Hybrid methods for a mixed equilibrium problem and fixed points of a countable family of multivalued nonexpansive mappings

Abstract

In this paper, we prove a strong convergence theorem for a new hybrid method, using shrinking projection method introduced by Takahashi and a fixed point method for finding a common element of the set of solutions of mixed equilibrium problem and the set of common fixed points of a countable family of multivalued nonexpansive mappings in Hilbert spaces. We also apply our main result to the convex minimization problem and the fixed point problem of a countable family of multivalued nonexpansive mappings.

MSC:47H09, 47H10.

1 Introduction

The mixed equilibrium problem (MEP) includes several important problems arising in optimization, economics, physics, engineering, transportation, network, Nash equilibrium problems in noncooperative games, and others. Variational inequalities and mathematical programming problems are also viewed as the abstract equilibrium problems (EP) (e.g., [1, 2]). Many authors have proposed several methods to solve the EP and MEP, see, for instance, [1–9] and the references therein.

Fixed point problems for multivalued mappings are more difficult than those of single-valued mappings and play very important role in applied science and economics. Recently, many authors have proposed their fixed point methods for finding a fixed point of both multivalued mapping and a family of multivalued mappings. All of those methods have only weak convergence.

It is known that Mann’s iterations have only weak convergence even in the Hilbert spaces. To overcome this problem, Takahashi [10] introduced a new method, known as shrinking projection method, which is a hybrid method of Mann’s iteration, and the projection method, and obtained strong convergence results of such method. In this paper, we use the shrinking projection method defined by Takahashi [10] and our new method to define a new hybrid method for MEP and a fixed point problem for a family of nonexpansive multivalued mappings.

An element $p\in K$ is called a fixed point of a single-valued mapping T if $p=Tp$ and of a multivalued mapping T if $p\in Tp$. The set of fixed points of T is denoted by $F(T)$.

Let X be a real Banach space. A subset K of X is called proximinal if for each $x\in X$, there exists an element $k\in K$ such that

$$d(x,k)=d(x,K),$$

where $d(x,K)=inf\{\parallel x-y\parallel :y\in K\}$ is the distance from the point x to the set K.

Let X be a uniformly convex real Banach space, and let K be a nonempty closed convex subset of X, and let $CB(K)$ be a family of nonempty closed bounded subsets of K, and let $P(K)$ be a nonempty proximinal bounded subsets of K.

For multivalued mappings $T:K\to P(K)$, define $P_T(x):=\{y\in T(x):\parallel x-y\parallel =d(x,T(x))\}$ for all $x\in K$.

The Hausdorff metric on $CB(X)$ is defined by

$$H(A,B)=max\{\underset{x\in A}{sup}d(x,B),\underset{y\in B}{sup}d(y,A)\}$$

for all $A,B\in CB(X)$.

A multivalued mapping $T:K\to CB(K)$ is said to be nonexpansive if $H(Tx,Ty)\le \parallel x-y\parallel$ for all $x,y\in K$.

Let H be a real Hilbert space with the inner product $〈\cdot ,\cdot 〉$ and the norm $\parallel \cdot \parallel$. Let D be a nonempty closed convex subset of H. Let $F:D\times D\to \mathbb{R}$ be a bifunction, and let $\phi :D\to \mathbb{R}\cup \{+\mathrm{\infty }\}$ be a function such that $D\cap dom\phi \ne \mathrm{\varnothing }$, where ℝ is the set of real numbers and $dom\phi =\{x\in H:\phi (x)<+\mathrm{\infty }\}$.

Flores-Bazán [11] introduced the following mixed equilibrium problem:

$$\text{Find }x\in D\text{ such that }F(x,y)+\phi (y)\ge \phi (x),\mathrm{\forall }y\in D.$$

(1.1)

The set of solutions of (1.1) is denoted by $MEP(F,\phi )$.

If $\phi \equiv 0$, then the mixed equilibrium problem (1.1) reduces to the following equilibrium problem:

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

(1.2)

The set of solutions of (1.2) is denoted by $EP(F)$ (see Combettes and Hirstoaga [12]).

If $F\equiv 0$, then the mixed equilibrium problem (1.1) reduces to the following convex minimization problem:

$$\text{Find }x\in D\text{ such that }\phi (y)\ge \phi (x),\mathrm{\forall }y\in D.$$

(1.3)

The set of solutions of (1.3) is denoted by $CMP(\phi )$.

In an infinite-dimensional Hilbert space, the Mann iteration algorithms have only a weak convergence. In 2003, Nakajo and Takahashi [13] introduced the method, called CQ method, to modify Mann’s iteration to obtain the strong convergence theorem for nonexpansive mapping in a Hilbert space. The CQ method has been studied extensively by many authors, for instance, Marino and Xu [14]; Zhou [15]; Zhang and Cheng [16].

In 2008, Takahashi et al. [10] introduced the following iteration scheme, which is usually called the shrinking projection method. Let $\{{\alpha }_n\}$ be a sequence in $(0,1)$ and $x_0\in H$. For $C_1=C$ and $x_1=P_{C_1}{x}_0$, define a sequence $\{x_n\}$ of D as follows:

$$\{\begin{array}{l}{y}_n=(1-{\alpha }_n)x_n+{\alpha }_n{T}_n{x}_n,\\ C_{n+1}=\{z\in C_n:\parallel y_n-z\parallel \le \parallel x_n-z\parallel \},\\ x_{n+1}=P_{C_{n+1}}{x}_0,n\ge 1,\end{array}$$

where $P_{C_n}$ is the metric projection of H onto $C_n$ and $\{T_n\}$ is a family of nonexpansive mappings. They proved that the sequence $\{x_n\}$ converges strongly to $z=P_{F(T)}{x}_0$, where $F(T)={\bigcap }_{n=1}^{\mathrm{\infty }}F(T_n)$. The shrinking projection method has been studied widely by many authors, for example, Tada and Takahashi [17]; Aoyama et al. [18]; Yao et al. [19]; Kang et al. [20]; Cholamjiak and Suantai [21]; Ceng et al. [22]; Tang et al. [23]; Cai and Bu [24]; Kumam et al. [25]; Kimura et al. [26]; Shehu [27, 28]; Wang et al. [29].

In 2009, Wangkeeree and Wangkeeree [30] proved a strong convergence theorem of an iterative algorithm based on extragradient method for finding a common element of the set of solutions of a mixed equilibrium problem, the set of common fixed points of a family of infinitely nonexpansive mappings and the set of the variational inequality for a monotone Lipschitz continuous mapping in a Hilbert space.

In 2011, Rodjanadid [31] introduced another iterative method modified from an iterative scheme of Klin-eam and Suantai [32] for finding a common element of the set of solutions of mixed equilibrium problems and the set of common fixed points of countable family of nonexpansive mappings in real Hilbert spaces. The mixed equilibrium problems have been studied by many authors, for instance, Peng and Yao [33]; Zeng et al. [34]; Peng et al. [35]; Wangkeeree and Kamraksa [36]; Jaiboon and Kumam [37]; Chamnarnpan and Kumam [38]; Cholamjiak et al. [39].

Nadler [40] started to study fixed points of multivalued contractions and nonexpansive mapping by using the Hausdorff metric.

Sastry and Babu [41] defined Mann and Ishikawa iterates for a multivalued map T with a fixed point p, and proved that these iterates converge strongly to a fixed point q of T under the compact domain in a real Hilbert space. Moreover, they illustrated that fixed point q may be different from p.

Panyanak [42] generalized results of Sastry and Babu [41] to uniformly convex Banach spaces and proved a strong convergence theorem of Mann iterates for a mapping defined on a noncompact domain and satisfying some conditions. He also obtained a strong convergence result of Ishikawa iterates for a mapping defined on a compact domain.

Hussain and Khan [43], in 2003, introduced the best approximation operator $P_T$ to find fixed points of ∗-nonexpansive multivalued mapping and proved strong convergence of its iterates on a closed convex unbounded subset of a Hilbert space, which is not necessarily compact.

Hu et al. [44] obtained common fixed point of two nonexpansive multivalued mappings satisfying certain contractive conditions.

Cholamjiak and Suantai [45] proved strong convergence theorems of two new iterative procedures with errors for two quasi-nonexpansive multivalued mappings by using the best approximation operator and the end point condition in uniformly convex Banach spaces. Later, Cholamjiak et al. [46] introduced a modified Mann iteration and obtained weak and strong convergence theorems for a countable family of nonexpansive multivalued mappings by using the best approximation operator in a Banach space. They also gave some examples of multivalued mappings T such that $P_T$ are nonexpansive.

Later, Eslamian and Abkar [47] generalized and modified the iteration of Abbas et al. [48] from two mappings to the infinite family of multivalued mappings $\{T_i\}$ such that each $P_{T_i}$ satisfies the condition (C).

In this paper, we introduce a new hybrid method for finding a common element of the set of solutions of a mixed equilibrium problem and the set of common fixed points of a countable family of multivalued nonexpansive mappings in Hilbert spaces. We obtain a strong convergence theorem for the sequences generated by the proposed method without the assumption of compactness of the domain and other conditions imposing on the mappings.

In Section 2, we give some preliminaries and lemmas, which will be used in proving the main results. In Section 3, we introduce a new hybrid method and a fixed point method defined by (3.1) and prove strong convergence theorem for finding a common element of the set of solutions between mixed equilibrium problem and common fixed point problems of a countable family of multivalued nonexpansive mappings in Hilbert spaces. We also give examples of the control sequences satisfying the control conditions in main results. In Section 4, we summarize the main results of this paper.

2 Preliminaries

Let D be a closed convex subset of H. For every point $x\in H$, there exists a unique nearest point in D, denoted by $P_{D}x$, such that

$$\parallel x-P_{D}x\parallel \le \parallel x-y\parallel ,\mathrm{\forall }y\in D.$$

$P_D$ is called the metric projection of H onto D. It is known that $P_D$ is a nonexpansive mapping of H onto D. It is also know that $P_D$ satisfies $〈x-y,P_{D}x-P_{D}y〉\ge {\parallel P_{D}x-P_{D}y\parallel }^2$ for every $x,y\in H$. Moreover, $P_{D}x$ is characterized by the properties: $P_{D}x\in D$ and $〈x-P_{D}x,P_{D}x-y〉\ge 0$ for all $y\in D$.

Lemma 2.1 [13]

Let D be a nonempty closed convex subset of a real Hilbert space H and $P_D:H\to D$ be the metric projection from H onto D. Then the following inequality holds:

$${\parallel y-P_{D}x\parallel }^2+{\parallel x-P_{D}x\parallel }^2\le {\parallel x-y\parallel }^2,\mathrm{\forall }x\in H,\mathrm{\forall }y\in D.$$

Lemma 2.2 [14]

Let H be a real Hilbert space. Then the following equations hold:

• (i) ${\parallel x-y\parallel }^2={\parallel x\parallel }^2-{\parallel y\parallel }^2-2〈x-y,y〉$, $\mathrm{\forall }x,y\in H$;

• (ii) ${\parallel tx+(1-t)y\parallel }^2=t{\parallel x\parallel }^2+(1-t){\parallel y\parallel }^2-t(1-t){\parallel x-y\parallel }^2$, $\mathrm{\forall }t\in [0,1]$ and $x,y\in H$.

Lemma 2.3 [21]

Let H be a real Hilbert space. Then for each $m\in \mathbb{N}$

$${\parallel \sum _{i=1}^m{t}_i{x}_i\parallel }^2=\sum _{i=1}^m{t}_i{\parallel x_i\parallel }^2-\sum _{i=1,i\ne j}^m{t}_i{t}_j{\parallel x_i-x_j\parallel }^2,$$

$x_i\in H$ and $t_i,t_j\in [0,1]$ for all $i,j=1,2,\dots ,m$ with ${\sum }_{i=1}^m{t}_i=1$.

Lemma 2.4 [49]

Let D be a nonempty closed and convex subset of a real Hilbert space H. Given $x,y,z\in H$ and also given $a\in \mathbb{R}$, the set

$$\{v\in D:{\parallel y-v\parallel }^2\le {\parallel x-v\parallel }^2+〈z,v〉+a\}$$

is convex and closed.

For solving the mixed equilibrium problem, we assume the bifunction F, φ and the set D satisfy the following conditions:

• (A1) $F(x,x)=0$ for all $x\in D$;

• (A2) F is monotone, that is, $F(x,y)+F(y,x)\le 0$ for all $x,y\in D$;

• (A3) for each $x,y,z\in D$, ${lim\hspace{0.17em}sup}_{t\downarrow 0}F(tz+(1-t)x,y)\le F(x,y)$;

• (A4) $F(x,\cdot )$ is convex and lower semicontinuous for each $x\in D$;

• (B1) for each $x\in H$ and $r>0$, there exist a bounded subset $D_x\subseteq D$ and $y_x\in D\cap dom\phi$ such that for any $z\in D\setminus D_x$,

  $$F(z,y_x)+\phi (y_x)+\frac{1}{r}〈y_x-z,z-x〉<\phi (z);$$

• (B2) D is a bounded set.

Lemma 2.5 [35]

Let D be a nonempty closed and convex subset of a real Hilbert space H. Let $F:D\times D\to \mathbb{R}$ be a bifunction satisfying conditions (A1)-(A4) and $\phi :D\to \mathbb{R}\cup \{+\mathrm{\infty }\}$ be a proper lower semicontinuous and convex function such that $D\cap dom\phi \ne \mathrm{\varnothing }$. For $r>0$ and $x\in D$, define a mapping $T_r:H\to D$ as follows:

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

for all $x\in H$. Assume that either (B1) or (B2) holds. Then the following conclusions hold:

• (1) for each $x\in H$, $T_r(x)\ne \mathrm{\varnothing }$;

• (2) $T_r$ is single-valued;

• (3) $T_r$ is firmly nonexpansive, that is, for any $x,y\in H$,

  $${\parallel T_r(x)-T_r(y)\parallel }^2\le 〈T_r(x)-T_r(y),x-y〉;$$

• (4) $F(T_r)=MEP(F,\phi )$;

• (5) $MEP(F,\phi )$ is closed and convex.

As in ([21], Lemma 2.7), the following lemma holds true for multivalued mapping. To avoid repetition, we omit the details of proof.

Lemma 2.6 Let D be a closed and convex subset of a real Hilbert space H. Let $T:D\to P(D)$ be a multivalued nonexpansive mapping with $F(T)\ne \mathrm{\varnothing }$ such that $P_T$ is nonexpansive. Then $F(T)$ is a closed and convex subset of D.

3 Main results

In the following theorem, we prove strong convergence of the sequence $\{x_n\}$ defined by (3.1) to a common element of the set of solutions of a mixed equilibrium problem and the set of common fixed points of a countable family of multivalued nonexpansive mappings.

Theorem 3.1 Let D be a nonempty closed and convex subset of a real Hilbert space H. Let F be a bifunction from $D\times D$ to ℝ satisfying (A1)-(A4), and let φ be a proper lower semicontinuous and convex function from D to $\mathbb{R}\cup \{+\mathrm{\infty }\}$ such that $D\cap dom\phi \ne \mathrm{\varnothing }$. Let $T_i:D\to P(D)$ be multivalued nonexpansive mappings for all $i\in \mathbb{N}$ with $\mathrm{\Omega }:={\bigcap }_{i=1}^{\mathrm{\infty }}F(T_i)\cap MEP(F,\phi )\ne \mathrm{\varnothing }$ such that all $P_{T_i}$ are nonexpansive. Assume that either (B1) or (B2) holds and $\{{\alpha }_{n,i}\}\subset [0,1)$ satisfies the condition ${lim\hspace{0.17em}inf}_{n\to \mathrm{\infty }}{\alpha }_{n,i}{\alpha }_{n,0}>0$ for all $i\in \mathbb{N}$. Define the sequence $\{x_n\}$ as follows: $x_1\in D=C_1$,

$$\{\begin{array}{l}F(u_n,y)+\phi (y)-\phi (u_n)+\frac{1}{r_n}〈y-u_n,u_n-x_n〉\ge 0,\mathrm{\forall }y\in D,\\ y_n={\alpha }_{n,0}{u}_n+{\sum }_{i=1}^n{\alpha }_{n,i}{x}_{n,i},\\ C_{n+1}=\{z\in C_n:\parallel y_n-z\parallel \le \parallel x_n-z\parallel \},\\ x_{n+1}=P_{C_{n+1}}{x}_0,n\ge 0,\end{array}$$

(3.1)

where the sequences $r_n\in (0,\mathrm{\infty })$ with ${lim\hspace{0.17em}inf}_{n\to \mathrm{\infty }}{r}_n>0$ and $\{{\alpha }_{n,i}\}\subset [0,1)$ satisfying ${\sum }_{i=0}^n{\alpha }_{n,i}=1$ and $x_{n,i}\in P_{T_i}{u}_n$ for $i\in \mathbb{N}$. Then the sequence $\{x_n\}$ converges strongly to $P_{\mathrm{\Omega }}{x}_0$.

Proof We split the proof into six steps.

Step 1. Show that $P_{C_{n+1}}{x}_0$ is well defined for every $x_0\in D$.

By Lemmas 2.5-2.6, we obtain that $MEP(F,\phi )$ and ${\bigcap }_{i=1}^{\mathrm{\infty }}F(T_i)$ is a closed and convex subset of D. Hence Ω is a closed and convex subset of D. It follows from Lemma 2.4 that $C_{n+1}$ is a closed and convex for each $n\ge 0$. Let $v\in \mathrm{\Omega }$. Then $P_{T_i}(v)=\{v\}$ for all $i\in \mathbb{N}$. Since $u_n=T_{r_n}{x}_n\in dom\phi$, we have

$$\parallel u_n-v\parallel =\parallel T_{r_n}{x}_n-T_{r_n}v\parallel \le \parallel x_n-v\parallel ,$$

for every $n\ge 0$. Then

$$\begin{array}{rl}\parallel y_n-v\parallel & =\parallel {\alpha }_{n,0}{u}_n+\sum _{i=1}^n{\alpha }_{n,i}{x}_{n,i}-v\parallel \\ \le {\alpha }_{n,0}\parallel u_n-v\parallel +\sum _{i=1}^n{\alpha }_{n,i}\parallel x_{n,i}-v\parallel \\ ={\alpha }_{n,0}\parallel u_n-v\parallel +\sum _{i=1}^n{\alpha }_{n,i}d(x_{n,i},P_{T_i}v)\\ \le {\alpha }_{n,0}\parallel u_n-v\parallel +\sum _{i=1}^n{\alpha }_{n,i}H(P_{T_i}{u}_n,P_{T_i}v)\\ \le {\alpha }_{n,0}\parallel u_n-v\parallel +\sum _{i=1}^n{\alpha }_{n,i}\parallel u_n-v\parallel \\ =\parallel u_n-v\parallel \le \parallel x_n-v\parallel .\end{array}$$

(3.2)

Hence $v\in C_{n+1}$, so that $\mathrm{\Omega }\subset C_{n+1}$. Therefore, $P_{C_{n+1}}{x}_0$ is well defined.

Step 2. Show that ${lim}_{n\to \mathrm{\infty }}\parallel x_n-x_0\parallel$ exists.

Since Ω is a nonempty closed convex subset of H, there exists a unique $v\in \mathrm{\Omega }$ such that $v=P_{\mathrm{\Omega }}{x}_0$. Since $x_n=P_{C_n}{x}_0$ and $x_{n+1}\in C_{n+1}\subset C_n$, $\mathrm{\forall }n\ge 0$, we have

$$\parallel x_n-x_0\parallel \le \parallel x_{n+1}-x_0\parallel ,\mathrm{\forall }n\ge 0.$$

On the other hand, as $v\in \mathrm{\Omega }\subset C_n$, we obtain

$$\parallel x_n-x_0\parallel \le \parallel v-x_0\parallel ,\mathrm{\forall }n\ge 0.$$

It follows that the sequence $\{x_n\}$ is bounded and nondecreasing. Therefore, ${lim}_{n\to \mathrm{\infty }}\parallel x_n-x_0\parallel$ exists.

Step 3. Show that ${lim}_{n\to \mathrm{\infty }}{x}_n=w\in D$.

For $m>n$, by the definition of $C_n$, we get $x_m=P_{C_m}{x}_0\in C_m\subset C_n$. By applying Lemma 2.1, we have

$${\parallel x_m-x_n\parallel }^2\le {\parallel x_m-x_0\parallel }^2-{\parallel x_n-x_0\parallel }^2.$$

Since ${lim}_{n\to \mathrm{\infty }}\parallel x_n-x_0\parallel$ exists, it follows that $\{x_n\}$ is Cauchy. Hence there exists $w\in D$ such that ${lim}_{n\to \mathrm{\infty }}{x}_n=w$.

Step 4. Show that $\parallel x_{n,i}-x_n\parallel \to 0$ as $n\to \mathrm{\infty }$ for every $i\in \mathbb{N}$.

From $x_{n+1}\in C_{n+1}$, we have

$$\begin{array}{rl}\parallel x_n-y_n\parallel & \le \parallel x_n-x_{n+1}\parallel +\parallel x_{n+1}-y_n\parallel \\ \le 2\parallel x_n-x_{n+1}\parallel \to 0\text{as }n\to \mathrm{\infty }.\end{array}$$

(3.3)

For $v\in \mathrm{\Omega }$, by Lemma 2.3 and (3.2), we get

$$\begin{array}{rl}{\parallel y_n-v\parallel }^2& ={\parallel {\alpha }_{n,0}(u_n-v)+\sum _{i=1}^n{\alpha }_{n,i}(x_{n,i}-v)\parallel }^2\\ \le {\alpha }_{n,0}{\parallel u_n-v\parallel }^2+\sum _{i=1}^n{\alpha }_{n,i}{\parallel x_{n,i}-v\parallel }^2-\sum _{i=1}^n{\alpha }_{n,i}{\alpha }_{n,0}{\parallel x_{n,i}-u_n\parallel }^2\\ ={\alpha }_{n,0}{\parallel u_n-v\parallel }^2+\sum _{i=1}^n{\alpha }_{n,i}d{(x_{n,i},P_{T_i}v)}^2-\sum _{i=1}^n{\alpha }_{n,i}{\alpha }_{n,0}{\parallel x_{n,i}-u_n\parallel }^2\\ \le {\alpha }_{n,0}{\parallel u_n-v\parallel }^2+\sum _{i=1}^n{\alpha }_{n,i}H{(P_{T_i}{u}_n,P_{T_i}v)}^2-\sum _{i=1}^n{\alpha }_{n,i}{\alpha }_{n,0}{\parallel x_{n,i}-u_n\parallel }^2\\ \le {\alpha }_{n,0}{\parallel u_n-v\parallel }^2+\sum _{i=1}^n{\alpha }_{n,i}{\parallel u_n-v\parallel }^2-\sum _{i=1}^n{\alpha }_{n,i}{\alpha }_{n,0}{\parallel x_{n,i}-u_n\parallel }^2\\ ={\parallel u_n-v\parallel }^2-\sum _{i=1}^n{\alpha }_{n,i}{\alpha }_{n,0}{\parallel x_{n,i}-u_n\parallel }^2\\ \le {\parallel x_n-v\parallel }^2-\sum _{i=1}^n{\alpha }_{n,i}{\alpha }_{n,0}{\parallel x_{n,i}-u_n\parallel }^2.\end{array}$$

This implies that

$$\begin{array}{rl}{\alpha }_{n,i}{\alpha }_{n,0}{\parallel x_{n,i}-u_n\parallel }^2& \le \sum _{i=1}^n{\alpha }_{n,i}{\alpha }_{n,0}{\parallel x_{n,i}-u_n\parallel }^2\\ \le {\parallel x_n-v\parallel }^2-{\parallel y_n-v\parallel }^2\\ \le M\parallel x_n-y_n\parallel ,\end{array}$$

where $M={sup}_{n\ge 0}\{\parallel x_n-v\parallel +\parallel y_n-v\parallel \}$. By the given control condition on $\{{\alpha }_{n,i}\}$ and (3.3), we obtain

$$\underset{n\to \mathrm{\infty }}{lim}\parallel x_{n,i}-u_n\parallel =0,\mathrm{\forall }i\in \mathbb{N}.$$

By Lemma 2.5, we have

$$\begin{array}{rl}{\parallel u_n-v\parallel }^2& ={\parallel T_{r_n}{x}_n-T_{r_n}v\parallel }^2\\ \le 〈T_{r_n}{x}_n-T_{r_n}v,x_n-v〉\\ =〈u_n-v,x_n-v〉\\ =\frac{1}{2}\{{\parallel u_n-v\parallel }^2+{\parallel x_n-v\parallel }^2-{\parallel x_n-u_n\parallel }^2\}.\end{array}$$

Hence ${\parallel u_n-v\parallel }^2\le {\parallel x_n-v\parallel }^2-{\parallel x_n-u_n\parallel }^2$. By Lemma 2.3, we get

$$\begin{array}{rl}{\parallel y_n-v\parallel }^2& ={\parallel {\alpha }_{n,0}{u}_n+\sum _{i=1}^n{\alpha }_{n,i}{x}_{n,i}-v\parallel }^2\\ ={\alpha }_{n,0}{\parallel u_n-v\parallel }^2+\sum _{i=1}^n{\alpha }_{n,i}{\parallel x_{n,i}-v\parallel }^2-\sum _{i=1}^n{\alpha }_{n,i}{\alpha }_{n,0}{\parallel x_{n,i}-u_n\parallel }^2\\ \le {\alpha }_{n,0}{\parallel u_n-v\parallel }^2+\sum _{i=1}^n{\alpha }_{n,i}{\parallel x_{n,i}-v\parallel }^2\\ ={\alpha }_{n,0}{\parallel u_n-v\parallel }^2+\sum _{i=1}^n{\alpha }_{n,i}d{(x_{n,i},P_{T_i}v)}^2\\ \le {\alpha }_{n,0}{\parallel u_n-v\parallel }^2+\sum _{i=1}^n{\alpha }_{n,i}H{(P_{T_i}{u}_n,P_{T_i}v)}^2\\ \le {\alpha }_{n,0}{\parallel u_n-v\parallel }^2+\sum _{i=1}^n{\alpha }_{n,i}{\parallel u_n-v\parallel }^2\\ ={\parallel u_n-v\parallel }^2\\ \le {\parallel x_n-v\parallel }^2-{\parallel x_n-u_n\parallel }^2.\end{array}$$

This implies that

$$\begin{array}{rl}{\parallel x_n-u_n\parallel }^2& \le {\parallel x_n-v\parallel }^2-{\parallel y_n-v\parallel }^2\\ \le M\parallel x_n-y_n\parallel ,\end{array}$$

where $M={sup}_{n\ge 0}\{\parallel x_n-v\parallel +\parallel y_n-v\parallel \}$. From (3.3), we get ${lim}_{n\to \mathrm{\infty }}\parallel x_n-u_n\parallel =0$. It follows that

$$\begin{array}{rl}\parallel x_{n,i}-x_n\parallel & \le \parallel x_{n,i}-u_n\parallel +\parallel u_n-x_n\parallel \\ \to 0\text{as }n\to \mathrm{\infty }.\end{array}$$

Step 5. Show that $w\in \mathrm{\Omega }$.

By ${lim\hspace{0.17em}inf}_{n\to \mathrm{\infty }}{r}_n>0$, we have

$$\parallel \frac{x_n-u_n}{r_n}\parallel =\frac{1}{r_n}\parallel x_n-u_n\parallel \to 0,n\to \mathrm{\infty }.$$

(3.4)

From ${lim}_{n\to \mathrm{\infty }}{x}_n=w$, we obtain ${lim}_{n\to \mathrm{\infty }}{u}_n=w$.

We will show that $w\in MEP(F,\phi )$. Since $u_n=T_{r_n}{x}_n\in dom\phi$, we have

$$F(u_n,y)+\phi (y)-\phi (u_n)+\frac{1}{r_n}〈y-u_n,u_n-x_n〉\ge 0,\mathrm{\forall }y\in D.$$

It follows by (A2) that

$$\phi (y)-\phi (u_n)+\frac{1}{r_n}〈y-u_n,u_n-x_n〉\ge F(y,u_n),\mathrm{\forall }y\in D.$$

Hence

$$\phi (y)-\phi (u_n)+〈y-u_n,\frac{u_n-x_n}{r_n}〉\ge F(y,u_n),\mathrm{\forall }y\in D.$$

It follows from (3.4), (A4) and the lower semicontinuous of φ that

$$F(y,w)+\phi (w)-\phi (y)\le 0,\mathrm{\forall }y\in D.$$

For t with $0<t\le 1$ and $y\in D$, let $y_t=ty+(1-t)w$. Since $y,w\in D$ and D is convex, then $y_t\in D$ and hence

$$F(y_t,w)+\phi (w)-\phi (y_t)\le 0.$$

This implies by (A1), (A4) and the convexity of φ, that

$$\begin{array}{rl}0& =F(y_t,y_t)+\phi (y_t)-\phi (y_t)\\ \le tF(y_t,y)+(1-t)F(y_t,w)+t\phi (y)+(1-t)\phi (w)-\phi (y_t)\\ \le t[F(y_t,y)+\phi (y)-\phi (y_t)].\end{array}$$

Dividing by t, we have

$$F(y_t,y)+\phi (y)-\phi (y_t)\ge 0,\mathrm{\forall }y\in D.$$

Letting $t\to 0$, it follows from the weakly semicontinuity of φ that

$$F(w,y)+\phi (y)-\phi (w)\ge 0,\mathrm{\forall }y\in D.$$

Hence $w\in MEP(F,\phi )$. Next, we will show that $w\in {\bigcap }_{i=1}^{\mathrm{\infty }}F(T_i)$. For each $i=1,2,\dots ,n$, we have

$$\begin{array}{rl}d(w,T_{i}w)& \le d(w,x_n)+d(x_n,x_{n,i})+d(x_{n,i},T_{i}w)\\ \le d(w,x_n)+d(x_n,x_{n,i})+H(T_i{u}_n,T_{i}w)\\ \le d(w,x_n)+d(x_n,x_{n,i})+d(u_n,w).\end{array}$$

By Steps 3-4, we have $d(w,T_{i}w)=0$. Hence $w\in T_{i}w$ for all $i=1,2,\dots ,n$.

Step 6. Show that $w=P_{\mathrm{\Omega }}{x}_0$.

Since $x_n=P_{C_n}{x}_0$, we get

$$〈z-x_n,x_0-x_n〉\le 0,\mathrm{\forall }z\in C_n.$$

Since $w\in \mathrm{\Omega }\subset C_n$, we have

$$〈z-w,x_0-w〉\le 0,\mathrm{\forall }z\in \mathrm{\Omega }.$$

Now, we obtain that $w=P_{\mathrm{\Omega }}{x}_0$.

This completes the proof. □

Setting $\phi \equiv 0$ in Theorem 3.1, we have the following result.

Corollary 3.2 Let D be a nonempty closed and convex subset of a real Hilbert space H. Let F be a bifunction from $D\times D$ to ℝ satisfying (A1)-(A4). Let $T_i:D\to P(D)$ be multivalued nonexpansive mappings for all $i\in \mathbb{N}$ with $\mathrm{\Omega }:={\bigcap }_{i=1}^{\mathrm{\infty }}F(T_i)\cap EP(F)\ne \mathrm{\varnothing }$ such that all $P_{T_i}$ are nonexpansive. Assume that $\{{\alpha }_{n,i}\}\subset [0,1)$ satisfies the condition ${lim\hspace{0.17em}inf}_{n\to \mathrm{\infty }}{\alpha }_{n,i}{\alpha }_{n,0}>0$ for all $i\in \mathbb{N}$. Define the sequence $\{x_n\}$ as follows: $x_1\in D=C_1$,

$$\{\begin{array}{l}F(u_n,y)+\frac{1}{r_n}〈y-u_n,u_n-x_n〉\ge 0,\mathrm{\forall }y\in D,\\ y_n={\alpha }_{n,0}{u}_n+{\sum }_{i=1}^n{\alpha }_{n,i}{x}_{n,i},\\ C_{n+1}=\{z\in C_n:\parallel y_n-z\parallel \le \parallel x_n-z\parallel \},\\ x_{n+1}=P_{C_{n+1}}{x}_0,n\ge 0,\end{array}$$

(3.5)

where the sequences $r_n\in (0,\mathrm{\infty })$ with ${lim\hspace{0.17em}inf}_{n\to \mathrm{\infty }}{r}_n>0$ and $\{{\alpha }_{n,i}\}\subset [0,1)$ satisfying ${\sum }_{i=0}^n{\alpha }_{n,i}=1$ and $x_{n,i}\in P_{T_i}{u}_n$ for $i\in \mathbb{N}$. Then the sequence $\{x_n\}$ converges strongly to $P_{\mathrm{\Omega }}{x}_0$.

Setting $F\equiv 0$ in Theorem 3.1, we have the following result.

Corollary 3.3 Let D be a nonempty closed and convex subset of a real Hilbert space H. Let φ be a proper lower semicontinuous and convex function from D to $\mathbb{R}\cup \{+\mathrm{\infty }\}$ such that $D\cap dom\phi \ne \mathrm{\varnothing }$. Let $T_i:D\to P(D)$ be multivalued nonexpansive mappings for all $i\in \mathbb{N}$ with $\mathrm{\Omega }:={\bigcap }_{i=1}^{\mathrm{\infty }}F(T_i)\cap CMP(\phi )\ne \mathrm{\varnothing }$ such that all $P_{T_i}$ are nonexpansive. Assume that either (B1) or (B2) holds, and $\{{\alpha }_{n,i}\}\subset [0,1)$ satisfies the condition ${lim\hspace{0.17em}inf}_{n\to \mathrm{\infty }}{\alpha }_{n,i}{\alpha }_{n,0}>0$ for all $i\in \mathbb{N}$. Define the sequence $\{x_n\}$ as follows: $x_1\in D=C_1$,

$$\{\begin{array}{l}\phi (y)-\phi (u_n)+\frac{1}{r_n}〈y-u_n,u_n-x_n〉\ge 0,\mathrm{\forall }y\in D,\\ y_n={\alpha }_{n,0}{u}_n+{\sum }_{i=1}^n{\alpha }_{n,i}{x}_{n,i},\\ C_{n+1}=\{z\in C_n:\parallel y_n-z\parallel \le \parallel x_n-z\parallel \},\\ x_{n+1}=P_{C_{n+1}}{x}_0,n\ge 0,\end{array}$$

(3.6)

where the sequences $r_n\in (0,\mathrm{\infty })$ with ${lim\hspace{0.17em}inf}_{n\to \mathrm{\infty }}{r}_n>0$ and $\{{\alpha }_{n,i}\}\subset [0,1)$ satisfying ${\sum }_{i=0}^n{\alpha }_{n,i}=1$ and $x_{n,i}\in P_{T_i}{u}_n$ for $i\in \mathbb{N}$. Then the sequence $\{x_n\}$ converges strongly to $P_{\mathrm{\Omega }}{x}_0$.

Remark 3.4

• (i) Let $\{{\alpha }_{n,i}\}$ be double sequence in $(0,1]$. Let (a) and (b) be the following conditions:

  • (a) ${lim\hspace{0.17em}inf}_{n\to \mathrm{\infty }}{\alpha }_{n,i}{\alpha }_{n,0}>0$ for all $i\in \mathbb{N}$,

  • (b) ${lim}_{n\to \mathrm{\infty }}{\alpha }_{n,i}$ exist and lie in $(0,1]$ for all $i=0,1,2,\dots$ .

  It is easy to see that if $\{{\alpha }_{n,i}\}$ satisfies the condition (a), then it satisfies the condition (b). So Theorem 3.1 and Corollaries 3.2-3.3 hold true when the control double sequence $\{{\alpha }_{n,i}\}$ satisfies the condition (a).

• (ii) The following double sequences are examples of the control sequences in Theorem 3.1 and Corollaries 3.2-3.3:

  • (1)
    $${\alpha }_{n,k}=\{\begin{array}{ll}\frac{1}{2^k}(\frac{n}{n+1}),& n\ge k;\\ 1-\frac{n}{n+1}({\sum }_{k=1}^n\frac{1}{2^k}),& n=k-1;\\ 0,& n<k-1,\end{array}$$

    that is,

    $${\alpha }_{n,k}=\left(\begin{array}{ccccccccc}\frac{1}{4}& \frac{3}{4}& 0& 0& 0& 0& \cdots & 0& \cdots \\ \frac{1}{3}& \frac{1}{6}& \frac{1}{2}& 0& 0& 0& \cdots & 0& \cdots \\ \frac{3}{8}& \frac{3}{16}& \frac{3}{32}& \frac{11}{32}& 0& 0& \cdots & 0& \cdots \\ \frac{2}{5}& \frac{1}{5}& \frac{1}{10}& \frac{1}{20}& \frac{1}{4}& 0& \cdots & 0& \cdots \\ ⋮& ⋮& ⋮& ⋮& ⋮& ⋮& ⋮\\ \frac{n}{2(n+1)}& \frac{n}{4(n+1)}& \frac{n}{8(n+1)}& \frac{n}{16(n+1)}& \frac{n}{32(n+1)}& \frac{n}{64(n+1)}& \cdots & \frac{n}{2^k(n+1)}& \cdots \\ ⋮& ⋮& ⋮& ⋮& ⋮& ⋮& ⋮\end{array}\right).$$

    We see that ${lim}_{n\to \mathrm{\infty }}{\alpha }_{n,k}=\frac{1}{2^k}$ and ${lim\hspace{0.17em}inf}_{n\to \mathrm{\infty }}{\alpha }_{n,0}{\alpha }_{n,k}=\frac{1}{2^{k+1}}$ for $k=1,2,3,\dots$ .

  • (2)
    $${\alpha }_{n,k}=\{\begin{array}{ll}\frac{1}{2^k}(\frac{n}{n+1}),& n\ge k\text{ and }n\text{ is odd};\\ \frac{1}{2^{k+1}}(\frac{n}{n+1}),& n\ge k\text{ and }n\text{ is even};\\ 1-\frac{n}{n+1}({\sum }_{k=1}^n\frac{1}{2^k}),& n=k-1\text{ and }n\text{ is odd};\\ 1-\frac{n}{n+1}({\sum }_{k=1}^n\frac{1}{2^{k+1}}),& n=k-1\text{ and }n\text{ is even};\\ 0,& n<k-1,\end{array}$$

    that is,

    $${\alpha }_{n,k}=\left(\begin{array}{ccccccccc}\frac{1}{4}& \frac{3}{4}& 0& 0& 0& 0& \cdots & 0& \cdots \\ \frac{1}{6}& \frac{1}{12}& \frac{3}{4}& 0& 0& 0& \cdots & 0& \cdots \\ \frac{3}{8}& \frac{3}{16}& \frac{3}{32}& \frac{11}{32}& 0& 0& \cdots & 0& \cdots \\ \frac{1}{5}& \frac{1}{10}& \frac{1}{20}& \frac{1}{40}& \frac{5}{8}& 0& \cdots & 0& \cdots \\ ⋮& ⋮& ⋮& ⋮& ⋮& ⋮& ⋮\\ \frac{2n-1}{2(2n)}& \frac{2n-1}{4(2n)}& \frac{2n-1}{8(2n)}& \frac{2n-1}{16(2n)}& \frac{2n-1}{32(2n)}& \frac{2n-1}{64(2n)}& \cdots & \frac{2n-1}{2^k(2n)}& \cdots \\ \frac{2n}{4(2n+1)}& \frac{2n}{8(2n+1)}& \frac{2n}{16(2n+1)}& \frac{2n}{32(2n+1)}& \frac{2n}{64(2n+1)}& \frac{2n}{128(2n+1)}& \cdots & \frac{2n}{2^{k+1}(2n+1)}& \cdots \\ ⋮& ⋮& ⋮& ⋮& ⋮& ⋮& ⋮\end{array}\right).$$

  We see that ${lim}_{n\to \mathrm{\infty }}{\alpha }_{n,k}$ does not exist and ${lim\hspace{0.17em}inf}_{n\to \mathrm{\infty }}{\alpha }_{n,0}{\alpha }_{n,k}=\frac{1}{2^{k+3}}$ for $k=1,2,3,\dots$ .

4 Conclusions

We use the shrinking projection method defined by Takahashi [10] together with our method for finding a common element of the set of solutions of mixed equilibrium problem and common fixed points of a countable family of multivalued nonexpansive mappings in Hilbert spaces. The main results of paper can be applied for solving convex minimization problems and fixed point problems.