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

Abstract EN

Let ℝ be the set of real numbers, ℝ+={x∈ℝ∣x>0}, ϵ∈ℝ+, and f,g,h:ℝ+→ℂ.

As classical and L∞ 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((1/x)+(1/y))|≤ϵ, and f(x+y)-g(xy)-h(1/x)+(1/y)L∞(Γd)≤ϵ in the sectors Γd={(x,y):x>0,y>0,(y/x)>d}.

As consequences of the results, we obtain asymptotic behaviors of the previous inequalities.

We also consider its distributional version u∘S-v∘Π-w∘RΓd≤ϵ, where u,v,w∈?'(ℝ+), S(x,y)=x+y, Π(x,y)=xy, R(x,y)=1/x+1/y, x,y∈ℝ+, and the inequality ·Γd≤ϵ means that |〈·,φ〉|≤ϵ∥φ∥L1 for all test functions φ∈Cc∞(Γd).