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2013 Stability of a Logarithmic Functional Equation in Distributions on a Restricted Domain
Jaeyoung Chung, Prasanna K. Sahoo
Abstr. Appl. Anal. 2013(SI58): 1-9 (2013). DOI: 10.1155/2013/751680

Abstract

Let be the set of real numbers, + = { x x > 0 } , ϵ + , and f , g , h : + . As classical and L versions of the Hyers-Ulam stability of the logarithmic type functional equation in a restricted domain, we consider the following inequalities: | f ( x + y ) - g ( x y ) - h ( (1 / x) + (1 / y) ) | ϵ , and f ( x + y ) - g ( x y ) - h (1 / x) + (1 / y) L ( Γ d ) ϵ in the sectors Γ d = { ( x , y ) : x > 0 , y > 0 , (y / x) > d } . As consequences of the results, we obtain asymptotic behaviors of the previous inequalities. We also consider its distributional version u S - v Π - w R Γ d ϵ , where u , v , w 𝒟 ' ( + ) , S ( x , y ) = x + y , Π ( x , y ) = x y , R ( x , y ) = 1 / x + 1 / y , x , y + , and the inequality · Γ d ϵ means that | · , φ | ϵ φ L 1 for all test functions φ C c ( Γ d ) .

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Jaeyoung Chung. Prasanna K. Sahoo. "Stability of a Logarithmic Functional Equation in Distributions on a Restricted Domain." Abstr. Appl. Anal. 2013 (SI58) 1 - 9, 2013. https://doi.org/10.1155/2013/751680

Information

Published: 2013
First available in Project Euclid: 26 February 2014

zbMATH: 07095327
MathSciNet: MR3095363
Digital Object Identifier: 10.1155/2013/751680

Rights: Copyright © 2013 Hindawi

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Vol.2013 • No. SI58 • 2013
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