## quartic functional equation (812)

### BibTeX

@MISC{Gordji812quarticfunctional,

author = {M. Eshaghi Gordji and S. Abbaszadeh},

title = {quartic functional equation},

year = {812}

}

### OpenURL

### Abstract

the stability of generalized mixed type quadratic and

### Citations

197 |
Geometric Nonlinear Functional Analysis American Mathematical Society Colloquim publications, Vol. 48. Alex Eskin and David Fisher [BCST
- Binyamini, Lindenstrauss
- 1992
(Show Context)
Citation Context ...ility of this functional equation whenever f is a function between two quasi-Banach spaces. We recall some basic facts concerning quasi-Banach space and some preliminary results. Definition 1.1. (See =-=[3, 18]-=-.) Let X be a real linear space. A quasi-norm is a real-valued function on X satisfying the following: (1) ‖x‖ ≥ 0 for all x ∈ X and ‖x‖ = 0 if and only if x = 0 . (2) ‖λ.x‖ = |λ|.‖x‖ for all λ ∈ R an... |

66 |
A generalization of the Hyers–Ulam–Rassias stability of approximately additive mappings
- Găvruta
- 1994
(Show Context)
Citation Context ... now on, let X and Y be a quasi-Banach space with quasi-norm ‖.‖X and a pBanach space with p-norm ‖.‖Y ,respectively.Let M be the modulus of concavity of ‖.‖Y .In this section using an idea of Gǎvruta=-=[8]-=- we prove the stability of Eq.(1.5) in the spirit of Hyers, Ulam and Rassias.For convenience we use the following abbreviation for a given function f : X −→ Y : △f(x, y) = f(nx+y)+f(nx−y)−n 2 f(x+y)−n... |

44 |
On stability of additive mappings
- Gajda
- 1991
(Show Context)
Citation Context ...t to some p-norm. Since it is much easier to work with p-norms, henceforth we restrict our attention mainly to p-norms. In [20], J. Tabor has investigated a version of Hyers-Rassias-Gajda theorem (see=-=[7,17]-=-) in quasi-Banach spaces. 2. General solution Throughout this section, X and Y will be real vector spaces. We here present the general solution of (1.5). Lemma 2.1. If a function f : X −→ Y satisfies ... |

19 |
Quadratic functional equation and inner product spaces
- Kannappan
- 1995
(Show Context)
Citation Context ...d a generalization of Hyers’ Theorem which allows the Cauchy difference to be unbounded. The functional equation f(x + y) + f(x − y) = 2f(x) + 2f(y), (1.1) is related to symmetric bi-additive function=-=[1,2,11,13]-=-. It is natural that this 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 ... |

16 |
The generalized Hyers-Ulam stability of a class of functional equations
- Grabiec
- 1996
(Show Context)
Citation Context ...[4] noticed that the Theorem of Skof is still true if relevant domain A is replaced an abelian group. In the paper [6] , Czerwik proved the Hyers-Ulam-Rassias stability of the equation (1.1). Grabiec =-=[9]-=- has generalized these result mentioned above. In [14], Won-Gil Prak and Jea Hyeong Bae, considered the following quartic functional equation: f(x + 2y) + f(x − 2y) = 4(f(x + y) + f(x − y) + 6f(y)) − ... |

16 |
On inner products in linear metric spaces
- Jordan, Neumann
- 1935
(Show Context)
Citation Context ...d a generalization of Hyers’ Theorem which allows the Cauchy difference to be unbounded. The functional equation f(x + y) + f(x − y) = 2f(x) + 2f(y), (1.1) is related to symmetric bi-additive function=-=[1,2,11,13]-=-. It is natural that this 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 ... |

13 |
Characterizations of inner product spaces, Operator Theory
- Amir
- 1986
(Show Context)
Citation Context ...d a generalization of Hyers’ Theorem which allows the Cauchy difference to be unbounded. The functional equation f(x + y) + f(x − y) = 2f(x) + 2f(y), (1.1) is related to symmetric bi-additive function=-=[1,2,11,13]-=-. It is natural that this 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 ... |

8 |
Moghimi, Stability of a functional equation deriving from quadratic and additive functions in quasi-Banach spaces
- Najati, B
(Show Context)
Citation Context ...tion for a given function f : X −→ Y : △f(x, y) = f(nx+y)+f(nx−y)−n 2 f(x+y)−n 2 f(x−y)−2f(nx)+2n 2 f(x)+2(n 2 −1)f(y) for all x,y ∈ X.We will use the following lemma in this section. Lemma 3.1. (see =-=[15]-=-.) Let 0 < p ≤ 1 and let x1, x2, . . . , xn be non-negative real numbers. Then nX ( xi) p nX ≤ i=1 i=1 xi p . Theorem 3.2. Let ϕq : X × X → [0, ∞) be a function such that for all x,y ∈ X and lim m→∞ 4... |

8 |
Propriet locali e approssimazione di operatori
- Skof
- 1983
(Show Context)
Citation Context ...y) − f(x − y)). (1.2) 4 A Hyers-Ulam-Rassias stability problem for the quadratic functional equation (1.1) was proved by Skof for functions f : A −→ B, where A is normed space and B Banach space (see =-=[19]-=-). Cholewa [4] noticed that the Theorem of Skof is still true if relevant domain A is replaced an abelian group. In the paper [6] , Czerwik proved the Hyers-Ulam-Rassias stability of the equation (1.1... |

6 |
On a bi-quadratic functional equation and its stability, Nonlinear Analysis 62(4
- Park, Bae
- 2005
(Show Context)
Citation Context ... relevant domain A is replaced an abelian group. In the paper [6] , Czerwik proved the Hyers-Ulam-Rassias stability of the equation (1.1). Grabiec [9] has generalized these result mentioned above. In =-=[14]-=-, Won-Gil Prak and Jea Hyeong Bae, considered the following quartic functional equation: f(x + 2y) + f(x − 2y) = 4(f(x + y) + f(x − y) + 6f(y)) − 6f(x). (1.3) In fact, they proved that a function f be... |

5 |
ZAMANI ESKANDANI, Stability of a mixed additive and cubic functional equation in quasi-Banach spaces
- NAJATI, G
(Show Context)
Citation Context ...2 n−2 1≤i<j≤n xj [ ] f(xi)] (1.4) for all n-variables x1, x2, ..., xn ∈ E1, where n > 2 and f : E1 −→ E2 be a function between two real linear spaces E1 and E2. Also A. Najati and G. Zamani Eskandani =-=[16]-=-, have established the general solution and the generalized Hyers-Ulam-Rassias stability for a mixed type of cubic and additive functional equation, whenever f is a mapping between two quasi-Banach sp... |

4 |
On the stability of the linear mapping in Banach spaces,Proc
- Rassias
- 1978
(Show Context)
Citation Context ...en there exists a unique additive mapping T : E −→ E ′ such that ‖f(x) − T(x)‖ ≤ δ for all x ∈ E. Moreover, if f(tx) is continuous in t for each fixed x ∈ E, then T is linear. In 1978, Th. M. Rassias =-=[17]-=- provided a generalization of Hyers’ Theorem which allows the Cauchy difference to be unbounded. The functional equation f(x + y) + f(x − y) = 2f(x) + 2f(y), (1.1) is related to symmetric bi-additive ... |

4 |
Metric Linear Spaces,PWN-Polish Sci
- Rolewicz
- 1984
(Show Context)
Citation Context ...ility of this functional equation whenever f is a function between two quasi-Banach spaces. We recall some basic facts concerning quasi-Banach space and some preliminary results. Definition 1.1. (See =-=[3, 18]-=-.) Let X be a real linear space. A quasi-norm is a real-valued function on X satisfying the following: (1) ‖x‖ ≥ 0 for all x ∈ X and ‖x‖ = 0 if and only if x = 0 . (2) ‖λ.x‖ = |λ|.‖x‖ for all λ ∈ R an... |

3 |
Functional Equations in Several Variables.—Cambridge etc.: Cambridge Univ
- Aczél, Dhombres
- 1989
(Show Context)
Citation Context |

3 |
Remarks on the stability of functional equations,Aequationes Math
- Cholewa
- 1984
(Show Context)
Citation Context ... (1.2) 4 A Hyers-Ulam-Rassias stability problem for the quadratic functional equation (1.1) was proved by Skof for functions f : A −→ B, where A is normed space and B Banach space (see [19]). Cholewa =-=[4]-=- noticed that the Theorem of Skof is still true if relevant domain A is replaced an abelian group. In the paper [6] , Czerwik proved the Hyers-Ulam-Rassias stability of the equation (1.1). Grabiec [9]... |

3 |
On the general solution of a quartic functional equation,Bull
- Chung, Sahoo
(Show Context)
Citation Context ... × X × X −→ Y such that f(x) = D(x, x, x,x) for all x. It is easy to show that the function f(x) = x 4 satisfies the functional equation (1.4), which is called a quartic functional equation (see also =-=[5]-=-). In addition H. Kim [12], has obtained the generalized Hyers-Ulam-Rassias stability for the following mixed type of quartic and quadratic functional equation: nX X n−1 ] x2,...,xn f(x1) + 2 n−1 (n −... |

3 |
On the stability of the linear functional equation,Proc
- Hyers
- 1941
(Show Context)
Citation Context ...ximate homomorphism? The concept of stability for functional equation arises when we replace the functional equation by an inequality which acts as a perturbation of the equation. In 1941, D.H. Hyers =-=[10]-=- gave a first affirmative answer to the question of Ulam for Banach spaces. 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, and for some δ >... |

3 |
On the stability problem for a mixed type of quartic and quadratic functional equation,J
- Kim
(Show Context)
Citation Context ...(x) = D(x, x, x,x) for all x. It is easy to show that the function f(x) = x 4 satisfies the functional equation (1.4), which is called a quartic functional equation (see also [5]). In addition H. Kim =-=[12]-=-, has obtained the generalized Hyers-Ulam-Rassias stability for the following mixed type of quartic and quadratic functional equation: nX X n−1 ] x2,...,xn f(x1) + 2 n−1 (n − 2) i=1 f(xi) = 2 n−2 1≤i<... |

2 |
On the stability of the quadratic mapping in normed spaces,Abh
- Czerwik
- 1992
(Show Context)
Citation Context ...functions f : A −→ B, where A is normed space and B Banach space (see [19]). Cholewa [4] noticed that the Theorem of Skof is still true if relevant domain A is replaced an abelian group. In the paper =-=[6]-=- , Czerwik proved the Hyers-Ulam-Rassias stability of the equation (1.1). Grabiec [9] has generalized these result mentioned above. In [14], Won-Gil Prak and Jea Hyeong Bae, considered the following q... |

2 |
stability of the Cauchy functional equation in quasi-Banach spaces
- Tabor
(Show Context)
Citation Context ...y the Aoki-Rolewicz Theorem [ 18](see also [3]), each quasi-norm is equivalent to some p-norm. Since it is much easier to work with p-norms, henceforth we restrict our attention mainly to p-norms. In =-=[20]-=-, J. Tabor has investigated a version of Hyers-Rassias-Gajda theorem (see[7,17]) in quasi-Banach spaces. 2. General solution Throughout this section, X and Y will be real vector spaces. We here presen... |