tailieunhanh - Báo cáo hóa học: "FIXED POINT VARIATIONAL SOLUTIONS FOR UNIFORMLY CONTINUOUS PSEUDOCONTRACTIONS 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: FIXED POINT VARIATIONAL SOLUTIONS FOR UNIFORMLY CONTINUOUS PSEUDOCONTRACTIONS IN BANACH SPACES | FIXED POINT VARIATIONAL SOLUTIONS FOR UNIFORMLY CONTINUOUS PSEUDOCONTRACTIONS IN BANACH SPACES ANIEFIOK UDOMENE Received 27 June 2005 Revised 21 November 2005 Accepted 28 November 2005 Let E be a reflexive Banach space with a uniformly Gateaux differentiable norm let K be a nonempty closed convex subset of E and let T K K be a uniformly continuous pseudocontraction. If f K K is any contraction map on K and if every nonempty closed convex and bounded subset of K has the fixed point property for nonexpansive self-mappings then it is shown under appropriate conditions on the sequences of real numbers an ỉpn that the iteration process Z1 e K zn 1 pn anTzn 1 - an zn 1 - pn f zn n e N strongly converges to the fixed point of T which is the unique solution of some variational inequality provided that K is bounded. Copyright 2006 Aniefiok Udomene. 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. 1. Introduction Let E be a real Banach space with dual E and K a nonempty closed convex subset of E. Let J E 2E denote the normalized duality mapping defined by J x f e E x f x 2 II f II x x e E where denotes the generalized duality pairing. Following Morales 6 a mapping T with domain D T and range T in E is called strongly pseudocontractive if for some constant k 1 and Vx y e D T A - k x - y AI - T x - AI - T y for all A k while T is called a pseudocontraction if holds for k 1. The mapping T is called Lipschitz if there exists L 0 such that II Tx - Ty II L x - y II Vx y e D T . The mapping T is called nonexpansive if L 1 and is called a strict contraction if L 1. Every nonexpansive mapping is a pseudocontraction. The converse is not true. The example T x 1 - x2 3 0 x 1 is a continuous pseudocontraction which is not nonexpansive. It follows from a result of Kato 3 that T is pseudocontractive if and only if there .