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

Lu-Chuan Ceng, Ching-Feng Wen, Yeong-Cheng Liou and Jen-Chih Yao
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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
Yeong-Cheng Liou: Department of Medical Research, Kaohsiung Medical University Hospital, Kaohsiung 80708, Taiwan
Jen-Chih Yao: Research Center for Interneural Computing, China Medical University Hospital, China Medical University, Taichung 40447, Taiwan

Mathematics, 2021, vol. 9, issue 24, 1-21

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 .

Keywords: systems of differential hemivariational inequalities; C0-semigroup; Rothe rule; Pseudomonotonicity; Partial Clarke’s generalized subdifferential (search for similar items in EconPapers)
JEL-codes: C (search for similar items in EconPapers)
Date: 2021
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