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.