Stability of Pexider Equations on Semigroup with No Neutral Element

Let S be a commutative semigroup with no neutral element, Y a Banach space, and ℂ the set of complex numbers. In this paper we prove the Hyers-Ulam stability for Pexider equation fx+y-gx-h(y)≤ϵ for all x,y∈S, where f,g,h:S→Y. Using Jung’s theorem we obtain a better bound than that usually obtained....

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Main Author: Jaeyoung Chung
Format: Article
Language:English
Published: Wiley 2014-01-01
Series:Journal of Function Spaces
Online Access:http://dx.doi.org/10.1155/2014/153610
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author Jaeyoung Chung
author_facet Jaeyoung Chung
author_sort Jaeyoung Chung
collection DOAJ
description Let S be a commutative semigroup with no neutral element, Y a Banach space, and ℂ the set of complex numbers. In this paper we prove the Hyers-Ulam stability for Pexider equation fx+y-gx-h(y)≤ϵ for all x,y∈S, where f,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 equation ft+s-gth(s)≤ϵ for all t,s∈S, where f,g,h:S→ℂ. As a direct consequence of the result we also obtain the general solutions of the Pexider-exponential equation ft+s=gth(s) for all t,s∈S, a closed form of which is not yet known.
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publishDate 2014-01-01
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spelling doaj-art-9b89dfdaba61473d9bb3a26502ba1c8c2025-02-03T07:25:21ZengWileyJournal of Function Spaces2314-88962314-88882014-01-01201410.1155/2014/153610153610Stability of Pexider Equations on Semigroup with No Neutral ElementJaeyoung Chung0Department of Mathematics, Kunsan National University, Kunsan 573-701, Republic of KoreaLet S be a commutative semigroup with no neutral element, Y a Banach space, and ℂ the set of complex numbers. In this paper we prove the Hyers-Ulam stability for Pexider equation fx+y-gx-h(y)≤ϵ for all x,y∈S, where f,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 equation ft+s-gth(s)≤ϵ for all t,s∈S, where f,g,h:S→ℂ. As a direct consequence of the result we also obtain the general solutions of the Pexider-exponential equation ft+s=gth(s) for all t,s∈S, a closed form of which is not yet known.http://dx.doi.org/10.1155/2014/153610
spellingShingle Jaeyoung Chung
Stability of Pexider Equations on Semigroup with No Neutral Element
Journal of Function Spaces
title Stability of Pexider Equations on Semigroup with No Neutral Element
title_full Stability of Pexider Equations on Semigroup with No Neutral Element
title_fullStr Stability of Pexider Equations on Semigroup with No Neutral Element
title_full_unstemmed Stability of Pexider Equations on Semigroup with No Neutral Element
title_short Stability of Pexider Equations on Semigroup with No Neutral Element
title_sort stability of pexider equations on semigroup with no neutral element
url http://dx.doi.org/10.1155/2014/153610
work_keys_str_mv AT jaeyoungchung stabilityofpexiderequationsonsemigroupwithnoneutralelement