In this study, we present a convergence analysis of a Newton-like midpoint method for solving nonlinear equations in a Banach space setting. The semilocal convergence is analyzed in two different ways. The first one is shown by replacing the existing conditions with weaker and tighter continuity conditions, thereby enhancing its applicability. The second one uses more general $\omega$-continuity conditions and the majorizing principle. This approach includes only the first order Fréchet derivative and is applicable for problems that were otherwise hard to solve by using approaches seen in the literature. Moreover, the local convergence is established along with the existence and uniqueness region of the solution. The method is useful for solving Engineering and Applied Science problems. The paper ends with numerical examples that show the applicability of our convergence theorems in cases not covered in earlier studies.

In this paper, we present a technique that improves the applicability of the result obtained by Cordero et al. in 2024 for solving nonlinear equations. Cordero et al. assumed the involved operator to be differentiable at least five times to extend a two-step p-order method to order $p+3$. We obtained the convergence order of Cordero et al.’s method by assuming only up to the third-order derivative of the operator. Our analysis is in a more general commutative Banach algebra setting and provides a radius of the convergence ball. Finally, we validate our theoretical findings with several numerical examples. Also, the concept of basin of attraction is discussed with examples.

Symmetries play an important role in the study of a plethora of physical phenomena, including the study of microworlds. These phenomena reduce to solving nonlinear equations in abstract spaces. Therefore, it is important to design iterative methods for approximating the solutions, since closed forms of them can be found only in special cases. Several iterative methods were developed whose convergence was established under very general conditions. Numerous applications are also provided to solve systems of nonlinear equations and differential equations appearing in the aforementioned areas. The ball convergence analysis was developed for the King-like and Jarratt-like families of methods to solve equations under the same set of conditions. Earlier studies have used conditions up to the fifth derivative, but they failed to show the fourth convergence order. Moreover, no error distances or results on the uniqueness of the solution were given either. However, we provide such results involving the derivative only appearing on these methods. Hence, we have expanded the usage of these methods. In the case of the Jarratt-like family of methods, our results also hold for Banach space-valued equations. Moreover, we compare the convergence ball and the dynamical features both theoretically and in numerical experiments.

The implementation of Newton’s method for solving nonlinear equations in abstract domains requires the inversion of a linear operator at each step. Such an inversion may be computationally very expensive or impossible to find. That is why alternative iterative methods are developed in this article that require no inversion or only one inversion of a linear operator at each step. The inverse of the operator is replaced by a frozen sum of linear operators depending on the Fréchet derivative of an operator. The numerical examples illustrate that for all practical purposes, the new methods are as effective as Newton’s but much cheaper to implement. The same methodology can be used to create similar alternatives to other methods using inversions of linear operators such as divided differences or other linear operators.

High-convergence order iterative methods play a major role in scientific, computational and engineering mathematics, as they produce sequences that converge and thereby provide solutions to nonlinear equations. The convergence order is calculated using Taylor Series extensions, which require the existence and computation of high-order derivatives that do not occur in the methodology. These results cannot, therefore, ensure that the method converges in cases where there are no such high-order derivatives. However, the method could converge. In this paper, we are developing a process in which both the local and semi-local convergence analyses of two related methods of the sixth order are obtained exclusively from information provided by the operators in the method. Numeric applications supplement the theory.

In this article, we propose a new methodology to construct and study generalized three-step numerical methods for solving nonlinear equations in Banach spaces. These methods are very general and include other methods already in the literature as special cases. The convergence analysis of the specialized methods is been given by assuming the existence of high-order derivatives which are not shown in these methods. Therefore, these constraints limit the applicability of the methods to equations involving operators that are sufficiently many times differentiable although the methods may converge. Moreover, the convergence is shown under a different set of conditions. Motivated by the optimization considerations and the above concerns, we present a unified convergence analysis for the generalized numerical methods relying on conditions involving only the operators appearing in the method. This is the novelty of the article. Special cases and examples are presented to conclude this article.