Journal of Applied MathematicsVolume 2012 (2012), Article ID 641479, 19 pagesdoi:10.1155/2012/641479

Research Article

Strong Convergence Theorems for Nonexpansive Semigroups and Variational Inequalities in Banach Spaces

Haiqing Wang,¹ Yongfu Su,¹ and Hong Zhang²

¹Department of Mathematics, Tianjin Polytechnic University, Tianjin 300160, China²Department of Mathematics, Tianjin No. 8 Middle School, Tianjin 300252, China

Received 11 November 2011; Accepted 17 December 2011

Academic Editor: Rudong Chen

Copyright © 2012 Haiqing Wang 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 a uniformly convex Banach space and be a nonexpansive semigroup such that . Consider the iterative method that generates the sequence by the algorithm , where , , and are three sequences satisfying certain conditions, is a contraction mapping. Strong convergence of the algorithm is proved assuming either has a weakly continuous duality map or has a uniformly Gâteaux differentiable norm.

1. Introduction

Let be a real Banach space and let be a nonempty closed convex subset of . A mapping of into itself is said to be nonexpansive if for each . We denote by the set of fixed points of . One classical way to study nonexpansive mappings is to use contractions to approximate a nonexpansive mapping (Browder [1] and Reich [2]). More precisely, take and define a contraction by

where is a fixed point. Banach’s contraction mapping principle guarantees that has a unique fixed point in . It is unclear, in general, what is the behavior of as , even if has a fixed point. In 1967, in the case of having a fixed point, Browder [3] proved that if is a Hilbert space, then converges strongly to the element of which is nearest to in as . Song and Xu [4] extended Browder’s result to the setting of Banach spaces and proved that if is a uniformly smooth Banach space, then converges strongly to a fixed point of and the limit defines the (unique) sunny nonexpansive retraction from onto .

Let be a contraction on such that , where is a constant. Let , and be the unique fixed point of the contraction , that is,

Concerning the convergence problem of the net , Moudafi [5] and Xu [6] by using the viscosity approximation method proved that the net converges strongly to a fixed point of T in C which is the unique solution to the following variational inequality:

Moreover, Xu [6] also studied the strong convergence of the following iterative sequence generated by

where is arbitrary, the sequence in satisfies the certain appropriate conditions.

A family of mappings of into itself is called a nonexpansive semigroup if it satisfies the following conditions:(i) for all ;(ii) for all and ;(iii) for all and ;(iv)for all , is continuous.

We denote by the set of all common fixed points of , that is, . It is known that is closed and convex.

It is an interesting problem to extend above (Moudafi’s [5], Xu’s [6], and so on) results to the nonexpansive semigroup case. Recently, for the nonexpansive semigroups , Plubtieng and Punpaeng [7] studied the continuous scheme defined by

where and is a positive real divergent net, and the iterative scheme defined by

where , , are a sequence in and is a positive real divergent real sequence in the setting of a real Hilbert space. They proved the continuous scheme defined by (1.5) and the iterative scheme defined by (1.6) converge strongly to a fixed point of which is the unique solution of the variational inequality (1.3). At this stage, the following question arises naturally.

Question 1. Do Plubtieng and Punpaeng’s results hold for the nonexpansive semigroups in a Banach space?

The purpose of this paper is to give affirmative answers of Question 1. One result of this paper says that Plubtieng and Punpaeng’s results hold in a uniformly convex Banach space which has a weakly continuous duality map.

On the other hand, Chen and Song [8] proved the following implicit and explicit viscosity iteration processes defined by (1.7) to nonexpansive semigroup case,

And they proved that converges strongly to a common fixed point of in a uniformly convex Banach space with a uniformly Gâteaux differentiable norm.

Motivated by the above results, the other result of this paper says that Plubtieng and Punpaeng’s results hold in the framework of uniformly convex Banach space with a uniformly Gâteaux differentiable norm. The results improve and extend the corresponding results of Plubtieng and Punpaeng [7], Chen and Song [8], Moudafi’s [5], Xu’s [6], and others.

2. Preliminaries

Let be a real Banach space with inner product and norm , respectively. Let denote the normalized duality mapping from into the dual space given by

In the sequel, we will denote the single valued duality mapping by . When is a sequence in , then will denote strong (weak) convergence of the sequence to .

Let . Then the norm of is said to be Gâteaux differentiable if exists for each . In this case, is called smooth. The norm of is said to be uniformly Gâteaux differentiable if for each , the limit (2.2) is attained uniformly for . It is well known that is smooth if and only if any duality mapping on is sigle valued. Also if has a uniformly Gâteaux differentiable norm, then the duality mapping is norm-to-weak* uniformly continuous on bounded sets. The norm of E is called Fréchet differentiable, if for each , the limit (2.2) is attained uniformly for . The norm of is called uniformly Fréchet differentiable, if the limit (2.2) is attained uniformly for . It is well known that (uniformly) Fréchet differentiability of the norm of implies (uniformly) Gâteaux differentiability of the norm of and is uniformly smooth if and only if the norm of is uniformly Fréchet differentiable.

A Banach space is said to be strictly convex if

A Banach space is said to be uniformly convex if for all , where is modulus of convexity of defined by

A uniformly convex Banach space is reflexive and strictly convex [9, Theorem , Theorem ].

Lemma 2.1 (Goebel and Reich [10], Proposition 5.3). Let be a nonempty closed convex subset of a strictly convex Banach space and a nonexpansive mapping with . Then is closed and convex.

Lemma 2.2 (see Xu [11]). In a smooth Banach space there holds the inequality

Lemma 2.3 (Browder [12]). Let be a uniformly convex Banach space, a nonempty closed convex subset of , and a nonexpansive mapping. Then is demi closed at zero.

Lemma 2.4 (see [8, Lemma 2.7]). Let be a nonempty bounded closed convex subset of a uniformly convex Banach space , and let be a nonexpansive semigroup on such that . For and . Then, for any ,

Recall that a gauge is a continuous strictly increasing function such that and as . Associated to a gauge is the duality map defined by

Following Browder [13], we say that a Banach space has a weakly continuous duality map if there exists a gauge for which the duality map is single valued and weak-to-weak* sequentially continuous (i.e., if is a sequence in weakly convergent to a point , then the sequence converges weakly* to ). It is known that has a weakly continuous duality map for all . Set

Then

where denotes the subdifferential in the sense of convex analysis. The next lemma is an immediate consequence of the subdifferential inequality.

Lemma 2.5 (Xu [11, Lemma 2.6]). Assume that has a weakly continuous duality map with gauge , for all , there holds the inequality

Lemma 2.6 (Xu [6]). Assume is a sequence of nonnegative real numbers such that where is a sequence in (0,1) and is a sequence in such that (i); (ii) or .Then .

Finally, we also need the following definitions and results [9, 14]. Let be a continuous linear functional on satisfying . Then we know that is a mean on if and only if for every . Occasionally, we will use instead of . A mean on is called a Banach limit if for every . Using the Hahn-Banach theorem, or the Tychonoff fixed point theorem, we can prove the existence of a Banach limit. We know that if is a Banach limit, then for every . So, if , , and (resp., ), as , we have

Subsequently, the following result was showed in [14, Lemma 1] and [9, Lemma ].

Lemma 2.7 (see [14, Lemma 1]). Let be a nonempty closed convex subset of a Banach space with a uniformly Gâteaux differentiable norm and a bounded sequence of . If , then if and only if

Lemma 2.8 (Song and Xu [4, Proposition 3.1]). Let be a reflexive strictly convex Banach space with a uniformly Gâteaux differentiable norm, and a nonempty closed convex subset of . Suppose is a bounded sequence in such that , an approximate fixed point of nonexpansive self-mapping on . Define the set If , then .

3. Implicit Iteration Scheme

Theorem 3.1. Let be a uniformly convex Banach space that has a weakly continuous duality map with gauge , and let be a nonempty closed convex subset of . Let be a nonexpansive semigroup from into itself such that and a contraction mapping with the contractive coefficient . Suppose is a net of positive real numbers such that , the sequence is given by the following equation: Then converges strongly to as , where is the unique solution in of the variational inequality

Proof. Note that is a nonempty closed convex set by Lemma 2.1. We first show that is bounded. Indeed, for any fixed , we have It follows that Thus is bounded, so are and for every . Furthermore, we note that for every . On the one hand, we observe that for every . On the other hand, let and , then is a nonempty closed bounded convex subset of which is -invariant for each and contains . It follows by Lemma 2.4 that Hence, by (3.5)–(3.7), we obtain for every . Assume is such that as . Put , , we will show that contains s subsequence converging strongly to , where . Since is a bounded sequence, there is a subsequence of which converges weakly to . By Lemma 2.3, we have . For each , we have Thus, by Lemma 2.5, we obtain This implies that In particular, we have Now observing that implies . And since is bounded, it follows from (3.12) that Hence .Next, we show that solves the variational inequality (3.2). Indeed, for , it is easy to see that However, we note that Thus, we get that for and Taking the limit through , we obtain This implies that since for .Finally, we show that the net convergence strong to . Assume that there is a sequence such that , where . we note by Lemma 2.3 that . It follows from the inequality (3.18) that Interchange and to obtain Adding (3.19) and (3.20) yields We must have and the uniqueness is proved. In a summary, we have shown that each cluster point of as equals . Therefore as .

Theorem 3.2. Let be a uniformly convex Banach space with a uniformly Gâteaux differentiable norm and be a nonempty closed convex subset of . Let be a nonexpansive semigroup from into itself such that and a contraction mapping with the contractive coefficient . Suppose is a net of positive real numbers such that , the sequence is given by the following equation: Then converges strongly to as , where is the unique solution in of the variational inequality

Proof. We include only those points in this proof which are different from those already presented in the proof of Theorem 3.1. As in the proof of Theorem 3.1, we obtain that there is a subsequence of which converges weakly to . For each , we have Thus, we have Therefore, We claim that the set is sequentially compact. Indeed, define the set By Lemma 2.8, we found . Using Lemma 2.7 we get that From (3.26), we get that is Hence, there exists a subsequence of converges strongly to as .Next we show that is a solution in to the variational inequality (3.23). In fact, for any fixed , there exists a constant such that , then Therefore, Since the duality mapping is single valued and norm topology to weak* topology uniformly continuous on any bounded subset of a Banach space with a uniformly Gâteaux differentiable norm, we have Taking limit as in two sides of (3.32), we get Finally we will show that the net convergence strong to . This section is similar to that of Theorem 3.1.

4. Explicit Iterative Scheme

Theorem 4.1. Let be a uniformly convex Banach space that has a weakly continuous duality map with gauge and be a nonempty closed convex subset of . Let be a nonexpansive semigroup from into itself such that and a contraction mapping with the contractive coefficient . Let and be the sequence in which satisfies , , and , and is a positive real divergent sequence such that . If the sequence defined by and Then converges strongly to as , where is the unique solution in of the variational inequality

Proof. Note that is a nonempty closed convex set. We first show that is bounded. Let . Thus, we compute that Therefore, is bounded, and for every are also bounded.Next we show as . Notice that Put and . Then is a nonempty closed bounded convex subset of which is -invariant for each and contains . So without loss of generality, we may assume that is a nonexpansive semigroup on . By Lemma 2.4, we get for every . On the other hand, since , , and are bounded, using the assumption that , , and (4.5) into (4.4), we get that and hence We now show that Let , where and satisfies the condition of Theorem 3.1. Then it follows from Theorem 3.1 that and be the unique solution in of the variational inequality (3.2). Clearly is a unique solution of (4.2). Take a subsequence of such that Since is uniformly convex and hence it is reflexive, we may further assume that . Moreover, we note that by Lemma 2.3 and (4.7). Therefore, from (4.9) and (3.17), we have That is (4.8) holds. Finally we will show that . For each , we have An application of Lemma 2.6, we can obtain , hence . That is, converges strongly to a fixed point of . This completes the proof.

Theorem 4.2. Let be a uniformly convex Banach space with a uniformly Gâteaux differentiable norm and be a nonempty closed convex subset of . Let be a nonexpansive semigroup from into itself such that and a contraction mapping with the contractive coefficient . Let and be the sequence in which satisfies , , , and , and is a positive real divergent sequence such that . If the sequence defined by and Then converges strongly to as , where is the unique solution in of the variational inequality

Proof. We also show only those points in this proof which are different from that already presented in the proof of Theorem 4.1. We now show that Let , where and satisfies the condition of Theorem 3.2. Then it follows from Theorem 3.2 that and is the unique solution in of the variational inequality (3.23). Clearly is a unique solution of (4.13). Take a subsequence of such that Since is uniformly convex and hence it is reflexive, we may further assume that . Moreover, we note that by Lemma 2.3 and (4.7). Therefore, from (4.15) and (3.23), we have That is, (4.14) holds.Finally we will show that . For each , by Lemma 2.2, we have which implies that where , , and . It is easily to see that , and by (4.14). Finally by using Lemma 2.6, we can obtain converges strongly to a fixed point . This completes the proof.

5. Applications

Theorem 5.1. Let be a uniformly convex Banach space that has a weakly continuous duality map with gauge and be a nonempty closed convex subset of . Let be a nonexpansive semigroup from into itself such that and a contraction mapping with the contractive coefficient . Let be the sequence in which satisfies and , and is a positive real divergent sequence such that . If the sequence defined by and Then converges strongly to as , where is the unique solution in of the variational inequality

Proof. Taking in the in Theorem 4.1, we get the desired conclusion easily.

Theorem 5.2. Let be a uniformly convex Banach space with a uniformly Gâteaux differentiable norm and be a nonempty closed convex subset of . Let be a nonexpansive semigroup from into itself such that and a contraction mapping with the contractive coefficient . Let be the sequence in which satisfies and , and is a positive real divergent sequence such that . If the sequence defined by and Then converges strongly to as , where is the unique solution in of the variational inequality

Proof. Taking in the in Theorem 4.2, we get the desired conclusion easily.

When is a Hilbert space, we can get the following corollary easily.

Corollary 5.3 (Reich [2]). Let be a nonempty closed convex subset of a real Hilbert space . Let be a strongly continuous semigroup of nonexpansive mapping on such that is nonempty. Let and be sequences of real numbers in which satisfies , , , and . Let be a contraction of into itself with a coefficient and be a positive real divergent sequence such that . Then the sequence defined by and Then converges strongly to , where is the unique solution in of the variational inequality or equivalent , where is a metric projection mapping from into .

Funding

This paper is supported by the National Science Foundation of China under Grants (10771050 and 11101305).

References

F. E. Browder, “Fixed-point theorems for noncompact mappings in Hilbert space,” Proceedings of the National Academy of Sciences of the United States of America, vol. 53, pp. 1272–1276, 1965.

S. Reich, “Strong convergence theorems for resolvents of accretive operators in Banach spaces,” Journal of Mathematical Analysis and Applications, vol. 75, no. 1, pp. 287–292, 1980.

F. E. Browder, “Convergence of approximants to fixed points of nonexpansive non-linear mappings in Banach spaces,” Archive for Rational Mechanics and Analysis, vol. 24, pp. 82–90, 1967.

Y. Song and S. Xu, “Strong convergence theorems for nonexpansive semigroup in Banach spaces,” Journal of Mathematical Analysis and Applications, vol. 338, no. 1, pp. 152–161, 2008.

A. Moudafi, “Viscosity approximation methods for fixed-points problems,” Journal of Mathematical Analysis and Applications, vol. 241, no. 1, pp. 46–55, 2000.

H.-K. Xu, “Viscosity approximation methods for nonexpansive mappings,” Journal of Mathematical Analysis and Applications, vol. 298, no. 1, pp. 279–291, 2004.

S. Plubtieng and R. Punpaeng, “Fixed-point solutions of variational inequalities for nonexpansive semigroups in Hilbert spaces,” Mathematical and Computer Modelling, vol. 48, no. 1-2, pp. 279–286, 2008.

R. Chen and Y. Song, “Convergence to common fixed point of nonexpansive semigroups,” Journal of Computational and Applied Mathematics, vol. 200, no. 2, pp. 566–575, 2007.

W. Takahashi, Nonlinear Functional Analysis—Fixed Point Theory and Its Applications, Yokohama Publishers, Yokohama, Japan, 2000.

K. Goebel and S. Reich, Uniform Convexity, Hyperbolic Geometry, and Nonexpansive Mappings, vol. 83 of Monographs and Textbooks in Pure and Applied Mathematics, Marcel Dekker, New York, NY, USA, 1984.

H.-K. Xu, “Strong convergence of an iterative method for nonexpansive and accretive operators,” Journal of Mathematical Analysis and Applications, vol. 314, no. 2, pp. 631–643, 2006.

F. E. Browder, “Semicontractive and semiaccretive nonlinear mappings in Banach spaces,” Bulletin of the American Mathematical Society, vol. 74, pp. 660–665, 1968.

F. E. Browder, “Convergence theorems for sequences of nonlinear operators in Banach spaces,” Mathematische Zeitschrift, vol. 100, pp. 201–225, 1967.

W. Takahashi and Y. Ueda, “On Reich's strong convergence theorems for resolvents of accretive operators,” Journal of Mathematical Analysis and Applications, vol. 104, no. 2, pp. 546–553, 1984.