scholarly journals Comments on the core of the direct method for proving Hyers–Ulam stability of functional equations

2004 ◽  
Vol 295 (1) ◽  
pp. 127-133 ◽  
Author(s):  
Gian-Luigi Forti
Keyword(s):  
Functional Equations ◽  
Direct Method ◽  
Ulam Stability ◽  
The Core ◽  
Stability Of Functional Equations

scholarly journals Solution and Hyers-Ulam-Rassias Stability of Generalized Mixed Type Additive-Quadratic Functional Equations in Fuzzy Banach Spaces

2012 ◽  
Vol 2012 ◽  
pp. 1-22 ◽  
Author(s):  
M. Eshaghi Gordji ◽  
H. Azadi Kenary ◽  
H. Rezaei ◽  
Y. W. Lee ◽  
G. H. Kim
Keyword(s):  
Functional Equation ◽  
Fixed Point ◽  
Banach Spaces ◽  
Functional Equations ◽  
Mixed Type ◽  
Direct Method ◽  
Quadratic Functional ◽  
Ulam Stability ◽  
Fixed Point Methods ◽  
Rassias Stability

By using fixed point methods and direct method, we establish the generalized Hyers-Ulam stability of the following additive-quadratic functional equationf(x+ky)+f(x−ky)=f(x+y)+f(x−y)+(2(k+1)/k)f(ky)−2(k+1)f(y)for fixed integerskwithk≠0,±1in fuzzy Banach spaces.

scholarly journals A General Uniqueness Theorem concerning the Stability of Additive and Quadratic Functional Equations

2015 ◽  
Vol 2015 ◽  
pp. 1-8 ◽  
Author(s):  
Yang-Hi Lee ◽  
Soon-Mo Jung
Keyword(s):  
Functional Equation ◽  
Uniqueness Theorem ◽  
Functional Equations ◽  
Additive Functional ◽  
Quadratic Functional ◽  
Ulam Stability ◽  
Additive Functional Equation ◽  
Additive Type ◽  
Stability Of Functional Equations ◽  
The Stability

We prove a general uniqueness theorem that can be easily applied to the (generalized) Hyers-Ulam stability of the Cauchy additive functional equation, the quadratic functional equation, and the quadratic-additive type functional equations. This uniqueness theorem can replace the repeated proofs for uniqueness of the relevant solutions of given equations while we investigate the stability of functional equations.

scholarly journals On a general Hyers-Ulam stability result

1995 ◽  
Vol 18 (2) ◽  
pp. 229-236 ◽  
Author(s):  
Costanz Borelli ◽  
Gian Luigi Forti
Keyword(s):  
Functional Equations ◽  
Stability Result ◽  
Quadratic Equation ◽  
Ulam Stability ◽  
Stability Of Functional Equations ◽  
The Stability

In this paper, we prove two general theorems about Hyers-Ulam stability of functional equations. As particular cases we obtain many of the results published in the last ten years on the stability of the Cauchy and quadratic equation.

scholarly journals On Ulam Stability of Functional Equations in 2-Normed Spaces—A Survey

Symmetry ◽  
2021 ◽  
Vol 13 (11) ◽  
pp. 2200
Author(s):  
Anna Bahyrycz ◽  
Janusz Brzdęk ◽  
El-sayed El-hady ◽  
Zbigniew Leśniak
Keyword(s):  
Exact Solutions ◽  
Approximation Theory ◽  
Functional Equations ◽  
Approximate Solutions ◽  
Main Issue ◽  
Ulam Stability ◽  
Normed Spaces ◽  
Stability Of Functional Equations

The theory of Ulam stability was initiated by a problem raised in 1940 by S. Ulam and concerning approximate solutions to the equation of homomorphism in groups. It is somehow connected to various other areas of investigation such as, e.g., optimization and approximation theory. Its main issue is the error that we make when replacing functions satisfying the equation approximately with exact solutions of the equation. This article is a survey of the published so far results on Ulam stability for functional equations in 2-normed spaces. We present and discuss them, pointing to the various pitfalls they contain and showing possible simple generalizations. In this way, in particular, we demonstrate that the easily noticeable symmetry between them and the analogous results obtained for the classical metric or normed spaces is in fact only apparent.

scholarly journals Stability of Functional Equations and Properties of Groups

2019 ◽  
Vol 33 (1) ◽  
pp. 77-96
Author(s):  
Gian Luigi Forti
Keyword(s):  
Additive Function ◽  
Functional Equations ◽  
Ulam Stability ◽  
Cauchy Equation ◽  
Stability Of Functional Equations ◽  
The Given

AbstractInvestigating Hyers–Ulam stability of the additive Cauchy equation with domain in a group G, in order to obtain an additive function approximating the given almost additive one we need some properties of G, starting from commutativity to others more sophisticated. The aim of this survey is to present these properties and compare, as far as possible, the classes of groups involved.

scholarly journals Ulam Stability of a Functional Equation in Various Normed Spaces

Symmetry ◽  
2020 ◽  
Vol 12 (7) ◽  
pp. 1119
Author(s):  
Krzysztof Ciepliński
Keyword(s):  
Functional Equation ◽  
Banach Spaces ◽  
Functional Equations ◽  
Direct Method ◽  
Ulam Stability ◽  
Normed Spaces ◽  
Quadratic Mappings ◽  
The Stability

In this note, we study the Ulam stability of a general functional equation in four variables. Since its particular case is a known equation characterizing the so-called bi-quadratic mappings (i.e., mappings which are quadratic in each of their both arguments), we get in consequence its stability, too. We deal with the stability of the considered functional equations not only in classical Banach spaces, but also in 2-Banach and complete non-Archimedean normed spaces. To obtain our outcomes, the direct method is applied.

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