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

For a delay difference equation x(n+1)=∑k=−dnA(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.