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Báo cáo hóa học: " EXISTENCE OF EXTREMAL SOLUTIONS FOR QUADRATIC FUZZY EQUATIONS"

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: EXISTENCE OF EXTREMAL SOLUTIONS FOR QUADRATIC FUZZY EQUATIONS | EXISTENCE OF EXTREMAL SOLUTIONS FOR QUADRATIC FUZZY EQUATIONS JUAN J. NIETO AND ROSANA RODRÍGUEZ-LÓPEZ Received 8 November 2004 and in revised form 8 March 2005 Some results on the existence of solution for certain fuzzy equations are revised and extended. In this paper we establish the existence of a solution for the fuzzy equation Ex2 Fx G x where E F G and x are positive fuzzy numbers satisfying certain conditions. To this purpose we use fixed point theory applying results such as the well-known fixed point theorem of Tarski presenting some results regarding the existence of extremal solutions to the above equation. 1. Preliminaries In 1 it is studied the existence of extremal fixed points for a map defined in a subset of the set E1 of fuzzy real numbers that is the family of elements x R 0 1 with the properties i x is normal there exists t0 e R with x t0 1. ii x is upper semicontinuous. iii x is fuzzy convex x Ab 1 - Ả t2 minlxlb x t2 Vt1 t2 e R A e 0 1 . 1.1 iv The support ofx supp x cl t e R x t 0 is a bounded subset of R. In the following for a fuzzy number x e E1 we denote the a-level set x a t e R x t a 1.2 by the interval xai xar for each a e 0 1 and x 0 cl Uae 0 1 x 0i x0r . 1.3 Note that this notation is possible since the properties of the fuzzy number x guarantee that x a is a nonempty compact convex subset of R for each a e 0 1 . Copyright 2005 Hindawi Publishing Corporation Fixed Point Theory and Applications 2005 3 2005 321-342 DOI 10.1155 FPTA.2005.321 322 Existence of extremal solutions for quadratic fuzzy equations We consider the partial ordering in E1 given by x y e E1 x y Xal yai Xar yar Va e 0 1 1.4 and the distance that provides E1 the structure of complete metric space is given by d- x y sup dtf x a y a for X y e E1 1.5 ae 0 1 being dH the Hausdorff distance between nonempty compact convex subsets of R that is compact intervals . For each fuzzy number x e E1 we define the functions xl 0 1 R xr 0 1 R given by xL a xai and xR a xar for each