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A class of differential hemivariational inequalities in Banach spaces

Author

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  • Stanisław Migórski

    (Qinzhou University
    Jagiellonian University in Krakow)

  • Shengda Zeng

    (Jagiellonian University in Krakow)

Abstract

In this paper we investigate an abstract system which consists of a hemivariational inequality of parabolic type combined with a nonlinear evolution equation in the framework of an evolution triple of spaces which is called a differential hemivariational inequality [(DHVI), for short]. A hybrid iterative system corresponding to (DHVI) is introduced by using a temporally semi-discrete method based on the backward Euler difference scheme, i.e., the Rothe method, and a feedback iterative technique. We apply a surjectivity result for pseudomonotone operators and properties of the Clarke subgradient operator to establish existence and a priori estimates for solutions to an approximate problem. Finally, through a limiting procedure for solutions of the hybrid iterative system, the solvability of (DHVI) is proved without imposing any convexity condition on the nonlinear function $$u\mapsto f(t,x,u)$$ u ↦ f ( t , x , u ) and compactness of $$C_0$$ C 0 -semigroup $$e^{A(t)}$$ e A ( t ) .

Suggested Citation

  • 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.
    • Handle: RePEc:spr:jglopt:v:72:y:2018:i:4:d:10.1007_s10898-018-0667-5
    DOI: 10.1007/s10898-018-0667-5
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    References listed on IDEAS

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    1. Xing Wang & Nan-jing Huang, 2014. "A Class of Differential Vector Variational Inequalities in Finite Dimensional Spaces," Journal of Optimization Theory and Applications, Springer, vol. 162(2), pages 633-648, August.
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    Cited by:

    1. 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.
    2. Jinjie Liu & Xinmin Yang & Shengda Zeng & Yong Zhao, 2022. "Coupled Variational Inequalities: Existence, Stability and Optimal Control," Journal of Optimization Theory and Applications, Springer, vol. 193(1), pages 877-909, June.
    3. Guo, Furi & Wang, JinRong & Han, Jiangfeng, 2022. "Dynamic viscoelastic unilateral constrained contact problems with thermal effects," Applied Mathematics and Computation, Elsevier, vol. 424(C).

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