On an alternative functional equation related to the Cauchy equation

1982 ◽  
Vol 24 (1) ◽  
pp. 119-120
Author(s):  
Gian Luigi Forti
Keyword(s):  
Functional Equation ◽  
Cauchy Equation ◽  
Alternative Functional Equation

On an alternative functional equation related to the Cauchy equation

1982 ◽  
Vol 24 (1) ◽  
pp. 195-206 ◽  
Author(s):  
Gian Luigi Forti
Keyword(s):  
Functional Equation ◽  
Cauchy Equation ◽  
Alternative Functional Equation

ON AN ALTERNATIVE FUNCTIONAL EQUATION IN Rn.

1994 ◽  
pp. 33-44 ◽  
Author(s):  
Costanza Borelli-Forti ◽  
Gian Luigi Forti
Keyword(s):  
Functional Equation ◽  
Alternative Functional Equation

scholarly journals Stability of an alternative functional equation

2008 ◽  
Vol 339 (1) ◽  
pp. 303-311 ◽  
Author(s):  
Bogdan Batko
Keyword(s):  
Functional Equation ◽  
Alternative Functional Equation

scholarly journals Alienation of the Jensen, Cauchy and d’Alembert Equations

2016 ◽  
Vol 30 (1) ◽  
pp. 181-191
Author(s):  
Barbara Sobek
Keyword(s):  
Functional Equation ◽  
Commutative Semigroup ◽  
Analogous Problem ◽  
Cauchy Equation ◽  
Jensen Equation

AbstractLet (S, +) be a commutative semigroup, σ : S → S be an endomorphism with σ2 = id and let K be a field of characteristic different from 2. Inspired by the problem of strong alienation of the Jensen equation and the exponential Cauchy equation, we study the solutions f, g : S → K of the functional equation $$f(x + y) + f(x + \sigma (y)) + g(x + y) = 2f(x) + g(x)g(y)\;\;\;\;{\rm for}\;\;x,y \in S.$$ We also consider an analogous problem for the Jensen and the d’Alembert equations as well as for the d’Alembert and the exponential Cauchy equations.

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 Stability of generalized additive Cauchy equations

2000 ◽  
Vol 24 (11) ◽  
pp. 721-727 ◽  
Author(s):  
Soon-Mo Jung ◽  
Ki-Suk Lee
Keyword(s):  
Functional Equation ◽  
Stability Problem ◽  
Cauchy Equation ◽  
Class Of Functions ◽  
Rassias Stability

A familiar functional equationf(ax+b)=cf(x)will be solved in the class of functionsf:ℝ→ℝ. Applying this result we will investigate the Hyers-Ulam-Rassias stability problem of the generalized additive Cauchy equationf(a1x1+⋯+amxm+x0)=∑i=1mbif(ai1x1+⋯+aimxm)in connection with the question of Rassias and Tabor.

A HYPERSTABILITY RESULT FOR THE CAUCHY EQUATION

2013 ◽  
Vol 89 (1) ◽  
pp. 33-40 ◽  
Author(s):  
JANUSZ BRZDĘK
Keyword(s):  
Functional Equation ◽  
Fixed Point ◽  
Fixed Point Theorem ◽  
Normed Space ◽  
Functional Equations ◽  
Cauchy Equation ◽  
Cauchy Functional Equation ◽  
Real Numbers ◽  
Formula One ◽  
Image Position

AbstractWe prove a hyperstability result for the Cauchy functional equation$f(x+ y)= f(x)+ f(y)$, which complements some earlier stability outcomes of J. M. Rassias. As a consequence, we obtain the slightly surprising corollary that for every function$f$, mapping a normed space${E}_{1} $into a normed space${E}_{2} $, and for all real numbers$r, s$with$r+ s\gt 0$one of the following two conditions must be valid:$$\begin{eqnarray*}\displaystyle \sup _{x, y\in E_{1}}\Vert f(x+ y)- f(x)- f(y)\Vert \hspace{0.167em} \mathop{\Vert x\Vert }\nolimits ^{r} \hspace{0.167em} \mathop{\Vert y\Vert }\nolimits ^{s} = \infty , &&\displaystyle\end{eqnarray*}$$$$\begin{eqnarray*}\displaystyle \sup _{x, y\in E_{1}}\Vert f(x+ y)- f(x)- f(y)\Vert \hspace{0.167em} \mathop{\Vert x\Vert }\nolimits ^{r} \hspace{0.167em} \mathop{\Vert y\Vert }\nolimits ^{s} = 0. &&\displaystyle\end{eqnarray*}$$In particular, we present a new method for proving stability for functional equations, based on a fixed point theorem.

scholarly journals On Approximate Solutions of Functional Equations in Vector Lattices

2014 ◽  
Vol 2014 ◽  
pp. 1-10 ◽  
Author(s):  
Bogdan Batko
Keyword(s):  
Functional Equation ◽  
Functional Equations ◽  
Spectral Representation ◽  
General Theorem ◽  
Approximate Solutions ◽  
Quadratic Functional ◽  
Cauchy Equation ◽  
Riesz Spaces ◽  
The Stability ◽  
Class Of Functions

We provide a method of approximation of approximate solutions of functional equations in the class of functions acting into a Riesz space (algebra). The main aim of the paper is to provide a general theorem that can act as a tool applicable to a possibly wide class of functional equations. The idea is based on the use of the Spectral Representation Theory for Riesz spaces. The main result will be applied to prove the stability of an alternative Cauchy functional equationF(x+y)+F(x)+F(y)≠0⇒F(x+y)=F(x)+F(y)in Riesz spaces, the Cauchy equation with squaresF(x+y)2=(F(x)+F(y))2inf-algebras, and the quadratic functional equationF(x+y)+F(x-y)=2F(x)+2F(y)in Riesz spaces.

scholarly journals Hyers-Ulam stability of an alternative functional equation of Jensen type

ScienceAsia ◽  
2019 ◽  
Vol 45 (3) ◽  
pp. 275
Author(s):  
Choodech Srisawat
Keyword(s):  
Functional Equation ◽  
Ulam Stability ◽  
Alternative Functional Equation
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