scholarly journals The Stability of Isometries on Restricted Domains

Symmetry ◽  
2021 ◽  
Vol 13 (2) ◽  
pp. 282
Author(s):  
Ginkyu Choi ◽  
Soon-Mo Jung
Keyword(s):  
Functional Equations ◽  
Ulam Stability ◽  
Orthogonality Equation ◽  
Restricted Domains ◽  
The Stability

We will prove the generalized Hyers–Ulam stability of isometries, with a focus on the stability for restricted domains. More precisely, we prove the generalized Hyers–Ulam stability of the orthogonality equation and we use this result to prove the stability of the equations ∥f(x)−f(y)∥=∥x−y∥ and ∥f(x)−f(y)∥2=∥x−y∥2 on the restricted domains. As we can easily see, these functional equations are symmetric in the sense that they become the same equations even if the roles of variables x and y are exchanged.

scholarly journals The new investigation of the stability of mixed type additive-quartic functional equations in non-Archimedean spaces

2020 ◽  
Vol 53 (1) ◽  
pp. 174-192
Author(s):  
Anurak Thanyacharoen ◽  
Wutiphol Sintunavarat
Keyword(s):  
Functional Equation ◽  
Normed Space ◽  
Functional Equations ◽  
Mixed Type ◽  
Additive Group ◽  
Ulam Stability ◽  
Quartic Functional Equation ◽  
The Stability

AbstractIn this article, we prove the generalized Hyers-Ulam stability for the following additive-quartic functional equation:f(x+3y)+f(x-3y)+f(x+2y)+f(x-2y)+22f(x)+24f(y)=13{[}f(x+y)+f(x-y)]+12f(2y),where f maps from an additive group to a complete non-Archimedean normed space.

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 Local stability of the additive functional equation and its applications

2003 ◽  
Vol 2003 (1) ◽  
pp. 15-26 ◽  
Author(s):  
Soon-Mo Jung ◽  
Byungbae Kim
Keyword(s):  
Functional Equation ◽  
Large Class ◽  
Local Stability ◽  
Unbounded Domains ◽  
Additive Functional ◽  
Ulam Stability ◽  
Additive Functional Equation ◽  
Jensen’S Functional Equation ◽  
Restricted Domains ◽  
The Stability

The main purpose of this paper is to prove the Hyers-Ulam stability of the additive functional equation for a large class of unbounded domains. Furthermore, by using the theorem, we prove the stability of Jensen's functional equation for a large class of restricted domains.

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 On the stability problem of quadratic functional equations in 2-Banach spaces

2017 ◽  
pp. 5054-5061
Author(s):  
Seong Sik Kim ◽  
Ga Ya Kim ◽  
Soo Hwan Kim
Keyword(s):  
Functional Equation ◽  
Banach Spaces ◽  
Stability Problem ◽  
Functional Equations ◽  
Quadratic Functional ◽  
Ulam Stability ◽  
Normed Spaces ◽  
The Stability ◽  
Stability Results

In this paper, we investigate the stability problem in the spirit of Hyers-Ulam, Rassias and G·avruta for the quadratic functional equation:f(2x + y) + f(2x ¡ y) = 2f(x + y) + 2f(x ¡ y) + 4f(x) ¡ 2f(y) in 2-Banach spaces. These results extend the generalized Hyers-Ulam stability results by thequadratic functional equation in normed spaces to 2-Banach spaces.

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.

A FIXED POINT APPROACH TO THE STABILITY OF GENERAL QUADRATIC EULER-LAGRANGE FUNCTIONAL EQUATIONS IN INTUITIONISTIC FUZZY SPACES

2016 ◽  
pp. 4430-4436
Author(s):  
Seong Sik Kim ◽  
Ga Ya Kim
Keyword(s):  
Fixed Point ◽  
Functional Equations ◽  
Fixed Point Method ◽  
Ulam Stability ◽  
Normed Spaces ◽  
Fixed Point Approach ◽  
Intuitionistic Fuzzy ◽  
The Stability ◽  
Lagrange Functional ◽  
Intuitionistic Fuzzy Normed Spaces

In this paper, we prove the generalized Hyers-Ulam stability of a general k-quadratic Euler-Lagrange functional equation:for any fixed positive integer in intuitionistic fuzzy normed spaces using a fixed point method.

scholarly journals On the Stability of Trigonometric Functional Equations in Distributions and Hyperfunctions

2013 ◽  
Vol 2013 ◽  
pp. 1-12 ◽  
Author(s):  
Jaeyoung Chung ◽  
Jeongwook Chang
Keyword(s):  
Functional Equations ◽  
Generalized Functions ◽  
Ulam Stability ◽  
Schwartz Distributions ◽  
The Stability

We consider the Hyers-Ulam stability for a class of trigonometric functional equations in the spaces of generalized functions such as Schwartz distributions and Gelfand hyperfunctions.

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.

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