Triple positive solutions and product-integration Nyström approximation for a doubly nonlocal fractional integro-boundary problem with weakly singular weights

A doubly nonlocal fractional integro-boundary problem with weakly singular weights is investigated for the order range $2<\alpha <3$. The governing equation contains a weakly singular Volterra memory operator, while the boundary condition involves a weighted integral functional with endpoint singularities. This configuration is relevant to anomalous transport and hereditary diffusion models in which interior memory and boundary feedback act simultaneously. An explicit Green representation is derived for the associated linear problem and is normalized by an endpoint-aware singular profile, which leads to two-sided kernel estimates and to a positive cone adapted to the doubly singular structure. Sufficient conditions for at least three positive solutions are then obtained by means of a Leggett–Williams framework. In parallel, a product-integration Nyström discretization is constructed for the weakly singular state operator and the weakly singular boundary operator. A uniform-grid convergence estimate is proved under a local contractivity condition, and a graded-mesh extension is formulated for endpoint-dominated regimes. Numerical experiments are reported for mesh refinement, boundary-parameter sensitivity, singularity-intensity variation, and graded-mesh comparison. The results confirm that the proposed approach provides a coherent analytic-computational framework for fractional models with simultaneous nonlocal singularities on the equation and boundary sides.