Abstract
<title>Abstract</title> <p>We study an implicit product-integration scheme for nonlinear Caputo initial-value problems written as weakly singular Volterra equations. The Abel kernel is integrated exactly against a piecewise-linear interpolant of the nonlinear term, producing a lower-triangular nonlinear system with a nonzero diagonal. A computable diagonal condition gives discrete existence and uniqueness without the stronger global Banach restriction. Under this condition, a positive triangular majorant gives convergence of global and nodewise Picard iterations and a componentwise a posteriori bound for any inexact discrete vector. The bound is obtained on the same mesh by forward substitution and separates iteration error from discretization error. A blockwise Volterra estimate yields a mesh-independent resolvent bound and second-order convergence on sufficiently fine meshes. If the solution has a power-type singularity near the initial point, the error on a graded mesh is of order $M^{-\min\{r\sigma,2\}}$; order two follows when $r\sigma\ge2$. Numerical tests compare the method with fractional Adams predictor--evaluate--correct--evaluate and left-endpoint product integration. The tests cover smooth, singular, and non-manufactured problems and report estimator effectivity, convergence rates, iteration counts, and execution times. Mathematics Subject Classification (2020): 34A08; 45D05; 65R20; 65L70; 47H10.</p>