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Journal of Applied Mathematics
Volume 2012, Article ID 759718, 10 pages
http://dx.doi.org/10.1155/2012/759718
Research Article

Strong Convergence Theorems for a Finite Family of -Strict Pseudocontractions in 2-Uniformly Smooth Banach Spaces by Metric Projections

1School of Mathematics, Yibin University, Sichuan,Yibin 644007, China
2College of Statistics and Mathematics, Yunnan University of Finance and Economics, Kunming, Yunnan 650221, China

Received 16 July 2012; Accepted 27 August 2012

Academic Editor: Nan-Jing Huang

Copyright © 2012 Xin-dong Liu 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

A new hybrid projection algorithm is considered for a finite family of -strict pseudocontractions. Using the metric projection, some strong convergence theorems of common elements are obtained in a uniformly convex and 2-uniformly smooth Banach space. The results presented in this paper improve and extend the corresponding results of Matsushita and Takahshi, 2008, Kang and Wang, 2011, and many others.

1. Introduction

Let be a real Banach space and let be the dual spaces of . Assume that is the normalized duality mapping from into defined by where is the generalized duality pairing between and .

Let be a closed convex subset of a real Banach space . A mapping is said to be nonexpansive if for all . Also a mapping is called a -strict pseudocontraction if there exists a constant such that for every and for some , the following holds:

From (1.3) we can prove that if is -strict pseudo-contractive, then is Lipschitz continuous with the Lipschitz constant .

It is well-known that the classes of nonexpansive mappings and pseudocontractions are two kinds important nonlinear mappings, which have been studied extensively by many authors (see [18]).

In [9] Reich considered the Mann iterative scheme for nonexpansive mappings, where is a sequence in . Under suitable conditions, the author proved that converges weakly to a fixed point of . In 2005, Kim and Xu [10] proved a strong convergence theorem for nonexpansive mappings by modified Mann iteration. In 2008, Zhou [11] extended and improved the main results of Kim and Xu to the more broad 2-uniformly smooth Banach spaces for -strict pseudocontractive mappings.

On the other hand, by using metric projection, Nakajo and Takahashi [12] introduced the following iterative algorithms for the nonexpansive mapping in the framework of Hilbert spaces: where , and is the metric projection from a Hilbert space onto . They proved that generated by (1.5) converges strongly to a fixed point of .

In 2006, Xu [13] extended Nakajo and Takahashi's theorem to Banach spaces by using the generalized projection.

In 2008, Matsushita and Takahashi [14] presented the following iterative algorithms for the nonexpansive mapping in the framework of Banach spaces: where denotes the convex closure of the set , is normalized duality mapping, is a sequence in (0, 1) with , and is the metric projection from onto . Then, they proved that generated by (1.6) converges strongly to a fixed point of nonexpansive mapping .

Recently, Kang and Wang [15] introduced the following hybrid projection algorithm for a pair of nonexpansive mapping in the framework of Banach spaces: where denotes the convex closure of the set , is a sequence in [0, 1], is a sequence in (0,1) with , and is the metric projection from onto . Then, they proved that generated by (1.7) converges strongly to a fixed point of two nonexpansive mappings and .

In this paper, motivated by the research work going on in this direction, we introduce the following iterative for finding fixed points of a finite family of -strict pseudocontractions in a uniformly convex and 2-uniformly smooth Banach space: where denotes the convex closure of the set , is sequences in [0,1] and for each , is a sequence in (0,1) with , and is the metric projection from onto . we prove defined by (1.8) converges strongly to a common fixed point of a finite family of -strictly pseudocontractions, the main results of Kang and Wang is extended and improved to strictly pseudocontractions.

2. Preliminaries

In this section, we recall the well-known concepts and results which will be needed to prove our main results. Throughout this paper, we assume that is a real Banach space and is a nonempty subset of . When is a sequence in , we denote strong convergence of to by and weak convergence by . We also assume that is the dual space of , and is the normalized duality mapping. Some properties of duality mapping have been given in [16].

A Banach space is said to be strictly convex if for all with . is said to be uniformly convex if for each there is a such that for with and holds. The modulus of convexity of is defined by is said to be smooth if the limit exists for all . The modulus of smoothness of is defined by A Banach space is said to be uniformly smooth if as . A Banach space is said to be -uniformly smooth, if there exists a fixed constant such that .

If is a reflexive, strictly convex, and smooth Banach space, then for any , there exists a unique point such that The mapping defined by is called the metric projection from onto . Let and . Then it is known that if and only if For the details on the metric projection, refer to [1720].

In the sequel, we make use the following lemmas for our main results.

Lemma 2.1 (see [21]). Let be a real 2-uniformly smooth Banach space with the best smooth constant . Then the following inequality holds for any .

Lemma 2.2 (see [11]). Let be a nonempty subset of a real 2-uniformly smooth Banach space with the best smooth constant and let be a -strict pseudocontraction. For , we define . Then is nonexpansive such that .

Lemma 2.3 (demiclosed principle, see [22]). Let be a real uniformly convex Banach space, let be a nonempty closed convex subset of , and let be a continuous pseudocontractive mapping. Then, is demiclosed at zero.

Lemma 2.4 (see [23]). Let be a closed convex subset of a uniformly convex Banach space. Then for each , there exists a strictly increasing convex continuous function such that and for all , , , and , where =  and , , and is the set of all nonexpansive mappings from into .

3. Main Results

Now we are ready to give our main results in this paper.

Lemma 3.1. Let be a closed convex subset of a uniformly convex and 2-uniformly smooth Banach space with the best smooth constant , and be a -strict pseudocontraction. Then for each , there exists a strictly increasing convex continuous function such that and for all , ,, where ,   and , .

Proof. Define the mapping as , for all . Then is nonexpansive. From Lemma 2.4, there exists a strictly increasing convex continuous function with and such that Hence This completes the proof.

Theorem 3.2. Let be a nonempty closed subset of a uniformly convex and 2-uniformly smooth Banach space with the best smooth constant , assume that for each , is a -strict pseudocontraction for some such that . Let be sequences in [0,1] with for each and be a sequence in (0,1) with . Let be a sequence generated by (1.8), where denotes the convex closure of the set and is the metric projection from onto . Then converges strongly to .

Proof. (I) First we prove that is well defined and bounded.
It is easy to check that is closed and convex and for all . Therefore is well defined.
Put . Since and , we have that for all . Hence is bounded.
(II) Now we prove that as for all .
Since , there exist some positive integer ( denotes the set of all positive integers), and such that for all . Put and . Take . It follows from Lemma 2.2 and (3.5) that for all . Moreover, (3.7) implies It follows from Lemma 3.1, (3.5)–(3.9) that This shows that for all .
(III) Finally, we prove that .
It follows from the boundedness of that there exists such that as . Since for each , is a -strict pseudocontraction, then is demiclosed. one has .
From the weakly lower semicontinuity of the norm and (3.4), we have This shows and hence as . Therefore, we obtain . Further, we have that Since is uniformly convex, we have . This shows that . This completes the proof.

Corollary 3.3. Let be a nonempty closed subset of a uniformly convex and 2-uniformly smooth Banach space with the best smooth constant , assume that is a -strict pseudocontraction for some such that . Let be a sequence generated by where is a sequence in (0,1) with . denotes the convex closure of the set and is the metric projection from onto . Then converges strongly to .

Proof. Set for all , and , for all in Theorem 3.2. The desired result can be obtained directly from Theorem 3.2.

Remark 3.4. At the end of the paper, we would like to point out that concerning the convergence problem of iterative sequences for strictly pseudocontractive mappings has been considered and studied by many authors. It can be consulted the references [2437].

Acknowledgment

The authors would like to express their thanks to the referees for their valuable suggestions and comments. This work is supported by the Scientific Research Fund of Sichuan Provincial Education Department (11ZA221) and the Scientific Research Fund of Science Technology Department of Sichuan Province 2011JYZ010.

References

  1. F. E. Browder and W. V. Petryshyn, “Construction of fixed points of nonlinear mappings in Hilbert spaces,” Journal of Mathematical Analysis and Applications, vol. 20, pp. 197–228, 1967. View at Google Scholar
  2. 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. View at Google Scholar
  3. F. E. Browder, “Convergence of approximants to fixed points of nonexpansive nonlinear mappings in Banach spaces,” Archive for Rational Mechanics and Analysis, vol. 24, pp. 82–90, 1967. View at Google Scholar
  4. R. E. Bruck,, “Nonexpansive projections on subsets of Banach spaces,” Pacific Journal of Mathematics, vol. 47, pp. 341–355, 1973. View at Google Scholar
  5. C. E. Chidume and S. A. Mutangadura, “An example on the Mann iteration method for Lipschitz pseudo-contractions,” Proceedings of the American Mathematical Society, vol. 129, pp. 2359–2363, 2001. View at Google Scholar
  6. K. Deimling, “Zeros of accretive operators,” Manuscripta Mathematica, vol. 13, pp. 365–374, 1974. View at Google Scholar
  7. K. Goebel and S. Reich, Uniform Convexity, Hyperbolic Geometry, and Nonexpansive Mappings, Marcel Dekker, New York, NY, USA, 1984.
  8. J. P. Gossez and E. L. Dozo, “Some geometric properties related to the fixed point theory for nonexpansive mappings,” Pacific Journal of Mathematics, vol. 40, pp. 565–573, 1972. View at Google Scholar
  9. S. Reich, “Weak convergence theorems for nonexpansive mappings in Banach spaces,” Journal of Mathematical Analysis and Applications, vol. 67, no. 2, pp. 274–276, 1979. View at Publisher · View at Google Scholar
  10. T.-H. Kim and H.-K. Xu, “Strong convergence of modified Mann iterations,” Nonlinear Analysis. Theory, Methods & Applications, vol. 61, no. 1-2, pp. 51–60, 2005. View at Publisher · View at Google Scholar
  11. H. Zhou, “Convergence theorems for λi-strict pseudo-contractions in 2-uniformly smooth Banach spaces,” Nonlinear Anal, vol. 69, pp. 3160–3173, 2008. View at Google Scholar
  12. K. Nakajo and W. Takahashi, “Strong convergence theorems for nonexpansive mappings and nonexpansive semigroups,” Journal of Mathematical Analysis and Applications, vol. 279, no. 2, pp. 372–379, 2003. View at Publisher · View at Google Scholar
  13. H.-K. Xu, “Strong convergence of approximating fixed point sequences for nonexpansive mappings,” Bulletin of the Australian Mathematical Society, vol. 74, no. 1, pp. 143–151, 2006. View at Publisher · View at Google Scholar
  14. S.-y. Matsushita and W. Takahashi, “Approximating fixed points of nonexpansive mappings in a Banach space by metric projections,” Applied Mathematics and Computation, vol. 196, no. 1, pp. 422–425, 2008. View at Publisher · View at Google Scholar
  15. S. M. Kang and S. Wang, “New hybrid algorithms for nonexpansive mappings in Banach spaces,” International Journal of Mathematical Analysis, vol. 5, no. 9-12, pp. 433–440, 2011. View at Google Scholar
  16. I. Cioranescu, Geometry of Banach Spaces, Duality Mappings and Nonlinear Problems, vol. 62 of Mathematics and Its Applications, Kluwer Academic Publishers, Dordrechtm The Netherlands, 1990. View at Publisher · View at Google Scholar
  17. Y. I. Alber, “Metric and generalized projection operators in Banach spaces: properties and applications,” in Theory and Applications of Nonlinear Operators of Accretive and Monotone Type, vol. 178 of Lecture Notes in Pure and Applied Mathematics, pp. 15–50, Marcel Dekker, New York, NY, USA, 1996. View at Google Scholar
  18. W. Takahashi, Nonlinear Functional Analysis, Fixed Point Theory and Its Applications, Yokohama Publishers, Yokohama, Japan, 2000.
  19. F. E. Browder, “Convergence theorems for sequences of nonlinear operators in Banach spaces,” Mathematische Zeitschrift, vol. 100, pp. 201–225, 1967. View at Google Scholar
  20. D. Pascali and S. Sburlan, Nonlinear Mappings of Monotone Type, Noordhoff, Leyden, The Netherlands, 1978.
  21. H. K. Xu, “Inequalities in Banach spaces with applications,” Nonlinear Analysis, vol. 16, pp. 1127–1138, 1991. View at Google Scholar
  22. H. Zhou, “Convergence theorems of common fixed points for a finite family of Lipschitz pseudo-contractions in Banach spaces,” Nonlinear Analysis: Theory, Methods and Applications, vol. 68, no. 10, pp. 2977–2983, 2008. View at Publisher · View at Google Scholar
  23. R. E. Bruck, “On the convex approximation property and the asymptotic behavior of nonlinear contractions in Banach spaces,” Israel Journal of Mathematics, vol. 38, no. 4, pp. 304–314, 1981. View at Publisher · View at Google Scholar
  24. Y. Su, M. Li, and H. Zhang, “New monotone hybrid algorithm for hemi-relatively nonexpansive mappings and maximal monotone operators,” Applied Mathematics and Computation, vol. 217, no. 12, pp. 5458–5465, 2011. View at Publisher · View at Google Scholar
  25. S. S. Chang, H. W. J. Lee, C. K. Chan, and J. K. Kim, “Approximating solutions of variational inequalities for asymptotically nonexpansive mappings,” Applied Mathematics and Computation, vol. 212, no. 1, pp. 51–59, 2009. View at Publisher · View at Google Scholar
  26. H. Zegeye and N. Shahzad, “Convergence of Mann's type iteration method for generalized asymptotically nonexpansive mappings,” Computers & Mathematics with Applications, vol. 62, no. 11, pp. 4007–4014, 2011. View at Publisher · View at Google Scholar
  27. X. Qin, S. Huang, and T. Wang, “On the convergence of hybrid projection algorithms for asymptotically quasi-nonexpansive mappings,” Computers & Mathematics with Applications, vol. 61, no. 4, pp. 851–859, 2011. View at Publisher · View at Google Scholar
  28. S.-s. Chang, L. Wang, Y.-K. Tang, B. Wang, and L.-J. Qin, “Strong convergence theorems for a countable family of quasi-ϕ-asymptotically nonexpansive nonself mappings,” Applied Mathematics and Computation, vol. 218, no. 15, pp. 7864–7870, 2012. View at Publisher · View at Google Scholar
  29. W. Nilsrakoo, “Halpern-type iterations for strongly relatively nonexpansive mappings in Banach spaces,” Computers & Mathematics with Applications, vol. 62, no. 12, pp. 4656–4666, 2011. View at Publisher · View at Google Scholar
  30. S.-S. Chang, H. W. J. Lee, C. K. Chan, and J. A. Liu, “Strong convergence theorems for countable families of asymptotically relatively nonexpansive mappings with applications,” Applied Mathematics and Computation, vol. 218, no. 7, pp. 3187–3198, 2011. View at Publisher · View at Google Scholar
  31. W. Nilsrakoo and S. Saejung, “Strong convergence theorems by Halpern-Mann iterations for relatively nonexpansive mappings in Banach spaces,” Applied Mathematics and Computation, vol. 217, no. 14, pp. 6577–6586, 2011. View at Publisher · View at Google Scholar
  32. S. Suantai, W. Cholamjiak, and P. Cholamjiak, “An implicit iteration process for solving a fixed point problem of a finite family of multi-valued mappings in Banach spaces,” Applied Mathematics Letters, vol. 25, no. 11, pp. 1656–1660, 2012. View at Google Scholar
  33. 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. View at Publisher · View at Google Scholar
  34. S. S. Chang, H. W. J. Lee, C. K. Chan, and W. B. Zhang, “A modified halpern-type iteration algorithm for totally quasi-ϕ-asymptotically nonexpansive mappings with applications,” Applied Mathematics and Computation, vol. 218, no. 11, pp. 6489–6497, 2012. View at Publisher · View at Google Scholar
  35. X. Qin, S. Y. Cho, and S. M. Kang, “Strong convergence of shrinking projection methods for quasi-ϕ-nonexpansive mappings and equilibrium problems,” Journal of Computational and Applied Mathematics, vol. 234, no. 3, pp. 750–760, 2010. View at Publisher · View at Google Scholar
  36. D. Wu, S.-s. Chang, and G. X. Yuan, “Approximation of common fixed points for a family of finite nonexpansive mappings in Banach space,” Nonlinear Analysis. Theory, Methods & Applications, vol. 63, no. 5–7, pp. 987–999, 2005. View at Publisher · View at Google Scholar
  37. C. Zhang, J. Li, and B. Liu, “Strong convergence theorems for equilibrium problems and relatively nonexpansive mappings in Banach spaces,” Computers & Mathematics with Applications, vol. 61, no. 2, pp. 262–276, 2011. View at Publisher · View at Google Scholar