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: FIXED POINTS AND COINCIDENCE POINTS FOR MULTIMAPS WITH NOT NECESSARILY BOUNDED IMAGES | FIXED POINTS AND COINCIDENCE POINTS FOR MULTIMAPS WITH NOT NECESSARILY BOUNDED IMAGES S. V. R. NAIDU Received 20 August 2003 and in revised form 24 February 2004 In metric spaces single-valued self-maps and multimaps with closed images are considered and fixed point and coincidence point theorems for such maps have been obtained without using the extended Hausdorff metric thereby generalizing many results in the literature including those on the famous conjecture of Reich on multimaps. 1. Introduction Many authors have been using the Hausdorff metric to obtain fixed point and coincidence point theorems for multimaps on a metric space. In most cases the metric nature of the Hausdorff metric is not used and the existence part of theorems can be proved without using the concept of Hausdorff metric under much less stringent conditions on maps. The aim of this paper is to illustrate this and to obtain fixed point and coincidence point theorems for multimaps with not necessarily bounded images. Incidentally we obtain improvements over the results of Chang 3 Daffer et al. 6 Jachymski 9 Mizoguchi and Takahashi 12 and Wegrzyk 17 on the famous conjecture of Reich on multimaps Conjecture . 2. Notation Throughout this paper unless otherwise stated X d is a metric space C X is the collection of all nonempty closed subsets of X B X is the collection of all nonempty bounded subsets of X CB X is the collection of all nonempty bounded closed subsets of X s T are self-maps on X I is the identity map on X F G are mappings from X into C X for a nonempty subset A of X and x E X d x A inf d x y y E A for nonempty subsets A B of X H A B max supd x B supd y A xeA yeB Copyright 2004 Hindawi Publishing Corporation Fixed Point Theory and Applications 2004 3 2004 221-242 2000 Mathematics Subject Classification 47H10 54H25 URL http S1687182004308090 222 Fixed points and coincidence points f g and p are functions on X defined as f x d Sx Fx g x d Tx Gx and p x d x Fx
