Faculty Publications
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Item Kantorovich-type results for generalized equations with applications(Springer Science and Business Media B.V., 2022) Regmi, S.; Argyros, I.K.; George, S.; Argyros, C.I.Kantorovich-type results for generalized equations are extended with no additional conditions using Newton procedures. Iterates are shown to belong in a smaller domain resulting to tighter Lipschitz constants and a finer convergence analysis than in earlier works. © 2022, The Author(s), under exclusive licence to The Forum D’Analystes.Item Extended Kantorovich theory for solving nonlinear equations with applications(Springer Nature, 2023) Regmi, S.; Argyros, I.K.; George, S.; Argyros, M.The Kantorovich theory plays an important role in the study of nonlinear equations. It is used to establish the existence of a solution for an equation defined in an abstract space. The solution is usually determined by using an iterative process such as Newton’s or its variants. A plethora of convergence results are available based mainly on Lipschitz-like conditions on the derivatives, and the celebrated Kantorovich convergence criterion. But there are even simple real equations for which this criterion is not satisfied. Consequently, the applicability of the theory is limited. The question there arises: is it possible to extend this theory without adding convergence conditions? The answer is, Yes! This is the novelty and motivation for this paper. Other extensions include the determination of better information about the solution, i.e. its uniqueness ball; the ratio of quadratic convergence as well as more precise error analysis. The numerical section contains a Hammerstein-type nonlinear equation and other examples as applications. © 2023, The Author(s) under exclusive licence to Sociedade Brasileira de Matemática Aplicada e Computacional.Item Extended convergence for two-step methods with non-differentiable parts in Banach spaces(Springer Science and Business Media B.V., 2024) Argyros, I.K.; George, S.; Senapati, K.In this study, we have extended the applicability of two-step methods with non-differentiable parts for solving nonlinear equations defined in Banach spaces. The convergence analysis uses conditions weaker than the ones in earlier studies. Other advantages include computable a priori error distances based on generalized conditions, an extended region of convergence as well as a better knowledge of the isolation for the solutions. By setting the divided differences equal to zero the results can be used to solve equations with differentiable part too. © The Author(s), under exclusive licence to The Forum D’Analystes 2023.Item Hybrid Newton-like Inverse Free Algorithms for Solving Nonlinear Equations(Multidisciplinary Digital Publishing Institute (MDPI), 2024) Argyros, I.K.; George, S.; Regmi, S.; Argyros, C.I.Iterative algorithms requiring the computationally expensive in general inversion of linear operators are difficult to implement. This is the reason why hybrid Newton-like algorithms without inverses are developed in this paper to solve Banach space-valued nonlinear equations. The inverses of the linear operator are exchanged by a finite sum of fixed linear operators. Two types of convergence analysis are presented for these algorithms: the semilocal and the local. The Fréchet derivative of the operator on the equation is controlled by a majorant function. The semi-local analysis also relies on majorizing sequences. The celebrated contraction mapping principle is utilized to study the convergence of the Krasnoselskij-like algorithm. The numerical experimentation demonstrates that the new algorithms are essentially as effective but less expensive to implement. Although the new approach is demonstrated for Newton-like algorithms, it can be applied to other single-step, multistep, or multipoint algorithms using inverses of linear operators along the same lines. © 2024 by the authors.Item A class of derivative free schemes for solving nondifferentiable Banach space valued equations(Springer Science and Business Media B.V., 2024) Argyros, I.K.; George, S.In this study, we have extended the applicability of two-step methods with non-differentiable parts for solving nonlinear equations defined in Banach spaces. The convergence analysis uses conditions weaker than the ones in earlier studies. Other advantages include computable a priori error distances based on generalized conditions, an extended region of convergence as well as a better knowledge of the isolation for the solutions. The earlier results use assumptions on the eighth derivative of the main operator. But there are no derivatives on the schemes. Moreover, the previous results cannot be used for nondifferentiable equations although the schemes may converge. Numerical examples validate further our approach. © The Author(s), under exclusive licence to The Forum D’Analystes 2024.