An Irregular Fractional BVP: Existence and Uniqueness Results via Rus’s Fixed Point Theorem with an Explicit Contraction Bound Obtained Using Hypergeometric ₃F₂(1) Identities

This paper studies the existence and uniqueness of solutions for an irregular fractional boundary value problem (IFBVP) of order \(q\in(3/2,2]\). In order to apply Rus's fixed point theorem with the maximum and \(L^2\)metrics, we derive new estimates for the associated Green's function. We evaluate the squared Green’s function in a double integral to obtain the constant in the \(L^2\)estimate. Evaluating the mixed terms of this integral makes the appearance of the hypergeometric function ₃F₂(1), in which we apply classical transformation formulas to obtain finite sum evaluations of these terms, yielding the explicit constant \(\mathcal{K}_2(q,\rho,\theta)\). Using these estimates, we establish existence and uniqueness of solutions to the IFBVP under a Lipschitz condition on the nonlinear function \(f\)with constant \(L>0\), provided that \(L\sqrt{\mathcal{K}_2(q,\rho,\theta)}<1\). Numerical comparisons with an existing condition involving \(\Lambda(q,\rho,\theta)\)show that our condition is less restrictive for some parameter values, while the existing condition is less restrictive for others.