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Extended Kantorovich theory for solving nonlinear equations with applications

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Abstract

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.

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Authors and Affiliations

  1. Department of Mathematics, University of Houston, Houston, TX, 772005, USA

    Samundra Regmi

  2. Department of Computing and Mathematical Sciences, Cameron University, Lawton, OK, 73505, USA

    Ioannis K. Argyros

  3. Department of Mathematical and Computational Sciences, National Institute of Technology Karnataka, Mangaluru, 575 025, India

    Santhosh George

  4. Department of Computer Science, University of Oklahoma, Norman, OK, 73019, USA

    Michael Argyros

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  1. Samundra Regmi
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  2. Ioannis K. Argyros
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  3. Santhosh George
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  4. Michael Argyros
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Correspondence to Ioannis K. Argyros.

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Communicated by Andreas Fischer.

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Regmi, S., Argyros, I.K., George, S. et al. Extended Kantorovich theory for solving nonlinear equations with applications. Comp. Appl. Math. 42, 76 (2023). https://doi.org/10.1007/s40314-023-02203-2

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