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: Generalizations of Caristi Kirk’s Theorem on Partial Metric Spaces | Karapinar Fixed Point Theory and Applications 2011 2011 4 http content 2011 1 4 Fixed Point Theory and Applications a SpringerOpen Journal RESEARCH Open Access Generalizations of Caristi Kirk s Theorem on Partial Metric Spaces Erdal Karapinar Correspondence erdalkarapinar@ Department of Mathematics Atilim University 06836 Incek Ankara Turkey SpringerOpen0 Abstract In this article lower semi-continuous maps are used to generalize Cristi-Kirk s fixed point theorem on partial metric spaces. First we prove such a type of fixed point theorem in compact partial metric spaces and then generalize to complete partial metric spaces. Some more general results are also obtained in partial metric spaces. 2000 Mathematics Subject Classification 47H10 54H25 Keywords Partial metric space Lower semi-continuous Fixed point theory 1. Introduction and preliminaries In 1992 Matthews 1 2 introduced the notion of a partial metric space which is a generalization of usual metric spaces in which d x x are no longer necessarily zero. After this remarkable contribution many authors focused on partial metric spaces and its topological properties see . 3 - 8 Let X be a nonempty set. The mapping p X X X 0 oo is said to be a partial metric on X if for any x y z e X the following conditions hold true PM1 p x y p y x symmetry PM2 If p x x p x y p y y then x y equality PM3 p x x p x y small self-distances PM4 p x z p y y p x y p y z triangularity for all x y z e X. The pair X pi is then called a partial metric space see . 1 2 . We use the abbreviation PMS for the partial metric space X p . Notice that for a partial metric p on X the function dp X X X 0 o given by dp x y 2p x y - p x x - p y y is a usual metric on X. Observe that each partial metric p on X generates a T0 topology Tp on X with a base of the family of open Ji-balls Bp x e x e X 0 where Bp x e y e X p x y p x x e for all x e X and e 0. Similarly closed Ji-ball is defined as Bp x
