Convergence theorems for equilibrium problem, variational inequality problem and countably infinite relatively quasi-nonexpansive mappings

Abstract

In this paper, we introduce an iterative process which converges strongly to a common element of set of common fixed points of countably infinite family of closed relatively quasi- nonexpansive mappings, the solution set of generalized equilibrium problem and the solution set of the variational inequality problem for a $\gamma$-inverse strongly monotone mapping in Banach spaces. Our theorems improve, generalize, unify and extend several results recently announced.

Introduction

Let C be a nonempty closed convex subset of a real Banach space E with dual $E^{\ast }$. We denote by J the normalized duality mapping from E to $2^{E^{\ast }}$ defined by

$$\mathit{Jx}:=\left\{f^{\ast }\in E^{\ast }:〈x\text{,}{f}^{\ast }〉=\Vert x{\Vert }^2=\Vert f^{\ast }{\Vert }^2\right\}\text{,}$$

where $〈.\text{,}.〉$ denotes the generalized duality pairing. It is well known that if $E^{\ast }$is strictly convex then J is single-valued and if E is uniformly smooth then J is uniformly continuous on bounded subsets of E. Moreover, if E is a reflexive and strictly convex Banach space with a strictly convex dual, then $J^{-1}$ is single-valued, one-to-one, surjective, and it is the duality mapping from $E^{\ast }$ into E and thus ${\mathit{JJ}}^{-1}=I_{E^{\ast }}$ and $J^{-1}J=I_E$ (see, [16]). We note that in a Hilbert space, $H\text{,}J$ is the identity operator.

A mapping $A:D(A)\subset E\to E^{\ast }$ is said to be monotone if for each $x\text{,}y\in D(A)$, the following inequality holds:

$$〈\mathit{Ax}-\mathit{Ay}\text{,}x-y〉⩾0\text{.}$$

A is said to be $\gamma$-inverse strongly monotone if there exists positive real number $\gamma$ such that

$$〈\mathit{Ax}-\mathit{Ay}\text{,}x-y〉⩾\gamma \Vert \mathit{Ax}-\mathit{Ay}{\Vert }^2\text{,}\text{for}\text{all}x\text{,}y\in D(A)\text{.}$$

If A is $\gamma$-inverse strongly monotone, then it is Lipschitz continuous with constant $\frac{1}{\gamma }$, i.e., $\Vert \mathit{Ax}-\mathit{Ay}\Vert ⩽\frac{1}{\gamma }\Vert x-y\Vert$, for all $x\text{,}y\in D(A)$, and hence uniformly continuous.

Let $A:C\to E^{\ast }$ be a monotone map. The variational inequality problem for A is to find $\hat{x}\in C$ such that

$$〈A\hat{x}\text{,}y-\hat{x}〉⩾0\text{,}\forall y\in C\text{.}$$

The set of solution of (1.3) is denoted by $\mathit{VI}(A\text{,}C)$.

Let $f:C\times C\to \mathbb{R}$ be a bifunction, where $\mathbb{R}$ is the set of real numbers. The equilibrium problem for f is to

$$\text{find}{x}^{\ast }\in C\text{such}\text{that}f(x^{\ast }\text{,}y)⩾0\text{,}\forall y\in C\text{.}$$

The set of solutions of (1.4) is denoted by $\mathit{EP}(f)$.

The equilibrium problem (1.4) is very general in the sense that it includes, as special cases, optimization problems, variational inequalities, the Nash equilibrium problems; (see, for instance, [2]). For solving the equilibrium problem for a bifunction $f:C\times C\to \mathbb{R}$, we assume that f satisfies the following conditions:

• $f(x\text{,}x)=0$ for all $x\in C$,

• f is monotone, i.e, $f(x\text{,}y)+f(y\text{,}x)⩽0$ for all $x\text{,}y\in C$,

• for each $x\text{,}y\text{,}z\in C\text{,}{\lim }_{t\to 0}f(\mathit{tz}+(1-t)x\text{,}y)⩽f(x\text{,}y)$,

• for each $x\in C$, the mapping $y\to f(x\text{,}y)$ is convex and lower semicontinuous.

Let E be a smooth Banach space. The function $\varphi :E\times E\to \mathbb{R}$ defined by

$$\varphi (x\text{,}y)=\Vert x{\Vert }^2-2〈x\text{,}\mathit{Jy}〉+\Vert y{\Vert }^2\text{for}x\text{,}y\in E\text{,}$$

is studied by Alber [1], Kamimula and Takahashi [5] and Riech [12]. It is obvious from the definition of the function $\varphi$ that

$$(\Vert x\Vert -\Vert y\Vert {)}^2⩽\varphi (x\text{,}y)⩽(\Vert x\Vert +\Vert y\Vert {)}^2\text{for}x\text{,}y\in E\text{.}$$

Observe that in a Hilbert space H, (1.5) reduces to $\varphi (x\text{,}y)=\Vert x-y{\Vert }^2$, for $x\text{,}y\in H$.

Let E be a reflexive, strictly convex and smooth Banach space and let C be a nonempty closed and convex subset of E. The generalized projection mapping, introduced by Alber [1], is a mapping ${\Pi }_C:E\to C$, that assigns to an arbitrary point $x\in E$ the minimum point of the functional $\varphi (y\text{,}x)$, that is, ${\Pi }_{C}x=\overline{x}$, where $\overline{x}$ is the solution to the minimization problem

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

In fact, we have the following result.

Lemma 1.1 [1]

Let C be a nonempty closed and convex subset of a real reflexive, strictly convex, and smooth Banach space E and let $x\in E$. Then, there exists a unique element $x_0\in C$ such that $\varphi (x_0\text{,}x)=\min \{\varphi (z\text{,}x):z\in C\}$.

Let C be a nonempty closed convex subset of E, and let T be a mapping from K into itself. We denote by $F(T)$ the set of fixed points of T. A point p in C is said to be an asymptotic fixed point of T (see e.g., [12]) if C contains a sequence $\{x_n\}$ which converges weakly to p such that ${\lim }_{n\to \infty }\Vert x_n-{\mathit{Tx}}_n\Vert =0$. The set of asymptotic fixed points of T will be denoted by $\stackrel{\sim }{F}(T)$. A mapping T from C into itself is said to be nonexpansive if $\Vert \mathit{Tx}-\mathit{Ty}\Vert ⩽\Vert x-y\Vert \forall x\text{,}y\in C$ and is called relatively nonexpansive (see e.g., [14], [19]) if $\stackrel{\sim }{F}(T)=F(T)$ and $\varphi (p\text{,}\mathit{Tx})⩽\varphi (p\text{,}x)$ for all $x\in C$ and $p\in F(T)$. The asymptotic behavior of relatively nonexpansive mapping was studied in [3]. T is said to be relatively quasi-nonexpansive if $F(T)\ne \varnothing$ and $\varphi (p\text{,}\mathit{Tx})⩽\varphi (p\text{,}x)$ for $x\in C$ and $p\in F(T)$.

Remark 1.2

The class of relatively quasi-nonexpansive mappings is more general than the class of relatively nonexpansive mappings (see [3]) which requires the strong restriction: $F(T)=\stackrel{\sim }{F}(T)$.

If E is smooth, strictly convex and reflexive Banach space, and $A\subset E\times E^{\ast }$ is a continuous monotone mapping with $A^{-1}(0)\ne \varnothing$ then it is proved in [6] that $J_r:=(J+\mathit{rA}{)}^{-1}J$, for $r>0$ is relatively quasi-nonexpansive. Moreover, if $T:E\to E$ is relatively quasi-nonexpansive then using the definition of $\varphi$ one can show that $F(T)$ is closed and convex (see, [10]).

In 1953, Mann [8] introduced the iteration sequence $\{x_n\}$ defined by

$$x_{n+1}={\alpha }_n{x}_n+(1-{\alpha }_n){\mathit{Tx}}_n\text{,}$$

where the initial guess element $x_0\in C$ is arbitrary and $\{{\alpha }_n\}$ is a real sequence in [0, 1], for nonexpansive mapping T. One of the fundamental convergence results is proved by Reich [11]. He proved that the sequence (1.7) converges weakly to a fixed point of T, under appropriate conditions on $\{{\alpha }_n\}$ and Banach space E. In an infinite dimensional Hilbert space, the Mann iteration can conclude only week convergence [4]. Attempts to modify the Mann iteration method (1.7) so that strong convergence is guaranteed have recently been made. Nakajo and Takahashi [9] proposed the following modification of the Mann iteration method (1.7):

$$\left\{\begin{array}{c}{x}_0\in C\text{chosen}\text{arbitrarily,}\hfill \\ y_n={\alpha }_n{x}_n+(1-{\alpha }_n){\mathit{Tx}}_n\text{,}\hfill \\ C_n=\{z\in C:\Vert y_n-z\Vert ⩽\Vert x_n-z\Vert \}\text{,}\hfill \\ Q_n=\{z\in C:〈x_n-z\text{,}{x}_0-x_n〉⩾0\}\text{,}\hfill \\ x_{n+1}=P_{C_n\cap Q_n}(x_0)\text{,}n⩾0\text{,}\hfill \end{array}\right.$$

and proved that $\{x_n\}$ converges strongly to $z=P_{F(T)}(x_0)$, where $P_C$ is the projection mapping from a Hilbert space H onto a closed convex subset C of H.

For finding an element of $\mathit{FP}(f)\cap F(T)$, Tada and Takahashi [15] introduced the following iterative scheme by the hybrid method in a Hilbert space:

$$\left\{\begin{array}{c}{x}_0\in C\text{chosen}\text{arbitrarily,}\hfill \\ u_n\in C\text{such}\text{that}f(u_n\text{,}y)+\frac{1}{r_n}〈y-u_n\text{,}{u}_n-x_n〉⩾0\text{,}\forall y\in C\text{,}\hfill \\ w_n=(1-{\alpha }_n)x_n+{\alpha }_n{\mathit{Tu}}_n\text{,}\hfill \\ C_n=\{z\in H:\Vert w_n-z\Vert ⩽\Vert x_n-z\Vert \}\text{,}\hfill \\ Q_n=\{z\in C:〈x_n-z\text{,}{x}_0-x_n〉⩾0\}\text{,}\hfill \\ x_{n+1}=P_{C_n\cap Q_n}(x_0)\text{,}n⩾0\text{,}\hfill \end{array}\right.$$

where $\{{\alpha }_n\}\subset [a\text{,}b]$ for some $a\text{,}b\in (0\text{,}1)$ and $\{r_n\}\subset (0\text{,}\infty )$ satisfies ${\mathrm{liminf}}_{n\to \infty }{r}_n>0$. Further, they proved that $\{x_n\}$ and $\{u_n\}$ converge strongly to $z\in \mathit{EP}(f)\cap F(T)$, where $z=P_{\mathit{EP}(f)\cap F(T)}(x_0)$.

For finding an element of $F(T)\cap \mathit{VI}(A\text{,}C)\cap \mathit{EP}(f)$, Kumam [7] introduced the following iterative scheme:

$$\left\{\begin{array}{c}{x}_0\in C\text{chosen}\text{arbitrarily,}\hfill \\ u_n\in C\text{such}\text{that}f(u_n\text{,}y)+\frac{1}{r_n}〈y-u_n\text{,}{u}_n-x_n〉⩾0\text{,}\forall y\in C\text{,}\hfill \\ w_n={\alpha }_n{x}_n+(1-{\alpha }_n){\mathit{TP}}_C(u_n-{\lambda }_n{\mathit{Au}}_n)\text{,}\hfill \\ C_n=\{z\in H:\Vert w_n-z\Vert ⩽\Vert x_n-z\Vert \}\text{,}\hfill \\ Q_n=\{z\in C:〈x_n-z\text{,}{x}_0-x_n〉⩾0\}\text{,}\hfill \\ x_{n+1}=P_{C_n\cap Q_n}(x_0)\text{,}n⩾0.\hfill \end{array}\right.$$

Under suitable conditions some strong convergence theorems, which extended and improved the result of Nakajo and Takahashi [9] and Tada and Takahashi [15], are proved in fact in the frame work of Hilbert spaces.

Recently, Takahashi and Zembayashi [17], introduced the following iterative scheme which is called the shrinking projection method:

$$\left\{\begin{array}{c}{x}_0\in C\text{chosen}\text{arbitrarily,}\hfill \\ y_n=J^{-1}({\alpha }_n{\mathit{Jx}}_n+(1-{\alpha }_n){\mathit{JTx}}_n)\text{,}\hfill \\ u_n\in C\text{such}\text{that}f(u_n\text{,}y)+\frac{1}{r_n}〈y-u_n\text{,}{\mathit{Ju}}_n-{\mathit{Jy}}_n〉⩾0\text{,}\forall y\in C\text{,}\hfill \\ C_{n+1}=\{z\in C_n:\varphi (z\text{,}{u}_n)⩽\varphi (z\text{,}{x}_n)\}\text{,}\hfill \\ x_{n+1}={\Pi }_{C_{n+1}}(x_0)\text{,}n⩾0\text{,}\hfill \end{array}\right.$$

for relatively nonexpansive mapping $T:C\to C$ and bifunction f from $C\times C$ to $\mathbb{R}$, where J is the duality mapping on E and ${\Pi }_C$ is the generalized projection from E onto C. They proved that the sequence $\{x_n\}$ converges strongly to $q={\Pi }_{F(T)\cap \mathit{EP}(f)}(x_0)$ under appropriate conditions.

Very recently, Qin et al. [10] extended the iteration process (1.11) from a single relatively nonexpansive mapping to two relatively nonexpansive mappings. In fact, they proved that the sequence $\{x_n\}$ generated by:

$$\left\{\begin{array}{c}{x}_0\in C_0=C\text{chosen}\text{arbitrarily,}\hfill \\ y_n=J^{-1}({\alpha }_n{\mathit{Jx}}_n+{\beta }_n{\mathit{JTx}}_n+{\gamma }_n{\mathit{JSx}}_n)\text{,}\hfill \\ u_n\in C\text{such}\text{that}f(u_n\text{,}y)+\frac{1}{r_n}〈y-u_n\text{,}{\mathit{Ju}}_n-{\mathit{Jy}}_n〉⩾0\text{,}\forall y\in C\text{,}\hfill \\ C_{n+1}=\{z\in C_n:\varphi (z\text{,}{u}_n)⩽\varphi (z\text{,}{x}_n)\}\text{,}\hfill \\ x_{n+1}={\Pi }_{C_{n+1}}(x_0)\text{,}n⩾0\text{,}\hfill \end{array}\right.$$

where T and S are relatively nonexpansive and $\{{\alpha }_n\}\text{,}\{{\beta }_n\}$ and $\{{\gamma }_n\}$ satisfy appropriate conditions, converges strongly to $q={\Pi }_{F(T)\cap F(S)\cap \mathit{EP}(f)}(x_0)$.

In this paper, inspired and motivated by the works mentioned above, we introduce an iterative process for finding a common element of set of common fixed points of countably infinite family of closed relatively quasi-nonexpansive mappings, the solution set of generalized equilibrium problem and the solution set of the variational inequality problem for a $\gamma$-inverse strongly monotone mapping in Banach spaces. The corresponding results of Qin et al. [10], Takahashi and Zembayashi [17], Kumam [7], Nakajo and Takahashi [9] and Tada and Takahashi [15] are extended to the case of countably infinite family of closed quasi-nonexpansive mappings in Banach spaces more general than Hilbert spaces.