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: Research Article A Kirk Type Characterization of Completeness for Partial Metric Spaces | Hindawi Publishing Corporation Fixed Point Theory and Applications Volume 2010 Article ID 493298 6 pages doi 2010 493298 Research Article A Kirk Type Characterization of Completeness for Partial Metric Spaces Salvador Romaguera Insitituto Universitario de Matemdtica Pura y Aplicada Universidad Politecnica de Valencia 46071 Valencia Spain Correspondence should be addressed to Salvador Romaguera sromague@ Received 1 October 2009 Accepted 25 November 2009 Academic Editor Mohamed A. Khamsi Copyright 2010 Salvador Romaguera. This is an open access article distributed under the Creative Commons Attribution License which permits unrestricted use distribution and reproduction in any medium provided the original work is properly cited. We extend the celebrated result of W. A. Kirk that a metric space X is complete if and only if every Caristi self-mapping for X has a fixed point to partial metric spaces. 1. Introduction and Preliminaries Caristi proved in 1 that if f is a selfmapping of a complete metric space X d such that there is a lower semicontinuous function ộ X 0 to satisfying d x fx ộ x - ộ fx for all x X then f has a fixed point. This classical result suggests the following notion. A selfmapping f of a metric space X d for which there is a function ộ X 0 to satisfying the conditions of Caristi s theorem is called a Caristi mapping for X d . There exists an extensive and well-known literature on Caristi s fixed point theorem and related results see . 2-10 etc. . In particular Kirk proved in 7 that a metric space X d is complete if and only if every Caristi mapping for X d has a fixed point. For other characterizations of metric completeness in terms of fixed point theory see 11-14 etc. and also 15 16 for recent contributions in this direction. In this paper we extend Kirk s characterization to a kind of complete partial metric spaces. 2 Fixed Point Theory and Applications Let us recall that partial metric spaces were introduced by Matthews .
