Existence and uniqueness of solutions for a class of discrete-time fractional equations of order 2 < α ≤ 3

Abstract

In this paper we consider a class of linear difference equations of fractional order $2 <\alpha< 3$ in the sense of Riemann–Liouville. The explicit solution for this model is provided in terms of a fractional resolvent sequence which allows to write the solution to this equation as a variation of constant formula. We also characterize the existence and uniqueness of solutions in $l_p(N0;X)$ spaces with X being a UMD-space in terms of the R-boundedness of the operator symbol of the model. Moreover, we are able to relax this condition in the case of Hilbert spaces. Finally, we illustrate our results with an example that involves the generator of a contraction semigroup.