## Abstract and Applied Analysis

### Stability of a Logarithmic Functional Equation in Distributions on a Restricted Domain

#### Abstract

Let $\Bbb R$ be the set of real numbers, ${\Bbb R}^{+}=\{x\in \Bbb R\mathrm{}\mid \mathrm{}x>\mathrm{0}\}$, $\epsilon\in {\Bbb R}_{+}$, and $f,g,h:{\Bbb R}^{+}\to \Bbb C$. As classical and ${L}^{\mathrm{\infty }}$ versions of the Hyers-Ulam stability of the logarithmic type functional equation in a restricted domain, we consider the following inequalities: $|f(x+y)-g(xy)-h(\mathrm{(1}/\mathrm{x)}+\mathrm{(1}/\mathrm{y)})|\le \epsilon$, and ${∥f(x+y)-g(xy)-h((1/\mathrm{x)}+(1/\mathrm{y)})∥}_{{L}^{\mathrm{\infty }}({\mathrm{\Gamma }}_{d})}\le \epsilon$ in the sectors ${\mathrm{\Gamma }}_{d}=\{(x,y):x>\mathrm{0},\mathrm{}y>\mathrm{0},\mathrm{}\mathrm{(y}/\mathrm{x)}>d\}$. As consequences of the results, we obtain asymptotic behaviors of the previous inequalities. We also consider its distributional version ${∥u\circ S-v\circ \mathrm{\Pi }-w\circ R∥}_{{\mathrm{\Gamma }}_{d}}\le \epsilon$, where $u,v,w\in \mathrm{\scr D}\text{'}({\Bbb R}^{+})$, $S(x,y)=x+y$, $\mathrm{\Pi }(x,y)=xy$, $R(x,y)=\mathrm{1}/x+\mathrm{1}/y$, $x,y\in {\Bbb R}^{+}$, and the inequality ${∥·∥}_{{\mathrm{\Gamma }}_{d}}\le \epsilon$ means that $|\langle·,\phi \rangle|\le \epsilon\parallel \phi {\parallel }_{{L}^{\mathrm{1}}}$ for all test functions $\phi \in {C}_{c}^{\mathrm{\infty }}({\mathrm{\Gamma }}_{d})$.

#### Article information

Source
Abstr. Appl. Anal., Volume 2013, Special Issue (2013), Article ID 751680, 9 pages.

Dates
First available in Project Euclid: 26 February 2014

https://projecteuclid.org/euclid.aaa/1393448678

Digital Object Identifier
doi:10.1155/2013/751680

Mathematical Reviews number (MathSciNet)
MR3095363

Zentralblatt MATH identifier
07095327

#### Citation

Chung, Jaeyoung; Sahoo, Prasanna K. Stability of a Logarithmic Functional Equation in Distributions on a Restricted Domain. Abstr. Appl. Anal. 2013, Special Issue (2013), Article ID 751680, 9 pages. doi:10.1155/2013/751680. https://projecteuclid.org/euclid.aaa/1393448678

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