Journal of Applied Mathematics
Volume 2012 (2012), Article ID 546819, 28 pages
http://dx.doi.org/10.1155/2012/546819
Research Article

Stabilities of Cubic Mappings in Various Normed Spaces: Direct and Fixed Point Methods

H. Azadi Kenary,1 H. Rezaei,1 S. Talebzadeh,2 and S. Jin Lee3

1Department of Mathematics, College of Sciences, Yasouj University, Yasouj 75914-353, Iran
2Department of Mathematics, Islamic Azad University, Firoozabad Branch, Firoozabad, Iran
3Department of Mathematics, Daejin University, Kyeonggi 487-711, Republic of Korea

Received 13 September 2011; Revised 5 November 2011; Accepted 6 November 2011

Academic Editor: Hui-Shen Shen

Copyright © 2012 H. Azadi Kenary et al. 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

In 1940 and 1964, Ulam proposed the general problem: “When is it true that by changing a little the hypotheses of a theorem one can still assert that the thesis of the theorem remains true or approximately true?”. In 1941, Hyers solved this stability problem for linear mappings. According to Gruber (1978) this kind of stability problems are of the particular interest in probability theory and in the case of functional equations of different types. In 1981, Skof was the first author to solve the Ulam problem for quadratic mappings. In 1982–2011, J. M. Rassias solved the above Ulam problem for linear and nonlinear mappings and established analogous stability problems even on restricted domains. The purpose of this paper is the generalized Hyers-Ulam stability for the following cubic functional equation: 𝑓 ( 𝑚 𝑥 + 𝑦 ) + 𝑓 ( 𝑚 𝑥 − 𝑦 ) = 𝑚 𝑓 ( 𝑥 + 𝑦 ) + 𝑚 𝑓 ( 𝑥 − 𝑦 ) + 2 ( 𝑚 3 − 𝑚 ) 𝑓 ( 𝑥 ) , 𝑚 ≥ 2 in various normed spaces.