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)|≤ϵ, x,y>0 will be proved, where f,g,h:ℝ+→ℂ. As a distributional analogue of the above inequality, the stability of inequality ∥u∘(x+y)-v∘(xy)-w∘(1/x+1/y)∥≤ϵ will be proved, where u,v,w∈𝒟'(ℝ+) and ∘ denotes the pullback of distributions.