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Báo cáo hóa học: " Research Article Generic Well-Posedness for a Class of Equilibrium Problems"

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 Generic Well-Posedness for a Class of Equilibrium Problems | Hindawi Publishing Corporation Journal of Inequalities and Applications Volume 2008 Article ID 581917 9 pages doi 10.1155 2008 581917 Research Article Generic Well-Posedness for a Class of Equilibrium Problems Alexander J. Zaslavski Department of Mathematics The Technion-Israel Institute of Technology 32000 Haifa Israel Correspondence should be addressed to Alexander J. Zaslavski ajzasl@tx.technion.ac.il Received 23 December 2007 Accepted 6 March 2008 Recommended by Simeon Reich We study a class of equilibrium problems which is identified with a complete metric space of functions. For most elements of this space of functions in the sense of Baire category we establish that the corresponding equilibrium problem possesses a unique solution and is well-posed. Copyright 2008 Alexander J. Zaslavski. 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 study of equilibriumproblems has recently been a rapidly growing area of research. See for example 1-3 and the references mentioned therein. Let X p be a complete metric space. In this paper we consider the following equilibrium problem To find x G X such that f x y 0 Vy G X P where f belongs to a complete metric space of functions A defined below. In this paper we show that for most elements of this space of functions A in the sense of Baire category the equilibrium problem P possesses a unique solution. In other words the problem P possesses a unique solution for a generic typical element of A 4-6 . Set pi xi yi X2 y2 p xi X2 p yi y2 xi X2 yi y2 G X. 1.1 Clearly X X X p1 is a complete metric space. Denote by A0 the set of all continuous functions f X X X R1 such that f x x 0 Vx G X. 1.2 2 Journal of Inequalities and Applications 1.4 1.5 We equip the set A0 with the uniformity determined by the base ư e f g A0 XA0 f z - g z e Vz X X X 1.3 where e