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 Fixed Points for Multivalued Mappings and the Metric Completeness | Hindawi Publishing Corporation Fixed Point Theory and Applications Volume 2009 Article ID 972395 15 pages doi 2009 972395 Research Article Fixed Points for Multivalued Mappings and the Metric Completeness S. Dhompongsa and H. Yingtaweesittikul Department of Mathematics Faculty of Science Chiang Mai University Chiang Mai 50200 Thailand Correspondence should be addressed to S. Dhompongsa sompongd@ Received 24 December 2008 Accepted 6 May 2009 Recommended by Wataru Takahashi We consider the equivalence of the existence of fixed points of single-valued mappings and multivalued mappings for some classes of mappings by proving some equivalence theorems for the completeness of metric spaces. Copyright 2009 S. Dhompongsa and H. Yingtaweesittikul. 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. 1. Introduction The Banach contraction principle 1 states that for a complete metric space X d every contraction T on X that is for some r e 0 1 d Tx Ty rd x y for all x y e X has a unique fixed point. Connell 2 gave an example of a noncomplete metric space X on which every contraction on X has a fixed point. Thus contractions cannot characterize the metric completeness of X. Theorem see 3 Kannan . Let X d be a complete metric space. Let T be a Kannan mapping on X that is for some a e 0 1 2 d Tx Ty ad x Tx ad y Ty for all x y e X. Then T has a unique fixed point. Subrahmanyam 4 proved that Kannan mappings can be used to characterize the completeness of the metric. That is a metric space X is complete if and only if every Kannan mapping on X has a fixed point. In 2008 Suzuki 5 introduced a new type of mappings and presented a generalization of the Banach contraction principle in which the completeness can also be characterized by the existence of fixed points of these mappings. Define a .