# Strong Convergence of an Iterative Method for Inverse Strongly Accretive Operators

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## Abstract

We study the strong convergence of an iterative method for inverse strongly accretive operators in the framework of Banach spaces. Our results improve and extend the corresponding results announced by many others.

### Keywords

Banach Space Variational Inequality Convex Subset Nonexpansive Mapping Strong Convergence## 1. Introduction and Preliminaries

for all Open image in new window (see [1, 2, 3, 4]). For such a case, Open image in new window is said to be Open image in new window -inverse strongly monotone.

for all Open image in new window It is known that if Open image in new window is a nonexpansive mapping of Open image in new window into itself, then Open image in new window is Open image in new window -inverse strongly monotone and Open image in new window , where Open image in new window denotes the set of fixed points of Open image in new window .

for Open image in new window Moreover, Open image in new window is characterized by the properties Open image in new window and Open image in new window for all Open image in new window

One can see that the variational inequality problem (1.1) is equivalent to some fixed-point problem. The element Open image in new window is a solution of the variational inequality (1.1) if and only if Open image in new window satisfies the relation Open image in new window where Open image in new window is a constant.

To find a solution of the variational inequality for an inverse strongly monotone operator, Iiduka et al. [2] proved the following weak convergence theorem.

Theorem 1.

for all Open image in new window , where Open image in new window is the metric projection from Open image in new window onto Open image in new window , Open image in new window is a sequence in Open image in new window and Open image in new window is a sequence in Open image in new window . If Open image in new window and Open image in new window are chosen so that Open image in new window for some Open image in new window Open image in new window with Open image in new window and Open image in new window for some Open image in new window Open image in new window with Open image in new window , then the sequence Open image in new window defined by (1.6) converges weakly to some element of Open image in new window .

- (1)
- (2)
- (3)

where Open image in new window is a function. It is known that Open image in new window is uniformly smooth if and only if Open image in new window . Let Open image in new window be a fixed real number with Open image in new window . A Banach space Open image in new window is said to be Open image in new window -uniformly smooth if there exists a constant Open image in new window such that Open image in new window for all Open image in new window .

- (1)is a uniformly smooth Banach space if and only if Open image in new window is single-valued and uniformly continuous on any bounded subset of Open image in new window ;
- (2)
all Hilbert spaces, Open image in new window (or Open image in new window ) spaces ( Open image in new window ), and the Sobolev spaces, Open image in new window ( Open image in new window ), are Open image in new window -uniformly smooth, while Open image in new window (or Open image in new window ) and Open image in new window spaces ( Open image in new window ) are Open image in new window -uniformly smooth.

for all Open image in new window .

for all Open image in new window . Evidently, the definition of the inverse strongly accretive operator is based on that of the inverse strongly monotone operator.

whenever Open image in new window for Open image in new window and Open image in new window . A mapping Open image in new window of Open image in new window into itself is called a retraction if Open image in new window . If a mapping Open image in new window of Open image in new window into itself is a retraction, then Open image in new window for all Open image in new window , where Open image in new window is the range of Open image in new window . A subset Open image in new window of Open image in new window is called a sunny nonexpansive retract of Open image in new window if there exists a sunny nonexpansive retraction from Open image in new window onto Open image in new window . We know the following lemma concerning sunny nonexpansive retraction.

Lemma 1.1 (see [5]).

for all Open image in new window and Open image in new window .

for all Open image in new window . In order to find a solution of the variational inequality (1.15), the authors proved the following theorem in the framework of Banach spaces.

Theorem 1.

converges weakly to some element Open image in new window of Open image in new window , where Open image in new window is the Open image in new window -uniformly smoothness constant of Open image in new window .

In this paper, motivated by Aoyama et al. [6], Iiduka et al. [2], Takahahsi and Toyoda [4], we introduce an iterative method to approximate a solution of variational inequality (1.15) for an Open image in new window -inverse strongly accretive operators. Strong convergence theorems are obtained in the framework of Banach spaces under appropriate conditions on parameters.

We also need the following lemmas for proof of our main results.

Lemma 1.2 (see [7]).

for all Open image in new window , where Open image in new window is the Open image in new window -uniformly smoothness constant of Open image in new window .

The following lemma is characterized by the set of solutions of variational inequality (1.15) by using sunny nonexpansive retractions.

Lemma 1.3 (see [6]).

Lemma 1.4 (see [8]).

Let Open image in new window be a nonempty bounded closed convex subset of a uniformly convex Banach space Open image in new window and let Open image in new window be nonexpansive mapping of Open image in new window into itself. If Open image in new window is a sequence of Open image in new window such that Open image in new window weakly and Open image in new window , then Open image in new window is a fixed point of Open image in new window .

Lemma 1.5 (see [9]).

Then Open image in new window .

Lemma 1.6 (see [10]).

for all Open image in new window , where Open image in new window is a sequence in Open image in new window and Open image in new window is a sequence in Open image in new window such that

(i) Open image in new window ;

(ii) Open image in new window or Open image in new window .

Then Open image in new window .

## 2. Main Results

Theorem 2.1.

Let Open image in new window be a uniformly convex and Open image in new window -uniformly smooth Banach space and Open image in new window a nonempty closed convex subset of Open image in new window . Let Open image in new window be a sunny nonexpansive retraction from Open image in new window onto Open image in new window , Open image in new window an arbitrarily fixed point, and Open image in new window an Open image in new window -inverse strongly accretive operator of Open image in new window into Open image in new window such that Open image in new window . Let Open image in new window and Open image in new window be two sequences in Open image in new window and let Open image in new window a real number sequence in Open image in new window for some Open image in new window satisfying the following conditions:

(i) Open image in new window and Open image in new window ;

(ii) Open image in new window ;

(iii) Open image in new window .

converges strongly to Open image in new window , where Open image in new window is a sunny nonexpansive retraction of Open image in new window onto Open image in new window .

Proof.

Applying Lemma 1.6 to (2.22), we can conclude the desired conclusion. This completes the proof.

As an application of Theorem 2.1, we have the following results in the framework of Hilbert spaces.

Corollary 2.2.

Let Open image in new window be a Hilbert space and Open image in new window a nonempty closed convex subset of Open image in new window . Let Open image in new window be a metric projection from Open image in new window onto Open image in new window , Open image in new window an arbitrarily fixed point, and Open image in new window an Open image in new window -inverse strongly monotone operator of Open image in new window into Open image in new window such that Open image in new window . Let Open image in new window and Open image in new window be two sequences in Open image in new window and let Open image in new window be a real number sequence in Open image in new window for some Open image in new window satisfying the following conditions:

(i) Open image in new window and Open image in new window ;

(ii) Open image in new window ;

(iii) Open image in new window .

converges strongly to Open image in new window .

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