On the Boundary Value Problems of Ψ-Hilfer Fractional Differential Equations

The paper formulates the nonlinear Ψ–Hilfer BVP with the nonlocal boundary I^{1−γ;Ψ}_{0+}y(0)=r I^{1−γ;Ψ}_{0+}y(T), constructs the Green-type operator A using the Mittag–Leffler kernels, and proves existence of extremal solutions by lower/upper solutions and monotone iterations, plus uniqueness under a one-sided Lipschitz with an explicit contraction constant Ω(Ψ(T)−Ψ(0))^αL̃<1. These are given explicitly in the operator formula (5.4), Theorem 5.1 (existence, extremal solutions), and Theorem 5.2 (uniqueness, rate) in the paper, and match the candidate’s representation and iteration scheme, including the ξ-calibration for boundary mismatch and the same contraction estimate. The small differences are stylistic (paper uses relative compactness; model uses monotone convergence), not substantive. See the BVP statement and assumptions, the operator A and its use, and the uniqueness constant Ω as stated in the paper’s Theorems 5.1–5.2 and related derivations . The lower/upper solution machinery with ξ aligns with Definitions 2.3–2.4 and Section 4’s linear theory and location-of-roots argument .