Intuitionistic random almost additive-quadratic mappings
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RESEARCH
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Intuitionistic random almost additive-quadratic mappings Choonkil Park1 , Madjid Eshaghi Gordji2* , Masumeh Ghanifard2 and Hamid Khodaei2 *
Correspondence: [email protected] 2 Department of Mathematics, Semnan University, P.O. Box 35195-363, Semnan, Iran Full list of author information is available at the end of the article
Abstract In this paper, we investigate the Hyers-Ulam stability of the additive-quadratic functional equation ni=1 f (xi – n1 nj=1 xj ) = ni=1 f (xi ) – nf ( n1 ni=1 xi ) (n ≥ 2) in intuitionistic random normed spaces. MSC: Primary 39B52; 34K36; 46S50; 47S50; 34Fxx Keywords: mixed functional equation; intuitionistic random normed space; Hyers-Ulam stability
1 Introduction The concept of stability of a functional equation arises when one replaces a functional equation by an inequality which acts as a perturbation of the equation. The first stability problem concerning group homomorphisms was raised by Ulam [] in and affirmatively solved by Hyers []. The result of Hyers was generalized by Aoki [] for approximate additive mappings and by Rassias [] for approximate linear mappings by allowing the difference Cauchy equation f (x + y) – f (x) – f (y) to be controlled by ε(xp + yp ). In , a generalization of the Th.M. Rassias’ theorem was obtained by Gˇavruta [], who replaced ε(xp + yp ) by a general control function ϕ(x, y). For more details about the results concerning such problems, the reader is referred to [–]. The functional equation f (x + y) + f (x – y) = f (x) + f (y)
(.)
is related to a symmetric bi-additive mapping [, ]. It is natural that this equation is called a quadratic functional equation. In particular, every solution of the quadratic equation (.) is said to be a quadratic mapping. It is well known that a mapping f between real vector spaces is quadratic if and only if there exists a unique symmetric bi-additive mapping B such that f (x) = B (x, x) for all x. The bi-additive mapping B is given by B (x, y) = (f (x + y) – f (x – y)). The Hyers-Ulam stability problem for the quadratic functional equation was solved by Skof []. In [], Czerwik proved the Hyers-Ulam stability of the function equation (.). Eshaghi Gordji and Khodaei [] have established the general solution and investigated the Hyers-Ulam stability for a mixed type of cubic, quadratic and additive functional equation f (x + ky) + f (x – ky) = k f (x + y) + k f (x – y) + – k f (x)
(.)
© 2012 Park et al.; licensee Springer. This is an Open Access article distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/2.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.
Park et al. Advances in Difference Equations 2012, 2012:152 http://www.advancesindifferenceequations.com/content/2012/1/152
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in quasi-Banach spaces, where k is a nonzero integer with k = ±. Obviously, the function f (x
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