The Split Common Fixed Point Problem for New Classes of Nonlinear Operators in Banach Spaces (The structure of function spaces and its environment)

Abstract

The aim of this article is to prove strong convergence theorems by the hybrid method and the shrinking projection method for finding common fixed points of families of new nonlinear mappings in Banach spaces. We first deal with basic properties of new nonlinear mappings. In particular, we prove that the common fixed point sets of new nonlinear mappings are closed and convex. Using these results and the hybrid method introduced by Nakajo and Takahashi [14], we prove a strong convergence theorem which solves the split common fixed point problem in two Banach Spaces. Furthermore, using the shrinking projection method introduced by Takahashi, Takeuchi and Kubota [28], we also prove another strong convergence theorem. Moreover, using these results, we obtain well-known and new strong convergence theorems in Hilbert spaces and Banach spaces.