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Abstract
The homotopy perturbation method applied to the Green’s-function (Hammerstein) form of a nonlinear two-point boundary value problem produces a series whose terms satisfy the boundary conditions and are obtained by single integrations. This coupling is known, and for the homotopy considered it coincides with the Adomian decomposition method in integral form. The present paper studies when, and to what, the resulting series converges. Three results of increasing sharpness are proved: a contraction theorem with a geometric error bound; a majorant theorem that reduces convergence to a scalar inequality and gives closed-form coefficient bounds of Catalan and Lambert- type; and, when the kernel, the source and the nonlinearity have fixed signs, a Pringsheim-type theorem showing that the series converges as far as the analytic solution branch extends and diverges beyond a fold of that branch. For the Bratu problem this yields convergence if and only if , where , whereas the first two results guarantee only and . For a Fredholm equation with the kernel , the linearisation at the intended solution becomes singular at ; numerical branch tracking indicates that beyond this value the series converges to a different solution. The computed coefficients are consistent with the decay expected at a quadratic fold, and extrapolation of 60 terms reproduces to seven significant digits. Comparisons with Picard iteration, finite differences and the Nyström method indicate that the approach is most useful in symbolic and parametric computations.