Some notes on convergence of homotopy based methods for functional equations

Abstract

Although homotopy-based methods, namely homotopy analysis method and<br />homotopy perturbation method, have largely been used to solve functional<br />equations, there are still serious questions on the convergence issue of these<br />methods. Some authors have tried to prove convergence of these methods, but<br />the researchers in this article indicate that some of those discussions are faulty.<br />Here, after criticizing previous works, a su cient condition for convergence of<br />homotopy methods is presented. Finally, examples are given to show that even<br />if the homotopy method leads to a convergent series, it may not converge to<br />the exact solution of the equation under consideration.