We provide necessary and sufficient conditions for the existence and uniqueness of solutions belonging to the vector‐valued space of sequences for equations that can be modeled in the form urn:x-wiley:mma:media:mma4757:mma4757-math-0002 where X is a Banach space, A is a closed linear operator with domain D ( A ) defined on X , and G is a nonlinear function. The operator Δ γ denotes the fractional difference operator of order γ >0 in the sense of Grünwald‐Letnikov. Our class of models includes the discrete time Klein‐Gordon, telegraph, and Basset equations, among other differential difference equations of interest. We prove a simple criterion that shows the existence of solutions assuming that f is small and that G is a nonlinear term.