International Journal of Applied Mathematical Research, 3 (2) (2014) 93-96 c Science Publishing Corporation www.sciencepubco.com/index.php/IJAMR doi: 10.14419/ijamr.v3i2.1936 Research Paper On the Mazur-Ulam problem in fuzzy anti-normed spaces Majid Abrishami-Moghaddam Department of Mathematics, Birjand Branch, Islamic Azad University, Birjand, Iran E-mail: m.abrishami.m@gmail.com Copyright c 2014 Majid Abrishami-Moghaddam. 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. Abstract The aim of this article is to proved a Mazur-Ulam type theorem in the strictly convex fuzzy anti-normed spaces. Keywords : Fuzzy anti-normed space, Mazur-Ulam theorem, strictly convex. 1. Introduction and preliminaries The theory of fuzzy sets was introduced by L. Zadeh [11] in 1965 and thereafter several authors applied it different branches of pure and applied mathematics. Many mathematicians considered the fuzzy normed spaces in several angels (see [1], [4], [10]). In [6] Iqbal H. Jebril and Samanta introduced fuzzy anti-norm on a linear space depending on the idea of fuzzy anti-norm was introduced by Bag and Samanta [2] and investigated their important properties. In 1932, the theory of isometric mappings was originated in the classical paper [8] by Mazur and Ulam. They proved that every isometry f of a normed real vector space X onto another normed real vector space X is a linear mapping up to translation, that is, x → f (x) − f (0) is linear, which amounts to the definition that f is affine. We call this the Mazur-Ulam theorem. The property is not true for normed complex vector spaces. The hypothesis of surjectivity is essential. Without this assumption, Baker [3] proved that every isometry from a normed real space into a strictly convex normed real space is linear up to translation. A number of mathematicians have had deal with the Mazur-Ulam theorem; see [7, 9] and references therein. In this paper, we prove that the Mazur-Ulam theorem holds under some conditions in the fuzzy anti-normed spaces. We establish a Mazur-Ulam type theorem in the framework of strictly convex normed spaces by using some ideas of [5]. Now we recall some notations and definitions used in this paper. Definition 1.1 [6] Let X be a linear space over a real field F . A fuzzy subset N of X ×R is called a fuzzy anti-norm on X if the following conditions are satisfied for all x, y ∈ X (a − N 1 ) For all t ∈R with t ≤ 0,N (x, t)=1, (a − N 2 ) For all t ∈R with t> 0,N (x, t)=0 if and only if x = ¯ 0, (a − N 3 ) For all t ∈R with t> 0,N (αx, t)= N (x, t/|α|), for all α =0,α ∈ F , (a − N 4 ) For all s, t ∈R,N (x + y,t + s) ≤ max{N (x, s),N (y,t)}, (a − N 5 ) N (x, t) is a non-increasing function of t ∈R and lim t→∞ N (x, t)=0. Then the pair (X, N ) is called a fuzzy anti-normed linear space.