Fixed point of strong duality pseudocontractive mappings and applications (Q1925402)

Summary: Let \(E\) be a smooth Banach space with the dual \(E^\ast\), an operator \(T : E \rightarrow E^\ast\) is said to be \(\alpha\)-strong duality pseudocontractive if \(\langle x - y, Tx - Ty\rangle \leq \langle x - y, Jx - Jy \rangle - \alpha||Jx - Jy - (Tx - Ty)||\) for all \(x, y \in E\), where \(\alpha\) 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 \(\alpha\)-strong duality pseudocontractive mappings and to study its fixed point problem and applications to operator equation and variational inequality problems.