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Volume 1, Issue 1

Local convergence of a higher-order method in Banach spaces

Pages: 68 – 80


Debasis Sharma, Sanjaya Kumar Parhi


We present the study of the local convergence for a fourth-order convergent nonlinear system solver under Lipschitz continuous Fr\'{e}chet derivative in Banach spaces. Our analysis only needs the first-order Fr\'{e}chet derivative to show the convergence and provides the radius of convergence ball, the computable error bounds and the uniqueness of the solution. This study is applicable in solving such problems for which higher-order derivative-based earlier studies fail. Furthermore, the generalization of this analysis using H\"{o}lder condition is studied. Finally, we solve a nonlinear system and various nonlinear integral equations to validate the usefulness of our theoretical outcomes.

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Received: 7 March 2020; Accepted: 20 March 2020.