Sub-super-solution method for a class of higher order evolution hemivariational inequalities. (English) Zbl 1162.49015

Xiao, Yi-Bin; Huang, Nan-Jing

Nonlinear Anal., Theory Methods Appl., Ser. A, Theory Methods 71, No. 1-2, 558-570 (2009).

Summary: We extend the extremality results for the variational inequality to a higher order evolution hemivariational inequality. More precisely, we give an existence theorem of solution for the higher order evolution hemivariational inequality by using the sub-super-solution method. We prove the compactness of the solution set within an interval formed by the sub-solution and super-solution. We also show an existence theorem of the extremal solution for the higher order evolution hemivariational inequality under consideration.

MSC:

49J40	Variational inequalities
35K90	Abstract parabolic equations
47J35	Nonlinear evolution equations
47J20	Variational and other types of inequalities involving nonlinear operators (general)

Keywords:

evolution hemivariational inequality; sub-solution; super-solution; extremal solution; sub-super-solution method

References:

[1]	Akô, K., On the Dirichlet problem for quasi-linear elliptic differential equations of the second order, J. math. soc. Japan, 13, 45-62, (1961) · Zbl 0102.09303
[2]	Berkovits, J.; Mustonen, V., Monotonicity methods for nonlinear evolution equations, Nonlinear anal. TMA, 27, 1397-1405, (1996) · Zbl 0894.34055
[3]	Carl, S.; Heikkilä, S., Nonlinear differential equations in ordered spaces, (2000), Chapman & Hall/CRC Boca Raton, FL · Zbl 0948.34001
[4]	Carl, S.; Le, V.K.; Motreanu, D., The sub – supersolution method and extremal solutions for quasilinear hemivariational inequalities, Differential integral equations, 17, 165-178, (2004) · Zbl 1164.35301
[5]	Carl, S.; Le, V.K.; Motreanu, D., Existence and comparison results for quasilinear evolution hemivariational inequalities, Electronic J. differential equations, 57, 1-17, (2004) · Zbl 1053.49005
[6]	Carl, S.; Le, V.K.; Motreanu, D., Existence and comparison principles for general quasilinear variational-hemivariational inequalities, J. math. anal. appl., 302, 65-83, (2005) · Zbl 1061.49007
[7]	Carl, S.; Motreanu, D., Extremal solutions of quasilinear parabolic inclusions with generalized clarke’s gradient, J. differential equations, 191, 206-233, (2003) · Zbl 1042.35092
[8]	Chipot, M.; Rodrigues, J.F., Comparison and stability of solutions to a class of quasilinear parabolic problems, Proc. roy. soc. Edinburgh, 110 A, 275-285, (1998)
[9]	Clarke, F.H., Optimization and nonsmooth analysis, (1990), SIAM Philadelphia · Zbl 0727.90045
[10]	Deuel, J.; Hess, P., A criterion for the existence of solutions of non-linear elliptic boundary value problems, Proc. roy. soc. Edinburgh sect. A, 74, 49-54, (1974-1975)
[11]	Deuel, J.; Hess, P., Nonlinear parabolic boundary value problems with upper and lower solutions, Israel J. math., 29, 92-104, (1978) · Zbl 0372.35045
[12]	Denkowski, Z.; Migorski, S.; Papageorgiou, N.S., An introduction to nonlinear analysis: applications, (2003), Kluwer Academic Publishers Boston, Dordrecht, London · Zbl 1030.35106
[13]	Le, V.K., Subsolution-supersolutions method in variational inequalities, Nonlinear anal. TMA, 45, 775-800, (2001) · Zbl 1040.49008
[14]	Le, V.K., Subsolution-supersolutions and the existence of extremal solutions in noncoercive variational inequalities, JIPAM J. inequal. pure appl. math., 2, 2, (2001), Article 20 (electronic)
[15]	Liu, Z.H., Browder – tikhonob regularization of non-coercive evolution hemivariational inequalities, Inverse problems, 21, 13-20, (2005) · Zbl 1078.49006
[16]	Liu, Z.H., A class of evolutional hemivariational inequalities, Nonlinear anal. TMA, 36, 91-100, (1999) · Zbl 0920.47056
[17]	Migorski, S., Boundary hemivariational inequalities of hyperbolic type and applications, J. global optim., 31, 505-533, (2005) · Zbl 1089.49014
[18]	Motreanu, D.; Panagiotopoulos, P.D., Minimax theorems and qualitative properties of the solutions of hemivariational inequalities and applications, () · Zbl 0829.49005
[19]	Naniewicz, Z.; Panagiotopoulos, P.D., Mathematical theory of hemivariational inequalities and applications, (1995), Marcel Dekker New York · Zbl 0968.49008
[20]	Ochal, A., Existence results for evolution hemivariational inequalities of second order, Nonlinear anal. TMA, 60, 1369-1391, (2005) · Zbl 1082.34052
[21]	Panagiotopoulos, P.D., Hemivariational inequalities: applications in mechanics and engineering, (1993), Springer-Verlag Berlin · Zbl 0826.73002
[22]	Panagiotopoulos, P.D., Coercive and semicoercive hemivariational inequalities, Nonlinear anal. TMA, 16, 209-231, (1991) · Zbl 0733.49012
[23]	Panagiotopoulos, P.D., Hemivariational inequality and Fan-variational inequality, new applications and results, Atti. sem. mat. fis. univ. modena, XLIII, 159-191, (1995) · Zbl 0843.49006
[24]	Panagiotopoulos, P.D.; Fundo, M.; Radulescu, V., Existence theorems of hartman – stampacchia type for hemivariational inequalities and applications, J. global optim., 15, 41-54, (1999) · Zbl 0951.49018
[25]	Sattinger, D.H., Monotone methods in nonlinear elliptic and parabolic boundary value problems, Indiana univ. math. J., 21, 979-1000, (1972) · Zbl 0223.35038
[26]	Showalter, R.E., Monotone operators in Banach spaces and nonlinear partial differential equations, (1997), Amer. Math. Soc Providence · Zbl 0870.35004
[27]	Verma, R.U., Nonlinear variational and constrained hemivariational inequalities involving relaxed operators, Z. angew. math. mech., 77, 5, 387-391, (1997) · Zbl 0886.49006
[28]	Xiao, Y.B.; Huang, N.J., Sub – supersolution method and extremal solutions for higher order quasi-linear elliptic hemi-variational inequalities, Nonlinear anal. TMA, 66, 1739-1752, (2007) · Zbl 1110.49012
[29]	Xiao, Y.B.; Huang, N.J., Generalized quasi-variational-like hemivariational inequalities, Nonlinear anal. TMA, 69, 637-646, (2008) · Zbl 1143.49009
[30]	Zeidler, E., Nonlinear functional analysis and its applications, vol. II B, (1990), Springer-Verlag Berlin