Nonlinear Operator Family And Semigroups Of Operators Of Public Fixed Point Iterative Approximation

Approximation to fixed points for nonlinear operator is the kernel of the study of the fixed point theory. In this paper, we study mainly the iterative approximating to the common fixed points of a pair of asymptotically nonexpansive mappings, finite nonexpansive mappings and nonexpansive semigroup.Let E be real Banach space, C be nonempty closed convex subset of E.Firstly, In uniformly convex Banach space E, let S, T : C → C be asymptotically nonexpansive mapping and C be bounded. On one hand, we proved the weak convergence to the common fixed points of S and T by modified generalization Ishikawa iterative with errors;On the other hand, we obtain the strong convergence theorem under a condition weaker than "C is compactness" .Secondly, let E be a uniformly smooth Banach space with a weakly sequentially continuous duality mapping, T₁, T₂, … , T_N : C → C be N nonexpansive mappings. Suppose that the set of common fixed points of T₁, T₂, … , T_N be nonempty, we prove that explicit viscosity iterative sequence for N nonexpansive mapping converges strongly to the common fixed point of T₁, T₂, … , T_N, which solves some variational inequality.Lastly, let E be a uniformly convex Banach space whose norm is uniformly G(a|^)teaux differentiable, S = {T(s) : s ≥ 0} be nonexpansive semigroup on C. Suppose the set of common fixed point of nonexpansive semigroup be nonempty, by the method of Banach limit, we prove that the implicit and explicit viscosity iterative for nonexpansive semigroup converges to some common fixed point of nonexpansive mapping, which solves some variational inequality.