Nonlinear systems with a partial Nash type equilibrium

Andrei Stan “Babeş-Bolyai” University,
Faculty of Mathematics and Computer Sciences
1, Kogălniceanu Street,
400084 Cluj-Napoca,
Romania

Abstract.

In this paper fixed point arguments and a critical point technique are combined leading to hybrid existence results for a system of three operator equations where only two of the equations have a variational structure. The components of the solution which are associated to the equations having a variational form represent a Nash-type equilibrium of the corresponding energy functionals. The result is achieved by an iterative scheme based on Ekeland’s variational principle.

Key words and phrases:

Nash-type equilibrium; Perov contraction; Ekeland variational principle; periodic solution.

1991 Mathematics Subject Classification:

47H10, 47J30, 34C25

1. Introduction

Many nonlinear equations can be seen as a problem of fixed point $N\left(u\right)=u$ , where $N$ is a certain operator. One says that the equation has a variational form if it is equivalent with a critical point equation $E^{\prime}(u)=0$ . In the paper [7], R. Precup studied systems of the form

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

(1.1)

in a Hilbert space, where each of the equations has a variational form, i.e. there are two $C^{1}$ functionals $E_{1}$ and $E_{2}$ such that $E_{11}\left(u,v\right)=u-N_{1}\left(u,v\right)$ and $E_{22}\left(u,v\right)=v-N_{2}\left(u,v\right)$ , where $E_{11}$ and $E_{22}$ are the partial Fréchet derivatives of $E_{1}$ and $E_{2}$ with respect to u and v, respectively. Sufficient conditions have been established for that the system admits a solution which is a Nash type equilibrium for the functionals $E_{1}$ and $E_{2}$ , that is

	$\displaystyle E_{1}(u,v)$	$\displaystyle=$	$\displaystyle\inf_{u}E(\cdot,v),$
	$\displaystyle E_{2}(u,v)$	$\displaystyle=$	$\displaystyle\inf_{v}E(u,\cdot).$

2. Main result

Let $(X_{1},d)$ be a complete metric space and $(X_{2},|\cdot|_{2}),$ $(X_{3},|\cdot|_{3})$ be two real Hilbert spaces which are identified with their duals. Denote $X:=X_{1}\times X_{2}\times X_{3}$ . Let $N_{i}\colon X\rightarrow X_{i}$ $\left(i=1,2,3\right)\$ be continuous and assume that $N_{2},N_{3}$ have a variational structure, i.e. there exist functionals $E_{2},E_{3}:X\rightarrow\mathbb{R}$ such that $E_{2}(u,\cdot,w)$ is Fréchet differentiable for every $(u,w)\in X_{1}\times X_{3},$ $E_{3}(u,v,\cdot)$ is Fréchet differentiable for every $(u,v)\in X_{1}\times X_{2}$ and

	$\displaystyle E_{22}(u,v,w)$	$\displaystyle=$	$\displaystyle v-N_{2}(u,v,w),$
	$\displaystyle E_{33}(u,v,w)$	$\displaystyle=$	$\displaystyle w-N_{3}(u,v,w).$

Here $E_{22}$ , $E_{33}$ are the Fréchet derivatives of $E_{2}(u,\cdot,w)$ and $E_{3}(u,v,\cdot),$ respectively.

We also assume that the operator $N:X\rightarrow X,$

N(u,v,w)=(N_{1}(u,v,w),\ N_{2}(u,v,w),\ N_{3}(u,v,w))

is a Perov contraction, i.e. there is a square matrix $A=[a_{ij}]_{1\leq i,j\leq 3}\in\mathbf{M}_{3}(\mathbb{R}_{+})$ such that $A^{k}$ tends to the zero matrix $0_{3}$ as $k\rightarrow\infty$ and the following vector Lipschitz condition is satisfied

\begin{bmatrix}d\left(N_{1}(u,v,w),\ N_{1}(\overline{u},\overline{v},\overline% {w})\right)\\ \left|N_{2}(u,v,w)-N_{2}(\overline{u},\overline{v},\overline{w})\right|_{2}\\ \left|N_{3}(u,v,w)-N_{3}(\overline{u},\overline{v},\overline{w})\right|_{3}% \end{bmatrix}\leq A\begin{bmatrix}d(u,\overline{u})\\ |v-\overline{v}|_{2}\\ |w-\overline{w}|_{3}\end{bmatrix}

(2.1)

for every $(u,v,w)\in X$ .

Note that the for a square matrix $A\in\mathbf{M}_{n}(\mathbb{R}_{+}),$ condition $A^{k}$ tends to the zero matrix $0_{n}$ as $k\rightarrow\infty$ is equivalent (see [6]) to each one of the following properties:

(i) The spectral radius of $A$ is less than one;

(ii) $I_{n}-A$ is invertible and $(I_{n}-A)^{-1}\in\mathbf{M}_{n}(\mathbb{R}_{+});$

(iii) $I_{n}-A$ is invertible and $I_{n}+M+M^{2}+\ ...\ =\left(I_{n}-A\right)^{-1}.$

Here $I_{n}$ stands for the unit matrix in $\mathbf{M}_{n}(\mathbb{R}).$

The main result is the following theorem.

Theorem 2.1.

Assume that the above conditions are satisfied. Moreover assume that $E_{2}(u,\cdot,w),\ E_{3}(u,v,\cdot)$ are bounded from below for every $(u,v,w)\in X$ and that are constants $R_{2},R_{3},a>0$ such that

E_{2}(u,v,w)\geq\inf_{X_{2}}E_{2}(u,\cdot,w)+a\ \ \text{ for all }(u,w)\in X_{% 1}\times X_{3}\text{ and }|v|_{2}\geq R_{2},

(2.2)

E_{3}(u,v,w)\geq\inf_{X_{3}}E_{3}(u,v,\cdot)+a\ \ \text{ for all }(u,v)\in X_{% 1}\times X_{2}\text{ and }|w|_{3}\geq R_{3}.

(2.3)

Then the unique fixed point $(u^{\ast},v^{\ast},w^{\ast})$ ensured by the Perov contraction theorem has the property that $(v^{\ast},w^{\ast})$ is a Nash type equilibrium for the pair of functionals $(E_{2},E_{3})$ , i.e.

	$\displaystyle E_{2}(u^{\ast},v^{\ast},w^{\ast})$	$\displaystyle=$	$\displaystyle\inf_{X_{2}}E_{2}(u^{\ast},\cdot,w^{\ast}),$
	$\displaystyle E_{3}(u^{\ast},v^{\ast},w^{\ast})$	$\displaystyle=$	$\displaystyle\inf_{X_{3}}E_{3}(u^{\ast},v^{\ast},\cdot).$

For the proof we need alternatively one of the following two auxiliary results.

Lemma 2.2.

Let $\left(A_{k,p}\right)_{k\geq 1},\ \left(B_{k,p}\right)_{k\geq 1}\ \$ be two sequences of vectors in $\mathbb{R}_{+}^{n}$ (column vectors) depending on a parameter $p,$ such that

A_{k,p}\leq MA_{k-1,p}+B_{k,p}

for all $k$ and $p,$ where $M\in\mathbf{M}_{n}(\mathbb{R}_{+})$ is a matrix with spectral radius less than one. If the sequence $\left(A_{k,p}\right)_{k\geq 1}$ is bounded uniformly with respect to $p$ and $B_{k,p}\rightarrow 0_{n}$ as $k\rightarrow\infty$ uniformly with respect to $p,$ then $A_{k,p}\rightarrow 0_{n}$ as $k\rightarrow\infty$ uniformly with respect to $p .$

Proof.

Since $B_{k,p}\rightarrow 0_{n}$ as $k\rightarrow\infty$ uniformly with respect to $p,$ for any fixed column vector $\epsilon\in\left(0,\infty\right)^{n}$ , we can find $k_{1}$ independent of $p$ such that $B_{k,p}\leq\epsilon$ for all $k\geq k_{1}$ and all $p .$ Then, for $k>k_{1}$ we have

$\displaystyle A_{k,p}$	$\displaystyle\leq$	$\displaystyle MA_{k-1,p}+\epsilon\leq M^{2}A_{k-2,p}+\epsilon+M\epsilon$
	$\displaystyle\leq$	$\displaystyle\ ...\$
	$\displaystyle\leq$	$\displaystyle M^{k-k_{1}}A_{k_{1},p}+\epsilon(I_{n}+M+...M^{k-k_{1}})$
	$\displaystyle\leq$	$\displaystyle M^{k-k_{1}}A_{k_{1},p}+\epsilon\left(I_{n}-M\right)^{-1}.$

The conclusion now follows since $\left(A_{k,p}\right)_{k\geq 1}$ is bounded uniformly with respect to $p$ and $M^{k-k_{1}}\rightarrow 0_{n}$ as $k\rightarrow\infty.$ ∎

Lemma 2.3.

Let $\left(x_{k,p}\right)_{k\geq 1},\ \left(y_{k,p}\right)_{k\geq 1}$ be two sequences of nonnegative real numbers depending on a parameter $p$ which are bounded uniformly with respect to $p .$ Assume that for all $k$ and $p,$

ax_{k,p}+by_{k,p}\leq a^{\prime}x_{k-1,p}+b^{\prime}y_{k-1,p}+q_{k,p}

where $0<a^{\prime}<a,\ 0\leq b^{\prime}<b,$ $\frac{b^{\prime}}{a^{\prime}}<\frac{b}{a},$ and $(q_{k,p})_{k\geq 1}$ is a sequence of positive real numbers converging to zero uniformly with respect to $p$ . Then $x_{k,p}\rightarrow 0$ and $y_{k,p}\rightarrow 0$ as $k\rightarrow\infty$ uniformly with respect to $p .$

Proof.

By the uniform convergence to zero of $q_{k,p\text{ }}$ , taking $\epsilon>0$ , we can find $k_{1}$ independent of $p$ , such that $\frac{q_{k,p}}{a}<\epsilon$ for all $k>k_{1}$ . Consider $k>k_{1}$ and assume $a>b$ . Then

	$\displaystyle x_{k,p}+\frac{b}{a}y_{k,p}$	$\displaystyle\leq\frac{a^{\prime}}{a}\left(x_{k-1,p}+\frac{b^{\prime}}{a^{% \prime}}y_{k-1,p}\right)+\epsilon\leq\frac{a^{\prime}}{a}\left(x_{k-1,p}+\frac% {b}{a}y_{k-1,p}\right)+\epsilon$
		$\displaystyle\leq\left(\frac{a^{\prime}}{a}\right)^{2}\left(x_{k-2,p}+\frac{b^% {\prime}}{a^{\prime}}y_{k-2,p}\right)+\epsilon\left(\frac{a^{\prime}}{a}+1\right)$
		$\displaystyle\cdot\cdot\cdot$
		$\displaystyle\leq\left(\frac{a^{\prime}}{a}\right)^{k-k_{1}}\left(x_{k_{1},p}+% \frac{b^{\prime}}{a^{\prime}}y_{k_{1},p}\right)+\epsilon\left(\left(\frac{a^{% \prime}}{a}\right)^{k-k_{1}-1}+...+1\right)$
		$\displaystyle\leq\left(\frac{a^{\prime}}{a}\right)^{k-k_{1}}\left(x_{k_{1},p}+% \frac{b^{\prime}}{a^{\prime}}y_{k_{1},p}\right)+\epsilon\frac{1}{1-\frac{a^{% \prime}}{a}}.$

Taking into account that $\frac{a^{\prime}}{a}<1$ and the boundedness of $x_{k,p}$ and $y_{k,p}$ , it is clear that $x_{k,p}+\frac{b}{a}y_{k,p}\rightarrow 0$ as $k\rightarrow\infty$ uniformly with respect to $p .$ This clearly gives the conclusion. The case $a<b$ can be treated analogously. ∎

Proof of the theorem.

First note that since the spectral radius of matrix $A$ is less than one, the elements $a_{ii}$ of the main diagonal are less than one. Consequently, for every $\left(v,w\right)\in X_{2}\times X_{3},$ the operator $N_{1}\left(\cdot,v,w\right):X_{1}\rightarrow X_{1}$ is a contraction. We now use an iterative procedure to construct an approximating sequence $(u_{k},v_{k},w_{k})$ . We start with some fixed element $(v_{0},w_{0})\in X_{2}\times X_{3}.$ Then, by Banach contraction principle, there exists $u_{1}\in X_{1}$ such that $N_{1}(u_{1},v_{0},w_{0})=u_{1}$ . Next, for fixed $\left(u_{1},w_{0}\right),$ according to Ekeland variational principle, there is $v_{1}\in X_{2}$ such that

E_{2}(u_{1},v_{1},w_{0})\leq\inf_{X_{2}}E_{2}(u_{1},\cdot,w_{0})+1,\ \ \ \ \ |% E_{22}(u_{1},v_{1},w_{0})|_{2}\leq 1.

Using again Ekeland variational principle for fixed $(u_{1},v_{1}),$ there is $w_{1}\in X_{3}$ with

E_{3}(u_{1},v_{1},w_{1})\leq\inf_{X_{3}}E_{2}(u_{1},v_{1},\cdot)+1,\ \ \ \ |E_% {33}(u_{1},v_{1},w_{1})|_{3}\leq 1.

At step $k,$ we find a triple $(u_{k},v_{k},w_{k})$ having the following proprieties:

	$\displaystyle N_{1}(u_{k},v_{k-1},w_{k-1})=u_{k},$		(2.4)
	$\displaystyle E_{2}(u_{k},v_{k},w_{k-1})\leq\inf_{X_{2}}E_{2}(u_{k},\cdot,w_{k% -1})+\frac{1}{k},\ \|E_{22}(u_{k},v_{k},w_{k-1})\|_{2}\leq\frac{1}{k},$
	$\displaystyle E_{3}(u_{k},v_{k},w_{k})\leq\inf_{X_{3}}E_{3}(u_{k},v_{k},\cdot)% +\frac{1}{k},\ \|E_{33}(u_{k},v_{k},w_{k]}\|_{3}\leq\frac{1}{k}.$

Our next task is to prove that the sequences $u_{k},v_{k},w_{k}$ are Cauchy, which will ensure their convergence. Since $N_{1}(u_{k},v_{k-1},w_{k-1})=u_{k},$ we have

	$\displaystyle d(u_{k+p},u_{k})$	$\displaystyle=d(N_{1}(u_{k+p},v_{k+p-1},w_{k+p-1}),\ N_{1}(u_{k},v_{k-1},w_{k-% 1}))$
		$\displaystyle\leq a_{11}d(u_{k+p},u_{k})+a_{12}\|v_{k+p-1}-v_{k-1}\|_{2}+a_{13}\|% w_{k+p-1}-w_{k-1}\|_{3},$

whence

d(u_{k+p},u_{k})\leq\frac{a_{12}}{1-a_{11}}|v_{k+p-1}-v_{k-1}|_{2}+\frac{a_{13% }}{1-a_{11}}|w_{k+p-1}-w_{k-1}|_{3}.

For the sequence $(v_{k})$ and $(w_{k})$ we have

	$\displaystyle\|v_{k+p}-v_{k}\|_{2}$	$\displaystyle\leq\|-N_{2}(u_{k+p},v_{k+p},w_{k+p-1})+v_{k+p}-v_{k}+N_{2}(u_{k},% v_{k},w_{k-1})\|_{2}$		(2.5)
		$\displaystyle+\|N_{2}(u_{k+p},v_{k+p},w_{k+p-1})-N_{2}(u_{k},v_{k},w_{k-1})\|_{2},$

	$\displaystyle\|w_{k+p}-w_{k}\|_{3}$	$\displaystyle\leq\|-N_{3}(u_{k+p},v_{k+p},w_{k+p})+w_{k+p}-w_{k}+N_{3}(u_{k},v_% {k},w_{k})\|_{3}$		(2.6)
		$\displaystyle+\|N_{3}(u_{k+p},v_{k+p},w_{k+p})-N_{3}(u_{k},v_{k},w_{k})\|_{3}.$

Denote

	$\displaystyle\beta_{k,p}$	$\displaystyle:=\|-N_{2}(u_{k+p},v_{k+p},w_{k+p-1})+v_{k+p}-v_{k}+N_{2}(u_{k},v_% {k},w_{k-1})\|_{2}$
		$\displaystyle=\|E_{22}(u_{k+p},v_{k+p},w_{k+p-1})-E_{22}(u_{k},v_{k},w_{k-1}))\|% _{2},$

	$\displaystyle\gamma_{k,p}$	$\displaystyle:=\|-N_{3}(u_{k+p},v_{k+p},w_{k+p})+w_{k+p}-w_{k}+N_{3}(u_{k},v_{k% },w_{k})\|_{3}$
		$\displaystyle=\|E_{33}(u_{k+p},v_{k+p},w_{k+p})-E_{33}(u_{k},v_{k},w_{k-}))\|_{3},$

x_{k,p}:=d(u_{u+p},u_{k}),\ \ \ y_{k,p}:=|v_{k+p}-v_{k}|_{2},\ \ \ z_{k,p}:=|w% _{k+p}-w_{k}|_{3}.

With these notations, using (2.5), (2.6) and the Perov contraction condition, we obtain

	$\displaystyle x_{k,p}\leq a_{11}x_{k,p}+a_{12}y_{k-1,p}+a_{13}z_{k-1,p},$		(2.7)
	$\displaystyle y_{k,p}\leq a_{21}x_{k,p}+a_{22}y_{k,p}+a_{23}z_{k-1,p}+\beta_{k% ,p},$		(2.8)
	$\displaystyle z_{k,p}\leq a_{31}x_{k,p}+a_{32}y_{k,p}+a_{33}z_{k,p}+\gamma_{k,% p}.$		(2.9)

For the continuation of the proof we may use either Lemma 2.2 or Lemma 2.3.

1) Use of Lemma 2.2. Letting

A^{\prime}=\begin{bmatrix}a_{11}&0&0\\ a_{21}&a_{22}&0\\ a_{31}&a_{32}&a_{33}\end{bmatrix}\ \text{ and \ \ }A^{\prime\prime}=A-A^{% \prime},

the following inequality holds

\begin{bmatrix}x_{k,p}\\ y_{k,p}\\ z_{k,p}\end{bmatrix}\leq A^{\prime}\begin{bmatrix}x_{k,p}\\ y_{k,p}\\ z_{k,p}\end{bmatrix}+A^{\prime\prime}\begin{bmatrix}x_{k-1,p}\\ y_{k-1,p}\\ z_{k-1,p}\end{bmatrix}+\begin{bmatrix}0\\ \beta_{k,p}\\ \gamma_{k,p}\end{bmatrix}.

(2.10)

Note that if $\rho(A)<1,$ than also $\rho(A^{\prime})<1$ . Indeed, one clearly has $A^{\prime k}<A^{k},$ and so if $A^{k}\rightarrow 0$ as $k\rightarrow\infty,$ then $A^{\prime k}\rightarrow 0$ too.
Rewriting (2.10) as

(I-A^{\prime})\begin{bmatrix}x_{k,p}\\ y_{k,p}\\ z_{k,p}\end{bmatrix}\leq A^{\prime\prime}\begin{bmatrix}x_{k-1,p}\\ y_{k-1,p}\\ z_{k-1,p}\end{bmatrix}+\begin{bmatrix}0\\ \beta_{k,p}\\ \gamma_{k,p}\end{bmatrix}

and using the fact that $I-A^{\prime}$ is invertible and its inverse has positive entries, we can multiply by $(I-A^{\prime})^{-1}$ to obtain

\begin{bmatrix}x_{k,p}\\ y_{k,p}\\ z_{k,p}\end{bmatrix}\leq(I-A^{\prime})^{-1}A^{\prime\prime}\begin{bmatrix}x_{k% -1,p}\\ y_{k-1,p}\\ z_{k-1,p}\end{bmatrix}+(I-A^{\prime})^{-1}\begin{bmatrix}0\\ \beta_{k,p}\\ \gamma_{k,p}\end{bmatrix}.

Observe that $M:=(I-A^{\prime})^{-1}A^{\prime\prime}$ has the spectral radius less than one. To prove this, it is enough to show that $I-M$ is invertible with the inverse has nonegative entries.
Is clear that

	$\displaystyle M$	$\displaystyle=(I-A^{\prime})^{-1}A^{\prime\prime}=(I-A^{\prime})^{-1}(A-A^{% \prime})=(I-A^{\prime})^{-1}(I-A^{\prime}+A-I)$
		$\displaystyle=I-(I-A^{\prime})^{-1}(I-A),$

hence $I-M=(I-A^{\prime})^{-1}(I-A)$ . Because $(I-A^{\prime})^{-1}$ and $I-A$ are invertible, by taking $Q:=(I-A)^{-1}(I-A^{\prime})$ , we have $Q(I-M)=(I-M)Q=I$ , hence $I-M$ is invertible and its inverse is $Q .$ One has

Q=(I-A)^{-1}(I-A^{\prime})=(I-A)^{-1}(I-A+A^{\prime\prime})=I+(I-A)^{-1}A^{% \prime\prime}

and since $(I-A)^{-1}A^{\prime\prime}$ and $I$ are positive matrices, it follows that $Q$ is also positive. Therefore, the spectral radius of $M$ is less than one.
From $(\ref{conditie_pentru_marginirea_lui_v_k})$ and $(\ref{conditie_pentru_marginirea_lui_w_k})$ we have that $y_{k,p}$ and $z_{k,p}$ are bounded uniformly with respect to $p$ . Because of this, is immediate that $x_{k,p}$ is also bounded uniformly with respect to $p$ . Moreover, it is clear that

\begin{bmatrix}0\\ \beta_{k,p}\\ \gamma_{k,p}\end{bmatrix}

converges to zero uniformly with respect to $p .$ Applying Lemma 2.2 we obtain that $x_{k,p},\ y_{k,p},\ z_{k,p}$ are convergent to zero uniformly with respect to $p .$ Hence the sequences $u_{k},v_{k}$ and $w_{k}$ are Cauchy as desired.

2) Use of Lemma 2.3. The relations $(\ref{conditie_de_marginire_pe_x_k_p}),(\ref{conditie_de_marginire_pe_y_k_p}),% (\ref{conditie_de_marginire_pe_z_k_p})$ can be rewritten under the form

	$\displaystyle x_{k,p}\leq a_{11}x_{k,p}+a_{12}y_{k,p}+a_{13}z_{k,p}+a_{12}(y_{% k-1,p}-y_{k,p})+a_{13}(z_{k-1,p}-z_{k,p}),$
	$\displaystyle y_{k,p}\leq a_{21}x_{k,p}+a_{22}y_{k,p}+a_{23}z_{k,p}+\beta_{k,p% }+a_{23}(z_{k-1,p}-z_{k,p}),$
	$\displaystyle z_{k,p}\leq a_{31}x_{k,p}+a_{32}y_{k,p}+a_{33}z_{k,p}+\gamma_{k,% p},$

which can be put under the vector form

\begin{bmatrix}x_{k,p}\\ y_{k,p}\\ z_{k,p}\end{bmatrix}\leq A\begin{bmatrix}x_{k,p}\\ y_{k,p}\\ z_{k,p}\end{bmatrix}+\begin{bmatrix}a_{12}(y_{k-1,p}-y_{k,p})+a_{13}(z_{k-1,p}% -z_{k,p})\\ \beta_{k,p}+a_{23}(z_{k-1,p}-z_{k,p})\\ \gamma_{k,p}\end{bmatrix}.

Denoting $(I-A)^{-1}=C=[c_{ij}]_{1\leq i,j\leq 3}$ we have

\begin{bmatrix}x_{k,p}\\ y_{k,p}\\ z_{k,p}\end{bmatrix}\leq C\begin{bmatrix}a_{12}(y_{k-1,p}-y_{k,p})+a_{13}(z_{k% -1,p}-z_{k,p})\\ \beta_{k,p}+a_{23}(z_{k-1,p}-z_{k,p})\\ \gamma_{k,p}\end{bmatrix},

whence

	$\displaystyle y_{k,p}$	$\displaystyle\leq c_{21}a_{12}(y_{k-1,p}-y_{k,p})+c_{21}a_{13}(z_{k-1,p}-z_{k,% p})$		(2.11)
		$\displaystyle+c_{22}a_{23}(z_{k-1,p}-z_{k,p})+c_{22}\beta_{k,p}+c_{33}\gamma_{% k,p},$

	$\displaystyle z_{k,p}$	$\displaystyle\leq c_{31}a_{12}(y_{k-1,p}-y_{k,p})+c_{31}a_{13}(z_{k-1,p}-w_{k,% p})$		(2.12)
		$\displaystyle+c_{32}a_{23}(z_{k-1,p}-z_{k,p})+c_{32}\beta_{k,p}+c_{33}\gamma_{% k,p}.$

We make the following notations

	$\displaystyle a^{\prime}=c_{12}a_{13}+c_{22}a_{2,3}+c_{31}a_{13}+c_{32}a_{23},$
	$\displaystyle b^{\prime}=c_{21}a_{12}+c_{31}a_{12}.$

Adding $(\ref{inegalitate_asupra_lui_y})$ and $(\ref{inegalitate_asupra_lui_z})$ we obtain

y_{k,p}+z_{k,p}\leq a^{\prime}y_{k-1,p}-a^{\prime}y_{k,p}+b^{\prime}z_{k-1,p}-% b^{\prime}z_{k,p}+c_{22}\beta_{k,p}+c_{33}\gamma_{k,p}+c_{32}\beta_{k,p}+c_{33% }\gamma_{k,p},

whence, with the notations $a=1+a^{\prime},$ $b=1+b^{\prime}$ and $q_{k,p}:=c_{22}\beta_{k,p}+c_{33}\gamma_{k,p}+c_{32}\beta_{k,p}+c_{33}\gamma_{% k,p},$ one has

ay_{k,p}+bz_{k,p}\leq a^{\prime}y_{k-1,p}+b^{\prime}z_{k-1,p}+q_{k,p}.

Note that the sequence $q_{k,p}:=c_{22}\beta_{k,p}+c_{33}\gamma_{k,p}+c_{32}\beta_{k,p}+c_{33}\gamma_{% k,p}$ converges to zero as $k\rightarrow\infty$ uniformly with respect to $p,$ and that from $(\ref{conditie_pentru_marginirea_lui_v_k})$ and $(\ref{conditie_pentru_marginirea_lui_w_k}),$ the sequences $\left(y_{k,p}\right)_{k\geq 1},\ \left(z_{k,p}\right)_{k\geq 1}$ are bounded uniformly with respect to $p .$ Also note that if $b^{\prime}<a^{\prime},$ then $\frac{b^{\prime}}{a^{\prime}}<\frac{b}{a}$ and from Lemma 2.3 we obtain that $y_{k,p}$ and $z_{k,p}$ converge to zero as $k\rightarrow\infty$ uniformly with respt to $p .$ Similarly, if $a^{\prime}<b^{\prime},$ then we obtain the same conclusion if we apply Lemma 2.3 by interchanging $a$ with $b$ and $y_{k,p}$ with $z_{k,p}.$ Next, from (2.7) we deduce that $x_{k,p}\rightarrow 0$ as $k\rightarrow\infty$ uniformly with respect to $p,$ and as above, that the sequences $u_{k},\ v_{k}$ and $w_{k}$ are Cauchy as desired.

Finally the limits $u^{\ast},\ v^{\ast},\ w^{\ast}$ of the sequences $u_{k},\ v_{k}$ and $w_{k}$ give the desired solution of the system after passing to the limit in (2.4). ∎

3. Application

Consider the system

\begin{cases}-u^{\prime\prime}+a_{1}^{2}u=f_{1}(t,u(t),v(t),w(t),u^{\prime}(t)% )\\ -v^{\prime\prime}+a_{2}^{2}v=\nabla_{y}f_{2}(t,u(t),v(t),w(t))\\ -w^{\prime\prime}+a_{3}^{2}w=\nabla_{z}f_{3}(t,u(t),v(t),w(t))\end{cases}

(3.1)

with the periodic conditions

$\displaystyle u(0)-u(T)$	$\displaystyle=$	$\displaystyle u^{\prime}(0)-u^{\prime}(T)=0,$
$\displaystyle v(0)-v(T)$	$\displaystyle=$	$\displaystyle v^{\prime}(0)-v^{\prime}(T)=0,$
$\displaystyle w(0)-w(T)$	$\displaystyle=$	$\displaystyle w^{\prime}(0)-w^{\prime}(T)=0,$

where $f_{2},f_{3}:(0,T)\times\mathbb{R}^{k_{1}}\times\mathbb{R}^{k_{2}}\times\mathbb% {R}^{k_{3}}\rightarrow\mathbb{R}$ and $f_{1}:(0,T)\times\mathbb{R}^{k_{1}}\times\mathbb{R}^{k_{2}}\times\mathbb{R}^{k% _{3}}\times\mathbb{R}^{k_{1}}\rightarrow\mathbb{R}^{k_{1}}$ . We will assume that $f_{1},f_{2},f_{3},\nabla_{y}f_{2}$ and $\nabla_{z}f_{3}$ are $L^{1}$ - Carathéodory functions.

For $i=1,2,3,$ let $H_{p}^{1}\left(0,T;\mathbb{R}^{k_{i}}\right)$ be the closure in $H^{1}\left(0,T;\mathbb{R}^{k_{i}}\right)$ of the space $\{u\in C^{1}\left(\left[0,T\right];\mathbb{R}^{k_{i}}\right):$ $u\left(0\right)=u\left(T\right),$ $u^{\prime}\left(0\right)=u^{\prime}\left(T\right)\}.$ We shall endow this space with the inner product

\left(u,v\right)_{i}:=(u^{\prime},v^{\prime})_{L^{2}\left(0,T;\mathbb{R}^{k_{i% }}\right)}+a_{i}^{2}(u,v)_{L^{2}\left(0,T;\mathbb{R}^{k_{i}}\right)}

and the corresponding norm

\left|u\right|_{i}=\left(\left|u^{\prime}\right|_{L^{2}\left(0,T;\mathbb{R}^{k% _{i}}\right)}^{2}+a_{i}^{2}\left|u\right|_{L^{2}\left(0,T;\mathbb{R}^{k_{i}}% \right)}^{2}\right)^{\frac{1}{2}}.

Also we consider the operator $J_{i}\colon(H_{p}^{1}\left(0,T;\mathbb{R}^{k_{i}}\right))^{\prime}\rightarrow H% _{p}^{1}\left(0,T;\mathbb{R}^{k_{i}}\right)$ given by $J_{i}h=u_{h}\ \left(h\in(H_{p}^{1}\left(0,T;\mathbb{R}^{k_{i}}\right))^{\prime% }\right),$ where $u_{h}\in H_{p}^{1}\left(0,T;\mathbb{R}^{k_{i}}\right)$ is the weak solution of the problem

\begin{cases}-u^{\prime\prime}+a_{i}^{2}u=h\ \ \ \text{on\ }\left(0,T\right)\\ u(0)-u(T)=u^{\prime}(0)-u^{\prime}(T)=0\end{cases}

(3.2)

For every $h\in L^{2}\left([0,T];\mathbb{R}^{k_{i}}\right)$ we have

|J_{i}h|_{i}^{2}=(J_{i}h,J_{i}h)_{i}=(h,J_{i}h)_{L^{2}}\leq|h|_{L^{2}}|J_{i}h|% _{L^{2}}\leq\frac{1}{a_{i}}|h|_{L^{2}}|J_{i}h|_{i},

(3.3)

hence

|J_{1}h_{1}|_{i}\leq\frac{1}{a_{i}}|h|_{L^{2}}.

Associate to the second and the third equation from (3.1) the functionals

E_{2},E_{3}\colon\ H_{p}^{1}(0,T;\mathbb{R}^{k_{1}})\times H_{p}^{1}(0,T;% \mathbb{R}^{k_{2}})\times H_{p}^{1}(0,T;\mathbb{R}^{k_{3}})\rightarrow\mathbb{R}

defined by

E_{2}(u,v,w)=\frac{1}{2}|v|_{2}^{2}-\int_{0}^{T}f_{2}(t,u(t),v(t),w(t))dt

and

E_{3}(u,v,w)=\frac{1}{2}|w|_{3}^{2}-\int_{0}^{T}f_{3}(t,u(t),v(t),w(t))dt.

According to [4, Theorem 1.4] we have

E_{22}(u,v,w)=L_{2}v-\nabla_{y}f_{2}(\cdot,u,v,w),

or equivalently, for any $\varphi\in H_{p}^{1}(0,T;\mathbb{R}^{k_{2}}),$

	$\displaystyle(E_{22}(u,v,w),\varphi)$	$\displaystyle=$	$\displaystyle(L_{2}v,\varphi)-(\nabla_{y}f_{2}(u,v,w,u^{\prime},w^{\prime}),\varphi)$
		$\displaystyle=$	$\displaystyle(v-J_{2}\nabla_{y}f_{2},\varphi)_{2}.$

Hence

E_{22}(u,v,w)=v-J_{2}\nabla_{y}f_{2}.

Similarly,

E_{33}(u,v,w)=w-J_{3}\nabla_{z}f_{3}.

On the other hand, system (3.1) is equivalent to the following fixed point equation

\begin{cases}N_{1}(u,v,w)=u\\ N_{2}(u,v,w)=v\\ N_{3}(u,v,w)=w\end{cases}

where

$\displaystyle N_{1}(u,v,w)$	$\displaystyle=$	$\displaystyle J_{1}f_{1}(\cdot,u,v,w,u^{\prime}),$
$\displaystyle N_{2}(u,v,w)$	$\displaystyle=$	$\displaystyle J_{2}\nabla_{y}f_{2}(\cdot,u,v,w),$
$\displaystyle N_{3}(u,v,w)$	$\displaystyle=$	$\displaystyle J_{3}\nabla_{z}f_{3}(\cdot,u,v,w).$

Related to $f_{1},f_{2},f_{3}$ we assume that the following Lipschitz conditions hold for some constants $a_{ij}:$

\left|f_{1}(t,x_{1},...,x_{4})-f_{1}(t,\overline{x_{1}},...,\overline{x_{4}})% \right|\leq\sum_{j=1}^{4}a_{1j}|x_{j}-\overline{x_{j}}|,

(3.4)

\left|\nabla_{y}f_{2}(t,x_{1},x_{2},x_{3})-\nabla_{y}f_{2}(t,\overline{x_{1}},% \overline{x_{2}},\overline{x_{3}})\right|\leq\sum_{j=1}^{3}a_{2j}|x_{j}-% \overline{x_{j}}|,

(3.5)

\left|\nabla_{z}f_{3}(t,x_{1},x_{2},x_{3})-\nabla_{z}f_{3}(\overline{x_{1}},% \overline{x_{2}},\overline{x_{3}})\right|\leq\sum_{j=1}^{3}a_{3j}|x_{j}-% \overline{x_{j}}|.

(3.6)

Then

	$\displaystyle\|N_{1}(u,v,w)-N_{1}(\overline{u},\overline{v},\overline{w})\|_{1}=% \|J_{1}\left(f_{1}(\cdot,u,v,w,u^{\prime})-f_{1}(\cdot,\overline{u},\overline{v% },\overline{w},\overline{u}^{\prime})\right)\|_{1}$
	$\displaystyle\leq\frac{1}{a_{1}}\|f_{1}(\cdot,u,v,w,u^{\prime})-f_{1}(\cdot,% \overline{u},\overline{v},\overline{w},\overline{u}^{\prime})\|_{L^{2}}$
	$\displaystyle\leq\frac{1}{a_{1}}\left(\int_{0}^{T}(a_{11}\|u(t)-\overline{u}(t)% \|+a_{14}\|u^{\prime}(t)-\overline{u}^{\prime}(t)\|)^{2}dt\right)^{\frac{1}{2}}$
	$\displaystyle+\frac{a_{12}}{a_{1}}\|v-\overline{v}\|_{L^{2}}+\frac{a_{13}}{a_{1}% }\|w-\overline{w}\|_{L^{2}}$
	$\displaystyle\leq\frac{1}{a_{1}}\left(\left(\frac{a_{11}}{a_{1}}\right)^{2}+a_% {14}^{2}\right)^{\frac{1}{2}}\left\|u-\overline{u}\right\|_{1}+\frac{a_{12}}{a_{% 1}}\|v-\overline{v}\|_{L^{2}}+\frac{a_{13}}{a_{1}}\|w-\overline{w}\|_{L^{2}}.$

Is clear that $|v-\overline{v}|_{L^{2}}\leq\frac{1}{a_{2}}|v-\overline{v}|_{2}$ and $|w-\overline{w}|_{L^{2}}\leq\frac{1}{a_{3}}|w-\overline{w}|_{3}.$ Hence, the above inequality becomes

			$\displaystyle\|N_{1}(u,v,w)-N_{1}(\overline{u},\overline{v},\overline{w})\|_{1}$
		$\displaystyle\leq$	$\displaystyle\frac{1}{a_{1}}\left(\left(\frac{a_{11}}{a_{1}}\right)^{2}+a_{14}% ^{2}\right)^{\frac{1}{2}}\|u-\overline{u}\|_{1}+\frac{a_{12}}{a_{1}a_{2}}\|v-% \overline{v}\|_{2}+\frac{a_{13}}{a_{1}a_{3}}\|w-\overline{w}\|_{3}.$

For $N_{2}(u,v,w)$ we obtain the following estimate

	$\displaystyle\|N_{2}(u,v,w)-N_{2}(\overline{u},\overline{v},\overline{w})\|_{2}$	$\displaystyle\leq\|J_{2}\nabla_{y}f_{2}(\cdot,u,v,w)-\nabla_{y}f_{2}(\cdot,% \overline{u},\overline{v},\overline{w})\|_{2}$
		$\displaystyle\leq\frac{1}{a_{2}}\|\nabla_{y}f_{2}(\cdot,u,v,w)-\nabla_{y}f_{2}(% \cdot,\overline{u},\overline{v},\overline{w})\|_{L^{2}}$
		$\displaystyle\leq\frac{a_{21}}{a_{2}}\|u-\overline{u}\|_{L^{2}}+\frac{a_{22}}{a_% {2}}\|v-\overline{v}\|_{L^{2}}+\frac{a_{23}}{a_{2}}\|w-\overline{w}\|_{L^{2}}$
		$\displaystyle\leq\frac{a_{21}}{a_{2}a_{1}}\|u-\overline{u}\|_{1}+\frac{a_{22}}{a% _{2}^{2}}\|v-\overline{v}\|_{2}+\frac{a_{23}}{a_{2}a_{3}}\|w-\overline{w}\|_{3}.$

Similarly

|N_{3}(u,v,w)-N_{3}(\overline{u},\overline{v},\overline{w})|_{3}\leq\frac{a_{3% 1}}{a_{3}a_{1}}|u-\overline{u}|_{1}+\frac{a_{32}}{a_{2}a_{3}}|v-\overline{v}|_% {2}+\frac{a_{33}}{a_{3}^{2}}|w-\overline{w}|_{3}.

Therefore, the condition related to (2.1) holds provided that the spectral radius of the matrix

A=\begin{bmatrix}\frac{1}{a_{1}}\left(\left(\frac{a_{11}}{a_{1}}\right)^{2}+a_% {14}^{2}\right)^{\frac{1}{2}}&\frac{a_{12}}{a_{1}a_{2}}&\frac{a_{13}}{a_{1}a_{% 3}}\\ \frac{a_{21}}{a_{2}a_{1}}&\frac{a_{22}}{a_{2}^{2}}&\frac{a_{23}}{a_{2}a_{3}}\\ \frac{a_{31}}{a_{3}a_{1}}&\frac{a_{32}}{a_{2}a_{3}}&\frac{a_{33}}{a_{3}^{2}}% \end{bmatrix}

(3.7)

is less than one.

In what follows we are trying to establish conditions for $E_{2}(u,\cdot,w)$ and $E_{3}(u,v,\cdot)$ to be bounded from below. To this aim, assume that for $i\in\{2,3\}\$ and $j\in\{1,2,3,4\},$ there are $\sigma_{ij}\in L^{1}(0,T;\mathbb{R}_{+})$ and $\gamma_{i}\in\mathbb{R}$ with $\gamma_{i}^{2}<\frac{a_{i}^{2}}{2}$ such that

f_{2}(t,x,y,z)\leq\gamma_{2}^{2}|y|^{2}+\sigma_{21}(t)|x|+\sigma_{22}(t)|y|+% \sigma_{23}(t)|z|+\sigma_{24}(t)

(3.8)

and

f_{3}(t,x,y,z)\leq\gamma_{3}^{2}|z|^{2}+\sigma_{31}(t)|x|+\sigma_{32}(t)|y|+% \sigma_{33}(t)|z|+\sigma_{34}(t).

(3.9)

Then taking into account the continuous embedding of $H_{p}^{1}\left(0,T;\mathbb{R}^{k_{i}}\right)$ into $C\left(\left[0,T\right];\mathbb{R}^{k_{i}}\right),$ we obtain

	$\displaystyle E_{2}(u,v,w)=\int_{0}^{T}\left(\frac{1}{2}\|v^{\prime}(t)\|^{2}+% \frac{a_{2}^{2}}{2}\|v^{2}(t)\|-f_{2}(t,u(t),v(t),w(t))\right)dt$
	$\displaystyle\geq\int_{0}^{T}\left(\frac{1}{2}\|v^{\prime}(t)\|^{2}+\frac{1}{2}(% a_{2}^{2}-2\gamma_{2}^{2})v^{2}(t)-\sigma_{21}(t)\|u(t)\|-\sigma_{22}(t)\|v(t)\|-% \sigma_{23}(t)\|w(t)\|-\sigma_{24}\left(t\right)\right)dt$
	$\displaystyle\geq\left(1-\frac{2\gamma_{2}^{2}}{a_{2}^{2}}\right)\left\|v\right% \|_{2}^{2}-C_{21}\|u\|_{1}-C_{22}\|v\|_{2}-C_{23}\left\|w\right\|_{3}-C_{24}$

for some constants $C_{2j},$ $j-1,2,3,4.$ This shows us that $E_{2}(u,v,w)\rightarrow\infty$ as $|v|_{2}\rightarrow\infty.$ Similarly, $E_{3}\left(u,v,w\right)\rightarrow\infty$ as $\left|w\right|_{3}\rightarrow\infty.$ Thus the functionals $E_{2}(u,\cdot,w)$ and $E_{3}(u,v,\cdot)$ are coercive. Then, as in [7, Lemma 4.1], these functionals are bounded from bellow.

Finally, assume that for $i\in\{2,3\},$ there are $L^{1}$ -Carathéodory functions $g_{i1},g_{i2}\colon(0,T)\times\mathbb{R}^{k_{i}}\rightarrow\mathbb{R}$ of coercive type such that

g_{21}(t,y)\leq f_{2}(t,x,y,z)\leq g_{22}(t,y)

(3.10)

and

g_{31}(t,z)\leq f_{3}(t,x,y,z)\leq g_{32}(t,z)

(3.11)

for all for all $(x,y,z)\in\mathbb{R}^{k_{1}}\times\mathbb{R}^{k_{2}}\times\mathbb{R}^{k_{3}}$ and $t\in\left(0,T\right).$ Here, for example, by the coercivity of $g_{21}\left(t,y\right)$ we mean that

\frac{1}{2}\left|v\right|_{2}^{2}-\int_{0}^{T}g_{21}\left(t,v\right)dt% \rightarrow\infty\ \ \ \text{as\ \ }\left|v\right|_{2}\rightarrow\infty.

Fix $a>0$ . Using the above assumption one has

\inf_{v\in H_{p}^{1}}E_{2}(u,\cdot,w)+a\leq\inf_{v\in H_{p}^{1}}\left(\frac{1}% {2}\left|v\right|_{2}^{2}-\int_{0}^{T}g_{21}\left(t,v\right)dt\right)+a.

By the coercivity of $g_{22}$ , there exists $R_{2}>0$ such that

\inf_{v\in H_{p}^{1}}\left(\frac{1}{2}\left|v\right|_{2}^{2}-\int_{0}^{T}g_{21% }\left(t,v\right)dt\right)+a\leq\frac{1}{2}\left|v\right|_{2}^{2}-\int_{0}^{T}% g_{22}\left(t,v\right)dt,

for all $|v|_{2}\geq R_{2}$ . Now, for $\left|v\right|_{2}\geq R_{2}$ and all $(u,w)\in H_{p}^{1}(0,T;\mathbb{R}^{k_{1}})\times H_{p}^{1}(0,T;\mathbb{R}^{k_{% 3}}),$ using again (3.10) we obtain

E_{2}\left(u,v,w\right)\geq\frac{1}{2}\left|v\right|_{2}^{2}-\int_{0}^{T}g_{22% }\left(t,v\right)dt\geq\inf_{v\in H_{p}^{1}}E_{2}(u,\cdot,w)+a,

as desired. The similar inequality for $E_{3}$ can be established analogously.

Under the assumptions (3.4), (3.5),(3.6), (3.8), (3.9), (3.10), (3.11) and if the spectral radius of matrix (3.7) is less than one, then all the hypotheses of Theorem 2.1 are fulfilled.

References

[1] Bełdzinski, M., Galewski, M., Nash–type equilibria for systems of non-potential equations, Appl. Math. Comput. 385(2020), 125456.
[2] Benedetti, I., Cardinali, T., Precup, R., Fixed point-critical point hybrid theorems and applications to systems with partial variational structure, submitted.
[3] Cournot, A., The mathematical principles of the theory of wealth, Economic J.,1838.
[4] Mawhin, J., Willem, M., Critical Point Theory and Hamiltonian Systems, Springer, Berlin, 1989.
[5] Nash, J., Non-cooperative games, Ann. of Math. 54(1951), 286-295.
[6] Precup, R., Methods in Nonlinear Integral Equations, Springer, Amsterdam, 2002.
[7] Precup, R., Nash-type equilibria and periodic solutions to nonvariational systems, Adv. Nonlinear Anal. 4(2014), 197-207.

	$\displaystyle\|v_{k+p}-v_{k}\|_{2}$	$\displaystyle\leq\|-N_{2}(u_{k+p},v_{k+p},w_{k+p-1})+v_{k+p}-v_{k}+N_{2}(u_{k},% v_{k},w_{k-1})\|_{2}$		(2.5)
		$\displaystyle+\|N_{2}(u_{k+p},v_{k+p},w_{k+p-1})-N_{2}(u_{k},v_{k},w_{k-1})\|_{2},$

	$\displaystyle\|w_{k+p}-w_{k}\|_{3}$	$\displaystyle\leq\|-N_{3}(u_{k+p},v_{k+p},w_{k+p})+w_{k+p}-w_{k}+N_{3}(u_{k},v_% {k},w_{k})\|_{3}$		(2.6)
		$\displaystyle+\|N_{3}(u_{k+p},v_{k+p},w_{k+p})-N_{3}(u_{k},v_{k},w_{k})\|_{3}.$

	$\displaystyle\|N_{1}(u,v,w)-N_{1}(\overline{u},\overline{v},\overline{w})\|_{1}=% \|J_{1}\left(f_{1}(\cdot,u,v,w,u^{\prime})-f_{1}(\cdot,\overline{u},\overline{v% },\overline{w},\overline{u}^{\prime})\right)\|_{1}$
	$\displaystyle\leq\frac{1}{a_{1}}\|f_{1}(\cdot,u,v,w,u^{\prime})-f_{1}(\cdot,% \overline{u},\overline{v},\overline{w},\overline{u}^{\prime})\|_{L^{2}}$
	$\displaystyle\leq\frac{1}{a_{1}}\left(\int_{0}^{T}(a_{11}\|u(t)-\overline{u}(t)% \|+a_{14}\|u^{\prime}(t)-\overline{u}^{\prime}(t)\|)^{2}dt\right)^{\frac{1}{2}}$
	$\displaystyle+\frac{a_{12}}{a_{1}}\|v-\overline{v}\|_{L^{2}}+\frac{a_{13}}{a_{1}% }\|w-\overline{w}\|_{L^{2}}$
	$\displaystyle\leq\frac{1}{a_{1}}\left(\left(\frac{a_{11}}{a_{1}}\right)^{2}+a_% {14}^{2}\right)^{\frac{1}{2}}\left\|u-\overline{u}\right\|_{1}+\frac{a_{12}}{a_{% 1}}\|v-\overline{v}\|_{L^{2}}+\frac{a_{13}}{a_{1}}\|w-\overline{w}\|_{L^{2}}.$

	$\displaystyle\|N_{2}(u,v,w)-N_{2}(\overline{u},\overline{v},\overline{w})\|_{2}$	$\displaystyle\leq\|J_{2}\nabla_{y}f_{2}(\cdot,u,v,w)-\nabla_{y}f_{2}(\cdot,% \overline{u},\overline{v},\overline{w})\|_{2}$
		$\displaystyle\leq\frac{1}{a_{2}}\|\nabla_{y}f_{2}(\cdot,u,v,w)-\nabla_{y}f_{2}(% \cdot,\overline{u},\overline{v},\overline{w})\|_{L^{2}}$
		$\displaystyle\leq\frac{a_{21}}{a_{2}}\|u-\overline{u}\|_{L^{2}}+\frac{a_{22}}{a_% {2}}\|v-\overline{v}\|_{L^{2}}+\frac{a_{23}}{a_{2}}\|w-\overline{w}\|_{L^{2}}$
		$\displaystyle\leq\frac{a_{21}}{a_{2}a_{1}}\|u-\overline{u}\|_{1}+\frac{a_{22}}{a% _{2}^{2}}\|v-\overline{v}\|_{2}+\frac{a_{23}}{a_{2}a_{3}}\|w-\overline{w}\|_{3}.$

Nonlinear systems with a partial Nash type equilibrium

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Nonlinear systems with a partial Nash type equilibrium

Abstract.

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1991 Mathematics Subject Classification:

1. Introduction

2. Main result

Theorem 2.1.

Lemma 2.2.

Proof.

Lemma 2.3.

Proof.

Proof of the theorem.

3. Application

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