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: Fixed point theorem for generalized weak contractions satisfying rational expressions in ordered metric spaces | Luong and Thuan Fixed Point Theory and Applications 2011 2011 46 http content 2011 1 46 Fixed Point Theory and Applications a SpringerOpen Journal RESEARCH Open Access Fixed point theorem for generalized weak contractions satisfying rational expressions in ordered metric spaces Nguyen Van Luong and Nguyen Xuan Thuan Correspondence luonghdu@ Department of Natural Sciences Hong Duc University Thanh Hoa Vietnam Springer Abstract In this paper we prove a fixed point theorem for generalized weak contractions satisfying rational expressions in partially ordered metric spaces. The result is a generalization of a recent result of Harjani et al. Abstr. Appl. Anal 1-8 2010 . An example is also given to show that our result is a proper generalization of the existing one. 2000 Mathematics Subject Classification 47H10 54H25. Keywords fixed point generalized weak contraction rational type ordered metric spaces 1 Introduction and preliminaries It is well known that the Banach contraction mapping principle is one of the pivotal results of analysis. Generalizations of this principle have been obtained in several directions. The following is an example of such generalizations. Jaggi in 1 proved the following theorem satisfying a contractive condition of rational type Theorem . 1 Let T be a continuous self-map defined on a complete metric space X d . Suppose that T satisfies the following condition ai d x Tx -d y T .na d ITx Ty a--- --------- pd x y d x y for all x y e X x y and for some a b 0 with a b 1 then T has a unique fixed point in X. Another generalization of the contraction principle was suggested by Alber and Guerre-Delabriere 2 in Hilbert spaces. Rhoades 3 has shown that their result is still valid in complete metric spaces. Definition . 3 Let X d be a metric space. A mapping T X X is said to be ộ-weak contraction if d Tx Ty d x y p d x y for all x y e X where Ộ 0 0 is a continuous and non-decreasing function
