Abstract:
In this paper, we prove generalized Hyres--Ulam--Rassias stability of the mixed type additive, quadratic and cubic functional equation $$f(x+ky)+f(x-ky)=k^2f(x+y)+k^2f(x-y)+2(1-k^2)f(x)$$ for fixed integers $k$ with $k\neq0,\pm1$ in non-Archimedean spaces.

Abstract:
In this paper, we achieve the general solution and the generalized Hyers-Ulam-Rassias stability of the following functional equation $$f(x+ky)+f(x-ky)=k^2f(x+y)+k^2f(x-y)+2(1-k^2)f(x)\eqno {2 cm}$$for fixed integers $k$ with $k\neq0,\pm1$ in the quasi-Banach spaces.

Abstract:
In this paper, we prove the generalized Hyers-Ulam stability of generalized mixed type cubic, quadratic, and additive functional equation, in fuzzy Banach spaces. 2010 Mathematics Subject Classification: 39B82; 39B52.

Abstract:
The objective of the present paper is to determine the generalized Hyers-Ulam stability of the mixed additive-cubic functional equation in n-Banach spaces by the direct method. In addition, we show under some suitable conditions that an approximately mixed additive-cubic function can be approximated by a mixed additive and cubic mapping.

Abstract:
In this paper, we consider the additive-cubic-quartic functional equation and prove the generalized Hyers-Ulam stability of the additive-cubic-quartic functional equation in Banach spaces.

Abstract:
We prove generalized Hyres-Ulam-Rassias stability of the cubic functional equation $f(kx+y)+f(kx-y)=k[f(x+y)+f(x-y)]+2(k^3-k)f(x)$ for all $k\in \Bbb N$ and the quartic functional equation $f(kx+y)+f(kx-y)=k^2[f(x+y)+f(x-y)]+2k^2(k^2-1)f(x)-2(k^2-1)f(y)$ for all $k\in \Bbb N$ in non-Archimedean normed spaces.

Abstract:
We study the following generalized additive functional inequality , associated with linear mappings in Banach spaces. Moreover, we prove the Hyers-Ulam-Rassias stability of the above generalized additive functional inequality, associated with linear mappings in Banach spaces.

Abstract:
We prove the Hyers-Ulam stability of the additive-cubic-quartic functional equation in multi-Banach spaces by using the fixed point alternative method. The first results on the stability in the multi-Banach spaces were presented in (Dales and Moslehian 2007). 1. Introduction Stability is investigated when one is asking whether a small error of parameters in one problem causes a large deviation of its solution. Given an approximate homomorphism, is it possible to approximate it by a true homomorphism? In other words, we are looking for situations when the homomorphisms are stable, that is, if a mapping is almost a homomorphism, then there exists a true homomorphism near it with small error as much as possible. This problem was posed by Ulam in 1940 (cf. [1]) and is called the stability of functional equations. For Banach spaces, the problem was solved by Hyers [2] in the case of approximately additive mappings. Later, Hyers' result was generalized by Aoki [3] for additive mappings and by Rassias [4] for linear mappings by allowing the Cauchy difference to be unbounded. During the last decade, stability of functional equations was studied by several mathematicians for mappings in various spaces including random normed spaces and fuzzy Banach spaces (cf. [5, 6]). For various other results on the stability of functional equations, one is referred to [7–26]. Most of the proofs of stability theorems in the Hyers-Ulam context have applied Hyers’ direct method. The exact solution of the functional equation is explicitly constructed as the limit of a sequence, which is originating from the given approximate solution. In 2003, Radu [27] proposed the fixed point alternative method for obtaining the existence of exact solutions and error estimations and noticed that a fixed point alternative method is essential for the solution of Ulam problem for approximate homomorphisms. Subsequently, some authors [28, 29] applied the fixed alternative method to investigate the stability problems of several functional equations. The notion of multi-normed space was introduced by Dales and Polyakov [30] (or see [31, 32]). This concept is somewhat similar to operator sequence space and has some connections with operator spaces and Banach lattices. Motivations for the study of multi-normed spaces and many examples were given in [30, 31]. In 2007, stability of mappings on multi-normed spaces was first given in [31], and asymptotic aspect of the quadratic functional equation in multi-normed spaces was investigated in [33]. In this paper, we consider the following functional equation

Abstract:
We study the following generalized additive functional inequality ￠ € –af(x)+bf(y)+cf(z) ￠ € – ￠ ‰ ¤ ￠ € –f( ±x+ 2y+ 3z) ￠ € –, associated with linear mappings in Banach spaces. Moreover, we prove the Hyers-Ulam-Rassias stability of the above generalized additive functional inequality, associated with linear mappings in Banach spaces.