On the Stability of Quadratic Functional Equations inF-Spaces

Quadratic Functional ◽

The Stability ◽

The Hyers-Ulam-Rassias stability of quadratic functional equationf(2x+y)+f(2x-y)=f(x+y)+f(x-y)+6f(x)and orthogonal stability of the Pexiderized quadratic functional equationf(x+y)+f(x-y)=2g(x)+2h(y)inF-spaces are proved.

On the Stability of Quadratic Functional Equations

Abstract and Applied Analysis ◽

10.1155/2008/628178 ◽

2008 ◽

Vol 2008 ◽

pp. 1-8 ◽

Cited By ~ 9

Author(s):

Jung Rye Lee ◽

Jong Su An ◽

Choonkil Park

Keyword(s):

Functional Equation ◽

Positive Integer ◽

Banach Spaces ◽

Functional Equations ◽

Vector Spaces ◽

Quadratic Functional ◽

The Stability ◽

Fixed Positive Integer ◽

LetX,Ybe vector spaces andka fixed positive integer. It is shown that a mappingf(kx+y)+f(kx-y)=2k2f(x)+2f(y)for allx,y∈Xif and only if the mappingf:X→Ysatisfiesf(x+y)+f(x-y)=2f(x)+2f(y)for allx,y∈X. Furthermore, the Hyers-Ulam-Rassias stability of the above functional equation in Banach spaces is proven.

On the generalized orthogonal stability of the pexiderized quadratic functional equations in modular spaces

Mathematica Slovaca ◽

10.1515/ms-2016-0255 ◽

2017 ◽

Vol 67 (1) ◽

Cited By ~ 4

Author(s):

Iz-iddine EL-Fassi ◽

Samir Kabbaj

Keyword(s):

Functional Equation ◽

Functional Equations ◽

Quadratic Functional ◽

Modular Spaces ◽

Generalized Orthogonal ◽

AbstractIn this paper, we establish the Hyers-Ulam-Rassias stability of the quadratic functional equation of Pexiderized type

On the stability of a quadratic functional equation over non-Archimedean spaces

Filomat ◽

10.2298/fil2108693b ◽

2021 ◽

Vol 35 (8) ◽

pp. 2693-2704

Author(s):

Gastão Bettencourt ◽

Sérgio Mendes

Keyword(s):

Banach Space ◽

Functional Equation ◽

Abelian Group ◽

Quadratic Functional ◽

The Stability ◽

Let G be an abelian group and suppose that X is a non-Archimedean Banach space. We study Hyers-Ulam-Rassias stability for the functional equation of quadratic type f(x+y+z)+ f(x)+f(y)+f(z)=f(x+y)+f(y+z)+ f(z+x) where f : G ? X is a map.

Fuzzy Stability of Quadratic Functional Equations in General Cases

ISRN Mathematical Analysis ◽

10.5402/2011/503164 ◽

2011 ◽

Vol 2011 ◽

pp. 1-8 ◽

Cited By ~ 1

Author(s):

Ehsan Movahednia

Keyword(s):

Functional Equation ◽

Functional Equations ◽

Quadratic Functional ◽

The aim of this paper is to investigate fuzzy Hyers-Ulam-Rassias stability of the general case of quadratic functional equation where and fixed integers with . These functional equations are equivalent. This has been proven by Ulam, 1964.

A Fixed Point Approach to the Stability of Quadratic Functional Equations in Modular Spaces

Journal of Advanced Physics ◽

10.1166/jap.2017.1292 ◽

2017 ◽

Vol 6 (1) ◽

pp. 171-175

Author(s):

Seong Sik Kim ◽

Soo Hwan Kim

Keyword(s):

Functional Equation ◽

Fixed Point ◽

Functional Equations ◽

Fixed Point Method ◽

Quadratic Functional ◽

Fixed Point Approach ◽

Modular Spaces ◽

The Stability ◽

Positive Integers ◽

In this paper, we investigate the generalized Hyers-Ulam-Rassias stability of the following quadratic functional equation f(kx + y) + f(kx – y) = 2k2f(x) + 2f(y) for any fixed positive integers k ∈ Ζ+ in modular spaces by using fixed point method.

Stability for the Mixed Type of Quartic and Quadratic Functional Equations

Journal of Function Spaces ◽

10.1155/2014/853743 ◽

2014 ◽

Vol 2014 ◽

pp. 1-10

Author(s):

Young-Su Lee ◽

Soomin Kim ◽

Chaewon Kim

Keyword(s):

Functional Equation ◽

Functional Equations ◽

Mixed Type ◽

Quadratic Functional ◽

General Solutions ◽

We establish the general solutions of the following mixed type of quartic and quadratic functional equation:f(2x+y)+f(2x-y)=4f(x+y)+4f(x-y)+2f(2x)-8f(x)-6f(y). Moreover we prove the Hyers-Ulam-Rassias stability of this equation under the approximately quartic and the approximately quadratic conditions.

Characterization and stability analysis of advanced multi-quadratic functional equations

Advances in Difference Equations ◽

10.1186/s13662-021-03541-3 ◽

2021 ◽

Vol 2021 (1) ◽

Author(s):

Abasalt Bodaghi ◽

Hossein Moshtagh ◽

Hemen Dutta

Keyword(s):

Functional Equation ◽

Fixed Point ◽

Stability Analysis ◽

Functional Equations ◽

Quadratic Functional ◽

Fixed Point Technique ◽

The Stability

AbstractIn this paper, we introduce a new quadratic functional equation and, motivated by this equation, we investigate n-variables mappings which are quadratic in each variable. We show that such mappings can be unified as an equation, namely, multi-quadratic functional equation. We also apply a fixed point technique to study the stability for the multi-quadratic functional equations. Furthermore, we present an example and a few corollaries corresponding to the stability and hyperstability outcomes.

On alienation of two functional equations of quadratic type

Aequationes Mathematicae ◽

10.1007/s00010-021-00809-7 ◽

2021 ◽

Author(s):

Roman Ger

Keyword(s):

Functional Equation ◽

Functional Equations ◽

Quadratic Functional ◽

Real Numbers ◽

Alpha Beta ◽

Beta 2 ◽

Special Case

Abstract We deal with an alienation problem for an Euler–Lagrange type functional equation $$\begin{aligned} f(\alpha x + \beta y) + f(\alpha x - \beta y) = 2\alpha ^2f(x) + 2\beta ^2f(y) \end{aligned}$$ f ( α x + β y ) + f ( α x - β y ) = 2 α 2 f ( x ) + 2 β 2 f ( y ) assumed for fixed nonzero real numbers $$\alpha ,\beta ,\, 1 \ne \alpha ^2 \ne \beta ^2$$ α , β , 1 ≠ α 2 ≠ β 2 , and the classic quadratic functional equation $$\begin{aligned} g(x+y) + g(x-y) = 2g(x) + 2g(y). \end{aligned}$$ g ( x + y ) + g ( x - y ) = 2 g ( x ) + 2 g ( y ) . We were inspired by papers of Kim et al. (Abstract and applied analysis, vol. 2013, Hindawi Publishing Corporation, 2013) and Gordji and Khodaei (Abstract and applied analysis, vol. 2009, Hindawi Publishing Corporation, 2009), where the special case $$g = \gamma f$$ g = γ f was examined.

Projective actions, invariant sigma-curves and quadratic functional equations

Mathematical Proceedings of the Cambridge Philosophical Society ◽

10.1017/s0305004100063416 ◽

1985 ◽

Vol 98 (2) ◽

pp. 195-212 ◽

Cited By ~ 1

Author(s):

Patrick J. McCarthy

Keyword(s):

Functional Equation ◽

Periodic Function ◽

Functional Equations ◽

Continuous Solution ◽

Quadratic Functional ◽

Continuous Solutions ◽

Linear Map

AbstractThe quadratic functional equation f(f(x)) *–Tf(x) + Dx = 0 is equivalent to the requirement that the graph be invariant under a certain linear map The induced projective map is used to show that the equation admits a rich supply of continuous solutions only when L is hyperbolic (T2 > 4D), and then only when T and D satisfy certain further conditions. The general continuous solution of the equation is given explicitly in terms of either (a) an expression involving an arbitrary periodic function, function additions, inverses and composites, or(b) suitable limits of such solutions.