Remote Access Proceedings of the American Mathematical Society
Green Open Access

Proceedings of the American Mathematical Society

ISSN 1088-6826(online) ISSN 0002-9939(print)

 
 

 

Nonlinear hemivariational inequalities of second order using the method of upper-lower solutions


Authors: Nikolaos C. Kourogenis and Nikolaos S. Papageorgiou
Journal: Proc. Amer. Math. Soc. 131 (2003), 2359-2369
MSC (2000): Primary 35J50, 35J85, 35R70
DOI: https://doi.org/10.1090/S0002-9939-03-06993-4
Published electronically: March 11, 2003
MathSciNet review: 1974632
Full-text PDF

Abstract | References | Similar Articles | Additional Information

Abstract: In this paper we examine a nonlinear hemivariational inequality of second order. The differential operator is set-valued, nonlinear and depends on both $x$ and its gradient $Dx$. The same is true for the zero order term $f$, while the right-hand side nonlinearity satisfies a one-sided Lipschitz condition. We use the method of upper and lower solutions, coupled with truncation and penalization techniques and the fixed point theory for multifunctions in an ordered Banach space.


References [Enhancements On Off] (What's this?)

  • 1. Ambrosetti, A - Badiale, M.: ``The dual variational principle and elliptic problems with discontinuous nonlinearities'' J. Math. Anal. Appl. 140 (1989), 363-373. MR 90k:35097
  • 2. Carl, S. - Dietrich, H.: ``The weak upper and lower solution method for quasilinear elliptic equations with generalized subdifferentiable perturbations'' Applic. Anal. 56 (1995), 263-278. MR 97f:35045
  • 3. Chang, K. C.: ``Variational methods for nondifferentiable functionals and their applications to partial differential equations'' J. Math. Anal. Appl. 80 (1981), 102-129. MR 82h:35025
  • 4. Deuel, J. - Hess, P.: ``A criterion for the existence of solutions for nonlinear elliptic boundary value problems'' Proc. Roy. Soc. Edinburgh Sect. A 74 (1974/75), 49-54. MR 55:13070
  • 5. Heikkila, S. - Hu, S.: ``On fixed points of multifunctions in ordered spaces'' Applic. Anal. 51 (1993), 115-127. MR 95a:54062
  • 6. Heikkila, S. - Lakshmikantham, V.: ``Monotone Iterative Techniques for Discontinuous Nonlinear Differential Equations'' Marcel-Dekker, New York (1994). MR 95d:34002
  • 7. Hu, S. - Papageorgiou, N. S.: ``Handbook of Multivalued Analysis. Volume I: Theory'' Kluwer, Dordrecht, The Netherlands (1997). MR 98k:47001
  • 8. Hu, S. - Papageorgiou, N. S.: ``Handbook of Multivalued Analysis. Volume II: Applications'' Kluwer, Dordrecht, The Netherlands (2000). MR 2001g:49001
  • 9. Kourogenis, N. C. - Papageorgiou, N. S.: ``Nonsmooth critical point theory and nonlinear elliptic equations at resonance'' J. Austral. Math. Soc. (Series A) 69 (2000), 245-271. MR 2001m:35078
  • 10. Le, V. K.: ``On some equivalent properties of sub- and supersolutions in second order quasilinear elliptic equations'' Hiroshima Math. Jour. 28 (1998), 373-380. MR 99h:35049
  • 11. Le, V. K.: ``Subsolution-supersolution method in variational inequalities'' Nonlin. Anal. 45 (2001), 775-800. MR 2002c:35126
  • 12. Naniewicz, Z. - Panagiotopoulos, P.: ``Mathematical Theory of Hemivariational Inequalities and Applications'' Marcel-Dekker, New York (1995). MR 96d:47067
  • 13. Panagiotopoulos, P.: ``Hemivariational Inequalities: Applications to Mechanics and Engineering'' Springer-Verlag, New York (1993).
  • 14. Papageorgiou, N. S. - Papalini, F. - Vercillo, S.: ``Minimal solutions for nonlinear parabolic problems with unilateral constraints'' Houston J. Math. 23 (1997), 189-201. MR 2000g:35124
  • 15. Showalter, R.: ``Monotone Operators in Banach Space and Nonlinear Partial Differential Equations'' Math. Surveys and Monographs, Vol. 49, AMS, Providence, R.I. (1997). MR 98c:47076
  • 16. Stuart, C.: ``Maximal and minimal solutions of elliptic differential equations with discontinuous nonlinearities'' Math Z. 163 (1978), 239-249. MR 80g:35059

Similar Articles

Retrieve articles in Proceedings of the American Mathematical Society with MSC (2000): 35J50, 35J85, 35R70

Retrieve articles in all journals with MSC (2000): 35J50, 35J85, 35R70


Additional Information

Nikolaos C. Kourogenis
Affiliation: Department of Mathematics, National Technical University, Zografou Campus, Athens 157 80, Greece
Address at time of publication: Department of Financial Management and Banking, University of Pireus, Pireus, Greece

Nikolaos S. Papageorgiou
Affiliation: Department of Mathematics, National Technical University, Zografou Campus, Athens 157 80, Greece
Email: npapg@math.ntua.gr

DOI: https://doi.org/10.1090/S0002-9939-03-06993-4
Keywords: Upper solution, lower solution, upper semicontinuous multifunction, lower semicontinuous multifunction, graph measurability, operator of type $S_+ $, pseudomonotone operator, truncation map, penalty map, coercive map, order interval
Received by editor(s): October 9, 2001
Published electronically: March 11, 2003
Additional Notes: The first author was supported by a grant of the National Scholarship Foundation of Greece (I.K.Y.)
Communicated by: David S. Tartakoff
Article copyright: © Copyright 2003 American Mathematical Society

American Mathematical Society