We provide necessary and sufficient conditions for the existence and uniqueness of solutions belonging to the vector‐valued space of sequences ℓpfalse(double-struckZ,Xfalse) for equations that can be modeled in the form normalΔαu(n)+λnormalΔβu(n)=Au(n)+G(u)(n)+f(n),n∈Z,α,β>0,λ≥0, where X is a Banach space, f∈ℓpfalse(double-struckZ,Xfalse), A is a closed linear operator with domain D(A) defined on X, and G is a nonlinear function. The operator Δγ 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.

In this work we provide a new and effective characterization for the existence and uniqueness of solutions for nonlocal time-discrete equations with delays, in the setting of vector-valued Lebesgue spaces of sequences. This characterization is given solely in terms of the R-boundedness of the data of the problem, and in the context of the class of UMD Banach spaces.

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