Generalized Krasnoselskii–Mann-Type Iteration for Nonexpansive Mappings in Banach Spaces
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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.
KeywordsKrasnoselskii–Mann-type iteration Nonexpansive mappings Weak convergence Accretive operator proximal point algorithm Banach spaces
Mathematics Subject Classification47H05 47H09
- 22.Rockafellar, R.T.: Monotone operators and the proximal point algorithm. SIAM J. Control Optim. 14(5), 97–116 (1976)Google Scholar
- 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)Google Scholar