Strong Convergence of an Iterative Method for Inverse Strongly Accretive Operators

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.

1. Introduction and Preliminaries

Let be a real Hilbert space with norm and inner product , a nonempty closed convex subset of and a monotone operator of into . The classical variational inequality problem is formulated as finding a point such that

(1.1)

for all . Such a point is called a solution of the variational inequality (1.1). Next, the set of solutions of the variational inequality (1.1) is denoted by . In the case when , holds, where

(1.2)

Recall that an operator of into is said to be inverse strongly monotone if there exists a positive real number such that

(1.3)

for all (see [1–4]). For such a case, is said to be -inverse strongly monotone.

Recall that is nonexpansive if

(1.4)

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

Let be the projection of onto the convex subset . It is known that projection operator is nonexpansive. It is also known that satisfies

(1.5)

for Moreover, is characterized by the properties and for all

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

Let be a nonempty closed convex subset of a real Hilbert space and let be an -inverse strongly monotone operator of into with . Let be a sequence defined as follows:

(1.6)

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

Next, we assume that is a nonempty closed and convex subset of a Banach space . Let be the dual space of and let denote the pairing between and . For , the generalized duality mapping is defined by

(1.7)

for all . In particular, is called the normalized duality mapping. It is known that for all . If is a Hilbert space, then . Further, we have the following properties of the generalized duality mapping :

• (1)
  

  for all with ;

• (2)
  

  for all and ;

• (3)
  

  for all .

Let . A Banach space is said to be uniformly convex if, for any , there exists such that, for any ,

(1.8)

It is known that a uniformly convex Banach space is reflexive and strictly convex. A Banach space is said to be smooth if the limit

(1.9)

exists for all . It is also said to be uniformly smooth if the limit (1.9) is attained uniformly for . The norm of is said to be Fréchet differentiable if, for any , the limit (1.9) is attained uniformly for all . The modulus of smoothness of is defined by

(1.10)

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

Note that

• (1)
  

  is a uniformly smooth Banach space if and only if is single-valued and uniformly continuous on any bounded subset of ;

• (2) all Hilbert spaces, (or ) spaces (), and the Sobolev spaces, (), are -uniformly smooth, while (or ) and spaces () are -uniformly smooth.

Recall that an operator of into is said to be accretive if there exists such that

(1.11)

for all .

For recall that an operator of into is said to be -inverse strongly accretive if

(1.12)

for all . Evidently, the definition of the inverse strongly accretive operator is based on that of the inverse strongly monotone operator.

Let be a subset of and let be a mapping of into . Then is said to be sunny if

(1.13)

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

Lemma 1.1 (see [5]).

Let be a closed convex subset of a smooth Banach space , let be a nonempty subset of , and let be a retraction from onto . Then is sunny and nonexpansive if and only if

(1.14)

for all and .

Recently, Aoyama et al. [6] first considered the following generalized variational inequality problem in a smooth Banach space. Let be an accretive operator of into . Find a point such that

(1.15)

for all . 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.

Let be a uniformly convex and -uniformly smooth Banach space and a nonempty closed convex subset of . Let be a sunny nonexpansive retraction from onto , and an -inverse strongly accretive operator of into with , where

(1.16)

If and are chosen such that for some and for some with , then the sequence defined by the following manners:

(1.17)

converges weakly to some element of , where is the -uniformly smoothness constant of .

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 -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]).

Let be a given real number with and let be a -uniformly smooth Banach space. Then

(1.18)

for all , where is the -uniformly smoothness constant of .

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]).

Let be a nonempty closed convex subset of a smooth Banach space . Let be a sunny nonexpansive retraction from onto and let be an accretive operator of into . Then, for all ,

(1.19)

Lemma 1.4 (see [8]).

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

Lemma 1.5 (see [9]).

Let , be bounded sequences in a Banach space and let be a sequence in which satisfies the following condition:

(1.20)

Suppose that

(1.21)

for all and

(1.22)

Then .

Lemma 1.6 (see [10]).

Assume that is a sequence of nonnegative real numbers such that

(1.23)

for all , where is a sequence in and is a sequence in such that

(i);

(ii) or .

Then .

2. Main Results

Theorem 2.1.

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

(i) and ;

(ii);

(iii).

Then the sequence defined by

(2.1)

converges strongly to , where is a sunny nonexpansive retraction of onto .

Proof.

First, we show that is nonexpansive for all . Indeed, for all and , from Lemma 1.2, one has

(2.2)

Therefore, one obtains that is a nonexpansive mapping for all . For all , it follows from Lemma 1.3 that . Put . Noticing that

(2.3)

one has

(2.4)

from which it follows that

(2.5)

Now, an induction yields

(2.6)

Hence, is bounded, and so is . On the other hand, one has

(2.7)

Put , that is,

(2.8)

Next, we compute Observing that

(2.9)

we have

(2.10)

Combining (2.7) with (2.10), one obtains

(2.11)

It follows that

(2.12)

Hence, from Lemma 1.5, we obtain . From (2.7) and the condition (ii), one arrives at

(2.13)

On the other hand, from (2.1), one has

(2.14)

which combines with (2.13), and from the conditions (i), (ii), one sees that

(2.15)

Next, we show that

(2.16)

To show (2.16), we choose a sequence of that converges weakly to such that

(2.17)

Next, we prove that . Since for some , it follows that is bounded and so there exists a subsequence of which converges to . We may assume, without loss of generality, that . Since is nonexpansive, it follows from that

(2.18)

It follows from (2.15) that

(2.19)

From Lemma 1.4, we have . It follows from Lemma 1.3 that . Now, from (2.17) and Lemma 1.1, we have

(2.20)

From (2.1), we have

(2.21)

It follows that

(2.22)

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

(i) and ;

(ii);

(iii).

Then the sequence defined by

(2.23)

converges strongly to .