scholarly journals Ulam's type stability of a functional equation deriving from quadratic and additive functions

2015 ◽  
pp. 73-84 ◽  
Author(s):  
Abasalt Bodaghi ◽  
Sang Og Kim
Keyword(s):  
Functional Equation ◽  
Additive Functions ◽  
Ulam’S Type Stability

scholarly journals Stability of a Functional Equation Deriving from Quadratic and Additive Functions in Non-Archimedean Normed Spaces

2013 ◽  
Vol 2013 ◽  
pp. 1-10 ◽  
Author(s):  
Abasalt Bodaghi ◽  
Sang Og Kim
Keyword(s):  
Functional Equation ◽  
Positive Integer ◽  
General Solution ◽  
Functional Equations ◽  
Quadratic Functional Equation ◽  
Quadratic Functional ◽  
Ulam Stability ◽  
Additive Functions ◽  
Normed Spaces

We obtain the general solution of the generalized mixed additive and quadratic functional equationfx+my+fx−my=2fx−2m2fy+m2f2y,mis even;fx+y+fx−y−2m2−1fy+m2−1f2y,mis odd, for a positive integerm. We establish the Hyers-Ulam stability for these functional equations in non-Archimedean normed spaces whenmis an even positive integer orm=3.

A special class of functional equations

2018 ◽  
Vol 68 (2) ◽  
pp. 397-404 ◽  
Author(s):  
Ahmed Charifi ◽  
Radosław Łukasik ◽  
Driss Zeglami
Keyword(s):  
Functional Equation ◽  
Finite Group ◽  
Abelian Group ◽  
Special Class ◽  
Functional Equations ◽  
Additive Functions ◽  
Group Of Automorphisms ◽  
Quadratic Equations ◽  
A Finite Group ◽  
Abelian Monoid

Abstract We obtain in terms of additive and multi-additive functions the solutions f, h: S → H of the functional equation $$\begin{array}{} \displaystyle \sum\limits_{\lambda \in \Phi }f(x+\lambda y+a_{\lambda })=Nf(x)+h(y),\quad x,y\in S, \end{array} $$ where (S, +) is an abelian monoid, Φ is a finite group of automorphisms of S, N = | Φ | designates the number of its elements, {aλ, λ ∈ Φ} are arbitrary elements of S and (H, +) is an abelian group. In addition, some applications are given. This equation provides a joint generalization of many functional equations such as Cauchy’s, Jensen’s, Łukasik’s, quadratic or Φ-quadratic equations.

scholarly journals ON SOME PEXIDER-TYPE FUNCTIONAL EQUATIONS CONNECTED WITH THE ABSOLUTE VALUE OF ADDITIVE FUNCTIONS. PART II

2012 ◽  
Vol 85 (2) ◽  
pp. 202-216 ◽  
Author(s):  
BARBARA PRZEBIERACZ
Keyword(s):  
Functional Equation ◽  
Abelian Group ◽  
Functional Equations ◽  
Additive Functions ◽  
Absolute Value ◽  
Real Functions ◽  
The Absolute ◽  
Image Position

AbstractWe investigate the Pexider-type functional equation where f, g, h are real functions defined on an abelian group G. We solve this equation under the assumptions G=ℝ and f is continuous.

Elementary volume and measurability properties of additive functions

2016 ◽  
Vol 23 (1) ◽  
Author(s):  
Tamar Kasrashvili ◽  
Aleks Kirtadze
Keyword(s):  
Functional Equation ◽  
Invariant Measure ◽  
Real Line ◽  
Elementary Volume ◽  
Additive Functions ◽  
Translation Invariant ◽  
One Dimensional ◽  
The One ◽  
Dimensional Lebesgue Measure ◽  
Theoretical Standpoint

AbstractThe paper is concerned with some aspects of the theory of elementary volume from the measure-theoretical standpoint. It is shown that there exists a nontrivial solution of Cauchy's functional equation, nonmeasurable with respect to every translation invariant measure on the real line, extending the one-dimensional Lebesgue measure.

scholarly journals Ulam’s Type Stability and Generalized Norms

Symmetry ◽  
2020 ◽  
Vol 12 (4) ◽  
pp. 502
Author(s):  
Laura Manolescu
Keyword(s):  
Functional Equation ◽  
Symmetric Spaces ◽  
Ulam Stability ◽  
Cauchy Equation ◽  
Cauchy Functional Equation ◽  
Type V ◽  
Ulam’S Type Stability

A symmetric functional equation is one whose form is the same regardless of the order of the arguments. A remarkable example is the Cauchy functional equation: f ( x + y ) = f ( x ) + f ( y ) . Interesting results in the study of the rigidity of quasi-isometries for symmetric spaces were obtained by B. Kleiner and B. Leeb, using the Hyers-Ulam stability of a Cauchy equation. In this paper, some results on the Ulam’s type stability of the Cauchy functional equation are provided by extending the traditional norm estimations to ther measurements called generalized norm of convex type (v-norm) and generalized norm of subadditive type (s-norm).

scholarly journals ON SOME PEXIDER-TYPE FUNCTIONAL EQUATIONS CONNECTED WITH THE ABSOLUTE VALUE OF ADDITIVE FUNCTIONS. PART I

2012 ◽  
Vol 85 (2) ◽  
pp. 191-201 ◽  
Author(s):  
BARBARA PRZEBIERACZ
Keyword(s):  
Functional Equation ◽  
Abelian Group ◽  
Functional Equations ◽  
Additive Functions ◽  
Absolute Value ◽  
Real Functions ◽  
The Absolute ◽  
Image Position

AbstractWe investigate the Pexider-type functional equation where f,g,h are real functions defined on an abelian group G.

scholarly journals On Ulam’s Type Stability of the Linear Equation and Related Issues

2014 ◽  
Vol 2014 ◽  
pp. 1-14 ◽  
Author(s):  
Janusz Brzdęk ◽  
Krzysztof Ciepliński ◽  
Zbigniew Leśniak
Keyword(s):  
Functional Equation ◽  
Linear Equation ◽  
Linear Functional ◽  
Linear Functional Equation ◽  
Single Variable ◽  
Survey Paper ◽  
Ulam’S Type Stability ◽  
Stability Results

This is a survey paper concerning stability results for the linear functional equation in single variable. We discuss issues that have not been considered or have been treated only briefly in other surveys concerning stability of the equation. In this way, we complement those surveys.

scholarly journals On Ulam's Type Stability of the Cauchy Additive Equation

2014 ◽  
Vol 2014 ◽  
pp. 1-7
Author(s):  
Janusz Brzdęk
Keyword(s):  
Functional Equation ◽  
General Result ◽  
Commutative Group ◽  
Ulam’S Type Stability ◽  
Additive Equation ◽  
Class Of Functions

We prove a general result on Ulam's type stability of the functional equationfx+y=fx+fy, in the class of functions mapping a commutative group into a commutative group. As a consequence, we deduce from it some hyperstability outcomes. Moreover, we also show how to use that result to improve some earlier stability estimations given by Isaac and Rassias.

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