The Existence of the pseu-Ky Fan’ Points and the Applications in Multiobjective Games

Abstract

In this paper, we first propose pseu-Ky Fan’ points for vector Ky Fan inequalities and prove the existence results under some relaxed assumptions by virtue of KKMF principle and Fan-Browder fixed point theorem. Mild continuity named pseudocontinuity is introduced for the existence results which is weaker than semicontinuity and generalizes the present results in the literature. As applications, we define pseu-weakly Pareto-Nash equillibrium for multiobjective games and obtain some existence theorems.

Supported by the National Natural Science Foundation of Guizhou Province (Grant No. QKH[2016]7425), and the Scientific Research Projects for the Introduced Talents of Guizhou University (Grant No. [2018]11).

1 Introduction

Let X be a nonempty set of Hausdorff topological space E and $\varPhi :X\times X\rightarrow R^n$ be a vector-valued function. The vector Ky Fan inequality which we will deal with is to find $x^*\in X$ such that

$$\varPhi (x^*,y)\notin intR^n_+,~~~~~\forall y\in X.$$

We call $x^*$ a solution or a Ky Fan point of the vector Ky Fan inequality $\varPhi$.

Vector Ky Fan inequalities are natural generalizations of the Ky Fan inequality to vector-valued functions. The important inequality was introduced by Ky Fan [1] which now has been called Ky Fan inequality and the solution $x^*$ of Ky Fan inequality was called Ky Fan’s points first by Tan, Yu and Yuan in 1995, to see [2]. It is well known that Ky Fan inequality plays a very important role in many fields such as game theory, fixed point theory, variational inequalities, control theory and mathematical economics, etc. A great deal of fruitful results have been achieved on how to improve and apply the important inequalities such as [3,4,5,6] and references therein. [6] proved some existence results of the solutions and the compactness of the solution sets for vector-valued functions with the cone semicontinuity and the cone quasiconvexity in infinite dimensional spaces. [7, 8] study the set-valued versions of Ky Fan inequality and [7]established the notion of weakly Ky Fan’s points of set-valued mapping and prove some existence theorems of weakly Ky Fan’s points while [8] obtained the two set-valued versions of Ky Fan inequalities and deduced Schauder’s and Kakutani’s fixed point theorems. In this paper, we will show the new version of vector Ky Fan inequalities with pseudocontinuous functions and established some the existence results of pseu-Ky Fan’s points, which generalize the present results in [6, 7, 9]. As applications, we will give the existence results of pseu-weakly Pareto-Nash equilibrium for multiobjective games.

The rest of the paper is organized as follows. In Sect. 2, we first recall some definitions including pseudocontinuity and their properties which are needed in the sequel. In Sect. 3, we propose the concept of pseu-Ky Fan’points and give some existence results for the vector Ky Fan inequalities, then show some other cases of vector Ky Fan inequalities with pseudocontinuity. Finally, we will obtain some existence results of pseu-weakly Pareto-Nash equilibrium for multiobjective games as applications in Sect. 4.

2 Preliminaries

Now let us begin with some definitions and lemmas which we will use in the later.

Definition 2.1

([10]).  Let X be a Hausdorff topological space and $f:X\rightarrow R$ be a function.

• (1) f is said to be upper pseudocontinuous at $x_{0}\in X$ if for all $x\in X$ such that $f(x_{0})<f(x)$, we have

  $$\limsup _{y\rightarrow x_{0}}f(y)<f(x);$$

• (2) f is said to be upper pseudocontinuous on X if it is upper pseudocontinuous at each x of X;

• (3) f is said to be lower pseudocontinuous at $x_{0}\in X$ if for all $x\in X$ such that $f(x)<f(x_{0})$, we have

  $$f(x)<\liminf _{y\rightarrow x_{0}}f(y);$$

• (4) f is said to be lower pseudocontinuous on X if it is lower pseudocontinuous at each x of X;

• (5) f is said to be pseudocontinuous at $x\in X$ if f is both upper pseudocontinuous and lower pseudocontinuous at x; f is said to be pseudocontinuous on X if f is pseudocontinuous at each x of X.

Remark 2.1

If f is upper pseudocontinuous on X, then $-f$ is lower pseudocontinuous on X. The converse is also true.

Remark 2.2

Each upper (resp. lower) semicontinuous function is also upper (resp. lower) pseudocontinuous. But the converse is not true. For example: let $X=[0,2]$, $f_{i}: X\rightarrow R, i=1,2$ be defined as fellows:

$$\begin{aligned} f_{1}(x):=\left\{ \begin{array}{l l l } -x, &{}&{} 0\le x<1, \\ -2, &{}&{} 1\le x<2. \\ \end{array};\right. f_{2}(x):=\left\{ \begin{array}{l l l} x,&{}&{} 0\le x<1, \\ 2,&{}&{} 1\le x<2.\\ \end{array}\right. \end{aligned}$$

One can easily check that $f_1$ is upper pseudocontinuous but not upper semicontinuous at $x=1$ and that $f_2$ is not lower semicontinuous but lower pseudocontinuous at $x=1$.

Lemma 2.1

([10]). Let X be a Hausdorff topological space and $f: X\rightarrow R$ be lower pseudocontinuous, then $\forall b \in f(X)$, the set $\{x\in X: f(x)\le b \}$ is closed.

Definition 2.2

Let X be a nonempty subset of Hausdorff topological space E, and $F=(f_1,\cdots ,f_k): X\rightarrow R^k$ be a vector-valued function.

• (1) F is said to be lower pseudocontinuous at $x\in X$ if and only if $f_i$ is lower pseudocontinuous at $x\in X$ for any $i=1,\cdots ,k;$

• (2) F is said to be lower pseudocontinuous on X if F is lower pseudocontinuous at each x of X; 

• (3) F is said to be upper pseudocontinuous on X if $-F$ is lower pseudoncontinuous on X.

Lemma 2.2

Let X be a nonempty subset of Hausdorff topological space E. Let the vector function $F:X\rightarrow R^k$ be lower pseudocontinuous on X and $F(0)=0$. Then $G=\{x\in X:F(x)\notin intR^k_+\}$ is a closed set in X.

Proof

Let $F(x)=(f_1(x),\cdots ,f_k(x))$, obviously, $G=\bigcup \limits _{i=1}^k G_i(x),$ where $G_i(x)=\{x\in X:f_i(x)\le 0\}, \forall i=1,\cdots ,k.$ From the Lemma 2.1 and $f_i(x)=0$, we can see that $G_i(x)$ is a closed set in X. Thus, G is closed in X.

Corollary 2.1

Let X be a nonempty subset of Hausdorff topological space E and the vector function $F:X\rightarrow R^k$ be lower pseudocontinuous on X and $F(0)=0$. Then $G=\{x\in X:F(x)\in intR^k_+\}$ is a open set in X.

Definition 2.3

([11]). Let X be a nonempty convex subset of Hausdorff topological space E, $f: X\rightarrow R$ be the real number function. $\forall x_1,x_2\in X, \forall \lambda \in (0,1),$

• (1) f is said to be convex function on X,  if there holds

  $$f(\lambda x_1+(1-\lambda )x_2)\le \lambda f(x_1)+(1-\lambda )f(x_2);$$

• (2) f is said to be concave function on X if $-f$ is convex function on X. That means

  $$f(\lambda x_1+(1-\lambda )x_2)\ge \lambda f(x_1)+(1-\lambda )f(x_2);$$

• (3) f is said to be quasi-concave function on X if there holds

  $$f(\lambda x_1+(1-\lambda )x_2)\ge \min \{f(x_1),f(x_2)\};$$

• (4) f is said to be quasi-convex function on X if $-f$ is quasi-concave function on X. That means

  $$f(\lambda x_1+(1-\lambda )x_2)\le \max \{f(x_1),f(x_2)\}.$$

Lemma 2.3

Let X be a nonempty convex subset of Hausdorff topological space E, the real number function $f: X\rightarrow R,$

• (1) f is quasi-concave function on X if and only if $\forall r\in R, \{x\in X: f(x)>r\}$ is convex;

• (2) f is quasi-convex function on X if and only if $\forall r\in R, \{x\in X: f(x)<r\}$ is convex.

Definition 2.4

Let X be a nonempty convex subset of Hausdorff topological space E and $F=(f_1,\cdots ,f_k): X\rightarrow R^k$ be a vector-valued function. Then F is said to be $R^k_+-$ quasi-convex on X if and only if $f_i$ is quasi-convex on X for any $i=1,\cdots ,k$ and F is said to be $R^k_+-$quasi-concave on X if $-F$ is $R^k_+-$quasi-convex on X.

Lemma 2.4

([6]). Let X be a convex subset of Hausdorff topological space E, let the vector function $F:X\rightarrow R^k$ be $R_+^k$-quasi-concave on X. Then $G=\{x\in X:F(x)\in intR^k_+\}$ is convex on X.

The following well-known KKMF lemma is an important generalization of KKM theorem to the infinite dimensional space by Ky Fan [12].

Lemma 2.5

(KKMF Lemma).  Let X be a nonempty convex subset of Hausdorff topological vector space E, and $F: X\rightarrow X$ be a set-valued mapping. For each $x\in X, F(x)$ is closed, and there exists some $x_0\in X$ such that $F(x_0)$ is compact. If $Co\{x_1,x_2,\cdots ,x_n\}\subset \bigcup _{i=1}^{n}F(x_i),$ where $Co\{x_1,x_2,\cdots ,x_n\}$ is the convex hull of $\{x_1,x_2,\cdots ,x_n\},$ then $\bigcap _{x\in X}F(x)\ne \emptyset .$

The following fixed theorem is Fan-Browder fixed point theorem.

Lemma 2.6

(Fan-Browder Fixed Point Theorem) ([13]). Let X be a nonempty convex and compact subset of Hausdorff topological vector space E. Suppose a set-valued mapping $F:X\rightarrow X$ has the following properties:

(1) $\forall x\in X, F(x)$ is nonempty and convex;

(2) $\forall y\in X,$ The inverse valued $F^{-1}(y)=\{x\in X: y\in F(x)\}$ is open in X.

Then F has at least one fixed point.

3 Pseu-Ky Fan’s Points

In the previous section, we discussed the existence of the solutions for vector Ky Fan inequalities with pseudocontinuity defined on a compact set.

Theorem 3.1

Let X be a nonempty convex and compact subset of Hausdorff topological space E. The vector function $\varPhi (x,y)=(\varPhi _1(x,y),\cdots ,\varPhi _k(x,y)): X\times X\rightarrow R^k,$ where $\varPhi _j:X\times X\rightarrow R$ for any $j=1,\cdots ,k,$ is satisfying:

• (1) $\forall y\in X, \forall j=1,\cdots ,k, x\rightarrow \varPhi _j(x,y)$ is lower pseudocontinuous on X;

• (2) $\forall x\in X, \forall j=1,\cdots ,k, y\rightarrow \varPhi _j(x,y)$ is quasi-concave on X;

• (3) $\forall x\in X, \forall j=1,\cdots ,k, \varPhi _j(x,x)=0.$

Then there exists $x^*\in X$ such that $\varPhi (x^*,y)\notin intR^k_+$ for any $y\in X$.

Proof

For any $y\in X,$ denote by $F(y)=\{x\in X:\varPhi (x,y)\notin intR^k_+\}$. By (3), $\varPhi (y,y)=0\notin intR^k_+,$ then $y\in F(y)$ and $F(y)\ne \emptyset .$ From the above Lemma 2.2, F(y) is compact.

Next we will proof that for any $\{y_1,y_2,\cdots ,y_n\}\subset X,$ there holds

$$Co\{y_1,y_2,\cdots ,y_n\}\subset \bigcup \limits _{i=1}^{n}F(y_i).$$

By the way of contradiction, there exists $y_0\in Co\{y_1,y_2,\cdots ,y_n\}\subset X$ and $y_0=\sum \limits _{i=1}^{n}\alpha _{i}y_i$ with $\alpha _i\ge 0, i=1,2,\cdots ,n,\sum \limits _{i=1}^{n}\alpha _i=1$ but $y_0\notin \bigcup _{i=1}^{n}F(y_i).$ Then for $i=1,\cdots ,n, y_0\notin F(y_i),$ i.e., $\varPhi (y_0,y_i)\in intR_+^k.$

Hence, for any $j=1,\cdots ,k,$ we have $\varPhi _j(y_0,y_i)>0.$ By (2), for any $j=1,\cdots ,k,$

$$\varPhi _j(y_0,y_0)\ge \min \limits _{1\le i\le n}\varPhi _j(y_0,y_i)>0,$$

Which is a contradiction with the condition (3).

Therefore, by KKMF lemma, we know $\bigcap _{y\in X}F(y)\ne \emptyset .$ We take $x^*\in \bigcap _{y\in X}f(y),$ which implies

$$\varPhi (x^*,y)\notin intR^k_+~~~~~~\forall y\in X.$$

Remark 3.1

We call $x^*$ as a pseu-Ky Fan’s point if $x^*$ is a solution of the vector Ky Fan inequality and the vector function $\varPhi (x,y)$ satisfies the condition(1) of Theorem 3.1.

Remark 3.2

We also obtain the existence result by the Fan-Browder fixed point theorem. Assume by contradiction that for any $x\in X,$ $F(x)=\{y\in X,\varPhi (x,y)\in intR_+^k\}\ne \emptyset .$ By (2), we know F(x) is convex. For any $y\in X, F^{-1}(y)=\{x\in X,\varPhi (x,y)\in intR_+^k\}.$ By Corollary 2.1, $F^{-1}(y)$ is open. These are sufficient condition for Fan-Browder fixed point theorem. Thus, there exists $x^*\in X$ such that $x^*\in F(x^*)$. That means $\varPhi (x^*,x^*)\in intR_+^k$, which is in contradiction with the condition (3).

Remark 3.3

If $k=1$, the vector Ky Fan inequality reduce the Ky Fan inequality as follows, to see [9].

Let X be a nonempty convex and compact subset of Hausdorff topological space E. The function $\phi (x,y)= X\times X\rightarrow R,$ is satisfying:

(1) $\forall y\in X, x\rightarrow \phi (x,y)$ is lower pseudocontinuous on X;

(2) $\forall x\in X, y\rightarrow \phi (x,y)$ is quasi-concave on X;

(3) $\forall x\in X, \phi (x,x)=0.$

Then there exists $x^*\in X$ such that $\phi (x^*,y)\le 0$ for any $y\in X$.

That means $x^*$ is a pseu-Ky Fan’s point of the function $\phi (x,y)$.

Let X be a nonempty set of Hausdorff topological space E,  and $\varPsi :X\times X\rightarrow R^n$ be a vector-valued function. If there exists $x^*\in X$ such that

$$\varPsi (x^*,y)\notin -intR^k_+,~~~~~\forall y\in X.$$

Then $x^*$ is called a solution of the vector equilibrium problem.

From the above Theorem 3.1, we can obtain an existence result of the solution for vector equilibrium problem as follows.

Theorem 3.2

Let X be a nonempty convex and compact subset of Hausdorff topological space E. The vector function $\varPsi (x,y)=(\varPsi _1(x,y),\cdots ,\varPsi _k(x,y)): X\times X\rightarrow R^k,$ where $\varPsi _i:X\times X\rightarrow R, \forall i=1,\cdots ,k,$ is satisfying:

• (1) $\forall y\in X, \forall j=1,\cdots ,k, x\rightarrow \varPsi _j(x,y)$ is upper pseudocontinuous on X;

• (2) $\forall x\in X, \forall j=1,\cdots ,k, y\rightarrow \varPsi _j(x,y)$ is quasi-convex on X;

• (3) $\forall x\in X, \forall j=1,\cdots ,k, \varPsi _j(x,x)=0.$

Then there exists $x^*\in X$ such that $\varPsi (x^*,y)\notin -intR^k_+$ for any $y\in X$.

Proof

 $\forall x\in X,\forall y\in X,$ Set $\varPhi (x,y)=-\varPsi (x,y).$ It is easy to check that

• (1) $\forall y\in X, \forall j=1,\cdots ,k, x\rightarrow \varPhi _j(x,y)$ is lower pseudocontinuous on X;

• (2) $\forall x\in X, \forall j=1,\cdots ,k, y\rightarrow \varPhi _j(x,y)$ is quasi-concave on X;

• (3) $\forall x\in X, \forall j=1,\cdots ,k, \varPhi _j(x,x)=0.$

By Theorem 3.1, there exists $x^*\in X$ such that $\varPhi (x^*,y)\notin intR^k_+$ for any $y\in X$. That implies $\varPsi (x^*,y)\notin -intR^k_+$ for any $y\in X$.

Remark 3.4

Theorem 3.1 is shown the new expression on the vector Ky Fan inequality without the $R_+^k$-lower semicontinuity and $R_+^k$ quasi-concavity of the vector function $\varPhi$ on the cone. The quasi-concavity is just needed for the every component of the vector function $\varPhi$ and the lower pseudocontinuity which is weaker than low semicontinuity is required for the components. So our judgment will be direct from the component of the vector function $\varPhi$.

Remark 3.5

Similarly, Theorem 3.2 is shown the new expression on the vector equilibrium inequality without the $R_+^k$-upper semicontinuity and $R_+^k$-quasi-convexity of the vector function $\varPsi$ on the cone. The quasi-concavity and the lower pseudocontinuity are just needed for the component of the vector function $\varPsi$ for the existence of the solution.

In the above theorems, we discussed the existence of the pseu-Ky Fan’s points of the vector Ky Fan inequalities in the case of nonempty convex compact set.

Theorem 3.3

Let X be a nonempty unbounded closed convex subset of Hausdorff linear topological space E, $\phi :X\times X\rightarrow R^k$ be a vector function with $\varPhi (x,y)=(\varPhi _1(x,y),\cdots ,\varPhi _k(x,y)$ where $\varPhi _i:X\times X\rightarrow R$ for any $i=1,\cdots ,k.$ Suppose $\varPhi$ satisfies the following conditions:

• (1) $\forall y\in X, \forall i=1,\cdots ,k, x\rightarrow \varPhi _i(x,y)$ is pseudocontinuous on X;

• (2) $\forall x\in X, \forall i=1,\cdots ,k, y\rightarrow \varPhi _i(x,y)$ is quasi-concave on X;

• (3) $\forall x\in X, \varPhi _i(x,x)=0;$

• (4) for any sequence $\{x^m\}$ with $\Vert x^m\Vert \rightarrow \infty ,$ there exist a positive integer $m_0$ and $y\in X$ such that $\Vert y\Vert \le \Vert x^m\Vert$ and $\varPhi (x^{m_0},y)\in intR_+^k$.

Then there exists $x^*\in X$ such that $\varPhi (x^*,y)\notin intR_+^k$ for any $y\in X$.

Proof

For each $m=1,2,\cdots ,$ set $C_m=\{x\in X:\Vert x\Vert \le m\}.$ We may assume that $C_m\ne \emptyset .$ $C_m$ is a bounded closed convex subset in X since X is closed convex set. By Theorem 3.1, there exist $x^m\in C_m$ such that $\varPhi (x^{m},y)\notin intR_+^k$ for any $y\in C_m.$

If the sequence $\{x^m\}$ is unbounded in X, we can suppose that $\Vert x^m\Vert \rightarrow \infty$ (otherwise subsequence). By (4), there exist a positive integer $m_0$ and $y\in X$ such that $\Vert y\Vert \le \Vert x^{m_0}\Vert$ and $\varPhi (x^{m_0},y)\in intR_+^k,$ which is a contradiction with $\Vert y\Vert \le \Vert x^{m_0}\Vert \le m_0, y\in C_{m_0}, \varPhi (x^{m_0},y)\notin intR_+^k.$ Thus $\{x^m\}$ is bounded in X and there is M such that $\Vert x^m\Vert \le M$. Since $C_M$ is bounded and we may assume $x^m\rightarrow x^*\in C_M\subset X.$

$\forall y\in X$, There is a positive integer K such that $y\in C_k$ and $C_k\subset C_M, y\in C_M, \varPhi (x^m,y)\notin intR_+^k$ when $m\ge k.$ Set $F(y)=\{x\in X:\varPhi (x,y)\notin intR_+^k\},$ by (1)  and Corollary 2.1, then F(y) is closed. Since $x^m\rightarrow x^*$, then $x^*\in F(y)$ which implies $\varPhi (x^*,y)\notin intR_+^k.$ The proof is thus complete.

Remark 3.6

Theorem 3.3 is shown the existences of the solution of the new version on vector Ky Fan inequalities with pseudocontinuity in the unbounded setting [14].

4 Applications

Let $N=\{1,2,\cdots ,n\}$ be the set of players. For each $i\in N$, let $X_{i}$ be the strategy set for player i, $X=\prod _{i=1}^{n}X_{i}$, $F^{i}=(f^{i}_{1},\cdots ,f^{i}_{k}):X\rightarrow R^{k}$ be the vector-valued payoff of player i, where k is a positive integer. This multiobjective game is denoted by $\varGamma ^*=\{X_i,F^{i}\}_{i\in N}$ . A strategy profile $x^*=(x^*_1,x^*_2,\cdots ,x^*_n)\in X$ is called a weakly Pareto-Nash equilibrium of a multiobjective game $\varGamma ^*$ if for each $i\in N$,

$$F^{i}(y_i,x^*_{\hat{i}})-F^{i}(x_i^*,x_{\hat{i}}^*)\not \in \mathrm{int}R^{k}_{+}, ~\forall y_i\in X_i,$$

where $x_{\hat{i}}^*=(x^*_1,\cdots ,x^*_{i-1},x^*_{i+1},\cdots ,x^*_n).$

For any $i\in N$, if $F^{i}$ has some pseucontinuous property, we say a strategy $x^*$ is a pseu-weakly Pareto-Nash equilibrium for multiobjective games.

From the above theorems, we can obtain some existence results of a pseu-weakly Pareto-Nash equilibrium of multiobjective games as applications.

Theorem 4.1

Let the multiobjective game $\varGamma ^*=\{X_i,F^{i}\}_{i\in N}$ satisfy the following conditions:

• (1) $\forall i\in N$, $X_i$ is a nonempty, convex and compact subset of a Hausdorff topological space $E_i$;

• (2) $\forall j=1,2,\cdots ,k$, $\forall y= (y_1,\cdots ,y_n)\in X, x\rightarrow \sum \limits _{i=1}^n[f^i_j(y_i,x_{\hat{i}})-f^i_j(x_i,x_{\hat{i}})]$ is lower pseudocontinuous on X;

• (3) $\forall j=1,2,\cdots ,k$, $\forall x=(x_1,\cdots ,x_n)\in X, y\rightarrow \sum \limits _{i=1}^n[f^i_j(y_i,x_{\hat{i}})-f^i_j(x_i,x_{\hat{i}})]$ is quasi-concave on X;

Then there exists at least a pseu-weakly Pareto-Nash equilibrium of $\varGamma ^*$.

Proof

 $\forall x=(x_1,\cdots ,x_n)\in X, \forall y=(y_1,\cdots ,y_n)\in X,$ Denote by

$$\varPhi (x,y)=\sum _{i=1}^n[F_i(y_i,x_{\hat{i}})-F_i(x_i,x_{\hat{i}})].$$

We notice that $\forall j=1,\cdots ,k,\varPhi _j(y,y)=0$ for any $y\in X.$ By the conditions (1)(2) and Theorem 3.1, there exists $x^*\in X$ such that $\varPhi (x^*,y)\notin intR^k_+$ for any $y\in X.$ $\forall i\in N, \forall y_i\in X_i,$ Set $y=(y_i,x^*_{\hat{i}})$, then $y\in X.$

Then

$$\varPhi (x^*,y)=\sum _{i=1}^n[F_i(y_i,x_{\hat{i}})-F_i(x_i,x_{\hat{i}})] =F_i(y_i,x^*_{\hat{i}})-F_i(x_i^*,x^*_{\hat{i}})\notin intR^k_+,$$

which implies $x^*\in X$ is a pseu-weakly Pareto-Nash equilibrium of a multiobjective game $\varGamma ^*.$

Assume that $k_1\le k_2 \le \cdots \le k_n$, by applying Theorem 4.1, we obtain the following existence theorem.

Theorem 4.2

Let $\varGamma ^{**}=\{X_i,F^{i}\}_{i\in N}$ be the multiobjective game, where $F^i=(f_1^i,\cdots ,f^i_{k_i}).$ Suppose that $\varGamma ^{**}$ satisfies the following conditions:

• (1) $\forall i\in N$, $X_i$ is a nonempty, convex and compact subset of a Hausdorff topological space $E_i$;

• (2) $\forall j=1, 2,\cdots , k_i$, $\forall y= (y_1,\cdots ,y_n)\in X, x\rightarrow \sum \limits _{i=1}^n[f^i_j(y_i,x_{\hat{i}})-f^i_j(x_i,x_{\hat{i}})]$ is lower pseudocontinuous on X;

• (3) $\forall j=1, 2,\cdots , k_i$, $\forall x=(x_1,\cdots ,x_n)\in X, y\rightarrow \sum \limits _{i=1}^n[f^i_j(y_i,x_{\hat{i}})-f^i_j(x_i,x_{\hat{i}})]$ is quasi-concave on X;

Then there exists a pseu-weakly Pareto-Nash equilibrium of $\varGamma ^{**}$.

Proof

$\forall x=(x_1,\cdots ,x_n)\in X, \forall y=(y_1,\cdots ,y_n)\in X,$ We define the vector-valued function $\varPsi :X\times X\rightarrow R^{k_n}$ by

$$\varPsi (x,y)=\sum \limits _{i=1}^n\varPsi _i(x,y)$$

where

$$\varPsi _i(x,y)=(\underbrace{F^i(y_i,x_{\hat{i}})-F^i(x_i,x_{\hat{i}})}_{k_i ~\text{ components }},\underbrace{\sum \limits _{i=1}^n[f^i_1(y_i,x_{\hat{i}})-f^i_1(x_i,x_{\hat{i}})],\cdots ,\sum \limits _{i=1}^n[f^i_1(y_i,x_{\hat{i}})-f^i_1(x_i,x_{\hat{i}})])}_{k_n-k_i ~ \text{ components }}$$

It is easy to check that

• (1) $\forall y= (y_1,\cdots ,y_n)\in X, x\rightarrow \varPsi (x,y)$ is lower pseudocontinuous on X;

• (2) $\forall x=(x_1,\cdots ,x_n)\in X, y\rightarrow \varPsi (x,y)$ is quasi-concave on X;

• (3) $\forall x\in X, \varPsi (x,x)=0\notin intR_+^{k_n}.$

Therefore, by Theorem 3.3, there exists $x^*\in X$ such that $\varPsi (x^*,y)\notin intR_+^{k_n}$ for any $y\in X.$ For each $i\in N$ and $y_i\in X_i,$ set $y=(y_i,x^*_i)\in X$, then

$$\varPsi _i(x^*,y)=\varPsi (x^*,y)\notin intR_+^{k_n}.$$

If $F^i(y_i,x^*_{\hat{i}})-F^i(x^*_i,x^*_{\hat{i}})\in intR_+^{k_i},$ then $f^i_j(y_i,x^*_{\hat{i}})-f_j^i(x^*_i,x^*_{\hat{i}})\in intR_+,$ for each $j=1,\cdots , k_i$ and $\varPsi _i(x^*,y)\in intR_+^{k_n},$ which contradicts that $\varPsi _i(x^*,y)\notin intR_+^{k_n}.$ Hence, $F^i(y_i,x^*_{\hat{i}})-F^i(x^*_i,x^*_{\hat{i}})\notin intR_+^{k_i},$ for each $i\in N$, i.e., $x^*$ is a pseu-weakly Pareto-Nash equilibrium point of the multiobjective game $\varGamma ^{**}.$ The proof is completed.

Theorem 4.3

$\forall i\in N,$ Let $X_i$ is a nonempty closed convex subset of Hausdorff linear topological space $E_i$, $X=\varPi _{i=1}^n$, $F^i=\{f_1^i,\cdots ,f_k^i\}:X\rightarrow R^k$ satisfy the following conditions:

• (1) $\forall j=1,2,\cdots ,k$, $\forall y= (y_1,\cdots ,y_n)\in X, x\rightarrow \sum \limits _{i=1}^n[f^i_j(y_i,x_{\hat{i}})-f^i_j(x_i,x_{\hat{i}})]$ is lower pseudocontinuous on X;

• (2) $\forall j=1,2,\cdots ,k$, $\forall x=(x_1,\cdots ,x_n)\in X, y\rightarrow \sum \limits _{i=1}^n[f^i_j(y_i,x_{\hat{i}})-f^i_j(x_i,x_{\hat{i}})]$ is quasi-concave on X;

• (3) $\forall x\in X, \phi _i(x,x)=0;$

• (4) For any sequence $\{x^m=(x_1^m,\cdots ,x_n^m)\}$ with $\Vert x^m\Vert =\sum \limits _{i=1}^n\Vert x_i^m\Vert _i\rightarrow \infty ,$ where $\Vert x_i^m\Vert _i$ means the norm of $x_i^m$ in $X_i$, there exist some $i\in N,$ a positive integer $m_0$ and $y\in X$ such that $\Vert y_i\Vert \le \Vert x_i^m\Vert _i$ and $F^i(y_i,x^{m_0}_{\hat{i}})-F^i(x_i^{m_0},x^{m_0}_{\hat{i}})\in intR_+^k$.

Then there exists a pseu-weakly Pareto-Nash equilibrium of multiobjective game.

Proof

 $\forall x=(x_1,\cdots ,x_n)\in X, \forall y=(y_1,\cdots ,y_n)\in X,$ Denote by

$$\phi (x,y)=\sum _{i=1}^n[F^i(y_i,x_{\hat{i}})-F^i(x_i,x_{\hat{i}})].$$

It is easy to check

• (1) $\forall y\in X, \forall i=1,\cdots ,k, x\rightarrow \phi _i(x,y)$ is pseudocontinuous on X;

• (2) $\forall x\in X, \forall i=1,\cdots ,k, y\rightarrow \phi _i(x,y)$ is quasi-concave on X;

• (3) $\forall x\in X, \phi _i(x,x)=0;$

By (4) , for any sequence $\{x^m=(x_1^m,\cdots ,x_n^m)\}$ with $\Vert x^m\Vert =\sum \limits _{i=1}^n\Vert x_i^m\Vert _i\rightarrow \infty ,$ there exist some $i\in N,$ a positive integer $m_0$ and $y\in X$ such that $\Vert y_i\Vert \le \Vert x_i^m\Vert _i$ and $F^i(y_i,x^{m_0}_{\hat{i}})-F^i(x_i^{m_0},x^{m_0}_{\hat{i}})\in intR_+^k$.

Set $y=(y_i,x^{m_0}_{\hat{i}}),$ then $y\in X, \Vert y\Vert \le \Vert x^{m_0}\Vert ,$ but

$$\phi (x^{m_0},y)=F^i(y_i,x^{m_0}_{\hat{i}})-F^i(x_i^{m_0},x^{m_0}_{\hat{i}})\in intR_+^k.$$

Thus, by Theorem 3.3, there exists $x^*\in X$ such that $\phi (x^*,y)\notin intR_+^k$ for any $y\in X.$

$\forall i\in N, \forall y_i\in X_i$, we take $y=(y_i,x^*_{\hat{i}})$, then $y\in X.$

Then

$$\phi (x^*,y)=\sum _{i=1}^n[F_i(y_i,x_{\hat{i}})-F_i(x_i,x_{\hat{i}})] =F_i(y_i,x^*_{\hat{i}})-F_i(x_i^*,x^*_{\hat{i}})\notin intR^k_+,$$

That means $x^*\in X$ is a pseu-weakly Pareto-Nash equilibrium of a multiobjective game.