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Convergence of a Time Discretization for a Nonlinear Second-Order Inclusion

Part of: Existence theories Equations and inequalities involving nonlinear operators

Published online by Cambridge University Press:  29 May 2017

Krzysztof Bartosz ,
Leszek Gasiński ,
Zhenhai Liu  and
Paweł Szafraniec
Krzysztof Bartosz*
Affiliation:
Jagiellonian University, Faculty of Mathematics and Computer Science, ul. Lojasiewicza 6, 30348 Krakow, Poland (krzysztof.bartosz@ii.uj.edu.pl)
Leszek Gasiński
Affiliation:
Jagiellonian University, Faculty of Mathematics and Computer Science, ul. Lojasiewicza 6, 30348 Krakow, Poland (krzysztof.bartosz@ii.uj.edu.pl)
Zhenhai Liu
Affiliation:
College of Sciences, Guangxi University for Nationalities, Nanning, Guangxi 530006, Peoples Republic of China
Paweł Szafraniec
Affiliation:
Jagiellonian University, Faculty of Mathematics and Computer Science, ul. Lojasiewicza 6, 30348 Krakow, Poland (krzysztof.bartosz@ii.uj.edu.pl)
*
*Corresponding author.
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Abstract

We study an abstract second order inclusion involving two nonlinear single-valued operators and a nonlinear multi-valued term. Our goal is to establish the existence of solutions to the problem by applying numerical scheme based on time discretization. We show that the sequence of approximate solution converges weakly to a solution of the exact problem. We apply our abstract result to a dynamic, second-order-in-time differential inclusion involving a Clarke subdifferential of a locally Lipschitz, possibly non-convex and non-smooth potential. In the two presented examples the Clarke subdifferential appears either in a source term or in a boundary term.

Keywords

time discretizationdifferential inclusionsnonconvex potentialweak solutionRothe method

MSC classification

Primary: 49J40: Variational methods including variational inequalities 47J20: Variational and other types of inequalities involving nonlinear operators (general) 47J22: Variational and other types of inclusions
Type
Research Article
Information
Proceedings of the Edinburgh Mathematical Society , Volume 61 , Issue 1 , February 2018 , pp. 93 - 120
Copyright
Copyright © Edinburgh Mathematical Society 2018 

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