On Katugampola Fractional Operator and Katugampola Pantograph Problems in Jα,β

This paper explores and establishes solvability results for a nonlinear Katugampola fractional pantograph initial value problem formulated in the integral-type Hölder space Jα,β. The constructed model blends generalized fractional dynamics with proportional delay terms, offering a unified setting for studying systems with memory-dependent behavior. To obtain the principal results, we begin by establishing several analytical properties of Katugampola-type fractional operators on Jα,β, such as continuity, boundedness, and their mapping behavior. These properties are subsequently combined with fixed-point techniques and the Hausdorff measure of noncompactness to establish existence and uniqueness of solutions under appropriate hypotheses. The theoretical findings are further illustrated through two representative examples and a numerical experiment for various choices of α and β, thereby confirming the practical relevance of the proposed framework.