Stability of an alternative functional equation

Abstract

Let f : S → X map an abelian semigroup ( S , + ) into a Banach space ( X ‖ ⋅ ‖ ) . We deal with stability of the following alternative functional equation f ( x + y ) + f ( x ) + f ( y ) ≠ 0 ⟹ f ( x + y ) = f ( x ) + f ( y ) . We assume that ‖ f ( x + y ) + f ( x ) + f ( y ) ‖ > Φ 1 ( x , y ) ⟹ ‖ f ( x + y ) − f ( x ) − f ( y ) ‖ ⩽ Φ 2 ( x , y ) for all x , y ∈ S , where Φ 1 , Φ 2 : S → R + are given functions and prove that, under some additional assumptions on Φ 1 , Φ 2 , there exists a unique additive mapping a : S → X such that ‖ f ( x ) − a ( x ) ‖ ⩽ Ψ ( x ) for x ∈ S , where Ψ : S → R + is a function which can be explicitly computed starting from Φ 1 and Φ 2 .