B^*\\|$$ and $$w\\begin{pmatrix} \nO&A\\\\ B&O\n\\end{pmatrix}\\geq \\left\\| \\Im\\begin{pmatrix} \nO&A\\\\ B&O\n\\end{pmatrix} \\right\\|=\\frac{1}{2}\\| A-B^*\\|.$$\nTherefore, we have\n\\begin{eqnarray*}\nw\\begin{pmatrix} \n\tO&A\\\\ B&O\n\\end{pmatrix} &\\geq & \\frac{1}{2} \\max \\Big \\{ \\|A+B^*\\|, \\|A-B^*\\|\\Big\\}\\\\\n&=& \\frac{1}{2} \\left ( \\frac{\\|A+B^*\\|+ \\|A-B^*\\| }{2}+ \\frac{|\\,\\|A+B^*\\|-\\|A-B^*\\|\\,| }{2} \\right) \\\\\n&=& \\frac{1}{2} \\left ( \\frac{\\|A+B^*\\|+ \\|A^*-B\\| }{2}+ \\frac{|\\,\\|A+B^*\\|-\\|A-B^*\\|\\,| }{2} \\right) \\\\\n&\\geq & \\frac{1}{2} \\left ( \\frac{\\|(A+B^*)+(A^*-B)\\| }{2}+ \\frac{|\\,\\|A+B^*\\|-\\|A-B^*\\|\\,| }{2} \\right) \\\\\n&=& \\frac{1}{2} \\left ( \\|\\Re(A)-{\\rm i}\\,\\Im(B) \\|+ \\frac{|\\,\\|A+B^*\\|-\\|A-B^*\\|\\,| }{2} \\right) \\\\\n&=& \\frac{1}{2} \\left \\|\\Re(A)-{\\rm i}\\,\\Im(B) \\right \\|+ \\frac{1 }{4}\\Big|\\,\\|A+B^*\\|-\\|A-B^*\\|\\,\\Big|.\n\\end{eqnarray*}\nThis implies the inequality (\\ref{lower1}). Interchanging $A$ and $B$ in (\\ref{lower1}) we get the inequality (\\ref{lower2}). \n\\end{proof}\nAs a consequence of Theorem \\ref{th2} we get the following corollaries.\n\\begin{cor}\\label{lowerbound1}\n\tLet $ A \\in \\mathbb{B}(\\mathscr{H}).$  Then \n\\begin{eqnarray}\\label{lowerLAA}\n\tw(A)\\geq \\frac{1}{2}\\big\\|A \\big\\|+\\frac{1}{2} \\Big|\\, \\|\\Re(A)\\|-\\|\\Im(A)\\|\\,\\Big|.\n\\end{eqnarray}\n\\end{cor}\n\\begin{proof} This follows clearly from Theorem \\ref{th2}  by considering $A=B.$ \n\\end{proof}\n\n\n\\begin{cor}\\label{cor_applLAA}\n\t Let $A,B\\in \\mathbb{B}(\\mathscr {H}).$ Then \n\t\\begin{eqnarray}\\label{applLAA}\n\t\tw\\begin{pmatrix} \n\t\t\tO&A\\\\ B&O\n\t\t\\end{pmatrix}\\geq \\frac{1}{2} \\max \\Big \\{\\|A\\|, \\|B\\|  \\Big \\}+\\frac{1}{4} \\Big | \\, \\|A+B^*\\|-\\|A-B^*\\|\\,  \\Big |.\n\t\\end{eqnarray}\n\\end{cor}\n\\begin{proof} Considering the operator $ \\begin{pmatrix} \n\t\tO&A\\\\ B&O\n\t\\end{pmatrix} $ and applying the inequality (\\ref{lowerLAA}) we get the desired inequality (\\ref{applLAA}).\n\\end{proof}\n\\begin{remark}\\label{rem1}\n(i) Consider the matrix  $A=\\begin{pmatrix} \n3&0\\\\ 0&0\n\\end{pmatrix}$ and $B=\\begin{pmatrix} \n2+3{\\rm i}&0\\\\ 0&0\n\\end{pmatrix}$. Then, $\\max \\Big \\{\\|A\\|, \\|B\\|  \\Big \\}=\\sqrt{13}$ and $\\left \\|\\Re(A)+{\\rm i}\\,\\Im(B) \\right \\|=\\sqrt{18}$. Again, if we assume that  $A=\\begin{pmatrix} \n3&0\\\\ 0&0\n\\end{pmatrix}$ and $B=\\begin{pmatrix} \n0&0\\\\ 0&2+3{\\rm i}\n\\end{pmatrix}$, then $\\max \\Big \\{\\|A\\|, \\|B\\|  \\Big \\}=\\sqrt{13}$ and $\\left \\|\\Re(A)+{\\rm i}\\,\\Im(B) \\right \\|=\\sqrt{9}$. Thus, we would like to remark that the inequalities obtained in Theorem \\ref{th2} and Corollary \\ref{cor_applLAA} are, in general, not comparable.\\\\\n(ii)  We observe that the inequalities in Theorem \\ref{th2} and Corollary \\ref{cor_applLAA} are sharp.\n\\end{remark}\n\n\nNext, we study the following necessary conditions for the equalities of $w\\begin{pmatrix} \nO&A\\\\ B&O\n\\end{pmatrix} = \\frac{1}{2} \\left \\|\\Re(A)+{\\rm i}\\,\\Im(B) \\right \\|$ and $w\\begin{pmatrix} \nO&A\\\\ B&O\n\\end{pmatrix} = \\frac{1}{2} \\left \\|\\Re(B)+{\\rm i}\\,\\Im(A) \\right \\|$.\n\n\\begin{proposition}\\label{prop1}\n\tLet $A,B\\in \\mathbb{B}(\\mathscr{H})$. Then the following results hold.\\\\\n(i)\tIf $w\\begin{pmatrix} \n\tO&A\\\\ B&O\n\t\\end{pmatrix} = \\frac{1}{2} \\left \\|\\Re(A)+{\\rm i}\\,\\Im(B) \\right \\|,$ then\n\t$\\|A+B^*\\|=\\|A-B^*\\|= \\left \\|\\Re(A)+{\\rm i}\\,\\Im(B) \\right \\|$.\\\\\n(ii) If $w\\begin{pmatrix} \n\tO&A\\\\ B&O\n\\end{pmatrix} = \\frac{1}{2} \\left \\|\\Re(B)+{\\rm i}\\,\\Im(A) \\right \\|$, then \n$\\|A+B^*\\|=\\|A-B^*\\|= \\left \\|\\Re(B)+{\\rm i}\\,\\Im(A) \\right \\|$. \n\n\\end{proposition}\n\n\n\\begin{proof}\n  Suppose $w\\begin{pmatrix} \n\tO&A\\\\ B&O\n\t\\end{pmatrix} = \\frac{1}{2} \\left \\|\\Re(A)+{\\rm i}\\,\\Im(B) \\right \\|.$  Then from the inequality (\\ref{lower1}) it follows that \n\t$\\|A+B^*\\|=\\|A-B^*\\|$. Therefore,\n\t\\begin{eqnarray*}\n\t\\frac{1}{2}\\|A+B^*\\|&\\leq& w\\begin{pmatrix} \n\t\tO&A\\\\ B&O\n\t\\end{pmatrix} \\\\\n& = & \\frac{1}{2} \\left \\|\\Re(A)+{\\rm i}\\,\\Im(B) \\right \\| \\\\\n&=& \\frac{1}{2} \\left \\| \\frac{A+A^*}{2}+{\\rm i}\\,\\frac{B-B^*}{2{\\rm i}} \\right \\| \\\\\n&\\leq&  \\frac{1}{2}  \\left (  \\frac{1}{2} \\|A-B^*\\|+ \\frac{1}{2} \\|A+B^*\\| \\right )\\\\\n&=&  \\frac{1}{2} \\|A+B^*\\| \\,\\,\\,\\,\\,\\,\\Big( \\textit{since}\\,\\,\\|A+B^*\\|=\\|A-B^*\\|\\Big).\n\t\\end{eqnarray*}\nThis completes the proof of (i). \nThe proof of (ii) follows from (i) by interchanging $A$ and $B$.\n\\end{proof}\n \n\n\nFinally, as a consequence of Theorem \\ref{th2} we obtain the following inequality.\n\n\\begin{cor}\\label{cor2}\n\tIf $A,B\\in \\mathbb{B}(\\mathscr{H})$, then \n\t$$ w\\begin{pmatrix} \n\tO&A\\\\ B&O\n\t\\end{pmatrix} \\geq \\frac{1}{4}\\|A+B\\|+\\frac{1}{4}|a-b|+\\frac{1}{4} \\Big| \\|A+B^*\\|-\\|A-B^*\\|\\Big|,$$\n\twhere $a= \\left \\|\\Re(A)+{\\rm i}\\,\\Im(B) \\right \\|,b= \\left \\|\\Re(B)+{\\rm i}\\,\\Im(A) \\right \\|.$\n\tThis inequality is sharp.\n\\end{cor}\n\\begin{proof}\n\tIt follows from (\\ref{lower1}) and (\\ref{lower2}) that \n\t\\begin{eqnarray}\\label{lower3}\n\tw\\begin{pmatrix} \n\tO&A\\\\ B&O\n\t\\end{pmatrix} \\geq \\frac{1}{2} \\max \\{a,b\\}+ \\frac{1}{4} \\Big| \\|A+B^*\\|-\\|A-B^*\\|\\Big|.\n\t\\end{eqnarray}\n\tNow, \n\\begin{eqnarray}\\label{lower4}\n\t\\max \\{a,b\\} & = & \\frac{1}{2}(a+b)+\\frac{1}{2}|a-b|\\geq \\frac{1}{2}\\|A+B\\|+\\frac{1}{2}|a-b|.\n\\end{eqnarray}\nTherefore, combining (\\ref{lower4}) with (\\ref{lower3}) we infer that the desired inequality. The proof of sharpness is trivial.\n\\end{proof}\n\nWe end this section with the following result.\n\\begin{proposition}\n\tLet $A,B\\in \\mathbb{B}(\\mathscr{H})$. If $w\\begin{pmatrix} \n\tO&A\\\\ B&O\n\t\\end{pmatrix} = \\frac{1}{4}\\|A+B\\|$, then the following results hold:\n\t\n\t(i) $\\|A\\|=\\|B\\|$.\n\t\n\t(ii) $\\|A+B^*\\|=\\|A-B^*\\|$.\n\t\n\t(iii) $\\left \\|\\Re(A)+{\\rm i}\\,\\Im(B) \\right \\|= \\left \\|\\Re(B)+{\\rm i}\\,\\Im(A) \\right \\|.$\n\t\n\\end{proposition}\n\n\\begin{proof}\n\tThe proof of (i) follows from (\\ref{applLAA}). The proofs of (ii) and (iii) follow from Corollary \\ref{cor2}.\n\\end{proof}\n\n\n\n\n\n\\bibliographystyle{amsplain}\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction}\n\\IEEEPARstart {T}{he} global mobile data traffics are predicted to be increased sevenfold by 2021, in which over three-forths will be multimedia \\cite{cisco}. The ever-increasing mobile data traffic has imposed huge burden to the networks. Recently, the cheaper cache with increasing availability provides us an alternative method to accommodate the explosive data traffic. In fact, by prefetching video contents at the end users, those locally cached contents can be directly served once they are requested, thus with caching the data traffic can be reduced, and this saving is referred to as the local caching gain.\n\nIn order to fully exploit the potential benefit of the cache, Maddah-Ali and Niesen proposed a $(K;M;N)$ centralized coded caching scheme (referred to as MN scheme for brevity) in \\cite{MN}, in which a single server containing $N$ files with equal length coordinates $K$ users over a shared link, and every user is assumed to be provisioned with an identical cache of size $M$ files \\cite{MN}. The coded caching scheme consists of two independent phases, \\emph{namely,} placement phase and delivery phase, which is referred to as one \\emph{round} in this paper. The placement phase occurs at off-peak hours, in which each user is able to access the server to fulfill its cache without the knowledge of users' demands. If user only prefetches the portions of the files at server, it is called uncoded placement. If the user fulfills its cache with some linear combinations of sub-packets from multiple files, it is the coded placement. Delivery phase follows in peak hours when users' demands are revealed. The server designs and multicasts coded messages through error-free and shared links to a set of users simultaneously, in which the global multicasting gain is maintained. By the end of the delivery phase, each user reconstructs its requested content on the basis of the received coded messages and its own caching contents. For this coded caching scheme, by jointly optimizing the placement and delivery phase, the system traffic load is expected to be minimized for all possible demands.\n\nMotivated by the MN scheme, how to further reduce the required transmission load has attracted many research attentions.\nAn improved lower bound of the transmission load was derived from the combinatorial problem by optimally labeling the leaves of a directed tree in \\cite{GR}. By interference elimination, a new scheme with smaller transmission load was disclosed for the case $K\\geq N$ in \\cite{T}.\nMore generally,  the transmission load for various demand patterns was derived by modifying the delivery phase of the MN scheme in \\cite{YMA}. Note that  it was shown in \\cite{WTP} and \\cite{JCLC} that the MN scheme can achieve the minimum transmission load via graph theory and an optimization framework under a specific uncoded placement rule when $K< N$. Moreover, the MN scheme has been extended to different scenario of networks, for instance, the  multi-server systems \\cite{Shariatpanahi_16,Mital_17}, D2D networks \\cite{Ji_16}, hierarchical networks \\cite{Karamchandani_16}, combination networks \\cite{Ji_15}, and heterogenous network \\cite{Daniel17}.\n\n\n\nIt should be addressed that, all the aforementioned works considered the coded caching scheme design within one round, namely, there is only one placement-then-delivery operation. Nonetheless, in practical applications, the coded caching system should be devised to operate within multiple rounds, in which the number of users $K$ may be time varying.\nFor instance, residents (fixed users) and visiting guests (mobile users) may coexist in one network. Intuitively, the residents may stay in network for a long time (multiple rounds)\\footnote{If some fixed users request the same file in previous round, the fixed users can be removed from the coded caching design since their traffic requirements have been fulfilled.}, while those visiting guests may dynamically move in or out in different round of coded caching operation. For such a dynamic network, when applying the coded caching scheme to all the users at each round separately, the variations in the participating users may lead to frequent update in both the content caching and the signal transmission in order to make both the placement and the delivery fit the variations. Sometimes, this may become undesirable\nand resource inefficient, especially when most of the users are fixed while only few users join or leave.\nIn this dynamic setup with multiple rounds of service request, how to tailor the coded caching design, such that the content updating in placement phase will be minimized and the full caching gain in delivery phase can be retained, will be a very interesting problem. This is exactly the motivation of our work in this paper.\n\nIn order to effectively handle the dynamic coded caching requirement, we need to rethink the content caching in placement phase and the coded signal generation in delivery phase for multiple rounds, such that the content updating at those fixed users can be minimized, while the coded caching gain for all participating users in delivery phase can be maximized. Intuitively the more users join the network in the same round, the possibility of the larger coding gain achieved should be higher. Therefore, all the set of fixed and mobile users should be considered when we design a coded caching scheme. However, in practice we do not have any knowledge of mobile users in the forthcoming rounds. To handle this issue, in this paper, we propose a Concatenating based placement and the Saturating Matching based delivery design (CSM) without the knowledge of mobile users. In the placement, the concatenating method is involved, in fact it has been widely used to cope with asynchronous problems. With this concatenating method we can keep the cache content unchanged for those fixed users who have already participated in the previous round of coded cooperation. For those newly joined mobile users, the server only needs to decide on the cache content placement by further sub-dividing the packets utilized by the fixed users. In this way, we can minimize the amount of the content updating. Since the matching over bipartite graph allows us to get the sum of the coded multicast transmissions from different groups (i.e., fixed and mobile users). Motivated by this, the saturating matching based delivery scheme is proposed. Our analysis reveals that the proposed CSM coded caching scheme is order-optimal.\n\nThe rest of the paper is organized as follows. In Section \\ref{sec-pre}, the system model and some results of the original coded caching system in \\cite{MN} are reviewed to introduce the $(K_1,K_2;M_1,M_2;N)$ dynamic coded caching design problem. Then the proposed CSM coded caching scheme and its order-optimality are presented in Section \\ref{sec-the main results-1} and \\ref{sec-first-performance}, respectively. Finally, we conclude our work in Section \\ref{sec-conclusion}.\n\n\n\\section{System Model and Problem Formulation}\\label{sec-pre}\n\\subsection{The Centralized Coded Caching Model}\nLet us consider the centralized coded caching system (Fig. \\ref{fig-origin-system}),\n\\begin{figure}[h]\n\\centering\n\\includegraphics[width=3.5in]{fig-origin-system}\n\\vskip 0.2cm\n\\caption{$(K;M;N)$ coded caching system}\\label{fig-origin-system}\n\\end{figure}in which a server containing $N$ files denoted by $\\mathcal{W}=\\{W_{0}, W_{1}, \\ldots, $ $W_{N-1}\\}$ connects through an error-free shared link to $K$ users $\\mathcal{K}=\\{0,1,\\ldots,K-1\\}$ with $K<N$, and every user has a cache of size $M$ files for $M\\in[0,N]$. The system contains two independent phases,\n\\begin{itemize}\n  \\item Placement phase: each file is divided into $F$ packets of equal size, and then  each user $k\\in\\mathcal{K}$ caches some packets out of each file, which is limited by its cache size $M$. \n  Let $\\mathcal{Z}_k$ denote the cache contents at user $k$, which is assumed to be known to the server.\n  \\item Delivery phase: each user randomly requests one file from the server. The requested file by user $k$ is represented by $W_{d_k}$, and the request vector by all users is denoted by $\\mathbf{d}=(d_0,d_1,\\ldots,d_{K-1})$. \n By the caching contents and requests from all the users, the server transmits some coded messages to all users such that each user's request can be satisfied.\n\\end{itemize}\nIn such system, the amount of worst-case transmissions for all possible requests is expected to be as small as possible, which is defined as\n\\begin{eqnarray*}\nR=\\sup_{\\tiny{\\mbox{$\\begin{array}{c}\n                 \\mathbf{d}\\in[N]^K\\\\\n               \\end{array}\n$}}} \\left\\{\\frac{S_{\\mathbf{d}}}{F}\\right\\}\n\\end{eqnarray*} where $S_{\\mathbf{d}}$ is the size of transmitted messages in the delivery phase for any given request $\\mathbf{d}$.\nThe above $(K;M;N)$ coded caching system was firstly proposed in \\cite{MN}, and its performance is given in Lemma \\ref{le-MN}\n\\begin{lemma}\\rm(MN scheme \\cite{MN})\n\\label{le-MN}\nFor any positive integers $K$, $M$ and $N$ with $t=\\frac{KM}{N}\\in \\mathbb{N}^+$, there exists a $(K;M;N)$ scheme with transmission rate $R_{MN}(K;M;N)=\\frac{K-t}{t+1}$.\n\\end{lemma}\nFor better understanding, the sketch of MN scheme is depicted by Algorithm \\ref{alg-MN}, in which placement and delivery phase are included.\n\\begin{algorithm}[htb]\n\\caption{MN Scheme \\cite{MN}}\\label{alg-MN}\n\\begin{algorithmic}[1]\n\\Procedure {Placement}{$\\mathcal{K}$, $\\mathcal{W}$}\n\\State $t\\leftarrow\\frac{KM}{N}$\n\\State $\\mathfrak{T}\\leftarrow\\{\\mathcal{T}| \\mathcal{T}\\subset \\mathcal{K},|\\mathcal{T}|=t\\}$\n\\For{$n\\in[0,N)$}\n\\State Split $W_n$ into $W_n=\\{W_{n,\\mathcal{T}}|\\mathcal{T}\\in\\mathfrak{T}\\}$ of equal sized packet\n\\EndFor\n\\For{$k\\in\\mathcal{K}$}\n\\State $\\mathcal{Z}_k\\leftarrow\\{W_{n,\\mathcal{T}}|n\\in[0,N),\\mathcal{T}\\in\\mathfrak{T},k\\in\\mathcal{T}\\}$\n\\EndFor\n\\EndProcedure\n\\Procedure{Delivery}{$\\mathcal{W}$, $\\mathbf{d}$}\n\\State $\\mathfrak{S}\\leftarrow\\{\\mathcal{S}|\\mathcal{S}\\subset\\mathcal{K},|\\mathcal{S}|=t+1\\}$\n\\State Server sends $\\{\\oplus_{k\\in\\mathcal{S}} W_{d_k,\\mathcal{S}\\backslash \\{k\\}}|\\mathcal{S}\\in\\mathfrak{S}\\}$\n\\EndProcedure\n\\end{algorithmic}\n\\end{algorithm}\n\nFrom Algorithm \\ref{alg-MN}, it is clear that each file is divided into $\\binom{K}{t}$ nonoverlapping equal-sized packets, and for a given $t$, there are in total $\\binom{K}{t+1}$ coded messages. To sum up, the transmission rate of MN scheme can thus be derived as,\n\\begin{eqnarray*}\n\\label{eq-MN-rate}\nR_{MN}(K;M;N)=\\frac{{K\\choose t+1}}{{K\\choose t}}=\\frac{K-t}{t+1}.\n\\end{eqnarray*}\n\n\\subsection{Dynamic Coded Caching Problem Formulation}\nAs illustrated in Fig.\\ref{Fig-sys}, now we consider a similar network configuration as that in the above $(K;M;N)$ model, except the fact that there are two sets of users, namely, $K_1$ fixed users $\\mathcal{K}_1=\\{0,1,\\ldots,K_1-1\\}$, each of which has a cache of size $M_{1}$ files, and $K_2$ mobile users $\\mathcal{K}_2=\\{K_1,K_1+1,\\ldots\\}$, each of which has a cache of size $M_{2}$ files.\nIt should be noted that, the cache sizes of two user sets $\\mathcal{K}_1$ and $\\mathcal{K}_2$ are not necessarily the same. For notation brevity, throughout the paper, we refer to this model as $(K_1,K_2;M_1,M_2;N)$ coded caching system.\n\\begin{figure}[h]\n\\centering\n\\includegraphics[width=4in]{system}\n\\vskip 0.2cm\n\\caption{$(K_1,K_2;M_{1},M_{2};N)$ coded caching model}\\label{Fig-sys}\n\\end{figure}\nThe above $(K_1,K_2;M_1,M_2;N)$ coded caching system can be utilized to characterize the dynamic network at any round $i$, in which $K_2$ mobile users join this system and want to perform coded caching with the $K_1$ fixed users that are already in the system. The first problem is related to the placement design. We aim to design a placement scheme intended only for the $K_2$ users $\\mathcal{K}_2$ in order to minimize the caching content update, which makes sense when there is no change in the files at server, and those $K_1$ fixed users have already filled their caches in previous rounds\\footnote{Note that if some new users move into network during round $i$, their cache will be filled in placement phase of the following round, \\emph{i.e.}, round $i+1$.}. On the basis of designed caching contents for users in $\\mathcal{K}_2$, how to design the coded multicasting among all the $K_1+K_2$ users to minimize the required transmission load will be the second design objective.\n\nIn order to derive the placement and delivery scheme suitable for the aforementioned dynamic coded caching applications, the bipartite graph representation of coded caching will be utilized, and the following symbol notations are utilized in the following analysis. Basically, a graph can be denoted by $\\mathbf{G}=(\\mathcal{V},\\mathcal{E})$, where $\\mathcal{V}$ is the set of vertices and $\\mathcal{E}$ is the set of edges, and a subset of edges $\\mathcal{M}\\subseteq \\mathcal{E}$ is a matching if no two edges have a common vertex. A bipartite graph, denoted by $\\mathbf{G}=(\\mathcal{X},\\mathcal{Y}; \\mathcal{E})$, is a graph whose vertices are divided into two disjoint parts $\\mathcal{X}$ and $\\mathcal{Y}$ such that every edge in $\\mathcal{E}$ connects a vertex in $\\mathcal{X}$ to one in $\\mathcal{Y}$. For a set $X\\subseteq \\mathcal{X}$, let $N_{\\mathbf{G}}(X)$ denote the set of all vertices in $\\mathcal{Y}$ adjacent to some vertices of $X$. The degree of a vertex is defined as the number of vertices adjacent to it. If every vertex of $\\mathcal{X}$ has the same degree, we  call it the degree of $\\mathcal{X}$ and denote as $d(\\mathcal{X})$.\n\n\n\n\\section{The Concatenating based Placement and the Saturating Matching based Delivery Scheme}\\label{sec-the main results-1}\nIn this section, we mainly focus on the $(K_1,K_2;M_1,M_2;N)$ coded caching design, and its achieved transmission rate for any $(M_1,M_2)$ pairs. Before introducing our proposed scheme, intuitively, the original MN scheme can be simply applied into our considered system by regarding $\\mathcal{K}_1$ and $\\mathcal{K}_2$ as two separate user groups. Thus, for the convenience of comparison, we firstly give the following trivial examples when directly utilizing MN scheme.\n\n\\subsection{Baseline Scheme}\nIntuitively, when adopting MN scheme without updating the cache contents of users in $\\mathcal{K}_1$, all the $K_1+K_2$ users can be classified into two groups, $\\mathcal{K}_1$ and  $\\mathcal{K}_2$, and perform MN coded caching scheme for two groups separately, which will be introduced in the following Example \\ref{ex-trivial}.\n\\begin{example}\\rm\n\\label{ex-trivial}\nConsider a $(K_1,K_2;M_1,M_2;N)$ system consisting of $K_1=4$ fixed users $\\mathcal{K}_1=\\{0,1,2,3\\}$, $K_2=3$ mobile user $\\mathcal{K}_2=\\{4,5,6\\}$ and $N=7$ files $\\mathcal{W}=\\{W_0,W_1,\\ldots,W_{6}\\}$. When assume that  $M_{1}=\\frac{21}{4}$, $M_{2}=\\frac{14}{3}$,  we have $t_1=3$, and $t_2=2$, respectively.\n\nFirstly, by applying the placement scheme in Algorithm \\ref{alg-MN} to all users in $\\mathcal{K}_1$, each file is split into $4$ packets of equal size, \\emph{i.e.},\n\\begin{eqnarray}\n\\label{exm-K=2-2}\nW_n=\\{W_{n,\\{0,1,2\\}}, W_{n,\\{0,1,3\\}}, W_{n,\\{0,2,3\\}}, W_{n,\\{1,2,3\\}}\\}, \\  n=0,1,\\ldots,6.\n\\end{eqnarray}\nand the four fixed users in $ \\mathcal{K}_1$ will cache the following contents in placement phase,\n\\begin{eqnarray}\n\\label{ex-k1-mn-p}\n\\begin{split}\n\\mathcal{Z}_{0}&=\\{ W_{n,\\{0,1,2\\}}, W_{n,\\{0,1,3\\}}, W_{n,\\{0,2,3\\}}\\}_{n=0}^6,\\ \\\n\\mathcal{Z}_{1}=\\{ W_{n,\\{0,1,2\\}}, W_{n,\\{0,1,3\\}}, W_{n,\\{1,2,3\\}}\\}_{n=0}^6,\\\\\n\\mathcal{Z}_{2}&=\\{ W_{n,\\{0,1,2\\}}, W_{n,\\{0,2,3\\}}, W_{n,\\{1,2,3\\}}\\}_{n=0}^6,\\ \\\n\\mathcal{Z}_{3}=\\{ W_{n,\\{0,1,3\\}}, W_{n,\\{0,2,3\\}}, W_{n,\\{1,2,3\\}}\\}_{n=0}^6.\n\\end{split}\n\\end{eqnarray}\nSimilarly for the three users in $\\mathcal{K}_2$, each file will be split into $3$ packets of the same size, \\emph{i.e.},\n\\begin{eqnarray*}\nW_n=\\{W_{n,\\{4,5\\}},  W_{n,\\{4,6\\}}, W_{n,\\{5,6\\}}\\},\\  n=0,1,\\ldots,6\n\\end{eqnarray*}\nand the users in $\\mathcal{K}_2$ will cache the following contents in placement phase,\n\\begin{eqnarray*}\n\\mathcal{Z}_{4}=\\{ W_{n,\\{4,5\\}}, W_{n,\\{4,6\\}}\\}_{n=0}^6,\n\\mathcal{Z}_{5}=\\{ W_{n,\\{4,5\\}}, W_{n,\\{5,6\\}}\\}_{n=0}^6,\n\\mathcal{Z}_{6}=\\{ W_{n,\\{4,6\\}}, W_{n,\\{5,6\\}}\\}_{n=0}^6.\n\\end{eqnarray*}\n\nWithout loss of generality, assume $\\mathbf{d}=\\{0,1,2,3,4,5,6\\}$. By using the delivery scheme in Algorithm \\ref{alg-MN}, we have $\\mathcal{S}_1=\\{0,1,2,3\\}$ and $\\mathcal{S}_2=\\{4,5,6\\}$, then the sever respectively sends\n    $$\\mathop{\\oplus}_{k_1\\in \\mathcal{S}_1}W_{d_{k_1},\\mathcal{S}_1\\setminus\\{k_1\\}}\\ \\ \\ \\hbox{and}\\ \\ \\ \\mathop{\\oplus}_{k_2\\in \\mathcal{S}_2}W_{d_{k_2},\\mathcal{S}_2\\setminus\\{k_2\\}}.$$\n    to the users in $\\mathcal{K}_1$ and $\\mathcal{K}_2$.\n\nBy the Lemma \\ref{le-MN}, we have $R_{MN}(4;\\frac{21}{4};7)=\\frac{1}{4}$ for $\\mathcal{K}_1$ and $R_{MN}(3;\\frac{14}{3};7)=\\frac{1}{4}$ for $\\mathcal{K}_2$ respectively. Then to sum up, the total transmission rate is $\\frac{1}{4}+\\frac{1}{3}=\\frac{7}{12}$.\n\\end{example}\n\nFrom the above Example \\ref{exm-K=2-2}, it can be observed that, when directly applying MN scheme to the $(K_1,K_2;M_1,M_2;N)$ system, users need to be divided into two groups, and MN scheme $(K_1;M_1;N)$ and $(K_2;M_2;N)$ are utilized in each group, respectively. Although, this strategy is applicable, during delivery, the multicasting opportunities between users in $\\mathcal{K}_1$ and $\\mathcal{K}_2$ are lost, which is less efficient. At this point, how to design a scheme to maximally exploit the multicasting gain among all the users would be an interesting problem, and is worth to be investigagted.\n\n\\subsection{The Proposed Dynamic Coded Caching Design and The Main Results}\nUnlike the application of the MN scheme to two user sets separately, for the $(K_1,K_2;M_{1},M_{2};N)$ caching system, we propose a new design by concatenating two user groups, such that the coding gain can be enlarged. More specifically, the coded multicasting gain in our proposed scheme is $\\frac{M_1K_1}{N}+\\frac{M_2K_2}{N}$. When $M_1=M_2$, our coded gain is only one gain less than the maximum gain of the MN scheme $(K_1+K_2;M_1;N)$. The transmission rate of our proposed scheme is given by Theorem \\ref{th-new-r}.\n\\begin{theorem}\\rm\n\\label{th-new-r}\nFor the positive integers $K_i$, $N$ and $M_{i}<N$ such that $t_i=\\frac{K_iM_{i}}{N}\\in [1,K_i)$, $i=1,2$, there exists a  $(K_1,K_2;M_{1},M_{2};N)$ coded caching scheme to achieve the following transmission rate\n\\begin{eqnarray*}\nR(K_1,K_2;M_1,M_2;N)\\leq\\begin{cases}\n\\frac{(K_1-t_1)(t_1t_2+t_1+1)}{(t_1+1)t_1t_2}+\\frac{(K_2-t_2)}{(t_1+1)t_2},&\\text{if $M_1<M_2$},\\\\[0.2cm]\n\\frac{(K_1-t_1)}{(t_2+1)t_1}+\\frac{(K_2-t_2)(t_1t_2+t_2+1)}{(t_2+1)t_1t_2},&\\text{if $M_1\\geq M_2$}.\n\\end{cases}\n\\end{eqnarray*}\n\\end{theorem}\n\nThe proof of Theorem \\ref{th-new-r} is presented in Appendix. The placement and the delivery scheme of the proposed CSM coded caching scheme are given in Algorithm \\ref{alg-NSP} and Algorithm \\ref{alg-NSD} respectively.\n\\begin{small}\n\\begin{algorithm}[htp]\n\\caption{Concatenating Placement Design of The $(K_1,K_2;M_1,M_2;N)$ Dynamic Coded Caching Scheme}\\label{alg-NSP}\n\\begin{algorithmic}[1]\n\\Procedure {Placement 1 } {$\\mathcal{K}_1$, $M_1$, $\\mathcal{W}$}  \n\\State $t_1 \\leftarrow \\frac{M_{1}K_1}{N}$\n\\State $\\mathfrak{A} \\leftarrow \\{\\mathcal{A}|\\mathcal{A}\\subset \\mathcal{K}_1, |\\mathcal{A}|=t_1\\}$\n\\State Split each file $W_i\\in\\mathcal{W}$ into ${K_1 \\choose t_1}$ packets, \\emph{i.e.}, $W_{i}=\\{W_{i,\\mathcal{A}}\\ |\\ \\mathcal{A}\\in \\mathfrak{A}\\}$.\n\\For{$k_1\\in \\mathcal{K}_1$}\n\\State $\\mathcal{Z}_{k_1}\\leftarrow\\left \\{W_{i,\\mathcal{A}}\\ |\\ k_1\\in\\mathcal{A},\\mathcal{A}\\in\\mathfrak{A},i\\in[0,N)\\right\\}$\n\\EndFor\n\\EndProcedure\n\\Procedure {Placement 2 }{$\\mathcal{K}_2$, $M_2$, $\\mathcal{W}$}\n\\State $t_2 \\leftarrow \\frac{M_{2}K_2}{N}$\n\\State $\\mathfrak{B}\\leftarrow \\{\\mathcal{B}|\\mathcal{B}\\subset \\mathcal{K}_2, |\\mathcal{B}|=t_2\\}$\n\\State Split each packet $W_{i,\\mathcal{A}}$ into $t_1t_2{K_2 \\choose t_2}$ sub-packets, \\emph{i.e.}, $$W_{i,\\mathcal{A}}=\\left \\{W_{i,\\mathcal{A}, \\mathcal{B}}^{(l_2,l_1)}\\ |\\ \\mathcal{A}\\in \\mathfrak{A}, \\mathcal{B}\\in\\mathfrak{B}, l_1\\in[0,t_1), l_2\\in[0,t_2)\\right\\}.$$\n\\For{$k_2\\in \\mathcal{K}_2$}\n\\State $\\mathcal{Z}_{k_2}\\leftarrow\\left \\{W_{i,\\mathcal{A},\\mathcal{B}}^{(l_2,l_1)}\\ |\\ k_2\\in\\mathcal{B}, \\mathcal{B}\\in\\mathfrak{B}, \\mathcal{A}\\in\\mathfrak{A}, i\\in[0,N), l_1\\in[0,t_1), l_2\\in[0,t_2)\\right\\}$\n\\EndFor\n\\EndProcedure\n\\end{algorithmic}\n\\end{algorithm}\n\\end{small}\n\nIn Algorithm \\ref{alg-NSP}, the PLACMENT 1 is for the users in $\\mathcal{K}_1$, which may be performed in any previous rounds, and the PLACMENT 2 is designed for those mobile users in $\\mathcal{K}_2$. It can be observed that PLACMENT 1 is independent of PLACMENT 2, while that for users in $\\mathcal{K}_2$ depends on the number of users in $\\mathcal{K}_1$ to maximally utilize the coded multicasting opportunities among all the users. After this placement phase, the cache contents of all the users denoted by $\\mathcal{Z}$ are assumed to be known by the server. Then, the delivery phase follows and is described in Algorithm \\ref{alg-NSD}.\n\n\n\\begin{small}\n\\begin{algorithm}[htp]\n\\caption{The Saturating Matching Delivery Design for The $(K_1,K_2;M_1,M_2;N)$ Dynamic Coded Caching Scheme}\\label{alg-NSD}\n\\begin{algorithmic}[1]\n \\Procedure{Find Vertices of Bipartite Graph }{$\\mathcal{Z}$, ${\\bf d}$, $\\mathcal{W}$}\n\\State $\\mathfrak{S}_1\\leftarrow \\{\\mathcal{S}_1| \\mathcal{S}_1\\subseteq \\mathcal{K}_1, |\\mathcal{S}_1|=t_1+1\\}$\n\\State $\\mathfrak{S}_2\\leftarrow \\{\\mathcal{S}_2| \\mathcal{S}_2\\subseteq \\mathcal{K}_2, |\\mathcal{S}_2|=t_2+1\\}$\n\\For{$\\mathcal{S}_1=\\{k_{1,0},k_{1,1},\\ldots,k_{1,t_1}\\}\\in \\mathfrak{S}_1$, $\\mathcal{B}\\in \\mathfrak{B}$}\n\\For{$l_2\\in [0,t_2)$, $m_1\\in [0,t_1]$}\n\\State\n{\\begin{align}\nX^{(l_2,m_1)}_{\\mathcal{S}_{1},\\mathcal{B}}\\triangleq\n\\left(\\bigoplus\\limits_{m'_1=0}^{m_1-1} W^{(l_2,m_1-1)}_{d_{k_{1,m_1'}},\\mathcal{S}_1\\setminus\\{k_{1,m'_1}\\},\\mathcal{B}}\\right)\n\\bigoplus\n\\left(\n\\bigoplus\\limits_{m'_1=m_1+1}^{t_1} W^{(l_2,m_1)}_{d_{k_{1,m_1'}},\\mathcal{S}_1\\setminus\\{k_{1,m'_1}\\},\\mathcal{B}}\\right)\n\\end{align}\\label{eq-ex-step1-0}}\n\\EndFor\n\\EndFor\n\n\n\\For{$\\mathcal{S}_2=\\{k_{2,0},k_{2,1},\\ldots,k_{2,t_1}\\}\\in \\mathfrak{S}_2$, $\\mathcal{A}\\in \\mathfrak{A}$ }\n\\For{ $l_1\\in[0,t_1)$, $m_2\\in [0,t_2]$ }\n\\begin{eqnarray}\n\\label{eq-ex-step1-1}\nY^{(m_2,l_1)}_{\\mathcal{A},\\mathcal{S}_2 }\\triangleq \\left(\\bigoplus\\limits_{m'_2=0}^{m_2-1} W^{(m_2-1,l_1)}_{d_{k_{2,m'_2}},\\mathcal{A},\\mathcal{S}_2\\setminus\\{k_{2,m'_2}\\}}\\right)\\bigoplus\\left(\n\\bigoplus\\limits_{m'_2=m_2+1}^{t_2} W^{(m_2,l_1)}_{d_{k_{2,m'_2}},\\mathcal{A},\\mathcal{S}_2\\setminus\\{k_{2,m'_2}\\}}\\right)\n\\end{eqnarray}\n\\EndFor\n\\EndFor\n\\EndProcedure\n\\Procedure{Define Bipartite Graph $\\mathbf{G}=(\\mathcal{X},\\mathcal{Y}; \\mathcal{E})$}{}\n\\State { Define $\\mathbf{G}=(\\mathcal{X},\\mathcal{Y}; \\mathcal{E})$ such that $X^{(l_2,m_1)}_{\\mathcal{S}_{1},\\mathcal{B}}\\in \\mathcal{X}$ is adjacent to $Y^{(m_2,l_1)}_{\\mathcal{A},\\mathcal{S}_2 }\\in \\mathcal{Y}$ if and only if $\\mathcal{S}_{1}\\setminus\\{k_{1,m_1}\\}=\\mathcal{A}$ and $\\mathcal{S}_{2}\\setminus\\{k_{2,m_2}\\}=\\mathcal{B}$, where}\\label{line:req}\n $$\\mathcal{X}=\\left\\{X^{(l_2,m_1)}_{\\mathcal{S}_{1},\\mathcal{B}}\\ |\\ \\mathcal{S}_1\\in \\mathfrak{S}_1, \\mathcal{B}\\in \\mathfrak{B},l_2\\in [0,t_2), m_1\\in [0,t_1]\\right\\}$$\n$$\\mathcal{Y}=\\left\\{Y^{(m_2,l_1)}_{\\mathcal{A},\\mathcal{S}_2 }\\ |\\ \\mathcal{S}_2\\in \\mathfrak{S}_2, \\mathcal{A}\\in \\mathfrak{A},l_1\\in [0,t_1), m_2\\in [0,t_2]\\right\\}$$\n\\EndProcedure\n\\Procedure {Delivery }{$\\mathbf{G}=(\\mathcal{X},\\mathcal{Y}; \\mathcal{E})$}\n\\State {Find an appropriate saturating matching of $\\mathbf{G}$ for $\\mathcal{X}$ (or $\\mathcal{Y}$), denoted by $\\mathbf{G}''=(\\mathcal{X},\\mathcal{Y}''; \\mathcal{E}'')$, such that the remaining vertices in $\\mathcal{Y}$ or ($\\mathcal{X}$) as in \\eqref{eq-ex-step1-1} can be sent with multicasting gain $t_2+1$ ($t_1+1$) as much as possible}\\label{line:match}\n\\For { each edge $e=\\left(X^{(l_2,m_1)}_{\\mathcal{S}_{1},\\mathcal{B}},Y^{(m_2,l_1)}_{\\mathcal{A},\\mathcal{S}_2 }\\right)\\in \\mathcal{E}''$}\n\\State Sends\n$X^{(l_2,m_1)}_{\\mathcal{S}_{1},\\mathcal{B}}\\bigoplus Y^{(m_2,l_1)}_{\\mathcal{A},\\mathcal{S}_2 }$\\label{line:send}\n\\EndFor\n\\State Sends remaining vertices in $\\mathcal{Y}$ ($\\mathcal{X}$) with $t_2$ ($t_1$) for every $t_2+1$ ($t_1+1$) vertices.\n\\EndProcedure\n\\end{algorithmic}\n\\end{algorithm}\n\\end{small}\n\nFrom the Algorithm \\ref{alg-NSD}, it can be observed that, the whole delivery phase can be realized by firstly finding the vertices of the bipartite graph, and then constructing the bipartite graph according to the requirements described in Line \\ref{line:req}. Finally based on the constructed graph we can find out the coded messages as described in Procedure Delivery. The working flow of the Algorithm \\ref{alg-NSD} can be briefly summarized as below: in Lines 2-13, we try to generate the vertices of a bipartite graph by means of the required sub-packets, i.e., to create the multicasting gains $\\frac{M_1K_1}{N}$ and $\\frac{M_2K_2}{N}$ respectively;  in Line \\ref{line:req} we can define its edge, i.e., to create the total coded multicasting gain $\\frac{M_1K_1}{N}+\\frac{M_2K_2}{N}$ according to the requests of users; Finally in Procedure Delivery, based on the saturating matching, the messages with multicasting gain $\\frac{M_1K_1}{N}+\\frac{M_2K_2}{N}$ are broadcast as much as possible.\n\nIt is worth noting that, in the Proposition \\ref{pro-baohe-X} in APPENDIX, we show that there always exists a saturating matching of $\\mathbf{G}$ for $\\mathcal{X}$ (or $\\mathcal{Y}$), denoted by $\\mathbf{G}''=(\\mathcal{X},\\mathcal{Y}''; \\mathcal{E}'')$, such that the remaining sub-packets in $\\mathcal{Y}$ (or $\\mathcal{X}$) as in \\eqref{eq-ex-step1-1} can be sent with multicasting gain of $t_2+1$ ($t_1+1$) as much as possible, i.e., Line 18-22 work.\n\\subsection{The Illustration of The Proposed Dynamic Coded Caching Design}\nFor clear illustration of our proposed scheme, in this subsection, we give the following example to illustrate the Algorithm  \\ref{alg-NSP} and Algorithm \\ref{alg-NSD}.\n\\begin{example}\\rm\n\\label{ex-neq}\nTo compare with the baseline scheme in Example \\ref{ex-trivial}, we assume the same system setup here. According to Algorithm \\ref{alg-NSP} and Algorithm \\ref{alg-NSD}, the placement and delivery phase are explicated as follows. It needs to be highlighted that, the placement for $\\mathcal{K}_1$ and $\\mathcal{K}_2$ are assumed to be performed in different round, just like we have addressed before.\n\\begin{itemize}\n\\item {\\bf Placement phase}: The placement phase can be completed by the following two steps.\n\\begin{itemize}\n\\item Step 1 (Placement for $\\mathcal{K}_1$): At any previous round,  the placement phase in Algorithm \\ref{alg-MN} is applied to users in $\\mathcal{K}_1$,which is the same as \\eqref{ex-k1-mn-p} in Example \\ref{ex-trivial}.\n\\item Step 2 (Placement for $\\mathcal{K}_2$): By Lines $10-12$ in Algorithm \\ref{alg-NSP}, for each packet in \\eqref{exm-K=2-2}, we further divide it into $t_1t_2{K_2 \\choose t_2}=3\\cdot 2 \\cdot{3\\choose 2}=18$ sub-packets with equal size, \\emph{i.e.,} for each $n=0,1,\\ldots,6$,\n\\begin{equation*}\n\\begin{split}\nW_{n,\\{0,1,2\\} }&=\\left\\{W_{n,\\{0,1,2\\},\\mathcal{B}}^{(l_2,l_1)}\\ |\\ \\mathcal{B}\\subset \\mathcal{K}_2, |\\mathcal{B}|=2, \\ l_1\\in [0,3),\\ l_2\\in[0,2)\\right\\},\\\\\nW_{n,\\{0,1,3\\} }&=\\left\\{W_{n,\\{0,1,3\\},\\mathcal{B}}^{(l_2,l_1)}\\ |\\ \\mathcal{B}\\subset \\mathcal{K}_2, |\\mathcal{B}|=2, \\ l_1\\in [0,3),\\ l_2\\in[0,2)\\right\\},\\\\\nW_{n,\\{0,2,3\\} }&=\\left\\{W_{n,\\{0,2,3\\},\\mathcal{B}}^{(l_2,l_1)}\\ |\\ \\mathcal{B}\\subset \\mathcal{K}_2, |\\mathcal{B}|=2, \\ l_1\\in [0,3),\\ l_2\\in[0,2)\\right\\},\\\\\nW_{n,\\{1,2,3\\} }&=\\left\\{W_{n,\\{1,2,3\\},\\mathcal{B}}^{(l_2,l_1)}\\ |\\ \\mathcal{B}\\subset \\mathcal{K}_2, |\\mathcal{B}|=2, \\ l_1\\in [0,3),\\ l_2\\in[0,2)\\right\\}.\n\\end{split}\n\\end{equation*}\nBy Lines $14$ in Algorithm \\ref{alg-NSP}, each users $k_2\\in \\mathcal{K}_2$ caches\n$$\\mathcal{Z}_{k_2}=\\left\\{W^{(l_2,l_1)}_{n,\\mathcal{A},\\mathcal{B}}\\ |\\ k_2\\in \\mathcal{B}, \\mathcal{A}\\subset{\\mathcal{K}_1},\\mathcal{B}\\subset{\\mathcal{K}_2}, |\\mathcal{A}|=3, |\\mathcal{B}|=2, l_1\\in [0,3), l_2\\in[0,2)\\right\\}.$$\nAfter this step, each user $k_2$ caches a total of $t_1t_2{K_2-1 \\choose t_2-1}=3\\cdot2\\cdot{2\\choose 1}=12$ sub-packets from each packet. Since each file is firstly divided into $\\binom{4}{3}$ packets, each\n  user $k_2$ caches $12\\cdot {4\\choose 3}\\cdot N=12\\cdot 4\\cdot N=48N$ sub-packets. Since the total number of sub-packets in each file is $18\\cdot{4\\choose 3}=72$, the user $k_2$ caches $\\frac{48N}{72}=\\frac{2}{3}N=\\frac{14}{3}$ files, which satisfies its cache size limitation.\n\\end{itemize}\n\\item {\\bf Delivery phase}: Also assume that $\\mathbf{d}=\\{0,1,2,3,4,5,6\\}$, and from the caching result in Algorithm \\ref{alg-NSP},\nwe have  $$\\mathcal{S}_1=\\{0,1,2,3\\},\\ \\ \\mathcal{A}_0=\\{1,2,3\\},\\ \\ \\mathcal{A}_1=\\{0,2,3\\},\\ \\ \\mathcal{A}_2=\\{0,1,3\\},\\ \\ \\mathcal{A}_3=\\{0,1,2\\}$$  $$\\mathcal{S}_2=\\{4,5,6\\},\\ \\ \\mathcal{B}_0=\\{5,6\\},\\ \\ \\mathcal{B}_1=\\{4,6\\},\\ \\ \\mathcal{B}_2=\\{4,5\\}.$$\n   By the first procedure in Algorithm \\ref{alg-NSD}, we have the following vertices, where the elements in the set of $\\mathcal{X}$ and $\\mathcal{Y}$ are labeled as $u_i, (i=0,\\ldots,23)$, and $v_j, (j=0,\\ldots,35)$ in short respectively.\n\\begin{small}\n\\begin{eqnarray*}\n\\begin{array}{cccccc}\n  u_0=X^{(0,0)}_{\\mathcal{S}_1,\\mathcal{B}_0}\n& u_1=X^{(0,1)}_{\\mathcal{S}_1,\\mathcal{B}_0}\n& u_2=X^{(0,2)}_{\\mathcal{S}_1,\\mathcal{B}_0}\n& u_3=X^{(0,3)}_{\\mathcal{S}_1,\\mathcal{B}_0}\n& u_4=X^{(0,0)}_{\\mathcal{S}_1,\\mathcal{B}_1}\n& u_5=X^{(0,1)}_{\\mathcal{S}_1,\\mathcal{B}_1}\\\\[0.2cm]\n  u_6=X^{(0,2)}_{\\mathcal{S}_1,\\mathcal{B}_1}\n& u_7=X^{(0,3)}_{\\mathcal{S}_1,\\mathcal{B}_1}\n& u_8=X^{(0,0)}_{\\mathcal{S}_1,\\mathcal{B}_2}\n& u_9=X^{(0,1)}_{\\mathcal{S}_1,\\mathcal{B}_2}\n& u_{10}=X^{(0,2)}_{\\mathcal{S}_1,\\mathcal{B}_2}\n& u_{11}=X^{(0,3)}_{\\mathcal{S}_1,\\mathcal{B}_2}\\\\[0.2cm]\n  u_{12}=X^{(1,0)}_{\\mathcal{S}_1,\\mathcal{B}_0}\n& u_{13}=X^{(1,1)}_{\\mathcal{S}_1,\\mathcal{B}_0}\n& u_{14}=X^{(1,2)}_{\\mathcal{S}_1,\\mathcal{B}_0}\n& u_{15}=X^{(1,3)}_{\\mathcal{S}_1,\\mathcal{B}_0}\n& u_{16}=X^{(1,0)}_{\\mathcal{S}_1,\\mathcal{B}_1}\n& u_{17}=X^{(1,1)}_{\\mathcal{S}_1,\\mathcal{B}_1}\\\\[0.2cm]\n  u_{18}=X^{(1,2)}_{\\mathcal{S}_1,\\mathcal{B}_1}\n& u_{19}=X^{(1,3)}_{\\mathcal{S}_1,\\mathcal{B}_1}\n& u_{20}=X^{(1,0)}_{\\mathcal{S}_1,\\mathcal{B}_2}\n& u_{21}=X^{(1,1)}_{\\mathcal{S}_1,\\mathcal{B}_2}\n& u_{22}=X^{(1,2)}_{\\mathcal{S}_1,\\mathcal{B}_2}\n& u_{23}=X^{(1,3)}_{\\mathcal{S}_1,\\mathcal{B}_2}\\\\[0.2cm]\n  v_0=Y^{(0,0)}_{\\mathcal{A}_0,\\mathcal{S}_2} &v_1=Y^{(1,0)}_{\\mathcal{A}_0,\\mathcal{S}_2}\n& v_2=Y^{(2,0)}_{\\mathcal{A}_0,\\mathcal{S}_2} &v_3=Y^{(0,0)}_{\\mathcal{A}_1,\\mathcal{S}_2}&\n  v_4=Y^{(1,0)}_{\\mathcal{A}_1,\\mathcal{S}_2} &v_5=Y^{(2,0)}_{\\mathcal{A}_1,\\mathcal{S}_2}\\\\[0.2cm]\n  v_6=Y^{(0,0)}_{\\mathcal{A}_2,\\mathcal{S}_2} &v_7=Y^{(1,0)}_{\\mathcal{A}_2,\\mathcal{S}_2}&\n  v_8=Y^{(2,0)}_{\\mathcal{A}_2,\\mathcal{S}_2} &v_9=Y^{(0,0)}_{\\mathcal{A}_3,\\mathcal{S}_2}\n& v_{10}=Y^{(1,0)}_{\\mathcal{A}_3,\\mathcal{S}_2}&v_{11}=Y^{(2,0)}_{\\mathcal{A}_3,\\mathcal{S}_2}\\\\[0.2cm]\n  v_{12}=Y^{(0,1)}_{\\mathcal{A}_0,\\mathcal{S}_2}&v_{13}=Y^{(1,1)}_{\\mathcal{A}_0,\\mathcal{S}_2}\n& v_{14}=Y^{(2,1)}_{\\mathcal{A}_0,\\mathcal{S}_2}&v_{15}=Y^{(0,1)}_{\\mathcal{A}_1,\\mathcal{S}_2}&\n  v_{16}=Y^{(1,1)}_{\\mathcal{A}_1,\\mathcal{S}_2}&v_{17}=Y^{(2,1)}_{\\mathcal{A}_1,\\mathcal{S}_2}\\\\[0.2cm]\n  v_{18}=Y^{(0,1)}_{\\mathcal{A}_2,\\mathcal{S}_2}&v_{19}=Y^{(1,1)}_{\\mathcal{A}_2,\\mathcal{S}_2}&\n  v_{20}=Y^{(2,1)}_{\\mathcal{A}_2,\\mathcal{S}_2}&v_{21}=Y^{(0,1)}_{\\mathcal{A}_3,\\mathcal{S}_2}\n& v_{22}=Y^{(1,1)}_{\\mathcal{A}_3,\\mathcal{S}_2}&v_{23}=Y^{(2,1)}_{\\mathcal{A}_3,\\mathcal{S}_2}\\\\[0.2cm]\n  v_{24}=Y^{(0,2)}_{\\mathcal{A}_0,\\mathcal{S}_2}&v_{25}=Y^{(1,2)}_{\\mathcal{A}_0,\\mathcal{S}_2}\n& v_{26}=Y^{(2,2)}_{\\mathcal{A}_0,\\mathcal{S}_2}&v_{27}=Y^{(0,2)}_{\\mathcal{A}_1,\\mathcal{S}_2}&\n  v_{28}=Y^{(1,2)}_{\\mathcal{A}_1,\\mathcal{S}_2}&v_{29}=Y^{(2,2)}_{\\mathcal{A}_1,\\mathcal{S}_2}\\\\[0.2cm]\n  v_{30}=Y^{(0,2)}_{\\mathcal{A}_2,\\mathcal{S}_2}&v_{31}=Y^{(1,2)}_{\\mathcal{A}_2,\\mathcal{S}_2}&\n  v_{32}=Y^{(2,2)}_{\\mathcal{A}_2,\\mathcal{S}_2}&v_{33}=Y^{(0,2)}_{\\mathcal{A}_3,\\mathcal{S}_2}&\n  v_{34}=Y^{(1,2)}_{\\mathcal{A}_3,\\mathcal{S}_2}&v_{35}=Y^{(2,2)}_{\\mathcal{A}_3,\\mathcal{S}_2}\n\\end{array}\n\\end{eqnarray*}\n\\end{small}\n\nBy the second prodecure in Algorithm \\ref{alg-NSD}, the following bipartite graph $\\mathbf{G}=(\\mathcal{X},\\mathcal{Y}; \\mathcal{E})$ depicted in Figure \\ref{Fig-exm-ebt} can be obtained.\n\\begin{figure}[h]\n\\centering\\includegraphics[width=0.9\\textwidth]{Figure4}\n\\caption{The bipartite graph $\\mathbf{G}=(\\mathcal{X},\\mathcal{Y}; \\mathcal{E})$ in Example \\ref{ex-neq}}\\label{Fig-exm-ebt}\n\\end{figure}\n\nThen, with this graph $\\mathbf{G}$, we can find a saturating matching for $\\mathcal{X}$ as  in Line \\ref{line:match}  described in the last precedure of Algorithm \\ref{alg-NSD}, which is depicted in Figure \\ref{Fig-exm-bhpp}.\n\\begin{figure}[h]\n\\centering\\includegraphics[width=0.9\\textwidth]{Figure5}\n\\caption{A saturating matching for $\\mathcal{X}$ in Example \\ref{ex-neq}}\\label{Fig-exm-bhpp}\n\\end{figure}\nFrom the saturating matching and the Line \\ref{line:send} in Algorithm \\ref{alg-NSD}, the server sends the following $24$ coded messages involving sub-packets from both $\\mathcal{X}$ and $\\mathcal{Y}$.\n\\begin{small}\n\\begin{eqnarray*}\n\\begin{array}{ccc}\nu_0\\bigoplus v_0=X^{(0,0)}_{\\mathcal{S}_1,\\mathcal{B}_0}\\bigoplus Y^{(0,0)}_{\\mathcal{A}_0,\\mathcal{S}_2}\n& u_1\\bigoplus v_3=X^{(0,1)}_{\\mathcal{S}_1,\\mathcal{B}_0}\\bigoplus Y^{(0,0)}_{\\mathcal{A}_1,\\mathcal{S}_2}\n& u_2\\bigoplus v_6=X^{(0,2)}_{\\mathcal{S}_1,\\mathcal{B}_0}\\bigoplus Y^{(0,0)}_{\\mathcal{A}_2,\\mathcal{S}_2}\\\\[0.2cm]\n  u_3\\bigoplus v_9=X^{(0,3)}_{\\mathcal{S}_1,\\mathcal{B}_0}\\bigoplus Y^{(0,0)}_{\\mathcal{A}_3,\\mathcal{S}_2}\n& u_4\\bigoplus v_1=X^{(0,0)}_{\\mathcal{S}_1,\\mathcal{B}_1}\\bigoplus Y^{(1,0)}_{\\mathcal{A}_0,\\mathcal{S}_2}\n& u_5\\bigoplus v_4=X^{(0,1)}_{\\mathcal{S}_1,\\mathcal{B}_1}\\bigoplus Y^{(1,0)}_{\\mathcal{A}_1,\\mathcal{S}_2}\\\\[0.2cm]\n  u_6\\bigoplus v_7=X^{(0,2)}_{\\mathcal{S}_1,\\mathcal{B}_1}\\bigoplus Y^{(1,0)}_{\\mathcal{A}_2,\\mathcal{S}_2}\n& u_7\\bigoplus v_{10}=X^{(0,3)}_{\\mathcal{S}_1,\\mathcal{B}_1}\\bigoplus Y^{(1,0)}_{\\mathcal{A}_3,\\mathcal{S}_2}\n& u_8\\bigoplus v_{2}=X^{(0,0)}_{\\mathcal{S}_1,\\mathcal{B}_2}\\bigoplus Y^{(2,0)}_{\\mathcal{A}_0,\\mathcal{S}_2}\\\\[0.2cm]\n  u_9\\bigoplus v_5=X^{(0,1)}_{\\mathcal{S}_1,\\mathcal{B}_2}\\bigoplus Y^{(2,0)}_{\\mathcal{A}_1,\\mathcal{S}_2}\n& u_{10}\\bigoplus v_{8}= X^{(0,2)}_{\\mathcal{S}_1,\\mathcal{B}_2}\\bigoplus Y^{(2,0)}_{\\mathcal{A}_2,\\mathcal{S}_2}\n& u_{11}\\bigoplus v_{11}=X^{(0,3)}_{\\mathcal{S}_1,\\mathcal{B}_2}\\bigoplus Y^{(2,0)}_{\\mathcal{A}_3,\\mathcal{S}_2}\\\\[0.2cm]\n  u_{12}\\bigoplus v_{12}=X^{(1,0)}_{\\mathcal{S}_1,\\mathcal{B}_0}\\bigoplus Y^{(0,1)}_{\\mathcal{A}_0,\\mathcal{S}_2}\n& u_{13}\\bigoplus v_{15}=X^{(1,1)}_{\\mathcal{S}_1,\\mathcal{B}_0}\\bigoplus Y^{(0,1)}_{\\mathcal{A}_1,\\mathcal{S}_2}\n& u_{14}\\bigoplus v_{18}=X^{(1,2)}_{\\mathcal{S}_1,\\mathcal{B}_0}\\bigoplus Y^{(0,1)}_{\\mathcal{A}_2,\\mathcal{S}_2}\\\\[0.2cm]\n  u_{15}\\bigoplus v_{21}=X^{(1,3)}_{\\mathcal{S}_1,\\mathcal{B}_0}\\bigoplus Y^{(0,1)}_{\\mathcal{A}_3,\\mathcal{S}_2}\n& u_{16}\\bigoplus v_{13}=X^{(1,0)}_{\\mathcal{S}_1,\\mathcal{B}_1}\\bigoplus Y^{(1,1)}_{\\mathcal{A}_0,\\mathcal{S}_2}\n& u_{17}\\bigoplus v_{16}=X^{(1,1)}_{\\mathcal{S}_1,\\mathcal{B}_1}\\bigoplus Y^{(1,1)}_{\\mathcal{A}_1,\\mathcal{S}_2}\\\\[0.2cm]\n  u_{18}\\bigoplus v_{19}=X^{(1,2)}_{\\mathcal{S}_1,\\mathcal{B}_1}\\bigoplus Y^{(1,1)}_{\\mathcal{A}_2,\\mathcal{S}_2}\n& u_{19}\\bigoplus v_{22}=X^{(1,3)}_{\\mathcal{S}_1,\\mathcal{B}_1}\\bigoplus Y^{(1,1)}_{\\mathcal{A}_3,\\mathcal{S}_2}\n& u_{20}\\bigoplus v_{14}=X^{(1,0)}_{\\mathcal{S}_1,\\mathcal{B}_2}\\bigoplus Y^{(2,1)}_{\\mathcal{A}_0,\\mathcal{S}_2}\\\\[0.2cm]\n  u_{21}\\bigoplus v_{17}=X^{(1,1)}_{\\mathcal{S}_1,\\mathcal{B}_2}\\bigoplus Y^{(2,1)}_{\\mathcal{A}_1,\\mathcal{S}_2}\n& u_{22}\\bigoplus v_{20}=X^{(1,2)}_{\\mathcal{S}_1,\\mathcal{B}_2}\\bigoplus Y^{(2,1)}_{\\mathcal{A}_2,\\mathcal{S}_2}\n& u_{23}\\bigoplus v_{23}=X^{(1,3)}_{\\mathcal{S}_1,\\mathcal{B}_2}\\bigoplus Y^{(2,1)}_{\\mathcal{A}_3,\\mathcal{S}_2}\n\\end{array}\n\\end{eqnarray*}\n\\end{small}Finally for the remaining vertices in $\\mathcal{Y}$ of $\\mathbf{G}=(\\mathcal{X},\\mathcal{Y};\\mathcal{E})$, the server sends the following $8$ coded messages,\n\\begin{small}\n\\begin{eqnarray*}\n\\begin{array}{cccccc}\nv_{24}\\bigoplus v_{25}=Y^{(0,2)}_{\\mathcal{A}_0,\\mathcal{S}_2}\\bigoplus Y^{(1,2)}_{\\mathcal{A}_0,\\mathcal{S}_2}&\nv_{25}\\bigoplus v_{26}=Y^{(1,2)}_{\\mathcal{A}_0,\\mathcal{S}_2}\\bigoplus Y^{(2,2)}_{\\mathcal{A}_0,\\mathcal{S}_2}\\\\[0.2cm]\nv_{27}\\bigoplus v_{28}=Y^{(0,2)}_{\\mathcal{A}_1,\\mathcal{S}_2}\\bigoplus Y^{(1,2)}_{\\mathcal{A}_1,\\mathcal{S}_2}&\nv_{28}\\bigoplus v_{29}=Y^{(1,2)}_{\\mathcal{A}_1,\\mathcal{S}_2}\\bigoplus Y^{(2,2)}_{\\mathcal{A}_1,\\mathcal{S}_2}\\\\[0.2cm]\nv_{30}\\bigoplus v_{31}=Y^{(0,2)}_{\\mathcal{A}_2,\\mathcal{S}_2}\\bigoplus Y^{(1,2)}_{\\mathcal{A}_2,\\mathcal{S}_2}&\nv_{31}\\bigoplus v_{32}=Y^{(1,2)}_{\\mathcal{A}_2,\\mathcal{S}_2}\\bigoplus Y^{(2,2)}_{\\mathcal{A}_2,\\mathcal{S}_2}\\\\[0.2cm]\nv_{33}\\bigoplus v_{34}=Y^{(0,2)}_{\\mathcal{A}_3,\\mathcal{S}_2}\\bigoplus Y^{(1,2)}_{\\mathcal{A}_3,\\mathcal{S}_2}&\nv_{34}\\bigoplus v_{35}=Y^{(1,2)}_{\\mathcal{A}_3,\\mathcal{S}_2}\\bigoplus Y^{(2,2)}_{\\mathcal{A}_3,\\mathcal{S}_2}\n\\end{array}\n\\end{eqnarray*}\n\\end{small}\n\\end{itemize}\nAfter receiving all the coded messages, and together with its cache content $\\mathcal{Z}_k$, each user is able to reconstruct requested file. Let us take the user $0$ as an example. By $\\mathcal{Z}_{0}$, user $0$ only requires the following $18$ sub-packets\n$$W^{(l_2,l_1)}_{0,\\mathcal{A}_0,\\mathcal{B}_0}, \\ \\ W^{(l_2,l_1)}_{0,\\mathcal{A}_0,\\mathcal{B}_1},\\ \\ W^{(l_2,l_1)}_{0,\\mathcal{A}_0,\\mathcal{B}_2},\\ \\ \\ \\ \\ \\  l_1\\in [0,3),\\ \\ l_2\\in[0,2).$$\nWe can check that the above sub-packets can be decoded from the following $18$ coded messages.\n\\begin{small}\n\\begin{eqnarray*}\n\\begin{array}{cccccc}\nu_1\\bigoplus v_3 & u_2\\bigoplus v_6 & u_3\\bigoplus v_9 & u_5\\bigoplus v_4 & u_6\\bigoplus v_7 & u_7\\bigoplus v_{10}\\\\[0.2cm]\nu_9\\bigoplus v_5 & u_{10}\\bigoplus v_8 & u_{11}\\bigoplus v_{11} & u_{13}\\bigoplus v_{15} & u_{14}\\bigoplus v_{18} & u_{15}\\bigoplus v_{21}\\\\[0.2cm]\nu_{17}\\bigoplus v_{16} & u_{18}\\bigoplus v_{19} & u_{19}\\bigoplus v_{22} & u_{21}\\bigoplus v_{17} & u_{22}\\bigoplus v_{20} & u_{23}\\bigoplus v_{23}\n\\end{array}\n\\end{eqnarray*}\n\\end{small}\nFurthermore, for each of the above coded message, user $0$ can decode a unique required sub-packet. For instance by\n\\begin{eqnarray*}\\begin{small}\n\\begin{array}{c}\nu_1\\bigoplus v_3=X^{(0,1)}_{\\mathcal{S}_1,\\mathcal{B}_0}\\bigoplus Y^{(0,0)}_{\\mathcal{A}_1,\\mathcal{S}_2}\n=\\left(W^{(0,0)}_{0,\\mathcal{A}_0,\\mathcal{B}_0}\\bigoplus W^{(0,1)}_{2,\\mathcal{A}_2,\\mathcal{B}_0}\\bigoplus W^{(0,1)}_{3,\\mathcal{A}_3,\\mathcal{B}_0}\\right)\\bigoplus \\left(W^{(0,0)}_{5,\\mathcal{A}_1,\\mathcal{B}_2}\\bigoplus W^{(0,0)}_{6,\\mathcal{A}_1,\\mathcal{B}_3}\\right)\n\\end{array}\\end{small}\n\\end{eqnarray*}\nclearly user $0$ can decode $W^{(0,0)}_{0,\\mathcal{A}_0,\\mathcal{B}_0}$, since it has cached all the other sub-packets in $u_1\\bigoplus v_3$. Thus finally, user $0$ can decode all the required sub-packets.\n\nSimilarly, let us take the  user $4$ as another example. After placement, user $4$ only needs the following $24$ sub-packets\n$$W^{(l_2,l_1)}_{4,\\mathcal{A}_0,\\mathcal{B}_0}, \\ \\ W^{(l_2,l_1)}_{4,\\mathcal{A}_1,\\mathcal{B}_0},\\ \\ W^{(l_2,l_1)}_{4,\\mathcal{A}_2,\\mathcal{B}_0},\\ \\ W^{(l_2,l_1)}_{4,\\mathcal{A}_3,\\mathcal{B}_0}\\ \\ \\ \\ \\ \\  l_1\\in [0,3),\\ \\ l_2\\in[0,2),$$\nwhich can be found in the following $24$ coded messages respectively.\n\\begin{small}\n\\begin{eqnarray}\n\\label{exm-2-k2-1}\n\\begin{array}{cccccccc}\nu_4\\bigoplus v_1 & u_5\\bigoplus v_4 & u_6\\bigoplus v_7 & u_7\\bigoplus v_{10} & u_8\\bigoplus v_2 & u_9\\bigoplus v_5 &\nu_{10}\\bigoplus v_8 & u_{11}\\bigoplus v_{11} \\\\[0.2cm]\nu_{16}\\bigoplus v_{13} & u_{17}\\bigoplus v_{16} & u_{18}\\bigoplus v_{19} & u_{19}\\bigoplus v_{22}\n&u_{20}\\bigoplus v_{14} & u_{21}\\bigoplus v_{17} & u_{22}\\bigoplus v_{20} & u_{23}\\bigoplus v_{23}\n\\end{array}\n\\end{eqnarray}\n\\end{small}\n\\begin{small}\n\\begin{eqnarray}\n\\label{exm-2-k2-2}\n\\begin{array}{cccccccc}\nv_{24}\\bigoplus v_{25} & v_{25}\\bigoplus v_{26} & v_{27}\\bigoplus v_{28} & v_{28}\\bigoplus v_{29} & v_{30}\\bigoplus v_{31} & v_{31}\\bigoplus v_{32} & v_{33}\\bigoplus v_{34} & v_{34}\\bigoplus v_{35}\n\\end{array}\n\\end{eqnarray}\n\\end{small} On the one hand, user $4$ can obtain the following sub-packets by \\eqref{exm-2-k2-1}.\n$$W^{(l_2,l_1)}_{4,\\mathcal{A}_0,\\mathcal{B}_0}, \\ \\ W^{(l_2,l_1)}_{4,\\mathcal{A}_1,\\mathcal{B}_0},\\ \\ W^{(l_2,l_1)}_{4,\\mathcal{A}_2,\\mathcal{B}_0},\\ \\ W^{(l_2,l_1)}_{4,\\mathcal{A}_3,\\mathcal{B}_0}\\ \\ \\ \\ \\ \\  l_1\\in [0,2),\\ \\ l_2\\in[0,2).$$\nand all the remaining\n$$W^{(l_2,2)}_{4,\\mathcal{A}_0,\\mathcal{B}_0},\\  W^{(l_2,2)}_{4,\\mathcal{A}_1,\\mathcal{B}_0},\\  W^{(l_2,2)}_{4,\\mathcal{A}_2,\\mathcal{B}_0},\\  W^{(l_2,2)}_{4,\\mathcal{A}_3,\\mathcal{B}_0}\\ \\ \\ l_2\\in[0,2)$$\ncan be obtained by \\eqref{exm-2-k2-2} on the other hand. For instance, by\n\\begin{small}\n\\begin{eqnarray}\n\\label{eq-ex-neq}\n\\begin{split}\nv_{24}\\bigoplus v_{25}=Y^{(0,2)}_{\\mathcal{A}_0,\\mathcal{S}_2}\\bigoplus Y^{(1,2)}_{\\mathcal{A}_0,\\mathcal{S}_2}=\\left(W^{(0,2)}_{5,\\mathcal{A}_0,\\mathcal{B}_1}\\bigoplus W^{(0,2)}_{6,\\mathcal{A}_0,\\mathcal{B}_2}\\right)\\bigoplus\\left(W^{(0,2)}_{4,\\mathcal{A}_0,\\mathcal{B}_0}\\bigoplus W^{(1,2)}_{6,\\mathcal{A}_0,\\mathcal{B}_2}\\right)\\\\\nv_{25}\\bigoplus v_{26}=Y^{(1,2)}_{\\mathcal{A}_0,\\mathcal{S}_2}\\bigoplus Y^{(2,2)}_{\\mathcal{A}_0,\\mathcal{S}_2}=\\left(W^{(0,2)}_{4,\\mathcal{A}_0,\\mathcal{B}_0}\\bigoplus W^{(1,2)}_{6,\\mathcal{A}_0,\\mathcal{B}_2}\\right)\\bigoplus \\left(W^{(1,2)}_{4,\\mathcal{A}_0,\\mathcal{B}_0}\\bigoplus W^{(1,2)}_{5,\\mathcal{A}_0,\\mathcal{B}_1}\\right)\n\\end{split}\\end{eqnarray}\n\\end{small}\nuser $4$ can firstly decode $W^{(0,2)}_{4,\\mathcal{A}_0,\\mathcal{B}_0}$ from $v_{24}\\bigoplus v_{25}$. Based on the decoded $W^{(0,2)}_{4,\\mathcal{A}_0,\\mathcal{B}_0}$ and $v_{25}\\bigoplus v_{26}$, user $4$ can further decode $W^{(1,2)}_{4,\\mathcal{A}_0,\\mathcal{B}_0}$. With the similar strategy, other users can also decode their requested sub-packets.\n\n\\end{example}\n\nIn this example, by using the proposed scheme, the transmission rate is $\\frac{24+8}{72}=\\frac{4}{9}$, which is smaller than $\\frac{7}{12}$ in Example \\ref{ex-trivial}. And this implies that, by using our proposed CSM scheme, the coded messages are designed to better exploit the coded multicasting opportunities among all the users in $\\mathcal{K}_1,\\mathcal{K}_2$. \n\n\n\n\n\\section{Performance Analysis}\\label{sec-first-performance}\nIn this section, we will show that the rate of our new scheme is order optimal. Firstly,  let us introduce the scheme in \\cite{SWXH}. In order to simplify, we denote the cache size of fixed  user and  group user as $M_{\\mathcal{K}_1}$, $M_{\\mathcal{K}_2}$ in this section.\n\\begin{lemma}\\rm(Order-optimal scheme in \\cite{SWXH})\n\\label{le-Decentral}\nFor a caching problem with $N$ files and $K$ users with $(K<N)$, without loss of generality, assume that the caching sizes satisfy $M_{0}\\leq M_{1}\\leq \\ldots \\leq M_{K-1}$ and denote $\\mathcal{M}=\\{M_{0}, M_{1}, \\ldots, M_{K-1}\\}$. A scheme can be obtained with transmission rate\n\\begin{eqnarray}\n\\label{eq-decentral-R}\nR_D(K;\\mathcal{M};N)=\\sum\\limits_{i=0}^{K-1}\\left[  \\prod\\limits_{j=0}^{i}\\left(1-\\frac{M_j}{N}\\right)\\right].\n\\end{eqnarray} Furthermore this scheme is order optimal since\n\\begin{eqnarray}\n\\label{eq-order-decentral}\n\\frac{R_D(K;\\mathcal{M};N)}{R^*(K;\\mathcal{M};N)}\\leq 6.\n\\end{eqnarray}\n\\end{lemma}\n\n Assume that $\\mathcal{M}=\\{M_{\\mathcal{K}_1},\\ldots,M_{\\mathcal{K}_1},M_{\\mathcal{K}_2},\\ldots,M_{\\mathcal{K}_2}\\}$. Let $\\lambda_1=\\frac{M_{\\mathcal{K}_1}}{N}$, $\\lambda_2=\\frac{M_{\\mathcal{K}_2}}{N}$ and $K=K_1+K_2$. Let us show the order optimality and smaller rate compared with the scheme from \\cite{SWXH} in the following separate cases.\n\n\\subsection{The case $\\lambda_1\\leq\\lambda_2$}\\label{subsec-M1<M2}\nWhen $\\lambda_1\\leq\\lambda_2$, the cache set can be denoted by $$\\mathcal{M}\n\\{\\underbrace{M_{\\mathcal{K}_1},\\ldots,M_{\\mathcal{K}_1}}_{K_1},\\underbrace{M_{\\mathcal{K}_2},\\ldots,M_{\\mathcal{K}_2}}_{K_2}\\},$$\nthen \\eqref{eq-decentral-R} can be written as\n\\begin{small}\n\\begin{eqnarray}\n\\begin{split}\n\\label{eq-decentral-R-K12-1}\nR_D(K;\\mathcal{M};N)\n=&\\frac{K_1-t_1}{t_1}\\left[1-\\left(1-\\frac{t_1}{K_1}\\right)^{K_1}\\right]+\\left(1-\\frac{t_1}{K_1}\\right)^{K_1}\\frac{K_2-t_2}{t_2}\\left[1-\\left(1-\\frac{t_2}{K_2}\\right)^{K_2}\\right]\\\\\n=&\\left(\\frac{1}{\\lambda_1}-1\\right)\\left[1-\\left(1-\\lambda_1\\right)^{K_1}\\right]+\\left(1-\\lambda_1\\right)^{K_1}\\left(\\frac{1}{\\lambda_2}-1\\right)\\left[1-\\left(1-\\lambda_2\\right)^{K_2}\\right].\n\\end{split}\\end{eqnarray}\n\\end{small} That is, from Lemma \\ref{le-Decentral} we have an order optimal scheme with rate in \\eqref{eq-decentral-R-K12-1}, and by Theorem \\ref{th-new-r} we have\n\\begin{small}\n\\begin{eqnarray}\n\\label{eq-th-new-Simplify-1}\n\\begin{split}\nR(K;\\mathcal{M};N)&=\\frac{(K_1-t_1)(t_1+t_2+1)}{(t_1+1)t_2t_1}+\\frac{(K_2-t_2)t_1}{t_2(t_1+1)}\\\\\n&=\\left(\\frac{1}{\\lambda_1}-1\\right)\\frac{K_1\\lambda_1+K_2\\lambda_2+1}{(K_1\\lambda_1+1)K_2\\lambda_2}+\\left(\\frac{1}{\\lambda_2}-1\\right)\\frac{K_1\\lambda_1}{(K_1\\lambda_1+1)}\\\\\n&=\\left(\\frac{1}{\\lambda_1}-1\\right)\\left(\\frac{1}{K_2\\lambda_2}+\\frac{1}{K_1\\lambda_1+1}\\right)+\\left(\\frac{1}{\\lambda_2}-1\\right)\\left(1-\\frac{1}{K_1\\lambda_1+1}\\right)\n\\end{split}\n\\end{eqnarray}\n\\end{small}\nBy \\eqref{eq-decentral-R-K12-1} and \\eqref{eq-th-new-Simplify-1}, we have\n\\begin{small}\n\\begin{eqnarray}\n\\begin{split}\n\\label{eq-minus-decentral-new-1}\n&\\frac{R(K;\\mathcal{M};N)}{R_D(K;\\mathcal{M};N)}\\\\\n=&\\frac{\\left(\\frac{1}{\\lambda_1}-1\\right)\\left(1-\\frac{1}{K_1\\lambda_1+1}+\\frac{1}{K_2\\lambda_2}\\right)+\\left(\\frac{1}{\\lambda_2}-1\\right)\\frac{1}{K_1\\lambda_1+1}}\n{\\left(\\frac{1}{\\lambda_1}-1\\right)\\left[1-(1-\\lambda_1)^{K_1}\\right]+(1-\\lambda_1)^{K_1}\\left(\\frac{1}{\\lambda_2}-1\\right)\\left[1-(1-\\lambda_2)^{K_2}\\right]}\\\\\n<&\\frac{\\left(\\frac{1}{\\lambda_1}-1\\right)\\left(1-\\frac{1}{K_1\\lambda_1+1}+\\frac{1}{K_2\\lambda_2}+\\frac{1}{K_1\\lambda_1+1}\\right)}\n{\\left(\\frac{1}{\\lambda_1}-1\\right)\\left[1-(1-\\lambda_1)^{K_1}\\right]}\\\\\n<&\\frac{\\left(\\frac{1}{\\lambda_1}-1\\right)\\left(1+\\frac{1}{K_2\\lambda_2}\\right)}{\\left(\\frac{1}{\\lambda_1}-1\\right)\\left(1-\\frac{1}{K_1\\lambda_1}\\right)}\\\\\n<&\\frac{1+\\frac{1}{K_2\\lambda_2}}{1-\\frac{1}{K_1\\lambda_1}}\\\\\n<&2\n\\end{split}\\end{eqnarray}\\end{small}\n\n\nNow we consider the parameter where the rate of our new scheme is smaller than the rate of the scheme from \\cite{SWXH}, i.e., $\\frac{R(K;\\mathcal{M};N)}{R_D(K;\\mathcal{M};N)}<1$. Similarly, according to \\eqref{eq-decentral-R-K12-1} and \\eqref{eq-th-new-Simplify-1}, we have\n\\begin{eqnarray*}\n\\begin{split}\n&\\frac{R(K;\\mathcal{M};N)}{R_D(K;\\mathcal{M};N)}\\\\[0.2cm]\n=&\\frac{\\left(\\frac{1}{\\lambda_1}-1\\right)\\left(1-\\frac{1}{K_1\\lambda_1+1}+\\frac{1}{K_2\\lambda_2}\\right)+\n\\left(\\frac{1}{\\lambda_2}-1\\right)\\frac{1}{K_1\\lambda_1+1}}\n{\\left(\\frac{1}{\\lambda_1}-1\\right)\\left[1-\\left(1-\\lambda_1\\right)^{K_1}\\right]+\n\\left(1-\\lambda_1\\right)^{K_1}\\left(\\frac{1}{\\lambda_2}-1\\right)\\left[1-\\left(1-\\lambda_2\\right)^{K_2}\\right]}\\\\[0.2cm]\n=&\\frac{\\left(\\frac{1}{\\lambda_1}-1\\right)\\left[-\\frac{1}{K_1\\lambda_1+1}+\\frac{1}{K_2\\lambda_2}+\\left(1-\\lambda_1\\right)^{K_1}\\right]+\n\\left(\\frac{1}{\\lambda_2}-1\\right)\\left[\\frac{1}{K_1\\lambda_1+1}-\\left(1-\\lambda_1\\right)^{K_1}+\\left(1-\\lambda_1\\right)^{K_1}\\left(1-\\lambda_2\\right)^{K_2}\\right]}\n{\\left(\\frac{1}{\\lambda_1}-1\\right)\\left[1-\\left(1-\\lambda_1\\right)^{K_1}\\right]+\n\\left(1-\\lambda_1\\right)^{K_1}\\left(\\frac{1}{\\lambda_2}-1\\right)\\left[1-\\left(1-\\lambda_2\\right)^{K_2}\\right]}+1\\\\[0.2cm]\n=&\\frac{\\left(\\frac{1}{\\lambda_1}-1\\right)x+\\left(\\frac{1}{\\lambda_2}-1\\right)y}{z}+1\n\\end{split}\\end{eqnarray*}\nwhere $x=-\\frac{1}{K_1\\lambda_1+1}+\\frac{1}{K_2\\lambda_2}+\\left(1-\\lambda_1\\right)^{K_1}$, $y=\\frac{1}{K_1\\lambda_1+1}-\\left(1-\\lambda_1\\right)^{K_1}+\\left(1-\\lambda_1\\right)^{K_1}\\left(1-\\lambda_2\\right)^{K_2}$ and $z=1\\left\/\\right.\\left(\\frac{1}{\\lambda_1}-1\\right)\\left[1-\\left(1-\\lambda_1\\right)^{K_1}\\right]+\n\\left(1-\\lambda_1\\right)^{K_1}\\left(\\frac{1}{\\lambda_2}-1\\right)\\left[1-\\left(1-\\lambda_2\\right)^{K_2}\\right]$. It is not difficult to check that $y>0$ and $z>0$ always hold. By calculating we can see that $\\frac{1}{K_2\\lambda_2+1}>\\left(1-\\lambda_2\\right)^{K_2}$ holds when $K_2$ is enough large. We claim that if $x<0$ and\n\\begin{eqnarray}\n\\label{eq-com-neq4}\nK_1(1-\\lambda_1)+K_2(1-\\lambda_2)+\\frac{1}{\\lambda_1}+\\frac{1}{\\lambda_2}<2+\\frac{K_1}{\\lambda_1},\\end{eqnarray}then\n\\begin{eqnarray}\\label{eq-com-neq1}\n\\left(\\frac{1}{\\lambda_1}-1\\right)x+\\left(\\frac{1}{\\lambda_2}-1\\right)y<0.\n\\end{eqnarray}\nThis implies that $\\frac{R(K;\\mathcal{M};N)}{R_D(K;\\mathcal{M};N)}<1$. Since\n\\begin{eqnarray}\n\\label{eq-com-neq3}\n\\begin{split}\n\\frac{-x}{y}\n=&\\frac{\\frac{1}{K_1\\lambda_1-1}+\\frac{1}{K_2\\lambda_2}-\\left(1-\\lambda_1\\right)^{K_1}}\n{\\frac{1}{K_1\\lambda_1+1}-\\left(1-\\lambda_1\\right)^{K_1}+\\left(1-\\lambda_1\\right)^{K_1}\\left(1-\\lambda_2\\right)^{K_2}}\\\\[0.2cm]\n\\geq&\\frac{\\frac{1}{K_1\\lambda_1-1}+\\frac{1}{K_2\\lambda_2}}\n{\\frac{1}{K_1\\lambda_1+1}+\\left(1-\\lambda_1\\right)^{K_1}\\left(1-\\lambda_2\\right)^{K_2}}\\\\[0.2cm]\n\\geq&\\frac{\\frac{1}{K_1\\lambda_1-1}+\\frac{1}{K_2\\lambda_2}}\n{\\frac{1}{K_1\\lambda_1+1}+\\frac{1}{K_1\\lambda_1+1}\\frac{1}{K_2\\lambda_2+1}}\\\\[0.2cm]\n=&\\frac{K_2\\lambda_2-K_1\\lambda_1-1}{K_2\\lambda_2+1},\n\\end{split}\n\\end{eqnarray} and \\eqref{eq-com-neq4} can be written as\n$$\\frac{\\frac{1}{\\lambda_2}-1}{\\frac{1}{\\lambda_1}-1}<\\frac{K_2\\lambda_2-K_1\\lambda_1-1}{K_2\\lambda_2+1},$$\nwe have \\begin{eqnarray}\n\\label{eq-com-neq2}\n\\frac{\\frac{1}{\\lambda_2}-1}{\\frac{1}{\\lambda_1}-1}<\\frac{-x}{y},\n\\end{eqnarray} i.e., \\eqref{eq-com-neq1} always holds.\n\\subsection{The case $\\lambda_1>\\lambda_2$}\\label{subsec-M1>M2}\nWhen $\\lambda_1>\\lambda_2$, the coresponding cache set can also be denoted by $$\\mathcal{M}'=\n\\{\\underbrace{M_{\\mathcal{K}_2},\\ldots,M_{\\mathcal{K}_2}}_{K_2},\\underbrace{M_{\\mathcal{K}_1},\\ldots,M_{\\mathcal{K}_1}}_{K_1}\\},$$\nSimilar to \\eqref{eq-decentral-R-K12-1}, \\eqref{eq-decentral-R} can be rewritten as\n\\begin{small}\n\\begin{eqnarray*}\n\\begin{split}\nR_D(K;\\mathcal{M}';N)=&\\left(\\frac{1}{\\lambda_2}-1\\right)\\left[1-\\left(1-\\lambda_2\\right)^{K_2}\\right]+\\left(1-\\lambda_2\\right)^{K_2}\\left(\\frac{1}{\\lambda_1}-1\\right)\\left[1-\\left(1-\\lambda_1\\right)^{K_1}\\right].\n\\end{split}\\end{eqnarray*}\n\\end{small}\nAccording to Theorem \\ref{th-new-r} we have\n\\begin{small}\n\\begin{eqnarray*}\n\\begin{split}\nR(K;\\mathcal{M}';N)&=\\frac{(K_1-t_1)t_2}{t_1(t_2+1)}+\\frac{(K_2-t_2)(t_1+t_2+1)}{(t_2+1)t_2t_1}\\\\\n&=\\left(\\frac{1}{\\lambda_1}-1\\right)\\frac{K_2\\lambda_2}{(K_2\\lambda_2+1)}+\\left(\\frac{1}{\\lambda_2}-1\\right)\\frac{K_1\\lambda_1+K_2\\lambda_2+1}{(K_2\\lambda_2+1)K_1\\lambda_1}\\\\\n&=\\left(\\frac{1}{\\lambda_1}-1\\right)\\left(1-\\frac{1}{K_2\\lambda_2+1}\\right)+\\left(\\frac{1}{\\lambda_2}-1\\right)\\left(\\frac{1}{K_1\\lambda_1}+\\frac{1}{K_2\\lambda_2+1}\\right)\n\\end{split}\n\\end{eqnarray*}\n\\end{small}\nSimilar to \\eqref{eq-minus-decentral-new-1}, we have\n\\begin{small}\n\\begin{eqnarray}\n\\label{eq-minus-decentral-new-2}\n\\begin{split}\n\\frac{R(K;\\mathcal{M}';N)}{R_D(K;\\mathcal{M}';N)}<2\\\\\n\\end{split}\\end{eqnarray}\\end{small} Then using the same analysis in the Subsection \\ref{subsec-M1<M2}, we claim that $\\frac{R(K;\\mathcal{M}';N)}{R_D(K;\\mathcal{M}';N)}<1$ always holds when $\\lambda_1$ and $\\lambda_2$ satisfy the following conditions.\n\\begin{eqnarray}\n\\label{eq-3}\nK_1(1-\\lambda_1)+K_2(1-\\lambda_2)+\\frac{1}{\\lambda_1}+\\frac{1}{\\lambda_2}<2+\\frac{K_2}{\\lambda_2}\n\\end{eqnarray}\n\\subsection{Order-optimality and Smaller Rate}\\label{sub-sec-Order-Smaller}\nFrom the discussions of Subsections \\ref{subsec-M1<M2} and \\ref{subsec-M1>M2}, the statements holds.\n\\begin{remark}\\rm\n\\label{rem:1}\n\\begin{eqnarray*}\n\\frac{R(K;\\mathcal{M};N)}{R^*(K;\\mathcal{M};N)}<\\frac{2R_D(K;\\mathcal{M};N)}{R^*(K;\\mathcal{M};N)}< 12.\n\\end{eqnarray*}\nTherefore, the scheme in Theorem \\ref{th-new-r} is order optimal.\n\\end{remark}\n\n\\begin{remark}\\label{rem:2}\nIf $\\lambda_1$ and $\\lambda_2$ satisfy the following conditions,\n\\begin{align}\\label{eq:rem2}\nK_1(1-\\lambda_1)+K_2(1-\\lambda_2)+\\frac{1}{\\lambda_1}+\\frac{1}{\\lambda_2}<\\left\\{\n             \\begin{array}{lr}\n             2+\\frac{K_1}{\\lambda_1},  &\\lambda_1 \\leq\\lambda_2\\\\\n             2+\\frac{K_2}{\\lambda_2},  &\\lambda_1 >\\lambda_2\n             \\end{array}\n\\right.,\n\\end{align}\nthe scheme in Theorem \\ref{th-new-r} has the smaller transmission rate than that of the scheme proposed in Lemma \\ref{le-Decentral}, by \\eqref{eq-com-neq4} and \\eqref{eq-3}.\n\\end{remark}\nIn fact, there is a large region of $\\lambda_1$ and $\\lambda_2$ satisfying \\eqref{eq:rem2}. For example, let $K_1=500$ and $K_2=100$. Assume that $1>\\lambda_1>\\lambda_2$. The range of $\\lambda_1$ and $\\lambda_2$ satisfying \\eqref{eq:rem2} is shown in Figure \\ref{fig-7}.\n\\begin{figure}[htbp]\n  \\centering\n  \\includegraphics[width=0.5\\textwidth]{Figure7.png}\n  \\caption{The range of $\\lambda_1$ and $\\lambda_2$ satisfying \\eqref{eq:rem2}.}\\label{fig-7}\n\\end{figure}\n\\section{Conclusion}\\label{sec-conclusion}\nIn this paper, we focus on the coded caching design for the system containing a set of fixed and mobile users. In order to avoid resource consumption resulted from unnecessary cache adaptation and updating for those fixed users in the placement phase, and to minimize the amount of transmission in the delivery phase, we propose a $(K_1,K_2;M_1,M_2;N)$ dynamic coded caching design framework through concatenating-based placement and the saturating matching based delivery. With the proposed scheme, once the cache of fixed users is fulfilled, there is no need to consider the cache content adaptation when there is no change in the files at server. Instead, the placement phase only needs to be performed for those mobile users. On the basis, the saturating matching based multicasting can be performed to minimize the transmission load in the delivery phase. We have also shown that the proposed scheme is order-optimal.\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}