Abstract
In this paper, we generalize an existing result regarding the existence of a Nash equilibrium for a system of fixed point equations. The problem is considered in a more general form and the initial conditions are also improved, without changing the final conclusion. This is achieved by combining the idea of a solution operator with monotone operator techniques and classical fixed point principles. An application to a coupled system with Dirichlet boundary conditions involving the pLaplacian is provided.
Authors
Andrei Stan
Faculty of Mathematics and Computer Science, BabesBolyai University, ClujNapoca, Romania
Tiberiu Popoviciu Institute of Numerical Analysis, Romanian Academy, ClujNapoca, Romania
Keywords
Dirichlet boundary condition; Monotone operator; Nash equilibrium.
Paper coordinates
A. Stan, Nash equilibria for componentwise variational systems, Journal of Nonlinear Functional Analysis, 2023 (2023), art. no. 6, http://jnfa.mathres.org/archives/3029
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About this paper
Journal
Journal of Nonlinear Functional Analysis
Publisher Name
Mathematical Research Press
DOI
http://doi.org/10.23952/jnfa.2023.6
(not working)
Print ISSN
2052532X
Online ISSN
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Paper (preprint) in HTML form
Nash equilibria for componentwise variational systems
^{1}Faculty of Mathematics and Computer Science, BabeşBolyai University, 400084 ClujNapoca, Romania
&
Tiberiu Popoviciu
Institute of Numerical Analysis, Romanian Academy, P.O. Box 681, 400110
ClujNapoca, Romania
Abstract. In this paper we generalize an existing result regarding the existence of a Nash equilibrium for a system of fixed point equations. The studied problem is considered in a more general form and the initial conditions are also improved, without changing the final conclusion. This is achieved combining the idea of a solution operator with monotone operators techniques and classical fixed point principles. An application to a coupled system with Dirichlet boundary conditions involving the $p$Laplacian is provided.
Keywords. Nash equilibrium, Monotone operator.
2010 Mathematics Subject Classification. 47J25, 47J30, 47H10.
1. Introduction
Numerous equations can be reduced to a fixed point equation $N(u)=u$, where $N$ is an operator. The equation is said to admit a variational structure if there exists a differentiable Fréchet functional $E$ (called the ”energy functional”) such that any solution $u$ is also a solution of the critical point equation ${E}^{\prime}(u)=0$, and vice versa.
In this paper we are concerned with an existence result for a system of type
$$\{\begin{array}{cc}{N}_{1}(u,v)={J}_{1}(u)\hfill & \\ {N}_{2}(u,v)={J}_{2}(v),\hfill & \end{array}$$  (1.1) 
where each of the equations admits a variational structure, i.e., there are two functionals ${E}_{1},{E}_{2}$ such that the system is equivalent with
$\{\begin{array}{cc}{E}_{11}(u,v)=0\hfill & \\ {E}_{22}(u,v)=0,\hfill & \end{array}$ 
where ${E}_{11}$ is the partial Fréchet derivative of ${E}_{1}$ with respect to the first variable and ${E}_{22}$ is the partial Fréchet derivative of ${E}_{2}$ with respect to the second variable. Here ${J}_{1}$, ${J}_{2}$ are two duality mappings and ${N}_{1}$, ${N}_{2}$ two continuous operators.
In what follows we are studding more than an existence result for (1.1), namely under what conditions we have a solution $({u}^{\ast},{v}^{\ast})$ which is also a Nash equilibrium for the corresponding energy functionals, i.e.,
$$\{\begin{array}{c}{E}_{1}({u}^{\ast},{v}^{\ast})=inf{E}_{1}(\cdot ,{v}^{\ast})\hfill \\ {E}_{2}({u}^{\ast},{v}^{\ast})=inf{E}_{2}({u}^{\ast},\cdot ).\hfill \end{array}$$ 
We aim to generalize the result from [1], where a similar system has been considered on Hilbert spaces. This was achieved imposing a Perov contraction condition and making use of Ekeland variational principle. Our contribution in the present paper aims to improve both the functional framework (real, separable and uniformly convex Banach spaces) as well as the initial conditions (less restrictive ones), while retaining the same conclusion. The result is obtained combining the idea of a solution operator, inspired from [2], with monotone operators techniques (MintyBrowder Theorem) and a fixed point principle (LeraySchauder Fixed Point Theorem).
2. Preliminaries
Let $X$ be a real, separable and uniformly convex Banach space, ${X}^{\ast}$ its dual and let $\u27e8\cdot ,\cdot \u27e9$ stand for the dual pairing between ${X}^{\ast}$ and $X$. We denote with $J$ the duality mapping corresponding to the gauge function $\phi (t):={t}^{p1}$, where $p>1$, i.e.,
$$Jx:=\{{x}^{\ast}\in {X}^{\ast}:\u27e8x,x\u27e9={x}^{p},{{x}^{\ast}}_{{X}^{\ast}}={x}^{p1}\}.$$  (2.1) 
Below (Lemma 2.1) some important properties of the duality mapping $J$ are stated. For proofs and further details we send to Dinca and Jebelean [3].
Lemma 2.1.
The duality mapping (2.1) has the following proprieties:

i)
$J$ is single valued and strictly monotone.

ii)
$J$ satisfies the ${(S)}_{+}$ condition, i.e., if ${x}_{n}\to x$ weakly and ${lim\; sup}_{n\to \mathrm{\infty}}\u27e8J{x}_{n},{x}_{n}x\u27e9\le 0$, then ${x}_{n}\to x$ strongly.

iii)
$J$ is demicontinuous, i.e., if ${x}_{n}\to x$ strongly, then $J{x}_{n}\to Jx$ weakly.

iv)
$J$ is bijective from $X$ to ${X}^{\ast}$.
A square matrix of nonnegative numbers $A={[{a}_{i,j}]}_{1\le i,j\le n}\in {\mathbb{M}}_{n,n}\left({\mathbb{R}}_{+}\right)$ is said to be convergent to zero if ${A}^{k}\to {O}_{n}$ as $k\to \mathrm{\infty},$ where ${O}_{n}$ is the zero matrix. In case $n=2$ we have the following equivalent characterization (see [4]).
Lemma 2.2.
Let $A={[{a}_{i,j}]}_{1\le i,j\le 2}$ be a square matrix of nonnegatie real numbers. Then $A$ is convergent to zero if and only if $$ and
$$ 
For the convenience of the reader, we present a list of theoretical results used throughout this paper. Because they are some classics in the theory of nonlinear analysis, we omit the proofs. However, further details can be found in the indicated sources.
Theorem 2.3 (MintyBrowder [5, Theorem 9.14]).
Let $X$ be a real, reflexive and separable Banach space. Assume $T:X\to {X}^{\ast}$ is a bounded, demicontinuous, coercive and monotone operator. Then for any given $v\in {X}^{\ast}$, there exist a unique $u\in X$ such that $T(u)=v$.
Theorem 2.4 (LeraySchauder [6, Theorem 2.11]).
Let $X$ be a Banach space and $T:X\to X$ a continuous compact mapping which satisfies the following condition: there exists $R>0$ such that the set ${\cup}_{\lambda \in [0,1]}\{x\in X:x=\lambda Tx\}$ lies in a ball of radius $R$, centered in the origin. Then $T$ admits at least one fixed point.
Theorem 2.5 ([6, Lemma 1.1]).
Let $X$ be a topological space and ${x}_{n}$ a sequence from $X$ with the following property: there exist $x\in X$ such that from any subsequence of ${x}_{n}$ we can extract a further subsequence converging to $x$. Then the whole sequence ${x}_{n}$ is convergent to $x$.
We conclude this preliminary section with some well known results related to the $p$Laplacian. For proofs and further details we refer to [3], [8] and [9]. Let $\mathrm{\Omega}$ be a bounded domain from ${\mathbb{R}}^{d}$ having Lipschitz boundary.
Consider the well known Sobolev space ${W}_{0}^{1,p}(\mathrm{\Omega}):=\{u\in {W}^{1,p}(\mathrm{\Omega}){u}_{\partial \mathrm{\Omega}}=0\}$ endowed with the norm
$${u}_{1,p}:={\nabla u}_{{L}^{p}}={\left({\int}_{\mathrm{\Omega}}{\nabla u}^{p}\right)}^{1/p}.$$ 
Is known that $({W}_{0}^{1,p}(\mathrm{\Omega}),\cdot {}_{1,p})$ is a separable and uniformly convex real Banach space (see [3, Theorem 6]). The notation ${W}^{1,{p}^{\prime}}(\mathrm{\Omega})$ stands for the dual of ${W}^{1,p}(\mathrm{\Omega})$, where $\frac{1}{p}+\frac{1}{{p}^{\prime}}=1$.
Proposition 2.6 ([3, Proposition 6]).
Let $f:\mathrm{\Omega}\times \mathbb{R}\to \mathbb{R}$ be a Carathéodory function which satisfies the following growth condition:
$$f(x,s)\le a{s}^{p1}+b(x),$$ 
where $b\in {L}^{{p}^{\prime}}$. Then the Nemytskii’s operator
$$\left({N}_{f}u\right)(x)=f(x,u(x)),$$ 
is continuous and bounded from ${L}^{p}(\mathrm{\Omega})$ to ${L}^{{p}^{\prime}}(\mathrm{\Omega}).$
One has the following diagram (see [3, p. 355]),
$${W}_{0}^{1,p}(\mathrm{\Omega})\hookrightarrow \hookrightarrow {L}^{p}(\mathrm{\Omega})\stackrel{{N}_{f}}{\to}{L}^{{p}^{\prime}}(\mathrm{\Omega})\hookrightarrow {W}^{1,p}(\mathrm{\Omega}),$$ 
and the Poincare’s inequalities: for any $u\in {W}_{0}^{1,p}(\mathrm{\Omega})$
${u}_{{L}^{p}}\le C{u}_{1,p}={\nabla u}_{{L}^{p}},$ 
and for any $f\in {L}^{{p}^{\prime}}(\mathrm{\Omega})$,
${f}_{{w}^{1,{p}^{\prime}}}\le C{f}_{{L}^{{p}^{\prime}}},$ 
where $C$ is a constant depending only on $\mathrm{\Omega}$ and $n$.
The following result establishes an equivalence between $p$Laplacian and the duality mapping corresponding to the gauge function $\phi (t)={t}^{p1}$ on $({W}_{0}^{1,p}(\mathrm{\Omega}),\cdot {}_{1,p})$.
Theorem 2.7 ([3, Theorem 7]).
The operator ${\mathrm{\Delta}}_{p}:{W}_{0}^{1,p}(\mathrm{\Omega})\to {W}^{1,{p}^{\prime}}(\mathrm{\Omega})$ is the Fréchet derivative of functional $\psi :{W}_{0}^{1,p}(\mathrm{\Omega})\to \mathbb{R}$ defined as $\psi (u)=\frac{1}{p}{u}_{1,p}^{p}$. More exactly,
$${\psi}^{\prime}={\mathrm{\Delta}}_{p}={J}_{\phi},$$ 
where ${J}_{\phi}$ represents the duality mapping corresponding to the gauge function $\phi (t)={t}^{p1}$.
3. Main result
In what follows, $({X}_{1},\cdot {}_{1})$ and $({X}_{2},\cdot {}_{2})$ are two separable and uniformly convex real Banach spaces. Further, ${X}_{1}^{\ast}$ and ${X}_{2}^{\ast}$ stand for the dual spaces of ${X}_{1}$, ${X}_{2}$ and ${\u27e8\cdot ,\cdot \u27e9}_{1}$, ${\u27e8\cdot ,\cdot \u27e9}_{2}$ for the dual pairing between ${X}_{1}^{\ast}$, ${X}_{1}$ and ${X}_{2}^{\ast}$, ${X}_{2}$.
We denote with ${J}_{i}$ ($i\in \{1,2\}$) the duality mapping corresponding to the gauge function $\phi (t):={t}^{p1}$, for some $p>1$. Clearly ${J}_{1}$ and ${J}_{2}$ satisfy all proprieties of duality mapping stated in Lemma 2.1.
We assume the operators ${N}_{i}:{X}_{1}\times {X}_{2}\to {X}_{i}^{\ast}$ ($i\in \{1,2\}$) are continuous and satisfies the following monotony conditions: there are real numbers ${a}_{11},{a}_{22}\in [0,1)$ such that
${\u27e8{N}_{1}(u,v){N}_{1}(\overline{u},v),u\overline{u}\u27e9}_{1}\le {a}_{11}\u27e8{J}_{1}(u){J}_{1}(\overline{u}),u\overline{u}\u27e9\text{, for all}u,\overline{u}\in {X}_{1}\text{and}v\in {X}_{2},$  (3.1)  
${\u27e8{N}_{2}(u,v){N}_{2}(u,\overline{v}),v\overline{v}\u27e9}_{1}\le {a}_{22}\u27e8{J}_{2}(v){J}_{2}(\overline{v}),u\overline{v}\u27e9\text{, for all}v,\overline{v}\in {X}_{2}\text{and}u\in {X}_{1}.$  (3.2) 
Below, we present an auxiliary result (Theorem 3.1) which allows us to split the problem of finding a solution which is a Nash equilibrium solution for system (1.1), as two individual problems: prove that any solution is a Nash equilibrium and prove that there is at least a solution.
3.1. Nash equilibria property
Theorem 3.1.
Under the previous assumptions, if the system (1.1) admits a solution $({u}^{\ast},{v}^{\ast})\in {X}_{1}\times {X}_{2}$ then it is a Nash equilibrium for the energy functionals $({E}_{1},{E}_{2})$, i.e.,
${E}_{1}({u}^{\ast},{v}^{\ast})=\underset{{X}_{1}}{inf}{E}_{1}(\cdot ,{v}^{\ast})$  (3.3)  
${E}_{2}({u}^{\ast},{v}^{\ast})=\underset{{X}_{2}}{inf}{E}_{2}({u}^{\ast},\cdot ).$ 
Proof.
In order to prove (3.3), is sufficient to show that for all $u\in {X}_{1}$ and $v\in {X}_{2}$, one has
${E}_{1}({u}^{\ast},{v}^{\ast})\le {E}_{1}({u}^{\ast}+u,{v}^{\ast})\text{and}{E}_{2}({u}^{\ast},{v}^{\ast})\le {E}_{2}({u}^{\ast},{v}^{\ast}+v).$ 
Let $u\in {X}_{1}$. Since ${E}_{1}(\cdot ,{v}^{\ast})$ is Fréchet differentiable and ${N}_{1}({u}^{\ast},{v}^{\ast})={J}_{1}({u}^{\ast})$, we obtain
${E}_{1}({u}^{\ast}+u,{v}^{\ast}){E}_{1}({u}^{\ast},{v}^{\ast})$  $={\displaystyle {\int}_{0}^{1}}{\u27e8{E}_{11}({u}^{\ast}+tu,{v}^{\ast}),u\u27e9}_{1}\mathit{d}t$  
$={\displaystyle {\int}_{0}^{1}}{\u27e8{J}_{1}({u}^{\ast}+tu){N}_{1}({u}^{\ast}+t,{v}^{\ast}),u\u27e9}_{1}\mathit{d}t$  
$={\displaystyle {\int}_{0}^{1}}{\u27e8{J}_{1}({u}^{\ast}+tu){J}_{1}({u}^{\ast}),u\u27e9}_{1}{\u27e8{N}_{1}({u}^{\ast}+tu,{v}^{\ast}){N}_{1}({u}^{\ast},{v}^{\ast}),u\u27e9}_{1}dt.$ 
Condition (3.1) yields
${E}_{1}({u}^{\ast}+u,{v}^{\ast}){E}_{1}({u}^{\ast},{v}^{\ast})$  $\ge {\displaystyle {\int}_{0}^{1}}{\displaystyle \frac{(1{a}_{11})}{t}}{\u27e8{J}_{1}({u}^{\ast}+tu,{v}^{\ast}){J}_{1}({u}^{\ast},{v}^{\ast}),{u}^{\ast}+tu{u}^{\ast}\u27e9}_{1}\mathit{d}t,$ 
and furthermore, from monotony of ${J}_{1}$, we deduce
$${E}_{1}({u}^{\ast}+u,{v}^{\ast}){E}_{1}({u}^{\ast},{v}^{\ast})\ge 0,\text{for all}u\in {X}_{1}.$$ 
Using a similar reasoning we find
$${E}_{2}({u}^{\ast},{v}^{\ast}+v){E}_{2}({u}^{\ast},{v}^{\ast})\ge 0,\text{for all}v\in {X}_{2}\text{.}$$ 
∎
Now, we are ready to state our main existence result. Taking into account Theorem 3.1, if one can prove the existence of a solution for system (1.1), then it is a Nash equilibrium for the associated energy functionals ${E}_{1},{E}_{2}$.
3.2. Exstence result
Theorem 3.2.
Under the previous mentioned setting, we additionally assume :

(h1)
The operator ${J}_{2}^{1}\circ {N}_{2}:{X}_{1}\times {X}_{2}\to {X}_{2}$ is compact.

(h2)
There are ${a}_{12},{a}_{21}\in (0,1)$ and ${M}_{1},{M}_{2}\in {\mathbb{R}}_{+}$ such that
$${N}_{1}(0,v)\le {a}_{12}{v}_{1}^{p1}+{M}_{1},\text{for all}v\in {X}_{2},$$ (3.4) $${N}_{2}(u,0)\le {a}_{21}{u}_{1}^{p1}+{M}_{2},\text{for all}u\in {X}_{1},$$ (3.5) and the matrix
$$A=\left[\begin{array}{cc}{a}_{11}& {a}_{12}\\ {a}_{21}& {a}_{22}\end{array}\right]$$ is convergent to zero.
Then there exists a solution $({u}^{\ast},{v}^{\ast})\in {X}_{1}\times {X}_{2}$ of the system (1.1).
Proof.
Our approach is inspired from a paper of Avramescu [2], where a solution operator is constructed from the first equation. This is then used, together with the second equation, to build a further operator, which is later shown to admit a fixed point.
Let $v\in {X}_{2}$ be arbitrarily chosen. First, note that from the monotony condition (3.4), the operator ${J}_{1}(\cdot ){N}_{1}(\cdot ,v)$ is monotone and coercive. Indeed, for any $u,\overline{u}\in {X}_{1}$, we have
${\u27e8{J}_{1}(u){N}_{1}(u,v){J}_{1}(\overline{u})+{N}_{1}(\overline{u},v),u\overline{u}\u27e9}_{1}\ge (1{a}_{11}){\u27e8{J}_{1}(u){J}_{1}(\overline{u}),u\overline{u}\u27e9}_{1}\ge 0,$ 
and
$\frac{{\u27e8{J}_{1}(u){N}_{1}(u,v),u\u27e9}_{1}}{{u}_{1}}$  $={\displaystyle \frac{{\u27e8{J}_{1}(u),u\u27e9}_{1}{\u27e8{N}_{1}(u,v){N}_{1}(0,v),u\u27e9}_{1}}{{u}_{1}}}{\displaystyle \frac{{\u27e8{N}_{1}(0,v),u\u27e9}_{1}}{{u}_{1}}}$  
$\ge (1{a}_{11}){\displaystyle \frac{{\u27e8{J}_{1}(u),u\u27e9}_{1}}{{u}_{1}}}{N}_{1}(0,v)$  
$=(1{a}_{11}){u}_{1}^{p1}{N}_{1}(0,v)\to \mathrm{\infty}\text{as}{u}_{1}\to \mathrm{\infty}.$ 
Moreover, since ${N}_{1}(\cdot ,v)$ is continuous and ${J}_{1}$ is bounded and demicontinuous, clearly ${J}_{1}(\cdot ){N}_{1}(\cdot ,v)$ is also bounded and demicontinuous. Now, in the virtue of Theorem 2.3 there exist a unique element $S(v)\in {X}_{1}$ such that
$${J}_{1}(S(v))={N}_{1}(S(v),v).$$  (3.6) 
Thus, we have defined by (3.6) the solution operator $S:{X}_{2}\to {X}_{1}$.
In what follows, we prove that $S$ is continuous. Let ${v}_{n}$ a sequence from ${X}_{2}$ convergent to some $v\in {X}_{2}$.
Boundedness of $S({v}_{n})$. From the monotony condition (3.1) and relation (3.6), we obtain
${S({v}_{n})}^{p}$  $={\u27e8{J}_{1}S({v}_{n}),S({v}_{n})\u27e9}_{1}$  
$={\u27e8{N}_{1}(S({v}_{n}),{v}_{n}){N}_{1}(0,{v}_{n}),S({v}_{n})\u27e9}_{1}+{\u27e8{N}_{1}(0,{v}_{n}),S({v}_{n})\u27e9}_{1}$  
$\le {a}_{11}{\u27e8{J}_{1}S({v}_{n}),S({v}_{n})\u27e9}_{1}+{N}_{1}(0,{v}_{n}){S({v}_{n})}_{1}$  
$={a}_{11}{S({v}_{n})}^{p}+{N}_{1}(0,{v}_{n}){S({v}_{n})}_{1}.$ 
Thus,
$${S({v}_{n})}^{p1}\le \frac{1}{1{a}_{11}}{N}_{1}(0,{v}_{n}),$$ 
which guarantees the boundedness of the sequence $S({v}_{n})$.
We intend to use Theorem 2.5. For this, let $S({v}_{n})$ be a subsequence of $S({v}_{n})$ (for simplicity, we keep the same indices). Since $S({v}_{n})$ is bounded, the is a further subsequence (also denoted with $S({v}_{n})$) and an element $w\in {X}_{1}$ such that $S({v}_{n})$ converges weakly to $w$ (see [10, Theorem 3.18]).
Strong convergence of $S({v}_{n})$ to w. One has,
${\u27e8{J}_{1}S({v}_{n}),S({v}_{n})w\u27e9}_{1}$  $={\u27e8{N}_{1}(S({v}_{n}),{v}_{n}),S({v}_{n})w\u27e9}_{1}$  
$={\u27e8{N}_{1}(S({v}_{n}),{v}_{n}){N}_{1}(w,{v}_{n}),S({v}_{n})w\u27e9}_{1}+{\u27e8{N}_{1}(w,{v}_{n}),S({v}_{n})w\u27e9}_{1}$  
$\le {a}_{11}{\u27e8{J}_{1}S({v}_{n}){J}_{1}w,S({v}_{n})w\u27e9}_{1}+{\u27e8{N}_{1}(w,{v}_{n}),S({v}_{n})w\u27e9}_{1}.$ 
Consequently,
${\u27e8{J}_{1}S({v}_{n}),S({v}_{n})w\u27e9}_{1}$  $\le {\displaystyle \frac{1}{1{a}_{11}}}\left({\u27e8{J}_{1}w,S({v}_{n})w\u27e9}_{1}+{\u27e8{N}_{1}(w,{v}_{n}),S({v}_{n})w\u27e9}_{1}\right)$  (3.7)  
$\le {\displaystyle \frac{1}{1{a}_{11}}}{\u27e8{J}_{1}w{N}_{1}(w,v),S({v}_{n})w\u27e9}_{1}$  
$+{\displaystyle \frac{S({v}_{n})+w}{1{a}_{11}}}{N}_{1}(w,{v}_{n}){N}_{1}(w,v)$ 
Since $S({v}_{n})$ is bounded, $S({v}_{n})\to w$ weakly and ${N}_{1}(w,{v}_{n})\to {N}_{1}(w,v)$ strongly, clearly
${\u27e8{J}_{1}w{N}_{1}(w,v),S({v}_{n})w\u27e9}_{1}\to 0\text{and}$  
${N}_{1}(w,{v}_{n}){N}_{1}(w,v)\left(S({v}_{n})+w\right)\to 0.$ 
Hence, passing to $lim\; sup$ in (3.7) we deduce
$$\underset{n\to \mathrm{\infty}}{lim\; sup}{\u27e8{J}_{1}S({v}_{n}),S({v}_{n})w\u27e9}_{1}\le 0,$$ 
which guarantees the strong convergence of $S({v}_{n})$ to $w$, based on the ${(S)}_{+}$ property of duality mapping ${J}_{1}$.
Prove of equatity w=S(v). Note that since ${N}_{1}(\cdot ,v)$ is continuous, one has
$$\underset{n\to \mathrm{\infty}}{lim}{J}_{1}S({v}_{n})={N}_{1}(w,v).$$  (3.8) 
Then
${\u27e8{J}_{1}w{J}_{1}S(v),wS(v)\u27e9}_{1}$  $={\u27e8{N}_{1}(w,v){N}_{1}(S(v),v),wS(v)\u27e9}_{1}+{\u27e8{J}_{1}w{N}_{1}(w,v),wS(v)\u27e9}_{1}$  
$\le {a}_{11}{\u27e8{J}_{1}w{J}_{1}S(v),wS(v)\u27e9}_{1}+{\u27e8{J}_{1}w{J}_{1}S({v}_{n}),wS(v)\u27e9}_{1}$  
$+{\u27e8{J}_{1}S({v}_{n}){N}_{1}(w,v),wS(v)\u27e9}_{1},$ 
that is,
$${\u27e8{J}_{1}w{J}_{1}S(v),wS(v)\u27e9}_{1}\le \frac{1}{1{a}_{11}}\left({\u27e8{J}_{1}w{J}_{1}S({v}_{n}),wS(v)\u27e9}_{1}+{\u27e8{J}_{1}S({v}_{n}){N}_{1}(w,v),wS(v)\u27e9}_{1}\right).$$  (3.9) 
From the demicontinuity of ${J}_{1}$, clearly one has
$${\u27e8{J}_{1}w{J}_{1}S({v}_{n}),wS(v)\u27e9}_{1}\to 0,$$ 
Thus, passing to limit in (3.9) we conclude that
$${\u27e8{J}_{1}w{J}_{1}S(v),wS(v)\u27e9}_{1}\le 0.$$ 
Now, from the strict monotony of the dual mapping ${J}_{1}$ we necessarily have $w=S(v)$.
Finally, putting all together we obtain the continuity of operator $S$.
Note that $S$ satisfies also the growth condition (3.10). Indeed, from (3.4), one has
${S(v)}_{1}^{p}$  $={\u27e8{J}_{1}S(v),S(v)\u27e9}_{1}$  
$={\u27e8{N}_{1}(S(v),v){N}_{1}(0,v),S(v)\u27e9}_{1}+{\u27e8{N}_{1}(0,v),S(v)\u27e9}_{1}$  
$\le {a}_{11}{\u27e8{J}_{1}S(v),S(v)\u27e9}_{1}+{N}_{1}(0,v){S(v)}_{1}$  
$={a}_{11}{S(v)}_{1}^{p}+{N}_{1}(0,v){S(v)}_{1},$ 
i.e,
$${S(v)}_{1}\le {\left(\frac{1}{1{a}_{11}}{N}_{1}(0,v)\right)}^{\frac{1}{p1}}.$$  (3.10) 
Until now, we proved that the solution operator is continuous and satisfies the growth condition (3.10). Next, via LeraySchauder theorem (Theorem 2.4), we show that the fixed point equation
$$v={J}_{2}^{1}{N}_{2}(S(v),v),$$ 
admits a fixed point. First, note that the operator ${J}_{2}^{1}\circ {N}_{2}\circ (S,I)$ is compact since ${J}_{2}^{1}\circ {N}_{2}$ is compact and $S,I$ are bounded and continuous operators.
Further, we show that there exists $R>0$ such that any fixed point of the operator $\lambda {J}_{2}^{1}{N}_{2}(S(\cdot ),\cdot )$ lies in the unit ball of radius $R$, for any $\lambda \in (0,1].$ Let $\lambda \le 1$ and $v$ a fixed point of the operator $\lambda {J}_{2}^{1}{N}_{2}(S(\cdot ),\cdot )$, i.e., $v$ satisfies
$$v=\lambda {J}_{2}^{1}{N}_{2}(S(v),v).$$ 
Since ${J}_{2}(\alpha v)={\alpha}^{p1}{J}_{2}(v)$, for any $\alpha \in {\mathbb{R}}_{+}$, one has
${v}_{2}^{p}$  $={\lambda}^{p}{\u27e8{J}_{2}\frac{1}{\lambda}v,\frac{1}{\lambda}v\u27e9}_{2}$  
$={\lambda}^{p1}{\u27e8{N}_{2}(S(v),v){N}_{2}(S(v),0),v\u27e9}_{2}+{\lambda}^{p1}{\u27e8{N}_{2}(S(v),0),v\u27e9}_{2}$  
$\le {a}_{22}{\lambda}^{p1}{\u27e8{J}_{2}v,v\u27e9}_{2}+{\lambda}^{p1}{v}_{2}{N}_{2}(S(v),0)$  
$\le {a}_{22}{\lambda}^{p1}{v}_{2}^{p}+{v}_{2}{N}_{2}(S(v),0).$ 
Now, from (3.10) and the growth conditions (3.43.5), we obtain
$(1{a}_{22}){v}_{2}^{p1}$  $\le {N}_{2}(S(v),0)$  
$\le {a}_{21}{S(v)}_{1}^{p1}+{M}_{2}$  
$\le {\displaystyle \frac{{a}_{12}{a}_{21}}{1{a}_{11}}}{v}^{p1}+{\displaystyle \frac{{a}_{21}}{1{a}_{11}}}{M}_{1}+{M}_{2},$ 
which gives
$${v}_{2}^{p1}\le {\left(1{a}_{22}\frac{{a}_{12}{a}_{21}}{1{a}_{11}}\right)}^{1}{M}^{\prime},$$  (3.11) 
where ${M}^{\prime}:=\frac{{a}_{21}}{1{a}_{11}}{M}_{1}+{M}_{2}$. Since matrix $A$ is convergent to zero, Lemma 2.2 yields that the solution $v$ lies in the ball $B(0,R)$, where
$$R={\left(1{a}_{22}\frac{{a}_{12}{a}_{21}}{1{a}_{11}}\right)}^{{\scriptscriptstyle \frac{1}{p}}}{\left({M}^{\prime}\right)}^{{\scriptscriptstyle \frac{1}{p}}}.$$ 
Now, LearySchauder fixed point theorem applies and guarantees the existence of a point ${v}^{\ast}$ such that
$${J}_{2}({v}^{\ast})={N}_{2}(S({v}^{\ast}),{v}^{\ast}).$$ 
One can see that $(S({v}^{\ast}),{v}^{\ast})$ is a solution of system (1.1), which ends our proof. ∎
4. Application
Consider the Dirichlet problem
$$\{\begin{array}{cc}{\mathrm{\Delta}}_{p}u={f}_{1}(\cdot ,u,v)\hfill & \\ {\mathrm{\Delta}}_{p}v={f}_{2}(\cdot ,u,v)\text{on}\mathrm{\Omega}\hfill & \\ {u}_{\partial \mathrm{\Omega}}={v}_{\partial \mathrm{\Omega}}=0,\hfill & \end{array}$$  (4.1) 
where $p>1$ and $\mathrm{\Omega}$ is some bounded domain from ${\mathbb{R}}^{n}$ with Lipschitz boundary. We seek for a pair of points $({u}^{\ast},{v}^{\ast})$, with ${u}^{\ast},{v}^{\ast}$ from the Sobolev space ${W}_{0}^{1,p}(\mathrm{\Omega})$ endowed with the usual norm ${u}_{1,p}:={\nabla u}_{{L}^{p}}$, which satisfies (4.1). Note that, in the light of Theorem 2.7, the dual mappings ${J}_{1},{J}_{2}$ are just the $p$Laplacian operator ${\mathrm{\Delta}}_{p}$.
Functions ${f}_{1},{f}_{2}:\mathrm{\Omega}\times {\mathbb{R}}^{n}\to \mathbb{R}$ are of Carathéodory type and satisfies the growth conditions
${f}_{1}(x,r,s)\le {C}_{1}{r}^{p1}+{C}_{2}{s}^{p1}+a(x),$  (4.2)  
${f}_{2}(x,r,s)\le {C}_{1}{r}^{p1}+{C}_{2}{r}^{p1}+b(x),$  (4.3) 
for all real numbers $r,s\in \mathbb{R}$, where ${C}_{1},{C}_{2}$ are constants and $a,b\in {L}^{{p}^{\prime}}(\mathrm{\Omega}).$
Since ${f}_{1}$ and ${f}_{2}$ satisfy the growth conditions (4.2), (4.3), the Nemytskii operators
${N}_{{f}_{1}}(u,v)(x):={f}_{1}(x,u(x),v(x))$ and ${N}_{{f}_{2}}(u,v)(x):={f}_{2}(x,u(x),v(x))$ 
are well defined from ${L}^{p}(\mathrm{\Omega})$ to ${L}^{{p}^{\prime}}(\mathrm{\Omega})$, continuous and bounded. Moreover, due to the compact embedding of ${W}_{0}^{1,p}(\mathrm{\Omega})$ in ${L}^{p}(\mathrm{\Omega})$, the operator
$$T={({\mathrm{\Delta}}_{p})}^{1}{N}_{{f}_{2}}(u,v):{W}_{0}^{1,p}(\mathrm{\Omega})\times {W}_{0}^{1,p}(\mathrm{\Omega})\to {W}_{0}^{1,p}(\mathrm{\Omega})$$ 
is compact (see Dinica and Jebelean [11]).
Note that the equations from (4.1) admit a variational structure, given by the energy functionals ${E}_{1},{E}_{2}:{W}^{1,p}(\mathrm{\Omega})\times {W}^{1,p}(\mathrm{\Omega})\to \mathbb{R}$,
${E}_{1}(u,v):={\displaystyle \frac{1}{p}}{u}_{1,p}^{p}{\displaystyle {\int}_{\mathrm{\Omega}}}{F}_{1}(\cdot ,u,v),$  
${E}_{2}(u,v):={\displaystyle \frac{1}{p}}{u}_{1,p}^{p}{\displaystyle {\int}_{\mathrm{\Omega}}}{F}_{2}(\cdot ,u,v),$ 
where
${F}_{1}(x,u(x),v(x)):={\displaystyle {\int}_{0}^{u(x)}}{f}_{1}(x,s,w(x))\mathit{d}s,$  
${F}_{2}(x,u(x),s):={\displaystyle {\int}_{0}^{v(x)}}{f}_{2}(x,u(x),s)\mathit{d}s.$ 
Theorem 4.1.
Let the above conditions be fulfilled. Additionally assume:

(H1)
There are nonnegative real numbers ${\overline{a}}_{11},{\overline{a}}_{22}$ such that
$(r\overline{r})({f}_{1}(\cdot ,r,s){f}_{1}(\cdot ,\overline{r},s))\le {\overline{a}}_{11}{r\overline{r}}^{p},$ (4.4) $(s\overline{s})({f}_{2}(\cdot ,r,s){f}_{2}(\cdot ,r,\overline{s}))\le {\overline{a}}_{22}{s\overline{s}}^{p},$ (4.5) for all real numbers $r,\overline{r},s,\overline{s}$.

(H2)
There are nonnegative real numbers ${\overline{a}}_{12},{\overline{a}}_{21},{M}_{1},{M}_{2}$ such that
$f(\cdot ,0,s)\le {\overline{a}}_{12}{s}^{p1}+{M}_{1},$ (4.6) $f(\cdot ,r,0)\le {\overline{a}}_{21}{r}^{p1}+{M}_{2},$ (4.7) for all real numbers $r,s$.

(H3)
The matrix
$$A:=\left[\begin{array}{cc}{\overline{a}}_{11}{C}^{p}& {\overline{a}}_{12}{C}^{p1}\\ {\overline{a}}_{21}{C}^{p1}& {\overline{a}}_{22}{C}^{p}\end{array}\right]$$ is convergent to zero.
Then there exist $({u}^{\ast},{v}^{\ast})\in {W}^{1,p}(\mathrm{\Omega})$ a solution of the system (4.1), which is a Nash equilibrium for the energy functionals ${E}_{1},{E}_{2}$.
For the proof we need the following lemma:
Lemma 4.2 ([9, Proposition 8]).
Under the growth conditions (4.64.7), the Nemytskii’s operators $({\overline{N}}_{{f}_{1}}v)(x):={f}_{1}(x,0,v(x))$ and $({\overline{N}}_{{f}_{2}}u)(x):={f}_{2}(x,u(x),0)$ satisfies
${{\overline{N}}_{{f}_{1}}v}_{{L}^{{p}^{\prime}}}\le {a}_{12}{v}_{{L}^{p}}^{p1}+{M}_{1}^{\prime}$  (4.8)  
${{\overline{N}}_{{f}_{2}}u}_{{L}^{{p}^{\prime}}}\le {a}_{21}{u}_{{L}^{p}}^{p1}+{M}_{2}^{\prime}.$ 
Proof of the Theorem.
We verify that all conditions of Theorem 3.2 are satisfied.
Check of conditions (3.1), (3.2). Let $u,\overline{u},v\in {W}_{0}^{1,p}(\mathrm{\Omega})$. Then, from (4.4), we obtain
${\u27e8{f}_{1}(\cdot ,u,v){f}_{1}(\cdot ,\overline{u},v),u\overline{u}\u27e9}_{{W}^{1,{p}^{\prime}}}$  $={\displaystyle {\int}_{\mathrm{\Omega}}}(u\overline{u})\left({f}_{1}(\cdot ,u,v){f}_{1}(\cdot ,\overline{u},v)\right)$  
$\le {a}_{12}{u\overline{u}}_{{L}^{p}}^{p}$  
$\le {a}_{12}{C}^{p}{u\overline{u}}_{1,p}$  
$={a}_{12}{C}^{p}{\u27e8({\mathrm{\Delta}}_{p})u({\mathrm{\Delta}}_{p})\overline{u},u\overline{u}\u27e9}_{{W}^{1,{p}^{\prime}}}.$ 
Similarly, (4.5) yields
$${\u27e8{f}_{2}(\cdot ,u,v){f}_{2}(\cdot ,u,\overline{u}),v\overline{v}\u27e9}_{{W}^{1,{p}^{\prime}}}\le {a}_{22}{C}^{p}{\u27e8({\mathrm{\Delta}}_{p})v({\mathrm{\Delta}}_{p})\overline{v},v\overline{v}\u27e9}_{{W}^{1,{p}^{\prime}}}.$$ 
Check of condition $(\text{h1)}$. The condition is trivially satisfied since the operator ${({\mathrm{\Delta}}_{p})}^{1}{N}_{{f}_{2}}(u,v)$ is compact.
Check of condition $(\text{h2)}$. Let $v\in {W}_{0}^{1,p}(\mathrm{\Omega})$. Then, Lemma 4.2 yields
${{f}_{1}(\cdot ,0,v)}_{{W}^{1,{p}^{\prime}}}$  $\le {{f}_{1}(\cdot ,0,v)}_{{L}^{\prime}}$  
$\le {a}_{12}{C}^{p1}{v}_{1,p}^{p1}+{M}_{1}^{\prime}.$ 
Similarly,
$${{f}_{2}(\cdot ,u,0)}_{{W}^{1,{p}^{\prime}}}\le {a}_{21}{C}^{p1}{u}_{1,p}^{p1}+{M}_{2}^{\prime}.$$ 
Finally, note that all assumptions from Theorem 3.2 are fulfilled, where
${a}_{11}={\overline{a}}_{11}{C}^{p};{a}_{22}={\overline{a}}_{22}{C}^{p}$  
${a}_{12}={\overline{a}}_{12}{C}^{p1};{a}_{21}={\overline{a}}_{21}{C}^{p1}.$ 
Therefore, there exist $({u}^{\ast},{v}^{\ast})\in {W}_{0}^{1,p}(\mathrm{\Omega})\times {W}_{0}^{1,p}(\mathrm{\Omega})$ a solution of the system (4.1). Moreover, from Theorem 3.1 it is a Nash equilibrium for the energy functionals $({E}_{1},{E}_{2})$. ∎
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