Sequential Fractional Hybrid Inclusions: A Theoretical Study via Dhage’s Technique and Special Contractions

Abstract

The most important objective of the present research is to establish some theoretical existence results on a novel combined configuration of a Caputo sequential inclusion problem and the hybrid integro-differential one in which the boundary conditions are also formulated as the hybrid multi-order integro-differential conditions. In this respect, firstly, some inequalities are proven in relation to the corresponding integral equation. Then, we employ some newly defined theoretical techniques with the help of the product operators on a Banach algebra and also with the aid of some special functions including α-ψ-contractions and α-admissible mappings to extract the existence criteria corresponding to the given mixed sequential hybrid BVPs. Some important useful properties such as the approximate endpoint property, (Cα)-property, and the compactness play a key role in this regard. The final part of the manuscript is devoted to formulating and computing two applicable examples to guarantee the correctness of the obtained results.