Multi-step methods for equations

This study is about a comprehensive convergence analysis of higher-order Newton-type iterative methods within the framework of Banach spaces. The primary objective is to ascertain locally unique solutions for systems of nonlinear equations. These Newton-type methods are notable for their reliance only on first-order derivative calculations. However, their conventional convergence analysis relies on Taylor expansions, which inherently assume the existence of higher-order derivatives, which are not present on the methods. This dependency limits their practicality. To overcome this limitation, we develop both local and semi-local convergence analysis by imposing hypotheses solely on first-order derivatives that are used by the methods. In the local analysis, our primary focus is to establish convergence domain boundaries while simultaneously estimating error approximations for successive iterates. In the semi-local analysis, we provide sufficient conditions based on arbitrarily chosen initial approximations within a given domain, ensuring the convergence of iterative sequence to a specific solution within that domain. Furthermore, we claim uniqueness of the solution by providing the requisite criteria within the specified domain.Therefore, with these actions, the applicability of these methods is extended in the cases not covered earlier, and under weak conditions. The same technique can be employed to extend the utilization of other methods relying on inverses of linear operators along the same lines. Finally, we validate our theoretical deductions by applying them to real-world problems and presenting the corresponding test results.