Some generalizations of common fixed point problems with applications

In three recent papers [1–3], by using some new concepts of generalized KKM mappings, the authors established common fixed point theorems for families of set-valued mappings in Hausdorff topological vector spaces. Recently, Agarwal et al. [4] established a common fixed point theorem for a family of self set-valued mappings on a compact and convex set in a locally convex topological vector space. As applications, an existence theorem of solutions for a variational inequality of Stampacchia type and some Ky Fan-type minimax inequalities were obtained.

It is well known that the equilibrium problems are unified models of several problems, namely, optimization problems, saddle point problems, variational inequalities, fixed point problems, Nash equilibrium problems etc. Recently, Luc [5] introduced a more general model of equilibrium problems, which is called a variational relation problem (in short, VR). The stability of the solution set of variational relation problems was studied in [6, 7]. Some various types of variational relation problems or systems of variational relation problems have been investigated in many recent papers (see [8–12]). Recently, Agarwal et al. [13] presented a unified approach in studying the existence of solutions for two types of variational relation problems, and Balaj and Lin [14] established existence criteria for the solutions of two very general types of variational relation problems (see also [15–22] for further studies of variational relation problems).

Motivated and inspired by research works mentioned above, in this paper, we establish a unified approach to study the existence of fixed points for common fixed point problems. As generalizations, mixed systems of common fixed point problems, systems of common fixed point problems, and common fixed point problems without convex assumptions are obtained.

In this short section, we recall some definitions and known results concerning set-valued mappings which will be needed throughout this paper.

Let X, Y, Z be three Hausdorff topological spaces. We adopt the following notations: for a set U and point x, ${\alpha }_1(x,U)$ and, respectively, ${\alpha }_2(x,U)$ means $\mathrm{\forall }x\in U$, and, respectively, $\mathrm{\exists }x\in U$. Denote $\hat{\alpha }$ by ${\hat{\alpha }}_1={\alpha }_2$ and ${\hat{\alpha }}_2={\alpha }_1$; for two sets A, B, ${\gamma }_1(A,B)$ and, respectively, ${\gamma }_2(A,B)$ means $A\subset B$, and, respectively, $A\cap B\ne \mathrm{\varnothing }$. Denote $\hat{\gamma }$ by ${\hat{\gamma }}_1={\gamma }_2$ and ${\hat{\gamma }}_2={\gamma }_1$. A set-valued mapping $F:X\rightrightarrows Y$ is said to be: (1) upper semicontinuous at $x\in X$ if, for any open subset O of Y with $O\supset F(x)$, there exists an open neighborhood $U(x)$ of x such that $O\supset F(x^{\prime })$ for any $x^{\prime }\in U(x)$; (2) upper semicontinuous on X, if it is upper semicontinuous at each $x\in X$; (3) lower semicontinuous at $x\in X$ if, for any open subset O of Y with $O\cap F(x)\ne \mathrm{\varnothing }$, there exists an open neighborhood $U(x)$ of x such that $O\cap F(x^{\prime })\ne \mathrm{\varnothing }$ for any $x^{\prime }\in U(x)$; (4 lower semicontinuous on X, if it is lower semicontinuous in each $x\in X$; (5) closed if $Graph(F)=\{(x,y)\in X\times Y|y\in F(x)\}$ is a closed subset of $X\times Y$.

Lemma 2.1 (Corollary 17.55 (Kakutani-Fan-Glicksberg fixed point theorem) of [23])

Let X be a nonempty, convex and compact subset of a locally convex topological linear space, and $F:X\rightrightarrows X$ be an upper semicontinuous set-valued mapping with nonempty convex compact values. Then there exists $x^{\ast }\in X$ such that $x^{\ast }\in F(x^{\ast })$.

Lemma 2.2 (Lemma 17.8, Theorem 17.10, Theorem 17.16, Theorem 17.19 of [23])

Then problem (CFP-α) has at least a solution.

Let $\{{\beta }_k|k=0,1,2,\dots ,n\}$ be the partition of unity subordinate to the open covering $\{U_0,U(y_k)|k=1,2,\dots ,n\}$ of X, i.e., $\{{\beta }_k|k=0,1,2,\dots ,n\}$ is a set of continuous functions with following properties: $0\le {\beta }_k(x)\le 1$, ${\sum }_{k=0}^n{\beta }_k(x)=1$, $\mathrm{\forall }x\in X$, $k=0,1,2,\dots ,n$; and if $x\notin U(y_k)$, for some $k\in \{1,\dots ,n\}$, then ${\beta }_k(x)=0$, and if $x\notin U_0$, then ${\beta }_0(x)=0$.

By (vii), there is $i_0\in I(x^{\ast })$ such that $\alpha (z,P(x^{\ast }))$, $z\in T(y_{i_0},z)$, which implies that $x^{\ast }\notin U(y_{i_0})$, i.e., ${\beta }_{i_0}(x^{\ast })=0$. It contradicts the fact that $i_0\in I(x^{\ast })$, that is, ${\beta }_{i_0}(x^{\ast })>0$. This completes the proof. □

Remark 3.1 By Proposition 3.3 of [14], when $\alpha ={\alpha }_1$, condition (iii) in Theorem 3.1 can be replaced by (a1) $P:X\rightrightarrows Z$ is lower semicontinuous; (b1) $\{z\in Z|z\in T(y,z)\}$ is closed in X for any $y\in Y$; (c1) $Q^{-1}(y)$ is open in X for any $y\in Y$. Thus we have the following theorem.

Theorem 3.2 Assume that the data of problem (CFP-${\alpha }_1$) satisfy the conditions (i), (ii), (iv)-(vi) of Theorem  3.1 and

(a1) $P:X\rightrightarrows Z$ is lower semicontinuous;

(b1) $\{z\in Z|z\in T(y,z)\}$ is closed in X for any $y\in Y$;

(c1) $Q^{-1}(y)$ is open in X for any $y\in Y$;

(d1) T is ${\alpha }_1$-$(F,P)$-KKM, i.e., for any finite set $\{y_1,\dots ,y_n\}$ of Y and any $x\in F(\mathit{co}\{y_1,\dots ,y_n\})$, there is $i_0\in \{1,\dots ,n\}$ such that $z\in T(y_{i_0},z)$ for any $z\in P(x)$.

Then problem (CFP-${\alpha }_1$) has at least a solution.

Remark 3.2 By Proposition 3.1 of [14], when $\alpha ={\alpha }_2$, condition (iii) in Theorem 3.1 can be replaced by (a2) $P:X\rightrightarrows Z$ is upper semicontinuous with nonempty compact values; (b2) $\{z\in Z|z\in T(y,z)\}$ is closed in X for any $y\in Y$; (c2) $Q^{-1}(y)$ is open in X for any $y\in Y$. Thus we have the following theorem.

Theorem 3.3 Assume that the data of problem (CFP-${\alpha }_2$) satisfy the conditions (i), (ii), (iv)-(vi) of Theorem  3.1 and

(a2) $P:X\rightrightarrows Z$ is upper semicontinuous with nonempty compact values;

(b2) $\{z\in Z|z\in T(y,z)\}$ is closed in X for any $y\in Y$;

(c2) $Q^{-1}(y)$ is open in X for any $y\in Y$;

(d2) T is ${\alpha }_2$-$(F,P)$-KKM, i.e., for any finite set $\{y_1,\dots ,y_n\}$ of Y and any $x\in F(\mathit{co}\{y_1,\dots ,y_n\})$, there is $i_0\in \{1,\dots ,n\}$ such that there is $z\in P(x)$ for which $z\in T(y_{i_0},z)$.

Then problem (CFP-${\alpha }_2$) has at least a solution.

If $X=Y$ and $F(x)=x$ for all $x\in X$, we have problem (CFP-α).

Then problem (CFP-α) has at least a solution.

When $X=Y=Z$, and $F(x)=x$, $P(x)=\{x\}$ for all $x\in X$, we obtain the following corollary.

Then there exists $\tilde{x}\in X$ such that $\tilde{x}\in S(\tilde{x})$ and $\tilde{x}\in {\bigcap }_{y\in Q(\tilde{x})}T(y,\tilde{x})$.

Remark 3.3 Theorems 3.1-3.4 generalize the results of [1–4]. When $X=Y$ and $P(x)=F(x)=x$ for all $x\in X$, and $S(x)=Q(x)=X$ for any $x\in X$, our Theorems 3.1-3.4 reduce to the results of [1–4].

Then problem (MSCFP) has at least a solution.

By (iii), (iv), and Remarks 3.1, 3.2, $A^{-1}(u)$, $B^{-1}(v)$ are open in $X\times Y$ for any $(u,v)\in X\times Y$.

Suppose there exists $(x,y)\in X\times Y$ such that $x\in \mathit{co}A(x,y)$, then there is a finite subset $\{u_1,\dots ,u_n\}$ of $A(x,y)$ such that $x\in \mathit{co}\{u_1,\dots ,u_n\}$. By (vi), there is $i_0\in \{1,\dots ,n\}$ such that $z\in T(u_{i_0},z)$ for any $z\in P(x,y)$, which contradicts the fact that $u_i\in A(x,y)$ for any $i\in \{1,\dots ,n\}$. Hence $x\notin \mathit{co}A(x,y)$ for any $(x,y)\in X\times Y$.

Suppose there exists $(x,y)\in X\times Y$ such that $y\in \mathit{co}B(x,y)$, then there is a finite subset $\{v_1,\dots ,v_n\}$ of $B(x,y)$ such that $y\in \mathit{co}\{v_1,\dots ,v_n\}$. By (vii), there is $i_0\in \{1,\dots ,n\}$ such that there exists $z\in P(x,y)$ for which $z\in F(v_{i_0},z)$, which contradicts the fact that $v_i\in B(x,y)$ for any $i\in \{1,\dots ,n\}$. Hence $y\notin \mathit{co}B(x,y)$ for any $(x,y)\in X\times Y$.

 □

Then problem (SCFP) has at least a solution.

By Lemma 2.1, there exists $x^{\ast }\in X$ such that $x^{\ast }\in F(x^{\ast })$, which implies that $x_i^{\ast }\in F_i(x^{\ast })$ for each $i\in I$. If $x\in W_{i_0}$ for some $i_0\in I$, $x_{i_0}^{\ast }=f_{i_0}(x^{\ast })\in \mathit{co}{G}_{i_0}(x^{\ast })$. Then there exists a finite set $\{y_{i_{0}1},\dots ,y_{i_{0}n}\}\subset G_{i_0}(x^{\ast })$ such that $x_{i_0}^{\ast }\in \mathit{co}\{y_{i_{0}1},\dots ,y_{i_{0}n}\}$. By (v), there is $j_0\in \{1,\dots ,n\}$ such that $\alpha (z_{i_0},P_{i_0}(x^{\ast }))$, $z_{i_0}\in T_{i_0}(y_{i_0{j}_0},z_{i_0})$, which contradicts the fact that $y_{i_0{j}_0}\in G_{i_0}(x^{\ast })$. Therefore, $x^{\ast }\notin W_i$ for any $i\in I$, which implies that, for each $i\in I$, $x_i^{\ast }\in S_i(x^{\ast })$ and, for any $y_i\in Q_i(x^{\ast })$, $\alpha (z_i,P_i(x^{\ast }))$, $z_i\in T_i(y_i,z_i)$. This completes the proof. □

As a generalization of Theorem 3.1, we derive the following existence result for the solution of problem (CFP-α) without convex assumptions.

(v1) for any $\lambda =({\lambda }_1,\dots ,{\lambda }_n)\in {\mathrm{\Delta }}_n$, there exists $i\in J(\lambda )$ such that $\alpha (z,P({\varphi }_n(\lambda )))$, $z\in T(y_i,z)$;

By (ii) and (iv), $G(y)$ is closed for any $y\in X$.

 □

By Remarks 3.1 and 3.2, we have the following results.

Theorem 3.8 If (iv) of Theorem  3.7 is replaced by

(a3) $P:X\rightrightarrows Z$ is lower semicontinuous;

(b3) $\{z\in Z|z\in T(y,z)\}$ is closed in X for any $y\in Y$, and $Q^{-1}(y)$ is open in X for any $y\in X$.

Theorem 3.9 If (iv) of Theorem  3.7 is replaced by

(a4) $P:X\rightrightarrows Z$ is upper semicontinuous with nonempty compact values;

(b4) $\{z\in Z|z\in T(y,z)\}$ is closed in X for any $y\in Y$, and $Q^{-1}(y)$ is open in X for any $y\in X$.

Then problem (CFP-${\alpha }_2$) has at least a solution, i.e., there exists $\tilde{x}\in X$ such that $\tilde{x}\in S(\tilde{x})$ and, for any $y\in Q(\tilde{x})$, there is $z\in P(\tilde{x})$ for which $z\in T(y,z)$.

Remark 3.4 In [1–4], convex assumptions or the KKM property played an important role in the proofs of common fixed points. In Theorems 3.7-3.9, the existence of common fixed points does not depend on convex assumptions.

Remark 3.5 This paper extends the research on common fixed point problems. The classical common fixed point problem (see [1–4]) is a special case of problem (CFP-α). Moreover, we introduce the system of common fixed points, and common fixed point problems without convex assumptions are obtained.