Fixed Point of Strong Duality Pseudocontractive Mappings and Applications

Abstract

Let E be a smooth Banach space with the dual $E^{\ast }$, an operator $T:E\to E^{\ast }$ is said to be α-strong duality pseudocontractive if ${〈x-y, Tx-Ty〉\le 〈x-y, Jx-Jy〉-\mathrm{\alpha }\parallel Jx-Jy-\left(Tx-Ty\right)\parallel }^2$, for all $x,y\in E$, where α is a nonnegative constant. An element $x\in E$ is called a duality fixed point of T if $Tx=Jx$. The purpose of this paper is to introduce the definition of α-strong duality pseudocontractive mappings and to study its fixed point problem and applications for operator equation and variational inequality problems.