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Quasi-Equilibrium Problems and Fixed Point Theorems of Separately l.s.c and u.s.c Mappings

Article title: Quasi-Equilibrium problems and fixed point theorems of separately l.s.c and u.s.c mappings

Author: Nguyen Xuan Tan

Journal: Numer. Funct. Anal. Optim.

Bibliometrics: Volume 39 (2018), Number. 02, pages. 233–255

DOI: 10.1080/01630563.2017.1360346

We formulated and proved the following theorem in the above article:

Theorem 3.1.

We assume that the following conditions hold:

1. D, K are nonempty convex compact sets;

2. $P:D\times K\to 2^D$ is a continuous multivalued mapping with nonempty closed convex values;

3. $Q:D\times K\to 2^K$ is a u.s.c multivalued mapping with nonempty closed convex values;

4. $F:D\times K\to 2^{X\times Z}$ is a separately l.s.c and u.s.c multivalued mapping;

5. For any $(x,y)\in P(x,y)\times Q(x,y),F(x,y)$ has nonempty convex closed values and $\varnothing \cancel{=}F(x,y)\cap T_{P(x,y)\times Q(x,y)}(x,y).$

Then there exists $(\overline{x},\overline{y})\in D\times K$ such that

1. $\overline{x}\in P(\overline{x},\overline{y});$

2. $\overline{y}\in Q(\overline{x},\overline{y});$

3. $0\in F(\overline{x},\overline{y}).$

We used in (3.5) $$\underset{u\in F(\overline{x},\overline{y})}{\inf }\underset{i\in I(\overline{x},\overline{y})}{\max }{p}_{j\left(i\right)}\left(u\right)=\underset{i\in I(\overline{x},\overline{y})}{\max }\underset{u\in F(\overline{x},\overline{y})}{\inf }{p}_{j\left(i\right)}\left(u\right).$$

But, the correct use the Sion’ minimax Theorem must be $$\underset{u\in F(\overline{x},\overline{y})}{\inf }\underset{p_{j\left(i\right)}\in M}{\max }{p}_{j\left(i\right)}\left(u\right)=\underset{p_{j\left(i\right)}\in M}{\max }\underset{u\in F(\overline{x},\overline{y})}{\inf }{p}_{j\left(i\right)}\left(u\right).$$

Here, ${\left\{p_{j(i)}\right\}}_{i\in I(\overline{x},\overline{y})}\cancel{=}M=\overline{\mathit{co}}\{p_{j(1)},\dots ,p_{j(s)}\}.$ Therefore, the proof of this theorem is not correct.

Now, we give a new proof for this theorem as follow:

Proof of Theorem 3.1.

We set $$B=\left\{\right(x,y)\in D\times K|x\in P(x,y), y\in Q(x,y)\}.$$

Since the multivalued mapping $T:D\times K\to 2^{D\times K},$ defined by $$T(x,y)=P(x,y)\times Q(x,y), (x,y)\in D\times K,$$ is upper semi-continuous with nonempty convex and closed values, by using the Ky Fan fixed point Theorem, we conclude that T has a fixed point in D × K. Therefore, B is a nonempty set. The upper semi-continuity and the closedness of values of T imply that B is a closed and then a compact set.

Assume that for any $(x,y)\in B, )0\notin F(x,y).$ Since F(x, y) is a nonempty closed convex set, by the Hahn-Banach Theorem, there exists $p\in {(X\times Z)}^{*}$ such that $$\underset{w\in F(x,y)}{\inf }p\left(w\right)<0.$$

Further, we define functions $c_p{}^1(.,.),c_p{}^2(.,.):D\times K\to R\cup \{-\infty \}$ by $$\begin{array}{l}{c}_p{}^1(x\prime ,y)=\underset{u\in F(x\prime ,y)}{\inf }p(u);\\ c_p{}^2(x,y\prime )=\underset{w\in F(x,y\prime )}{\sup }p(w).\end{array}$$

By Proposition 2.2, the functions $c_p^1$ and $c_p^2$ are u.s.c on $D\times K,$ therefore, the sets $$\begin{array}{c}{U}_p\left(x\right)=\{x\prime \in D|c_p{}^1(x\prime ,y)<0\},\\ U_p\left(y\right)=\{y\in K|c_p{}^2(x,y\prime )<0\},\end{array}$$ are open. Since $(x,y)\in U_p=U_p(x)\times U_p(y),$ and so U_p is a nonempty open neighborhood of $(x,y).$

Thus, for any $(x,y)\in B$ there is $p\in {(X\times Z)}^{*}$ such that $$U_p(x,y)=\left\{\right(x\prime ,y\prime )\in D\times K|c_p^1(x\prime ,y)<0,c_p^2(x,y\prime )<0\}$$ is nonempty and open and hence ${\left\{U_p\right\}}_{p\in {(X\times Z)}^{*}}$ is an open cover of B. Since B is compact, there exist finite $p_1,\dots ,p_s\in {(X\times Z)}^{*}$ such that $$B\subseteq \underset{j=1}{\overset{s}{\cup }}{U}_{p_j}.$$

Further, since B is closed in $D\times K,U_{p_0}=D\times K\setminus B$ is open in D × K and hence $\{U_{p_0},U_{p_1},\dots ,U_{p_s}\}$ is an open cover of the compact set $D\times K.$ By Theorem 2.8 in Section 2 in [1], there exist continuous functions ${\psi }_i:D\times K\to R,(i=0,1,\dots ,s)$ such that

1. $0\le {\psi }_i(x,y)\le 1;$

2. ${\sum }_{i=1}^s{\psi }_i(x,y)=1, \text{for} \text{all} (x,y)\in D\times K;$

3. $\text{for} \text{any} i\in \{0,1,\dots ,s\}, \text{there} \text{exists} j(i)\in \{0,\dots ,s\}$ such that $\mathit{supp}{\psi }_{j(i)}\subset U_{p_{j(i)}}.$

We define the function $\varphi :D\times K\times D\times K\to R$ by $$\varphi \left(\right(x,y),(t,z\left)\right)=\sum _{i=0}^s{\psi }_{j\left(i\right)}(x,y).p_{j\left(i\right)}(t-x,z-y), \text{for} \text{any} t,x\in D,z,y\in K.$$

Then, $\varphi$ is a continuous function on $(D\times K)\times (D\times K).$ Moreover, for any fixed $(x,y)\in D\times K,\varphi ((x,y),.):D\times K\to R$ is an affine function and $\varphi ((x,y),(x,y))=0$ for all $(x,y)\in D\times K.$ Further, we define the multivalued mapping $N:D\times K\to 2^D$ by $$\begin{array}{c}N(x,y)=\{(v,w)\in P(x,y)\times Q(x,y\left)\right|\varphi \left(\right(x,y),(v,w\left)\right)\le \varphi \left(\right(x,y),(t,z\left)\right),\\ \text{for} \text{all} (t,z)\in P(x,y)\times Q(x,y)\}.\end{array}$$

Since $\varphi$ is a continuous function and $P(x,y)\times Q(x,y)$ is a compact set, $\varphi (((x,y),.):D\times K\to R$ is an affine function,we conclude that N(x, y) is a nonempty convex and closed subset, for any fixed $(x,y)\in D\times K.$ It is also easy to prove that N is a closed mapping and then is a upper semi-continuous multivalued mapping with nonempty convex compact values. Therefore, the multivalued mapping $N:D\times K\to 2^{D\times K}$ is a u.s.c multivalued mapping with nonempty convex compact values. Applying the Ky Fan fixed point Theorem again, we conclude that there exists $(\overline{x},\overline{y})\in D\times K$ such that $(\overline{x},\overline{y})\in N(\overline{x},\overline{y}).$ This follows $\varphi ((\overline{x},\overline{y}),(t,z))\ge 0,$ for all $(t,z)\in P(\overline{x},\overline{y})\times Q(\overline{x},\overline{y}).$ This gives (1) $$\sum _{i=0}^s{\psi }_i(\overline{x},\overline{y}).p_{j\left(i\right)}(t-\overline{x},z-\overline{y}))\ge 0, \text{for} \text{all} (t,z)\in P(\overline{x},\overline{y})\times Q(\overline{x},\overline{y}).$$(1)

Setting $p^{*}={\sum }_{i=0}^s{\psi }_i(\overline{x},\overline{y}).p_{j(i)},$ we get from (1)(1) $$\sum _{i=0}^s{\psi }_i(\overline{x},\overline{y}).p_{j\left(i\right)}(t-\overline{x},z-\overline{y}))\ge 0, \text{for} \text{all} (t,z)\in P(\overline{x},\overline{y})\times Q(\overline{x},\overline{y}).$$(1) , $p^{*}(t-\overline{x},z-\overline{y})) \ge 0,$ for all $(t,z)\in P(\overline{x},\overline{y})\times Q(\overline{x},\overline{y})$ and hence $p^{*}(u)\ge 0,$ for all $u\in T_{P(\overline{x},\overline{y})\times Q(\overline{x},\overline{y})}(\overline{x},\overline{y}).$ By assumption $(v),F(x,y)\cap T_{P(x,y)\times Q(\overline{x},\overline{y})}(\overline{x},\overline{y})\cancel{=}\varnothing ,$ we conclude that $$p^{*}\left(v\right)\ge 0, \text{for} \text{all} v\in F(\overline{x},\overline{y})\cap T_{P(\overline{x},\overline{y})\times Q(\overline{x},\overline{y})}(\overline{x},\overline{y}).$$

This yields $$\underset{v\in F(\overline{x},\overline{y})\cap T_{P(\overline{x},\overline{y})\times Q(\overline{x},\overline{y})}(\overline{x},\overline{y})}{\inf }{p}^{*}\left(v\right)\ge 0$$ and (2) $$\underset{v\in F(\overline{x},\overline{y})\cap T_{P(\overline{x},\overline{y})\times Q(\overline{x},\overline{y})}(\overline{x},\overline{y})}{\sup }{p}^{*}\left(v\right)\ge 0.$$(2)

Now, we assert that for any convex subset $A\subset D\times K,$ the following equality holds (3) $$p^{*}\left(A\right)=\sum _{i=0}^s{\psi }_i(\overline{x},\overline{y}).p_{j\left(i\right)}\left(A\right).$$(3)

Indeed, for any $v\in A,p^{*}(v)={\sum }_{i=0}^s{\psi }_i(\overline{x},\overline{y}).p_{j(i)}(v).$ This shows $$p^{*}\left(A\right)\subseteq \sum _{i=0}^s{\psi }_i(\overline{x},\overline{y}).p_{j\left(i\right)}\left(A\right).$$

We need to verify $$\sum _{i=0}^s{\psi }_i(\overline{x},\overline{y}).p_{j\left(i\right)}\left(A\right)\subseteq p^{*}\left(A\right).$$

Take $w\in {\sum }_{i=0}^s{\psi }_i(\overline{x},\overline{y}).p_{j(i)}(A),$ we can write $w={\sum }_{i=0}^s{\psi }_i(\overline{x},\overline{y}).p_{j(i)}(v_i)$ with $v_i\in A,$ for $i=0,\dots ,s.$ By the definition of $p^{*},$ we have $$p^{*}\left({\psi }_m\right(\overline{x},\overline{y}\left)v_m\right)=\sum _{i=0}^s{\psi }_i(\overline{x},\overline{y}).p_{j\left(i\right)}\left({\psi }_m\right(\overline{x},\overline{y}\left)v_m\right),m=0,\dots ,s.$$

Taking the sum of the both sides of these equalities, we obtain $$p^{*}\left(\sum _{m=0}^s{\psi }_m\right(\overline{x},\overline{y}\left)v_m\right)=\sum _{m=0}^s{\psi }_m(\overline{x},\overline{y})w=w.$$

Setting $v={\sum }_{m=0}^s{\psi }_m(\overline{x},\overline{y})v_m$ and using the convexity of A, we conclude $w=p^{*}(v)\in p^{*}(A).$ Thus, we have the proof of (3)(3) $$p^{*}\left(A\right)=\sum _{i=0}^s{\psi }_i(\overline{x},\overline{y}).p_{j\left(i\right)}\left(A\right).$$(3) . Inserting $A=F(\overline{x},\overline{y})\cap T_{P(\overline{x},\overline{y})\times Q(\overline{x},\overline{y})}(\overline{x},\overline{y})$ in (3)(3) $$p^{*}\left(A\right)=\sum _{i=0}^s{\psi }_i(\overline{x},\overline{y}).p_{j\left(i\right)}\left(A\right).$$(3) , we deduce $$p^{*}\left(F\right(\overline{x},\overline{y})\cap T_{P(\overline{x},\overline{y})\times Q(\overline{x},\overline{y})}(\overline{x},\overline{y}\left)\right)=\sum _{i=0}^s{\psi }_i(\overline{x},\overline{y}).p_{j\left(i\right)}\left(F\right(\overline{x},\overline{y})\cap T_{P(\overline{x},\overline{y})\times Q(\overline{x},\overline{y})}(\overline{x},\overline{y}\left)\right).$$

This gives $$\begin{array}{c}\underset{w\in F(\overline{x},\overline{y})\cap T_{P(\overline{x},\overline{y})\times Q(\overline{x},\overline{y})}(\overline{x},\overline{y}))}{\inf }{p}^{*}(w)=\sum _{i=0}^s\underset{w\in (F(\overline{x},\overline{y})\cap T_{P(\overline{x},\overline{y})\times Q(\overline{x},\overline{y})}(\overline{x},\overline{y}))}{\inf }{\psi }_i(\overline{x},\overline{y}).p_{j(i)}(w)\\ \le \underset{i=0,\dots s}{\max }\underset{w\in F(\overline{x},\overline{y})\cap T_{P(\overline{x},\overline{y})\times Q(\overline{x},\overline{y})}(\overline{x},\overline{y}))}{\inf }{p}_{j(i)}(w)\}\end{array}$$ and (4) $$\begin{array}{c}\underset{w\in F(\overline{x},\overline{y})\cap T_{P(\overline{x},\overline{y})\times Q(\overline{x},\overline{y})}(\overline{x},\overline{y}))}{\sup }{p}^{*}(w)=\sum _{i=0}^s\underset{w\in (F(\overline{x},\overline{y})\cap T_{P(\overline{x},\overline{y})\times Q(\overline{x},\overline{y})}(\overline{x},\overline{y})){\psi }_i(\overline{x},\overline{y}).p_{j(i)}}{\sup }(w)\\ \le \underset{i=0,\dots ,s}{\max }\underset{w\in F(\overline{x},\overline{y})\cap T_{P(\overline{x},\overline{y})\times Q(\overline{x},\overline{y})}(\overline{x},\overline{y}))}{\sup }{p}_{j(i)}(w)\}.\end{array}$$(4)

Put $I(\overline{x},\overline{y})=\{i\in \{0,1,\dots ,s\left\}\right|{\psi }_i(\overline{x},\overline{y})>0\}.$ Since ${\psi }_i(\overline{x},\overline{y})\ge 0$ and $(\text{Tex} \text{translation} \text{failed})$ we deduce $I(\overline{x},\overline{y})\cancel{=}\varnothing .$ So, for any $i\in I(\overline{x},\overline{y}),(\overline{x},\overline{y})\in \mathit{supp}{\psi }_{j(i)}\subset U_{p_{j(i)}}.$ This yields $$\begin{array}{c}\underset{w\in F(\overline{x},\overline{y})\cap T_{P(\overline{x},\overline{y})\times Q(\overline{x},\overline{y})}(\overline{x},\overline{y})}{\inf }{p}_{j\left(i\right)}\left(w\right)=c_{p_{j\left(i\right)}}^1(\overline{x},\overline{y})<0,\\ \underset{w\in F(\overline{x},\overline{y})\cap T_{P(\overline{x},\overline{y})\times Q(\overline{x},\overline{y})}(\overline{x},\overline{y})}{\sup }{p}_{j\left(i\right)}\left(w\right)=c_{p_{j\left(i\right)}}^2(\overline{x},\overline{y})<0,\end{array}$$ and then $$\underset{i\in I(\overline{x},\overline{y})}{\max }\max \left\{c_{p_{j\left(i\right)}}^1\right(\overline{x},\overline{y});c_{p_{j\left(i\right)}}^2(\overline{x},\overline{y}\left)\right\}<0.$$

Therefore, together with (4)(4) $$\begin{array}{c}\underset{w\in F(\overline{x},\overline{y})\cap T_{P(\overline{x},\overline{y})\times Q(\overline{x},\overline{y})}(\overline{x},\overline{y}))}{\sup }{p}^{*}(w)=\sum _{i=0}^s\underset{w\in (F(\overline{x},\overline{y})\cap T_{P(\overline{x},\overline{y})\times Q(\overline{x},\overline{y})}(\overline{x},\overline{y})){\psi }_i(\overline{x},\overline{y}).p_{j(i)}}{\sup }(w)\\ \le \underset{i=0,\dots ,s}{\max }\underset{w\in F(\overline{x},\overline{y})\cap T_{P(\overline{x},\overline{y})\times Q(\overline{x},\overline{y})}(\overline{x},\overline{y}))}{\sup }{p}_{j(i)}(w)\}.\end{array}$$(4) , we have $$\begin{array}{c}\underset{w\in F(\overline{x},\overline{y})\cap T_{P(\overline{x},\overline{y})\times Q(\overline{x},\overline{y})}(\overline{x},\overline{y})}{\inf }{p}^{*}\left(w\right)\\ \le \underset{i\in I(\overline{x},\overline{y})}{\inf }{\psi }_i(\overline{x},\overline{y})\left\{\underset{w\in F(\overline{x},\overline{y})\cap T_{P(\overline{x},\overline{y})\times Q(\overline{x},\overline{y})}(\overline{x},\overline{y})}{\inf }{p}_{j\left(i\right)}\right(w\left)\right\}\\ \le \underset{i\in I(\overline{x},\overline{y})}{\max }{c}_{p_{j\left(i\right)}}^1(\overline{x},\overline{y})<0\end{array}$$ and (5) $$\begin{array}{c}\underset{w\in F(\overline{x},\overline{y})\cap T_{P(\overline{x},\overline{y})\times Q(\overline{x},\overline{y})}(\overline{x},\overline{y})}{\sup }{p}^{*}\left(w\right)\\ \le \underset{i\in I(\overline{x},\overline{y})}{\max }{\psi }_i(\overline{x},\overline{y})\left\{\underset{w\in F(\overline{x},\overline{y})\cap T_{P(\overline{x},\overline{y})\times Q(\overline{x},\overline{y})}(\overline{x},\overline{y})}{\sup }{p}_{j\left(i\right)}\right(w\left)\right\}\\ \le \underset{i\in I(\overline{x},\overline{y})}{\max }{c}_{p_{j\left(i\right)}}^2(\overline{x},\overline{y})<0.\end{array}$$(5)

A combination of (2)(2) $$\underset{v\in F(\overline{x},\overline{y})\cap T_{P(\overline{x},\overline{y})\times Q(\overline{x},\overline{y})}(\overline{x},\overline{y})}{\sup }{p}^{*}\left(v\right)\ge 0.$$(2) and (5)(5) $$\begin{array}{c}\underset{w\in F(\overline{x},\overline{y})\cap T_{P(\overline{x},\overline{y})\times Q(\overline{x},\overline{y})}(\overline{x},\overline{y})}{\sup }{p}^{*}\left(w\right)\\ \le \underset{i\in I(\overline{x},\overline{y})}{\max }{\psi }_i(\overline{x},\overline{y})\left\{\underset{w\in F(\overline{x},\overline{y})\cap T_{P(\overline{x},\overline{y})\times Q(\overline{x},\overline{y})}(\overline{x},\overline{y})}{\sup }{p}_{j\left(i\right)}\right(w\left)\right\}\\ \le \underset{i\in I(\overline{x},\overline{y})}{\max }{c}_{p_{j\left(i\right)}}^2(\overline{x},\overline{y})<0.\end{array}$$(5) gives a contradiction. This completes the proof of the theorem.

Acknowledgments

This work was supported by the Viet Nam Academy of Science and Technologies under the code NVCC01.14/19-19. The author is a member of the project “Joint study in Analysis and Geometr” conducted at the International Centre for Research and Training in Mathematics, Institute of Mathematics, VAST, ICRTM01–2019.05.