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

Abstract

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