This paper investigates the existence of solutions for initial value problems (IVPs) involving implicit fractional differential inclusions defined via the Hilfer-Katugampola fractional derivative. The Hilfer-Katugampola operator, recently introduced as a generalization of Katugampola and Caputo-Katugampola derivatives, encompasses a wide class of fractional operators. We establish existence results for the multivalued fractional differential problem under convexity and compactness assumptions on the multivalued right-hand side, leveraging Bohnenblust-Karlin fixed point theorem and contraction principles for multivalued maps. An illustrative example is provided to demonstrate the applicability of the main theoretical results. Our work contributes to the emerging theory of fractional differential inclusions governed by fractional derivatives of generalized type.
By examining the properties of a certain linear transformation of functionals, we present applications of Cauchy, Aczel, Callebaut, and Beckenbach type inequalities. Additionally, we provide results for complex functionals.
