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 .
