Hostname: page-component-78c5997874-s2hrs Total loading time: 0 Render date: 2024-11-04T19:02:01.314Z Has data issue: false hasContentIssue false
  • English
  • Français

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 ([email protected])
Leszek Gasiński
Affiliation:
Jagiellonian University, Faculty of Mathematics and Computer Science, ul. Lojasiewicza 6, 30348 Krakow, Poland ([email protected])
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 ([email protected])
*
*Corresponding author.
Get access Rights & Permissions [Opens in a new window]

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 

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

1. Aubin, J. P. and Frankowska, H., Set-Valued Analysis (Birkhäuser, 1990).Google Scholar
2. Bartosz, K., Numerical methods for evolution hemivariational inequalities, In Advances in variational and hemivariational inequalities: theory, numerical analysis and applications (ed. Han, W., Migórski, S. and Sofonea, M.), Advances in Mechanics and Mathematics, Volume 33, pp. 111–144 (Springer, 2015).Google Scholar
3. Bartosz, K. and Sofonea, M., The Rothe method for variational-hemivariational inequalities with applications to contact mechanics, SIAM J. Math. Anal. 48 (2016), 861–863.Google Scholar
4. Bartosz, K., Cheng, X., Kalita, P., Yu, Y. and Zheng, C., Rothe method for evolution variational-hemivariational inequalities, J. Math. Anal. Appl. 423 (2015), 841–862.Google Scholar
5. Emrich, E. and Thalhammer, M., Convergence of a time discretisation for doubly nonlinear evolution equations of second order, Found. Comput. Math. 10 (2010), 171–190.Google Scholar
6. Kalita, P., Regularity and Rothe method error estimates for parabolic hemivariational inequality, J. Math. Anal. Appl. 389 (2012), 618–631.Google Scholar
7. Kalita, P., Convergence of Rothe scheme for hemivariational inequalities of parabolic type, Int. J. Numer. Anal. Mod. 10 (2013) 445–465.Google Scholar
8. Kalita, P., Semidiscrete variable time-step θ-scheme for nonmonotone evolution inclusion, Preprint (arXiv:1402.3721 [math.AP]; 2014).Google Scholar
9. Lupulescu, V., A viability result for non convex second order differential inclusions, Nonlin. Funct. Analysis Applic. 9(3) (2004), 495–512.Google Scholar
10. Mahmudov, E. N., Approximation and optimization of discrete and differential inclusions (Elsevier, 2011).Google Scholar
11. Mahmudov, E. N., Optimization of second order discrete approximation inclusions, Numer. Func. Anal. Opt. 76(5) (2015), 624–643.Google Scholar
12. Peng, Z. and Liu, Z., Evolution hemivariational inequality problems with doubly nonlinear operators, J. Glob. Optim. 51 (2011), 413–427.Google Scholar
13. Peng, Z. and Xiao, C., Existence and convergence theorems for evolutionary hemivariational inequalities of second order, Electron. J. Diff. Eqns 2015 (2015), No. 65.Google Scholar
14. Peng, Z., Liu, Z. and Liu, X., Boundary hemivariational inequality problems with doubly nonlinear operators, Math. Annalen 356 (2013), 1339–1358.Google Scholar
15. Roubiček, T., Nonlinear Partial Differential Equations with Applications (Birkhäuser, 2005).Google Scholar