Convergence of rothe scheme for hemivariational inequalities of parabolic type

International Journal of Numerical Analysis and Modeling

Volumn 10, Issue 2, 2013, Pages 445-465

  Kalita, Piotr ^a  

^a JAGIELLONIAN UNIVERSITY   (Poland)

Author keywords

Convergence; Existence; Hemivariational inequality; Rothe method

Indexed keywords

EID: 84877003956     PISSN: 17055105     EISSN: None     Source Type: Journal    
DOI: None     Document Type: Article

References (36)

1. N.U. Ahmed, K.L. Teo and S.H. Hou, Nonlinear impulsive systems on infinite dimensions, Nonlinear Anal., 54 (2003) 907-925.
2. J.P. Aubin and A. Cellina, Differential Inclusions, Springer, 1984.
3. J. Berkovits and V. Mustonen, Monotonicity methods for nonlinear evolution equations, Nonlinear Anal., 27 (1996) 1397-1405.
4. S. Carl, Enclosure of solutions for quasilinear dynamic hemivariational inequalities, Nonlinear World, 3 (1996) 281-298.
5. S. Carl and R.P. Gilbert, Extremal solutions of a class of dynamic boundary hemivariational inequalities, J. Inequal. Appl., 7 (2002) 479-502.
6. S. Carl and D. Motreanu, Extremal solutions of quasilinear parabolic inclusions with generalized Clarkes gradient, J. Differential Equations, 191 (2003) 206-233.
7. S. Carl and D. Motreanu, Extremality in solving general quasilinear parabolic inclusions, J. Optim. Theory and Appl., 123 (2004) 463-477.
8. S. Carl, Existence and comparison results for noncoercive and nonmonotone multivalued elliptic problems, Nonlinear Anal., 65 (2006), 1532-1546.
9. S. Carl, V.K. Le and D. Motreanu, Evolutionary variational-hemivariational inequalities: existence and comparison results, J. Math. Anal. Appl., 345 (2008) 545-558.
10. F.H. Clarke, Optimization and Nonsmooth Analysis, SIAM, Philadelphia, 1990.
11. Z. Denkowski, S. Migórski and N.S. Papageorgiou, An Introduction to Nonlinear Analysis, Theory, Kluwer, 2003.
12. Z. Denkowski, S. Migórski and N.S. Papageorgiou, An Introduction to Nonlinear Analysis, Applications, Kluwer, 2003.
13. J. Francu, Weakly continuous operators. Applications to differential equations, Appl. Math., 39 (1994), 45-56.
14. D. Goeleven, D. Motreanu, Y. Dumonte and M. Rochdi, Variational and Hemivariational Inequalities: Theory, Methods and Applications, Volume II: Unilateral Problems, Kluwer, 2003.
15. G. Wang and X. Yang, Finite difference approximation of a parabolic hemivariational inequalities arising from temperature control problem, Int. J. Numer. Anal. Model., 7 (2010) 108-124.
16. J. Haslinger, M. Miettinen and P.D. Panagiotopoulos, Finite element method for hemivariational inequalities: theory, methods andapplications, Kluwer, 1999.
17. P. Kalita, Regularity and Rothe method error estimates for parabolic hemivariational inequality, J. Math. Anal. Appl., 389 (2012) 618-631.
18. Z. Liu and S. Zhang, On the degree theory for multivalued (S+) type mappings, Appl. Math. Mech., 1 (1998) 1141-1149.
19. Z. Liu, A class of evolution hemivariational inequalities, Nonlinear Anal., 36 (1999) 91-100.
20. Z. Liu, A class of parabolic hemivariational inequalities, Appl. Math. Mech., 21 (2000) 1045-1052.
21. Z. Liu, Some existence theorems for evolution hemivariational inequalities, Indian J. Pure Appl. Math., 34 (2003)1165-1176.
22. Z. Liu, Browder-Tikhonov regularization of non-coercive evolution hemivariational inequalities, Inverse Problems, 21 (2005) 13-20.
23. M. Miettinen, A parabolic hemivariational inequality, Nonlinear Anal., 26 (1996) 725-734.
24. M. Miettinen and P.D. Panagiotopoulos, Hysteresis and hemivariational inequalities: semilinear case, J. Global Optim., 13 (1998) 269-298.
25. M. Miettinen and P.D. Panagiotopoulos, On parabolic hemivariational inequalities and applications, Nonlinear Anal., 35 (1999) 885-915.
26. M. Miettinen and P.D. Panagiotopoulos, Hysteresis and hemivariational inequalities: quasilinear case, Appl. Anal., 82 (2003) 535-560.
27. S. Migórski, On existence of solutions for parabolic hemivariational inequalities, J. Comput. Appl. Math., 129 (2001) 77-87.
28. S. Migórski and A. Ochal, Boundary hemivariational inequality of parabolic type, Nonlinear Anal., 57 (2004) 579-596.
29. S. Migórski, Dynamic hemivariational inequalities in contact mechanics, Nonlinear Anal., 63 (2005) e77-e86.
30. Z. Naniewicz and P.D. Panagiotopoulos, Mathematical Theory of Hemivariational Inequalities and Applications, Dekker, New York, 1995.
31. N.S. Papageorgiou, On the existence of solutions for nonlinear parabolic problems with nonmonotone discontinuities, J. Math. Anal. Appl., 205 (1997) 434-453.
32. Z. Peng and Z. Liu, Evolution hemivariational inequality problems with doubly nonlinear operators, J. Global Optim., 51 (2011) 413-427.
33. R. Rossi, Compactness results for evolution equations, Istit. Lombardo Accad. Sci. Lett. Rend. A, 135 (2001) 19-30.
34. R. Rossi and G. Savare, Tightness, integral equicontinuity and compactness for evolution problems in Banach spaces, Ann. Sc. Norm. Super. Pisa Cl. Sci. (5), (2003) 395-431.
35. T. Roubiček, Nonlinear Partial Differential Equations with Applications, Birkhäuser Verlag, Basel, Boston, Berlin, 2005.
36. p(0; T;B), Ann. Mat. Pura Appl., 146 (1987) 65-96.