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A General Class of Differential Hemivariational Inequalities Systems in Reflexive Banach Spaces

Author

Listed:
  • Lu-Chuan Ceng

    (Department of Mathematics, Shanghai Normal University, Shanghai 200234, China)

  • Ching-Feng Wen

    (Department of Medical Research, Kaohsiung Medical University Hospital, Kaohsiung 80708, Taiwan
    Center for Fundamental Science, and Nonlinear Analysis and Optimization, Kaohsiung Medical University, Kaohsiung 80708, Taiwan)

  • Yeong-Cheng Liou

    (Department of Medical Research, Kaohsiung Medical University Hospital, Kaohsiung 80708, Taiwan
    Research Center of Nonlinear Analysis and Optimization, Department of Healthcare Administration and Medical Informatics, Kaohsiung Medical University, Kaohsiung 807, Taiwan)

  • Jen-Chih Yao

    (Research Center for Interneural Computing, China Medical University Hospital, China Medical University, Taichung 40447, Taiwan)

Abstract

We consider an abstract system consisting of the parabolic-type system of hemivariational inequalities (SHVI) along with the nonlinear system of evolution equations in the frame of the evolution triple of product spaces, which is called a system of differential hemivariational inequalities (SDHVI). A hybrid iterative system is proposed via the temporality semidiscrete technique on the basis of the Rothe rule and feedback iteration approach. Using the surjective theorem for pseudomonotonicity mappings and properties of the partial Clarke’s generalized subgradient mappings, we establish the existence and priori estimations for solutions to the approximate problem. Whenever studying the parabolic-type SHVI, the surjective theorem for pseudomonotonicity mappings, instead of the KKM theorems exploited by other authors in recent literature for a SHVI, guarantees the successful continuation of our demonstration. This overcomes the drawback of the KKM-based approach. Finally, via the limitation process for solutions to the hybrid iterative system, we derive the solvability of the SDHVI with no convexity of functions u ↦ f l ( t , x , u ) , l = 1 , 2 and no compact property of C 0 -semigroups e A l ( t ) , l = 1 , 2 .

Suggested Citation

  • Lu-Chuan Ceng & Ching-Feng Wen & Yeong-Cheng Liou & Jen-Chih Yao, 2021. "A General Class of Differential Hemivariational Inequalities Systems in Reflexive Banach Spaces," Mathematics, MDPI, vol. 9(24), pages 1-21, December.
    • Handle: RePEc:gam:jmathe:v:9:y:2021:i:24:p:3173-:d:698593
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    References listed on IDEAS

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    1. Stanisław Migórski & Shengda Zeng, 2018. "A class of differential hemivariational inequalities in Banach spaces," Journal of Global Optimization, Springer, vol. 72(4), pages 761-779, December.
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