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 Kannappanf(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>0will be proved, wheref,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)∥≤ϵwill be proved, whereu,v,w∈𝒟'(ℝ+)and∘denotes the pullback of distributions.