Steepest-descent Ishikawa iterative methods for a class of variational inequalities in Banach spaces

Abstract

In this paper, for finding a fixed point of a nonexpansive mapping in either uniformly smooth or reflexive nad strictly convex Banach space with a uniformly Gateaux differentiable norm, we present a new explicit iterative method, based ona combination of the steepest-descent method with the Ishikawa iterative one. We also show its several particular cases one of which is the composite Halpern iterative method in literature. The explicit iterative methodis also extended to the case of infinite family of nonexpansive mappings. Numerical experiment are given for illustration.