## Abstract and Applied Analysis

### On a Stability of Logarithmic-Type Functional Equation in Schwartz Distributions

Jae-Young Chung

#### Abstract

We prove the Hyers-Ulam stability of the logarithmic functional equation of Heuvers and Kannappan $f(x+y)-g(xy)-h(1/x+1/y)=0,\mathrm{ }x,y>0$, in both classical and distributional senses. As a classical sense, the Hyers-Ulam stability of the inequality $|f(x+y)-g(xy)-h(1/x+1/y)|\le {\epsilon},\mathrm{ }x,y>0$ will be proved, where $f,g,h:{\Bbb R}_{+}\to \Bbb C$. As a distributional analogue of the above inequality, the stability of inequality $\parallel u\circ (x+y)-v\circ (xy)-w\circ (1/x+1/y)\parallel \le {\epsilon}$ will be proved, where $u,v,w\in \mathrm{\scr D}\text{'}({\Bbb R}_{+})$ and $\circ$ denotes the pullback of distributions.

#### Article information

Source
Abstr. Appl. Anal., Volume 2012 (2012), Article ID 435310, 15 pages.

Dates
First available in Project Euclid: 28 March 2013

https://projecteuclid.org/euclid.aaa/1364475944

Digital Object Identifier
doi:10.1155/2012/435310

Mathematical Reviews number (MathSciNet)
MR2999887

Zentralblatt MATH identifier
1259.39020

#### Citation

Chung, Jae-Young. On a Stability of Logarithmic-Type Functional Equation in Schwartz Distributions. Abstr. Appl. Anal. 2012 (2012), Article ID 435310, 15 pages. doi:10.1155/2012/435310. https://projecteuclid.org/euclid.aaa/1364475944

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