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Convergence of Dissipative-Like Dynamics and Algorithms Governed by Set-Valued Nonexpansive Mappings

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Abstract

In this paper, we consider a differential inclusion governed by a set-valued nonexpansive mapping and study the asymptotic behavior (weak and strong convergence) of its solutions with various assumptions on this mapping. Then for a set-valued nonexpansive mapping, we define the corresponding resolvent (proximal) operator as a set-valued mapping and study some of its elementary properties. Subsequently, we apply the resolvent operator to state the implicit discretization of the differential inclusion and study the asymptotic behavior of its solutions which yields similar convergence results as in the continuous case. This provides an algorithm for approximating a fixed point of a set-valued nonexpansive mapping which extends the classical proximal point algorithm. An application to variational inequalities and a numerical comparison with another iterative method for approximating a fixed point of set-valued nonexpansive mappings are also presented.

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Acknowledgements

The authors are grateful to the anonymous referee for their careful observations and valuable comments leading to the improvement of the paper.

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Authors and Affiliations

  1. Department of Mathematics, University of Zanjan, P.O. Box 45195-313, Zanjan, Iran

    Hadi Khatibzadeh

  2. Department of Mathematics, Institute for Advanced Studies in Basic Sciences, P.O. Box 45195-1159, Zanjan, Iran

    Mohsen Rahimi Piranfar & Jamal Rooin

Authors
  1. Hadi Khatibzadeh
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  2. Mohsen Rahimi Piranfar
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  3. Jamal Rooin
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Corresponding author

Correspondence to Hadi Khatibzadeh.

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Communicated by Rosihan M. Ali.

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Khatibzadeh, H., Rahimi Piranfar, M. & Rooin, J. Convergence of Dissipative-Like Dynamics and Algorithms Governed by Set-Valued Nonexpansive Mappings. Bull. Malays. Math. Sci. Soc. 44, 1101–1121 (2021). https://doi.org/10.1007/s40840-020-00997-6

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  • DOI: https://doi.org/10.1007/s40840-020-00997-6

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