tailieunhanh - Báo cáo hóa học: " Research Article Diametrically Contractive Multivalued Mappings"

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 Diametrically Contractive Multivalued Mappings | Hindawi Publishing Corporation Fixed Point Theory and Applications Volume 2007 Article ID 19745 7 pages doi 2007 19745 Research Article Diametrically Contractive Multivalued Mappings S. Dhompongsa and H. Yingtaweesittikul Received 1 March 2007 Accepted 2 May 2007 Recommended by Tomas Dominguez Benavides Diametrically contractive mappings on a complete metric space are introduced by V. I. Istratescu. We extend and generalize this idea to multivalued mappings. An easy example shows that our fixed point theorem is more applicable than a former one obtained by H. K. Xu. A convergence theorem of Picard iteratives is also provided for multivalued mappings on hyperconvex spaces thereby extending a Proinov s result. Copyright 2007 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. I. Introduction Let X d be a complete metric space. A mapping T X X is a contraction if for some a e 0 1 d Tx Ty ad x y Vx y e X. The mapping T is said to be contractive if d Tx Ty d x y Vx y e X x y. By the well-known Banach s contraction principle every contraction has a unique fixed point x0 and the Picard iteration Tnx converges to x0 for every x e X. Examples in 1 2 show that a contractive mapping may fail to have a fixed point. However a question of the existence of a fixed point is of interest. In fact it has been left open the following question. Question 3 . Let M be a weakly compact subset of a Banach space and let T M M be contractive. Does T have a fixed point 2 Fixed Point Theory and Applications Istratescu 4 introduced a proper subclass of the class of the contractive mappings whose elements are called the diametrically contractive mappings. Xu 2 proved in the framework of Banach spaces the following theorem. Theorem 2 Theorem . Let M be a weakly compact .