Strong convergence theorems for equilibrium problems and fixed point problem of multivalued nonexpansive mappings via hybrid projection method

Abstract

In this paper, a new iterative process by the hybrid projection method is constructed. Strong convergence of the iterative process to a common element of the set of common fixed points of a finite family of generalized nonexpansive multivalued mappings and the solution set of two equilibrium problems in a Hilbert space is proved. Our results extend some important recent results.

MSC:47H10, 47H09.

1 Introduction

Let C be a nonempty closed convex subset of a Hilbert space H. A subset $C\subset H$ is called proximal if for each $x\in H$ there exists an element $y\in C$ such that

$$\parallel x-y\parallel =dist(x,C)=inf\{\parallel x-z\parallel :z\in C\}.$$

We denote by $\mathit{CB}(C)$ and $P(C)$ the collection of all nonempty closed bounded subsets and nonempty proximal bounded subsets of C, respectively. The Hausdorff metric H on $\mathit{CB}(H)$ is defined by

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

for all $A,B\in \mathit{CB}(H)$.

Let $T:H\to 2^H$ be a multivalued mapping. An element $x\in H$ is said to be a fixed point of T, if $x\in Tx$. The set of fixed points of T will be denoted by $F(T)$.

Definition 1.1 A multivalued mapping $T:H\to \mathit{CB}(H)$ is called

• (i) nonexpansive if

  $$H(Tx,Ty)\le \parallel x-y\parallel ,x,y\in H.$$

• (ii) quasi-nonexpansive if $F(T)\ne \mathrm{\varnothing }$ and $H(Tx,Tp)\le \parallel x-p\parallel$ for all $x\in H$ and all $p\in F(T)$.

Recently, J. Garcia-Falset, E. Llorens-Fuster and T. Suzuki [1] introduced a new condition on singlevalued mappings, called condition (E), which is weaker than nonexpansiveness.

Definition 1.2 A mapping $T:H\to H$ is said to satisfy the condition $(E_{\mu })$ provided that

$$\parallel x-Ty\parallel \le \mu \parallel x-Tx\parallel +\parallel x-y\parallel ,x,y\in H.$$

We say that T satisfies the condition (E) whenever T satisfies $(E_{\mu })$ for some $\mu \ge 1$.

Now we modify this condition for multivalued mappings as follows (see also [2]):

Definition 1.3 A multivalued mapping $T:H\to \mathit{CB}(H)$ is said to satisfy the condition $(P)$ provided that

$$H(Tx,Ty)\le \mu dist(x,Tx)+\eta \parallel x-y\parallel ,x,y\in H,$$

for some $\mu ,\eta \ge 1$.

It is obvious that every nonexpansive multivalued mapping satisfies the condition $(P)$. The theory of multivalued mappings has applications in control theory, convex optimization, differential equations and economics. Theory of nonexpansive multivalued mappings is harder than the corresponding theory of nonexpansive single valued mappings. Different iterative processes have been used to approximate fixed points of multivalued nonexpansive mappings (see [3–8]). Let Φ be a bifunction from $C\times C$ into $\mathbb{R}$, where $\mathbb{R}$ is the set of real numbers. The equilibrium problem for $\mathrm{\Phi }:C\times C\to \mathbb{R}$ is to find $x\in C$ such that

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

The set of solutions is denoted by $EP(\mathrm{\Phi })$. It is well known that this problem is closely related to minimax inequalities (see [9] and [10]). The equilibrium problem includes fixed point problems, optimization problems and variational inequality problems as special cases. Some methods have been proposed to solve the equilibrium problem, see, for example, [11–14].

Recently, many authors have studied the problems of finding a common element of the set of fixed points of nonexpansive single valued mappings and the set of solutions of an equilibrium problem in the framework of Hilbert spaces: see, for instance, [15–29] and the references therein. In this paper, a new iterative process by the hybrid projection method is constructed. Strong convergence of the iterative process to a common element of a set of common fixed points of a finite family of multivalued mappings satisfying the condition (P) and the solution set of two equilibrium problems in a Hilbert space is proved. Our results generalize some results of Tada, Takahashi [15] and many others.

2 Preliminaries

Let us recall the following definitions and results which will be used in the sequel.

Lemma 2.1 ([6])

Let H be a real Hilbert space. Then for$i,j=1,2,\dots ,k$we have

$${\parallel a_1{x}_1+a_2{x}_2+\cdots +a_k{x}_k\parallel }^2\le a_1{\parallel x_1\parallel }^2+a_2{\parallel x_2\parallel }^2+\cdots +a_k{\parallel x_k\parallel }^2-a_i{a}_j{\parallel x_i-x_j\parallel }^2$$

for all$x_i,x_j\in H$and$a_i,a_j\in [0,1]$with${\sum }_{i=1}^k{a}_i=1$.

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

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

$P_C$ is called the metric projection of H onto C. It is well known that $P_C$ is a nonexpansive mapping.

Lemma 2.2 ([16])

Let C be a closed convex subset of H. Given$x\in H$and a point$z\in C$. Then$z=P_{C}x$if and only if

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

Lemma 2.3 ([30])

Let C be a closed convex subset of H. Then for all$x\in H$and$y\in C$we have

$${\parallel y-P_{C}x\parallel }^2+{\parallel x-P_{C}x\parallel }^2\le {\parallel x-y\parallel }^2.$$

For solving the equilibrium problem, we assume that the bifunction Φ satisfies the following conditions:

(A1) $\mathrm{\Phi }(x,x)=0$ for any $x\in C$,

(A2) Φ is monotone, i.e., $\mathrm{\Phi }(x,y)+\mathrm{\Phi }(y,x)\le 0$ for any $x,y\in C$,

(A3) Φ is upper-hemicontinuous, i.e., for each $x,y,z\in C$,

$$\underset{t\to 0^{+}}{lim\hspace{0.17em}sup}\mathrm{\Phi }(tz+(1-t)x,y)\le \mathrm{\Phi }(x,y),$$

(A4) $\mathrm{\Phi }(x,\cdot )$ is convex and lower semicontinuous for each $x\in C$.

The following lemma was proved in [11].

Lemma 2.4 Let C be a nonempty closed convex subset of H and let Φ be a bifunction of$C\times C$into$\mathbb{R}$satisfying (A 1)-(A 4). Let$r>0$and$x\in H$. Then, there exists$z\in C$such that

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

The following lemma was given in [14].

Lemma 2.5 Assume that$\mathrm{\Phi }:C\times C\to \mathbb{R}$satisfies (A 1)-(A 4). For$r>0$and$x\in H$, define a mapping$T_r:H\to C$as follows:

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

Then, the following hold:

• (i) $T_r$ is single valued;

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

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

• (iii) $F(T_r)=EP(\mathrm{\Phi })$;

• (iv) $EP(\mathrm{\Phi })$ is closed and convex.

The following lemma was proved in [31] for nonexpansive multivalued mappings. The statement is true for quasi-nonexpansive multivalued mappings as well. To avoid repetition, we omit the details of the proof.

Lemma 2.6 Let C be a closed convex subset of a real Hilbert space H. Let$T:C\to \mathit{CB}(C)$be a quasi-nonexpansive multivalued mapping such that$T(p)=\{p\}$for all$p\in F(T)$. Then$F(T)$is closed and convex.

Now, following Shahzad and Zegeye [3], we remove the restriction $T(p)=\{p\}$ for all $p\in F(T)$. Let $T:C\to P(C)$ be a multivalued mapping and

$$P_T(x)=\{y\in Tx:\parallel x-y\parallel =dist(x,Tx)\}.$$

We use a similar argument as in the proof of Lemma 3.1 in [31] to obtain the following lemma.

Lemma 2.7 Let C be a closed convex subset of a real Hilbert space H. Let$T:C\to P(C)$be a multivalued mapping such that$P_T$is quasi-nonexpansive. Then$F(T)$is closed and convex.

Note that for all $p\in F(T)$, $P_T(p)=\{p\}$. We remark that there exist some examples of multivalued mappings for which $P_T$ is nonexpansive (see [3] for details), so that the assumption on T is not artificial.

3 The main result

Theorem 3.1 Let C be a nonempty closed convex subset of a real Hilbert space H, ${\mathrm{\Phi }}_1$and${\mathrm{\Phi }}_2$be two bifunctions of$C\times C$into$\mathbb{R}$satisfying (A 1)-(A 4). Let$T_i:C\to \mathit{CB}(C)$, ($i=1,2,\dots ,m$), be a finite family of quasi-nonexpansive multivalued mappings, each satisfying the condition (P). Assume further that$\mathcal{F}={\bigcap }_{i=1}^{m}F(T_i)\cap EP({\mathrm{\Phi }}_1)\cap EP({\mathrm{\Phi }}_2)\ne \mathrm{\varnothing }$and$T_i(p)=\{p\}$, ($i=1,2,\dots ,m$), for each$p\in \mathcal{F}$. For$C_0=C$, let$\{x_n\}$and$\{u_n\}$be sequences generated by the following algorithm:

$$\{\begin{array}{c}{x}_0\in C,\hfill \\ u_n\in C\mathit{\text{such that}}{\mathrm{\Phi }}_1(u_n,y)+\frac{1}{r_n}〈y-u_n,u_n-x_n〉\ge 0;\mathrm{\forall }y\in C,\hfill \\ u_n^{\prime }\in C\mathit{\text{such that}}{\mathrm{\Phi }}_2(u_n^{\prime },y)+\frac{1}{s_n}〈y-u_n^{\prime },u_n^{\prime }-x_n〉\ge 0;\mathrm{\forall }y\in C,\hfill \\ v_n={\delta }_n{u}_n+(1-{\delta }_n)u_n^{\prime },\hfill \\ w_n=a_{n,0}{v}_n+a_{n,1}{z}_{n,1}+\cdots +a_{n,m}{z}_{n,m},\hfill \\ y_n=b_{n,0}{v}_n+b_{n,1}{z}_{n,1}^{\prime }+\cdots +b_{n,m}{z}_{n,m}^{\prime },\hfill \\ C_{n+1}=\{v\in C_n:\parallel y_n-v\parallel \le \parallel x_n-v\parallel \},\hfill \\ x_{n+1}=P_{C_{n+1}}{x}_0:\mathrm{\forall }n\ge 0,\hfill \end{array}$$

where$z_{n,i}\in T_i{v}_n$, $z_{n,i}^{\prime }\in T_i{w}_n$for$i=1,2,\dots ,m$. Assume that$\{a_{n,j}\}$, $\{b_{n,j}\}$, $\{{\delta }_n\}$and$\{r_n\}$, $\{s_n\}$satisfy the following conditions:

• (i) $\{a_{n,j}\},\{b_{n,j}\},\{{\delta }_n\}\subset [a,b]\subset (0,1)$ ($j=0,1,2,\dots ,m$),

• (ii) $\{r_n\},\{s_n\}\subset (0,\mathrm{\infty })$, and ${lim\hspace{0.17em}inf}_{n\to \mathrm{\infty }}{r}_n>0$ and ${lim\hspace{0.17em}inf}_{n\to \mathrm{\infty }}{s}_n>0$.

Then, the sequences$\{x_n\}$and$\{v_n\}$converge strongly to$P_{\mathcal{F}}{x}_0$.

Proof First, we show that $\mathcal{F}\subset C_n$ for all $n\ge 0$. Fix $q\in \mathcal{F}$. We set

$$T_r^{{\mathrm{\Phi }}_1}(x)=\{z\in C:{\mathrm{\Phi }}_1(z,y)+\frac{1}{r}〈y-z,z-x〉\ge 0\mathrm{\forall }y\in C\}.$$

Hence we have $u_n=T_{r_n}^{{\mathrm{\Phi }}_1}{x}_n$ and $u_n^{\prime }=T_{s_n}^{{\mathrm{\Phi }}_2}{x}_n$. By Lemma 2.5, we have

$$\parallel u_n-q\parallel =\parallel T_{r_n}^{{\mathrm{\Phi }}_1}{x}_n-T_{r_n}^{{\mathrm{\Phi }}_1}q\parallel \le \parallel x_n-q\parallel$$

and

$$\parallel u_n^{\prime }-q\parallel =\parallel T_{s_n}^{{\mathrm{\Phi }}_2}{x}_n-T_{s_n}^{{\mathrm{\Phi }}_2}q\parallel \le \parallel x_n-q\parallel ,$$

which implies that

$$\parallel v_n-q\parallel \le {\delta }_n\parallel u_n-q\parallel +(1-{\delta }_n)\parallel u_n^{\prime }-q\parallel \le \parallel x_n-q\parallel .$$

Since $T_i$ is quasi-nonexpansive, for $i=1,2,\dots ,m$, we have

$$\begin{array}{rcl}\parallel w_n-q\parallel & =& \parallel a_{n,0}{v}_n+a_{n,1}{z}_{n,1}+\cdots +a_{n,m}{z}_{n,m}-q\parallel \\ \le & a_{n,0}\parallel v_n-q\parallel +a_{n,1}\parallel z_{n,1}-q\parallel +\cdots +a_{n,m}\parallel z_{n,m}-q\parallel \\ \le & a_{n,0}\parallel x_n-q\parallel +a_{n,1}dist(z_{n,1},T_{1}q)+\cdots +a_{n,m}dist(z_{n,m},T_{m}q)\\ \le & a_{n,0}\parallel x_n-q\parallel +a_{n,1}H(T_1{v}_n,T_{1}q)+\cdots +a_{n,m}H(T_m{v}_n,T_{m}q)\\ \le & a_{n,0}\parallel x_n-q\parallel +a_{n,1}\parallel v_n-q\parallel +\cdots +a_{n,m}\parallel v_n-q\parallel \\ \le & a_{n,0}\parallel x_n-q\parallel +a_{n,1}\parallel x_n-q\parallel +\cdots +a_{n,m}\parallel x_n-q\parallel \le \parallel x_n-q\parallel ,\end{array}$$

and also

$$\begin{array}{rcl}\parallel y_n-q\parallel & =& \parallel b_{n,0}{v}_n+b_{n,1}{z}_{n,1}^{\prime }+\cdots +b_{n,m}{z}_{n,m}^{\prime }-q\parallel \\ \le & b_{n,0}\parallel v_n-q\parallel +b_{n,1}\parallel z_{n,1}^{\prime }-q\parallel +\cdots +b_{n,m}\parallel z_{n,m}^{\prime }-q\parallel \\ \le & b_{n,0}\parallel x_n-q\parallel +b_{n,1}dist(z_{n,1}^{\prime },T_{1}q)+\cdots +b_{n,m}dist(z_{n,m}^{\prime },T_{m}q)\\ \le & b_{n,0}\parallel x_n-q\parallel +b_{n,1}H(T_1{w}_n,T_{1}q)+\cdots +b_{n,m}H(T_m{w}_n,T_{m}q)\\ \le & b_{n,0}\parallel x_n-q\parallel +b_{n,1}\parallel w_n-q\parallel +\cdots +b_{n,m}\parallel w_n-q\parallel \\ \le & b_{n,0}\parallel x_n-q\parallel +b_{n,1}\parallel x_n-q\parallel +\cdots +b_{n,m}\parallel x_n-q\parallel \le \parallel x_n-q\parallel .\end{array}$$

Therefore $q\in C_n$, which implies that

$$\mathcal{F}=\bigcap _{i=1}^{m}F(T_i)\cap EP({\mathrm{\Phi }}_1)\cap EP({\mathrm{\Phi }}_2)\subset C_n,\text{for all }n\ge 0.$$

We observe that $C_n$ is closed and convex (see [32]). Now we show that ${lim}_{n\to \mathrm{\infty }}\parallel x_n-x_0\parallel$ exists. By Lemma 2.6 we have $\mathcal{F}$ is closed and convex. Put $w=P_{\mathcal{F}}{x}_0$. From $x_n=P_{C_n}{x}_0$ and $x_{n+1}\in C_{n+1}\subset C_n$ we have

$$\parallel x_n-x_0\parallel \le \parallel x_{n+1}-x_0\parallel .$$

Also from $w\in \mathcal{F}\subset C_n$ and $x_n=P_{C_n}{x}_0$ for all $n\ge 0$, we get that

$$\parallel x_n-x_0\parallel \le \parallel w-x_0\parallel .$$

It follows that the sequence $\{x_n\}$ is bounded and nondecreasing. Hence the limit ${lim}_{n\to \mathrm{\infty }}\parallel x_n-x_0\parallel$ exists. We show that ${lim}_{n\to \mathrm{\infty }}{x}_n=u\in C$. For $k>n$ we have $x_k=P_{C_k}{x}_0\in C_k\subset C_n$. Now by applying Lemma 2.3 we have

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

Since ${lim}_{n\to \mathrm{\infty }}\parallel x_n-x_0\parallel$ exists, it follows that $\{x_n\}$ is a Cauchy sequence, and hence there exists $u\in C$ such that ${lim}_{n\to \mathrm{\infty }}{x}_n=u$. Putting $k=n+1$, in the above inequality we have

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

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

$$\parallel y_n-x_{n+1}\parallel \le \parallel x_n-x_{n+1}\parallel ,$$

so that ${lim}_{n\to \mathrm{\infty }}\parallel y_n-x_{n+1}\parallel =0$. This implies that ${lim}_{n\to \mathrm{\infty }}{y}_n=u$. Take $q\in \mathcal{F}$. By Lemma 2.1, for each $1\le i\le m$, we have

$$\begin{array}{rcl}{\parallel w_n-q\parallel }^2& =& {\parallel a_{n,0}{v}_n+a_{n,1}{z}_{n,1}+\cdots +a_{n,m}{z}_{n,m}-q\parallel }^2\\ \le & a_{n,0}{\parallel v_n-q\parallel }^2+a_{n,1}{\parallel z_{n,1}-q\parallel }^2\\ +\cdots +a_{n,m}{\parallel z_{n,m}-q\parallel }^2-a_{n,i}{a}_{n,o}{\parallel v_n-z_{n,i}\parallel }^2\\ \le & a_{n,0}{\parallel v_n-q\parallel }^2+a_{n,1}dist{(z_{n,1},T_{1}q)}^2\\ +\cdots +a_{n,m}dist{(z_{n,m},T_{m}q)}^2-a_{n,i}{a}_{n,o}{\parallel v_n-z_{n,i}\parallel }^2\\ \le & a_{n,0}{\parallel v_n-q\parallel }^2+a_{n,1}H{(T_1{v}_n,T_{1}q)}^2\\ +\cdots +a_{n,m}H{(T_m{v}_n,T_{m}q)}^2-a_{n,i}{a}_{n,o}{\parallel v_n-z_{n,i}\parallel }^2\\ \le & a_{n,0}{\parallel v_n-q\parallel }^2+a_{n,1}{\parallel v_n-q\parallel }^2\\ +\cdots +a_{n,m}{\parallel v_n-q\parallel }^2-a_{n,i}{a}_{n,o}{\parallel v_n-z_{n,i}\parallel }^2\\ \le & {\parallel v_n-q\parallel }^2-a_{n,i}{a}_{n,o}{\parallel v_n-z_{n,i}\parallel }^2,\end{array}$$

and also

$$\begin{array}{rcl}{\parallel y_n-q\parallel }^2& =& {\parallel b_{n,0}{v}_n+b_{n,1}{z}_{n,1}^{\prime }+\cdots +b_{n,m}{z}_{n,m}^{\prime }-q\parallel }^2\\ \le & b_{n,0}{\parallel v_n-q\parallel }^2+b_{n,1}{\parallel z_{n,1}^{\prime }-q\parallel }^2+\cdots +b_{n,m}{\parallel z_{n,m}^{\prime }-q\parallel }^2\\ \le & b_{n,0}{\parallel v_n-q\parallel }^2+b_{n,1}dist{(z_{n,1}^{\prime },T_{1}q)}^2+\cdots +b_{n,m}dist{(z_{n,m}^{\prime },T_{m}q)}^2\\ \le & b_{n,0}{\parallel v_n-q\parallel }^2+b_{n,1}H{(T_1{w}_n,T_{1}q)}^2+\cdots +b_{n,m}H{(T_m{w}_n,T_{m}q)}^2\\ \le & b_{n,0}{\parallel v_n-q\parallel }^2+b_{n,1}{\parallel w_n-q\parallel }^2+\cdots +b_{n,m}{\parallel w_n-q\parallel }^2\\ \le & b_{n,0}{\parallel v_n-q\parallel }^2+b_{n,1}{\parallel v_n-q\parallel }^2+\cdots +b_{n,m}{\parallel v_n-q\parallel }^2\\ -b_{n,i}{a}_{n,i}{a}_{n,o}{\parallel v_n-z_{n,i}\parallel }^2\\ \le & {\parallel x_n-q\parallel }^2-b_{n,i}{a}_{n,i}{a}_{n,o}{\parallel v_n-z_{n,i}\parallel }^2.\end{array}$$

(1)

So we have that

$$\begin{array}{rcl}{a}^{2}b{\parallel v_n-z_{n,i}\parallel }^2& \le & b_{n,i}{a}_{n,i}{a}_{n,o}{\parallel v_n-z_{n,i}\parallel }^2\\ \le & {\parallel v_n-q\parallel }^2-{\parallel y_n-q\parallel }^2\\ \le & {\parallel x_n-q\parallel }^2-{\parallel y_n-q\parallel }^2,\end{array}$$

which implies that

$$\underset{n\to \mathrm{\infty }}{lim}\parallel v_n-z_{n,i}\parallel =0,\text{for }i=1,2,\dots ,m.$$

Hence

$$\underset{n\to \mathrm{\infty }}{lim}dist(v_n,T_i{v}_n)\le \underset{n\to \mathrm{\infty }}{lim}\parallel v_n-z_{n,i}\parallel =0(i=1,2,\dots ,m).$$

As above $u_n=T_{r_n}^{{\mathrm{\Phi }}_1}{x}_n$ so that

$$\begin{array}{rcl}{\parallel u_n-q\parallel }^2& =& {\parallel T_{r_n}^{{\mathrm{\Phi }}_1}{x}_n-T_{r_n}^{{\mathrm{\Phi }}_1}q\parallel }^2\\ \le & 〈T_{r_n}^{{\mathrm{\Phi }}_1}{x}_n-T_{r_n}^{{\mathrm{\Phi }}_1}q,x_n-q〉\\ =& 〈u_n-q,x_n-q〉\\ =& \frac{1}{2}({\parallel u_n-q\parallel }^2+{\parallel x_n-q\parallel }^2-{\parallel x_n-u_n\parallel }^2)\end{array}$$

and hence

$${\parallel u_n-q\parallel }^2\le {\parallel x_n-q\parallel }^2-{\parallel x_n-u_n\parallel }^2.$$

(2)

And also by $u_n^{\prime }=T_{s_n}^{{\mathrm{\Phi }}_2}{x}_n$ we have

$$\begin{array}{rcl}{\parallel u_n^{\prime }-q\parallel }^2& =& {\parallel T_{s_n}^{{\mathrm{\Phi }}_2}{x}_n-T_{s_n}^{{\mathrm{\Phi }}_2}q\parallel }^2\\ \le & 〈T_{s_n}^{{\mathrm{\Phi }}_2}{x}_n-T_{s_n}^{{\mathrm{\Phi }}_2}q,x_n-q〉\\ =& 〈u_n^{\prime }-q,x_n-q〉\\ =& \frac{1}{2}({\parallel u_n^{\prime }-q\parallel }^2+{\parallel x_n-q\parallel }^2-{\parallel x_n-u_n^{\prime }\parallel }^2)\end{array}$$

and hence

$${\parallel u_n^{\prime }-q\parallel }^2\le {\parallel x_n-q\parallel }^2-{\parallel x_n-u_n^{\prime }\parallel }^2.$$

(3)

Now we use (2) and (3) to obtain

$$\begin{array}{rcl}{\parallel v_n-q\parallel }^2& \le & {\delta }_n{\parallel u_n-q\parallel }^2+(1-{\delta }_n){\parallel u_n^{\prime }-q\parallel }^2\\ \le & {\parallel x_n-q\parallel }^2-{\delta }_n{\parallel x_n-u_n\parallel }^2-(1-{\delta }_n){\parallel x_n-u_n^{\prime }\parallel }^2.\end{array}$$

It follows from (1) and the last inequality that

$${\parallel y_n-q\parallel }^2\le {\parallel v_n-q\parallel }^2\le {\parallel x_n-q\parallel }^2-{\delta }_n{\parallel x_n-u_n\parallel }^2-(1-{\delta }_n){\parallel x_n-u_n^{\prime }\parallel }^2.$$

So we have

$$a{\parallel x_n-u_n\parallel }^2\le {\delta }_n{\parallel x_n-u_n\parallel }^2\le {\parallel x_n-q\parallel }^2-{\parallel y_n-q\parallel }^2,$$

and

$$(1-b){\parallel x_n-u_n^{\prime }\parallel }^2\le (1-{\delta }_n){\parallel x_n-u_n^{\prime }\parallel }^2\le {\parallel x_n-q\parallel }^2-{\parallel y_n-q\parallel }^2.$$

Since ${lim}_{n\to \mathrm{\infty }}{x}_n={lim}_{n\to \mathrm{\infty }}{y}_n=u$, we obtain that

$$\underset{n\to \mathrm{\infty }}{lim}\parallel u_n-x_n\parallel =\underset{n\to \mathrm{\infty }}{lim}\parallel u_n^{\prime }-x_n\parallel =0,$$

which implies that

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

Since ${lim}_{n\to \mathrm{\infty }}\parallel v_n-x_n\parallel =0$, for $i=1,2,\dots ,m$, we obtain that

$$\begin{array}{rcl}dist(x_n,T_i{x}_n)& \le & \parallel x_n-v_n\parallel +dist(v_n,T_i{v}_n)+H(T_i{v}_n,T_i{x}_n)\\ \le & (\eta +1)\parallel x_n-v_n\parallel +(\mu +1)dist(v_n,T_i{v}_n)\to 0\text{as }n\to \mathrm{\infty }.\end{array}$$

(4)

We observe that $u\in {\bigcap }_{i=1}^{m}F(T_i)$. Indeed,

$$\begin{array}{rcl}dist(u,T_{i}u)& \le & \parallel u-x_n\parallel +dist(x_n,T_i{x}_n)+H(T_i{x}_n,T_{i}u)\\ \le & (\eta +1)\parallel u-x_n\parallel +(\mu +1)dist(x_n,T_i{x}_n)\to 0\text{as }n\to \mathrm{\infty },\end{array}$$

which implies that $u\in {\bigcap }_{i=1}^{m}F(T_i)$. Let us show that $u\in EP({\mathrm{\Phi }}_1)\cap EP({\mathrm{\Phi }}_2)$. From ${lim}_{n\to \mathrm{\infty }}{x}_n=u$ and ${lim}_{n\to \mathrm{\infty }}\parallel x_n-u_n\parallel =0$, we have $u_n\to u$ as $n\to \mathrm{\infty }$. Since $u_n=T_{r_n}^{{\mathrm{\Phi }}_1}{x}_n$ we obtain

$${\mathrm{\Phi }}_1(u_n,y)+\frac{1}{r_n}〈y-u_n,u_n-x_n〉\ge 0\mathrm{\forall }y\in C.$$

From (A2), we have

$$\frac{1}{r_n}〈y-u_n,u_n-x_n〉\ge {\mathrm{\Phi }}_1(y,u_n),$$

and hence

$$〈y-u_n,\frac{u_n-x_n}{r_n}〉\ge {\mathrm{\Phi }}_1(y,u_n).$$

Since

$$\frac{u_n-x_n}{r_n}\to 0,$$

and $u_n\to u$, from (A4) we have

$$0\ge {\mathrm{\Phi }}_1(y,u),\mathrm{\forall }y\in C.$$

For $t\in (0,1]$ and $y\in C$, let $y_t=ty+(1-t)u$. Since $y,u\in C$, and C is convex we have $y_t\in C$ and hence ${\mathrm{\Phi }}_1(y_t,u)\le 0$. So, from (A1) and (A4), we have

$$0={\mathrm{\Phi }}_1(y_t,y_t)\le t{\mathrm{\Phi }}_1(y_t,y)+(1-t){\mathrm{\Phi }}_1(y_t,u)\le t{\mathrm{\Phi }}_1(y_t,y),$$

which gives ${\mathrm{\Phi }}_1(y_t,y)\ge 0$. From (A3) we have $0\le {\mathrm{\Phi }}_1(u,y)$, $\mathrm{\forall }y\in C$ and hence $u\in EP({\mathrm{\Phi }}_1)$. Similarly, we have $u\in EP({\mathrm{\Phi }}_2)$. Now we show that $u=P_{\mathcal{F}}{x}_0$. Since $x_n=P_{C_n}{x}_0$, by Lemma 2.2 we have

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

Since $u\in \mathcal{F}\subset C_n$ we get

$$〈z-u,x_0-u〉\le 0,\mathrm{\forall }z\in \mathcal{F}.$$

Now by Lemma 2.2 we obtain that $u=P_{\mathcal{F}}{x}_0$. □

By substituting $P_{T_i}$ by $T_i$ and using a similar argument as in Theorem 3.1, we obtain the following result.

Theorem 3.2 Let C be a nonempty closed convex subset of a real Hilbert space H, ${\mathrm{\Phi }}_1$and${\mathrm{\Phi }}_2$be two bifunctions of$C\times C$into$\mathbb{R}$satisfying (A 1)-(A 4). Let$T_i:C\to P(C)$, ($i=1,2,\dots ,m$), be a finite family of multivalued mappings such that each$P_{T_i}$is quasi-nonexpansive and satisfies the condition (P). Assume further that$\mathcal{F}={\bigcap }_{i=1}^{m}F(T_i)\cap EP({\mathrm{\Phi }}_1)\cap EP({\mathrm{\Phi }}_2)\ne \mathrm{\varnothing }$. For$C_0=C$, let$\{x_n\}$and$\{v_n\}$be sequences generated by the following algorithm:

$$\{\begin{array}{c}{x}_0\in C,\hfill \\ u_n\in C\mathit{\text{such that}}{\mathrm{\Phi }}_1(u_n,y)+\frac{1}{r_n}〈y-u_n,u_n-x_n〉\ge 0;\mathrm{\forall }y\in C,\hfill \\ u_n^{\prime }\in C\mathit{\text{such that}}{\mathrm{\Phi }}_2(u_n^{\prime },y)+\frac{1}{s_n}〈y-u_n^{\prime },u_n^{\prime }-x_n〉\ge 0;\mathrm{\forall }y\in C,\hfill \\ v_n={\delta }_n{u}_n+(1-{\delta }_n)u_n^{\prime },\hfill \\ w_n=a_{n,0}{v}_n+a_{n,1}{z}_{n,1}+\cdots +a_{n,m}{z}_{n,m},\hfill \\ y_n=b_{n,0}{v}_n+b_{n,1}{z}_{n,1}^{\prime }+\cdots +b_{n,m}{z}_{n,m}^{\prime },\hfill \\ C_{n+1}=\{v\in C_n:\parallel y_n-v\parallel \le \parallel x_n-v\parallel \},\hfill \\ x_{n+1}=P_{C_{n+1}}{x}_0:\mathrm{\forall }n\ge 0,\hfill \end{array}$$

where$z_{n,i}\in P_{T_i}(v_n)$, $z_{n,i}^{\prime }\in P_{T_i}(w_n)$for$i=1,2,\dots ,m$. Assume that$\{a_{n,j}\}$, $\{b_{n,j}\}$, $\{{\delta }_n\}$and$\{r_n\}$, $\{s_n\}$satisfy the following conditions:

• (i) $\{a_{n,j}\},\{b_{n,j}\},\{{\delta }_n\}\subset [a,b]\subset (0,1)$ ($j=0,1,2,\dots ,m$),

• (ii) $\{r_n\},\{s_n\}\subset (0,\mathrm{\infty })$, and ${lim\hspace{0.17em}inf}_{n\to \mathrm{\infty }}{r}_n>0$ and ${lim\hspace{0.17em}inf}_{n\to \mathrm{\infty }}{s}_n>0$.

Then, the sequences$\{x_n\}$and$\{v_n\}$converge strongly to$P_{\mathcal{F}}{x}_0$.

As a result, for single valued mappings we obtain the following theorem.

Theorem 3.3 Let C be a nonempty closed convex subset of a real Hilbert space H and${\mathrm{\Phi }}_1$and${\mathrm{\Phi }}_2$be two bifunctions of$C\times C$into$\mathbb{R}$satisfying (A 1)-(A 4). Let$T_i:C\to C$ ($i=1,2,\dots ,m$), be a finite family of quasi-nonexpansive mappings, each satisfying the condition (P). Assume further that$\mathcal{F}={\bigcap }_{i=1}^{m}F(T_i)\cap EP({\mathrm{\Phi }}_1)\cap EP({\mathrm{\Phi }}_2)\ne \mathrm{\varnothing }$. For$C_0=C$, let$\{x_n\}$and$\{v_n\}$be sequences generated by the following algorithm:

$$\{\begin{array}{c}{x}_0\in C,\hfill \\ u_n\in C\mathit{\text{such that}}{\mathrm{\Phi }}_1(u_n,y)+\frac{1}{r_n}〈y-u_n,u_n-x_n〉\ge 0;\mathrm{\forall }y\in C,\hfill \\ u_n^{\prime }\in C\mathit{\text{such that}}{\mathrm{\Phi }}_2(u_n^{\prime },y)+\frac{1}{s_n}〈y-u_n^{\prime },u_n^{\prime }-x_n〉\ge 0;\mathrm{\forall }y\in C,\hfill \\ v_n={\delta }_n{u}_n+(1-{\delta }_n)u_n^{\prime },\hfill \\ w_n=a_{n,0}{v}_n+a_{n,1}{T}_1{v}_n+\cdots +a_{n,m}{T}_m{v}_n,\hfill \\ y_n=b_{n,0}{v}_n+b_{n,1}{T}_1{w}_n+\cdots +b_{n,m}{T}_m{w}_n,\hfill \\ C_{n+1}=\{v\in C_n:\parallel y_n-v\parallel \le \parallel x_n-v\parallel \},\hfill \\ x_{n+1}=P_{C_{n+1}}{x}_0:\mathrm{\forall }n\ge 0.\hfill \end{array}$$

Assume that$\{a_{n,j}\}$, $\{b_{n,j}\}$, $\{{\delta }_n\}$and$\{r_n\}$, $\{s_n\}$satisfy the following conditions:

• (i) $\{a_{n,j}\}$, $\{b_{n,j}\}$, $\{{\delta }_n\}\subset [a,b]\subset (0,1)$ ($j=0,1,2,\dots ,m$),

• (ii) $\{r_n\},\{s_n\}\subset (0,\mathrm{\infty })$, and ${lim\hspace{0.17em}inf}_{n\to \mathrm{\infty }}{r}_n>0$ and ${lim\hspace{0.17em}inf}_{n\to \mathrm{\infty }}{s}_n>0$.

Then, the sequences$\{x_n\}$and$\{v_n\}$converge strongly to$P_{\mathcal{F}}{x}_0$.

Theorem 3.4 Let C be a nonempty closed convex subset of a real Hilbert space H. Let$T_i:C\to P(C)$ ($i=1,2,\dots ,m$), be a finite family of multivalued mappings such that$P_{T_i}$is quasi-nonexpansive and satisfies the condition (P). Assume further that$\mathcal{F}={\bigcap }_{i=1}^{m}F(T_i)\ne \mathrm{\varnothing }$. For$C_0=C$, let$\{x_n\}$be the sequence generated by the following algorithm:

$$\{\begin{array}{c}{x}_0\in C,\hfill \\ w_n=a_{n,0}{x}_n+a_{n,1}{z}_{n,1}+\cdots +a_{n,m}{z}_{n,m},\hfill \\ y_n=b_{n,0}{x}_n+b_{n,1}{z}_{n,1}^{\prime }+\cdots +b_{n,m}{z}_{n,m}^{\prime },\hfill \\ C_{n+1}=\{v\in C_n:\parallel y_n-v\parallel \le \parallel x_n-v\parallel \},\hfill \\ x_{n+1}=P_{C_{n+1}}{x}_0:\mathrm{\forall }n\ge 0,\hfill \end{array}$$

where$z_{n,i}\in P_{T_i}(x_n)$, $z_{n,i}^{\prime }\in P_{T_i}(w_n)$for$i=1,2,\dots ,m$. Assume that$\{a_{n,j}\},\{b_{n,j}\}\subset [a,b]\subset (0,1)$ ($j=0,1,2,\dots ,m$), Then, the sequence$\{x_n\}$converges strongly to$P_{\mathcal{F}}{x}_0$.

Proof Putting ${\mathrm{\Phi }}_1(x,y)={\mathrm{\Phi }}_2(x,y)=0$ for all $x,y\in C$ and $r_n=s_n=1$ in Theorem 3.2, we have $u_n=u_n^{\prime }=x_n$ and hence $v_n=x_n$. Now, the desired conclusion follows directly from Theorem 3.2. □

Now, we supply an example to illustrate the main result of this paper.

Example 3.5 We consider the nonempty closed convex subset $C=[0,5]$ of the Hilbert space $\mathbb{R}$. Define two mappings $T_1$ and $T_2$ on C as follows:

$$T_1(x)=[\frac{x}{6},\frac{x}{2}],T_2(x)=\{\begin{array}{cc},\hfill & x\ne 5,\hfill \\ \{1\},\hfill & x=5.\hfill \end{array}$$

We note that $T_1$ and $T_2$ are quasi-nonexpansive mappings satisfying the condition (P), (for details, see [2]). Also we define two bifunctions ${\mathrm{\Phi }}_1$ and ${\mathrm{\Phi }}_2$ as follows:

$$\left\{\begin{array}{c}{\mathrm{\Phi }}_1:C\times C\to \mathbb{R},\hfill \\ {\mathrm{\Phi }}_1(x,y)=y-x,\hfill \end{array}\right\{\begin{array}{c}{\mathrm{\Phi }}_2:C\times C\to \mathbb{R},\hfill \\ {\mathrm{\Phi }}_2(x,y)=y^2+xy-2x^2.\hfill \end{array}$$

It is easy to see that ${\mathrm{\Phi }}_1$ and ${\mathrm{\Phi }}_2$ satisfy the conditions (A1)-(A4). If we put $r_n=5$ and $s_n=1$, then $u_n=T_{r_n}^{{\mathrm{\Phi }}_1}{x}_n=0$ and $u_n^{\prime }=T_{s_n}^{{\mathrm{\Phi }}_1}{x}_n=\frac{x_n}{3s_n+1}=\frac{x_n}{4}$ (for details, see [26]). Put $a_{n,i}=b_{n,i}=\frac{1}{3}$ for $i=0,1,2$ and ${\delta }_n=\frac{1}{2}$. For any arbitrary $x_0\in C$ we have

$$C_1=\{v\in C:|y_0-v|\le |x_0-v|\}=[0,\frac{x_0+y_0}{2}].$$

Since $\frac{x_0+y_0}{2}\le x_0$, we obtain that

$$x_1=P_{C_1}{x}_0=\frac{x_0+y_0}{2}.$$

By continuing this process we obtain

$$C_{n+1}=\{v\in C_n:|y_n-v|\le |x_n-v|\}=[0,\frac{x_n+y_n}{2}],$$

and hence

$$x_{n+1}=P_{C_{n+1}}{x}_0=\frac{x_n+y_n}{2}.$$

Now, we have the following algorithm:

$$\{\begin{array}{c}{x}_0\in C,\hfill \\ v_n=\frac{x_n}{8},\hfill \\ z_{n,i}\in T_i(v_n),i=1,2,\hfill \\ w_n=\frac{1}{3}{v}_n+\frac{1}{3}{z}_{n,1}+\frac{1}{3}{z}_{n,2},\hfill \\ z_{n,i}^{\prime }\in T_i{w}_n,i=1,2,\hfill \\ y_n=\frac{1}{3}{v}_n+\frac{1}{3}{z}_{n,1}^{\prime }+\frac{1}{3}{z}_{n,2}^{\prime },\hfill \\ x_{n+1}=\frac{x_n+y_n}{2}:\mathrm{\forall }n\ge 0.\hfill \end{array}$$

Putting $z_{n,1}=z_{n,2}=\frac{v_n}{6}$ and $z_{n,1}^{\prime }=z_{n,2}^{\prime }=\frac{w_n}{6}$ we get that

$$x_{n+1}=\frac{679}{1,296}{x}_n={\left(\frac{679}{1,296}\right)}^{n+1}{x}_0,\mathrm{\forall }n\ge 0.$$

We observe that for an arbitrary $x_0\in C$, $x_n$ is convergent to zero. We note that $\mathcal{F}=F(T_1)\cap F(T_2)\cap EP({\mathrm{\Phi }}_1)\cap EP({\mathrm{\Phi }}_2)=\{0\}$.

Remark 3.6 Since every nonexpansive mapping is quasi-nonexpansive and satisfies the condition (P), our results hold for nonexpansive mappings.

Remark 3.7 Our results generalize the results of Tada and Takahashi [15], of a nonexpansive single valued mapping to a finite family of generalized nonexpansive multivalued mappings.