Viscosity approximative methods to Cesàro mean iterations for nonexpansive nonself-mappings in Banach spaces

Abstract

Let E be a real uniformly convex Banach space which admits a weakly sequentially continuous duality mapping from E to $E^{\ast }$, C a closed convex subset of E which is also a sunny nonexpansive retract of E. Let $T:C\to E$ be a nonexpansive nonself-mapping with $F(T)\ne \varnothing$, and $f:C\to C$ be a fixed contractive mapping. The implicit viscosity iterative sequence $\{z_m\}$ is defined by $\{t_m\}\subset (0\text{,}1)$, $z_m=\frac{1}{m+1}{\sum }_{j=0}^{m}P(t_{m}f(z_m)+(1-t_m)(\mathit{TP}{)}^j{z}_m)$, for $m⩾0$ and two explicit viscosity iterative sequences $\{x_n\}$ and $\{y_n\}$ are given by $x_0\in C\text{,}{x}_{n+1}={\alpha }_{n}f(x_n)+(1-{\alpha }_n)\frac{1}{n+1}{\sum }_{j=0}^n(\mathit{PT}{)}^j{x}_n$ and $y_0\in C\text{,}{y}_{n+1}=\frac{1}{n+1}{\sum }_{j=0}^{n}P({\alpha }_{n}f(y_n)+(1-{\alpha }_n)(\mathit{TP}{)}^j{y}_n)$ for $n⩾0$, where $\{{\alpha }_n\}$ is a sequence in $(0\text{,}1)$ and P is a sunny nonexpansive retraction of E onto C. We prove that under appropriate conditions imposed on $\{t_m\}$ and $\{{\alpha }_n\}$, the sequences $\{z_m\}$, $\{x_n\}$ and $\{y_n\}$ converge strongly to some fixed point of T which solves some variational inequalities. The results presented extend and improve the corresponding results of Matsushita and Kuroiwa [S. Matsushita, D. Kuroiwa, Strong convergence of averaging iterations of nonexpansive nonself-mappings, J. Math. Anal. Appl. 294 (2004) 206–214], Song and Chen [Y. Song, R. Chen, Viscosity approximation methods to Cesàro means for nonexpansive mappings, Appl. Math. Comput. 186 (2007) 1120–1128] and many authors.

Introduction

Let C be a nonempty closed convex subset of a Hilbert space H and T a nonexpansive mapping of C into itself that is $\Vert \mathit{Tx}-\mathit{Ty}\Vert ⩽\Vert x-y\Vert$ for all $x\text{,}y\in C$. Recall that a self-mapping $f:C\to C$ is a contraction on C if there exists a constant $\alpha \in (0\text{,}1)$ such that $\Vert f(x)-f(y)\Vert ⩽\alpha \Vert x-y\Vert \forall x\text{,}y\in C$. Shimizu and Takahashi [10] studied the convergence of an iterative process for a family of nonexpansive mappings in the framework of a Hilbert space. They proved that the sequence $\{x_n\}$ defined by the iterative method below, with the initial guesses $x\text{,}{x}_0\in C$ chosen arbitrarily,

$$x_{n+1}={\alpha }_{n}x+(1-{\alpha }_n)\frac{1}{n+1}\sum _{j=0}^n{T}^j{x}_n\text{,}n⩾0\text{,}$$

converges strongly to an element of fixed point of T which is nearest to x provided the sequence $\{{\alpha }_n\}\subseteq [0\text{,}1]$ satisfies certain conditions. Shioji and Takahashi [9] extended the result of Shimizu and Takahashi [10] to a uniformly convex Banach space whose norm is uniformly Gâteaux differentiable. But this approximation method is not suitable for some nonexpansive nonself-mappings. On the other hand, Matsushita and Kuroiwa [7] introduced the following approximation method for nonexpansive nonself-mappings in a real Hilbert space. Let P be the metric projection of H onto C and $T:C\to H$ a nonexpansive nonself-mapping. Starting with arbitrary initial points $x_0\text{,}x\text{,}{y}_0\text{,}y\in C$, define the sequences $\{x_n\}$ and $\{y_n\}$, respectively, by

$$x_{n+1}={\alpha }_{n}x+(1-{\alpha }_n)\frac{1}{n+1}\sum _{j=0}^n(\mathit{PT}{)}^j{x}_n\text{,}n⩾0$$

and

$$y_{n+1}=\frac{1}{n+1}\sum _{j=0}^{n}P({\alpha }_{n}y+(1-{\alpha }_n)(\mathit{TP}{)}^j{y}_n)\text{,}n⩾0\text{,}$$

where $\{{\alpha }_n\}$ is a sequence in (0, 1). Using the nowhere-normal outward condition for such mapping and appropriate conditions on $\{{\alpha }_n\}$, they proved that $\{x_n\}$ generated by (1.2) converges strongly to a fixed point of T which is nearest to x, further they proved that $\{y_n\}$ generated by (1.3) converges strongly to a fixed point of T which is nearest to y when $F(T)$ is nonempty. Very recently, using Cesàro means, Song and Chen [11] introduced the viscosity approximation method for nonexpansive self-mappings in a real Banach space. Let f be a contraction of C into itself and T nonexpansive mapping of C into itself with $F(T)\ne \varnothing$. They proposed the following implicit viscosity iterative method:

$$z_m=t_{m}f(z_m)+(1-t_m)\frac{1}{m+1}\sum _{j=0}^m{T}^j{z}_m\text{,}m⩾0\text{,}$$

where $\{t_m\}$ is an appropriate sequence in (0, 1), and the following explicit viscosity iterative method:

$$x_0\in C\text{,}{x}_{n+1}={\alpha }_{n}f(x_n)+(1-{\alpha }_n)\frac{1}{n+1}\sum _{j=0}^n{T}^j{x}_n\text{,}n⩾0\text{,}$$

where $\{{\alpha }_n\}$ is an appropriate sequence in (0, 1), and then they proved the following theorems.

Theorem 1.1 [11, Theorem 3.2]

Let E be a real uniformly convex Banach space which admits a weakly sequentially continuous duality mapping J from E to $E^{\ast }$ and C a nonempty closed convex subset of E. Suppose that T is a nonexpansive mapping from C into C with $F(T)$ is nonempty, and $f:C\to C$ is a fixed contractive mapping with coefficient $\beta \in (0\text{,}1)$. Let $\{z_m\}$ be defined by (1.4) and ${\lim }_{m\to \infty }{t}_m=0$. Then $\{z_m\}$ converges strongly to some fixed point p of T, where p is the unique solution in $F(T)$ to the following variational inequality:

$$〈(f-I)p\text{,}j(u-p)〉⩽0\text{for all}u\in F(T)\text{.}$$

Theorem 1.2 [11, Theorem 4.1]

Let E be a real uniformly convex Banach space which admits a weakly sequentially continuous duality mapping J from E to $E^{\ast }$ and C a nonempty closed convex subset of E. Suppose that T is a nonexpansive mapping from C into C with $F(T)$ is nonempty, and $f:C\to C$ is a fixed contractive mapping with coefficient $\beta \in (0\text{,}1)$. Let $\{x_n\}$ be defined by (1.5), where $\{{\alpha }_n\}$ is a sequence of real numbers satisfying $0⩽{\alpha }_n⩽1\text{,}$ ${\lim }_{n\to \infty }{\alpha }_n=0$ and ${\sum }_{n=0}^{\infty }{\alpha }_n=\infty$. Then $\{x_n\}$ converges strongly to some fixed point p of T, where p is the unique solution in $F(T)$ to the variational inequality (1.6).

Motivated and inspired by the above results, in this paper, we study the strong convergence of the viscosity iterative processes $\{z_m\}$, $\{x_n\}$ and $\{y_n\}$ by, respectively, Eqs. (1.7), (1.8), (1.9) which are mixed iteration processes of (1.2), (1.3), (1.4), (1.5). To do this, we will use similar method given in [11] with different approximation from obtained many results. We consider the case $T:C\to E$ is a nonexpansive nonself-mapping with $F(T)\ne \varnothing$, $f:C\to C$ is a fixed contractive self-mapping, and P is the sunny nonexpansive retraction of E onto C, and define the implicit viscosity iterative method as follows:

$$\{t_m\}\subseteq (0\text{,}1)\text{,}{z}_m=\frac{1}{m+1}\sum _{j=0}^{m}P(t_{m}f(z_m)+(1-t_m)(\mathit{TP}{)}^j{z}_m)\text{,}m⩾0$$

and two explicit viscosity iterative methods by

$$x_0\in C\text{,}{x}_{n+1}={\alpha }_{n}f(x_n)+(1-{\alpha }_n)\frac{1}{n+1}\sum _{j=0}^n(\mathit{PT}{)}^j{x}_n\text{,}n⩾0$$

and

$$y_0\in C\text{,}{y}_{n+1}=\frac{1}{n+1}\sum _{j=0}^{n}P({\alpha }_{n}f(y_n)+(1-{\alpha }_n)(\mathit{TP}{)}^j{y}_n)\text{,}n⩾0\text{,}$$

where $\{{\alpha }_n\}$ is an appropriate sequence in $(0\text{,}1)$. In a real uniformly convex Banach space E which admits a weakly sequentially continuous duality mapping J from E to $E^{\ast }$, we will prove that $\{z_m\}$, $\{x_n\}$ and $\{y_n\}$ converge strongly to some $p\in F(T)$, where p is a unique solution to the following variational inequality:

$$〈(f-I)p\text{,}j(u-p)〉⩽0\text{for all}u\in F(T)\text{.}$$

Our results extend and improve the corresponding ones announced by Matsushita and Kuroiwa [7], Song and Chen [11] and others.