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Strong convergence of composite iterative schemes for a countable family of nonexpansive mappings in Banach spaces. (English) Zbl 1226.47080
Summary: We propose a new modified viscosity approximation method for approximating common fixed points for a countable family of nonexpansive mappings in a Banach space. We prove strong convergence theorems for a countable family nonexpansive mappings in a reflexive Banach space with uniformly Gâteaux differentiable norm under some control conditions. These results improve and extend the results of J. S. Jung [Fixed Point Theory Appl. 2008, Article ID 167535 (2008; Zbl 1203.47053)]. Further, we apply our result to the problem of finding a zero of an accretive operator and extend the results of T.-H. Kim and H.-K. Xu [Nonlinear Anal., Theory Methods Appl. 61, No. 1–2, A, 51–60 (2005; Zbl 1091.47055)], L.-C. Ceng, A. R. Khan, Q. H. Ansari and J.-C. Yao [ibid. 70, No. 5, A, 1830–1840 (2009; Zbl 1226.47069)] and R.-D. Chen and Z.-C. Zhu [ibid. 69, No. 4, A, 1356–1363 (2008; Zbl 1196.47045)].

MSC:
47J25 Iterative procedures involving nonlinear operators
47H06 Nonlinear accretive operators, dissipative operators, etc.
47H09 Contraction-type mappings, nonexpansive mappings, \(A\)-proper mappings, etc.
47H10 Fixed-point theorems
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