tailieunhanh - Báo cáo hóa học: " On relaxed and contraction-proximal point algorithms in hilbert 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: On relaxed and contraction-proximal point algorithms in hilbert spaces | Wang and Wang Journal of Inequalities and Applications 2011 2011 41 http content 2011 1 41 Journal of Inequalities and Applications a SpringerOpen Journal RESEARCH Open Access On relaxed and contraction-proximal point algorithms in hilbert spaces Shuyu Wang and Fenghui Wang Correspondence shyuwang@163. com Department Of Mathematics Luoyang Normal University Luoyang 471022 China Springer Abstract We consider the relaxed and contraction-proximal point algorithms in Hilbert spaces. Some conditions on the parameters for guaranteeing the convergence of the algorithm are relaxed or removed. As a result we extend some recent results of Ceng-Wu-Yao and Noor-Yao. Keywords maximal monotone operator proximal point algorithm firmly nonexpan-sive operator 1. Introduction Throughout H denotes a real Hilbert space and A a multi-valued operator with domain D A . We know that A is called monotone if u - v x - y 0 for any u e Ax v e Ay maximal monotone if its graph G A x y x e D A y e Ax is not properly contained in the graph of any other monotone operator. Denote by S x e D A 0 e Ax the zero set and by Jc I cA -1 the resolvent of A. It is well known that Jc is single valued and D Jc H for any c 0. A fundamental problem of monotone operators is that of finding an element x so that 0 e Ax. This problem is essential because it includes many concrete examples such as convex programming and monotone variational inequalities. A successful and powerful algorithm for solving this problem is the well-known proximal point algorithm PPA which generates for any initial guess x0 e H an iterative sequence as xn 1 Jcn xn en 1 - 1 II where cn is a positive real sequence and en is the error sequence see 1 . To guarantee the convergence of PPA there are two kinds of accuracy criterion posed on the error sequence X I IK II X 52 en X or n 0 X II lien II Vn IIXn - Xn II 52 Vn X n 0 where xn JCn xn en . In 2001 Han and He 2 proved that in finite dimensional .