Nash equilibria for componentwise variational systems

Stan Andrei^1,∗

¹Faculty of Mathematics and Computer Science, Babeş-Bolyai University, 400084 Cluj-Napoca, Romania
&
Tiberiu Popoviciu Institute of Numerical Analysis, Romanian Academy, P.O. Box 68-1, 400110 Cluj-Napoca, 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.

^†^†footnotetext: ^∗Corresponding author. E-mail addresses: andrei.stan@ubbcluj.ro.

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{cases}N_{1}(u,v)=J_{1}(u)\\ N_{2}(u,v)=J_{2}(v),\end{cases}

(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

\displaystyle\begin{cases}E_{11}(u,v)=0\\ E_{22}(u,v)=0,\end{cases}

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.,

\left\{\begin{array}[]{l}E_{1}(u^{\ast},v^{\ast})=\inf E_{1}(\cdot,v^{\ast})\\ E_{2}(u^{\ast},v^{\ast})=\inf E_{2}(u^{\ast},\cdot)\ .\end{array}\right.

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 (Minty-Browder Theorem) and a fixed point principle (Leray-Schauder Fixed Point Theorem).

2. Preliminaries

Let $X$ be a real, separable and uniformly convex Banach space, $X^{\ast}$ its dual and let $\langle\cdot,\cdot\rangle$ stand for the dual pairing between $X^{\ast}$ and $X$ . We denote with $J$ the duality mapping corresponding to the gauge function $\varphi(t):=t^{p-1}$ , where $p>1$ , i.e.,

Jx:=\{x^{\ast}\in X^{\ast}\,:\,\langle x\,,\,x\rangle=|x|^{p}\,,\,|x^{\ast}|_{% X^{\ast}}=|x|^{p-1}\}.

(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 $\limsup_{n\to\infty}\langle J\,x_{n}\,,\,x_{n}-x\rangle\leq 0$ , then $x_{n}\to x$ strongly.
iii)

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

$J$ is bijective from $X$ to $X^{\ast}$ .

A square matrix of non-negative numbers $A=[a_{i,j}]_{1\leq i,j\leq n}\in\mathbb{M}_{n,n}\left(\mathbb{R}_{+}\right)$ is said to be convergent to zero if $A^{k}\to O_{n}$ as $k\to\infty,$ where $O_{n}$ is the zero matrix. In case $n=2$ we have the following equivalent characterization (see [4]).

Lemma 2.2.

Let $A=[a_{i,j}]_{1\leq i,j\leq 2}$ be a square matrix of non-negatie real numbers. Then $A$ is convergent to zero if and only if $a_{11},a_{22}<1$ and

a_{11}+a_{22}<1+a_{11}a_{22}-a_{12}a_{21}.

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 (Minty-Browder [5, Theorem 9.14]).

Let $X$ be a real, reflexive and separable Banach space. Assume $T\colon X\to X^{\ast}$ 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 (Leray-Schauder [6, Theorem 2.11]).

Let $X$ be a Banach space and $T\colon X\to X$ 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 $\Omega$ be a bounded domain from $\mathbb{R}^{d}$ having Lipschitz boundary.

Consider the well known Sobolev space $W^{1,p}_{0}(\Omega):=\{u\in W^{1,p}(\Omega)\,|\,u_{\partial\Omega}=0\}$ endowed with the norm

|u|_{1,p}:=|\nabla u|_{L^{p}}=\left(\int_{\Omega}|\nabla u|^{p}\right)^{% \nicefrac}.

Is known that $\left(W^{1,p}_{0}(\Omega)\,,\,|\cdot|_{1,p}\right)$ is a separable and uniformly convex real Banach space (see [3, Theorem 6]). The notation $W^{-1,p^{\prime}}(\Omega)$ stands for the dual of $W^{1,p}(\Omega)$ , where $\tfrac{1}{p}+\tfrac{1}{p^{\prime}}=1$ .

Proposition 2.6 ([3, Proposition 6]).

Let $f\colon\Omega\times\mathbb{R}\to\mathbb{R}$ be a Carathéodory function which satisfies the following growth condition:

|f(x,s)|\leq a|s|^{p-1}+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 $L^{p}(\Omega)$ to $L^{p^{\prime}}(\Omega).$

One has the following diagram (see [3, p. 355]),

W_{0}^{1,p}(\Omega)\mathrel{\ensurestackMath{\stackanchor[.1ex]{% \hookrightarrow}{\hookrightarrow}}}L^{p}(\Omega)\xrightarrow{N_{f}}L^{p^{% \prime}}(\Omega)\hookrightarrow W^{-1,p}(\Omega),

and the Poincare’s inequalities: for any $u\in W_{0}^{1,p}(\Omega)$

\displaystyle|u|_{L^{p}}\leq C\,|u|_{1,p}=|\nabla u|_{L^{p}},

and for any $f\in L^{p^{\prime}}(\Omega)$ ,

\displaystyle|f|_{w^{-1,p^{\prime}}}\leq C\,|f|_{L^{p^{\prime}}},

where $C$ is a constant depending only on $\Omega$ and $n$ .

The following result establishes an equivalence between $p$ -Laplacian and the duality mapping corresponding to the gauge function $\varphi(t)=t^{p-1}$ on $\left(W^{1,p}_{0}(\Omega)\,,\,|\cdot|_{1,p}\right)$ .

Theorem 2.7 ([3, Theorem 7]).

The operator $-\Delta_{p}\colon W^{1,p}_{0}(\Omega)\to W^{-1,p^{\prime}}(\Omega)$ is the Fréchet derivative of functional $\psi\colon W^{1,p}_{0}(\Omega)\to\mathbb{R}$ defined as $\psi(u)=\tfrac{1}{p}|u|_{1,p}^{p}$ . More exactly,

\psi^{\prime}=-\Delta_{p}=J_{\varphi},

where $J_{\varphi}$ represents the duality mapping corresponding to the gauge function $\varphi(t)=t^{p-1}$ .

3. Main result

In what follows, $\left(X_{1},|\cdot|_{1}\right)$ and $\left(X_{2},|\cdot|_{2}\right)$ 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 $\langle\cdot\,,\,\cdot\rangle_{1}$ , $\langle\cdot\,,\,\cdot\rangle_{2}$ 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 $\varphi(t):=t^{p-1}$ , 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 $N_{i}\colon X_{1}\times X_{2}\to X_{i}^{\ast}$ ( $i\in\{1,2\}$ ) are continuous and satisfies the following monotony conditions: there are real numbers $a_{11},a_{22}\in[0,1)$ such that

		$\displaystyle\langle N_{1}(u,v)-N_{1}(\overline{u},v)\,,\,u-\overline{u}% \rangle_{1}\leq a_{11}\langle J_{1}(u)-J_{1}(\overline{u})\,,\,u-\overline{u}% \rangle\text{, for all }u,\overline{u}\in X_{1}\text{ and }v\in X_{2}\,,$		(3.1)
		$\displaystyle\langle N_{2}(u,v)-N_{2}(u,\overline{v})\,,\,v-\overline{v}% \rangle_{1}\leq a_{22}\langle J_{2}(v)-J_{2}(\overline{v})\,,\,u-\overline{v}% \rangle\text{, for all }v,\overline{v}\in X_{2}\text{ and }u\in X_{1}.$		(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.,

		$\displaystyle E_{1}(u^{\ast},v^{\ast})=\inf_{X_{1}}E_{1}(\cdot,v^{\ast})$		(3.3)
		$\displaystyle E_{2}(u^{\ast},v^{\ast})=\inf_{X_{2}}E_{2}(u^{\ast},\cdot)\,.$		(3.3)

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

\displaystyle E_{1}(u^{\ast},v^{\ast})\leq E_{1}(u^{\ast}+u,v^{\ast})\ \text{ % and }\ E_{2}(u^{\ast},v^{\ast})\leq 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

	$\displaystyle E_{1}(u^{\ast}+u,v^{\ast})-E_{1}(u^{\ast},v^{\ast})$	$\displaystyle=\int_{0}^{1}\langle E_{11}(u^{\ast}+t\,u,v^{\ast})\,,\,u\rangle_% {1}dt$
		$\displaystyle=\int_{0}^{1}\langle J_{1}(u^{\ast}+t\,u)-N_{1}(u^{\ast}+t,v^{% \ast})\,,\,u\rangle_{1}dt$
		$\displaystyle=\int_{0}^{1}\langle J_{1}(u^{\ast}+t\,u)-J_{1}(u^{\ast})\,,\,u% \rangle_{1}-\langle N_{1}(u^{\ast}+t\,u,v^{\ast})-N_{1}(u^{\ast},v^{\ast})\,,% \,u\rangle_{1}dt.$

Condition (3.1) yields

\displaystyle E_{1}(u^{\ast}+u,v^{\ast})-E_{1}(u^{\ast},v^{\ast})

\displaystyle\geq\int_{0}^{1}\frac{(1-a_{11})}{t}\langle J_{1}(u^{\ast}+t\,u,v% ^{\ast})-J_{1}(u^{\ast},v^{\ast})\,,\,u^{\ast}+t\,u-u^{\ast}\rangle_{1}dt,

and furthermore, from monotony of $J_{1}$ , we deduce

E_{1}(u^{\ast}+u,v^{\ast})-E_{1}(u^{\ast},v^{\ast})\geq 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})\geq 0,\text{ for all $v\in X% _{2}$.}

∎

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 $J_{2}^{-1}\circ N_{2}\colon X_{1}\times X_{2}\to X_{2}$ 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)|\leq a_{12}|v|_{1}^{p-1}+M_{1},\quad\text{ for all }v\in X_{2},

(3.4)

|N_{2}(u,0)|\leq a_{21}|u|_{1}^{p-1}+M_{2},\quad\text{ for all }u\in X_{1},

(3.5)

and the matrix

A=\begin{bmatrix}a_{11}&a_{12}\\ a_{21}&a_{22}\end{bmatrix}

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

\displaystyle\langle J_{1}(u)-N_{1}(u,v)-J_{1}(\overline{u})+N_{1}(\overline{u% },v)\,,\,u-\overline{u}\rangle_{1}\geq(1-a_{11})\langle J_{1}(u)-J_{1}(% \overline{u})\,,\,u-\overline{u}\rangle_{1}\geq 0,

and

	$\displaystyle\frac{\langle J_{1}(u)-N_{1}(u,v)\,,\,u\rangle_{1}}{\|u\|_{1}}$	$\displaystyle=\frac{\langle J_{1}(u)\,,\,u\rangle_{1}-\langle N_{1}(u,v)-N_{1}% (0,v)\,,\,u\rangle_{1}}{\|u\|_{1}}-\frac{\langle N_{1}(0,v)\,,\,u\rangle_{1}}{\|u% \|_{1}}$
		$\displaystyle\geq(1-a_{11})\frac{\langle J_{1}(u),u\rangle_{1}}{\|u\|_{1}}-\|N_{1% }(0,v)\|$
		$\displaystyle=(1-a_{11})\|u\|_{1}^{p-1}-\|N_{1}(0,v)\|\to\infty\,\,\text{ as }\|u\|_% {1}\to\infty.$

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 $S\colon X_{2}\to X_{1}$ .

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

	$\displaystyle\|S(v_{n})\|^{p}$	$\displaystyle=\langle J_{1}\,S(v_{n})\,,\,S(v_{n})\rangle_{1}$
		$\displaystyle=\langle N_{1}\left(S(v_{n}),v_{n}\right)-N_{1}(0,v_{n})\,,\,S(v_% {n})\rangle_{1}+\langle N_{1}(0,v_{n})\,,\,S(v_{n})\rangle_{1}$
		$\displaystyle\leq a_{11}\langle J_{1}\,S(v_{n})\,,\,S(v_{n})\rangle_{1}+\|N_{1}% (0,v_{n})\|\,\|S(v_{n})\|_{1}$
		$\displaystyle=a_{11}\|S(v_{n})\|^{p}+\|N_{1}(0,v_{n})\|\,\|S(v_{n})\|_{1}.$

Thus,

|S(v_{n})|^{p-1}\leq\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,

	$\displaystyle\langle J_{1}\,S(v_{n})\,,\,S(v_{n})-w\rangle_{1}$	$\displaystyle=\langle N_{1}\left(S(v_{n})\,,\,v_{n}\right)\,,\,S(v_{n})-w% \rangle_{1}$
		$\displaystyle=\langle N_{1}\left(S(v_{n})\,,\,v_{n}\right)-N_{1}(w,v_{n})\,,\,% S(v_{n})-w\rangle_{1}+\langle N_{1}(w,v_{n})\,,\,S(v_{n})-w\rangle_{1}$
		$\displaystyle\leq a_{11}\langle J_{1}\,S(v_{n})-J_{1}w\,,\,S(v_{n})-w\rangle_{% 1}+\langle N_{1}(w,v_{n})\,,\,S(v_{n})-w\rangle_{1}.$

Consequently,

$\displaystyle\langle J_{1}\,S(v_{n})\,,\,S(v_{n})-w\rangle_{1}$	$\displaystyle\leq\frac{1}{1-a_{11}}\bigg{(}\langle J_{1}w\,,\,S(v_{n})-w% \rangle_{1}+\langle N_{1}(w,v_{n})\,,\,S(v_{n})-w\rangle_{1}\bigg{)}$	(3.7)
	$\displaystyle\leq\frac{1}{1-a_{11}}\langle J_{1}w-N_{1}(w,v)\,,\,S(v_{n})-w% \rangle_{1}$
	$\displaystyle\quad+\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

	$\displaystyle\langle J_{1}w-N_{1}(w,v)\,,\,S(v_{n})-w\rangle_{1}\to 0\text{ % and }$
	$\displaystyle\|N_{1}(w,v_{n})-N_{1}(w,v)\|\,\left(\|S(v_{n})\|+\|w\|\right)\to 0.$

Hence, passing to $\limsup$ in (3.7) we deduce

\limsup_{n\to\infty}\langle J_{1}\,S(v_{n})\,,\,S(v_{n})-w\rangle_{1}\leq 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

\lim_{n\to\infty}J_{1}\,S(v_{n})=N_{1}(w,v).

(3.8)

Then

	$\displaystyle\langle J_{1}w-J_{1}S(v)\,,\,w-S(v)\rangle_{1}$	$\displaystyle=\langle N_{1}(w,v)-N_{1}(S(v),v)\,,\,w-S(v)\rangle_{1}+\langle J% _{1}w-N_{1}(w,v)\,,\,w-S(v)\rangle_{1}$
		$\displaystyle\leq a_{11}\langle J_{1}w-J_{1}\,S(v)\,,\,w-S(v)\rangle_{1}+% \langle J_{1}w-J_{1}\,S(v_{n})\,,\,w-S(v)\rangle_{1}$
		$\displaystyle\quad+\langle J_{1}\,S(v_{n})-N_{1}(w,v)\,,\,w-S(v)\rangle_{1},$

that is,

\langle J_{1}w-J_{1}S(v)\,,\,w-S(v)\rangle_{1}\leq\frac{1}{1-a_{11}}\bigg{(}% \langle J_{1}w-J_{1}\,S(v_{n})\,,\,w-S(v)\rangle_{1}+\langle J_{1}\,S(v_{n})-N% _{1}(w,v)\,,\,w-S(v)\rangle_{1}\bigg{)}.

(3.9)

From the demicontinuity of $J_{1}$ , clearly one has

\langle J_{1}w-J_{1}\,S(v_{n})\,,\,w-S(v)\rangle_{1}\to 0,

Thus, passing to limit in (3.9) we conclude that

\langle J_{1}w-J_{1}S(v)\,,\,w-S(v)\rangle_{1}\leq 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

	$\displaystyle\|S(v)\|^{p}_{1}$	$\displaystyle=\langle J_{1}\,S(v)\,,\,S(v)\rangle_{1}$
		$\displaystyle=\langle N_{1}(S(v),v)-N_{1}(0,v)\,,\,S(v)\rangle_{1}+\langle N_{% 1}(0,v)\,,\,S(v)\rangle_{1}$
		$\displaystyle\leq a_{11}\langle J_{1}\,S(v)\,,\,S(v)\rangle_{1}+\|N_{1}(0,v)\|\,% \|S(v)\|_{1}$
		$\displaystyle=a_{11}\|S(v)\|^{p}_{1}+\|N_{1}(0,v)\|\,\|S(v)\|_{1},$

i.e,

|S(v)|_{1}\leq\left(\frac{1}{1-a_{11}}|N_{1}(0,v)|\right)^{\frac{1}{p-1}}.

(3.10)

Until now, we proved that the solution operator is continuous and satisfies the growth condition (3.10). Next, via Leray-Schauder 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 $\lambda\leq 1$ 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 $J_{2}(\alpha\,v)=\alpha^{p-1}J_{2}(v)$ , for any $\alpha\in\mathbb{R}_{+}$ , one has

	$\displaystyle\|v\|^{p}_{2}$	$\displaystyle=\lambda^{p}\langle J_{2}\tfrac{1}{\lambda}v\,,\,\tfrac{1}{% \lambda}v\rangle_{2}$
		$\displaystyle=\lambda^{p-1}\langle N_{2}(S(v),v)-N_{2}(S(v),0)\,,\,v\rangle_{2% }+\lambda^{p-1}\langle N_{2}(S(v),0)\,,\,v\rangle_{2}$
		$\displaystyle\leq a_{22}\lambda^{p-1}\langle J_{2}v\,,\,v\rangle_{2}+\lambda^{% p-1}\,\|v\|_{2}\,\|N_{2}(S(v),0)\|$
		$\displaystyle\leq a_{22}\lambda^{p-1}\|v\|^{p}_{2}+\|v\|_{2}\,\|N_{2}(S(v),0)\|.$

Now, from (3.10) and the growth conditions (3.4-3.5), we obtain

	$\displaystyle(1-a_{22})\|v\|_{2}^{p-1}$	$\displaystyle\leq\|N_{2}(S(v),0)\|$
		$\displaystyle\leq a_{21}\|S(v)\|^{p-1}_{1}+M_{2}$
		$\displaystyle\leq\frac{a_{12}a_{21}}{1-a_{11}}\|v\|^{p-1}+\frac{a_{21}}{1-a_{11}% }M_{1}+M_{2},$

which gives

|v|_{2}^{p-1}\leq\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)^{-\tfrac{1}{p}}\left(M^{% \prime}\right)^{\tfrac{1}{p}}.

Now, Leary-Schauder 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{cases}-\Delta_{p}\;u=f_{1}(\cdot,u,v)\\ -\Delta_{p}\;v=f_{2}(\cdot,u,v)\text{ on }\Omega\\ u|_{\partial\Omega}=v|_{\partial\Omega}=0\,,\end{cases}

(4.1)

where $p>1$ and $\Omega$ 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 $W^{1,p}_{0}(\Omega)$ 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 $-\Delta_{p}$ .

Functions $f_{1},f_{2}\colon\Omega\times\mathbb{R}^{n}\to\mathbb{R}$ are of Carathéodory type and satisfies the growth conditions

	$\displaystyle\|f_{1}(x,r,s)\|\leq C_{1}\|r\|^{p-1}+C_{2}\|s\|^{p-1}+a(x)\,,$		(4.2)
	$\displaystyle\|f_{2}(x,r,s)\|\leq C_{1}\|r\|^{p-1}+C_{2}\|r\|^{p-1}+b(x)\,,$		(4.3)

for all real numbers $r,s\in\mathbb{R}$ , where $C_{1},C_{2}$ are constants and $a,b\in L^{p^{\prime}}(\Omega).$

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 $L^{p}(\Omega)$ to $L^{p^{\prime}}(\Omega)$ , continuous and bounded. Moreover, due to the compact embedding of $W_{0}^{1,p}(\Omega)$ in $L^{p}(\Omega)$ , the operator

T=(-\Delta_{p})^{-1}N_{f_{2}}(u,v)\colon W_{0}^{1,p}(\Omega)\times W_{0}^{1,p}% (\Omega)\to W_{0}^{1,p}(\Omega)

is compact (see Dinica and Jebelean [11]).

Note that the equations from (4.1) admit a variational structure, given by the energy functionals $E_{1},E_{2}\colon W^{1,p}(\Omega)\times W^{1,p}(\Omega)\to\mathbb{R}$ ,

		$\displaystyle E_{1}(u,v):=\frac{1}{p}\|u\|_{1,p}^{p}-\int_{\Omega}F_{1}(\cdot,u,% v),$
		$\displaystyle E_{2}(u,v):=\frac{1}{p}\|u\|_{1,p}^{p}-\int_{\Omega}F_{2}(\cdot,u,% v),$

where

		$\displaystyle F_{1}(x,u(x),v(x)):=\int_{0}^{u(x)}f_{1}(x,s,w(x))ds,$
		$\displaystyle F_{2}(x,u(x),s):=\int_{0}^{v(x)}f_{2}(x,u(x),s)ds.$

Theorem 4.1.

Let the above conditions be fulfilled. Additionally assume:

(H1)

There are non-negative real numbers $\overline{a}_{11},\overline{a}_{22}$ such that

		$\displaystyle(r-\overline{r})(f_{1}(\cdot,r,s)-f_{1}(\cdot,\overline{r},s))% \leq\overline{a}_{11}\|r-\overline{r}\|^{p},$		(4.4)
		$\displaystyle(s-\overline{s})(f_{2}(\cdot,r,s)-f_{2}(\cdot,r,\overline{s}))% \leq\overline{a}_{22}\|s-\overline{s}\|^{p},$		(4.5)

for all real numbers $r,\overline{r},s,\overline{s}$ .

(H2)

There are non-negative real numbers $\overline{a}_{12},\overline{a}_{21},M_{1},M_{2}$ such that

		$\displaystyle\|f(\cdot,0,s)\|\leq\overline{a}_{12}\|s\|^{p-1}+M_{1},$		(4.6)
		$\displaystyle\|f(\cdot,r,0)\|\leq\overline{a}_{21}\|r\|^{p-1}+M_{2},$		(4.7)

for all real numbers $r, s$ .

(H3)

The matrix

A:=\begin{bmatrix}\overline{a}_{11}\,C^{p}&\overline{a}_{12}\,C^{p-1}\\ \overline{a}_{21}\,C^{p-1}&\overline{a}_{22}\,C^{p}\end{bmatrix}

is convergent to zero.

Then there exist $(u^{\ast},v^{\ast})\in W^{1,p}(\Omega)$ 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 (4.6-4.7), 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

		$\displaystyle\|\overline{N}_{f_{1}}v\|_{L^{p^{\prime}}}\leq a_{12}\|v\|^{p-1}_{L^{% p}}+M_{1}^{\prime}$		(4.8)
		$\displaystyle\|\overline{N}_{f_{2}}u\|_{L^{p^{\prime}}}\leq a_{21}\|u\|^{p-1}_{L^{% p}}+M_{2}^{\prime}.$		(4.8)

Proof of the Theorem.

We verify that all conditions of Theorem 3.2 are satisfied.

Check of conditions (3.1), (3.2). Let $u,\overline{u},v\in W^{1,p}_{0}(\Omega)$ . Then, from (4.4), we obtain

	$\displaystyle\langle f_{1}(\cdot,u,v)-f_{1}(\cdot,\overline{u},v)\,,\,u-% \overline{u}\rangle_{W^{-1,p^{\prime}}}$	$\displaystyle=\int_{\Omega}(u-\overline{u})\,\left(f_{1}(\cdot,u,v)-f_{1}(% \cdot,\overline{u},v)\right)$
		$\displaystyle\leq a_{12}\|u-\overline{u}\|_{L^{p}}^{p}$
		$\displaystyle\leq a_{12}\,C^{p}\,\|u-\overline{u}\|_{1,p}$
		$\displaystyle=a_{12}\,C^{p}\,\langle(-\Delta_{p})u-(-\Delta_{p})\overline{u}\,% ,\,u-\overline{u}\rangle_{W^{-1,p^{\prime}}}.$

Similarly, (4.5) yields

\langle f_{2}(\cdot,u,v)-f_{2}(\cdot,u,\overline{u})\,,\,v-\overline{v}\rangle% _{W^{-1,p^{\prime}}}\leq a_{22}\,C^{p}\,\langle(-\Delta_{p})v-(-\Delta_{p})% \overline{v}\,,\,v-\overline{v}\rangle_{W^{-1,p^{\prime}}}.

Check of condition $(\textbf{h1)}$ . The condition is trivially satisfied since the operator $(-\Delta_{p})^{-1}N_{f_{2}}(u,v)$ is compact.

Check of condition $(\textbf{h2)}$ . Let $v\in W^{1,p}_{0}(\Omega)$ . Then, Lemma 4.2 yields

	$\displaystyle\|f_{1}(\cdot,0,v)\|_{W^{-1,p^{\prime}}}$	$\displaystyle\leq\|f_{1}(\cdot,0,v)\|_{L^{\prime}}$
		$\displaystyle\leq a_{12}\,C^{p-1}\,\|v\|^{p-1}_{1,p}+M_{1}^{\prime}.$

Similarly,

|f_{2}(\cdot,u,0)|_{W^{-1,p^{\prime}}}\leq a_{21}\,C^{p-1}\,|u|^{p-1}_{1,p}+M_% {2}^{\prime}.

Finally, note that all assumptions from Theorem 3.2 are fulfilled, where

		$\displaystyle a_{11}=\overline{a}_{11}C^{p}\,;\,a_{22}=\overline{a}_{22}C^{p}$
		$\displaystyle a_{12}=\overline{a}_{12}C^{p-1}\,;\,a_{21}=\overline{a}_{21}C^{p% -1}.$

Therefore, there exist $(u^{\ast},v^{\ast})\in W^{1,p}_{0}(\Omega)\times W^{1,p}_{0}(\Omega)$ a solution of the system (4.1). Moreover, from Theorem 3.1 it is a Nash equilibrium for the energy functionals $(E_{1},E_{2})$ . ∎

References

[1] R. Precup, Nash-type equilibria and periodic solutions to nonvariational systems, Adv. Nonlinear Anal. 3(2014), 197–207.
[2] C. Avramescu, On a fixed point theorem (in Romanian), St. Cerc. Mat. 22(1970), no. 2, 215–221.
[3] G. Dinca, P. Jebelean, J. Mawhin, Variational and topological methods for Dirichlet problems with p-Laplacian, Portugal. Math. 58(2001), 339-378.
[4] A. Berman, R. J. Plemmons Nonnegative Matrices in the Mathematical Sciences, Academic Press (1997).
[5] G. Ciarlet, Linear and Nonlinear Functional Analysis with Applications, SIAM, Philadelphia, 2013.
[6] H. Le Dret, Nonlinear Elliptic Partial Differential Equations, Springer, Berlin, 2018.
[7] D.G. Costa, C.A. Magalhaes, Variational elliptic problems which are nonquadratic at infinity, Nonlinear Analysis: Theory, Methods and Applications. 23(1994).
[8] J. Diestel, Geometry of Banach Spaces-Selected Topics, Springer-Verlag, New-York, 1975.
[9] G. Dinca, P. Jebelean, Some existence results for a class of nonlinear equations involving a duality mapping, Nonlinear Anal. 46(2001), 347-363.
[10] H. Brezis, Functional Analysis, Sobolev Spaces and Partial Differential Equations, Springer, New York, 2011.
[11] G. Dinca, P. Jebelean, J. Mawhin, Une m´ethode de point fixe pour le p-Laplacien, C. R. Acad. Sci. Paris, Ser. I 324 (1997), 165–168

	$\displaystyle\frac{\langle J_{1}(u)-N_{1}(u,v)\,,\,u\rangle_{1}}{\|u\|_{1}}$	$\displaystyle=\frac{\langle J_{1}(u)\,,\,u\rangle_{1}-\langle N_{1}(u,v)-N_{1}% (0,v)\,,\,u\rangle_{1}}{\|u\|_{1}}-\frac{\langle N_{1}(0,v)\,,\,u\rangle_{1}}{\|u% \|_{1}}$
		$\displaystyle\geq(1-a_{11})\frac{\langle J_{1}(u),u\rangle_{1}}{\|u\|_{1}}-\|N_{1% }(0,v)\|$
		$\displaystyle=(1-a_{11})\|u\|_{1}^{p-1}-\|N_{1}(0,v)\|\to\infty\,\,\text{ as }\|u\|_% {1}\to\infty.$

	$\displaystyle\|S(v_{n})\|^{p}$	$\displaystyle=\langle J_{1}\,S(v_{n})\,,\,S(v_{n})\rangle_{1}$
		$\displaystyle=\langle N_{1}\left(S(v_{n}),v_{n}\right)-N_{1}(0,v_{n})\,,\,S(v_% {n})\rangle_{1}+\langle N_{1}(0,v_{n})\,,\,S(v_{n})\rangle_{1}$
		$\displaystyle\leq a_{11}\langle J_{1}\,S(v_{n})\,,\,S(v_{n})\rangle_{1}+\|N_{1}% (0,v_{n})\|\,\|S(v_{n})\|_{1}$
		$\displaystyle=a_{11}\|S(v_{n})\|^{p}+\|N_{1}(0,v_{n})\|\,\|S(v_{n})\|_{1}.$

	$\displaystyle\|S(v)\|^{p}_{1}$	$\displaystyle=\langle J_{1}\,S(v)\,,\,S(v)\rangle_{1}$
		$\displaystyle=\langle N_{1}(S(v),v)-N_{1}(0,v)\,,\,S(v)\rangle_{1}+\langle N_{% 1}(0,v)\,,\,S(v)\rangle_{1}$
		$\displaystyle\leq a_{11}\langle J_{1}\,S(v)\,,\,S(v)\rangle_{1}+\|N_{1}(0,v)\|\,% \|S(v)\|_{1}$
		$\displaystyle=a_{11}\|S(v)\|^{p}_{1}+\|N_{1}(0,v)\|\,\|S(v)\|_{1},$

	$\displaystyle\|v\|^{p}_{2}$	$\displaystyle=\lambda^{p}\langle J_{2}\tfrac{1}{\lambda}v\,,\,\tfrac{1}{% \lambda}v\rangle_{2}$
		$\displaystyle=\lambda^{p-1}\langle N_{2}(S(v),v)-N_{2}(S(v),0)\,,\,v\rangle_{2% }+\lambda^{p-1}\langle N_{2}(S(v),0)\,,\,v\rangle_{2}$
		$\displaystyle\leq a_{22}\lambda^{p-1}\langle J_{2}v\,,\,v\rangle_{2}+\lambda^{% p-1}\,\|v\|_{2}\,\|N_{2}(S(v),0)\|$
		$\displaystyle\leq a_{22}\lambda^{p-1}\|v\|^{p}_{2}+\|v\|_{2}\,\|N_{2}(S(v),0)\|.$

	$\displaystyle(1-a_{22})\|v\|_{2}^{p-1}$	$\displaystyle\leq\|N_{2}(S(v),0)\|$
		$\displaystyle\leq a_{21}\|S(v)\|^{p-1}_{1}+M_{2}$
		$\displaystyle\leq\frac{a_{12}a_{21}}{1-a_{11}}\|v\|^{p-1}+\frac{a_{21}}{1-a_{11}% }M_{1}+M_{2},$

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Nash equilibria for componentwise variational systems

1. Introduction

2. Preliminaries

Lemma 2.1.

Lemma 2.2.

Theorem 2.3 (Minty-Browder [5, Theorem 9.14]).

Theorem 2.4 (Leray-Schauder [6, Theorem 2.11]).

Theorem 2.5 ([6, Lemma 1.1]).

Proposition 2.6 ([3, Proposition 6]).

Theorem 2.7 ([3, Theorem 7]).

3. Main result

3.1. Nash equilibria property

Theorem 3.1.

Proof.

3.2. Exstence result

Theorem 3.2.

Proof.

4. Application

Theorem 4.1.

Lemma 4.2 ([9, Proposition 8]).

Proof of the Theorem.

References

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