Journal of Applied MathematicsVolume 2012 (2012), Article ID 563438, 18 pagesdoi:10.1155/2012/563438

Research Article

On the Convergence of Iterative Processes for Generalized Strongly Asymptotically -Pseudocontractive Mappings in Banach Spaces

Vittorio Colao

Dipartimento di Matematica, Universitá della Calabria, 87036 Arcavacata di Rende (CS), Italy

Received 5 October 2011; Accepted 11 October 2011

Academic Editor: Yonghong Yao

Copyright © 2012 Vittorio Colao. 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.

We prove the equivalence and the strong convergence of iterative processes involving generalized strongly asymptotically -pseudocontractive mappings in uniformly smooth Banach spaces.

1. Introduction

Throughout this paper, we assume that is a uniformly convex Banach space and is the dual space of . Let denote the normalized duality mapping form into given by for all , where denotes the generalized duality pairing. It is well known that if is uniformly smooth, then is single valued and is norm to norm uniformly continuous on any bounded subset of . In the sequel, we will denote the single valued duality mapping by .

In 1967, Browder [1] and Kato [2], independently, introduced accretive operators (see, for details, Chidume [3]). Their interest is connected with the existence of results in the theory of nonlinear equations of evolution in Banach spaces.

In 1972, Goebel and Kirk [4] introduced the class of asymptotically nonexpansive mappings as follows.

Definition 1.1. Let be a subset of a Banach space . A mapping is said to be asymptotically nonexpansive if for each where is a sequence of real numbers converging to 1.

This class is more general than the class of nonexpansive mappings as the following example clearly shows.

Example 1.2 (see [4]). If is the unit ball of and is defined as where is such that , it satisfies.

In 1974, Deimling [5], studying the zeros of accretive operators, introduced the class of -strongly accretive operators.

Definition 1.3. An operator defined on a subset of a Banach space is said, -strongly accretive if where is a strictly increasing function such that .

Note that in the special case in which ,  , we obtain a strongly accretive operator.

Osilike [6], among the others, proved that in is -strongly accretive where but not strongly accretive.

Since an operator is a strongly accretive operator if and only if is a strongly pseudocontractive mapping (i.e., ,  ), taking in to account Definition 1.3, it is natural to study the class of -pseudocontractive mappings, that is, the maps such that where is a strictly increasing function such that . Of course, the set of fixed points for these mappings contains, at most, only one point.

Recently, has been also studied the following class of maps.

Definition 1.4. A mapping is a generalized -strongly pseudocontractive mapping if where is a strictly increasing function such that .

Choosing , we obtain Definition 1.3. In [7], Xiang remarked that it is a open problem if every generalized -strongly pseudocontractive mapping is -pseudocontractive mapping. In the same paper, Xiang obtained a fixed-point theorem for continuous and generalized -strongly pseudocontractive mappings in the setting of the Banach spaces.

In 1991, Schu [8] introduced the class of asymptotically pseudocontractive mappings.

Definition 1.5 (see [8]). Let be a normed space, and . A mapping is said to be asymptotically pseudocontractive with the sequence if and only if , and for all and all , there exists such that where is the normalized duality mapping.

Obviously every asymptotically nonexpansive mapping is asymptotically pseudocontractive, but the converse is not valid; it is well known that defined by is not Lipschitz but asymptotically pseudocontractive [9].

In [8], Schu proved the following.

Theorem 1.6 (see [8]). Let be a Hilbert space and closed and convex; ; completely continuous, uniformly Lipschitzian, and asymptotically pseudocontractive with sequence ;   for all ; ; ; for all , some and some ; ; for all , define then converges strongly to some fixed point of .

Until 2009, no results on fixed-point theorems for asymptotically pseudocontractive mappings have been proved. First, Zhou in [10] completed this lack in the setting of Hilbert spaces proving a fixed-point theorem for an asymptotically pseudocontractive mapping that is also uniformly -Lipschitzian and uniformly asymptotically regular and that the set of fixed points of is closed and convex. Moreover, Zhou proved the strong convergence of a CQ-iterative method involving this kind of mappings.

In this paper, our attention is on the class of the generalized strongly asymptotically -pseudocontraction defined as follows.

Definition 1.7. If is a Banach space and is a subset of , a mapping is said to be a generalized asymptotically -strongly pseudocontraction if where is converging to one and is strictly increasing and such that .

One can note that (i)if has fixed points, then it is unique. In fact, if are fixed points for , then for every , so passing to , it results that Since is strictly increasing and , then . (ii)the mapping , where , is generalized asymptotically strongly -pseudocontraction with , for all and . However, is not strongly pseudocontractive; see [6].

We study the equivalence between three kinds of iterative methods involving the generalized asymptotically strongly -pseudocontractions.

Moreover, we prove that these methods are equivalent and strongly convergent to the unique fixed point of the generalized strongly asymptotically -pseudocontraction , under suitable hypotheses.

We will briefly introduce some of the results in the same line of ours. In 2001, [11] Chidume and Osilike proved the strong convergence of the iterative method where ,   ( a -strongly accretive operator), and , to a solution of the equation .

In 2003, Chidume and Zegeye [12] studied the following iterative method: where is a Lipschitzian pseudocontractive map with fixed points. The authors proved the strong convergence of the method to a fixed point of under suitable hypotheses on the control sequences .

Taking in to account Chidume and Zegeye [12] and Chang [13], we introduce the modified Mann and Ishikawa iterative processes as follows: for any given , the sequence is defined by where , , , and are four sequences in satisfying the conditions and for all .

In particular, if for all , we can define a sequence by which is called the modified Mann iteration sequence.

We also introduce an implicit iterative process as follows: where are two real sequences in satisfying and for all , is a sequence in , and is an initial point.

The algorithm is well defined. Indeed, if is a asymptotically strongly -pseudocontraction, one can observe that, for every fixed , the mapping defined by is such that that is, is a strongly pseudocontraction, for every fixed , then (see Theorem  13.1 in [14]) there exists a unique fixed point of for each .

These kind of iterative processes (also called by Chang iterative processes with errors) have been developed in [15–18], while equivalence theorem for Mann and Ishikawa methods has been studied, in [19, 20], among the others.

In [21], Huang established equivalences between convergence of the modified Mann iteration process with errors (1.15) and convergence of modified Ishikawa iteration process with errors (1.14) for strongly successively -pseudocontractive mappings in uniformly smooth Banach space.

In the next section, we prove that, in the setting of the uniformly smooth Banach space, if is an asymptotically strongly -pseudocontraction, not only (1.14) and (1.15) are equivalent but also (1.16) is equivalent to the others. Moreover, we prove also that (1.14), (1.15), and (1.16) strongly converge to the unique fixed point of , if it exists.

2. Preliminaries

We recall some definitions and conclusions.

Definition 2.1. is said to be a uniformly smooth Banach space if the smooth module of satisfies .

Lemma 2.2 (see [22]). Let be a Banach space, and let be the normalized duality mapping, then for any , one has

The next lemma is one of the main tools for our proofs.

Lemma 2.3 (see [21]). Let be a strictly increasing function with , and let , and be nonnegative real sequences such that Suppose that there exists an integer such that then .

Proof. The proof is the same as in [21], but we substitute with , in (2.4).

Lemma 2.4 (see [23]). Let , , , and be sequences such that for all . Assume that , then the following results hold: (1)if (where ), then is a bounded sequence;(2)if one has and , then as .

Remark 2.5. If in Lemma 2.3 choosing , for all , (), then the inequality (2.4) becomes Setting and and by the hypotheses of Lemma 2.3, we get as , , and . That is, we reobtain Lemma 2.4 in the case of .

3. Main Results

The ideas of the proofs of our main Theorems take in to account the papers of Chang and Chidume et al. [11, 13, 24].

Theorem 3.1. Let be a uniformly smooth Banach space, and let be generalized strongly asymptotically -pseudocontractive mapping with fixed point and bounded range. Let and be the sequences defined by (1.14) and (1.15), respectively, where ,,, satisfy (H1) and ,(H2),and the sequences ,, are bounded in , then for any initial point , the following two assertions are equivalent: (i)the modified Ishikawa iteration sequence with errors (1.14) converges to ; (ii)the modified Mann iteration sequence with errors (1.15) converges to .

Proof. First of all, we note that by boundedness of the range of , of the sequences and by Lemma 2.4, it results that and are bounded sequences. So, we can set By Lemma 2.2, we have where . Using (1.14) and (1.15), we have In view of the uniformly continuity of , we obtain that as . Furthermore, it follows from the definition of that for all so Therefore, we have where . By (H1), we have that as . If for an infinite number of indices, we can extract a subsequence such that . For this subsequence, , as .In this case, we can prove that , that is, the thesis.Firstly, we note that substituting (3.4) into (3.2), we have where .Moreover, we observe that Thus, for every fixed , there exists such that for all Since , , , , and are null sequences (and in particular ), for the previous fixed , there exists an index such that, for all , for all .Take such that for a certain .We prove, by induction, that , for every . Let . Suppose that .By (3.6), we have Thus, . Since is strictly increasing, . From (3.8), we obtain that One can note that hence In the same manner, Thus, So we have , which contradicts . By the same idea, we can prove that and then, by inductive step, , for all . This is enough to ensure that .If there are only finite indices for which , then definitively . By the strict increasing function , we have definitively Again substituting (3.4) and (3.18) into (3.2) and simplifying, we have Suppose that . It follows from , , , and the hypothesis that we have, and , as . By virtue of Lemma 2.3, we obtain that . Hence, .

Theorem 3.2. Let be a uniformly smooth Banach space, and let be generalized strongly asymptotically -pseudocontractive mapping with fixed point and bounded range. Let and be the sequences defined by (1.15) and (1.16), respectively, where , are null sequences satisfying (H1) and , (H2), of Theorem 3.1 and such that , for every . Suppose moreover that the sequences , are bounded in , then for any initial point , the following two assertions are equivalent: (i)the modified Mann iteration sequence with errors (1.15) converges to the fixed point , (ii)the implicit iteration sequence with errors (1.16) converges to the fixed point .

Proof. As in Theorem 3.1, by the boundedness of the range of and by Lemma 2.4, one obtains that our schemes are bounded. We define By the iteration schemes (1.15) and (1.16), we have where . By (1.15), we get It follows from (H1) that as , which implies that as . Moreover, for all , Again by the boundedness of all components, we have that and so Hence, we have that , where . Note that as . As in proof of Theorem 3.1, we distinguish two cases:(i)the set of indices for which contains infinite terms; (ii)the set of indices for which contains finite terms. In the first case, (i) we can extract a subsequence such that , as . Substituting (3.23) in (3.21), we have that where . Again by (3.23), for every , there exists an index such that if , By hypotheses on the control sequence, with the same , there exists an index such that definitively So take with for a certain .We can prove that as proving that, for every , the result is .Let . If we suppose that , it results that so . In consequence of this, .In (3.26), we note that so hence in (3.26) remains as in Theorem 3.1. This is a contradiction. By the same idea, and using the inductive hypothesis, we obtain that , for every . This ensures that . In the second case (ii), definitively, , then from the strictly increasing function , we have Substituting (3.33) and (3.23) into (3.21) and simplifying, we have By virtue of Lemma 2.3, we obtain that .

Theorem 3.3. Let be a uniformly smooth Banach space, and let be generalized strongly asymptotically -pseudocontractive mapping with fixed point and bounded range. Let be the sequences defined by (1.15) where satisfy (i), (ii), .and the sequence is bounded on , then for any initial point , the sequence strongly converges to .

Proof. Firstly, we observe that, by the boundedness of the range of , of the sequence , and by Lemma 2.4, we have that is bounded.By Lemma 2.2, we observe that where . Let We have so we can observe that(1) as . Indeed from the inequality and since is norm to norm uniformly continuous, then , as , (2). Indeed, if we supposed that , by the monotonicity of , Thus, by (1) and by the hypotheses on and , the value is definitively negative. In this case, we conclude that there exists such that for every , and so In the same way we obtain that By the hypotheses and , the previous is a contradiction, and it follows that .Then, there exists a subsequence of that strongly converges to . This implies that for every , there exists an index such that, for all , .Now, we will prove that the sequence converges to . Since the sequences in (3.37) are null sequences and , but , then, for every , there exists an index such that for all , it results that So, fixing , let with for a certain . We will prove, by induction, that for every . Let . If not, it results that . Thus, that is, . By the strict increasing of , .By (3.37), it results that We can note that so Moreover, , so it results that This is a contradiction. Thus, .In the same manner, by induction, one obtains that, for every , . So .

Corollary 3.4. Let be a uniformly smooth Banach space, and let be generalized strongly asymptotically -pseudocontractive mapping with bounded range and fixed point . The sequences , , and are defined by (1.14), (1.15), and (1.16), respectively, where the sequences , , , satisfy (i),(ii),and the sequences , , , and are bounded in X. Then for any initial point , the following two assertions are equivalent and true: (i)the modified Ishikawa iteration sequence with errors (1.14) converges to the fixed point ;(ii)the modified Mann iteration sequence with errors (1.15) converges to the fixed point ;(iii)the implicit iteration sequence with errors (1.16) converges to the fixed point .

References

F. E. Browder, “Nonlinear mappings of nonexpansive and accretive type in Banach spaces,” Bulletin of the American Mathematical Society, vol. 73, pp. 875–882, 1967.

T. Kato, “Nonlinear semigroups and evolution equations,” Journal of the Mathematical Society of Japan, vol. 19, pp. 508–520, 1967.

C. Chidume, Geometric properties of Banach spaces and nonlinear iterations, vol. 1965 of Lecture Notes in Mathematics, Springer, London, UK, 2009.

K. Goebel and W. A. Kirk, “A fixed point theorem for asymptotically nonexpansive mappings,” Proceedings of the American Mathematical Society, vol. 35, pp. 171–174, 1972.

K. Deimling, “Zeros of accretive operators,” Manuscripta Mathematica, vol. 13, pp. 365–374, 1974.

M. O. Osilike, “Iterative solution of nonlinear equations of the -strongly accretive type,” Journal of Mathematical Analysis and Applications, vol. 200, no. 2, pp. 259–271, 1996.

C. H. Xiang, “Fixed point theorem for generalized -pseudocontractive mappings,” Nonlinear Analysis, vol. 70, no. 6, pp. 2277–2279, 2009.

J. Schu, “Iterative construction of fixed points of asymptotically nonexpansive mappings,” Journal of Mathematical Analysis and Applications, vol. 158, no. 2, pp. 407–413, 1991.

E. U. Ofoedu, “Strong convergence theorem for uniformly L-Lipschitzian asymptotically pseudocontractive mapping in real Banach space,” Journal of Mathematical Analysis and Applications, vol. 321, no. 2, pp. 722–728, 2006.

H. Zhou, “Demiclosedness principle with applications for asymptotically pseudo-contractions in Hilbert spaces,” Nonlinear Analysis, vol. 70, no. 9, pp. 3140–3145, 2009.

C. E. Chidume and M. O. Osilike, “Equilibrium points for a system involving m-accretive operators,” Proceedings of the Edinburgh Mathematical Society. Series II, vol. 44, no. 1, pp. 187–199, 2001.

C. E. Chidume and H. Zegeye, “Approximate fixed point sequences and convergence theorems for Lipschitz pseudocontractive maps,” Proceedings of the American Mathematical Society, vol. 132, no. 3, pp. 831–840, 2004.

S. S. Chang, “Some results for asymptotically pseudo-contractive mappings and asymptotically nonexpansive mappings,” Proceedings of the American Mathematical Society, vol. 129, no. 3, pp. 845–853, 2001.

K. Deimling, Nonlinear Functional Analysis, Springer, Berlin, Germany, 1985.

S. S. Chang, K. K. Tan, H. W. J. Lee, and C. K. Chan, “On the convergence of implicit iteration process with error for a finite family of asymptotically nonexpansive mappings,” Journal of Mathematical Analysis and Applications, vol. 313, no. 1, pp. 273–283, 2006.

F. Gu, “The new composite implicit iterative process with errors for common fixed points of a finite family of strictly pseudocontractive mappings,” Journal of Mathematical Analysis and Applications, vol. 329, no. 2, pp. 766–776, 2007.

Z. Huang and F. Bu, “The equivalence between the convergence of Ishikawa and Mann iterations with errors for strongly successively pseudocontractive mappings without Lipschitzian assumption,” Journal of Mathematical Analysis and Applications, vol. 325, no. 1, pp. 586–594, 2007.

L. S. Liu, “Ishikawa and Mann iterative process with errors for nonlinear strongly accretive mappings in Banach spaces,” Journal of Mathematical Analysis and Applications, vol. 194, no. 1, pp. 114–125, 1995.

B. E. Rhoades and S. M. Soltuz, “The equivalence between the convergences of Ishikawa and Mann iterations for an asymptotically nonexpansive in the intermediate sense and strongly successively pseudocontractive maps,” Journal of Mathematical Analysis and Applications, vol. 289, no. 1, pp. 266–278, 2004.

B. E. Rhoades and S. M. Soltuz, “The equivalence between Mann-Ishikawa iterations and multistep iteration,” Nonlinear Analysis, vol. 58, no. 1-2, pp. 219–228, 2004.

Z. Huang, “Equivalence theorems of the convergence between Ishikawa and Mann iterations with errors for generalized strongly successively -pseudocontractive mappings without Lipschitzian assumptions,” Journal of Mathematical Analysis and Applications, vol. 329, no. 2, pp. 935–947, 2007.

Y. Xu, “Ishikawa and Mann iterative processes with errors for nonlinear strongly accretive operator equations,” Journal of Mathematical Analysis and Applications, vol. 224, no. 1, pp. 91–101, 1998.

P.-E. Maingé, “Approximation methods for common fixed points of nonexpansive mappings in Hilbert spaces,” Journal of Mathematical Analysis and Applications, vol. 325, no. 1, pp. 469–479, 2007.

C. E. Chidume and C. O. Chidume, “Convergence theorem for zeros of generalized lipschitz generalized phi-quasi-accretive operators,” Proceedings of the American Mathematical Society, vol. 134, no. 1, pp. 243–251, 2006.