Maximal ℓ p ‐regularity of multiterm fractional equations with delay

We provide a characterization for the existence and uniqueness of solutions in the space of vector‐valued sequencesℓ p ( ℤ , X ) for the multiterm fractional delayed model in the formΔ α u ( n ) + λ Δ β u ( n ) = A u ( n ) + u ( n − τ ) + f ( n ) , n ∈ ℤ , α , β ∈ ℝ + , τ ∈ ℤ , λ ∈ ℝ , where X is a Banach space, A is a closed linear operator with domain D ( A ) defined on X , f ∈ ℓ p ( ℤ , X ) and Δ Γ denotes the Grünwald–Letkinov fractional derivative of order Γ  > 0. We also give some conditions to ensure the existence of solutions when adding nonlinearities. Finally, we illustrate our results with an example given by a general abstract nonlinear model that includes the fractional Fisher equation with delay.