A Class of Generalized Evolutionary Problems Driven by Variational Inequalities and Fractional Operators

Abstract

This paper is devoted to a generalized evolution system called fractional partial differential variational inequality which consists of a mixed quasi-variational inequality combined with a fractional partial differential equation in a Banach space. Invoking the pseudomonotonicity of multivalued operators and a generalization of the Knaster-Kuratowski-Mazurkiewicz theorem, first, we prove that the solution set of the mixed quasi-variational inequality involved in system is nonempty, closed and convex. Next, the measurability and upper semicontinuity for the mixed quasi-variational inequality with respect to the time variable and state variable are established. Finally, the existence of mild solutions for the system is delivered. The approach is based on the theory of operator semigroups, the Bohnenblust-Karlin fixed point principle for multivalued mappings, and theory of fractional operators.

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Acknowledgements

The authors would like to express their thanks to the Editors and the Reviewers for their helpful comments and advices.

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Correspondence to Shengda Zeng.

Additional information

This project has received funding from the European Union’s Horizon 2020 Research and Innovation Programme under the Marie Skł odowska-Curie grant agreement No. 823731 – CONMECH. It is supported by the National Science Center of Poland under Maestro Project No. UMO-2012/06/A/ST1/00262, and National Science Center of Poland under Preludium Project No. 2017/25/N/ST1/00611. The first author is also supported by Qinzhou University Project No. 2018KYQD06, and the International Project co-financed by the Ministry of Science and Higher Education of Republic of Poland under Grant No. 3792/GGPJ/H2020/2017/0.

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Migórski, S., Zeng, S. A Class of Generalized Evolutionary Problems Driven by Variational Inequalities and Fractional Operators. Set-Valued Var. Anal 27, 949–970 (2019). https://doi.org/10.1007/s11228-018-0502-7

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Keywords

  • Fractional partial differential variational inequalities
  • Caputo derivative
  • Knaster-Kuratowski-Mazurkiewicz theorem
  • Bohnenblust-Karlin fixed point principle
  • ϕ-pseudomonotonicity
  • Mixed quasi-variational inequalities

Mathematics Subject Classification (2010)

  • 47J20
  • 49J40
  • 35J88
  • 26A33
  • 34A08