Some remarks on generalized Kirk’s process in Banach spaces and application

In this work, we establish a common fixed point result for mappings satisfying a controllable punctualinequality and we study the convergence (resp. weak convergence) of the generalized Kirk’s process associated withthem. In addition, our results are applied to investigate the convergence (resp. weak convergence) of Kuhfittig’s iterativeprocess to the solution of a nonlinear system of functional equations.

PDF

___

[1] Alspach D. A fixed point free nonexpansive map. Proceedings of the American Mathematical Society 1981; 82: 423-424.
[2] Browder FE. Semicontractive and semiaccretive nonlinear mapping in Banach spaces. Bulletin of the American Mathematical Society 1968; 74: 660-665.
[3] Browder FE. Nonexpansive nonlinear operators in a Banach space. Proceedings of the National Academy of Sciences of the United States of America 1965; 54: 1041-1044.
[4] Browder FE, Petryshyn WV. The solution by iteration of linear functional equations in Banach spaces. Bulletin of the American Mathematical Society 1966; 72: 566-570.
[5] Goebel K, Kirk WA. Topics in Metric Fixed Point Theory. Cambridge Studies in Advanced Mathematics. 1st ed. Cambridge, UK: Cambridge University Press, 1990.
[6] Göhde D. Zum Prinzip der kontraktiven Abbildungen. Mathematische Nachrichten 1966; 30: 251-258 (in German).
[7] Groetsch CW. A nonstationary iterative process for nonexpansive mappings. Proceedings of the American Mathematical Society 1974; 1: 155-158.
[8] Khamsi MA, Kirk WA. An Introduction to Metric Spaces and Fixed Point Theory. Pure and Applied Mathematics: A Wiley-Interscience Series of Texts. New York, NY, USA: John Wiley and Sons, Inc., 2001.
[9] Khan AR. Common fixed point and solution of nonlinear functional equations. Fixed Point Theory and Applications 2013. doi: 10.1186/1687-2013-290.
[10] Kirk WA. On successive approximations for nonexpansive mappings. Glasgow Mathematical Journal J 1971; 2: 6-9.
[11] Kitahara S, Takahashi W. Image recovery by convex combinations of sunny nonexpansive retractions. Topological Methods in Nonlinear Analysis 1993; 2: 333-342.
[12] Kuhfittig PF. Common fixed points of nonexpansive mappings by iteration. Pacific Journal of Mathematics 1981; 97: 137-139.
[13] Lau AT. Normal structure and common fixed point properties for semigroups of nonexpansive mappings in Banach spaces. Fixed Point Theory and Applications 2010. doi:10.1155/2010/580956.
[14] Ray BK, Singh SP. Fixed point theorems in Banach spaces. Indian Journal of Pure and Applied Mathematics 1978; 9: 216-221.