Abstract and Applied Analysis

VolumeΒ 2012Β (2012), Article IDΒ 453452, 14 pages

http://dx.doi.org/10.1155/2012/453452

## Viscosity Methods of Asymptotically Pseudocontractive and Asymptotically Nonexpansive Mappings for Variational Inequalities

^{1}Department of Mathematics, Tianjin Polytechnic University, Tianjin 300387, China^{2}Department of Information Management, Cheng Shiu University, Kaohsiung 833, Taiwan

Received 27 June 2012; Accepted 21 August 2012

Academic Editor: YonghongΒ Yao

Copyright Β© 2012 Xionghua Wu et al. 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 be such that as , let and be two positive numbers such that , and let be a contraction. If be a continuous asymptotically pseudocontractive self-mapping of a nonempty bounded closed convex subset of a real reflexive Banach space with a uniformly Gateaux differentiable norm, under suitable conditions on the sequence , we show the existence of a sequence satisfying the relation and prove that converges strongly to the fixed point of , which solves some variational inequality provided is uniformly asymptotically regular. As an application, if be an asymptotically nonexpansive self-mapping of a nonempty bounded closed convex subset of a real Banach space with a uniformly Gateaux differentiable norm and which possesses uniform normal structure, we prove that the iterative process defined by converges strongly to the fixed point of .

#### 1. Introduction

Let be a real Banach space with dual and a nonempty closed convex subset of . Recall that a mapping is said to be asymptotically pseudocontractive if, for each and , there exist and a constant with such that where denote the normalized duality mapping defined by The class of asymptotically pseudocontractive mappings is essentially wider than the class of asymptotically nonexpansive mappings. A mapping is called asymptotically nonexpansive if there exists a sequence with such that for all integers and all . A mapping is called a contraction if there exists a constant such that It is clear that every contraction is nonexpansive, every nonexpansive mapping is asymptotically nonexpansive, and every asymptotically nonexpansive mapping is asymptotically pseudocontractive. The converses do not hold. The asymptotically nonexpansive mappings are important generalizations of nonexpansive mappings. For details, you may see [1].

The mapping is called uniformly asymptotically regular (in short u.a.r.) if for each there exists such that for all and and it is called uniformly asymptotically regular with sequence (in short u.a.r.s.) if for all integers and all , where as .

The viscosity approximation method of selecting a particular fixed point of a given nonexpansive mapping was proposed by Moudafi [2] who proved the strong convergence of both the implicit and explicit methods in Hilbert spaces, see [2, Theorems 2.1 and 2.2]. Xu [3] studied the viscosity approximation methods proposed by Moudafi [2] for a nonexpansive mapping in a uniformly smooth Banach space.

Very recently, Shahzad and Udomene [4] obtained fixed point solutions of variational inequalities for an asymptotically nonexpansive mapping defined on a real Banach space with uniformly Gateaux differentiable norm possessing uniform normal structure. They proved the following theorem.

Theorem 1.1. *Let be a real Banach space with a uniformly Gateaux differentiable norm possessing uniform normal structure, let be a nonempty closed convex and bounded subset of , let be an asymptotically nonexpansive mapping with sequence , and let be a contraction with constant . Let be such that ,β , and , where . For an arbitrary let the sequence be iteratively defined by
**
Then*(i)* for each integer , there is a unique such that
*β*if in addition
*β*then*(ii)* the sequence converges strongly to the unique solution of the variational inequality:
*

*Remark 1.2. *We note that , then the condition (1.9) and imply that
respectively. In other words, if an asymptotically nonexpansive mapping satisfies the condition (1.9) then must be u.a.r.s.

Inspired by the works in [4β8], in this paper, we suggest and analyze a modification of the iterative algorithm.

Let , let and be two positive numbers such that , and let be a contraction on , a sequence iteratively defined by: ,

*Remark 1.3. *The algorithm (1.12) includes the algorithm (1.7) of Chidume et al. [5] and Shahzad and Udomene [4] as a special case.

We show the convergence of the proposed algorithm (1.12) to the unique solution of some variational inequality (some related works on VI, please see [9β12]). In this respect, our results can be considered as a refinement and improvement of the known results of Chidume et al. [5], Shahzad and Udomene [4], and Lim and Xu [13].

#### 2. Preliminaries

Let denote the unit sphere of the Banach space . The space is said to have a Gateaux differentiable norm if the limit exists for each , and we call smooth; is said to have a uniformly Gateaux differentiable norm if for each the limit (2.1) is attained uniformly for . Further, is said to be uniformly smooth if the limit (2.1) exists uniformly for . It is well known [14] that if is smooth then any duality mapping on is single-valued, and if has a uniformly Gateaux differentiable norm then the duality mapping is norm-to-weak* uniformly continuous on bounded sets.

Let be a nonempty closed convex and bounded subset of the Banach space and let the diameter of be defined by . For each , let and let denote the Chebyshev radius of relative to itself. The normal structure coefficient of is defined by A space such that is said to have uniform normal structure. It is known that every space with a uniform normal structure is reflexive, and that all uniformly convex and uniformly smooth Banach spaces have uniform normal structure (see [13]).

We will let LIM be a Banach limit. Recall that such that , , and for all . Let be a bounded sequence of . Then we can define the real-valued continuous convex function on by for all .

Let be a nonlinear mapping and . is said to satisfy the property (S) if for any bounded sequence in , implies .

Lemma 2.1 (see [15]). *Let be a Banach space with the uniformly Gateaux differentiable norm and . Then
**
if and only if
**
for all .*

Lemma 2.2 (see [16]). * Assume is a sequence of nonnegative real numbers such that
**
where is a sequence in and is a sequence such that *(1)*; *(2)* or . ** Then .*

Lemma 2.3 (see [17]). *Let and be bounded sequences in a Banach space and let be a sequence in with . Suppose that
**
for all and
**
Then .*

Lemma 2.4. *Let be an arbitrary real Banach space. Then
**
for all and for all .*

Lemma 2.5 (see [5]). *Let be a Banach space with uniform normal structure, a nonempty closed convex and bounded subset of , and an asymptotically nonexpansive mapping. Then has a fixed point.*

#### 3. Main Results

Theorem 3.1. *Let be a real reflexive Banach space with a uniformly Gateaux differentiable norm, a nonempty closed convex and bounded subset of , a continuous asymptotically pseudocontractive mapping with sequence , and a contraction with constant . Let be such that and . Suppose satisfies the property (S). Then *(i)* for each integer , there is a unique such that
*β*if is u.a.r.s., then *(ii)* the sequence converges strongly to the unique solution of the variational inequality:
*

*Proof. *By the conditions on , implies for each integer , then the mapping defined for each by is a strictly pseudocontractive mapping.

Indeed, for all , we have
It follows [18, Corollary 1] that possesses exactly one fixed point in such that .

From (3.1), we have
Notice that
Therefore, from (3.4), (3.5), and which is u.a.r.s., we obtain as .

Define a function by
for all . Since is continuous and convex, as , and is reflexive, attains it infimum over . Let such that and let . Then is nonempty because . Since satisfies the property (S), it follows that . Suppose that . Then, by Lemma 2.1, we have
for all . In particular, we have
On the other hand, from (3.1), we have
Now, for any , we have
for some and it follows from (3.9) that
which implies that
Consequently, similar to the lines of the proof of [4, Theorem 3.1], Theorem 3.1 is easily proved. This completes the proof.

Corollary 3.2. *Let be a real Banach space with a uniformly Gateaux differentiable norm possessing uniform normal structure, a nonempty closed convex and bounded subset of , be an asymptotically nonexpansive mapping with sequence , and a contraction with constant . Let be such that and . Then *(i)* for each integer , there is a unique such that
*β*if is u.a.r.s., then *(ii)* the sequence converges strongly to the unique solution of the variational inequality:
*

Theorem 3.3. *Let be a real Banach space with a uniformly Gateaux differentiable norm possessing uniform normal structure, a nonempty closed convex and bounded subset of , an asymptotically nonexpansive mapping with sequence , and a contraction with constant . Let be such that , and , where . For an arbitrary , let the sequence be iteratively defined by (1.12). Then *(i)* for each integer , there is a unique such that
*β*if is u.a.r.s., then *(ii)* the sequence converges strongly to the unique solution of the variational inequality:
*

*Proof. *Set , then as . Define
Observe that
It follows that
We note that
It follows that
which implies that
From (3.19) and (3.22), we obtain
Hence, by Lemma 2.3 we know
consequently

On the other hand,
which implies that
Hence, we have
From (3.15), . Applying Lemma 2.4, we estimate as follows:
Since is bounded, for some constant , it follows that
so that
By Corollary 3.2, , which solve the variational inequality (3.16). Since is norm to weak* continuous on bounded sets, in the limit as , we obtain that
From Lemma 2.4, we estimate as follows:
so that
Let
Consequently, following the lines of the proof of [4, Theorem 3.3], Theorem 3.3 is easily proved.

From the lines of the proof of Theorem 3.3, we can obtain the following corollary.

Corollary 3.4. *Let be a real Banach space with a uniformly Gateaux differentiable norm possessing uniform normal structure, a nonempty closed convex and bounded subset of , an asymptotically nonexpansive mapping with sequence , and a contraction with constant . Let be such that , and , where . For an arbitrary , let the sequence be iteratively defined by (1.12). Then*(i)* for each integer , there is a unique such that
*β*if satisfies and then*(ii)* the sequence converges strongly to the unique solution of the variational inequality:
*

*Remark 3.5. *Since every nonexpansive mapping is asymptotically nonexpansive, our theorems hold for the case when is simply nonexpansive. In this case, the boundedness requirement on can be removed from the above results.

*Remark 3.6. *Our results can be viewed as a refinement and improvement of the corresponding results by Shahzad and Udomene [4], Chidume et al. [5], and Lim and Xu [13].

*Example 3.7. *Let be a nonexpansive mapping. Let the iterative sequence be defined by
It is easy to see that converges strongly to some fixed point of .

In particular, let and define by
and take that is a fix element in . It is obvious that is a nonexpansive mapping with a unique fixed point . In this case, (3.38) becomes
It is clear that the complex number sequence converges strongly to a fixed point .

#### Acknowledgment

Y.-C. Liou was partially supported by NSC 101-2628-E-230-001-MY3.

#### References

- K. Goebel and W. A. Kirk, βA fixed point theorem for asymptotically nonexpansive mappings,β
*Proceedings of the American Mathematical Society*, vol. 35, pp. 171β174, 1972. View at Publisher Β· View at Google Scholar Β· View at Zentralblatt MATH - A. Moudafi, βViscosity approximation methods for fixed-points problems,β
*Journal of Mathematical Analysis and Applications*, vol. 241, no. 1, pp. 46β55, 2000. View at Publisher Β· View at Google Scholar Β· View at Zentralblatt MATH - H.-K. Xu, βViscosity approximation methods for nonexpansive mappings,β
*Journal of Mathematical Analysis and Applications*, vol. 298, no. 1, pp. 279β291, 2004. View at Publisher Β· View at Google Scholar Β· View at Zentralblatt MATH - N. Shahzad and A. Udomene, βFixed point solutions of variational inequalities for asymptotically nonexpansive mappings in Banach spaces,β
*Nonlinear Analysis*, vol. 64, no. 3, pp. 558β567, 2006. View at Publisher Β· View at Google Scholar Β· View at Zentralblatt MATH - C. E. Chidume, J. Li, and A. Udomene, βConvergence of paths and approximation of fixed points of asymptotically nonexpansive mappings,β
*Proceedings of the American Mathematical Society*, vol. 133, no. 2, pp. 473β480, 2005. View at Publisher Β· View at Google Scholar Β· View at Zentralblatt MATH - Y. Yao and Y.-C. Liou, βStrong convergence to common fixed points of a finite family of asymptotically nonexpansive map,β
*Taiwanese Journal of Mathematics*, vol. 11, no. 3, pp. 849β865, 2007. View at Google Scholar Β· View at Zentralblatt MATH - L.-C. Ceng, H.-K. Xu, and J.-C. Yao, βThe viscosity approximation method for asymptotically nonexpansive mappings in Banach spaces,β
*Nonlinear Analysis*, vol. 69, no. 4, pp. 1402β1412, 2008. View at Publisher Β· View at Google Scholar Β· View at Zentralblatt MATH - S. Wang, L. Yu, and B. Guo, βAn implicit iterative scheme for an infinite countable family of asymptotically nonexpansive mappings in Banach spaces,β
*Fixed Point Theory and Applications*, vol. 2008, Article ID 350483, 10 pages, 2008. View at Google Scholar Β· View at Zentralblatt MATH - Y. Yao, Y. C. Liou, and S. M. Kang, βTwo-step projection methods for a system of variational inequality problems in Banach spaces,β
*Journal of Global Optimization*, pp. 1β11, 2011. View at Publisher Β· View at Google Scholar - Y. Yao, M. A. Noor, and Y.-C. Liou, βStrong convergence of a modified extragradient method to the minimum-norm solution of variational inequalities,β
*Abstract and Applied Analysis*, vol. 2012, Article ID 817436, 9 pages, 2012. View at Publisher Β· View at Google Scholar Β· View at Zentralblatt MATH - Y. Yao, M. A. Noor, Y.-C. Liou, and S. M. Kang, βIterative algorithms for general multivalued variational inequalities,β
*Abstract and Applied Analysis*, vol. 2012, Article ID 768272, 10 pages, 2012. View at Publisher Β· View at Google Scholar Β· View at Zentralblatt MATH - Y. Yao, R. Chen, and H.-K. Xu, βSchemes for finding minimum-norm solutions of variational inequalities,β
*Nonlinear Analysis*, vol. 72, no. 7-8, pp. 3447β3456, 2010. View at Publisher Β· View at Google Scholar Β· View at Zentralblatt MATH - T.-C. Lim and H. K. Xu, βFixed point theorems for asymptotically nonexpansive mappings,β
*Nonlinear Analysis*, vol. 22, no. 11, pp. 1345β1355, 1994. View at Publisher Β· View at Google Scholar Β· View at Zentralblatt MATH - I. Cioranescu,
*Geometry of Banach Spaces, Duality Mappings and Nonlinear Problems*, vol. 62 of*Mathematics and its Applications*, Kluwer Academic, Dordrecht, The Netherlands, 1990. View at Publisher Β· View at Google Scholar - K. S. Ha and J. S. Jung, βStrong convergence theorems for accretive operators in Banach spaces,β
*Journal of Mathematical Analysis and Applications*, vol. 147, no. 2, pp. 330β339, 1990. View at Publisher Β· View at Google Scholar Β· View at Zentralblatt MATH - H.-K. Xu, βIterative algorithms for nonlinear operators,β
*Journal of the London Mathematical Society*, vol. 66, no. 1, pp. 240β256, 2002. View at Publisher Β· View at Google Scholar Β· View at Zentralblatt MATH - T. Suzuki, βStrong convergence of Krasnoselskii and Mann's type sequences for one-parameter nonexpansive semigroups without Bochner integrals,β
*Journal of Mathematical Analysis and Applications*, vol. 305, no. 1, pp. 227β239, 2005. View at Publisher Β· View at Google Scholar Β· View at Zentralblatt MATH - K. Deimling, βZeros of accretive operators,β
*Manuscripta Mathematica*, vol. 13, pp. 365β374, 1974. View at Publisher Β· View at Google Scholar Β· View at Zentralblatt MATH