A class of differential hemivariational inequalities in Banach spaces

Stanisław Migórski () and Shengda Zeng ()
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Stanisław Migórski: Qinzhou University
Shengda Zeng: Jagiellonian University in Krakow

Journal of Global Optimization, 2018, vol. 72, issue 4, No 8, 779 pages

Abstract: 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 ) .

Keywords: Differential hemivariational inequality; $$C_0$$ C 0 -semigroup; Rothe method; Pseudomonotone; Clarke subdifferential; 35L15; 35L86; 35L87; 74Hxx; 74M10 (search for similar items in EconPapers)
Date: 2018
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Citations: View citations in EconPapers (3)

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DOI: 10.1007/s10898-018-0667-5

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