Journal of Applied Mathematics

  • J. Appl. Math.
  • Volume 2012, Special Issue (2012), Article ID 474031, 15 pages.

Convergence of Implicit and Explicit Schemes for an Asymptotically Nonexpansive Mapping in q -Uniformly Smooth and Strictly Convex Banach Spaces

Meng Wen, Changsong Hu, and Zhiyu Wu

Full-text: Open access

Abstract

We introduce a new iterative scheme with Meir-Keeler contractions for an asymptotically nonexpansive mapping in q -uniformly smooth and strictly convex Banach spaces. We also proved the strong convergence theorems of implicit and explicit schemes. The results obtained in this paper extend and improve many recent ones announced by many others.

Article information

Source
J. Appl. Math., Volume 2012, Special Issue (2012), Article ID 474031, 15 pages.

Dates
First available in Project Euclid: 3 January 2013

Permanent link to this document
https://projecteuclid.org/euclid.jam/1357179976

Digital Object Identifier
doi:10.1155/2012/474031

Mathematical Reviews number (MathSciNet)
MR2959975

Zentralblatt MATH identifier
1325.47134

Citation

Wen, Meng; Hu, Changsong; Wu, Zhiyu. Convergence of Implicit and Explicit Schemes for an Asymptotically Nonexpansive Mapping in $q$ -Uniformly Smooth and Strictly Convex Banach Spaces. J. Appl. Math. 2012, Special Issue (2012), Article ID 474031, 15 pages. doi:10.1155/2012/474031. https://projecteuclid.org/euclid.jam/1357179976


Export citation

References

  • J. Diestel, Geometry of Banach Spaces: Selected Topics, vol. 485 of Lecture Notes in Mathematics, Springer, Berlin, Germany, 1975.
  • 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.
  • S. Banach, “Surles operations dans les ensembles abstraits et leur application aux equations integrales,” Fundamenta Mathematicae, vol. 3, pp. 133–181, 1922.
  • A. Meir and E. Keeler, “A theorem on contraction mappings,” Journal of Mathematical Analysis and Applications, vol. 28, pp. 326–329, 1969.
  • B. Ali and G. C. Ugwunnadi, “Convergence of implicit and explicit schemes for common fixed points for finite families of asymptotically nonexpansive mappings,” Nonlinear Analysis, vol. 5, no. 3, pp. 492–501, 2011.
  • H. K. Xu, “Inequalities in Banach spaces with applications,” Nonlinear Analysis A, vol. 16, no. 12, pp. 1127–1138, 1991.
  • T. Suzuki, “Moudafi's viscosity approximations with Meir-Keeler contractions,” Journal of Mathematical Analysis and Applications, vol. 325, no. 1, pp. 342–352, 2007.
  • T. Suzuki, “Strong convergence of Krasnoselskii and Mann's type sequences for one-parameter nonexpansive semigroups without Bochner integrals,” Journal of Mathematical Analysis and Applications, vol. 305, no. 1, pp. 227–239, 2005.
  • 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.
  • H.-K. Xu, “Iterative algorithms for nonlinear operators,” Journal of the London Mathematical Society, vol. 66, no. 1, pp. 240–256, 2002.
  • F. Gu, “Convergence of implicit iterative process with errors for a finite family of asymptotically non-expansive mappings,” Acta Mathematica Scientia A, vol. 26, no. 7, pp. 1131–1143, 2006.
  • D. S. Mitrinović, Analytic Inequalities, Springer, New York, NY, USA, 1970.