Ulam’s Type Stability of Involutional-Exponential Functional Equations

Abstract EN

Let S be a commutative semigroup, f , g : S → C and σ : S → S an involution.

In this paper we consider the stability of involution-exponential functional equations f x + σ y - g x f y ≤ ϕ x resp .,   ϕ y ,   | f ( x + σ y ) - f ( x ) g ( y ) | ≤ ϕ ( x )   [ resp .,   ϕ ( y ) ] for all x , y ∈ S , where ϕ : S → R + satisfies the growth condition: there exists C > 1 such that lim k → ∞ C - k ϕ ( k x ) = 0 for each x ∈ S .

We also consider the stability of L ∞ -version | f ( x + σ y ) - f ( x ) f ( y ) | L ∞ ( R 2 n ) ≤ ϵ , where f : R n → C is a locally integrable function.