Fixed Point Theorems for an Elastic Nonlinear Mapping in Banach Spaces - Hiroko Manaka (Abstract and Applied Analysis)
Let E be a smooth Banach space with a norm ·. Let V(x,y)=x2+y2-2 x,Jy for any x,y∈E, where ·,· stands for the duality pair and J is the normalized duality mapping. We define a V-strongly nonexpansive mapping by V(·,·). This nonlinear mapping is nonexpansive in a Hilbert space. However, we show that there exists a V-strongly nonexpansive mapping with fixed points which is not nonexpansive in a Banach space. In this paper, we show a weak convergence theorem and strong convergence theorems for fixed points of this elastic nonlinear mapping and give the existence theorem.