Acta mathematica scientia,Series B ›› 2017, Vol. 37 ›› Issue (2): 342-354.doi: 10.1016/S0252-9602(17)30006-1

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IMPROVED GRAOIENT METHOD FOR MONOTONE AND LIPSCHITZ CONTINUOUS MAPPINGS IN BANACH SPACES

Kazuhide NAKAJO   

  1. Sundai Preparatory School, Surugadai, Kanda, Chiyoda-ku, Tokyo 101-8313, Japan
  • Received:2015-06-05 Online:2017-04-25 Published:2017-04-25

Abstract:

Let C be a nonempty closed convex subset of a 2-uniformly convex and uniformly smooth Banach space C and {An}n∈N 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.

Key words: Variational inequality problem, gradient method, monotone operators, 2-uniformly convex Banach space, hybrid method

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