IJSRSET15145 | Received: 11 July 2015 | Accepted: 18 July 2015 | July-August 2015 [(1)4: 59-77] © 2015 IJSRSET | Volume 1 | Issue 4 | Print ISSN : 2395-1990 | Online ISSN : 2394-4099 Themed Section: Science 59 Ulam-Hyers Stability of Additive and Reciprocal Functional Equations: Direct and Fixed Point Methods M. Arunkumar 1 , A. Vijayakumar 2 , S. Karthikeyan 3 1 Department of Mathematics, Government Arts College, Tiruvannamalai, Tamilnadu, India 2,3 Department of Mathematics, R.M.K. Engineering College, Kavarapettai, Tamilnadu, India ABSTRACT In this paper, the authors established the generalized Ulam - Hyers stability of additive functional equation     1 () 2 n l l l f x ly f x ly fx n               which is originating from arithmetic mean of n consecutive terms of an arithmetic progression in Intuitionistic fuzzy normed spaces and reciprocal functional equation 1 ( )( ) 2 ( ) ( ) n l l l l l hx ly h x ly x h n hx ly hx ly                     originating from n-consecutive terms of a harmonic progression in Non - Archimedean Fuzzy 2    normed spaces using direct and fixed point methods. Applications of the above functional equations are also given. Keywords: Additive functional equation, Reciprocal functional equation, generalized Ulam-Hyers stability, Intuitionistic fuzzy normed spaces, Non - Archimedean Fuzzy 2    normed spaces, fixed point method. 2010 hematics Subject Classification: 39B52, 32B72, 32B82. 1. INTRODUCTION In 1940, S.M. Ulam [47] introduced the stability of functional equations. Next year 1941, D. H. Hyers [16] gave first confirmatory answer to the Ulam question for Banach spaces. In 1978, Hyers theorem was generalized by Th.M. Rassias [37]. Gajda [12] answered the question for the case 1 p  in the year 1991, which was raised by Rassias. This stability results is known as generalized Hyers-Ulam stability of functional equations (see [1, 2, 14, 20, 22, 26, 38]). During the years 1982–1994, Rassias [32-36] investigated the Ulam stability problem for different mappings involving a product of different powers of norms. Recently, Rassias gave the mixed product sum of powers of norms control function [39]. We also refer the readers to the books: P. Czerwik [7] and D.H. Hyers, G. Isac and Th.M. Rassias [17]. In 2003, V. Radu [31] introduced a new method, successively developed in [8-10], to obtaining the existence of the exact solutions and the error estimations, based on the fixed point alternative. The stability of several functional equations have been extensively investigated by a number of mathematicians and there are many interesting results concerning this problem (see [3, 4, 21, 23-25, 40, 41]). In this paper, the authors proved the generalized Ulam - Hyers stability of an additive functional equation     1 () 2 n l l l f x ly f x ly fx n               (1.1) which is originating from arithmetic mean of n consecutive terms of an arithmetic progression in Intuitionistic fuzzy normed spaces, and reciprocal functional equation