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Newton-type method for solving generalized inclusion

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Abstract

In this paper, we propose and study the problem of solve non-linear inclusion problem in Banach spaces, where the involved operator can be written as sum of a Fréchet differentiable function with a continuous perturbation. We use a specific technique introduced by Robinson (Numer. Math. 19, 341–347, 1972) to obtain Newton-Kantorovich theorem, which extends the results of Rokne (Numer. Math. 18, 401–412, 1971), for instance. In our main convergence result, we assume a kind of Hölder condition. Thus, one of the major difficulties to obtain our main result is to show that the sequence of scalars associated with the Newton sequence is convergent. Numerical examples are given to justify the theoretical results.

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Acknowledgements

We are thankful to the anonymous reviewers, and to the Associate Editor, whose comments and suggestions improved the presentation of our work.

Funding

P.S.M. Santos was supported in part by CNPq Grant 311825/2019-2. G.N. Silva was supported in part by CNPq Grant 434796/2018-2. R.C.M. Silva was supported in part by CAPES/FAPEAM Grant 062.01818/2015.

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Authors and Affiliations

  1. Campus Ministro Reis Velloso, Universidade Federal do Delta do Parnaíba, Parnaíba, PI, Brazil

    P. S. M. Santos

  2. Centro das Ciências Exatas e das Tecnologias, Universidade Federal do Oeste da Bahia, CEP 47808-021, Barreiras, BA, Brazil

    G. N. Silva

  3. Departamento de Matemática, Universidade Federal do Amazonas, Manaus, AM, Brazil

    R. C. M. Silva

Authors
  1. P. S. M. Santos
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  2. G. N. Silva
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  3. R. C. M. Silva
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Correspondence to P. S. M. Santos.

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Santos, P.S.M., Silva, G.N. & Silva, R.C.M. Newton-type method for solving generalized inclusion. Numer Algor 88, 1811–1829 (2021). https://doi.org/10.1007/s11075-021-01096-8

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