tailieunhanh - Báo cáo hóa học: " WEAK AND STRONG CONVERGENCE THEOREMS FOR RELATIVELY NONEXPANSIVE MAPPINGS 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: WEAK AND STRONG CONVERGENCE THEOREMS FOR RELATIVELY NONEXPANSIVE MAPPINGS IN BANACH SPACES | WEAK AND STRONG CONVERGENCE THEOREMS FOR RELATIVELY NONEXPANSIVE MAPPINGS IN BANACH SPACES SHIN-YA MATSUSHITA AND WATARU TAKAHASHI Received 29 October 2003 We first introduce an iterative sequence for finding fixed points of relatively nonexpansive mappings in Banach spaces and then prove weak and strong convergence theorems by using the notion of generalized projection. We apply these results to the convex feasibility problem and a proximal-type algorithm for monotone operators in Banach spaces. 1. Introduction Let E be a real Banach space and let A be a maximal monotone operator from E to E where E is the dual space of E. It is well known that many problems in nonlinear analysis and optimization can be formulated as follows find u G E such that 0 G Au. A well-known method for solving in a Hilbert space H is the proximal point algorithm x0 G H and Xn 1 JrnXn n 0 1 2 . where rn G 0 to and Jr I rA 1 for all r 0. This algorithm was first introduced by Martinet 9 . In 16 Rockafellar proved that if liminfn TO rn 0 and A-10 0 then the sequence xn defined by converges weakly to an element of solutions of . On the other hand Kamimura and Takahashi 4 considered an algorithm to generate a strong convergent sequence in a Hilbert space. Further Kamimura and Takahashi s result was extended to more general Banach spaces by Kohsaka and Takahashi 7 . They introduced and studied the following iteration sequence x x0 G E and Xn 1 J-1 ttnJx 1 - an jJr Xn n 0 1 2 . where J is the duality mapping on E and Jr J rA -1J for all r 0. Kohsaka and Takahashi 7 proved that if A-10 0 limi to an 0 to 0 an to and limi to rn to then the sequence generated by converges strongly to an element of A-10. Copyright 2004 Hindawi Publishing Corporation Fixed Point Theoryand Applications 2004 1 2004 37-47 2000 Mathematics Subject Classification 47H09 47H05 47J25 URL http S1687182004310089 38 Weak and strong convergence theorems On the other hand Reich 13 .