Refined Green-Function Estimates for a Caputo Fractional Three-Point Boundary Value Problem: Sharper Existence, Uniqueness, and Ulam–Hyers Stability Conditions

We study a Caputo fractional three-point boundary value problem of order α∈(c−1,c] and establish three interrelated contributions, all resting on a single refined pointwise L2 estimate for the associated Green function Gr(t,s). First, we derive a tighter upper bound for sup0<t<1∫01Gr2(t,s)ds by retaining a sign-definite negative mixed term that the classical L1-based analysis of Shivanian discards. The resulting admissible Lipschitz constant κB is explicit in α, a, b, c and exceeds Shivanian’s constant κS under an explicit algebraic condition; a closed-form refinement κB∗≥κB follows by maximising the pointwise bound in closed form. On Shivanian’s benchmark, the admissible constant rises from 14.646 to 23.220 and then to 32.165. Second, the same contraction constant yields an explicit Ulam–Hyers stability theorem for this problem. While a stability estimate already follows from the classical L1 condition, the refined constant both enlarges the range of admissible Lipschitz constants for which stability is certified and yields a strictly smaller stability constant. Third, we establish quantitative continuous-dependence bounds with respect to the nonlinearity h and the boundary parameter a, and characterize the deterioration of the contraction-based boundary-parameter estimate as the problem approaches resonance. For a fixed Lipschitz constant, this estimate becomes singular as the perturbed contraction factor approaches one and ceases to apply once the contraction condition fails. A more accurate analysis of the Green function thus simultaneously sharpens solvability conditions, stability estimates, and sensitivity bounds for nonlinear Caputo fractional boundary value problems.