## Abstract

Let $\mathbb{R}$ be the set of real numbers, ${\mathbb{R}}^{+}=\{x\in \mathbb{R}\mathrm{}\mid \mathrm{}x>\mathrm{0}\}$, $\u03f5\in {\mathbb{R}}_{+}$, and $f,g,h:{\mathbb{R}}^{+}\to \u2102$. 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: $|f(x+y)-g(xy)-h(\mathrm{(1}/\mathrm{x)}+\mathrm{(1}/\mathrm{y)})|\le \u03f5$, and ${\u2225f(x+y)-g(xy)-h\left(\mathrm{(1}/\mathrm{x)}+\mathrm{(1}/\mathrm{y)}\right)\u2225}_{{L}^{\mathrm{\infty}}({\mathrm{\Gamma}}_{d})}\le \u03f5$ 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 ${\u2225u\circ S-v\circ \mathrm{\Pi}-w\circ R\u2225}_{{\mathrm{\Gamma}}_{d}}\le \u03f5$, where $u,v,w\in \mathcal{D}\text{'}({\mathbb{R}}^{+})$, $S(x,y)=x+y$, $\mathrm{\Pi}(x,y)=xy$, $R(x,y)=\mathrm{1}/x+\mathrm{1}/y$, $x,y\in {\mathbb{R}}^{+}$, and the inequality ${\u2225\xb7\u2225}_{{\mathrm{\Gamma}}_{d}}\le \u03f5$ means that $|\langle \xb7,\phi \rangle |\le \u03f5\parallel \phi {\parallel}_{{L}^{\mathrm{1}}}$ for all test functions $\phi \in {C}_{c}^{\mathrm{\infty}}({\mathrm{\Gamma}}_{d})$.

## Citation

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