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Báo cáo hóa học: " Some new fixed point theorems for set-valued contractions in complete metric spaces"

Tuyển tập các báo cáo nghiên cứu về hóa học được đăng trên tạp chí sinh học đề tài : Some new fixed point theorems for set-valued contractions in complete metric spaces | Chen Fixed Point Theory and Applications 2011 2011 72 http www.fixedpointtheoryandapplications.eom content 2011 1 72 Fixed Point Theory and Applications a SpringerOpen Journal RESEARCH Open Access Some new fixed point theorems for set-valued contractions in complete metric spaces Chi-Ming Chen Correspondence ming@mail.nhcue. edu.tw Department of Applied Mathematics National Hsinchu University of Education Taiwan Springer Abstract In this article we obtain some new fixed point theorems for set-valued contractions in complete metric spaces. Our results generalize or improve many recent fixed point theorems in the literature. MSC 47H10 54C60 54H25 55M20. Keywords fixed point theorem set-valued contraction 1 Introduction and preliminaries Let X d be a metric space D a subset of X and f D X be a map. We say f is contractive if there exists a e 0 1 such that for all x y e D d fx fy a d x y . The well-known Banach s fixed point theorem asserts that if D X f is contractive and X d is complete then f has a unique fixed point in X. It is well known that the Banach contraction principle 1 is a very useful and classical tool in nonlinear analysis. Also this principle has many generalizations. For instance a mapping f X X is called a quasi-contraction if there exists k 1 such that d fx fy k max d x y d x fx d y fy d x fy d y fx for any x y e X. In 1974 C iric 2 introduced these maps and proved an existence and uniqueness fixed point theorem. Throughout we denote the family of all nonempty closed and bounded subsets of X by CB X . The existence of fixed points for various multi-valued contractive mappings had been studied by many authors under different conditions. In 1969 Nadler 3 extended the famous Banach Contraction Principle from single-valued mapping to multi-valued mapping and proved the below fixed point theorem for multi-valued contraction. Theorem 1 3 Let X d be a complete metric space and T X CB X . Assume that there exists c e 0 1 such that H Tx Ty cd x y for all x y