On exponential dichotomy, Bohl–Perron type theorems and stability of difference equations

For a delay difference equation x(n+ 1) = n k=−d A(n, k)x(k)+ f (n) in a Banach space the following result is proved: if for any f ∈ lp the solution is x ∈ lp then the solution of the homogeneous equation (f ≡ 0) is exponentially stable. This result is applied to obtain new explicit conditions for exponential stability of a scalar nonautonomous delay difference equation.  2004 Elsevier Inc. All rights reserved.