tailieunhanh - Báo cáo hóa học: " Research Article Strong Convergence Theorem for a New General System of Variational Inequalities in Banach Spaces"

Tuyển tập báo cáo các nghiên cứu khoa học quốc tế ngành hóa học dành cho các bạn yêu hóa học tham khảo đề tài: Research Article Strong Convergence Theorem for a New General System of Variational Inequalities in Banach Spaces | Hindawi Publishing Corporation Fixed Point Theory and Applications Volume 2010 Article ID 246808 13 pages doi 2010 246808 Research Article Strong Convergence Theorem for a New General System of Variational Inequalities in Banach Spaces S. Imnang1 2 and S. Suantai2 3 1 Department of Mathematics Faculty of Science Thaksin University Phatthalung Campus Phatthalung 93110 Thailand 2 Centre of Excellence in Mathematics CHE Si Ayutthaya Road Bangkok 10400 Thailand 3 Department of Mathematics Faculty of Science Chiang Mai University Chiang Mai 50200 Thailand Correspondence should be addressed to S. Suantai scmti005@ Received 26 July 2010 Revised 7 December 2010 Accepted 30 December 2010 Academic Editor S. Reich Copyright 2010 S. Imnang and S. Suantai. This is an open access article distributed under the Creative Commons Attribution License which permits unrestricted use distribution and reproduction in any medium provided the original work is properly cited. We introduce a new system of general variational inequalities in Banach spaces. The equivalence between this system of variational inequalities and fixed point problems concerning the nonexpansive mapping is established. By using this equivalent formulation we introduce an iterative scheme for finding a solution of the system of variational inequalities in Banach spaces. Our main result extends a recent result acheived by Yao Noor Noor Liou and Yaqoob. 1. Introduction Let X be a real Banach space and X be its dual space. Let U x e X xH 1 denote the unit sphere of X. X is said to be uniformly convex if for each e e 0 2 there exists a constant Ỗ 0 such that for any x y e U x - y II e implies x y 1 2 The norm on X is said to be Gateaux differentiable if the limit x tyịị- IIx lim-------f------- t-X 0 t 2 Fixed Point Theory and Applications exists for each x y e U and in this case X is said to have a uniformly Frechet differentiable norm if the limit is attained uniformly for x y e U and .