Algorithms for treating equilibrium and fixed point problems

Abstract

In this paper, a common solution problem is investigated based on a projection algorithm. Strong convergence theorems for common solutions of a system of equilibrium problems and a family of asymptotically quasi-ϕ-nonexpansive mappings are established in a uniformly smooth and strictly convex Banach space which also enjoys the Kadec-Klee property.

1 Introduction

Bifunction equilibrium problems, which include many important problems in nonlinear analysis and optimization, such as the Nash equilibrium problem, variational inequalities, complementarity problems, vector optimization problems, fixed point problems, saddle point problems and game theory, recently have been studied as an effective and powerful tool for studying many real world problems which arise in economics, finance, image reconstruction, ecology, transportation, and network; see [1–23] and the references therein. The theory of fixed points as an important branch of functional analysis is a bridge between nonlinear functional analysis and optimization. Indeed, lots of problems arising in economics, engineering, and physics can be studied by fixed point techniques. The study of fixed point approximation algorithms for computing fixed points constitutes now a topic of intensive research efforts. Many well-known problems can be studied by using algorithms which are iterative in their nature. As an example, in computer tomography with limited data, each piece of information implies the existence of a convex set in which the required solution lies. The problem of finding a point in the intersection of these convex subsets is then of crucial interest, and it cannot be usually solved directly. Therefore, an iterative algorithm must be used to approximate such a point. The well known convex feasibility problem, which captures applications in various disciplines such as image restoration and radiation therapy treatment planning, is to find a point in the intersection of common fixed point sets of a family of nonlinear mappings. Krasnoselskii-Mann iteration, which is also known as a one-step iteration, is a classic algorithm to study fixed points of nonlinear operators. However, Krasnoselskii-Mann iteration only enjoys weak convergence for nonexpansive mappings; see [24] and the references therein. There are a lot of real world problems which exist in infinite dimension spaces. In such problems, strong convergence or norm convergence is often much more desirable than weak convergence. To guarantee the strong convergence of Krasnoselskii-Mann iteration, many authors use different regularization methods. The projection technique which was first introduced by Haugazeau [25] has been considered for the approximation of fixed points of nonexpansive mappings. The advantage of projection methods is that strong convergence of iterative sequences can be guaranteed without any compact assumptions.

In this paper, we study a common solution problem based on a projection algorithm. Strong convergence theorems of common solutions are established in a uniformly smooth and strictly convex Banach space which also enjoys the Kadec-Klee property.

2 Preliminaries

Let E be a real Banach space, $E^{\ast }$ be the dual space of E and C be a nonempty subset of E. Let f be a bifunction from $C\times C$ to ℝ, where ℝ denotes the set of real numbers. Recall the following equilibrium problem: find $\overline{x}\in C$ such that

$$f(\overline{x}y)\ge 0,\mathrm{\forall }y\in C.$$

(2.1)

We use $\mathit{EP}(f)$ to denote the solution set of equilibrium problem (2.1). That is,

$$\mathit{EP}(f)=\{x\in C:f(x,y)\ge 0,\mathrm{\forall }y\in C\}.$$

Given a mapping $A:C\to E^{\ast }$, let

$$f(x,y)=〈Ax,y-x〉,\mathrm{\forall }x,y\in C.$$

Then $\overline{x}\in \mathit{EP}(f)$ iff $\overline{x}$ is a solution of the following variational inequality: find $\overline{x}$ such that

$$〈A\overline{x}y-\overline{x}〉\ge 0,\mathrm{\forall }y\in C.$$

(2.2)

In order to study the solution of problem (2.1), we assume that f satisfies the following conditions:

• (A1) $f(x,x)=0$, $\mathrm{\forall }x\in C$;

• (A2) f is monotone, i.e., $f(x,y)+f(y,x)\le 0$, $\mathrm{\forall }x,y\in C$;

• (A3)
  $$\underset{t\downarrow 0}{lim\hspace{0.17em}sup}f(tz+(1-t)x,y)\le f(x,y),\mathrm{\forall }x,y,z\in C;$$

• (A4) for each $x\in C$, $y\mapsto f(x,y)$ is convex and weakly lower semi-continuous.

Recall that a Banach space E is said to be strictly convex iff $\parallel \frac{x+y}{2}\parallel <1$ for all $x,y\in E$ with $\parallel x\parallel =\parallel y\parallel =1$ and $x\ne y$. It is said to be uniformly convex iff ${lim}_{n\to \mathrm{\infty }}\parallel x_n-y_n\parallel =0$ for any two sequences $\{x_n\}$ and $\{y_n\}$ in E such that $\parallel x_n\parallel =\parallel y_n\parallel =1$ and ${lim}_{n\to \mathrm{\infty }}\parallel \frac{x_n+y_n}{2}\parallel =1$. Let $U_E=\{x\in E:\parallel x\parallel =1\}$ be the unit sphere of E. Then the Banach space E is said to be smooth iff

$$\underset{t\to 0}{lim}\frac{\parallel x+ty\parallel -\parallel x\parallel }{t}$$

exists for each $x,y\in U_E$. It is also said to be uniformly smooth iff the above limit is attained uniformly for $x,y\in U_E$. It is well known that if E is uniformly smooth, then J is uniformly norm-to-norm continuous on each bounded subset of E. It is also well known that E is uniformly smooth if and only if $E^{\ast }$ is uniformly convex.

Recall that a Banach space E enjoys the Kadec-Klee property if for any sequence $\{x_n\}\subset E$, and $x\in E$ with $x_n\rightharpoonup x$, and $\parallel x_n\parallel \to \parallel x\parallel$, then $\parallel x_n-x\parallel \to 0$ as $n\to \mathrm{\infty }$. It is well known that if E is a uniformly convex Banach space, then E enjoys the Kadec-Klee property. Let $T:C\to C$ be a mapping. In this paper, we use $F(T)$ to denote the fixed point set of T. T is said to be closed if for any sequence $\{x_n\}\subset C$ such that ${lim}_{n\to \mathrm{\infty }}{x}_n=x_0$ and ${lim}_{n\to \mathrm{\infty }}T{x}_n=y_0$, then $T{x}_0=y_0$. In this paper, we use → and ⇀ to denote strong convergence and weak convergence, respectively.

Recall that the normalized duality mapping J from E to $2^{E^{\ast }}$ is defined by

$$Jx=\{f^{\ast }\in E^{\ast }:〈x,f^{\ast }〉={\parallel x\parallel }^2={\parallel f^{\ast }\parallel }^2\},$$

where $〈\cdot ,\cdot 〉$ denotes the generalized duality pairing. Next, we assume that E is a smooth Banach space. Consider the functional defined by

$$\varphi (x,y)={\parallel x\parallel }^2-2〈x,Jy〉+{\parallel y\parallel }^2,\mathrm{\forall }x,y\in E.$$

Observe that, in a Hilbert space H, the equality is reduced to $\varphi (x,y)={\parallel x-y\parallel }^2$, $x,y\in H$. As we all know, if C is a nonempty closed convex subset of a Hilbert space H and $P_C:H\to C$ is the metric projection of H onto C, then $P_C$ is nonexpansive. This fact actually characterizes Hilbert spaces and, consequently, it is not available in more general Banach spaces. In this connection, Alber [26] recently introduced a generalized projection operator ${\mathrm{\Pi }}_C$ in a Banach space E which is an analogue of the metric projection $P_C$ in Hilbert spaces. Recall that the generalized projection ${\mathrm{\Pi }}_C:E\to C$ is a map that assigns to an arbitrary point $x\in E$ the minimum point of the functional $\varphi (x,y)$, that is, ${\mathrm{\Pi }}_{C}x=\overline{x}$, where $\overline{x}$ is the solution to the minimization problem

$$\varphi (\overline{x},x)=\underset{y\in C}{min}\varphi (y,x).$$

Existence and uniqueness of the operator ${\mathrm{\Pi }}_C$ follows from the properties of the functional $\varphi (x,y)$ and strict monotonicity of the mapping J. In Hilbert spaces, ${\mathrm{\Pi }}_C=P_C$. It is obvious from the definition of a function ϕ that

$${(\parallel x\parallel -\parallel y\parallel )}^2\le \varphi (x,y)\le {(\parallel y\parallel +\parallel x\parallel )}^2,\mathrm{\forall }x,y\in E,$$

(2.3)

and

$$\varphi (x,y)=\varphi (x,z)+\varphi (z,y)+2〈x-z,Jz-Jy〉,\mathrm{\forall }x,y,z\in E.$$

(2.4)

Remark 2.1 If E is a reflexive, strictly convex, and smooth Banach space, then $\varphi (x,y)=0$ if and only if $x=y$.

Recall that a point p in C is said to be an asymptotic fixed point of a mapping T [27] iff C contains a sequence $\{x_n\}$ which converges weakly to p such that ${lim}_{n\to \mathrm{\infty }}\parallel x_n-T{x}_n\parallel =0$. The set of asymptotic fixed points of T will be denoted by $\tilde{F}(T)$.

Recall that a mapping T is said to be relatively nonexpansive iff

$$\tilde{F}(T)=F(T)\ne \mathrm{\varnothing },\varphi (p,Tx)\le \varphi (p,x),\mathrm{\forall }x\in C,\mathrm{\forall }p\in F(T).$$

Recall that a mapping T is said to be relatively asymptotically nonexpansive iff

$$\tilde{F}(T)=F(T)\ne \mathrm{\varnothing },\varphi (p,T^{n}x)\le (1+{\mu }_n)\varphi (p,x),\mathrm{\forall }x\in C,\mathrm{\forall }p\in F(T),\mathrm{\forall }n\ge 1,$$

where $\{{\mu }_n\}\subset [0,\mathrm{\infty })$ is a sequence such that ${\mu }_n\to 0$ as $n\to \mathrm{\infty }$.

Remark 2.2 The class of relatively asymptotically nonexpansive mappings was considered in [28] and [29]; see the references therein.

Recall that a mapping T is said to be quasi-ϕ-nonexpansive iff

$$F(T)\ne \mathrm{\varnothing },\varphi (p,Tx)\le \varphi (p,x),\mathrm{\forall }x\in C,\mathrm{\forall }p\in F(T).$$

Recall that a mapping T is said to be asymptotically quasi-ϕ-nonexpansive iff there exists a sequence $\{{\mu }_n\}\subset [0,\mathrm{\infty })$ with ${\mu }_n\to 0$ as $n\to \mathrm{\infty }$ such that

$$F(T)\ne \mathrm{\varnothing },\varphi (p,T^{n}x)\le (1+{\mu }_n)\varphi (p,x),\mathrm{\forall }x\in C,\mathrm{\forall }p\in F(T),\mathrm{\forall }n\ge 1.$$

Remark 2.3 The class of asymptotically quasi-ϕ-nonexpansive mappings was considered in Zhou et al. [30] and Qin et al. [31]; see also [32].

Remark 2.4 The class of quasi-ϕ-nonexpansive mappings and the class of asymptotically quasi-ϕ-nonexpansive mappings are more general than the class of relatively nonexpansive mappings and the class of relatively asymptotically nonexpansive mappings. Quasi-ϕ-nonexpansive mappings and asymptotically quasi-ϕ-nonexpansive mappings do not require the restriction $F(T)=\tilde{F}(T)$.

Remark 2.5 The class of quasi-ϕ-nonexpansive mappings and the class of asymptotically quasi-ϕ-nonexpansive mappings are generalizations of the class of quasi-nonexpansive mappings and the class of asymptotically quasi-nonexpansive mappings in Banach spaces.

In order to get our main results, we also need the following lemmas.

Lemma 2.6 [26]

Let E be a reflexive, strictly convex, and smooth Banach space, let C be a nonempty, closed, and convex subset of E, and let $x\in E$. Then

$$\varphi (y,{\mathrm{\Pi }}_{C}x)+\varphi ({\mathrm{\Pi }}_{C}x,x)\le \varphi (y,x),\mathrm{\forall }y\in C.$$

Lemma 2.7 [26]

Let C be a nonempty, closed, and convex subset of a smooth Banach space E, and let $x\in E$. Then $x_0={\mathrm{\Pi }}_{C}x$ if and only if

$$〈x_0-y,Jx-J{x}_0〉\ge 0,\mathrm{\forall }y\in C.$$

Lemma 2.8 [33]

Let E be a smooth and uniformly convex Banach space, and let $r>0$. Then there exists a strictly increasing, continuous, and convex function $g:[0,2r]\to R$ such that $g(0)=0$ and

$${\parallel tx+(1-t)y\parallel }^2\le t{\parallel x\parallel }^2+(1-t){\parallel y\parallel }^2-t(1-t)g(\parallel x-y\parallel )$$

for all $x,y\in B_r=\{x\in E:\parallel x\parallel \le r\}$ and $t\in [0,1]$.

Lemma 2.9 Let C be a closed convex subset of a smooth, strictly convex, and reflexive Banach space E. Let f be a bifunction from $C\times C$ to ℝ satisfying (A1)-(A4). Let $r>0$ and $x\in E$. Then

• (a) [1]There exists $z\in C$ such that

  $$f(z,y)+\frac{1}{r}〈y-z,Jz-Jx〉\ge 0,\mathrm{\forall }y\in C.$$

• (b) [31, 34]Define a mapping $T_r:E\to C$ by

  $$S_{r}x=\{z\in C:f(z,y)+\frac{1}{r}〈y-z,Jz-Jx〉,\mathrm{\forall }y\in C\}.$$

  Then the following conclusions hold:

  • (1) $S_r$ is single-valued;

  • (2) $S_r$ is a firmly nonexpansive-type mapping, i.e., for all $x,y\in E$,

    $$〈S_{r}x-S_{r}y,J{S}_{r}x-J{S}_{r}y〉\le 〈S_{r}x-S_{r}y,Jx-Jy〉;$$

  • (3) $F(S_r)=\mathit{EP}(f)$;

  • (4) $S_r$ is quasi-ϕ-nonexpansive;

  • (5)
    $$\varphi (q,S_{r}x)+\varphi (S_{r}x,x)\le \varphi (q,x),\mathrm{\forall }q\in F(S_r);$$

  • (6) $\mathit{EP}(f)$ is closed and convex.

Lemma 2.10 [35]

Let E be a uniformly smooth and strictly convex Banach space which also enjoys the Kadec-Klee property. Let C be a nonempty closed and convex subset of E. Let $T:C\to C$ be a closed asymptotically quasi-ϕ-nonexpansive mapping. Then $F(T)$ is a closed convex subset of C.

3 Main results

Theorem 3.1 Let E be a uniformly smooth and strictly convex Banach space which also enjoys the Kadec-Klee property. Let C be a nonempty closed and convex subset of E. Let Λ be an index set. Let $f_i$ be a bifunction from $C\times C$ to ℝ satisfying (A1)-(A4), and let $T_i:C\to C$ be an asymptotically quasi-ϕ-nonexpansive mapping for every $i\in \mathrm{\Lambda }$. Assume that $T_i$ is closed and uniformly asymptotically regular on C for every $i\in \mathrm{\Lambda }$. Assume that ${\bigcap }_{i\in \mathrm{\Lambda }}F(T_i)\cap {\bigcap }_{i\in \mathrm{\Lambda }}\mathit{EF}(f_i)$ is nonempty and bounded. Let $\{x_n\}$ be a sequence generated in the following manner:

$$\{\begin{array}{c}{x}_0\in E\mathit{\text{chosen arbitrarily}},\hfill \\ C_{(1,i)}=C,\hfill \\ C_1={\bigcap }_{i\in \mathrm{\Lambda }}{C}_{(1,i)},\hfill \\ x_1={\mathrm{\Pi }}_{C_1}{x}_0,\hfill \\ y_{(n,i)}=J^{-1}({\alpha }_{(n,i)}J{x}_n+(1-{\alpha }_{(n,i)})J{T}_i^n{x}_n),\hfill \\ u_{(n,i)}\in C\mathit{\text{ such that }}{f}_i(u_{(n,i)},y)+\frac{1}{r_{(n,i)}}〈y-u_{(n,i)},J{u}_{(n,i)}-J{y}_{(n,i)}〉\ge 0,\mathrm{\forall }y\in C,\hfill \\ C_{(n+1,i)}=\{z\in C_{(n,i)}:\varphi (z,u_{(n,i)})\le \varphi (z,x_n)+{\mu }_{(n,i)}{M}_{(n,i)}\},\hfill \\ C_{n+1}={\bigcap }_{i\in \mathrm{\Lambda }}{C}_{(n+1,i)},\hfill \\ x_{n+1}={\mathrm{\Pi }}_{C_{n+1}}{x}_1,\hfill \end{array}$$

where $M_{(n,i)}=sup\{\varphi (p,x_n):p\in {\bigcap }_{i\in \mathrm{\Lambda }}F(T_i)\cap {\bigcap }_{i\in \mathrm{\Lambda }}\mathit{EF}(f_i)\}$, $\{{\alpha }_{(n,i)}\}$ is a real sequence in $[0,1]$ such that ${lim\hspace{0.17em}inf}_{n\to \mathrm{\infty }}{\alpha }_{(n,i)}(1-{\alpha }_{(n,i)})>0$, and $\{r_{(n,i)}\}$ is a real sequence in $[a_i,\mathrm{\infty })$, where $\{a_i\}$ is a positive real number sequence, for every $i\in \mathrm{\Lambda }$. Then the sequence $\{x_n\}$ converges strongly to ${\mathrm{\Pi }}_{{\bigcap }_{i\in \mathrm{\Lambda }}F(T_i)\cap {\bigcap }_{i\in \mathrm{\Lambda }}\mathit{EF}(f_i)}{x}_1$.

Proof We divide the proof into six steps.

Step 1. We prove that ${\bigcap }_{i\in \mathrm{\Lambda }}F(T_i)\cap {\bigcap }_{i\in \mathrm{\Lambda }}\mathit{EF}(f_i)$ is closed and convex.

In the light of Lemma 2.9 and 2.10, we easily find the conclusion.

Step 2. We prove that $C_n$ is closed and convex.

To show that $C_n$ is closed and convex, it suffices to show that for each fixed but arbitrary $i\in \mathrm{\Lambda }$, $C_{(n,i)}$ is closed and convex. This can be proved by induction on n. It is obvious that $C_{(1,i)}=C$ is closed and convex. Assume that $C_{(k,i)}$ is closed and convex for some $k\ge 1$. For $z_1,z_2\in C_{(k+1,i)}$, we see that $z_1,z_2\in C_{(k,i)}$. It follows that $z=t{z}_1+(1-t)z_2\in C_{(k,i)}$, where $t\in (0,1)$. Notice that

$$\varphi (z_1,u_{(k,i)})\le \varphi (z_1,x_k)+{\mu }_{(k,i)}{M}_{(k,i)}$$

and

$$\varphi (z_2,u_{(k,i)})\le \varphi (z_2,x_k)+{\mu }_{(k,i)}{M}_{(k,i)}.$$

The above inequalities are equivalent to

$$2〈z_1,J{x}_k-J{u}_{(k,i)}〉\le {\parallel x_k\parallel }^2-{\parallel u_{(k,i)}\parallel }^2+{\mu }_{(k,i)}{M}_{(k,i)}$$

(3.1)

and

$$2〈z_2,J{x}_k-J{u}_{(k,i)}〉\le {\parallel x_k\parallel }^2-{\parallel u_{(k,i)}\parallel }^2+{\mu }_{(k,i)}{M}_{(k,i)}.$$

(3.2)

Multiplying t and $(1-t)$ on the both sides of (3.1) and (3.2), respectively, yields that

$$2〈z,J{x}_k-J{u}_{(k,i)}〉\le {\parallel x_k\parallel }^2-{\parallel u_{(k,i)}\parallel }^2+{\mu }_{(k,i)}{M}_{(k,i)}.$$

That is,

$$\varphi (z,u_{(k,i)})\le \varphi (z,x_k)+{\mu }_{(k,i)}{M}_{(k,i)},$$

where $z\in C_{(k,i)}$. This finds that $C_{(k+1,i)}$ is closed and convex. We conclude that $C_{(n,i)}$ is closed and convex. This in turn implies that $C_n={\bigcap }_{i\in \mathrm{\Lambda }}{C}_{(n,i)}$ is closed and convex. This implies that ${\mathrm{\Pi }}_{C_{n+1}}{x}_1$ is well defined.

Step 3. We prove that ${\bigcap }_{i\in \mathrm{\Lambda }}F(T_i)\cap {\bigcap }_{i\in \mathrm{\Lambda }}\mathit{EF}(f_i)\subset C_n$.

${\bigcap }_{i\in \mathrm{\Lambda }}F(T_i)\cap {\bigcap }_{i\in \mathrm{\Lambda }}\mathit{EF}(f_i)\subset C_1=C$ is clear. Suppose that ${\bigcap }_{i\in \mathrm{\Lambda }}F(T_i)\cap {\bigcap }_{i\in \mathrm{\Lambda }}\mathit{EF}(f_i)\subset C_{(k,i)}$ for some positive integer k. For any $w\in {\bigcap }_{i\in \mathrm{\Lambda }}F(T_i)\cap {\bigcap }_{i\in \mathrm{\Lambda }}\mathit{EF}(f_i)\subset C_{(k,i)}$, we see that

$$\begin{array}{r}\varphi (w,u_{(k,i)})\\ =\varphi (w,S_{r_{(k,i)}}{y}_{(k,i)})\\ \le \varphi (w,y_{(k,i)})\\ =\varphi (w,J^{-1}({\alpha }_{(k,i)}J{x}_k+(1-{\alpha }_{(k,i)})J{T}_i^k{x}_k))\\ ={\parallel w\parallel }^2-2〈w,{\alpha }_{(k,i)}J{x}_k+(1-{\alpha }_{(k,i)})J{T}_i^k{x}_k〉\\ +{\parallel {\alpha }_{(k,i)}J{x}_k+(1-{\alpha }_{(k,i)})J{T}_i^k{x}_k\parallel }^2\\ \le {\parallel w\parallel }^2-2{\alpha }_{(k,i)}〈w,J{x}_k〉-2(1-{\alpha }_{(k,i)})〈w,J{T}_i^k{x}_k〉\\ +{\alpha }_{(k,i)}{\parallel x_k\parallel }^2+(1-{\alpha }_{(k,i)}){\parallel T_i^k{x}_k\parallel }^2\\ ={\alpha }_{(k,i)}\varphi (w,x_k)+(1-{\alpha }_{(k,i)})\varphi (w,T_i^k{x}_k)\\ \le {\alpha }_{(k,i)}\varphi (w,x_k)+(1-{\alpha }_{(k,i)})\varphi (w,x_k)+(1-{\alpha }_{(k,i)}){\mu }_{(k,i)}\varphi (w,x_k)\\ \le \varphi (w,x_k)+{\mu }_{(k,i)}\varphi (w,x_k),\end{array}$$

(3.3)

which shows that $w\in C_{(k+1,i)}$. This implies that ${\bigcap }_{i\in \mathrm{\Lambda }}F(T_i)\cap {\bigcap }_{i\in \mathrm{\Lambda }}\mathit{EF}(f_i)\subset C_{(n,i)}$. This in turn implies that ${\bigcap }_{i\in \mathrm{\Lambda }}F(T_i)\cap {\bigcap }_{i\in \mathrm{\Lambda }}\mathit{EF}(f_i)\subset {\bigcap }_{i\in \mathrm{\Lambda }}{C}_{(n,i)}$. This completes the proof that ${\bigcap }_{i\in \mathrm{\Lambda }}F(T_i)\cap {\bigcap }_{i\in \mathrm{\Lambda }}\mathit{EF}(f_i)\subset C_n$.

Step 4. We prove that the sequence $\{x_n\}$ is bounded.

In the light of the construction $x_n={\mathrm{\Pi }}_{C_n}{x}_1$, we find from Lemma 2.7 that $〈x_n-z,J{x}_1-J{x}_n〉\ge 0$ for any $z\in C_n$.

Since ${\bigcap }_{i\in \mathrm{\Lambda }}F(T_i)\cap {\bigcap }_{i\in \mathrm{\Lambda }}\mathit{EF}(f_i)\subset C_n$, we find that

$$〈x_n-w,J{x}_1-J{x}_n〉\ge 0,\mathrm{\forall }w\in \bigcap _{i\in \mathrm{\Lambda }}F(T_i)\cap \bigcap _{i\in \mathrm{\Lambda }}\mathit{EF}(f_i).$$

(3.4)

On the other hand, we find from Lemma 2.6 that

$$\begin{array}{rl}\varphi (x_n,x_1)& \le \varphi ({\mathrm{\Pi }}_{{\bigcap }_{i\in \mathrm{\Lambda }}F(T_i)\cap {\bigcap }_{i\in \mathrm{\Lambda }}\mathit{EF}(f_i)}{x}_1,x_1)-\varphi ({\mathrm{\Pi }}_{{\bigcap }_{i\in \mathrm{\Lambda }}F(T_i)\cap {\bigcap }_{i\in \mathrm{\Lambda }}\mathit{EF}(f_i)}{x}_1,x_n)\\ \le \varphi ({\mathrm{\Pi }}_{{\bigcap }_{i\in \mathrm{\Lambda }}F(T_i)\cap {\bigcap }_{i\in \mathrm{\Lambda }}\mathit{EF}(f_i)}{x}_1,x_1).\end{array}$$

This implies that the sequence $\{\varphi (x_n,x_1)\}$ is bounded. It follows from (2.3) that the sequence $\{x_n\}$ is also bounded. Since the space is reflexive, we may assume that $x_n\rightharpoonup \overline{x}$.

Step 5. We prove that $\overline{x}\in {\bigcap }_{i\in \mathrm{\Lambda }}F(T_i)\cap {\bigcap }_{i\in \mathrm{\Lambda }}\mathit{EF}(f_i)$.

Since $C_n$ is closed and convex, we find that $\overline{x}\in C_n$. This implies that $\varphi (x_n,x_1)\le \varphi (\overline{x},x_1)$. On the other hand, we see from the weak lower semicontinuity of the norm that

$$\begin{array}{rl}\varphi (\overline{x},x_1)& ={\parallel \overline{x}\parallel }^2-2〈\overline{x},J{x}_1〉+{\parallel x_1\parallel }^2\\ \le \underset{n\to \mathrm{\infty }}{lim\hspace{0.17em}inf}({\parallel x_n\parallel }^2-2〈x_n,J{x}_1〉+{\parallel x_1\parallel }^2)\\ =\underset{n\to \mathrm{\infty }}{lim\hspace{0.17em}inf}\varphi (x_n,x_1)\\ \le \underset{n\to \mathrm{\infty }}{lim\hspace{0.17em}sup}\varphi (x_n,x_1)\\ \le \varphi (\overline{x},x_1),\end{array}$$

which implies that ${lim}_{n\to \mathrm{\infty }}\varphi (x_n,x_1)=\varphi (\overline{x},x_1)$. Hence, we have ${lim}_{n\to \mathrm{\infty }}\parallel x_n\parallel =\parallel \overline{x}\parallel$. In view of the Kadec-Klee property of E, we find that $x_n\to \overline{x}$ as $n\to \mathrm{\infty }$. Since $x_n={\mathrm{\Pi }}_{C_n}{x}_1$, and $x_{n+1}={\mathrm{\Pi }}_{C_{n+1}}{x}_1\in C_{n+1}\subset C_n$, we find that $\varphi (x_n,x_1)\le \varphi (x_{n+1},x_1)$. This shows that $\{\varphi (x_n,x_1)\}$ is nondecreasing. We find from its boundedness that ${lim}_{n\to \mathrm{\infty }}\varphi (x_n,x_1)$ exists. It follows that

$$\begin{array}{rl}\varphi (x_{n+1},x_n)& =\varphi (x_{n+1},{\mathrm{\Pi }}_{C_n}{x}_1)\le \varphi (x_{n+1},x_1)-\varphi ({\mathrm{\Pi }}_{C_n}{x}_1,x_1)\\ =\varphi (x_{n+1},x_1)-\varphi (x_n,x_1).\end{array}$$

This implies that

$$\underset{n\to \mathrm{\infty }}{lim}\varphi (x_{n+1},x_n)=0.$$

(3.5)

In the light of $x_{n+1}={\mathrm{\Pi }}_{C_{n+1}}{x}_1\in C_{n+1}$, we find that

$$\varphi (x_{n+1},u_{(n,i)})\le \varphi (x_{n+1},x_n)+{\mu }_{(n,i)}{M}_{(n,i)}.$$

This implies from (3.5) that

$$\underset{n\to \mathrm{\infty }}{lim}\varphi (x_{n+1},u_{(n,i)})=0.$$

(3.6)

In view of (2.3), we see that ${lim}_{n\to \mathrm{\infty }}(\parallel x_{n+1}\parallel -\parallel u_{(n,i)}\parallel )=0$. This implies that

$$\underset{n\to \mathrm{\infty }}{lim}\parallel u_{(n,i)}\parallel =\parallel \overline{x}\parallel .$$

That is,

$$\underset{n\to \mathrm{\infty }}{lim}\parallel J{u}_{(n,i)}\parallel =\underset{n\to \mathrm{\infty }}{lim}\parallel u_{(n,i)}\parallel =\parallel \overline{x}\parallel =\parallel J\overline{x}\parallel .$$

(3.7)

This implies that $\{J{u}_{(n,i)}\}$ is bounded. Note that both E and $E^{\ast }$ are reflexive. We may assume that $J{u}_{(n,i)}\rightharpoonup u^{(\ast ,i)}\in E^{\ast }$. In view of the reflexivity of E, we see that $J(E)=E^{\ast }$. This shows that there exists an element $u^i\in E$ such that $J{u}^i=u^{(\ast ,i)}$. It follows that

$$\begin{array}{rl}\varphi (x_{n+1},u_{(n,i)})& ={\parallel x_{n+1}\parallel }^2-2〈x_{n+1},J{u}_{(n,i)}〉+{\parallel u_{(n,i)}\parallel }^2\\ ={\parallel x_{n+1}\parallel }^2-2〈x_{n+1},J{u}_{(n,i)}〉+{\parallel J{u}_{(n,i)}\parallel }^2.\end{array}$$

Taking ${lim\hspace{0.17em}inf}_{n\to \mathrm{\infty }}$ on the both sides of the equality above from (3.5) yields that

$$\begin{array}{rl}0& \ge {\parallel \overline{x}\parallel }^2-2〈\overline{x},u^{(\ast ,i)}〉+{\parallel u^{(\ast ,i)}\parallel }^2\\ ={\parallel \overline{x}\parallel }^2-2〈\overline{x},J{u}^i〉+{\parallel J{u}^i\parallel }^2\\ ={\parallel \overline{x}\parallel }^2-2〈\overline{x},J{u}^i〉+{\parallel u^i\parallel }^2\\ =\varphi (\overline{x},u^i).\end{array}$$

That is, $\overline{x}=u^i$, which in turn implies that $u^{(\ast ,i)}=J\overline{x}$. It follows that $J{u}_{(n,i)}\rightharpoonup J\overline{x}\in E^{\ast }$. Since $E^{\ast }$ enjoys the Kadec-Klee property, we obtain from (3.7) that ${lim}_{n\to \mathrm{\infty }}J{u}_{(n,i)}=J\overline{x}$. Since $J^{-1}:E^{\ast }\to E$ is demicontinuous and E enjoys the Kadec-Klee property, we obtain that $u_{(n,i)}\to \overline{x}$, as $n\to \mathrm{\infty }$. Note that $\parallel x_n-u_{(n,i)}\parallel \le \parallel x_n-\overline{x}\parallel +\parallel \overline{x}-u_{(n,i)}\parallel$. It follows that

$$\underset{n\to \mathrm{\infty }}{lim}\parallel x_n-u_{(n,i)}\parallel =0.$$

(3.8)

On the other hand, we have

$$\begin{array}{rl}\varphi (w,x_n)-\varphi (w,u_{(n,i)})& ={\parallel x_n\parallel }^2-{\parallel u_{(n,i)}\parallel }^2-2〈w,J{x}_n-J{u}_{(n,i)}〉\\ \le \parallel x_n-u_{(n,i)}\parallel (\parallel x_n\parallel +\parallel u_{(n,i)}\parallel )+2\parallel w\parallel \parallel J{x}_n-J{u}_{(n,i)}\parallel .\end{array}$$

In view of (3.8), we find that

$$\underset{n\to \mathrm{\infty }}{lim}(\varphi (w,x_n)-\varphi (w,u_{(n,i)}))=0.$$

(3.9)

Since E is uniformly smooth, we know that $E^{\ast }$ is uniformly convex. In view of Lemma 2.8, we find that

$$\begin{array}{r}\varphi (w,u_{(n,i)})\\ =\varphi (w,S_{r_{(n,i)}}{y}_{(n,i)})\\ \le \varphi (w,y_{(n,i)})\\ =\varphi (w,J^{-1}({\alpha }_{(n,i)}J{x}_n+(1-{\alpha }_{(n,i)})J{T}_i^n{x}_n))\\ ={\parallel w\parallel }^2-2〈w,{\alpha }_{(n,i)}J{x}_n+(1-{\alpha }_{(n,i)})J{T}_i^n{x}_n〉+{\parallel {\alpha }_{(n,i)}J{x}_n+(1-{\alpha }_{(n,i)})J{T}_i^n{x}_n\parallel }^2\\ \le {\parallel w\parallel }^2-2{\alpha }_{(n,i)}〈w,J{x}_n〉-2(1-{\alpha }_{(n,i)})〈w,J{T}_i^n{x}_n〉\\ +{\alpha }_{(n,i)}{\parallel x_n\parallel }^2+(1-{\alpha }_{(n,i)}){\parallel T_i^n{x}_n\parallel }^2-{\alpha }_{(n,i)}(1-{\alpha }_{(n,i)})g\left(\parallel J{x}_n-J{T}_i^n{x}_n\parallel \right)\\ ={\alpha }_{(n,i)}\varphi (w,x_n)+(1-{\alpha }_{(n,i)})\varphi (w,T_i^n{x}_n)-{\alpha }_{(n,i)}(1-{\alpha }_{(n,i)})g\left(\parallel J{x}_n-J{T}_i^n{x}_n\parallel \right)\\ \le \varphi (w,x_n)+(1-{\alpha }_{(n,i)}){\mu }_{(n,i)}\varphi (w,x_n)-{\alpha }_{(n,i)}(1-{\alpha }_{(n,i)})g\left(\parallel J{x}_n-J{T}_i^n{x}_n\parallel \right)\\ \le \varphi (w,x_n)+{\mu }_{(n,i)}{M}_{(n,i)}-{\alpha }_{(n,i)}(1-{\alpha }_{(n,i)})g\left(\parallel J{x}_n-J{T}_i^n{x}_n\parallel \right).\end{array}$$

This implies that

$${\alpha }_{(n,i)}(1-{\alpha }_{(n,i)})g\left(\parallel J{x}_n-J{T}_i^n{x}_n\parallel \right)\le \varphi (w,x_n)-\varphi (w,u_{(n,i)})+{\mu }_{(n,i)}{M}_{(n,i)}.$$

In view of the restrictions on the sequence $\{{\alpha }_{(n,i)}\}$, we find from (3.9) that

$$\underset{n\to \mathrm{\infty }}{lim}\parallel J{x}_n-J{T}_i^n{x}_n\parallel =0.$$

Notice that $\parallel J{T}_i^n{x}_n-J\overline{x}\parallel \le \parallel J{T}_i^n{x}_n-J{x}_n\parallel +\parallel J{x}_n-J\overline{x}\parallel$. It follows that

$$\underset{n\to \mathrm{\infty }}{lim}\parallel J{T}_i^n{x}_n-J\overline{x}\parallel =0.$$

(3.10)

The demicontinuity of $J^{-1}:E^{\ast }\to E$ implies that $T_i^n{x}_n\rightharpoonup \overline{x}$. Note that

$$|\parallel T_i^n{x}_n\parallel -\parallel \overline{x}\parallel |=|\parallel J{T}_i^n{x}_n\parallel -\parallel J\overline{x}\parallel |\le \parallel J{T}_i^n{x}_n-J\overline{x}\parallel .$$

This implies from (3.10) that ${lim}_{n\to \mathrm{\infty }}\parallel T_i^n{x}_n\parallel =\parallel \overline{x}\parallel$. Since E has the Kadec-Klee property, we obtain that ${lim}_{n\to \mathrm{\infty }}\parallel T_i^n{x}_n-\overline{x}\parallel =0$. On the other hand, we have

$$\parallel T_i^{n+1}{x}_n-\overline{x}\parallel \le \parallel T_i^{n+1}{x}_n-T_i^n{x}_n\parallel +\parallel T_i^n{x}_n-\overline{x}\parallel .$$

It follows from the uniformly asymptotic regularity of $T_i$ that

$$\underset{n\to \mathrm{\infty }}{lim}\parallel T_i^{n+1}{x}_n-\overline{x}\parallel =0.$$

That is, $T_i{T}_i^n{x}_n\to \overline{x}$. From the closedness of $T_i$, we find $\overline{x}=T_i\overline{x}$ for each $i\in \mathrm{\Lambda }$. This proves $\overline{x}\in {\bigcap }_{i\in \mathrm{\Lambda }}F(T_i)$. Next, we show that $\overline{x}\in {\bigcap }_{i\in \mathrm{\Lambda }}\mathit{EF}(f_i)$. In view of Lemma 2.6, we find that

$$\begin{array}{rl}\varphi (u_{(n,i)},y_{(n,i)})& \le \varphi (w,y_{(n,i)})-\varphi (w,u_{(n,i)})\\ \le \varphi (w,x_n)+{\mu }_{(n,i)}\varphi (w,x_n)-\varphi (w,u_{(n,i)}).\end{array}$$

It follows from (3.9) that ${lim}_{n\to \mathrm{\infty }}\varphi (u_{(n,i)},y_{(n,i)})=0$. This implies that ${lim}_{n\to \mathrm{\infty }}(\parallel u_{(n,i)}\parallel -\parallel y_{(n,i)}\parallel )=0$. In view of (3.8), we see that $u_{(n,i)}\to \overline{x}$ as $n\to \mathrm{\infty }$. This implies that $\parallel y_{(n,i)}\parallel -\parallel \overline{x}\parallel \to 0$, as $n\to \mathrm{\infty }$. It follows that ${lim}_{n\to \mathrm{\infty }}\parallel J{y}_{(n,i)}\parallel =\parallel J\overline{x}\parallel$. This shows that $\{J{y}_{(n,i)}\}$ is bounded. Since $E^{\ast }$ is reflexive, we may assume that $J{y}_{(n,i)}\rightharpoonup y^{(\ast ,i)}\in E^{\ast }$. In view of $J(E)=E^{\ast }$, we see that there exists $y^i\in E$ such that $J{y}^i=y^{(\ast ,i)}$. It follows that

$$\begin{array}{rl}\varphi (u_{(n,i)},y_{(n,i)})& ={\parallel u_{(n,i)}\parallel }^2-2〈u_{(n,i)},J{y}_{(n,i)}〉+{\parallel y_{(n,i)}\parallel }^2\\ ={\parallel u_{(n,i)}\parallel }^2-2〈u_{(n,i)},J{y}_{(n,i)}〉+{\parallel J{y}_{(n,i)}\parallel }^2.\end{array}$$

Taking ${lim\hspace{0.17em}inf}_{n\to \mathrm{\infty }}$ on the both sides of the equality above yields that

$$\begin{array}{rl}0& \ge {\parallel \overline{x}\parallel }^2-2〈\overline{x},y^{(\ast ,i)}〉+{\parallel y^{(\ast ,i)}\parallel }^2\\ ={\parallel \overline{x}\parallel }^2-2〈\overline{x},J{y}^i〉+{\parallel J{y}^i\parallel }^2\\ ={\parallel \overline{x}\parallel }^2-2〈\overline{x},J{y}^i〉+{\parallel y^i\parallel }^2\\ =\varphi (\overline{x},y^i).\end{array}$$

That is, $\overline{x}=y^i$, which in turn implies that $y^{(\ast ,i)}=J\overline{x}$. It follows that $J{y}_{(n,i)}\rightharpoonup J\overline{x}\in E^{\ast }$. Since $E^{\ast }$ enjoys the Kadec-Klee property, we obtain that $J{y}_{(n,i)}-J\overline{x}\to 0$ as $n\to \mathrm{\infty }$. Note that $J^{-1}:E^{\ast }\to E$ is demicontinuous. It follows that $y_{(n,i)}\rightharpoonup \overline{x}$. Since E enjoys the Kadec-Klee property, we obtain that $y_{(n,i)}\to \overline{x}$ as $n\to \mathrm{\infty }$. Note that

$$\parallel u_{(n,i)}-y_{(n,i)}\parallel \le \parallel u_{(n,i)}-\overline{x}\parallel +\parallel \overline{x}-y_{(n,i)}\parallel .$$

This implies that

$$\underset{n\to \mathrm{\infty }}{lim}\parallel u_{(n,i)}-y_{(n,i)}\parallel =0.$$

Since J is uniformly norm-to-norm continuous on any bounded sets, we have

$$\underset{n\to \mathrm{\infty }}{lim}\parallel J{u}_{(n,i)}-J{y}_{(n,i)}\parallel =0.$$

From the assumption $r_{(n,i)}\ge a_i$, we see that

$$\underset{n\to \mathrm{\infty }}{lim}\frac{\parallel J{u}_{(n,i)}-J{y}_{(n,i)}\parallel }{r_{(n,i)}}=0.$$

(3.11)

In view of $u_{(n,i)}=S_{r_{(n,i)}}{y}_{(n,i)}$, we see that

$$f_i(u_{(n,i)},y)+\frac{1}{r_{(n,i)}}〈y-u_{(n,i)},J{u}_{(n,i)}-J{y}_{(n,i)}〉\ge 0,\mathrm{\forall }y\in C.$$

It follows from (A2) that

$$\begin{array}{rl}\parallel y-u_{(n,i)}\parallel \frac{\parallel J{u}_{(n,i)}-J{y}_{(n,i)}\parallel }{r_n}& \ge \frac{1}{r_{(n,i)}}〈y-u_{(n,i)},J{u}_{(n,i)}-J{y}_{(n,i)}〉\\ \ge f_i(y,u_{(n,i)}),\mathrm{\forall }y\in C.\end{array}$$

In view of (A4), we find from (3.11) that

$$f_i(y,\overline{x})\le 0,\mathrm{\forall }y\in C.$$

For $0<t_i<1$ and $y\in C$, define $y_{(t,i)}=t_{i}y+(1-t_i)\overline{x}$. It follows that $y_{(t,i)}\in C$, which yields that $f_i(y_{(t,i)},\overline{x})\le 0$. It follows from the (A1) and (A4) that

$$0=f_i(y_{(t,i)},y_{(t,i)})\le t_i{f}_i(y_{(t,i)},y)+(1-t_i)f_i(y_{(t,i)},\overline{x})\le t_i{f}_i(y_{(t,i)},y).$$

That is,

$$f_i(y_{(t,i)},y)\ge 0.$$

Letting $t_i\downarrow 0$, we obtain from (A3) that $f_i(\overline{x},y)\ge 0$, $\mathrm{\forall }y\in C$. This implies that $\overline{x}\in \mathit{EP}(f_i)$ for every $i\in \mathrm{\Lambda }$. This shows that $\overline{x}\in {\bigcap }_{i\in \mathrm{\Lambda }}\mathit{EF}(f_i)$. This completes the proof that $\overline{x}\in {\bigcap }_{i\in \mathrm{\Lambda }}F(T_i)\cap {\bigcap }_{i\in \mathrm{\Lambda }}\mathit{EF}(f_i)$.

Step 6. We prove that $\overline{x}={\mathrm{\Pi }}_{{\bigcap }_{i\in \mathrm{\Lambda }}F(T_i)\cap {\bigcap }_{i\in \mathrm{\Lambda }}\mathit{EF}(f_i)}{x}_1$.

Letting $n\to \mathrm{\infty }$ in (3.4), we see that

$$〈\overline{x}-w,J{x}_1-J\overline{x}〉\ge 0,\mathrm{\forall }w\in \mathit{CSS}.$$

In view of Lemma 2.7, we find that $\overline{x}={\mathrm{\Pi }}_{{\bigcap }_{i\in \mathrm{\Lambda }}F(T_i)\cap {\bigcap }_{i\in \mathrm{\Lambda }}\mathit{EF}(f_i)}{x}_1$. This completes the proof. □

Remark 3.2 Theorem 3.1 mainly improves Theorem 2.1 of Qin et al. [29] in the following aspects:

• (1) improves the mappings from a finite family of mappings to an infinite family of mappings;

• (2) extends the framework of the space from a uniformly smooth and uniformly convex space to a uniformly smooth and strictly convex Banach space which also enjoys the Kadec-Klee property.

If T is a quasi-ϕ-nonexpansive mapping, we find from Theorem 2.1 the following.

Corollary 3.3 Let E be a uniformly smooth and strictly convex Banach space which also enjoys the Kadec-Klee property. Let C be a nonempty closed and convex subset of E. Let Λ be an index set. Let $f_i$ be a bifunction from $C\times C$ to ℝ satisfying (A1)-(A4), and let $T_i:C\to E$ be a closed quasi-ϕ-nonexpansive mapping for every $i\in \mathrm{\Lambda }$. Assume that ${\bigcap }_{i\in \mathrm{\Lambda }}F(T_i)\cap {\bigcap }_{i\in \mathrm{\Lambda }}\mathit{EF}(f_i)$ is nonempty. Let $\{x_n\}$ be a sequence generated in the following manner:

$$\{\begin{array}{c}{x}_0\in E\mathit{\text{chosen arbitrarily}},\hfill \\ C_{(1,i)}=C,\hfill \\ C_1={\bigcap }_{i\in \mathrm{\Lambda }}{C}_{(1,i)},\hfill \\ x_1={\mathrm{\Pi }}_{C_1}{x}_0,\hfill \\ y_{(n,i)}=J^{-1}({\alpha }_{(n,i)}J{x}_n+(1-{\alpha }_{(n,i)})J{T}_i{x}_n),\hfill \\ u_{(n,i)}\in C\mathit{\text{ such that }}{f}_i(u_{(n,i)},y)+\frac{1}{r_{(n,i)}}〈y-u_{(n,i)},J{u}_{(n,i)}-J{y}_{(n,i)}〉\ge 0,\mathrm{\forall }y\in C,\hfill \\ C_{(n+1,i)}=\{z\in C_{(n,i)}:\varphi (z,u_{(n,i)})\le \varphi (z,x_n)\},\hfill \\ C_{n+1}={\bigcap }_{i\in \mathrm{\Lambda }}{C}_{(n+1,i)},\hfill \\ x_{n+1}={\mathrm{\Pi }}_{C_{n+1}}{x}_1,\hfill \end{array}$$

where $\{{\alpha }_{(n,i)}\}$ is a real sequence in $[0,1]$ such that ${lim\hspace{0.17em}inf}_{n\to \mathrm{\infty }}{\alpha }_{(n,i)}(1-{\alpha }_{(n,i)})>0$, and $\{r_{(n,i)}\}$ is a real sequence in $[a_i,\mathrm{\infty })$, where $\{a_i\}$ is a positive real number sequence for every $i\in \mathrm{\Lambda }$. Then the sequence $\{x_n\}$ converges strongly to ${\mathrm{\Pi }}_{{\bigcap }_{i\in \mathrm{\Lambda }}F(T_i)\cap {\bigcap }_{i\in \mathrm{\Lambda }}\mathit{EF}(f_i)}{x}_1$.

Remark 3.4 Corollary 3.3 can be viewed as an extension of the corresponding results announced in [36] and [37].

Since every uniformly convex Banach space is a strictly convex Banach space which also enjoys the Kadec-Klee property, we see that Theorem 3.1 is still valid in a uniformly smooth and uniformly convex Banach space. Theorem 3.1 improves the corresponding results in Qin et al. [31].

For a single bifunction and mapping, we find from Theorem 3.1 the following.

Corollary 3.5 Let E be a uniformly smooth and strictly convex Banach space which also enjoys the Kadec-Klee property. Let C be a nonempty closed and convex subset of E. Let f be a bifunction from $C\times C$ to ℝ satisfying (A1)-(A4), and let $T:C\to E$ be an asymptotically quasi-ϕ-nonexpansive mapping. Assume that T is closed and uniformly asymptotically regular on C. Assume that $F(T)\cap \mathit{EF}(f)$ is nonempty. Let $\{x_n\}$ be a sequence generated in the following manner:

$$\{\begin{array}{c}{x}_0\in E\mathit{\text{chosen arbitrarily}},\hfill \\ C_1=C,\hfill \\ x_1={\mathrm{\Pi }}_{C_1}{x}_0,\hfill \\ y_n=J^{-1}({\alpha }_{n}J{x}_n+(1-{\alpha }_n)J{T}^n{x}_n),\hfill \\ u_n\in C\mathit{\text{ such that }}f(u_n,y)+\frac{1}{r_n}〈y-u_n,J{u}_n-J{y}_n〉\ge 0,\mathrm{\forall }y\in C,\hfill \\ C_{n+1}=\{z\in C_n:\varphi (z,u_n)\le \varphi (z,x_n)+{\mu }_n{M}_n\},\hfill \\ x_{n+1}={\mathrm{\Pi }}_{C_{n+1}}{x}_1,\hfill \end{array}$$

where $M_n=sup\{\varphi (p,x_n):p\in F(F)\cap \mathit{EF}(f)\}$, $\{{\alpha }_n\}$ is a real sequence in $[0,1]$ such that ${lim\hspace{0.17em}inf}_{n\to \mathrm{\infty }}{\alpha }_n(1-{\alpha }_n)>0$, and $\{r_n\}$ is a real sequence in $[a,\mathrm{\infty })$, where a is a positive real number. Then the sequence $\{x_n\}$ converges strongly to ${\mathrm{\Pi }}_{F(T)\cap \mathit{EF}(f)}{x}_1$.

If T is the identity mapping, then Theorem 3.1 is reduced to the following.

Corollary 3.6 Let E be a uniformly smooth and strictly convex Banach space which also enjoys the Kadec-Klee property. Let C be a nonempty closed and convex subset of E. Let Λ be an index set. Let $f_i$ be a bifunction from $C\times C$ to ℝ satisfying (A1)-(A4) for every $i\in \mathrm{\Lambda }$. Assume that ${\bigcap }_{i\in \mathrm{\Lambda }}\mathit{EF}(f_i)$ is nonempty and bounded. Let $\{x_n\}$ be a sequence generated in the following manner:

$$\{\begin{array}{c}{x}_0\in E\mathit{\text{chosen arbitrarily}},\hfill \\ C_{(1,i)}=C,\hfill \\ C_1={\bigcap }_{i\in \mathrm{\Lambda }}{C}_{(1,i)},\hfill \\ x_1={\mathrm{\Pi }}_{C_1}{x}_0,\hfill \\ u_{(n,i)}\in C\mathit{\text{ such that }}{f}_i(u_{(n,i)},y)+\frac{1}{r_{(n,i)}}〈y-u_{(n,i)},J{u}_{(n,i)}-J{x}_n〉\ge 0,\mathrm{\forall }y\in C,\hfill \\ C_{(n+1,i)}=\{z\in C_{(n,i)}:\varphi (z,u_{(n,i)})\le \varphi (z,x_n)+{\mu }_{(n,i)}{M}_{(n,i)}\},\hfill \\ C_{n+1}={\bigcap }_{i\in \mathrm{\Lambda }}{C}_{(n+1,i)},\hfill \\ x_{n+1}={\mathrm{\Pi }}_{C_{n+1}}{x}_1,\hfill \end{array}$$

where $M_{(n,i)}=sup\{\varphi (p,x_n):p\in {\bigcap }_{i\in \mathrm{\Lambda }}\mathit{EF}(f_i)\}$, $\{{\alpha }_{(n,i)}\}$ is a real sequence in $[0,1]$ such that ${lim\hspace{0.17em}inf}_{n\to \mathrm{\infty }}{\alpha }_{(n,i)}(1-{\alpha }_{(n,i)})>0$, and $\{r_{(n,i)}\}$ is a real sequence in $[a_i,\mathrm{\infty })$, where $\{a_i\}$ is a positive real number sequence, for every $i\in \mathrm{\Lambda }$. Then the sequence $\{x_n\}$ converges strongly to ${\mathrm{\Pi }}_{{\bigcap }_{i\in \mathrm{\Lambda }}\mathit{EF}(f_i)}{x}_1$.