tailieunhanh - Báo cáo toán học: " Weak convergence theorem for the three-step iterations of non-Lipschitzian nonself mappings in Banach spaces"

Tuyển tập các báo cáo nghiên cứu khoa học ngành toán học được đăng trên tạp chí toán học quốc tế đề tài: Weak convergence theorem for the three-step iterations of non-Lipschitzian nonself mappings in Banach spaces | Zhu et al. Fixed Point Theory and Applications 2011 2011 106 http content 2011 1 106 Fixed Point Theory and Applications a SpringerOpen Journal RESEARCH Open Access Weak convergence theorem for the three-step iterations of non-Lipschitzian nonself mappings in Banach spaces Lanping Zhu Qianglian Huang and Xiaoru Chen Correspondence qlhmath@ College of Mathematics Yangzhou University Yangzhou 225002 China Springer Abstract In this article we introduce a new three-step iterative scheme for the mappings which are asymptotically nonexpansive in the intermediate sense in Banach spaces. Weak convergence theorem is established for this three-step iterative scheme in a uniformly convex Banach space that satisfies Opial s condition or whose dual space has the Kadec-Klee property. Furthermore we give an example of the nonself mapping which is asymptotically nonexpansive in the intermediate sense but not asymptotically nonexpansive. The results obtained in this article extend and improve many recent results in this area. AMS classification 47H10 47H09 46B20. Keywords asymptotically nonexpansive in the intermediate sense non-self mapping Kadec-Klee property Opial s condition common fixed point 1 Introduction Fixed-point iterations process for nonexpansive and asymptotically nonexpansive mappings in Banach spaces have been studied extensively by various authors 1-13 . In 1991 Schu 4 considered the following modified Mann iteration process for an asymptotically nonexpansive map T on C and a sequence an in 0 1 x1 e c xn 1 anxn 1 an T xn n 1. 1 1 Since then Schu s iteration process has been widely used to approximate fixed points of asymptotically nonexpansive mappings in Hilbert spaces or Banach spaces 7 8 10-13 . Noor in 2000 introduced a three-step iterative scheme and studied the approximate solutions of variational inclusion in Hilbert spaces 6 . Later Xu and Noor 7 Cho et al. 8 Suantai 9 Plubtieng et al. 12 studied the .