Mediterranean Journal of Mathematics Generalized Hyers-Ulam Stability for a General Mixed Functional Equation in Quasi-β -normed Spaces G. Zamani Eskandani * , Pasc Gavruta, John M. Rassias and Ramazan Zarghami Abstract. In this paper, we establish the general solution and investigate the generalized Hyers-Ulam stability of the following mixed additive and quadratic functional equation f (λx + y)+ f (λx - y)= f (x + y)+ f (x - y)+(λ - 1)[(λ + 2)f (x) + λf (-x)], (λ ∈ N,λ = 1) in quasi-β-normed spaces. Mathematics Subject Classification (2010). 39B72, 39B82, 46B03, 47Jxx. Keywords. Generalized Hyers-Ulam stability, Contractively subadditive, Expansively superadditive, quasi-β-normed space, (β,p)-Banach space. 1. Introduction and preliminaries In 1940, S.M. Ulam [41] asked the first question on the stability problem. In 1941, D. H. Hyers [16] solved the problem of Ulam. This result was general- ized by Aoki [1] for additive mappings and independently by Th. M. Rassias [32] for linear mappings by considering an unbounded Cauchy difference. In 1994, a further generalization was obtained by P. G˘ avruta [14]. J.M. Rassias [27]-[30] treated the Ulam-Gavruta-Rassias stability on linear and nonlin- ear mappings and generalized Hyers result. During the last two decades, a number of papers and research monographs have been published on various generalizations and applications of the generalized Hyers-Ulam stability to a number of functional equations and mappings (see [9]-[13],[18]-[26],[33]-[35]). We also refer the readers to the books: P. Czerwik [7] and D.H. Hyers, G. Isac and Th.M. Rassias [17]. The functional equation f (x + y)+ f (x - y)=2f (x)+2f (y) (1.1) * Corresponding author. Mediterr. J. Math. 8 (2011), 331–348 DOI 10.1007/s00009-010-0082-8 1660-5446/11/030331-18, published online May 2, 2010 © 2010 Springer Basel AG