Abstract
The significance of this study is that the applicability of a Newton-Simpson-like scheme is extended using weaker conditions for convergence analysis. In earlier works, the local convergence of the scheme was established applying third-order Frechet derivative in finite-dimensional Euclidean spaces. In addition, no information about the error bounds, convergence radii and uniqueness of the solution was provided. Therefore, the applicability of the scheme is restricted. In this work, all these issues are addressed using Lipschitz continuous first derivative in Banach spaces. The convergence radii, uniqueness results and the error estimates are also discussed. Furthermore, the generalization of this analysis is studied under Holder condition. At last, using several problems, it is shown that our findings produce better results and effective enough to address such equations where earlier works fail.
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