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Stability of generalized mixed type additive-quadratic-cubic functional equation in non-Archimedean spaces  [PDF]
M. Eshaghi Gordji,M. Bavand Savadkouhi,Th. M. Rassias
Mathematics , 2009,
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.
Solution and stability of generalized mixed type cubic, quadratic and additive functional equation in quasi-Banach spaces  [PDF]
M. Eshaghi Gordji,H. Khodaei
Mathematics , 2008,
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.
Fuzzy Stability of Generalized Mixed Type Cubic, Quadratic, and Additive Functional Equation  [cached]
Gordji Madjid,Kamyar Mahdie,Khodaei Hamid,Shin Dong
Journal of Inequalities and Applications , 2011,
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.
On the Hyers-Ulam Stability of a General Mixed Additive and Cubic Functional Equation in n-Banach Spaces
Tian Zhou Xu,John Michael Rassias
Abstract and Applied Analysis , 2012, DOI: 10.1155/2012/926390
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.
Stability of an Additive-Cubic-Quartic Functional Equation  [cached]
Eshaghi-Gordji M,Kaboli-Gharetapeh S,Park Choonkil,Zolfaghari Somayyeh
Advances in Difference Equations , 2009,
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.
Stability of cubic and quartic functional equations in non-Archimedean spaces  [PDF]
M. Eshaghi Gordji,M. Bavand Savadkouhi
Mathematics , 2008,
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.
On the Stability of Generalized Additive Functional Inequalities in Banach Spaces
Lee JungRye,Park Choonkil,Shin DongYun
Journal of Inequalities and Applications , 2008,
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.
Stability of an Additive-Cubic-Quartic Functional Equation in Multi-Banach Spaces  [PDF]
Zhihua Wang,Xiaopei Li,Themistocles M. Rassias
Abstract and Applied Analysis , 2011, DOI: 10.1155/2011/536520
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
On the Stability of a General Mixed Additive-Cubic Functional Equation in Random Normed Spaces  [cached]
Xu TianZhou,Rassias JohnMichael,Xu WanXin
Journal of Inequalities and Applications , 2010,
Abstract: We prove the generalized Hyers-Ulam stability of the following additive-cubic equation in the setting of random normed spaces.
On the Stability of Generalized Additive Functional Inequalities in Banach Spaces
Jung Rye Lee,Choonkil Park,Dong Yun Shin
Journal of Inequalities and Applications , 2008, DOI: 10.1155/2008/210626
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.
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