Fixed Point Theorems for an Elastic Nonlinear Mapping in Banach Spaces

LetEbe a smooth Banach space with a norm·. LetV(x,y)=x2+y2-2 x,Jyfor anyx,y∈E, where·,·stands for the duality pair andJis the normalized duality mapping. We define aV-strongly nonexpansive mapping byV(·,·). This nonlinear mapping is nonexpansive in a Hilbert space. However, we show that there exists aV-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.

Convergence Theorems for a Maximal Monotone Operator and a -Strongly Nonexpansive Mapping in a Banach Space

LetEbe a smooth Banach space with a norm . Let for any , where stands for the duality pair andJis the normalized duality mapping. With respect to this bifunction , a generalized nonexpansive mapping and a -strongly nonexpansive mapping are defined in . In this paper, using the properties of generalized nonexpansive mappings, we prove convergence theorems for common zero points of a maximal monotone operator and a -strongly nonexpansive mapping.