Existence and Uniqueness of Positive Solutions for Hilfer–Hadamard-Type Fractional Differential Equations with γ-Concave and Sub-Homogeneous Operators

In this research, we present necessary and sufficient conditions for the existence and uniqueness of positive solutions for a class of Hilfer–Hadamard-type fractional differential equations with boundary value problems, including those with integral boundary conditions. The obtained results are conditional on a specific set of strong assumptions, which substantially narrow the class of admissible nonlinearities, coefficients, and boundary data. Thus, the present work extends the Hadamard-type framework to the Hilfer–Hadamard setting only within this restrictive regime, rather than providing a full extension to all Hilfer–Hadamard systems. We utilize the properties of γ-concave and sub-homogeneous operators along with two Banach fixed point theorems to achieve our results. An illustrative example is also provided to demonstrate the applicability of the methodology and highlight the main findings. The novelty of our work in this paper is twofold: First, we have expanded the scope by considering Hilfer–Hadamard-type fractional differential equations. Second, we employed a new fixed point theorem to improve our results within this particular framework.