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Journal of Applied Mathematics
Volume 2012 (2012), Article ID 294718, 11 pages
http://dx.doi.org/10.1155/2012/294718
Research Article

Implicit Mann Type Iteration Method Involving Strictly Hemicontractive Mappings in Banach Spaces

1Hajvery University, 43-52 Industrial Area, Gulberg-III, Lahore, Pakistan
2Department of Mathematics and RINS, Gyeongsang National University, Jinju 660-701, Republic of Korea

Received 20 April 2012; Accepted 2 May 2012

Academic Editor: Yonghong Yao

Copyright © 2012 Arif Rafiq and Shin Min Kang. 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 proved that the modified implicit Mann iteration process can be applied to approximate the fixed point of strictly hemicontractive mappings in smooth Banach spaces.

1. Introduction

Let be a nonempty subset of an arbitrary Banach space and let be its dual space. The symbols and stand for the domain and the set of fixed points of (for a single-valued mapping is called a fixed point of iff ). We denote by the normalized duality mapping from to defined by where denotes the duality pairing. In a smooth Banach space, is singlevalued (we denoted by ).

Remark 1.1. (1) is called uniformly smooth if is uniformly convex.
(2) In a uniformly smooth Banach space, is uniformly continuous on bounded subsets of .

Let be a mapping.

Definition 1.2. The mapping is called Lipshitz if there exists a constant such that
for all . If , then is called nonexpansive and if , then is called contractive.

Definition 1.3 (see [1, 2]). (1) The mapping is said to be pseudocontractive if for all and .
(2) The mapping is said to be strongly pseudocontractive if there exists a constant such that for all and .
(3) The mapping is said to be local strongly pseudocontractive if for each there exists a constant such that for all and .
(4) The mapping is said to be strictly hemicontractive if and if there exists a constant such that for all ,   and .

Clearly, each strongly pseudocontractive mapping is local strongly pseudocontractive.

Chidume [1] established that the Mann iteration sequence converges strongly to the unique fixed point of in case is a Lipschitz strongly pseudocontractive mapping from a bounded closed convex subset of (or ) into itself. Schu [3] generalized the result in [1] to both uniformly continuous strongly pseudocontractive mappings and real smooth Banach spaces. Park [4] extended the result in [1] to both strongly pseudocontractive mappings and certain smooth Banach spaces. Rhoades [5] proved that the Mann and Ishikawa iteration methods may exhibit different behaviors for different classes of nonlinear mappings. Afterwards, several generalizations have been made in various directions (see, e.g., [613]).

In 2001, Xu and Ori [14] introduced the following implicit iteration process for a finite family of nonexpansive mappings (here ) with a real sequence in and an initial point : which can be written in the following compact form: where (here the function takes values in ). Xu and Ori [14] proved the weak convergence of this process to a common fixed point of the finite family defined in a Hilbert space. They further remarked that it is yet unclear what assumptions on the mappings and/or the parameters are sufficient to guarantee the strong convergence of the sequence .

In [11], Osilike proved the following results.

Theorem 1.4. Let be a real Banach space and let be a nonempty closed convex subset of . Let be strictly pseudocontractive mappings from to with . Let be a real sequence satisfying the following conditions: (i), (ii), (iii).
From arbitrary , define the sequence by the implicit iteration process (1.8). Then converges strongly to a common fixed point of the mappings if and only if .

Remark 1.5. One can easily see that for ,  . Hence the results of Osilike [11] are needed to be improved.

Let be a nonempty closed bounded convex subset of an arbitrary smooth Banach space and let be a continuous strictly hemicontractive mapping. We proved that the implicit Mann type iteration method converges strongly to a unique fixed point of .

The results presented in this paper extend and improve the corresponding results particularly in [1, 3, 4, 7, 8, 10, 11, 13, 15].

2. Preliminaries

We need the following results.

Lemma 2.1 (see [4]). Let be a smooth Banach space. Suppose that one of the following holds: (a) is uniformly continuous on any bounded subsets of ,(b) for all in ,(c)for any bounded subset of , there is a function such that
for all , where satisfies .
Then for any and any bounded subset , there exists such that for all and .

Remark 2.2. (1) If is uniformly smooth, then in Lemma 2.1 holds.
(2) If is a Hilbert space, then in Lemma 2.1 holds.

Lemma 2.3 (see [8]). Let be a mapping with . Then is strictly hemicontractive if and only if there exists a constant such that for all and , there exists satisfying

Lemma 2.4 (see [10]). Let be an arbitrary normed linear space and let be a mapping. (a)If is a local strongly pseudocontractive mapping and , then is a singleton and is strictly hemicontractive.(b)If is strictly hemicontractive, then is a singleton.

Lemma 2.5 (see [10]). Let ,, and be nonnegative real sequences and let be a constant satisfying where ,   for all and . Then .

3. Main Results

We now prove our main results.

Lemma 3.1. Let be a smooth Banach space. Suppose that one of the following holds:(a) is uniformly continuous on any bounded subsets of ,(b) for all , in ,(c)for any bounded subset of , there is a function such that
for all , where satisfies .
Then for any and any bounded subset , there exists such that for all and ;.Proof. For ;, by using (2.2), consider This completes the proof.

Theorem 3.2. Let be a smooth Banach space satisfying any one of the Axioms of Lemma 3.1. Let be a nonempty closed bounded convex subset of and let be a continuous strictly hemicontractive mapping. Let , and be real sequences in satisfying conditions (iv), , (v), (vi) and .
For a sequence in , suppose that is the sequence generated from an arbitrary by satisfying .
Then the sequence converges strongly to a unique fixed point of  .

Proof. By [2, Corollary 1], has a unique fixed point in . It follows from Lemma 2.4 that is a singleton. That is, for some .
Set . It is easy to verify that Also Consider where the first inequality holds by the convexity of .
Now we put , where satisfies (2.3). Using (3.4) and Lemma 3.1, we infer that where Also, we have implies as , and consequently as . Since is uniformly continuous on any bounded subsets of , we have For any given and the bounded subset , there exists a satisfying (2.2). Note that (3.13) and (vi) ensure that there exists an such that Now substituting (3.6) in (3.8) to obtain by using (3.7), implies for all .
Put and we have from (3.16) For , set . Because , we imply and . Now observe that , for all and . It follows from Lemma 2.5 that Letting , we obtain that , which implies that as . This completes the proof.

Corollary 3.3. Let be a smooth Banach space satisfying any one of the Axioms of Lemma 3.1. Let be a nonempty closed bounded convex subset of and let be a Lipschitz strictly hemicontractive mapping. Let , and be real sequences in satisfying the conditions (iv)–(vi).
From arbitrary , define the sequence by the implicit iteration process (3.4). Then the sequence converges strongly to a unique fixed point of .

Corollary 3.4. Let be a smooth Banach space satisfying any one of the Axioms of Lemma 3.1. Let be a nonempty closed bounded convex subset of and let be a continuous strictly hemicontractive mapping. Suppose that be a real sequence in satisfying the conditions (v) and .
From arbitrary , define the sequence by the implicit iteration process (1.8). Then the sequence converges strongly to a unique fixed point of .

Corollary 3.5. Let be a smooth Banach space satisfying any one of the Axioms of Lemma 3.1. Let be a nonempty closed bounded convex subset of and let be a Lipschitz strictly hemicontractive mapping. Suppose that be a real sequence in satisfying the conditions (v) and .
From arbitrary , define the sequence by the implicit iteration process (1.8). Then the sequence converges strongly to a unique fixed point of .

Remark 3.6. Similar results can be found for the iteration processes involved error terms; we omit the details.

Remark 3.7. Theorem 3.2 and Corollary 3.3 extend and improve Theorem 1.4 in the following directions.
We do not need the assumption as in Theorem 1.4.

4. Applications for Multistep Implicit Iterations

Let be a nonempty closed convex subset of a smooth Banach space and let , be a family of mappings.

Algorithm 4.1. For a given , compute the sequence by the implicit iteration process of arbitrary fixed order which is called the multistep implicit iteration process, where , ,  , and , are real sequences in and , .

For , we obtain the following three-step implicit iteration process.

Algorithm 4.2. For a given , compute the sequence by the iteration process where , ,  , and are real sequences in satisfying some certain conditions.
For , we obtain the following two-step implicit iteration process.

Algorithm 4.3. For a given , compute the sequence by the iteration process where , ,   and are real sequences in satisfying some certain conditions.
If ,   and in (4.3), we obtain the following implicit Mann iteration process.

Algorithm 4.4. For any given , compute the sequence by the iteration process where is a real sequence in satisfying some certain conditions.

Theorem 4.5. Let be a smooth Banach space satisfying any one of the Axioms of Lemma 3.1. Let be a nonempty closed bounded convex subset of and let be mappings. Let be continuous strictly hemicontractive mappings. Let , ,   and , be real sequences in satisfying the conditions (iv)–(vi) and . For arbitrary , define the sequence by (4.1). Then converges strongly to the common fixed point of .

Proof. By applying Theorem 3.2 under assumption that and are continuous strictly hemicontractive mappings, we obtain Theorem 4.5 which proves strong convergence of the iteration process defined by (4.1). Consider by taking and , From (4.5) and the condition , we obtain This completes the proof.

Corollary 4.6. Let be a smooth Banach space satisfying any one of the Axioms of Lemma 3.1. Let be a nonempty closed bounded convex subset of and let be mappings. Let be Lipschitz strictly hemicontractive mappings. Let , ,    and , be real sequences in satisfying the conditions (iv)-(vi) and . For arbitrary , define the sequence by (4.1). Then converges strongly to the common fixed point of .

References

  1. C. E. Chidume, “Iterative approximation of fixed points of Lipschitzian strictly pseudocontractive mappings,” Proceedings of the American Mathematical Society, vol. 99, no. 2, pp. 283–288, 1987. View at Publisher · View at Google Scholar
  2. K. Deimling, “Zeros of accretive operators,” Manuscripta Mathematica, vol. 13, pp. 365–374, 1974. View at Publisher · View at Google Scholar · View at Zentralblatt MATH
  3. J. Schu, “Iterative construction of fixed points of strictly pseudocontractive mappings,” Applicable Analysis, vol. 40, no. 2-3, pp. 67–72, 1991. View at Publisher · View at Google Scholar · View at Zentralblatt MATH
  4. J. A. Park, “Mann-iteration process for the fixed point of strictly pseudocontractive mapping in some Banach spaces,” Journal of the Korean Mathematical Society, vol. 31, no. 3, pp. 333–337, 1994. View at Zentralblatt MATH
  5. B. E. Rhoades, “Comments on two fixed point iteration methods,” Journal of Mathematical Analysis and Applications, vol. 56, no. 3, pp. 741–750, 1976. View at Publisher · View at Google Scholar · View at Zentralblatt MATH
  6. S. S. Chang, “Some problems and results in the study of nonlinear analysis,” Nonlinear Analysis A, vol. 30, pp. 4197–4208, 1997.
  7. S. S. Chang, Y. J. Cho, B. S. Lee, J. S. Jung, and S. M. Kang, “Iterative approximations of fixed points and solutions for strongly accretive and strongly pseudo-contractive mappings in Banach spaces,” Journal of Mathematical Analysis and Applications, vol. 224, no. 1, pp. 149–165, 1998. View at Publisher · View at Google Scholar · View at Zentralblatt MATH
  8. C. E. Chidume and M. O. Osilike, “Fixed point iterations for strictly hemi-contractive maps in uniformly smooth Banach spaces,” Numerical Functional Analysis and Optimization, vol. 15, no. 7-8, pp. 779–790, 1994. View at Publisher · View at Google Scholar · View at Zentralblatt MATH
  9. Z. Liu and S. M. Kang, “Stability of Ishikawa iteration methods with errors for strong pseudocontractions and nonlinear equations involving accretive operators in arbitrary real Banach spaces,” Mathematical and Computer Modelling, vol. 34, no. 3-4, pp. 319–330, 2001. View at Publisher · View at Google Scholar · View at Zentralblatt MATH
  10. Z. Liu, S. M. Kang, and S. H. Shim, “Almost stability of the Mann iteration method with errors for strictly hemi-contractive operators in smooth Banach spaces,” Journal of the Korean Mathematical Society, vol. 40, no. 1, pp. 29–40, 2003. View at Publisher · View at Google Scholar · View at Zentralblatt MATH
  11. M. O. Osilike, “Implicit iteration process for common fixed points of a finite family of strictly pseudocontractive maps,” Journal of Mathematical Analysis and Applications, vol. 294, no. 1, pp. 73–81, 2004. View at Publisher · View at Google Scholar · View at Zentralblatt MATH
  12. S. Reich, “Constructive techniques for accretive and monotone operators,” in Applied Nonlinear Analysis, pp. 335–345, Academic Press, New York, NY, USA, 1979. View at Zentralblatt MATH
  13. X. Weng, “Fixed point iteration for local strictly pseudo-contractive mapping,” Proceedings of the American Mathematical Society, vol. 113, no. 3, pp. 727–731, 1991. View at Publisher · View at Google Scholar · View at Zentralblatt MATH
  14. H.-K. Xu and R. G. Ori, “An implicit iteration process for nonexpansive mappings,” Numerical Functional Analysis and Optimization, vol. 22, no. 5-6, pp. 767–773, 2001. View at Publisher · View at Google Scholar · View at Zentralblatt MATH
  15. W. R. Mann, “Mean value methods in iteration,” Proceedings of the American Mathematical Society, vol. 4, pp. 506–510, 1953. View at Publisher · View at Google Scholar · View at Zentralblatt MATH