## FUZZY STABILITY OF ADDITIVE–QUADRATIC FUNCTIONAL EQUATIONS (2009)

### Cached

### Download Links

### BibTeX

@MISC{Gordji09fuzzystability,

author = {M. Eshaghi Gordji and N. Ghobadipour and J. M. Rassias},

title = {FUZZY STABILITY OF ADDITIVE–QUADRATIC FUNCTIONAL EQUATIONS},

year = {2009}

}

### OpenURL

### Abstract

In this paper we investigate the generalized Hyers- Ulam stability of the functional equation in fuzzy Banach spaces. f(2x + y) + f(2x − y) = f(x + y) + f(x − y) + 2f(2x) − 2f(x) in

### Citations

227 |
On the stability of the linear functional equation
- Hyers
- 1941
(Show Context)
Citation Context ...mate homomorphism? The concept of stability for functional equations arises when we replace the functional equation by an inequality which acts as a perturbation of the equation. In 1941, D. H. Hyers =-=[13]-=- gave the first affirmative answer to the question of Ulam for Banach spaces E and E ′ . Let f : E −→ E ′ be a mapping between Banach spaces such that ‖f(x + y) − f(x) − f(y)‖ ≤ δ for all x, y ∈ E, an... |

175 |
On the stability of the linear mapping in Banach spaces
- Rassias
- 1978
(Show Context)
Citation Context ...x ∈ E, and that there exist δ ≥ 0 and p ̸= 1 such that ‖f(x + y) − f(x) − f(y)‖ ≤ δ(‖x‖ p + ‖y‖ p ) 3 for all x, y ∈ E. Then there exists a unique linear map T : E → E ′ such that for all x ∈ E. (see =-=[27]-=-). ‖f(x) − T(x)‖ ≤ 2δ‖x‖p |2 p − 2| On the other hand J. M. Rassias [23, 24, 25, 26] generalized the Hyers stability result by presenting a weaker condition controlled by a product of different powers... |

113 | A generalization of the Hyers-Ulam-Rassias stability of approximately additive mappings - Găvruţa - 1994 |

86 |
Hyers-Ulam-Rassias Stability of Functional Equations
- Jung
- 2001
(Show Context)
Citation Context ...ility of functional equations. During the last decades several stability problems for various functional equations have been investigated by many mathematicians; we refer the reader to the monographs =-=[3, 5, 6, 7, 11, 12, 13, 16, 21, 24]-=-. The functional equation f(x+ y) + f(x− y) = 2f(x) + 2f(y), (1.1) is called the quadratic functional equation and every solution of the quadratic equation (1.1) is said to be a quadratic function. It... |

84 |
On stability of additive mappings
- Gajda
- 1991
(Show Context)
Citation Context ...ded (see also [20]). In 1990, Th.M. Rassias [20] during the 27th International Symposium on Functional Equations asked the question whether such a theorem can also be proved for p ≥ 1. In 1991, Gajda =-=[8]-=- gave an affirmative solution to this question for p > 1 by following the same approach as in Rassias’ paper [19]. It was proved by Gajda [8], as well as by Th.M. Rassias and Šemrl [22] that one cann... |

59 |
Remarks on the stability of functional equations
- CHOLEWA
- 1984
(Show Context)
Citation Context ...1.2) 4 A Hyers-Ulam stability problem for the quadratic functional equation (1.1) was proved by Skof for functions f : E1 −→ E2 where E1 is a normed space and E2 is a Banach space (see [29]). Cholewa =-=[8]-=- noticed that the theorem of Skof is still true if the relevant domain E1 is replaced by an Abelian group. In the paper [9], Czerwik proved the generalized HyersUlam stability of the quadratic functio... |

56 |
On the behavior of mappings which do not satisfy Hyers-Ulam stability
- Rassias, Šemrl
- 1992
(Show Context)
Citation Context ...In 1991, Gajda [8] gave an affirmative solution to this question for p > 1 by following the same approach as in Rassias’ paper [19]. It was proved by Gajda [8], as well as by Th.M. Rassias and Šemrl =-=[22]-=- that one cannot prove a Rassias type theorem when p = 1. In 1994, P. Găvruta [9] provided a generalization of Rassias’ theorem in which he replaced the bound ε(‖x‖p + ‖y‖p) in ([19]) by a general co... |

50 |
A Collection of
- Ulam
- 1960
(Show Context)
Citation Context ... is convergent, then the fuzzy norm is said to be complete and the fuzzy normed space is called a fuzzy Banach space. The study of the stability problem of functional equations was introduced by Ulam =-=[30]-=- Let (G1, .) be a group and let (G2, ∗) be a metric group with the metric d(., .). Given ǫ > 0, does there exist a δ > 0, such that if a mapping h : G1 −→ G2 satisfies the inequality d(h(x.y), h(x) ∗ ... |

46 |
On the stability of quadratic mapping in normed spaces
- CZERWIK
- 1992
(Show Context)
Citation Context ...→ E2 where E1 is a normed space and E2 is a Banach space (see [29]). Cholewa [8] noticed that the theorem of Skof is still true if the relevant domain E1 is replaced by an Abelian group. In the paper =-=[9]-=-, Czerwik proved the generalized HyersUlam stability of the quadratic functional equation (1.1). Grabiec [12] has generalized these results mentioned above. Jun and Lee [15] proved the generalized Hye... |

46 |
On approximation of approximately linearmappings by linearmappings
- Rassias
- 1982
(Show Context)
Citation Context ...− f(y)‖ ≤ δ(‖x‖ p + ‖y‖ p ) 3 for all x, y ∈ E. Then there exists a unique linear map T : E → E ′ such that for all x ∈ E. (see [27]). ‖f(x) − T(x)‖ ≤ 2δ‖x‖p |2 p − 2| On the other hand J. M. Rassias =-=[23, 24, 25, 26]-=- generalized the Hyers stability result by presenting a weaker condition controlled by a product of different powers of norms. According to J. M. Rassias Theorem: Theorem 1.4. If it is assumed that th... |

28 |
Functional Equations and Inequalities in Several Variables, World Scientific
- Czerwik
- 2002
(Show Context)
Citation Context ...ility of functional equations. During the last decades several stability problems for various functional equations have been investigated by many mathematicians; we refer the reader to the monographs =-=[3, 5, 6, 7, 11, 12, 13, 16, 21, 24]-=-. The functional equation f(x+ y) + f(x− y) = 2f(x) + 2f(y), (1.1) is called the quadratic functional equation and every solution of the quadratic equation (1.1) is said to be a quadratic function. It... |

25 |
Proprieta locali e approssimazione di operat6ri
- SKOF
- 1983
(Show Context)
Citation Context ...tive function B such that f(x) = B(x, x) for all x, where B(x, y) = 1 4 (f(x+ y)− f(x− y)), (1.2) (see [17]). The Hyers–Ulam stability problem for the quadratic functional equation was solved by Skof =-=[25]-=- and, independently, by Cholewa [4]. An analogous result for quadratic stochastic processes was obtained by Nikodem [18]. In [2], Czerwik proved the generalized Hyers–Ulam stability of the quadratic f... |

24 |
Quadratic functional equation and inner product spaces
- Kannappan
- 1995
(Show Context)
Citation Context ...isfies the important parallelogram equality ‖x + y‖ 2 + ‖x − y‖ 2 = 2(‖x‖ 2 + ‖y‖ 2 ). The functional equation f(x + y) + f(x − y) = 2f(x) + 2f(y) (1.1) is related to a symmetric bi-additive function =-=[1, 16]-=-. It is natural that each equation is called a quadratic functional equation. In particular, every solution of the quadratic equation (1.1) is said to be a quadratic function. It is well known that a ... |

20 |
On inner products in linear metric spaces
- Jordan, Neumann
- 1935
(Show Context)
Citation Context ..., such that for all x ∈ E. If in addition for every x ∈ E, f(tx) is continuous in t ∈ R for each fixed x, then T is linear. Quadratic functional equation was used to characterize inner product spaces =-=[1, 2, 14]-=-. Several other functional equations were also used to characterize inner product spaces. A square norm on an inner product space satisfies the important parallelogram equality ‖x + y‖ 2 + ‖x − y‖ 2 =... |

18 |
The generalized Hyers–Ulam stability of a class of functional equations
- Grabiec
- 1996
(Show Context)
Citation Context ... Skof is still true if the relevant domain E1 is replaced by an Abelian group. In the paper [9], Czerwik proved the generalized HyersUlam stability of the quadratic functional equation (1.1). Grabiec =-=[12]-=- has generalized these results mentioned above. Jun and Lee [15] proved the generalized Hyers-Ulam stability of the pexiderized quadratic equation (1.1). A. Najati and M.B. Moghimi [22], have obtained... |

18 |
On the Hyers–Ulam–Rassias stability of a pexiderized quadratic inequality
- Jun, Lee
- 2001
(Show Context)
Citation Context ...Abelian group. In the paper [9], Czerwik proved the generalized HyersUlam stability of the quadratic functional equation (1.1). Grabiec [12] has generalized these results mentioned above. Jun and Lee =-=[15]-=- proved the generalized Hyers-Ulam stability of the pexiderized quadratic equation (1.1). A. Najati and M.B. Moghimi [22], have obtained the generalized Hyers- Ulam stability for a functional equation... |

17 |
Fuzzy metric and statistical metric spaces, Kybernetika
- KRAMOSIL, J
(Show Context)
Citation Context ... [5, 10, 19, 28, 31]. In 1994, Cheng and Mordeson introduced a definition of fuzzy norm on a linear space in such a manner that the corresponding induced fuzzy metric is of Kramosil and Michalek type =-=[18]-=-. In 2003, Bag and Samanta [5] modified the definition of Cheng and Mordeson [7] by removing a regular condition. They also established a decomposition theorem of a fuzzy norm into a family of crisp n... |

17 |
On the Hyers-Ulam stability of ψ-additive mappings
- RASSIAS
- 1993
(Show Context)
Citation Context ...ility of functional equations. During the last decades several stability problems for various functional equations have been investigated by many mathematicians; we refer the reader to the monographs =-=[3, 5, 6, 7, 11, 12, 13, 16, 21, 24]-=-. The functional equation f(x+ y) + f(x− y) = 2f(x) + 2f(y), (1.1) is called the quadratic functional equation and every solution of the quadratic equation (1.1) is said to be a quadratic function. It... |

17 |
of the quadratic equation of Pexider type
- Jung, Stability
(Show Context)
Citation Context ...holewa [4]. An analogous result for quadratic stochastic processes was obtained by Nikodem [18]. In [2], Czerwik proved the generalized Hyers–Ulam stability of the quadratic functional equation. Jung =-=[15]-=- dealt with stability problems for the quadratic functional equation of Pexider type. Jun and Kim [14] introduced the following functional equation f(2x+ y) + f(2x− y) = 2f(x+ y) + 2f(x− y) + 12f(x), ... |

16 |
Rassias, “Solution of a problem of Ulam
- M
- 1989
(Show Context)
Citation Context ...− f(y)‖ ≤ δ(‖x‖ p + ‖y‖ p ) 3 for all x, y ∈ E. Then there exists a unique linear map T : E → E ′ such that for all x ∈ E. (see [27]). ‖f(x) − T(x)‖ ≤ 2δ‖x‖p |2 p − 2| On the other hand J. M. Rassias =-=[23, 24, 25, 26]-=- generalized the Hyers stability result by presenting a weaker condition controlled by a product of different powers of norms. According to J. M. Rassias Theorem: Theorem 1.4. If it is assumed that th... |

15 |
The generalized Hyers-Ulam-Rassias stability of a cubic functional equation
- Jun, Kim
- 2002
(Show Context)
Citation Context ...], Czerwik proved the generalized Hyers–Ulam stability of the quadratic functional equation. Jung [15] dealt with stability problems for the quadratic functional equation of Pexider type. Jun and Kim =-=[14]-=- introduced the following functional equation f(2x+ y) + f(2x− y) = 2f(x+ y) + 2f(x− y) + 12f(x), (1.3) and established the general solution and the generalized Hyers–Ulam–Rassias stability for functi... |

14 |
Characterizations of Inner Product Spaces, in
- Amir
- 1986
(Show Context)
Citation Context ..., such that for all x ∈ E. If in addition for every x ∈ E, f(tx) is continuous in t ∈ R for each fixed x, then T is linear. Quadratic functional equation was used to characterize inner product spaces =-=[1, 2, 14]-=-. Several other functional equations were also used to characterize inner product spaces. A square norm on an inner product space satisfies the important parallelogram equality ‖x + y‖ 2 + ‖x − y‖ 2 =... |

14 |
Local properties and approximations of operators
- SKOF
- 1983
(Show Context)
Citation Context ... − f(x − y)). (1.2) 4 A Hyers-Ulam stability problem for the quadratic functional equation (1.1) was proved by Skof for functions f : E1 −→ E2 where E1 is a normed space and E2 is a Banach space (see =-=[29]-=-). Cholewa [8] noticed that the theorem of Skof is still true if the relevant domain E1 is replaced by an Abelian group. In the paper [9], Czerwik proved the generalized HyersUlam stability of the qua... |

13 |
Stability of a functional equation deriving from quadratic and additive functions in quasi-Banach spaces
- Najati, Moghimi
(Show Context)
Citation Context ...(1.1). Grabiec [12] has generalized these results mentioned above. Jun and Lee [15] proved the generalized Hyers-Ulam stability of the pexiderized quadratic equation (1.1). A. Najati and M.B. Moghimi =-=[22]-=-, have obtained the generalized Hyers- Ulam stability for a functional equation deriving from quadratic and additive functions in quasi-Banach spaces. In this paper, we deal with the the following fun... |

11 |
On the stability of the linear transformationin Banach spaces
- Aoki
- 1950
(Show Context)
Citation Context ...the space is complete with respect to this metric. The Lp spaces are F−spaces for all p > 0 and for p = 1 they are locally convex and thus Frchet spaces and even Banach spaces. So for example, L 1 2 (=-=[0, 1]-=-) is a F−space, which is not a Banach space. 2. Main results We start our work with the following result, which explain the relation between additive–quadratic maps and cubic maps. Theorem 2.1. Let X,... |

9 |
Finite dimensional fuzzy normed linear spaces
- Bag, Samanta
(Show Context)
Citation Context ...on [7] by removing a regular condition. They also established a decomposition theorem of a fuzzy norm into a family of crisp norms and investigated some properties of fuzzy norms (see [4]). Following =-=[3]-=-, we give the employing notion of a fuzzy norm. Let X be a real linear space. A function N : X × R −→ [0, 1](the so-called fuzzy subset) is said to be a fuzzy norm on X if for all x, y ∈ X and all a, ... |

8 |
Solution and stability of generalized mixed type cubic, quadratic and additive functional equation in quasi-Banach spaces
- Gordji, Khodaei
(Show Context)
Citation Context |

7 | Fuzzy versions of HyersUlamRassias theorem - Mirmostafee, Moslehian |

6 |
Fuzzy bounded linear operators, Fuzzy Sets and Systems
- Bag, Samanta
(Show Context)
Citation Context ...Cheng and Mordeson [7] by removing a regular condition. They also established a decomposition theorem of a fuzzy norm into a family of crisp norms and investigated some properties of fuzzy norms (see =-=[4]-=-). Following [3], we give the employing notion of a fuzzy norm. Let X be a real linear space. A function N : X × R −→ [0, 1](the so-called fuzzy subset) is said to be a fuzzy norm on X if for all x, y... |

6 |
Finite dimensional fuzzy normed linear space
- Felbin
- 1999
(Show Context)
Citation Context ...space. In [6], Biswas defined and studied fuzzy inner product spaces in linear space. Since then some mathematicians have defined fuzzy metrics and norms on a linear space from various points of view =-=[5, 10, 19, 28, 31]-=-. In 1994, Cheng and Mordeson introduced a definition of fuzzy norm on a linear space in such a manner that the corresponding induced fuzzy metric is of Kramosil and Michalek type [18]. In 2003, Bag a... |

6 |
Fuzzy topological vector spaces II, Fuzzy Sets and Systems 12 282
- Katsaras
- 1984
(Show Context)
Citation Context ...e generalized Hyers- Ulam stability of the functional equation in fuzzy Banach spaces. f(2x + y) + f(2x − y) = f(x + y) + f(x − y) + 2f(2x) − 2f(x) 1. Introduction and preliminaries In 1984, Katsaras =-=[17]-=- defined a fuzzy norm on a linear space and at the same year Wu and Fang [32] also introduced a notion of fuzzy normed space and gave the generalization of the Kolmogoroff normalized theorem for fuzzy... |

6 |
Seperation of fuzzy normed linear spaces, Fuzzy Sets and Systems
- Krishna, Sarma
- 1994
(Show Context)
Citation Context ...space. In [6], Biswas defined and studied fuzzy inner product spaces in linear space. Since then some mathematicians have defined fuzzy metrics and norms on a linear space from various points of view =-=[5, 10, 19, 28, 31]-=-. In 1994, Cheng and Mordeson introduced a definition of fuzzy norm on a linear space in such a manner that the corresponding induced fuzzy metric is of Kramosil and Michalek type [18]. In 2003, Bag a... |

5 |
Fuzzy inner product spaces and fuzzy normed functions
- Biswas
- 1991
(Show Context)
Citation Context ...ear space and at the same year Wu and Fang [32] also introduced a notion of fuzzy normed space and gave the generalization of the Kolmogoroff normalized theorem for fuzzy topological linear space. In =-=[6]-=-, Biswas defined and studied fuzzy inner product spaces in linear space. Since then some mathematicians have defined fuzzy metrics and norms on a linear space from various points of view [5, 10, 19, 2... |

5 |
Fuzzy linear operators and fuzzy normed linear spaces
- Cheng, Mordeson
- 1994
(Show Context)
Citation Context ...y norm on a linear space in such a manner that the corresponding induced fuzzy metric is of Kramosil and Michalek type [18]. In 2003, Bag and Samanta [5] modified the definition of Cheng and Mordeson =-=[7]-=- by removing a regular condition. They also established a decomposition theorem of a fuzzy norm into a family of crisp norms and investigated some properties of fuzzy norms (see [4]). Following [3], w... |

5 |
Infinite Fuzzy Relation Equations with Continuous t-norms
- Shieh
- 2008
(Show Context)
Citation Context ...space. In [6], Biswas defined and studied fuzzy inner product spaces in linear space. Since then some mathematicians have defined fuzzy metrics and norms on a linear space from various points of view =-=[5, 10, 19, 28, 31]-=-. In 1994, Cheng and Mordeson introduced a definition of fuzzy norm on a linear space in such a manner that the corresponding induced fuzzy metric is of Kramosil and Michalek type [18]. In 2003, Bag a... |

5 |
Fuzzy normed space of operators and its completeness, Fuzzy Sets and Systems 133
- Xiao, Zhu
- 2003
(Show Context)
Citation Context |

5 |
Fuzzy generalization of klomogoroffs theorem
- Congxin, Fang
- 1984
(Show Context)
Citation Context ...h spaces. f(2x + y) + f(2x − y) = f(x + y) + f(x − y) + 2f(2x) − 2f(x) 1. Introduction and preliminaries In 1984, Katsaras [17] defined a fuzzy norm on a linear space and at the same year Wu and Fang =-=[32]-=- also introduced a notion of fuzzy normed space and gave the generalization of the Kolmogoroff normalized theorem for fuzzy topological linear space. In [6], Biswas defined and studied fuzzy inner pro... |

5 |
Stability of a functional equation deriving from cubic and quartic functions, Abstract and Applied Analysis Volume 2008
- Gordji, Ebadian, et al.
(Show Context)
Citation Context |

4 |
Distance and Similarity measures for fuzzy operators
- Balopoulos, Hatzimichailidis, et al.
(Show Context)
Citation Context |

2 |
On some properties of quadratic stochastic processes
- Nikodem
- 1990
(Show Context)
Citation Context ...Ulam stability problem for the quadratic functional equation was solved by Skof [25] and, independently, by Cholewa [4]. An analogous result for quadratic stochastic processes was obtained by Nikodem =-=[18]-=-. In [2], Czerwik proved the generalized Hyers–Ulam stability of the quadratic functional equation. Jung [15] dealt with stability problems for the quadratic functional equation of Pexider type. Jun a... |

2 |
A generalized cubic functional equation, Acta
- Sahoo
(Show Context)
Citation Context ... AN ADDITIVE-QUADRATIC FUNCTIONAL EQUATION 253 fixed one variable and is additive for fixed two variables. Later a number of mathematicians worked on the stability of some types of the cubic equation =-=[23]-=-. Let X, Y and Z be vector spaces on R or C. We say that a mapping f : X × Y → Z is additive–quadratic if f satisfies the following system of functional equations: f(x1 + x2, y) = f(x1, y) + f(x2... |

1 | Fuzzy almost quadratic functions, Results Math - Mirmostafaee, Moslehian |

1 |
Stability of a mixed type additive, quadratic, cubic and quartic functional equation
- Eshaghi, Moslehian, et al.
(Show Context)
Citation Context |

1 |
A generalization of the Hyers–Ulam–Rassias stability of approximately additive mappings
- Guavruta
- 1994
(Show Context)
Citation Context ...wing the same approach as in Rassias’ paper [19]. It was proved by Gajda [8], as well as by Th.M. Rassias and Šemrl [22] that one cannot prove a Rassias type theorem when p = 1. In 1994, P. Găvruta =-=[9]-=- provided a generalization of Rassias’ theorem in which he replaced the bound ε(‖x‖p + ‖y‖p) in ([19]) by a general control function ϕ(x, y). The paper of Th.M. Rassias [19] has provided a lot of infl... |

1 |
Functional inequalities for approximately additive mappings, Stability of mappings of Hyers-Ulam type
- Isac, Rassias
- 1994
(Show Context)
Citation Context |

1 | Intuitionistic fuzzy stability of Jensen type mapping
- Shakeri
(Show Context)
Citation Context |

1 |
E-mail address: madjid.eshaghi@gmail.com
- Ulam
- 1964
(Show Context)
Citation Context ...1, y) + f(x2, y), f(x, y1 + y2) + f(x, y1 − y2) = 2f(x, y1) + 2f(x, y2) in F-spaces. 1. Introduction and preliminaries The stability problem of functional equations originated from a question of Ulam =-=[26]-=- in 1940, concerning the stability of group homomorphisms. Let (G1, .) be a group and let (G2, ∗) be a metric group with the metric d(., .). Given > 0, dose there exist a δ > 0, such that if a mappi... |