The stability of certain functional equations

Abstract

The aim of this paper is to prove the stability (in the sense of Ulam) of the functional equation: \[ f ( t ) = α ( t ) + β ( t ) f ( ϕ ( t ) ) , f(t) = \alpha (t) + \beta (t)f(\phi (t)), \] where α \alpha and β \beta are given complex valued functions defined on a nonempty set S S such that sup { | β ( t ) | : t ∈ S } > 1 \sup \{ |\beta (t)|:t \in S\} > 1 and ϕ \phi is a given mapping of S S into itself.