A class of evolution hemivariational inequalities. (English) Zbl 0920.47056

Liu, Zhenhai

Nonlinear Anal., Theory Methods Appl. 36, No. 1, A, 91-100 (1999).

The author studies the existence of solutions to the problem: Find \(u\in X= L^p(I,V)\) such that \(u(0)= 0\) and \[ \Biggl\langle{du\over dt}, v\Biggr\rangle_X+ \langle Au, v\rangle_X+ \int_I g^0(t, u,v)dt\geq \langle f,v\rangle_X,\quad\forall v\in X, \] where \(I= [0,T]\), \(V\subset X\subset V'\) forms an evolution triple, \(g^0(t,\cdot,\cdot)\) denotes the Clarke’s directional derivative and \(A\) is a map of class \((S_+)\) with respect to \(D(L)\) with \(Lu= {du\over dt}\).

Reviewer: V.Mustonen (Oulu)

MSC:

47J20	Variational and other types of inequalities involving nonlinear operators (general)
34G20	Nonlinear differential equations in abstract spaces

Keywords:

evolution hemivariational inequalities; Clarke’s directional derivative

References:

[1]	Berkovits, J.; Mustonen, V., Monotone methods for nonlinear evolution equations,, Nonlinear Anal., 27, 12, 1397-1405 (1996) · Zbl 0894.34055
[2]	Browder, F. E.; Hess, P., Nonlinear mappings of monotone type in Banach spaces,, J. Funct. Anal., 11, 251-294 (1972) · Zbl 0249.47044
[4]	Clarke, F. H., Optimization and Nonsmooth Analysis (1983), Wiley: Wiley New York · Zbl 0727.90045
[5]	Liu, Z., Hemivariational inequalities of quasilinear elliptic systems,, Systems Engineering, 14, 63-65 (1996)
[7]	Miettinen, M., A parabolic hemivariational inequalities,, Nonlinear Anal., 26, 725-734 (1996) · Zbl 0858.35072
[8]	Panagiotopoulos, P. D., Nonconvex superpotentials in sense of F.H. Clarke and applications,, Mech. Res. Comm., 8, 335-340 (1981) · Zbl 0497.73020
[9]	Panagiotopoulos, P. D., Hemivariational inequalities Applications in Mechanics and Engineering, (1993), Springer: Springer Berlin · Zbl 0826.73002