Some Kinds Of Generalized Viscosity Iterative Approximation Algorithms For Hierarchical Fixed Points In Banach Spaces

This paper first introduce a generalized viscosity implicit continuous iterative method to approximate a hierarchical fixed point of nonexpansive mappings:The first result in this paper is to introduce a new generalized viscosity implicit continuous iterative method in Banach space E xt＝tγf(xt)＋(I-tμF)T(xt), where f is a contraction mapping, T is a nonexpansive mapping, F is a k-Lipschitzian and η- strongly monotone operator on E. Under certain conditions, we prove that the sequence{xt} converges strongly to a hierarchical fixed point x of T, which also solves the variational inequality ((μF-γf)x, Jφ(x, z))≤0, z ∈E Fix(T). where φ:[0,1] (?) [0,1] is a gauge function, E is a reflexive Banach space which admits a weakly continuous duality mapping Jφ.The second result is that we combine Mann iteration and viscosity iteration to construct a new modified Ishikawa iterative algorithm for the common fixed point of a family of nonexpansive mappings in reflexive Banach spaces where f is a contraction mapping, Tn is a family of nonexpansive mappings, F is a k - Lipschitzian and η - strongly monotone operator on E. Under certain conditions, we prove that the sequence {xn} converges strongly to a fixed point x of Tn.In a sense the results here extend and improve the corresponding results reported by some other authors recently.The structure is that:in the first chapter we introduce some related research back-grounds, some relevant concepts and lemmas; in the second chapter, we prove the convergence of a generalized viscosity iterative approximate sequence for a hierarchi-cal fixed point; in the third chapter, we study a new modified Ishikawa iterative for the common fixed point of a family of nonexpansive mappings.