scholarly journals Hyperstability of Some Functional Equations on Restricted Domain

2017 ◽  
Vol 2017 ◽  
pp. 1-6 ◽  
Author(s):  
Anna Bahyrycz ◽  
Jolanta Olko
Keyword(s):  
Functional Equations ◽  
Linear Functional ◽  
Stability Result ◽  
Linear Functional Equation ◽  
Ulam Stability ◽  
Restricted Domain ◽  
Stability Studies ◽  
Control Functions ◽  
Restricted Domains ◽  
Fréchet Equation

The paper concerns functions which approximately satisfy, not necessarily on the whole linear space, a generalization of linear functional equation. A Hyers-Ulam stability result is proved and next applied to give conditions implying the hyperstability of the equation. The results may be used as tools in stability studies on restricted domains for various functional equations. We use the main theorem to obtain a few hyperstability results of Fréchet equation on restricted domain for different control functions.

scholarly journals General Solutions of Two Quadratic Functional Equations of Pexider Type on Orthogonal Vectors

2012 ◽  
Vol 2012 ◽  
pp. 1-10 ◽  
Author(s):  
Margherita Fochi
Keyword(s):  
Functional Equations ◽  
Inner Product ◽  
Quadratic Functional ◽  
Ulam Stability ◽  
Restricted Domain ◽  
Product Spaces ◽  
General Solutions ◽  
Inner Product Spaces ◽  
Restricted Domains

Based on the studies on the Hyers-Ulam stability and the orthogonal stability of some Pexider-quadratic functional equations, in this paper we find the general solutions of two quadratic functional equations of Pexider type. Both equations are studied in restricted domains: the first equation is studied on the restricted domain of the orthogonal vectors in the sense of Rätz, and the second equation is considered on the orthogonal vectors in the inner product spaces with the usual orthogonality.

scholarly journals Stability of Functional Equations in Dislocated Quasi-Metric Spaces

2018 ◽  
Vol 32 (1) ◽  
pp. 215-225 ◽  
Author(s):  
Beata Hejmej
Keyword(s):  
Functional Equation ◽  
Functional Equations ◽  
Linear Functional ◽  
Metric Spaces ◽  
Linear Functional Equation ◽  
Ulam Stability ◽  
Cauchy Functional Equation ◽  
Complete Metric Spaces ◽  
Stability Of Functional Equations ◽  
Quasimetric Space

Abstract We present a result on the generalized Hyers-Ulam stability of a functional equation in a single variable for functions that have values in a complete dislocated quasi-metric space. Next, we show how to apply it to prove stability of the Cauchy functional equation and the linear functional equation in two variables, also for functions taking values in a complete dislocated quasimetric space. In this way we generalize some earlier results proved for classical complete metric spaces.

scholarly journals Applications of Banach Limit in Ulam Stability

Symmetry ◽  
2021 ◽  
Vol 13 (5) ◽  
pp. 841
Author(s):  
Roman Badora ◽  
Janusz Brzdęk ◽  
Krzysztof Ciepliński
Keyword(s):  
Functional Equation ◽  
Functional Equations ◽  
Linear Functional ◽  
Linear Functional Equation ◽  
Ulam Stability ◽  
Single Variable ◽  
Cauchy Equation ◽  
Banach Limit

We show how to get new results on Ulam stability of some functional equations using the Banach limit. We do this with the examples of the linear functional equation in single variable and the Cauchy equation.

scholarly journals Additive and Fréchet functional equations on restricted domains with some characterizations of inner product spaces

2021 ◽  
Vol 7 (3) ◽  
pp. 3379-3394
Author(s):  
Choonkil Park ◽  
◽  
Abbas Najati ◽  
Batool Noori ◽  
Mohammad B. Moghimi ◽  
...  
Keyword(s):  
Functional Equations ◽  
Inner Product ◽  
Ulam Stability ◽  
Product Spaces ◽  
Asymptotic Behaviors ◽  
Inner Product Spaces ◽  
Restricted Domains ◽  
Different Types

<abstract><p>In this paper, we investigate the Hyers-Ulam stability of additive and Fréchet functional equations on restricted domains. We improve the bounds and thus the results obtained by S. M. Jung and J. M. Rassias. As a consequence, we obtain asymptotic behaviors of functional equations of different types. One of the objectives of this paper is to bring out the involvement of functional equations in various characterizations of inner product spaces.</p></abstract>

scholarly journals Stability of Jensen-Type Functional Equations on Restricted Domains in a Group and Their Asymptotic Behaviors

2012 ◽  
Vol 2012 ◽  
pp. 1-12 ◽  
Author(s):  
Jae-Young Chung ◽  
Dohan Kim ◽  
John Michael Rassias
Keyword(s):  
Asymptotic Behavior ◽  
Functional Equations ◽  
Functional Inequality ◽  
Ulam Stability ◽  
Asymptotic Behaviors ◽  
Stability Problems ◽  
Restricted Domains

We consider the Hyers-Ulam stability problems for the Jensen-type functional equations in general restricted domains. The main purpose of this paper is to find the restricted domains for which the functional inequality satisfied in those domains extends to the inequality for whole domain. As consequences of the results we obtain asymptotic behavior of the equations.

ON HYPERSTABILITY OF GENERALISED LINEAR FUNCTIONAL EQUATIONS IN SEVERAL VARIABLES

2015 ◽  
Vol 92 (2) ◽  
pp. 259-267 ◽  
Author(s):  
DONG ZHANG
Keyword(s):  
Functional Equation ◽  
Exact Solutions ◽  
Normed Space ◽  
Functional Equations ◽  
Linear Functional ◽  
Approximate Solutions ◽  
Linear Functional Equation ◽  
Several Variables ◽  
Linear Functional Equations

We obtain some results on approximate solutions of the generalised linear functional equation $\sum _{i=1}^{m}L_{i}f(\sum _{j=1}^{n}a_{ij}x_{j})=0$ for functions mapping a normed space into a normed space. We show that, under suitable assumptions, the approximate solutions are in fact exact solutions. The theorems correspond to and complement recent results on the hyperstability of generalised linear 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 Stability of linear functional equations in Banach modules

2004 ◽  
Vol 35 (1) ◽  
pp. 29-36
Author(s):  
Chun-Gil Park
Keyword(s):  
Functional Equation ◽  
Banach Algebra ◽  
Functional Equations ◽  
Linear Functional ◽  
Linear Functional Equation ◽  
Linear Functional Equations ◽  
Banach Modules ◽  
Unital Banach Algebra ◽  
Rassias Stability

We prove the Hyers-Ulam-Rassias stability of the linear functional equation in Banach modules over a unital Banach algebra.

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