On the stability of Jensen functional equation in Felbin’s type fuzzy normed linear spaces

Abstract

In this paper,we investigate the generalized Hyers-Ulam-Rassias stability of Jensen functional equation in Felbin’s type fuzzy normed linear spaces . 1.Introduction In 1940, Ulam[1] proposed the general Ulam stability problem.Next year, Hyers[2] solved this problem.In 1978, Rassias[3] took account of the unbounded Cauchy difference in Hyers’ theorem and obtained the results for linear mappings. The stability problems of several functional equations have been extensively investigated by a number of authors (see [4,5] and references therein).In 1989,Kominek[6] proved the stability of Jensen functional equation on a restricted domain.In1998, Jung[7] proved the Hyers-Ulam-Rassias stability of Jensen functional equation. In 2014, Eskandani and Rassias[8] investigated the stability of a general cubic functional equation in Felbin’s type fuzzy normed linear spaces. In this paper,we investigate the generalized Hyers-Ulam-Rassias stability of Jensen functional equation in Felbin’s type fuzzy normed linear spaces . We consider some basic concepts concerning in the theory of fuzzy real numbers. Let be a fuzzy subset on R, i.e., a mapping associating with each real number t its grade of membership . Definition1.1 ] 9 [ A fuzzy subset on R is called a fuzzy real number,whose -level set is denoted by ,i.e., ,if it satisfies two axioms: (1)There exists such that . (2)For each ; where . The set of all fuzzy real numbers denoted by .If and whenever ,then is called a nonnegative fuzzy real number and denotes the set of all non-negative fuzzy real numbers. Definition1.2 ] 9 [ Let X be a real linear space, L and R (respectively, left norm and right norm) be 3rd International Conference on Mechatronics, Robotics and Automation (ICMRA 2015) © 2015. The authors Published by Atlantis Press 563 symmetric and non-decreasing mappings in both arguments from into satisfying and .The mapping from X into is called a fuzzy norm if for and : (1) if and only if , (2) for all and (3)For all , (a)if , and then     t s y x    ), , ( t y s x L (b)if , and then   t s y x       ) , ( t y s x R . The quaternary (X, ,L,R) is called a fuzzy normed linear space. Definition1.3 ] 9 [ Let (X, ,L,R) be a fuzzy normed linear space and .A sequence in X is said to converge to ,denoted by ,if for every and is called a Cauchy sequence if for every .A subset in X is said to be complete if every Cauchy sequence in A converges in A.The fuzzy normed space (X, ,L,R) is said to be a fuzzy Banach space if it is complete. Theorem1.4 ] 10 [ Let (X,‖.‖,L,R) be a fuzzy normed linear space, if ,then for any , for all A mapping is called a Jensen function if f satisfies the functional equation for .For a given mapping we define the difference operator for Then f is a Jensen function if for all