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On a hybrid method for generalized mixed equilibrium problem and fixed point problem of a family of quasi-\(\varphi \)-asymptotically nonexpansive mappings in Banach spaces. (English) Zbl 1205.47062

Summary: We prove a strong convergence theorem by using a hybrid method for finding a common element of the set of solutions for generalized mixed equilibrium problems, the set of fixed points of a family of quasi-\(\varphi \)-asymptotically nonexpansive mappings in strictly convex reflexive Banach space with the Kadec-Klee property and a Fréchet differentiable norm under weaker conditions. The method of the proof is different from S.Takahashi and W.Takahashi [Nonlinear Anal., Theory Methods Appl.69, No.3 (A), 1025–1033 (2008; Zbl 1142.47350)] and W.Takahashi and K.Zembayashi [Fixed Point Theory Appl.2008, Article ID 528476 (2008; Zbl 1187.47054)]. It also shows that the type of projection used in the iterative method is independent of the properties of the mappings. The results presented in the paper improve and extend some recent results.
Editorial remark: For related results, see the authors’ paper [J. Inequal.Appl.2010, Article ID 101690 (2010; Zbl 1187.47052)].

MSC:

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