Improved gradient method for monotone and lipschitz continuous mappings in banach spaces

Abstract

Let C be a nonempty closed convex subset of a 2-uniformly convex and uniformly smooth Banach space E and {An}nɛN be a family of monotone and Lipschitz continuos mappings of C into E*. In this article, we consider the improved gradient method by the hybrid method in mathematical programming [10] for solving the variational inequality problem for {An} and prove strong convergence theorems. And we get several results which improve the well-known results in a real 2-uniformly convex and uniformly smooth Banach space and a real Hilbert space.