## Journal of Applied Mathematics

• J. Appl. Math.
• Volume 2013, Special Issue (2013), Article ID 284937, 5 pages.

### On the Convergence of Implicit Picard Iterative Sequences for Strongly Pseudocontractive Mappings in Banach Spaces

#### Abstract

We study the convergence of implicit Picard iterative sequences for strongly accretive and strongly pseudocontractive mappings. We have also improved the results of Ćirić et al. (2009).

#### Article information

Source
J. Appl. Math., Volume 2013, Special Issue (2013), Article ID 284937, 5 pages.

Dates
First available in Project Euclid: 14 March 2014

https://projecteuclid.org/euclid.jam/1394807694

Digital Object Identifier
doi:10.1155/2013/284937

Mathematical Reviews number (MathSciNet)
MR3045403

Zentralblatt MATH identifier
1266.47094

#### Citation

Kang, Shin Min; Rafiq, Arif; Cho, Sun Young. On the Convergence of Implicit Picard Iterative Sequences for Strongly Pseudocontractive Mappings in Banach Spaces. J. Appl. Math. 2013, Special Issue (2013), Article ID 284937, 5 pages. doi:10.1155/2013/284937. https://projecteuclid.org/euclid.jam/1394807694

#### References

• T. Kato, “Nonlinear semigroups and evolution equations,” Journal of the Mathematical Society of Japan, vol. 19, pp. 508–520, 1967.
• 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.
• 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.
• C. E. Chidume, “An iterative process for nonlinear Lipschitzian strongly accretive mappings in ${L}_{p}$ spaces,” Journal of Mathematical Analysis and Applications, vol. 151, no. 2, pp. 453–461, 1990.
• K. Deimling, Nonlinear Functional Analysis, Springer, Berlin, Germany, 1985.
• L. Deng, “On Chidume's open questions,” Journal of Mathematical Analysis and Applications, vol. 174, no. 2, pp. 441–449, 1993.
• L. Deng, “An iterative process for nonlinear Lipschitzian and strongly accretive mappings in uniformly convex and uniformly smooth Banach spaces,” Acta Applicandae Mathematicae, vol. 32, no. 2, pp. 183–196, 1993.
• L. Deng, “Iteration processes for nonlinear Lipschitzian strongly accretive mappings in ${L}_{p}$ spaces,” Journal of Mathematical Analysis and Applications, vol. 188, no. 1, pp. 128–140, 1994.
• L. Deng and X. P. Ding, “Iterative approximation of Lipschitz strictly pseudocontractive mappings in uniformly smooth Banach spaces,” Nonlinear Analysis, vol. 24, no. 7, pp. 981–987, 1995.
• N. Hussain, L. B. Ćirić, Y. J. Cho, and A. Rafiq, “On Mann-type iteration method for a family of hemicontractive mappings in Hilbert spaces,” Journal of Inequalities and Applications, vol. 2013, article 41, 11 pages, 2013.
• N. Hussain, A. Rafiq, and L. B. Ćirić, “Stability of the Ishikawa iteration scheme with errors for two strictly hemicontractive operators in Banach spaces,” Fixed Point Theory and Applications, vol. 2012, article 160, 14 pages, 2012.
• N. Hussain, A. Rafiq, L. B. Ćirić, and S. Al-Mezel, “Almost stability of the Mann type iteration method with error term involving strictly hemicontractive mappings in smooth Banach spaces,” Journal of Inequalities and Applications, vol. 2012, article 207, 11 pages, 2012.
• N. Hussain, A. Rafiq, B. Damjanović, and R. Lazović, “On rate of convergence of various iterative schemes,” Fixed Point Theory and Applications, vol. 2011, article 45, 6 pages, 2011.
• L. Liu, “Approximation of fixed points of a strictly pseudocontractive mapping,” Proceedings of the American Mathematical Society, vol. 125, no. 5, pp. 1363–1366, 1997.
• Q. H. Liu, “The convergence theorems of the sequence of Ishikawa iterates for hemicontractive mappings,” Journal of Mathematical Analysis and Applications, vol. 148, no. 1, pp. 55–62, 1990.
• A. Rafiq, “On Mann iteration in Hilbert spaces,” Nonlinear Analysis, vol. 66, no. 10, pp. 2230–2236, 2007.
• A. Rafiq, “Implicit fixed point iterations for pseudocontractive mappings,” Kodai Mathematical Journal, vol. 32, no. 1, pp. 146–158, 2009.
• K. P. R. Sastry and G. V. R. Babu, “Approximation of fixed points of strictly pseudocontractive mappings on arbitrary closed, convex sets in a Banach space,” Proceedings of the American Mathematical Society, vol. 128, no. 10, pp. 2907–2909, 2000.
• X. L. Weng, “Fixed point iteration for local strictly pseudo-contractive mapping,” Proceedings of the American Mathematical Society, vol. 113, no. 3, pp. 727–731, 1991.
• W. R. Mann, “Mean value methods in iteration,” Proceedings of the American Mathematical Society, vol. 4, pp. 506–510, 1953.
• S. Ishikawa, “Fixed points by a new iteration method,” Proceedings of the American Mathematical Society, vol. 44, pp. 147–150, 1974.
• C. E. Chidume, “Picard iteration for strongly accretive and strongly pseudocontractive Lipschitz maps,” International Centre for Theoretical Physics. In press.
• C. E. Chidume, “Iterative algorithms for nonexpansive mappings and some of their generalizations,” in Nonlinear Analysis and Applications: to V. Lakshmikantham on His 80th Birthday, vol. 1,2, pp. 383–429, Kluwer Academic, Dordrecht, The Netherlands, 2003.
• L. Ćirić, A. Rafiq, and N. Cakić, “On Picard iterations for strongly accretive and strongly pseudo-contractive Lipschitz mappings,” Nonlinear Analysis, vol. 70, no. 12, pp. 4332–4337, 2009.