On the Generalized Hyers–Ulam Stability of the Pexider Equation on Restricted Domains

2014 ◽  
pp. 279-299
Author(s):  
Youssef Manar ◽  
Elhoucien Elqorachi ◽  
Themistocles M. Rassias
Keyword(s):  
Ulam Stability ◽  
Pexider Equation ◽  
Restricted Domains

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 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.

scholarly journals Stability of Pexider Equations on Semigroup with No Neutral Element

2014 ◽  
Vol 2014 ◽  
pp. 1-7
Author(s):  
Jaeyoung Chung
Keyword(s):  
Banach Space ◽  
Closed Form ◽  
Direct Consequence ◽  
Neutral Element ◽  
Complex Numbers ◽  
Ulam Stability ◽  
Exponential Equation ◽  
Pexider Equation ◽  
Pexider Equations ◽  
Jung’S Theorem

LetSbe a commutative semigroup with no neutral element,Ya Banach space, andℂthe set of complex numbers. In this paper we prove the Hyers-Ulam stability for Pexider equationfx+y-gx-h(y)≤ϵfor allx,y∈S, wheref,g,h:S→Y. Using Jung’s theorem we obtain a better bound than that usually obtained. Also, generalizing the result of Baker (1980) we prove the superstability for Pexider-exponential equationft+s-gth(s)≤ϵfor allt,s∈S, wheref,g,h:S→ℂ. As a direct consequence of the result we also obtain the general solutions of the Pexider-exponential equationft+s=gth(s)for allt,s∈S, a closed form of which is not yet known.

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.

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