Journal of Applied Mathematics

  • J. Appl. Math.
  • Volume 2013, Special Issue (2013), Article ID 305068, 9 pages.

A System of Generalized Variational-Hemivariational Inequalities with Set-Valued Mappings

Zhi-bin Liu, Jian-hong Gou, Yi-bin Xiao, and Xue-song Li

Full-text: Open access

Abstract

By using surjectivity theorem of pseudomonotone and coercive operators rather than the KKM theorem and fixed point theorem used in recent literatures, we obtain some conditions under which a system of generalized variational-hemivariational inequalities concerning set-valued mappings, which includes as special cases many problems of hemivariational inequalities studied in recent literatures, is solvable. As an application, we prove an existence theorem of solutions for a system of generalized variational-hemivariational inequalities involving integrals of Clarke's generalized directional derivatives.

Article information

Source
J. Appl. Math., Volume 2013, Special Issue (2013), Article ID 305068, 9 pages.

Dates
First available in Project Euclid: 14 March 2014

Permanent link to this document
https://projecteuclid.org/euclid.jam/1394807474

Digital Object Identifier
doi:10.1155/2013/305068

Mathematical Reviews number (MathSciNet)
MR3108939

Zentralblatt MATH identifier
06950605

Citation

Liu, Zhi-bin; Gou, Jian-hong; Xiao, Yi-bin; Li, Xue-song. A System of Generalized Variational-Hemivariational Inequalities with Set-Valued Mappings. J. Appl. Math. 2013, Special Issue (2013), Article ID 305068, 9 pages. doi:10.1155/2013/305068. https://projecteuclid.org/euclid.jam/1394807474


Export citation

References

  • S. Carl, V. K. Le, and D. Motreanu, Nonsmooth Variational Problems and Their Inequalities, Springer, Berlin, Germany, 2007.
  • S. Migórski, A. Ochal, and M. Sofonea, Nonlinear Inclusions and Hemivariational Inequalities, vol. 26 of Advances in Mechanics and Mathematics, Springer, New York, NY, USA, 2013, Models and analysis of contact problems.
  • Z. Naniewicz and P. D. Panagiotopoulos, Mathematical Theory of Hemivariational Inequalities and Applications, vol. 188 of Monographs and Textbooks in Pure and Applied Mathematics, Marcel Dekker, New York, NY, USA, 1995.
  • P. D. Panagiotopoulos, Hemivariational Inequalities, Springer, Berlin, Germany, 1993, Applications in mechanics and engineering.
  • Z. Denkowski and S. Migórski, “A system of evolution hemivariational inequalities modeling thermoviscoelastic frictional contact,” Nonlinear Analysis. Theory, Methods & Applications, vol. 60, no. 8, pp. 1415–1441, 2005.
  • D. Repovš and C. Varga, “A Nash type solution for hemivariational inequality systems,” Nonlinear Analysis. Theory, Methods & Applications, vol. 74, no. 16, pp. 5585–5590, 2011.
  • S. Carl, “Parameter-dependent variational-hemivariational inequalities and an unstable degenerate elliptic free boundary problem,” Nonlinear Analysis, vol. 12, no. 6, pp. 3185–3198, 2011.
  • S. Carl and D. Motreanu, “Extremal solutions of quasilinear parabolic inclusions with generalized Clarke's gradient,” Journal of Differential Equations, vol. 191, no. 1, pp. 206–233, 2003.
  • Z. Denkowski and S. Migórski, “Hemivariational inequalities in thermoviscoelasticity,” Nonlinear Analysis. Theory, Methods & Applications, vol. 63, pp. 87–97, 2005.
  • Z. Liu, “Generalized quasi-variational hemi-variational inequalities,” Applied Mathematics Letters, vol. 17, no. 6, pp. 741–745, 2004.
  • Z. Liu and D. Motreanu, “A class of variational-hemivariational inequalities of elliptic type,” Nonlinearity, vol. 23, no. 7, pp. 1741–1752, 2010.
  • S. Migórski and P. Szafraniec, “A class of dynamic čommentComment on ref. [10?]: Please update the information of this reference, if possible.frictional contact problems governed by a system of hemivariational inequalities in thermoviscoelasticity,” Nonlinear Analysis. In press.
  • P. D. Panagiotopoulos, “Nonconvex energy functions. Hemivariational inequalities and substationarity principles,” Acta Mechanica, vol. 48, no. 3-4, pp. 111–130, 1983.
  • Y. B. Xiao and N. J. Huang, “Generalized quasi-variational-like hemivariational inequalities,” Nonlinear Analysis. Theory, Methods & Applications, vol. 69, no. 2, pp. 637–646, 2008.
  • Y. B. Xiao and N. J. Huang, “Browder-Tikhonov regularization for a class of evolution second order hemivariational inequalities,” Journal of Global Optimization, vol. 45, no. 3, pp. 371–388, 2009.
  • Y. B. Xiao and N. J. Huang, “Well-posedness for a class of variational-hemivariational inequalities with perturbations,” Journal of Optimization Theory and Applications, vol. 151, no. 1, pp. 33–51, 2011.
  • F. H. Clarke, Optimization and Nonsmooth Analysis, vol. 5, Society for Industrial and Applied Mathematics, Philadelphia, Pa, USA, 2nd edition, 1990.
  • E. Zeidler, Nonlinear Functional Analysis and Its Applications, Springer, Berlin, Germany, 1990.
  • P. D. Panagiotopoulos, M. Fundo, and V. Rădulescu, “Existence theorems of Hartman-Stampacchia type for hemivariational inequalities and applications,” Journal of Global Optimization, vol. 15, no. 1, pp. 41–54, 1999.