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A Multivalued History-Dependent Operator and Implicit Evolution Inclusions. I

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Abstract

On the space of continuous functions from a line segment to a reflexive Banach space, we consider some operator whose values are closed convex subsets of the space. If the values are singletons, the operator becomes a well-known single-valued history-dependent operator. We study the properties of the operator, prove a fixed-point theorem analogous to the fixed-point theorem for single-valued history-dependent operators, and provide some examples. The results are applied to study implicit (unresolved for derivatives) evolution inclusions with maximal monotone operators and with perturbations in a Hilbert space. These perturbations are single-valued and multivalued history-dependent operators.

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References

  1. Michael E., “Continuous selections,” Ann. Math., vol. 63, no. 2, 361–382 (1956).

    Article  MathSciNet  Google Scholar 

  2. Sofonea M. and Migórski St., Variational-Hemivariational Inequalities with Applications, Boca Raton, London and New York (2018).

    MATH  Google Scholar 

  3. Migórski St., Sofonea M., and Zeng Sh., “Well-posedness of history-dependent sweeping processes,” SIAM J. Math. Anal., vol. 51, no. 2, 1082–1107 (2019).

    Article  MathSciNet  Google Scholar 

  4. Himmelberg C. J., “Measurable relations,” Fund. Math., vol. 87, 53–72 (1975).

    Article  MathSciNet  Google Scholar 

  5. Attouch H., Variational Convergence for Functions and Operators, Pitman, Boston, London, and Melbourne (1984).

    MATH  Google Scholar 

  6. Aumann R. J., “Integrals of set-valued functions,” J. Math. Anal. Appl., vol. 12, no. 1, 1–12 (1965).

    Article  MathSciNet  Google Scholar 

  7. Hiai F. and Umegaki H., “Integrals, conditional expectations, and martingales of multivalued functions,” J. Multivariate Anal., vol. 7, 149–182 (1977).

    Article  MathSciNet  Google Scholar 

  8. Diestel J., Geometry of Banach Spaces: Selected Topics, Springer, Berlin etc. (1975).

    Book  Google Scholar 

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Authors and Affiliations

  1. Matrosov Institute of Systems Dynamics and Control, Irkutsk, Russia

    A. A. Tolstonogov

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  1. A. A. Tolstonogov
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Correspondence to A. A. Tolstonogov.

Additional information

Translated from Sibirskii Matematicheskii Zhurnal, 2021, Vol. 62, No. 3, pp. 672–682. https://doi.org/10.33048/smzh.2021.62.317

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Tolstonogov, A.A. A Multivalued History-Dependent Operator and Implicit Evolution Inclusions. I. Sib Math J 62, 545–553 (2021). https://doi.org/10.1134/S0037446621030174

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  • DOI: https://doi.org/10.1134/S0037446621030174

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