tailieunhanh - Báo cáo toán học: " Kannan-type contractions and fixed points in uniform spaces"

Tuyển tập các báo cáo nghiên cứu khoa học ngành toán học được đăng trên tạp chí toán học quốc tế đề tài: Kannan-type contractions and fixed points in uniform spaces | Wlodarczyk and Plebaniak Fixed Point Theory and Applications 2011 2011 90 http content 2011 1 90 Fixed Point Theory and Applications a SpringerOpen Journal RESEARCH Open Access Kannan-type contractions and fixed points in uniform spaces Kazimierz Wtodarczyk and Robert Plebaniak Correspondence wlkzxa@math. Department of Nonlinear Analysis Faculty of Mathematics and Computer Science University of tódz Banacha 22 90-238 tódz Poland Springer Abstract In uniform spaces using J -families of generalized pseudodistances we construct four kinds of contractions of Kannan type and by techniques based on these generalized pseudodistances we prove fixed point theorems for such contractions. The results are new in uniform and locally convex spaces and even in metric spaces. Examples are given. MSC 47H10 47H09 54E15 46A03 54E35. Keywords uniform space metric space J-family of generalized pseudodistances contractions of Kannan type fixed point iterative approximation 1 Introduction Given a space X Fix T denotes the set of fixed points of T X X . Fix T w e X w T w . For v0 any point of X by vm m e 0 u N we mean the sequence of iteration of T X X starting at v0 . Vme 0 UN vm T m v0 . Recall that maps satisfying the conditions B and K that are presented in Theorems and below are called in literature Banach contractions and Kannan contractions respectively and first arose in works 1 2 and 3 4 respectively. Theorem 1 2 Let X d be a complete metric space. IfT X X satisfies B 3xe 0 i Vx yex d T x T y kd x y then a T has a unique fixed point w in X and b Vu0 x lim um w . Theorem 3 Let X d be a complete metric space. If T X X satisfies K 3he 0 1 2 Vx X d T x T y h d T x x d T y y then a T has a unique fixed point w in X and b Vu0 X limm . um w . Theorem 4 Let X d be a metric space. Assume that i T X X satisfies K ii there exists w e X such that T is continuous at a point w and iii there exists a point v0e X such