Existence of an Almost Periodic Solution in a Difference Equation with Infinite Delay

In order to obtain the existence of an almost periodic functional difference equation x(n + 1) = ƒ(n,xn ),n ∈ Z + and where xn is defined by xn (s) = x(n + s) for s ∈ Z −, on an axiomatic phase space B, we consider a certain stability property, which is referred to as BS-stable under disturbances from Ω(ƒ) with respect to K, this stability implies ρ-stable under disturbances from Ω(ƒ) with respect to compact set K.