The Krasnoselskii-Mann iteration plays an important role in the approximation of fixed points of nonexpansive mappings, and it is well known that the classic Krasnoselskii-Mann iteration is weakly convergent in Hilbert spaces. The weak convergence is also known even in Banach spaces. Recently, Kanzow and Shehu proposed a generalized Krasnoselskii-Mann-type iteration for nonexpansive mappings and established its convergence in Hilbert spaces. In this paper, we show that the generalized Krasnoselskii-Mann-type iteration proposed by Kanzow and Shehu also converges in Banach spaces. As applications, we proved the weak convergence of generalized proximal point algorithm in the uniformly convex Banach spaces.
You-Cai Zhang, Ke Guo, Tao Wang
. Generalized Krasnoselskii–Mann-Type Iteration for Nonexpansive Mappings in Banach Spaces[J]. Journal of the Operations Research Society of China, 2021
, 9(1)
: 195
-206
.
DOI: 10.1007/s40305-018-0235-1
[1] Browder, F.E.:Nonexpansive nonlinear operators in a Banach space. Proc. Natl. Acad. Sci. USA 54(4), 1041-1044(1965)
[2] Kirk, W.A.:A fixed point theorem for mappings which do not increase distances. Am. Math. Mon. 72(9), 1004-1006(1965)
[3] Krasnoselskii, M.A.:Two remarks on the method of successive approximations. Uspekhi Mat. Nauk. 10, 123-127(1955)
[4] Mann, W.R.:Mean value methods in iteration. Proc. Am. Math. Soc. 4(3), 506-510(1953)
[5] Reich, S.:Weak convergence theorems for nonexpansive mappings in Banach spaces. J. Math. Anal. Appl. 67(2), 274-276(1979)
[6] Kirk, W.A.:Krasnoselskii's iteration process in hyperbolic space. Numer. Funct. Anal. Optim. 4(4), 371-381(1982)
[7] Reich, S., Zaslavski, A.J.:Convergence of Krasnoselskii-Mann iterations of nonexpansive operators. Math. Comput. Model. 32(11), 1423-1431(2000)
[8] Combettes, P.L.:Solving monotone inclusions via compositions of nonexpansive averaged operators. Optimization 53(5-6), 475-504(2004)
[9] Sahu, D.R., Ansari, Q.H., Yao, J.C.:Convergence of inexact Mann iterations generated by nearly nonexpansive sequences and applications. Numer. Funct. Anal. Optim. 37(10), 1312-1338(2016)
[10] Kim, T., Xu, H.K.:Robustness of Mann's algorithm for nonexpansive mappings. J. Math. Anal. Appl. 327(2), 1105-1115(2007)
[11] Opial, Z.:Weak convergence of the successive approximation for non-expansive mappings in Banach spaces. Bull. Am. Math. Soc. 73(4), 591-597(1967)
[12] Lindenstrauss, J., Tzafriri, L.:Classical Banach Spaces Ⅱ:Function Spaces. Springer, Berlin (1979)
[13] Chilin, V.I., Dodds, P.G., Sedaev, A.A.:Characterizations of Kadec-Klee properties in symmetric spaces of measurable functions. Trans. Am. Math. Soc. 348(2), 4895-4918(1996)
[14] Kanzow, C., Shehu, Y.:Generalized Krasnoselskii-Mann-type iterations for nonexpansive mappings in Hilbert spaces. Comput. Optim. Appl. 67(3), 593-620(2017)
[15] Falset, J.G., Kaczor, W.A., Kuczumow, T.:Weak convergence theorems for asymptotically nonexpansive mappings and semigroups. Nonlinear Anal. 43(3), 377-401(2001)
[16] Dulst, D.V.:Equivalent norms and the fixed point property for nonexpansive mappings. J. Lond. Math. Soc. 2(1), 139-144(1982)
[17] Tan, K.K., Xu, H.K.:Approximating fixed points of nonexpansive mappings by the Ishikawa iteration process. J. Math. Anal. Appl. 178(2), 301-308(1993)
[18] Xu, H.K.:Inequalities in Banach spaces with applications. Nonlinear Anal. 16(12), 1127-1138(1991)
[19] Browder, F.E.:Convergence theorems for sequences of nonlinear operators in Banach spaces. Math. Z. 100(3), 201-225(1967)
[20] Martinet, B.:Regularization d'inequations variationelles par approximations successives. Rev. Fr. d'Inf. et de Rech. Oper. 4(3), 154-159(1970). (In French)
[21] Moreau, J.J.:Proximité et dualité dans un espace hilbertien. Bull. Soc. Math. Fr. 93, 273-299(1965). (In French)
[22] Rockafellar, R.T.:Monotone operators and the proximal point algorithm. SIAM J. Control Optim. 14(5), 97-116(1976)
[23] López, G., Martínmárquez, V., Wang, F., Xu, H.K.:Forward-backward splitting methods for accretive operators in Banach spaces. Abstr. Appl. Anal. 2012(5), 933-947(2014)
