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Abstract and Applied Analysis
Volume 2012 (2012), Article ID 623625, 7 pages
http://dx.doi.org/10.1155/2012/623625
Research Article

Fixed Point of Strong Duality Pseudocontractive Mappings and Applications

Department of Mathematics, Cangzhou Normal University, Cangzhou 061001, China

Received 16 July 2012; Accepted 22 July 2012

Academic Editor: Yongfu Su

Copyright © 2012 Baowei Liu. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Abstract

Let E be a smooth Banach space with the dual , an operator is said to be α-strong duality pseudocontractive if , for all , where α is a nonnegative constant. An element is called a duality fixed point of T if . The purpose of this paper is to introduce the definition of α-strong duality pseudocontractive mappings and to study its fixed point problem and applications for operator equation and variational inequality problems.

1. Introduction and Preliminaries

Let be a real Banach space with the dual : let be an operator from into . We consider the first operator equation problem of finding an element such that We also consider the second variational inequality problem of finding an element such that

Let be a real Banach space with the dual . Let be a given real number with . The generalized duality mapping from into is defined by where denotes the generalized duality pairing. In particular, is called the normalized duality mapping and for all . If is a Hilbert space, then , where is the identity mapping. The duality mapping has the following properties: (i)if is smooth, then is single valued;(ii)if is strictly convex, then is one to one;(iii)if is reflexive, then is a mapping of onto ;(iv)if is uniformly smooth, then is uniformly norm-to-norm continuous on each bounded subset of ;(v)if is uniformly convex, then is uniformly continuous on each bounded subsets of and is single valued and also one to one.

For more details, see [1, 2].

Let be a smooth Banach space with the dual . Let be an operator; an element is called a duality fixed point of , if .

We also consider the third variational inequality problem of finding an element such that where is a closed convex subset of . The set of solutions of the variational inequality problem (1.4) is denoted by .

We also consider the fourth variational inequality problem of finding an element such that where is a closed convex subset of . The set of solutions of the variational inequality problem (1.5) is denoted by .

Conclusion. If is a duality fixed point of , then must be a solution of problem (1.1).

Proof. If is a normalized fixed point of , then , so that This completes the proof.

Conclusion. If is a duality fixed point of , then must be a solution of variational inequality problem (1.2).

Proof. Suppose is a duality fixed point of ; then Obverse that for all . This completes the proof.

Let . A Banach space is said to be strictly convex if for any , implies . It is also said to be uniformly convex if for each , there exists such that for any , implies . It is known that a uniformly convex Banach space is reflexive and strictly convex. And we define a function called the modulus of convexity of as follows: It is known that is uniformly convex if and only if for all . Let be a fixed real number with . Then is said to be -uniformly convex if there exists a constant such that for all . For example, see [3, 4] for more details. The constant is said to be uniformly convexity constant of .

A Banach space E is said to be smooth if the limit exists for all . It is also said to be uniformly smooth if the above limit is attained uniformly for . One should note that no Banach space is -uniformly convex for ; see [5] for more details. It is well known that the Hilbert and the Lebesgue spaces are -uniformly convex and uniformly smooth. Let be a Banach space, and let be the Lebesgue-Bochner space on an arbitrary measure space . Let , and let . Then is -uniformly convex if and only if is -uniformly convex; see [4].

In this paper, we first propose the definition of generalized -strongly pseudocontractive mappings from a smooth Banach into its dual as follows. We also discuss the problem of fixed point for generalized -strongly pseudocontractive mappings and its applications.

Let be a smooth Banach space and denote the dual of . An operator is said to be

(1) -inverse-strongly monotone if there exists nonnegative real number such that

(2) -strong duality pseudocontractive mapping, if there exists a nonnegative real number such that for all .

It is easy to show that is -strong duality pseudocontractive if and only if is -inverse-strongly monotone.

Let be a smooth Banach space and denote the dual of . Let be an operator. The set of zero points of is defined by . The set of duality fixed points of is defined by . It is also easy to show that, an element is a zero point of an -inverse-strongly monotone operator if and only if is a duality fixed point of the -strong duality pseudocontractive mapping .

2. Main Results and Applications

Recently, Zegeye and Shahzad [6] proved the following result.

Theorem 2.1 (see, [6]). Let be a uniformly smooth and 2-uniformly convex real Banach space with the dual : let be a -inverse-strongly monotone mapping and a relatively weak nonexpansive mapping with . Assume that , where is the uniformly convexity constant. Define a sequence in by the following algorithm: where is the duality mapping on . Then converges strongly to , where is the generalized projection from onto .

IF taking , then Theorem 2.1 reduces to the following result.

Theorem 2.2. Let be a uniformly smooth and 2-uniformly convex real Banach space with the dual , let be a -inverse strongly monotone mapping with . Assume that , where is the uniformly convexity constant. Define a sequence in by the following algorithm: where is the duality mapping on . Then converges strongly to , where is the generalized projection from onto .

Theorem 2.3. Let be a uniformly smooth and 2-uniformly convex real Banach space; let be an -strong duality pseudocontractive mapping with nonempty set of duality fixed points . Let be a relatively weak nonexpansive mapping and . Assume . Define a sequence in by the following algorithm: where is the duality mapping on . Then converges strongly to a common element . This element is also a common solution of operator equation (1.1) and variational inequality (1.2).

Proof. Let , then is -inverse-strongly monotone and -strongly monotone, so that has only one element. On the other hand, we have By using Theorem 2.1 and Conclusions 1 and 2, we obtain the conclusion of Theorem 2.3.
Taking in Theorem 2.3, we get the following result.

Theorem 2.4. Let be a uniformly smooth and 2-uniformly convex real Banach space; let be a -Lipschitz and -strongly duality pseudocontractive mapping with nonempty set of duality fixed points . Assume . Define a sequence in by the following algorithm: where is the duality mapping on . Then converges strongly to a duality fixed point . This element is also a common solution of operator equation (1.1) and variational inequality (1.2).

Iiduka and Takahashi [7] introduce an iterative scheme for finding a solution of the variational inequality problem for an operator that satisfies the following conditions (i)–(iii) in a -uniformly convex and uniformly smooth Banach space :

(i) is -inverse-strongly monotone;

(ii) ;

(iii) for all and .

They proved the following convergence theorem.

Theorem 2.5 (see, [7]). Let be a -uniformly convex and uniformly smooth Banach space, whose duality mapping is weakly sequentially continuous, and a nonempty, closed convex subset of . Assume that is an operator of into , that satisfies the conditions (i)–(iii). Suppose that and is given by for every ,…, where is a sequence of positive numbers. If is chosen so that for some with , then the sequence converges weakly to some element , where is the -uniformly convexity constant of . Further .

In this paper, we introduce an iterative scheme for finding a solution of the variational inequality problem for an operator that satisfies the following conditions (iv)–(vi) in a -uniformly convex and uniformly smooth Banach space :

(iv) is -strong duality pseudocontractive,

(v) ,

(vi) for all and .

By using Theorem 2.5, we prove the following convergence theorem.

Theorem 2.6. Let be a -uniformly convex and uniformly smooth Banach space, whose duality mapping is weakly sequentially continuous, and a nonempty, closed convex subset of . Assume that is an operator of into . that satisfies the conditions (iv)–(vi). Suppose that and is given by for every ,…, where is a sequence of positive numbers. If is chosen so that for some with , then the sequence converges weakly to some element , where is the -uniformly convexity constant of . Further .

Proof. Let , then is -inverse-strongly monotone, so that . On the other hand, we have By using Theorem 2.5, we obtain the conclusion of Theorem 2.6.

In fact, from condition (vi), we have , so that under the conditions of Theorem 2.6, the converges strongly to a duality fixed point . This element is also a common solution of operator equation (1.1) and variational inequality (1.2). where is defined by Algorithm (2.7).

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