Well-posed hemivariational inequalities. (English) Zbl 0848.49013

Goeleven, D.; Mentagui, D.

Numer. Funct. Anal. Optimization 16, No. 7-8, 909-921 (1995).

Let \(V\) be a real reflexive Banach space with topological dual \(V^*\). Let \(K\) be a closed convex subjset of \(V\) and \(f\in V^*\). In this paper, the authors have obtained some basic results concerning the well-posedness for hemivariational inequalities of finding \(u\in K\) such that \[ \langle Au+ Tu- \rho, \nu- u\rangle+ \int_\Omega j^0 (x, u(x); \nu(x)- u(\alpha)) d\Omega \geq 0, \qquad \text{for all } \nu\in K, \] where \(j^0 (x,u(x); \nu(x)- u(x))\) denotes the generalized directional derivative of the function \(j(x)\) at \(u(x)\) in the direction \(\nu (x)- u(x)\) and \(T,A: V\to V^*\) are nonlinear operators.

Reviewer: M.A.Noor (Riyadh)

MSC:

49J40	Variational inequalities
90C33	Complementarity and equilibrium problems and variational inequalities (finite dimensions) (aspects of mathematical programming)

Keywords:

existence results; well-posedness; hemivariational inequalities; generalized directional derivative

References:

[1]	Adams R. A., Sobolev spaces (1975)
[2]	Attouch H., Variational convergence for functions and operators (1984) · Zbl 0561.49012
[3]	Banatiotopoulos C. C., ZAMM 69 pp 489– (1989)
[4]	Clarke F. H., Nonsmooth analysis and optimization (1984) · Zbl 0727.90045
[5]	Dontchev A. L., lectures notes in mathematics (1543)
[6]	Goeleven D., FUNDP Research-Report (1994)
[7]	Karamanlis H. N., Journal Mech. (1994)
[8]	DOI: 10.1080/01630568108816100 · Zbl 0479.49025
[9]	DOI: 10.1080/01630568308816145 · Zbl 0517.49007
[10]	Mosco U., lecture notes in mathematics 543 pp 83-156– (1976)
[11]	Naniewicz Z., Journal of Optimization Theory and Applications 543 (1976)
[12]	Naniewicz, Z. and Panagiotopoulos, P. D. 1995. ”Mathematical theory of hemivariational inequalities and applications”. New York: Dekker. · Zbl 0968.49008
[13]	DOI: 10.1016/0362-546X(91)90224-O · Zbl 0733.49012
[14]	Panagiotopoulos P. D., C.R. Acad. Sci. Paris, t. 307 pp 735– (1988)
[15]	DOI: 10.1007/BF00041685 · Zbl 0712.73059
[16]	Panagiotopoulos P. D., Quaterly of Applied Mathematics 3 pp 409– (1988) · Zbl 0672.73011
[17]	Panagiotopoulos P. D., Quaterly of Applied Mathematics pp 611– (1989) · Zbl 0693.73007
[18]	DOI: 10.1002/zamm.19850650608
[19]	Panagiotopoulos P. D., Topics in Nonsmooth Mechanics (1988) · Zbl 0646.00014
[20]	DOI: 10.1016/0093-6413(81)90064-1 · Zbl 0497.73020
[21]	DOI: 10.1007/BF01170410 · Zbl 0538.73018
[22]	Panagiotopoulos P. D., Hemivariational inequalities, applications in mechanics and engineering (1993) · Zbl 0826.73002
[23]	Pascali D., Nonlinear mappings of monotone type (1978) · Zbl 0423.47021
[24]	Revalski, J. 1985. Variational inequalities with unique solution. Mathematics and Education in Mathematics. 14th Spring Conference of the Union of Bulgarian Mathematicians Sofia. 1985. pp.534–541.