×

New iterative algorithm for two infinite families of multivalued quasi-nonexpansive mappings in uniformly convex Banach spaces. (English) Zbl 1266.47101

Summary: We introduce a new iterative scheme for finding a common fixed point of two countable families of multivalued quasi-nonexpansive mappings and prove a weak convergence theorem under the suitable control conditions in a uniformly convex Banach space. We also give a new proof method to the iteration in [M. Abbas et al., Appl. Math. Lett. 24, No. 2, 97–102 (2011; Zbl 1223.47068)].

MSC:

47J25 Iterative procedures involving nonlinear operators
47H09 Contraction-type mappings, nonexpansive mappings, \(A\)-proper mappings, etc.
47H04 Set-valued operators

Citations:

Zbl 1223.47068
PDFBibTeX XMLCite
Full Text: DOI

References:

[1] R. P. Agarwal, D. O’Regan, and D. R. Sahu, Fixed Point Theory for Lipschitzian-Type Mappings with Applications, vol. 6 of Topological Fixed Point Theory and Its Applications, Springer, New York, NY, USA, 2009. · Zbl 1254.34126 · doi:10.1007/978-0-387-75818-3
[2] C. E. Chidume, Geometric Properties of Banach Spaces and Nonlinear Iterations, Lecture Notes in Mathematics, Springer, New York, NY, USA, 2009. · Zbl 1167.47002
[3] P. Sunthrayuth and P. Kumam, “A general iterative algorithm for the solution of variational inequalities for a nonexpansive semigroup in Banach spaces,” Journal of Nonlinear Analysis and Optimization, vol. 1, no. 1, pp. 139-150, 2010. · Zbl 1413.47141
[4] P. Sunthrayuth and P. Kumam, “Iterative methods for variational inequality problems and fixed point problems of a countable family of strict pseudo-contractions in a q-uniformly smooth Banach space,” Fixed Point Theory and Applications, vol. 2012, article 65, 2012. · Zbl 1475.47076 · doi:10.1186/1687-1812-2012-65
[5] P. Sunthrayuth and P. Kumam, “Viscosity approximation methods base on generalized contraction mappings for a countable family of strict pseudo-contractions, a general system of variational inequalities and a generalized mixed equilibrium problem in Banach spaces,” Mathematical and Computer Modelling, 2013. · Zbl 1327.47057 · doi:10.1016/j.mcm.2013.02.010
[6] P. Katchang and P. Kumam, “Strong convergence of the modified Ishikawa iterative method for infinitely many nonexpansive mappings in Banach spaces,” Computers & Mathematics with Applications, vol. 59, no. 4, pp. 1473-1483, 2010. · Zbl 1189.65116 · doi:10.1016/j.camwa.2010.01.025
[7] P. Sunthrayuth and P. Kumam, “Strong convergence theorems of a general iterative process for two nonexpansive mappings in Banach spaces,” Journal of Computational Analysis and Applications, vol. 14, no. 3, pp. 446-457, 2012. · Zbl 1272.47085
[8] P. Sunthrayuth and P. Kumam, “A new composite general iterative scheme for nonexpansive semigroups in Banach spaces,” International Journal of Mathematics and Mathematical Sciences, vol. 2011, Article ID 560671, 18 pages, 2011. · Zbl 1221.47127 · doi:10.1155/2011/560671
[9] P. Phuangphoo and P. Kumam, “An iterative procedure for solving the common solution of two total quasi-\varphi -asymptotically nonexpansive multi-valued mappings in Banach spaces,” Journal of Applied Mathematics and Computing, 2012. · Zbl 1475.47093 · doi:10.1007/s12190-012-0630-4
[10] S. B. Nadler Jr., “Multi-valued contraction mappings,” Pacific Journal of Mathematics, vol. 30, pp. 475-488, 1969. · Zbl 0187.45002 · doi:10.2140/pjm.1969.30.475
[11] J. T. Markin, “Continuous dependence of fixed point sets,” Proceedings of the American Mathematical Society, vol. 38, pp. 545-547, 1973. · Zbl 0278.47036 · doi:10.2307/2038947
[12] T. Hu, J. C. Huang, and B. E. Rhoades, “A general principle for Ishikawa iterations for multi-valued mappings,” Indian Journal of Pure and Applied Mathematics, vol. 28, no. 8, pp. 1091-1098, 1997. · Zbl 0898.47046
[13] K. P. R. Sastry and G. V. R. Babu, “Convergence of Ishikawa iterates for a multi-valued mapping with a fixed point,” Czechoslovak Mathematical Journal, vol. 55(130), no. 4, pp. 817-826, 2005. · Zbl 1081.47069 · doi:10.1007/s10587-005-0068-z
[14] M. Abbas, S. H. Khan, A. R. Khan, and R. P. Agarwal, “Common fixed points of two multivalued nonexpansive mappings by one-step iterative scheme,” Applied Mathematics Letters, vol. 24, no. 2, pp. 97-102, 2011. · Zbl 1223.47068 · doi:10.1016/j.aml.2010.08.025
[15] S. S. Chang, J. K. Kim, and X. R. Wang, “Modified block iterative algorithm for solving convex feasibility problems in Banach spaces,” Journal of Inequalities and Applications, vol. 2010, Article ID 869684, 14 pages, 2010. · Zbl 1187.47045 · doi:10.1155/2010/869684
[16] A. Bunyawat and S. Suantai, “Convergence theorems for infinite family of multivalued quasi-nonexpansive mappings in uniformly convex Banach spaces,” Abstract and Applied Analysis, vol. 2012, Article ID 435790, 6 pages, 2012. · Zbl 1237.47068 · doi:10.1155/2012/435790
[17] L. Górniewicz, Topological Fixed Point Theory of Multivalued Mappings, vol. 495 of Mathematics and Its Applications, Kluwer Academic, Dordrecht, The Netherlands, 1999. · Zbl 0937.55001
[18] M. Abbas and B. E. Rhoades, “Fixed point theorems for two new classes of multivalued mappings,” Applied Mathematics Letters, vol. 22, no. 9, pp. 1364-1368, 2009. · Zbl 1173.47311 · doi:10.1016/j.aml.2009.04.002
[19] N. Shahzad and H. Zegeye, “On Mann and Ishikawa iteration schemes for multi-valued maps in Banach spaces,” Nonlinear Analysis: Theory, Methods & Applications, vol. 71, no. 3-4, pp. 838-844, 2009. · Zbl 1218.47118 · doi:10.1016/j.na.2008.10.112
[20] W. Cholamjiak and S. Suantai, “A hybrid method for a countable family of multivalued maps, equilibrium problems, and variational inequality problems,” Discrete Dynamics in Nature and Society, vol. 2010, Article ID 349158, 14 pages, 2010. · Zbl 1194.90104 · doi:10.1155/2010/349158
[21] S. Hong, “Fixed points of multivalued operators in ordered metric spaces with applications,” Nonlinear Analysis: Theory, Methods & Applications, vol. 72, no. 11, pp. 3929-3942, 2010. · Zbl 1184.54041 · doi:10.1016/j.na.2010.01.013
[22] S. H. Khan, I. Yildirim, and B. E. Rhoades, “A one-step iterative process for two multivalued nonexpansive mappings in Banach spaces,” Computers & Mathematics with Applications, vol. 61, no. 10, pp. 3172-3178, 2011. · Zbl 1236.47072 · doi:10.1016/j.camwa.2011.08.067
[23] S. Saewan and P. Kumam, “Modified hybrid block iterative algorithm for convex feasibility problems and generalized equilibrium problems for uniformly quasi-\varphi -asymptotically nonexpansive mappings,” Abstract and Applied Analysis, vol. 2010, Article ID 357120, 22 pages, 2010. · Zbl 1206.47084 · doi:10.1155/2010/357120
[24] J. Schu, “Weak and strong convergence to fixed points of asymptotically nonexpansive mappings,” Bulletin of the Australian Mathematical Society, vol. 43, no. 1, pp. 153-159, 1991. · Zbl 0709.47051 · doi:10.1017/S0004972700028884
[25] 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. · Zbl 0872.47031 · doi:10.1006/jmaa.1995.1289
[26] H. K. Xu, “Iterative algorithms for nonlinear operators,” Journal of the London Mathematical Society, vol. 66, no. 1, pp. 240-256, 2002. · Zbl 1013.47032 · doi:10.1112/S0024610702003332
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.