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 . 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 by the interval xai xar for each a e 0 1 and x 0 cl Uae 0 1 x 0i x0r . 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 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 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 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
