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

Abstract

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