Open Access Research Article

A new composite implicit iterative process for a finite family of nonexpansive mappings in Banach spaces

Feng Gu* and Jing Lu

Author Affiliations

Department of Mathematics, Institute of Applied Mathematics, Hangzhou Teacher's College, Hangzhou, Zhejiang 310036, China

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Fixed Point Theory and Applications 2006, 2006:82738  doi:10.1155/FPTA/2006/82738


The electronic version of this article is the complete one and can be found online at: http://www.fixedpointtheoryandapplications.com/content/2006/1/82738


Received: 18 January 2006
Revisions received: 22 August 2006
Accepted: 23 August 2006
Published: 7 November 2006

© 2006 Gu and Lu

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.

The purpose of this paper is to study the weak and strong convergence of implicit iteration process with errors to a common fixed point for a finite family of nonexpansive mappings in Banach spaces. The results presented in this paper extend and improve the corresponding results of Chang and Cho (2003), Xu and Ori (2001), and Zhou and Chang (2002).

References

  1. Bauschke, HH: The approximation of fixed points of compositions of nonexpansive mappings in Hilbert space. Journal of Mathematical Analysis and Applications. 202(1), 150–159 (1996). Publisher Full Text OpenURL

  2. Browder, FE: Nonlinear operators and nonlinear equations of evolution in Banach spaces. Nonlinear Functional Analysis (Proc. Sympos. Pure Math., Vol. 18, Part 2, Chicago, Ill., 1968), pp. 1–308. American Mathematical Society, Rhode Island (1976)

  3. Chang, S-S, Cho, YJ: The implicit iterative processes for asymptotically nonexpansive mappings. Nonlinear Analysis and Applications: to V. Lakshmikantham on His 80th Birthday. Vol. 1, 2, pp. 369–382. Kluwer Academic, Dordrecht (2003)

  4. Chang, S-S, Cho, YJ, Zhou, H: Demi-closed principle and weak convergence problems for asymptotically nonexpansive mappings. Journal of the Korean Mathematical Society. 38(6), 1245–1260 (2001)

  5. Chidume, CE, Zegeye, H, Shahzad, N: Convergence theorems for a common fixed point of a finite family of nonself nonexpansive mappings. Fixed Point Theory and Applications. 2005(2), 233–241 (2005). Publisher Full Text OpenURL

  6. Goebel, K, Kirk, WA: A fixed point theorem for asymptotically nonexpansive mappings. Proceedings of the American Mathematical Society. 35(1), 171–174 (1972). Publisher Full Text OpenURL

  7. Górnicki, J: Weak convergence theorems for asymptotically nonexpansive mappings in uniformly convex Banach spaces. Commentationes Mathematicae Universitatis Carolinae. 30(2), 249–252 (1989)

  8. Halpern, B: Fixed points of nonexpanding maps. Bulletin of the American Mathematical Society. 73, 957–961 (1967). Publisher Full Text OpenURL

  9. Lami Dozo, E: Multivalued nonexpansive mappings and Opial's condition. Proceedings of the American Mathematical Society. 38(2), 286–292 (1973)

  10. Lions, P-L: Approximation de points fixes de contractions. Comptes Rendus de l'Académie des Sciences de Paris Série A. 284(21), 1357–1359 (1977)

  11. Opial, Z: Weak convergence of the sequence of successive approximations for nonexpansive mappings. Bulletin of the American Mathematical Society. 73, 591–597 (1967). Publisher Full Text OpenURL

  12. Reich, S: Strong convergence theorems for resolvents of accretive operators in Banach spaces. Journal of Mathematical Analysis and Applications. 75(1), 287–292 (1980). Publisher Full Text OpenURL

  13. Rhoades, BE: Quadratic optimization of fixed points for a family of nonexpansive mappings in Hilbert space. Fixed Point Theory and Applications. 2004(2), 135–147 (2004). Publisher Full Text OpenURL

  14. Schu, J: Weak and strong convergence to fixed points of asymptotically nonexpansive mappings. Bulletin of the Australian Mathematical Society. 43(1), 153–159 (1991). Publisher Full Text OpenURL

  15. Shioji, N, Takahashi, W: Strong convergence theorems for asymptotically nonexpansive semigroups in Hilbert spaces. Nonlinear Analysis. Theory, Methods & Applications. 34(1), 87–99 (1998). PubMed Abstract | Publisher Full Text OpenURL

  16. Tan, K-K, Xu, H-K: The nonlinear ergodic theorem for asymptotically nonexpansive mappings in Banach spaces. Proceedings of the American Mathematical Society. 114(2), 399–404 (1992). Publisher Full Text OpenURL

  17. Tan, K-K, Xu, H-K: Approximating fixed points of nonexpansive mappings by the Ishikawa iteration process. Journal of Mathematical Analysis and Applications. 178(2), 301–308 (1993). Publisher Full Text OpenURL

  18. Wittmann, R: Approximation of fixed points of nonexpansive mappings. Archiv der Mathematik. 58(5), 486–491 (1992). Publisher Full Text OpenURL

  19. Xu, H-K, Ori, RG: An implicit iteration process for nonexpansive mappings. Numerical Functional Analysis and Optimization. 22(5-6), 767–773 (2001). Publisher Full Text OpenURL

  20. Zhou, Y, Chang, S-S: Convergence of implicit iteration process for a finite family of asymptotically nonexpansive mappings in Banach spaces. Numerical Functional Analysis and Optimization. 23(7-8), 911–921 (2002). Publisher Full Text OpenURL