Fuzzy Sets and Systems 159 (2008) 730 – 738 www.elsevier.com/locate/fss Fuzzy stability of the Jensen functional equation A.K. Mirmostafaee a, b, d , M. Mirzavaziri b, c , M.S. Moslehian b, d , ∗ a Department of Mathematics, Damghan University of Basic Sciences, Damghan, Iran b Department of Mathematics, Ferdowsi University, P.O. Box 1159, Mashhad 91775, Iran c Banach Mathematical Research Group (BMRG), Mashhad, Iran d Centre of Excellence in Analysis on Algebraic Structures (CEAAS), Ferdowsi University, Iran Received 11 January 2007; received in revised form 13 May 2007; accepted 18 July 2007 Available online 7 August 2007 Abstract We establish a generalized Hyers–Ulam–Rassias stability theorem in the fuzzy sense. In particular, we introduce the notion of fuzzy approximate Jensen mapping and prove that if a fuzzy approximate Jensen mapping is continuous at a point, then we can approximate it by an everywhere continuous Jensen mapping. As a fuzzy version of a theorem of Schwaiger, we also show that if every fuzzy approximate Jensen type mapping from the natural numbers into a fuzzy normed space can be approximated by an additive mapping, then the fuzzy norm is complete. © 2007 Elsevier B.V. All rights reserved. MSC: Primary, 46S40; Secondary, 39B52; 39B82; 26E50; 46S50 Keywords: Fuzzy normed space; Jensen functional equation; Additive mapping; Hyers–Ulam–Rassias stability; Continuity; -approximately Jensen type mapping; Completeness 1. Introduction and preliminaries In 1984, Katrasas [11] defined a fuzzy norm on a linear space to construct a fuzzy vector topological structure on the space. Later, some mathematicians have defined fuzzy norms on a linear space from various points of view [6,14,25]. In particular, in 2003, Bag and Samanta [2], following Cheng and Mordeson [4], gave an idea of a fuzzy norm in such a manner that the corresponding fuzzy metric is of Kramosil and Michalek type [13]. They also established a decomposition theorem of a fuzzy norm into a family of crisp norms and investigated some properties of fuzzy normed spaces [3]. Defining, in some way, the class of approximate solutions of the given functional equation, one can ask whether each mapping from this class can be somehow approximated by an exact solution of the considered equation; cf. [5,8,10,22]. In 1940 Ulam [24] posed the first stability problem. In the next year, Hyers [7] gave a partial affirmative answer to the question of Ulam. Hyers’ theorem was generalized by Aoki [1] for additive mappings and by Rassias [20] for linear mappings by considering an unbounded Cauchy difference. The paper of Rassias [20] has provided a lot of influence in the development of what we now call Hyers–Ulam–Rassias stability of functional equations. ∗ Corresponding author. Tel./fax: +98 511 8828606. E-mail addresses: mirmostafaei@ferdowsi.um.ac.ir (A.K. Mirmostafaee), mirzavaziri@math.um.ac.ir (M. Mirzavaziri), moslehian@ferdowsi.um.ac.ir (M.S. Moslehian). 0165-0114/$ - see front matter © 2007 Elsevier B.V. All rights reserved. doi:10.1016/j.fss.2007.07.011