Journal of Applied Mathematics

  • J. Appl. Math.
  • Volume 2012, Special Issue (2012), Article ID 804032, 18 pages.

Well-Posedness by Perturbations for Variational-Hemivariational Inequalities

Shu Lv, Yi-bin Xiao, Zhi-bin Liu, and Xue-song Li

Full-text: Open access

Abstract

We generalize the concept of well-posedness by perturbations for optimization problem to a class of variational-hemivariational inequalities. We establish some metric characterizations of the well-posedness by perturbations for the variational-hemivariational inequality and prove their equivalence between the well-posedness by perturbations for the variational-hemivariational inequality and the well-posedness by perturbations for the corresponding inclusion problem.

Article information

Source
J. Appl. Math., Volume 2012, Special Issue (2012), Article ID 804032, 18 pages.

Dates
First available in Project Euclid: 3 January 2013

Permanent link to this document
https://projecteuclid.org/euclid.jam/1357179985

Digital Object Identifier
doi:10.1155/2012/804032

Mathematical Reviews number (MathSciNet)
MR2959983

Zentralblatt MATH identifier
1251.49010

Citation

Lv, Shu; Xiao, Yi-bin; Liu, Zhi-bin; Li, Xue-song. Well-Posedness by Perturbations for Variational-Hemivariational Inequalities. J. Appl. Math. 2012, Special Issue (2012), Article ID 804032, 18 pages. doi:10.1155/2012/804032. https://projecteuclid.org/euclid.jam/1357179985


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References

  • A. N. Tykhonov, “On the stability of the functional optimization problem,” USSR Computational Mathematics and Mathematical Physics, vol. 6, pp. 631–634, 1966.
  • E. S. Levitin and B. T. Polyak, “Convergence of minimizing sequences in conditional extremum problems,” Soviet Mathematics-Doklady, vol. 7, pp. 764–767, 1966.
  • M. B. Lignola and J. Morgan, “Well-posedness for optimization problems with constraints defined by variational inequalities having a unique solution,” Journal of Global Optimization, vol. 16, no. 1, pp. 57–67, 2000.
  • B. Lemaire, C. Ould Ahmed Salem, and J. P. Revalski, “Well-posedness by perturbations of variational problems,” Journal of Optimization Theory and Applications, vol. 115, no. 2, pp. 345–368, 2002.
  • T. Zolezzi, “Well-posedness criteria in optimization with application to the calculus of variations,” Nonlinear Analysis A, vol. 25, no. 5, pp. 437–453, 1995.
  • T. Zolezzi, “Extended well-posedness of optimization problems,” Journal of Optimization Theory and Applications, vol. 91, no. 1, pp. 257–266, 1996.
  • A. L. Dontchev and T. Zolezzi, Well-posed Optimization Problems, vol. 1543 of Lecture Notes in Mathematics, Springer, Berlin, Germany, 1993.
  • R. Lucchetti and F. Patrone, “A characterization of Tyhonov well-posedness for minimum problems, with applications to variational inequalities,” Numerical Functional Analysis and Optimization, vol. 3, no. 4, pp. 461–476, 1981.
  • X. X. Huang, X. Q. Yang, and D. L. Zhu, “Levitin-Polyak well-posedness of variational inequality problems with functional constraints,” Journal of Global Optimization, vol. 44, no. 2, pp. 159–174, 2009.
  • Y.-P. Fang, N.-J. Huang, and J.-C. Yao, “Well-posedness by perturbations of mixed variational inequalities in Banach spaces,” European Journal of Operational Research, vol. 201, no. 3, pp. 682–692, 2010.
  • Y.-P. Fang and R. Hu, “Parametric well-posedness for variational inequalities defined by bifunctions,” Computers & Mathematics with Applications, vol. 53, no. 8, pp. 1306–1316, 2007.
  • Y.-P. Fang, N.-J. Huang, and J.-C. Yao, “Well-posedness of mixed variational inequalities, inclusion problems and fixed point problems,” Journal of Global Optimization, vol. 41, no. 1, pp. 117–133, 2008.
  • R. Hu and Y.-P. Fang, “Levitin-Polyak well-posedness by perturbations of inverse variational inequalities,” Optimization Letters. In press.
  • M. B. Lignola and J. Morgan, “Approximating solutions and $\alpha $-well-posedness for variational inequalities and Nash equilibria,” in Decision and Control in Management Science, pp. 367–378, Kluwer Academic Publishers, Dordrecht, The Netherlands, 2002.
  • R. Lucchetti and J. Revalski, Eds., Recent Developments in Well-Posed Variational Problems, vol. 331, Kluwer Academic Publishers, Dordrecht, The Netherlands, 1995.
  • P. D. Panagiotopoulos, “Nonconvex energy functions. Hemivariational inequalities and substationarity principles,” Acta Mechanica, vol. 42, pp. 160–183, 1983.
  • F. H. Clarke, Optimization and Nonsmooth Analysis, vol. 5, SIAM, Philadelphia, Pa, USA, 2nd edition, 1990.
  • D. Motreanu and P. D. Panagiotopoulos, Minimax Theorems and Qualitative Properties of the Solutions of Hemivariational Inequalities and Applications, vol. 29 of Nonconvex Optimization and its Applications, Kluwer Academic Publishers, Dordrecht, The Netherlands, 1999.
  • Z. Naniewicz and P. D. Panagiotopoulos, Mathematical Theory of Hemivariational Inequalities and Applications, vol. 188, Marcel Dekker, New York, NY, USA, 1995.
  • P. D. Panagiotopoulos, Hemivariational Inequalities: Applications in Mechanics and Engineering, Springer, Berlin, Germany, 1993.
  • S. Carl, V. K. Le, and D. Motreanu, Nonsmooth Variational Problems and Their Inequalities, Comparison Principles and Applications, Springer, Berlin, Germany, 2005.
  • S. Carl, V. K. Le, and D. Motreanu, “Evolutionary variational-hemivariational inequalities: existence and comparison results,” Journal of Mathematical Analysis and Applications, vol. 345, no. 1, pp. 545–558, 2008.
  • D. Goeleven and D. Mentagui, “Well-posed hemivariational inequalities,” Numerical Functional Analysis and Optimization, vol. 16, no. 7-8, pp. 909–921, 1995.
  • Z. Liu, “Existence results for quasilinear parabolic hemivariational inequalities,” Journal of Differential Equations, vol. 244, no. 6, pp. 1395–1409, 2008.
  • Z. Liu, “Browder-Tikhonov regularization of non-coercive evolution hemivariational inequalities,” Inverse Problems, vol. 21, no. 1, pp. 13–20, 2005.
  • S. Migórski and A. Ochal, “Dynamic bilateral contact problem for viscoelastic piezoelectric materials with adhesion,” Nonlinear Analysis A, vol. 69, no. 2, pp. 495–509, 2008.
  • Y.-B. Xiao and N.-J. Huang, “Sub-supersolution method and extremal solutions for higher order quasi-linear elliptic hemi-variational inequalities,” Nonlinear Analysis A, vol. 66, no. 8, pp. 1739–1752, 2007.
  • Y.-B. Xiao and N.-J. Huang, “Generalized quasi-variational-like hemivariational inequalities,” Nonlinear Analysis A, vol. 69, no. 2, pp. 637–646, 2008.
  • Y.-B. Xiao and N.-J. Huang, “Sub-super-solution method for a class of higher order evolution hemivariational inequalities,” Nonlinear Analysis A, vol. 71, no. 1-2, pp. 558–570, 2009.
  • 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, N.-J. Huang, and M.-M. Wong, “Well-posedness of hemivariational inequalities and inclusion problems,” Taiwanese Journal of Mathematics, vol. 15, no. 3, pp. 1261–1276, 2011.
  • 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.
  • R. T. Rockafellar, Convex Analysis, Princeton University Press, Princeton, NJ, USA, 1997.
  • E. Zeidler, Nonlinear Functional Analysis and Its Applications, vol. 2, Springer, Berlin, Germany, 1990.
  • K. Kuratowski, Topology, vol. 1-2, Academic, New York, NY, USA.
  • F. Giannessi and A. A. Khan, “Regularization of non-coercive quasi variational inequalities,” Control and Cybernetics, vol. 29, no. 1, pp. 91–110, 2000.