×

Convergence of infinite family of multivalued quasi-nonexpansive mappings using multistep iterative processes. (English) Zbl 1473.47033

Summary: We prove strong and weak convergence results using multistep iterative sequences for countable family of multivalued quasi-nonexpansive mappings by using some conditions in uniformly convex real Banach space. The results presented extended and improved the corresponding result of F. Zhang et al. [J. Appl. Math. 2013, Article ID 649537, 7 p. (2013; Zbl 1266.47101)], A. Bunyawat and S. Suantai [Abstr. Appl. Anal. 2012, Article ID 435790, 6 p. (2012; Zbl 1237.47068)], and some others from finite family, one countable family, and two countable families to \(k\)-number of countable families of multivalued quasi-nonexpansive mappings. Also we use a numerical example in C++ computational programs to prove that the iterative scheme we used has better rate of convergence than other existing iterative schemes.

MSC:

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

References:

[1] Nadler, S. B., Multi-valued contraction mappings, Pacific Journal of Mathematics, 30, 475-488 (1969) · Zbl 0187.45002 · doi:10.2140/pjm.1969.30.475
[2] Markin, J. T., Continuous dependence of fixed point sets, Proceedings of the American Mathematical Society, 38, 545-547 (1973) · Zbl 0278.47036 · doi:10.1090/S0002-9939-1973-0313897-4
[3] Hu, T.; Huang, J.-C.; Rhoades, B. E., A general principle for Ishikawa iterations for multi-valued mappings, Indian Journal of Pure and Applied Mathematics, 28, 8, 1091-1098 (1997) · Zbl 0898.47046
[4] Sastry, K. P.; Babu, G. V., Convergence of Ishikawa iterates for a multi-valued mapping with a fixed point, Czechoslovak Mathematical Journal, 55, 4, 817-826 (2005) · Zbl 1081.47069 · doi:10.1007/s10587-005-0068-z
[5] Abbas, M.; Khan, S. H.; Khan, A. R.; Agarwal, R. P., Common fixed points of two multivalued nonexpansive mappings by one-step iterative scheme, Applied Mathematics Letters, 24, 2, 97-102 (2011) · Zbl 1223.47068 · doi:10.1016/j.aml.2010.08.025
[6] Bunyawat, A.; Suantai, S., Convergence theorems for infinite family of multivalued quasi-nonexpansive mappings in uniformly convex Banach spaces, Abstract and Applied Analysis, 2012 (2012) · Zbl 1237.47068 · doi:10.1155/2012/435790
[7] Sunthrayuth, P.; Kumam, P., A general iterative algorithm for the solution of variational inequalities for a nonexpansive semigroup in Banach spaces, Journal of Nonlinear Analysis and Optimization, 1, 1, 139-150 (2010) · Zbl 1413.47141
[8] Sunthrayuth, P.; Kumam, P., 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, 2012, article 65 (2012) · Zbl 1475.47076 · doi:10.1186/1687-1812-2012-65
[9] Sunthrayuth, P.; Kumam, P., Viscosity approximation methods based 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, 58, 11-12, 1814-1828 (2013) · Zbl 1327.47057 · doi:10.1016/j.mcm.2013.02.010
[10] Katchang, P.; Kumam, P., Strong convergence of the modified Ishikawa iterative method for infinitely many nonexpansive mappings in Banach spaces, Computers and Mathematics with Applications, 59, 4, 1473-1483 (2010) · Zbl 1189.65116 · doi:10.1016/j.camwa.2010.01.025
[11] Sunthrayuth, P.; Kumam, P., Strong convergence theorems of a general iterative process for two nonexpansive mappings in Banach spaces, Journal of Computational Analysis and Applications, 14, 3, 446-457 (2012) · Zbl 1272.47085
[12] Sunthrayuth, P.; Kumam, P., A new composite general iterative scheme for nonexpansive semigroups in Banach spaces, International Journal of Mathematics and Mathematical Sciences, 2011 (2011) · Zbl 1221.47127 · doi:10.1155/2011/560671
[13] Phuangphoo, P.; Kumam, P., An iterative procedure for solving the common solution of two total quasi-\(ϕ\)-asymptotically nonexpansive multi-valued mappings in Banach spaces, Journal of Applied Mathematics and Computing, 42, 1-2, 321-338 (2013) · Zbl 1364.47037 · doi:10.1007/s12190-012-0630-4
[14] Górniewicz, L., Topological Fixed Point Theory of Multivalued Mappings. Topological Fixed Point Theory of Multivalued Mappings, Mathematics and Its Applications, 495 (1999), Dordrecht, The Netherlands: Kluwer Academic Publishers, Dordrecht, The Netherlands · Zbl 0937.55001
[15] Abbas, M.; Rhoades, B. E., Fixed point theorems for two new classes of multivalued mappings, Applied Mathematics Letters, 22, 9, 1364-1368 (2009) · Zbl 1173.47311 · doi:10.1016/j.aml.2009.04.002
[16] Shahzad, N.; Zegeye, H., On Mann and Ishikawa iteration schemes for multi-valued maps in Banach spaces, Nonlinear Analysis: Theory, Methods & Applications, 71, 3-4, 838-844 (2009) · Zbl 1218.47118 · doi:10.1016/j.na.2008.10.112
[17] Cholamjiak, W.; Suantai, S., A hybrid method for a countable family of multivalued maps, equilibrium problems, and variational inequality problems, Discrete Dynamics in Nature and Society, 2010 (2010) · Zbl 1194.90104 · doi:10.1155/2010/349158
[18] Hong, S., Fixed points of multivalued operators in ordered metric spaces with applications, Nonlinear Analysis: Theory, Methods & Applications, 72, 11, 3929-3942 (2010) · Zbl 1184.54041 · doi:10.1016/j.na.2010.01.013
[19] Khan, S. H.; Yildirim, I.; Rhoades, B. E., A one-step iterative process for two multivalued nonexpansive mappings in Banach spaces, Computers and Mathematics with Applications, 61, 10, 3172-3178 (2011) · Zbl 1223.47081 · doi:10.1016/j.camwa.2011.04.011
[20] Chang, S.-S.; Kim, J. K.; Wang, X. R., Modified block iterative algorithm for solving convex feasibility problems in Banach spaces, Journal of Inequalities and Applications, 2010 (2010) · Zbl 1187.47045 · doi:10.1155/2010/869684
[21] Zhang, F.; Zhang, H.; Zhang, Y., New iterative algorithm for two infinite families of multivalued quasi-nonexpansive mappings in uniformly convex Banach spaces, Journal of Applied Mathematics, 2013 (2013) · Zbl 1266.47101 · doi:10.1155/2013/649537
[22] Bunyawat, A.; Suantai, S., Common fixed points of a countable family of multivalued quasi-nonexpansive mappings in uniformly convex Banach spaces, International Journal of Computer Mathematics, 89, 16, 2274-2279 (2012) · Zbl 1255.47062 · doi:10.1080/00207160.2012.705280
[23] Schu, J., Weak and strong convergence to fixed points of asymptotically nonexpansive mappings, Bulletin of the Australian Mathematical Society, 43, 1, 153-159 (1991) · Zbl 0709.47051 · doi:10.1017/S0004972700028884
[24] Liu, L. S., Ishikawa and Mann iterative process with errors for nonlinear strongly accretive mappings in Banach spaces, Journal of Mathematical Analysis and Applications, 194, 1, 114-125 (1995) · Zbl 0872.47031 · doi:10.1006/jmaa.1995.1289
[25] Xu, H.-K., Iterative algorithms for nonlinear operators, Journal of the London Mathematical Society, 66, 1, 240-256 (2002) · Zbl 1013.47032 · doi:10.1112/S0024610702003332
[26] Ahmed, R.; Altwqi, S., Convergence theorems for three finite families of multivalued nonexpansive mappings, Journal of the Egyptian Mathematical Society, 22, 3, 459-465 (2014) · Zbl 1318.47068 · doi:10.1016/j.joems.2013.10.008
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.