Tuyển tập các báo cáo nghiên cứu về hóa học được đăng trên tạp chí hóa hoc quốc tế đề tài : proximal point algorithm for finding a common fixed point of a finite family of nonexpansive mappings in Banach spaces | Kim and Tuyen Fixed Point Theory and Applications 2011 2011 52 http content 2011 1 52 Fixed Point Theory and Applications a SpringerOpen Journal RESEARCH Open Access Regularization proximal point algorithm for finding a common fixed point of a finite family of nonexpansive mappings in Banach spaces Jong Kyu Kim 1 and Truong Minh Tuyen2 Correspondence jongkyuk@ department of Mathematics Education Kyungnam University Masan Kyungnam 631-701 Korea Full list of author information is available at the end of the article Springer Abstract We study the strong convergence of a regularization proximal point algorithm for finding a common fixed point of a finite family of nonexpansive mappings in a uniformly convex and uniformly smooth Banach space. 2010 Mathematics Subject Classification 47H09 47J25 47J30. Keywords accretive operators uniformly smooth and uniformly convex Banach space sunny nonexpansive retraction weak sequential continuous mapping regularization 1 Introduction Let E be a Banach space with its dual space E . For the sake of simplicity the norms of E and E are denoted by the symbol . We write x x instead of x x for x e E and x e E. We denote as and the weak convergence and strong convergence respectively. A Banach space E is reflexive if E E . The problem of finding a fixed point of a nonexpansive mapping is equivalent to the problem of finding a zero of the following operator equation 0 e A x involving the accretive mapping A. One popular method of solving equation 0 e A x is the proximal point algorithm of Rockafellar 1 which is recognized as a powerful and successful algorithm for finding a zero of monotone operators. Starting from any initial guess x0 e H this proximal point algorithm generates a sequence xn given by Xn 1 JA Xn en where JA I rA -1 Vr 0 is the resolvent of A in a Hilbert space H. Rockafellar 1 proved the weak convergence of the algorithm provided that the regularization .
