Using the Schauder fixed point theorem, we prove the existence of impulsive fractional differential equations using Hilfer fractional derivative and nearly sectorial operators in this paper. We’ve gone over the two scenarios where the related semigroup is compact and noncompact for this purpose. We also go over an example to back up the main points.

<abstract><p>In this article, we study the existence and uniqueness results for a sequential nonlinear Caputo fractional sum-difference equation with fractional difference boundary conditions by using the Banach contraction principle and Schaefer's fixed point theorem. Furthermore, we also show the existence of a positive solution. Our problem contains different orders and four fractional difference operators. Finally, we present an example to display the importance of these results.</p></abstract>