Weak convergence of algorithms for asymptotically strict pseudocontractions in the intermediate sense and equilibrium problems

Abstract

In this paper, equilibrium problems and fixed-point problems based on an iterative method are investigated. It is proved that the sequence generated in the purposed iterative process weakly converges to a common element of the fixed-point set of an asymptotically strict pseudocontraction in the intermediate sense and the solution set of a system of equilibrium problems in the framework of real Hilbert spaces.

MSC:47H05, 47H09, 47J25, 90C33.

1 Introduction

Approximating solutions of nonlinear operator equations based on iterative methods is now a hot topic of intensive research efforts. Indeed, 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 $C_m$ in which the required solution lies. The problem of finding a point in the intersection ${\bigcap }_{m=1}^N{C}_m$, where $N\ge 1$ is some positive integer is of crucial interest, and it cannot be usually solved directly. Therefore, an iterative algorithm must be used to approximate such 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 (see [1–7]). There many classic algorithms, for example, the Picard iterative algorithm, the Mann iterative algorithm, the Ishikawa iterative algorithm, steepest descent iterative algorithms, hybrid projection algorithms, and so on. In this paper, we shall investigate fixed-point and equilibrium problems based on a Mann-like iterative algorithm.

The organization of this paper is as follows. In Section 2, we provide some necessary preliminaries. In Section 3, equilibrium problems and fixed-point problems of asymptotically strict pseudocontractions in the intermediate sense are discussed based on a Mann iterative algorithm. Weak convergence theorems are established in Hilbert spaces. And some deduced results are also obtained.

2 Preliminaries

Throughout this paper, we always assume that H is a real Hilbert space with an inner product $〈\cdot ,\cdot 〉$ and norm $\parallel \cdot \parallel$. Let C be a nonempty, closed, and convex subset of H and F a bifunction of $C\times C$ into $\mathbb{R}$, where $\mathbb{R}$ stands for the set of real numbers. In this paper, we consider the following equilibrium problem.

$$\text{Find }x\in C\text{ such that }F(x,y)\ge 0,\mathrm{\forall }y\in C.$$

(2.1)

The set of such an $x\in C$ is denoted by $EP(F)$, i.e.,

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

Given a mapping $A:C\to H$, let $F(x,y)=〈Ax,y-x〉$ for all $x,y\in C$. Then $z\in EP(F)$ if and only if

$$〈Az,y-z〉\ge 0,\mathrm{\forall }y\in C,$$

that is, z is a solution of the classical variational inequality. In this paper, we use $VI(C,A)$ to stand for the set of solutions of the variational inequality. Numerous problems in physics, optimization, and economics reduce to find a solution of the equilibrium problem (2.1).

To study the equilibrium problem (2.1), we may assume that F satisfies the following conditions:

(A1) $F(x,x)=0$ for all $x\in C$;

(A2) F is monotone, i.e., $F(x,y)+F(y,x)\le 0$ for all $x,y\in C$;

(A3) for each $x,y,z\in C$,

$$\underset{t\downarrow 0}{lim\hspace{0.17em}sup}F(tz+(1-t)x,y)\le F(x,y);$$

(A4) for each $x\in C$, $y\mapsto F(x,y)$ is convex and lower semicontinuous.

Let $S:C\to C$ be a mapping. In this paper, we use $F(S)$ to denote the fixed-point set of S. Recall the following definitions.

S is said to be nonexpansive if

$$\parallel Sx-Sy\parallel \le \parallel x-y\parallel ,\mathrm{\forall }x,y\in C.$$

If C is a bounded, closed, and convex subset of H, then $F(S)$ is nonempty, closed, and convex. Recently, many beautiful convergence theorems for fixed points of nonexpansive mappings (semigroups) have been established in Banach spaces (see [8–10] and the references therein).

S is said to be asymptotically nonexpansive if there exists a sequence $\{k_n\}\subset [1,\mathrm{\infty })$ with $k_n\to 1$ as $n\to \mathrm{\infty }$ such that

$$\parallel S^{n}x-S^{n}y\parallel \le k_n\parallel x-y\parallel ,\mathrm{\forall }x,y\in C,n\ge 1.$$

It is known that if C is a nonempty, bounded, and closed convex subset of a Hilbert space H, then every asymptotically nonexpansive self-mapping has a fixed point. Further, the set $F(S)$ of fixed points of S is closed and convex. Since 1972, a host of authors have studied the weak and strong convergence problems of iterative processes for such a class of mappings.

S is said to be asymptotically nonexpansive in the intermediate sense if it is continuous and the following inequality holds:

$$\underset{n\to \mathrm{\infty }}{lim\hspace{0.17em}sup}\underset{x,y\in C}{sup}(\parallel S^{n}x-S^{n}y\parallel -\parallel x-y\parallel )\le 0.$$

(2.2)

Putting

$${\xi }_n=max\{0,\underset{x,y\in C}{sup}(\parallel S^{n}x-S^{n}y\parallel -\parallel x-y\parallel )\},$$

(2.3)

we see that ${\xi }_n\to 0$ as $n\to \mathrm{\infty }$. Then (2.2) is reduced to the following:

$$\parallel S^{n}x-S^{n}y\parallel \le \parallel x-y\parallel +{\xi }_n,\mathrm{\forall }x,y\in C.$$

The class of asymptotically nonexpansive mappings in the intermediate sense was considered in [11] and [12] as a generalization of the class of asymptotically nonexpansive mappings. It is known that if C is a nonempty, closed, and convex bounded subset of a real Hilbert space, then every asymptotically nonexpansive self-mapping in the intermediate sense has a fixed point.

S is said to be strictly pseudocontractive if there exists a constant $\kappa \in [0,1)$ such that

$${\parallel Sx-Sy\parallel }^2\le {\parallel x-y\parallel }^2+\kappa {\parallel (I-S)x-(I-S)y\parallel }^2,\mathrm{\forall }x,y\in C.$$

For such a case, S is also said to be a κ-strict pseudocontraction. It is clear that every nonexpansive mapping is a 0-strict pseudocontraction. We also remark that if $\kappa =1$, then S is said to be pseudocontractive.

S is said to be an asymptotically strict pseudocontraction if there exist a sequence $\{k_n\}\subset [1,\mathrm{\infty })$ with $k_n\to 1$ as $n\to \mathrm{\infty }$ and a constant $\kappa \in [0,1)$ such that

$${\parallel S^{n}x-S^{n}y\parallel }^2\le k_n{\parallel x-y\parallel }^2+\kappa {\parallel (I-S^n)x-(I-S^n)y\parallel }^2,\mathrm{\forall }x,y\in C,n\ge 1.$$

For such a case, S is also said to be an asymptotically κ-strict pseudocontraction. It is clear that every asymptotically nonexpansive mapping is an asymptotically 0-strict pseudocontraction. We also remark here that if $\kappa =1$, then S is said to be an asymptotically pseudocontractive mapping which was introduced by Schu [13] in 1991.

S is said to be an asymptotically strict pseudocontraction in the intermediate sense if there exist a sequence $\{k_n\}\subset [1,\mathrm{\infty })$ with $k_n\to 1$ as $n\to \mathrm{\infty }$ and a constant $\kappa \in [0,1)$ such that

$$\underset{n\to \mathrm{\infty }}{lim\hspace{0.17em}sup}\underset{x,y\in C}{sup}({\parallel S^{n}x-S^{n}y\parallel }^2-k_n{\parallel x-y\parallel }^2-\kappa {\parallel (I-S^n)x-(I-S^n)y\parallel }^2)\le 0.$$

(2.4)

For such a case, S is also said to be an asymptotically κ-strict pseudocontraction in the intermediate sense. Putting

$${\xi }_n=max\{0,\underset{x,y\in C}{sup}({\parallel S^{n}x-S^{n}y\parallel }^2-k_n{\parallel x-y\parallel }^2-\kappa {\parallel (I-S^n)x-(I-S^n)y\parallel }^2)\},$$

(2.5)

then we see that ${\xi }_n\to 0$ as $n\to \mathrm{\infty }$. We know that (2.4) is reduced to the following:

$${\parallel S^{n}x-S^{n}y\parallel }^2\le k_n{\parallel x-y\parallel }^2+\kappa {\parallel (I-S^n)x-(I-S^n)y\parallel }^2+{\xi }_n,\mathrm{\forall }x,y\in C,n\ge 1.$$

The class of asymptotically strict pseudocontractions in the intermediate sense was introduced by Sahu, Xu, and Yao [14] as a generalization of the class of asymptotically strict pseudocontractions; see [14] for more details.

Recently, many authors considered the weak convergence of iterative sequences for the classical variational inequality, the equilibrium problem (2.1), and fixed-point problems based on iterative methods (see [15–24]).

In 2003, Takahashi and Toyoda [23] considered the classical variational inequality and a nonexpansive mapping. To be more precise, they proved the following theorem.

Theorem 1 Let C be a closed convex subset of a real Hilbert space H. Let A be an α-inverse strongly-monotone mapping of C into H, and let S be a nonexpansive mapping of C into itself such that$F(S)\cap VI(C,A)\ne \mathrm{\varnothing }$. Let$\{x_n\}$be a sequence generated by

$$x_0\in C,x_{n+1}={\alpha }_n{x}_n+(1-{\alpha }_n)S{P}_C(x_n-{\lambda }_{n}A{x}_n),\mathrm{\forall }n\ge 0,$$

for every$n\ge 0$, where${\lambda }_n\in [a,b]$for some$a,b\in (0,2\alpha )$and${\alpha }_n\in [c,d]$for some$c,d\in (0,1)$. Then$\{x_n\}$converges weakly to$z\in F(S)\cap VI(C,A)$, where$z={lim}_{n\to \mathrm{\infty }}{P}_{F(S)\cap VI(C,A)}{x}_n$.

In 2007, Tada and Takahashi [24] considered the equilibrium problem (2.1) and a nonexpansive mapping. To be more precise, they proved the following result.

Theorem 2 Let C be a nonempty, closed, and convex subset of H. Let F be a bifunction from$C\times C$to$\mathbb{R}$satisfying (A 1)-(A 4) and let S be a nonexpansive mapping of C into H, such that$F(S)\cap EP(F)\ne \mathrm{\varnothing }$. Let$\{x_n\}$and$\{u_n\}$be sequences generated by$x_1=x\in H$and let

$$\{\begin{array}{c}{u}_n\in C\mathit{\text{such that}}F(u_n,u)+\frac{1}{r_n}〈u-u_n,u_n-x_n〉\ge 0,\mathrm{\forall }u\in C,\hfill \\ x_{n+1}={\alpha }_n{x}_n+(1-{\alpha }_n)S{u}_n,\hfill \end{array}$$

for each$n\ge 1$, where$\{{\alpha }_n\}\subset [a,b]$for some$a,b\in (0,1)$and$\{r_n\}\subset (0,\mathrm{\infty })$satisfies${lim\hspace{0.17em}inf}_{n\to \mathrm{\infty }}{r}_n>0$. Then$\{x_n\}$converges weakly to$w\in F(S)\cap EP(F)$, where$w={lim}_{n\to \mathrm{\infty }}{P}_{F(S)\cap EP(F)}{x}_n$.

In this paper, motivated by the results announced in [23] and [24], we consider the equilibrium problem (2.1) and an asymptotically strict pseudocontraction in the intermediate sense based on a Mann-like iterative process. We show that the sequence generated in the purposed iterative process converges weakly to a common element of the fixed-point set of an asymptotically strict pseudocontraction in the intermediate sense and the solution set of the equilibrium problem (2.1). The results presented in this paper improve and extend the corresponding results announced by Takahashi and Toyoda [23], and Tada and Takahashi [24].

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

The following lemma can be found in [25] and [26].

Lemma 2.1 Let C be a nonempty, closed, and convex subset of H, and let$F:C\times C\to \mathbb{R}$be a bifunction satisfying (A 1)-(A 4). Then for any$r>0$and$x\in H$, there exists$z\in C$such that

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

Further, define

$$T_{r}x=\{z\in C:F(z,y)+\frac{1}{r}〈y-z,z-x〉\ge 0,\mathrm{\forall }y\in C\}$$

for all$r>0$and$x\in H$. Then the following statements hold:

• (a) $T_r$ is single-valued;

• (b) $T_r$ is firmly nonexpansive, i.e., for any $x,y\in H$,

  $${\parallel T_{r}x-T_{r}y\parallel }^2\le 〈T_{r}x-T_{r}y,x-y〉;$$

• (c) $F(T_r)=EP(F)$;

• (d) $EP(F)$ is closed and convex.

Lemma 2.2 ([14])

Let H be a real Hilbert space, C a nonempty, closed, and convex subset of H, and$S:C\to C$a uniformly continuous and asymptotically strict pseudocontraction in the intermediate sense. If$\{x_n\}$is a sequence in C such that$x_n\rightharpoonup x$and${lim\hspace{0.17em}sup}_{m\to \mathrm{\infty }}{lim\hspace{0.17em}sup}_{n\to \mathrm{\infty }}\parallel x_n-T^m{x}_n\parallel =0$, then$x=Tx$.

Lemma 2.3 ([27])

Let H be a real Hilbert space, and$0<p\le t_n\le q<1$, for all$n\ge 1$. Suppose that$\{x_n\}$and$\{y_n\}$are sequences in H such that

$$\underset{n\to \mathrm{\infty }}{lim\hspace{0.17em}sup}\parallel x_n\parallel \le r,\underset{n\to \mathrm{\infty }}{lim\hspace{0.17em}sup}\parallel y_n\parallel \le r$$

and for some$r\ge 0$,

$$\underset{n\to \mathrm{\infty }}{lim}\parallel t_n{x}_n+(1-t_n)y_n\parallel =r.$$

Then${lim}_{n\to \mathrm{\infty }}\parallel x_n-y_n\parallel =0$.

Lemma 2.4 ([28])

Let$\{a_n\}$, $\{b_n\}$, and$\{c_n\}$be three nonnegative sequences satisfying the following condition:

$$a_{n+1}\le (1+b_n)a_n+c_n,\mathrm{\forall }n\ge n_0,$$

where$n_0$is some nonnegative integer, ${\sum }_{n=1}^{\mathrm{\infty }}{b}_n<\mathrm{\infty }$and${\sum }_{n=1}^{\mathrm{\infty }}{c}_n<\mathrm{\infty }$. Then the limit${lim}_{n\to \mathrm{\infty }}{a}_n$exists.

Lemma 2.5 ([29])

Let H be a real Hilbert space. Let${\{a_n\}}_{n=1}^N$be real numbers in$[0,1]$such that${\sum }_{n=1}^N{a}_n=1$. Then we have the following:

$${\parallel \sum _{i=1}^N{a}_i{x}_i\parallel }^2\le \sum _{i=1}^N{a}_i{\parallel x_i\parallel }^2,$$

for any given bounded sequence${\{x_n\}}_{n=1}^N$in H.

3 Main results

Theorem 3.1 Let C be a nonempty, closed, and convex subset of a real Hilbert space H. Let$F_m$be a bifunction from$C\times C$to$\mathbb{R}$which satisfies (A 1)-(A 4) for each$1\le m\le N$, where$N\ge 1$is some positive integer. Let$S:C\to C$be a uniformly continuous and asymptotically κ-strict pseudocontraction in the intermediate sense. Assume that$\mathcal{F}:=F(S)\cap {\bigcap }_{m=1}^{N}EP(F_m)$is nonempty. Let$\{{\alpha }_n\}$, $\{{\beta }_n\}$, $\{{\delta }_n\}$, and$\{{\lambda }_n\}$be sequences in$[0,1]$, and$\{e_n\}$a bounded sequence in C. Let$\{r_{n,m}\}$be a positive sequence such that${lim\hspace{0.17em}inf}_{n\to \mathrm{\infty }}{r}_{n,m}>0$and$\{{\gamma }_{n,m}\}$a sequence in$[0,1]$for each$1\le m\le N$. Let$\{x_n\}$be a sequence generated in the following manner:

$$\{\begin{array}{c}{x}_1\in H,\hfill \\ u_{n,m}\in C\mathit{\text{such that}}{F}_m(u_{n,m},u_m)+\frac{1}{r_{n,m}}〈u_m-u_{n,m},u_{n,m}-x_n〉\ge 0,\mathrm{\forall }{u}_m\in C,\hfill \\ z_n={\sum }_{m=1}^N{\gamma }_{n,m}{u}_{n,m},\hfill \\ x_{n+1}={\alpha }_n{x}_n+{\beta }_n({\delta }_n{z}_n+(1-{\delta }_n)S^n{z}_n)+{\lambda }_n{e}_n,n\ge 1.\hfill \end{array}$$

Assume that the following restrictions are satisfied:

• (a) ${\alpha }_n+{\beta }_n+{\lambda }_n=1$;

• (b) ${\sum }_{n=1}^{\mathrm{\infty }}(k_n-1)<\mathrm{\infty }$ and ${\sum }_{n=1}{\xi }_n<\mathrm{\infty }$, where ${\xi }_n$ is defined in (2.5);

• (c) $0<a\le {\beta }_n\le b<1$, $0\le \kappa \le {\delta }_n\le c<1$ and ${\sum }_{n=1}^{\mathrm{\infty }}{\lambda }_n<\mathrm{\infty }$;

• (d) ${\sum }_{m=1}^N{\gamma }_{n,m}=1$ and $0<d\le {\gamma }_{n,m}\le 1$ for each $1\le m\le N$,

where a, b, c, and d are some real constants. Then the sequence$\{x_n\}$weakly converges to some point in$\mathcal{F}$.

Proof Let $p\in \mathcal{F}$. Then we see that

$$\parallel z_n-p\parallel \le \sum _{m=1}^N{\gamma }_{n,m}\parallel u_{n,m}-p\parallel \le \parallel x_n-p\parallel .$$

(3.1)

Put $y_n={\delta }_n{z}_n+(1-{\delta }_n)S^n{z}_n$ for each $n\ge 1$. In view of Lemma 2.5, we see from the restriction (c) that

$$\begin{array}{rcl}{\parallel y_n-p\parallel }^2& =& {\parallel {\delta }_n{z}_n+(1-{\delta }_n)S^n{z}_n-p\parallel }^2\\ =& {\delta }_n{\parallel z_n-p\parallel }^2+(1-{\delta }_n){\parallel S^n{z}_n-p\parallel }^2-{\delta }_n(1-{\delta }_n){\parallel z_n-S^n{z}_n\parallel }^2\\ \le & {\delta }_n{\parallel z_n-p\parallel }^2+(1-{\delta }_n)k_n{\parallel z_n-p\parallel }^2+(\kappa -{\delta }_n){\parallel z_n-S^n{z}_n\parallel }^2+{\xi }_n\\ \le & k_n{\parallel z_n-p\parallel }^2+{\xi }_n\\ \le & k_n{\parallel x_n-p\parallel }^2+{\xi }_n.\end{array}$$

(3.2)

It follows that

$$\begin{array}{rcl}{\parallel x_{n+1}-p\parallel }^2& \le & {\alpha }_n{\parallel x_n-p\parallel }^2+{\beta }_n{\parallel y_n-p\parallel }^2+{\lambda }_n{\parallel e_n-p\parallel }^2\\ \le & {\alpha }_n{\parallel x_n-p\parallel }^2+{\beta }_n(k_n{\parallel x_n-p\parallel }^2+{\xi }_n)+{\lambda }_n{\parallel e_n-p\parallel }^2\\ \le & k_n{\parallel x_n-p\parallel }^2+{\xi }_n+{\lambda }_n{\parallel e_n-p\parallel }^2.\end{array}$$

(3.3)

From Lemma 2.4, we obtain the existence of the limit of the sequence $\{\parallel x_n-p\parallel \}$. Notice that

$$\begin{array}{rcl}{\parallel u_{n,m}-p\parallel }^2& =& {\parallel T_{r_{n,m}}{x}_n-T_{r_{n,m}}p\parallel }^2\\ \le & 〈T_{r_{n,m}}{x}_n-T_{r_{n,m}}p,x_n-p〉\\ =& 〈u_{n,m}-p,x_n-p〉\\ =& \frac{1}{2}({\parallel u_{n,m}-p\parallel }^2+{\parallel x_n-p\parallel }^2-{\parallel u_{n,m}-x_n\parallel }^2),\mathrm{\forall }1\le m\le N.\end{array}$$

This implies that

$${\parallel u_{n,m}-p\parallel }^2\le {\parallel x_n-p\parallel }^2-{\parallel u_{n,m}-x_n\parallel }^2,\mathrm{\forall }1\le m\le N.$$

(3.4)

In view of (3.4) and $z_n={\sum }_{m=1}^N{\gamma }_{n,m}{u}_{n,m}$, where ${\sum }_{m=1}^N{\gamma }_{n,m}=1$, we see from Lemma 2.5 that

$$\begin{array}{rcl}{\parallel z_n-p\parallel }^2& \le & \sum _{m=1}^N{\gamma }_{n,m}{\parallel u_{n,m}-p\parallel }^2\\ \le & \sum _{m=1}^N{\gamma }_{n,m}({\parallel x_n-p\parallel }^2-{\parallel u_{n,m}-x_n\parallel }^2)\\ \le & \sum _{m=1}^N{\gamma }_{n,m}{\parallel x_n-p\parallel }^2-\sum _{m=1}^N{\gamma }_{n,m}{\parallel u_{n,m}-x_n\parallel }^2\\ =& {\parallel x_n-p\parallel }^2-\sum _{m=1}^N{\gamma }_{n,m}{\parallel u_{n,m}-x_n\parallel }^2.\end{array}$$

(3.5)

In view of Lemma 2.5, we obtain from restriction (c) that

(3.6)

This implies from (3.5) that

$${\parallel x_{n+1}-p\parallel }^2\le {\parallel x_n-p\parallel }^2+(k_n-1){\parallel x_n-p\parallel }^2-{\beta }_n{k}_n\sum _{m=1}^N{\gamma }_{n,m}{\parallel u_{n,m}-x_n\parallel }^2+{\xi }_n+{\lambda }_n{\parallel e_n-p\parallel }^2.$$

It follows that

$${\beta }_n{k}_n{\gamma }_{n,m}{\parallel u_{n,m}-x_n\parallel }^2\le {\parallel x_n-p\parallel }^2-{\parallel x_{n+1}-p\parallel }^2+(k_n-1){\parallel x_n-p\parallel }^2+{\xi }_n+{\lambda }_n{\parallel e_n-p\parallel }^2.$$

In view of the restrictions (b), (c), and (d), we obtain that

$$\underset{n\to \mathrm{\infty }}{lim}\parallel u_{n,m}-x_n\parallel =0,\mathrm{\forall }1\le m\le N.$$

(3.7)

Since $\{x_n\}$ is bounded, we see that there exists a subsequence $\{x_{n_i}\}$ of $\{x_n\}$ which converges weakly to ω. We can get from (3.7) that $\{u_{n_i,m}\}$ converges weakly to ω for each $1\le m\le N$. Note that

$$F_m(u_{n,m},u_m)+\frac{1}{r_{n,m}}〈u_m-u_{n,m},u_{n,m}-x_n〉\ge 0,\mathrm{\forall }{u}_m\in C.$$

From the assumption (A2), we see that

$$\frac{1}{r_{n,m}}〈u_m-u_{n,m},u_{n,m}-x_n〉\ge F_m(u_m,u_{n,m}),\mathrm{\forall }{u}_m\in C.$$

Replacing n by $n_i$, we arrive at

$$\frac{1}{r_{n_i,m}}〈u_m-u_{n_i,m},u_{n_i,m}-x_{n_i}〉\ge F_m(u_m,u_{n_i,m}),\mathrm{\forall }{u}_m\in C.$$

(3.8)

In view of (3.7) and the assumption (A4), we get that

$$F_m(u_m,\omega )\le 0,\mathrm{\forall }{u}_m\in C.$$

For any $t_m$ with $0<t_m\le 1$ and $u_m\in C$, let $u_{t_m}=t_m{u}_m+(1-t_m)\omega$ for each $1\le m\le N$. Since $u_m\in C$ and $\omega \in C$, we have $u_{t_m}\in C$, and hence $F_m(u_{t_m},\omega )\le 0$ for each $1\le m\le N$. It follows that

$$\begin{array}{rcl}0& =& F_m(u_{t_m},u_{t_m})\\ \le & t_m{F}_m(u_{t_m},u_m)+(1-t_m)F_m(u_{t_m},\omega )\\ \le & t_m{F}_m(u_{t_m},u_m),\mathrm{\forall }1\le m\le N,\end{array}$$

which yields that

$$F_m(u_{t_m},u_m)\ge 0,\mathrm{\forall }{u}_m\in C.$$

Letting $t_m\downarrow 0$ for each $1\le m\le N$, we obtain from the assumption (A3) that

$$F_m(\omega ,u_m)\ge 0,\mathrm{\forall }{u}_m\in C.$$

This implies that $\omega \in EP(F_m)$ for each $1\le m\le N$. This proves that $\omega \in {\bigcap }_{m=1}^{N}EP(F_m)$.

Next, we show that $\omega \in F(S)$. Put ${lim}_{n\to \mathrm{\infty }}\parallel x_n-p\parallel =d$, we obtain that

$$\underset{n\to \mathrm{\infty }}{lim\hspace{0.17em}sup}\parallel x_n-p+{\lambda }_n(e_n-x_n)\parallel \le d.$$

From (3.2), we see that

$$\underset{n\to \mathrm{\infty }}{lim\hspace{0.17em}sup}\parallel y_n-p+{\lambda }_n(e_n-x_n)\parallel \le d.$$

On the other hand, we have

$$\begin{array}{rcl}\underset{n\to \mathrm{\infty }}{lim}\parallel x_{n+1}-p\parallel & =& \underset{n\to \mathrm{\infty }}{lim}\parallel (1-{\beta }_n)(x_n-p+{\lambda }_n(e_n-x_n))+{\beta }_n(y_n-p+{\lambda }_n(e_n-x_n))\parallel \\ =& d.\end{array}$$

In view of Lemma 2.3, we obtain that

$$\underset{n\to \mathrm{\infty }}{lim}\parallel y_n-x_n\parallel =0.$$

(3.9)

Note that

$$\parallel z_n-x_n\parallel \le \sum _{m=1}^N{\gamma }_{n,m}\parallel u_{n,m}-x_n\parallel .$$

This implies from (3.7) that

$$\underset{n\to \mathrm{\infty }}{lim}\parallel z_n-x_n\parallel =0.$$

(3.10)

On the other hand, we have

It follows from (3.9) and (3.10) that

$$\underset{n\to \mathrm{\infty }}{lim}\parallel x_n-({\delta }_n{x}_n+(1-{\delta }_n)S^n{x}_n)\parallel =0.$$

(3.11)

Note that

$$\begin{array}{rcl}\parallel S^n{x}_n-x_n\parallel & \le & \parallel S^n{x}_n-({\delta }_n{x}_n+(1-{\delta }_n)S^n{x}_n)\parallel +\parallel ({\delta }_n{x}_n+(1-{\delta }_n)S^n{x}_n)-x_n\parallel \\ \le & {\delta }_n\parallel S^n{x}_n-x_n\parallel +\parallel ({\delta }_n{x}_n+(1-{\delta }_n)S^n{x}_n)-x_n\parallel ,\end{array}$$

which yields that

$$(1-{\delta }_n)\parallel S^n{x}_n-x_n\parallel \le \parallel ({\delta }_n{x}_n+(1-{\delta }_n)S^n{x}_n)-x_n\parallel .$$

This implies from the restriction (c) and (3.11) that

$$\underset{n\to \mathrm{\infty }}{lim}\parallel S^n{x}_n-x_n\parallel =0.$$

(3.12)

Notice that

$$\parallel x_{n+1}-x_n\parallel \le {\beta }_n\parallel y_n-x_n\parallel +{\lambda }_n\parallel e_n-x_n\parallel .$$

This implies from the restriction (c) and (3.9) that

$$\underset{n\to \mathrm{\infty }}{lim}\parallel x_{n+1}-x_n\parallel =0.$$

(3.13)

On the other hand, we have

$$\begin{array}{rcl}\parallel x_n-S{x}_n\parallel & \le & \parallel x_n-x_{n+1}\parallel +\parallel x_{n+1}-S^{n+1}{x}_{n+1}\parallel \\ +\parallel S^{n+1}{x}_{n+1}-S^{n+1}{x}_n\parallel +\parallel S^{n+1}{x}_n-S{x}_n\parallel .\end{array}$$

Since S is uniformly continuous, we obtain from (3.12) and (3.13) that

$$\underset{n\to \mathrm{\infty }}{lim}\parallel S{x}_n-x_n\parallel =0.$$

(3.14)

In view of Lemma 2.2, we obtain that $\omega \in F(S)$. This proves that $\omega \in \mathcal{F}$. Let $\{x_{n_j}\}$ be another subsequence of $\{x_n\}$ converging weakly to ${\omega }^{\prime }$, where ${\omega }^{\prime }\ne \omega$. In the same way, we can show that ${\omega }^{\prime }\in \mathcal{F}$. Notice that we have proved that ${lim}_{n\to \mathrm{\infty }}\parallel x_n-p\parallel$ exists for each $p\in \mathcal{F}$. Assume that ${lim}_{n\to \mathrm{\infty }}\parallel x_n-\omega \parallel =Q$, where Q is a nonnegative number. By virtue of Opial’s condition of H, we have

$$Q=\underset{i\to \mathrm{\infty }}{lim\hspace{0.17em}inf}\parallel x_{n_i}-\omega \parallel <\underset{i\to \mathrm{\infty }}{lim\hspace{0.17em}inf}\parallel x_{n_i}-{\omega }^{\prime }\parallel =\underset{j\to \mathrm{\infty }}{lim\hspace{0.17em}inf}\parallel x_j-{\omega }^{\prime }\parallel <\underset{j\to \mathrm{\infty }}{lim\hspace{0.17em}inf}\parallel x_j-\omega \parallel =Q.$$

This is a contradiction. Hence, $\omega ={\omega }^{\prime }$. This completes the proof. □

If $N=1$, then Theorem 3.1 is reduced to the following.

Corollary 3.2 Let C be a nonempty, closed, and convex subset of a real Hilbert space H. Let F be a bifunction from$C\times C$to$\mathbb{R}$which satisfies (A 1)-(A 4). Let$S:C\to C$be a uniformly continuous and asymptotically κ-strict pseudocontraction in the intermediate sense. Assume that$\mathcal{F}:=F(S)\cap EP(F)$is nonempty. Let$\{{\alpha }_n\}$, $\{{\beta }_n\}$, $\{{\delta }_n\}$, and$\{{\lambda }_n\}$be sequences in$[0,1]$, and$\{e_n\}$a bounded sequence in C. Let$\{r_n\}$be a positive sequence such that${lim\hspace{0.17em}inf}_{n\to \mathrm{\infty }}{r}_n>0$. Let$\{x_n\}$be a sequence generated in the following manner:

$$\{\begin{array}{c}{x}_1\in H,\hfill \\ u_n\in C\mathit{\text{such that}}F(u_n,u)+\frac{1}{r_n}〈u-u_n,u_n-x_n〉\ge 0,\mathrm{\forall }u\in C,\hfill \\ x_{n+1}={\alpha }_n{x}_n+{\beta }_n({\delta }_n{u}_n+(1-{\delta }_n)S^n{u}_n)+{\lambda }_n{e}_n,n\ge 1.\hfill \end{array}$$

Assume that the following restrictions are satisfied:

• (a) ${\alpha }_n+{\beta }_n+{\lambda }_n=1$;

• (b) ${\sum }_{n=1}^{\mathrm{\infty }}(k_n-1)<\mathrm{\infty }$ and ${\sum }_{n=1}{\xi }_n<\mathrm{\infty }$, where ${\xi }_n$ is defined in (2.5);

• (c) $0<a\le {\beta }_n\le b<1$, $0\le \kappa \le {\delta }_n\le c<1$ and ${\sum }_{n=1}^{\mathrm{\infty }}{\lambda }_n<\mathrm{\infty }$;

where a, b, and c are some real constants. Then the sequence$\{x_n\}$converges weakly to some point in$\mathcal{F}$.

If $F(x,y)=0$ for all $x,y\in C$ and $r_n=1$ for all $n\ge 1$, then Corollary 3.2 is reduced to the following.

Corollary 3.3 Let C be a nonempty, closed, and convex subset of a real Hilbert space H. Let$S:C\to C$be a uniformly continuous and asymptotically κ-strict pseudocontraction in the intermediate sense with$F(S)\ne \mathrm{\varnothing }$. Let$\{{\alpha }_n\}$, $\{{\beta }_n\}$, $\{{\delta }_n\}$, and$\{{\lambda }_n\}$be sequences in$[0,1]$, and$\{e_n\}$a bounded sequence in C. Let$\{r_n\}$be a positive sequence such that${lim\hspace{0.17em}inf}_{n\to \mathrm{\infty }}{r}_n>0$. Let$\{x_n\}$be a sequence generated in the following manner:

$$\{\begin{array}{c}{x}_1\in H,\hfill \\ x_{n+1}={\alpha }_n{x}_n+{\beta }_n({\delta }_n{P}_C{x}_n+(1-{\delta }_n)S^n{P}_C{x}_n)+{\lambda }_n{e}_n,n\ge 1.\hfill \end{array}$$

Assume that the following restrictions are satisfied:

• (a) ${\alpha }_n+{\beta }_n+{\lambda }_n=1$;

• (b) ${\sum }_{n=1}^{\mathrm{\infty }}(k_n-1)<\mathrm{\infty }$ and ${\sum }_{n=1}{\xi }_n<\mathrm{\infty }$, where ${\xi }_n$ is defined in (2.5);

• (c) $0<a\le {\beta }_n\le b<1$, $0\le \kappa \le {\delta }_n\le c<1$ and ${\sum }_{n=1}^{\mathrm{\infty }}{\lambda }_n<\mathrm{\infty }$;

where a, b, and c are some real constants. Then the sequence$\{x_n\}$converges weakly to some point in$F(S)$.

For the class of asymptotically nonexpansive mappings in the intermediate sense, we can obtain from Theorem 3.1 the following results immediately.

Corollary 3.4 Let C be a nonempty, closed, and convex subset of a real Hilbert space H. Let$F_m$be a bifunction from$C\times C$to$\mathbb{R}$which satisfies (A 1)-(A 4) for each$1\le m\le N$, where$N\ge 1$is some positive integer. Let$S:C\to C$be a uniformly continuous and asymptotically nonexpansive mapping in the intermediate sense. Assume that$\mathcal{F}:=F(S)\cap {\bigcap }_{m=1}^{N}EP(F_m)$is nonempty. Let$\{{\alpha }_n\}$, $\{{\beta }_n\}$, $\{{\delta }_n\}$, and$\{{\lambda }_n\}$be sequences in$[0,1]$, and$\{e_n\}$a bounded sequence in C. Let$\{r_{n,m}\}$be a positive sequence such that${lim\hspace{0.17em}inf}_{n\to \mathrm{\infty }}{r}_{n,m}>0$and$\{{\gamma }_{n,m}\}$a sequence in$[0,1]$for each$1\le m\le N$. Let$\{x_n\}$be a sequence generated in the following manner:

$$\{\begin{array}{c}{x}_1\in H,\hfill \\ u_{n,m}\in C\mathit{\text{such that}}{F}_m(u_{n,m},u_m)+\frac{1}{r_{n,m}}〈u_m-u_{n,m},u_{n,m}-x_n〉\ge 0,\mathrm{\forall }{u}_m\in C,\hfill \\ z_n={\sum }_{m=1}^N{\gamma }_{n,m}{u}_{n,m},\hfill \\ x_{n+1}={\alpha }_n{x}_n+{\beta }_n({\delta }_n{z}_n+(1-{\delta }_n)S^n{z}_n)+{\lambda }_n{e}_n,n\ge 1.\hfill \end{array}$$

Assume that the following restrictions are satisfied:

• (a) ${\alpha }_n+{\beta }_n+{\lambda }_n=1$;

• (b) ${\sum }_{n=1}^{\mathrm{\infty }}(k_n-1)<\mathrm{\infty }$ and ${\sum }_{n=1}{\xi }_n<\mathrm{\infty }$, where

  $${\xi }_n=max\{0,\underset{x,y\in C}{sup}({\parallel S^{n}x-S^{n}y\parallel }^2-k_n{\parallel x-y\parallel }^2)\};$$

• (c) $0<a\le {\beta }_n\le b<1$, $0\le \kappa \le {\delta }_n\le c<1$ and ${\sum }_{n=1}^{\mathrm{\infty }}{\lambda }_n<\mathrm{\infty }$;

• (d) ${\sum }_{m=1}^N{\gamma }_{n,m}=1$ and $0<d\le {\gamma }_{n,m}\le 1$ for each $1\le m\le N$,

where a, b, c, and d are some real constants. Then the sequence$\{x_n\}$converges weakly to some point in$\mathcal{F}$.