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Journal of Applied Mathematics
Volume 2012 (2012), Article ID 563438, 18 pages
http://dx.doi.org/10.1155/2012/563438
Research Article

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

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.

Linked References

  1. 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. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  2. T. Kato, “Nonlinear semigroups and evolution equations,” Journal of the Mathematical Society of Japan, vol. 19, pp. 508–520, 1967. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  3. C. Chidume, Geometric properties of Banach spaces and nonlinear iterations, vol. 1965 of Lecture Notes in Mathematics, Springer, London, UK, 2009.
  4. 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. View at Publisher · View at Google Scholar · View at Zentralblatt MATH
  5. 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 · View at MathSciNet
  6. M. O. Osilike, “Iterative solution of nonlinear equations of the ϕ-strongly accretive type,” Journal of Mathematical Analysis and Applications, vol. 200, no. 2, pp. 259–271, 1996. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  7. C. H. Xiang, “Fixed point theorem for generalized ϕ-pseudocontractive mappings,” Nonlinear Analysis, vol. 70, no. 6, pp. 2277–2279, 2009. View at Publisher · View at Google Scholar
  8. 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. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  9. 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. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  10. H. Zhou, “Demiclosedness principle with applications for asymptotically pseudo-contractions in Hilbert spaces,” Nonlinear Analysis, vol. 70, no. 9, pp. 3140–3145, 2009. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  11. 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. View at Publisher · View at Google Scholar
  12. 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. View at Google Scholar · View at Zentralblatt MATH
  13. 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. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  14. K. Deimling, Nonlinear Functional Analysis, Springer, Berlin, Germany, 1985.
  15. 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 · View at Zentralblatt MATH · View at MathSciNet
  16. 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. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  17. 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. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  18. 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. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  19. 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. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  20. 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. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  21. Z. Huang, “Equivalence theorems of the convergence between Ishikawa and Mann iterations with errors for generalized strongly successively ϕ-pseudocontractive mappings without Lipschitzian assumptions,” Journal of Mathematical Analysis and Applications, vol. 329, no. 2, pp. 935–947, 2007. View at Publisher · View at Google Scholar · View at MathSciNet
  22. 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. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  23. 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. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  24. 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. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet