Skip to main content
SpringerLink
Log in

Existence theorems of the variational-hemivariational inequalities

  • Published:
Journal of Global Optimization Aims and scope Submit manuscript

Abstract

This paper is devoted to the existence of solutions for the variational-hemivariational inequalities in reflexive Banach spaces. Using the notion of the stable \({\phi}\) -quasimonotonicity and the properties of Clarke’s generalized directional derivative and Clarke’s generalized gradient, some existence results of solutions are proved when the constrained set is nonempty, bounded (or unbounded), closed and convex. Moreover, a sufficient condition to the boundedness of the solution set and a necessary and sufficient condition to the existence of solutions are also derived. The results presented in this paper generalize and improve some known results.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

References

  1. Carl S.: Existence of extremal solutions of boundary hemivariational inequalities. J. Differ. Equ. 171, 370–396 (2001)

    Article  Google Scholar 

  2. Carl S., Le V.K., Motreanu D.: Existence and comparison principles for general quasilinear variational-hemivariational inequalities. J. Math. Anal. Appl. 302, 65–83 (2005)

    Article  Google Scholar 

  3. Carl S., Le V.K., Motreanu D.: Nonsmooth Variational Problems and Their Inequalities, Comparison Principles and Applications. Springer, New York (2007)

    Google Scholar 

  4. Clarke F.H.: Optimization and Nonsmooth Analysis. Wiley, New York (1983)

    Google Scholar 

  5. Costea N., Rădulescu V.: Hartman-Stampacchia results for stably pseudomonotone operators and nonlinear hemivariational inequalities. Appl. Anal. 89, 175–188 (2010)

    Article  Google Scholar 

  6. Costea N., Lupu C.: On a class of variational-hemivariational inequalities involving set valued mappings. Adv. Pure Appl. Math. 1, 233–246 (2010)

    Google Scholar 

  7. Costea N.: Existence and uniqueness results for a class of quasi-hemivariational inequalities. J. Math. Anal. Appl. 373, 305–315 (2011)

    Article  Google Scholar 

  8. Fan K.: Some properties of convex sets related to fixed point theorems. Math. Ann. 266, 519–537 (1984)

    Article  Google Scholar 

  9. Giannessi F., Maugeri A., Pardalos P.M.: Equilibrium Problems: Nonsmooth Optimization and Variational Inequality Models. Kluwer, Dordrecht (2002)

    Google Scholar 

  10. Gilbert R.P., Panagiotopoulos P.D., Pardalos P.M.: From Convexity to Nonconvexity. Kluwer, Dordrecht (2001)

    Book  Google Scholar 

  11. Goeleven D., Théra M.: Semicoercive variational inequalities. J. Glob. Optim. 6, 367–381 (1995)

    Article  Google Scholar 

  12. Goeleven D., Motreanu D., Panagiotopoulos P.D.: Eigenvalue problems for variational-hemivariational inequalities at resonance. Nonlinear Anal. 33, 161–180 (1998)

    Article  Google Scholar 

  13. Hadjisavvas N., Schaible S.: Quasimonotone variational inequalities in Banach spaces. J. Optim. Theory Appl. 90(1), 95–111 (1996)

    Article  Google Scholar 

  14. Hadjisavvas, N., Schaible, S., Wong, N.C.: Pseudomonotone operators: a survey of the theory and its applications. J. Optim. Theory Appl. doi:10.1007/s10957-011-9912-5

  15. He, Y.R.: Asymptotic analysis and error bounds in optimization, Ph.D. Thesis, The Chinese University of Hong Kong (2001)

  16. He Y.R.: A relationship between pseudomonotone and monotone mappings. Appl. Math. Lett. 17, 459–461 (2004)

    Article  Google Scholar 

  17. He Y.R.: A new projection algorithm for mixed variational inequalities. Acta Math. Sci. 27, 215–220 (2007)

    Google Scholar 

  18. He Y.R.: Stable pseudomonotone variational inequality in reflexive Banach spaces. J. Math. Anal. Appl. 330, 352–363 (2007)

    Article  Google Scholar 

  19. He Y.R., Ng K.F.: Strict feasibility of generalized complementarity problems. J. Aust. Math. Soc. Ser. A 81(1), 15–20 (2006)

    Article  Google Scholar 

  20. Huang Y.S., Zhou Y.Y.: Existence of solutions for a class of hemivariational inequality problems. Comput. Math. Appl. 57, 1456–1462 (2009)

    Article  Google Scholar 

  21. Kinderlehrer D., Stampacchia G.: An Introduction to Variational Inequalities and Their Applications. Academic Press, New York (1980)

    Google Scholar 

  22. Liu Z.H.: Elliptic variational hemivariational inequalities. Appl. Math. Lett. 16, 871–876 (2003)

    Article  Google Scholar 

  23. Liu Z.H.: Existence results for quasilinear parabolic hemivariational inequalities. J. Differ. Equ. 244, 1395–1409 (2008)

    Article  Google Scholar 

  24. Liu Z.H., Motreanu D.: A class of variational-hemivariational inequalities of elliptic type. Nonlinearity 23, 1741–1752 (2010)

    Article  Google Scholar 

  25. Migórski S., Ochal A.: Boundary hemivariational inequality of parabolic type. Nonlinear Anal. 57, 579–596 (2004)

    Article  Google Scholar 

  26. Marano S.A., Papageorgiou N.S.: On some elliptic hemivariational and variational-hemivariational inequalities. Nonlinear Anal. 62, 757–774 (2005)

    Article  Google Scholar 

  27. Motreanu D., Rădulescu V.: Existence results for inequality problems with lack of convexity. Numer. Funct. Anal. Optim. 21, 869–884 (2000)

    Article  Google Scholar 

  28. Naniewicz Z., Panagiotopoulos P.D.: Mathematical Theory of Hemivariational Inequalities and Applications. Marcel Dekker, New York (1995)

    Google Scholar 

  29. Panagiotopoulos P.D.: Nonconvex energy functions, hemivariational inequalities and substationarity principles. Acta Mech. 42, 160–183 (1983)

    Google Scholar 

  30. Panagiotopoulos P.D.: Inequality Problems in Mechanics and Applications, Convex and Nonconvex Energy Functions. Birkhäser, Basel (1985)

    Book  Google Scholar 

  31. Panagiotopoulos P.D.: Coercive and semicoercive hemivariational inequalities. Nonlinear Anal. 16, 209–231 (1991)

    Article  Google Scholar 

  32. Panagiotopoulos P.D.: Hemivariational Inequalities, Applications in Mechnics and Engineering. Springer, Berlin (1993)

    Book  Google Scholar 

  33. Panagiotopoulos P.D., Fundo M., Rădulescu V.: Existence theorems of Hartman-Stampacchia type for hemivariational inequalities and applications. J. Glob. Optim. 15, 41–54 (1999)

    Article  Google Scholar 

  34. Park J.Y., Ha T.G.: Existence of antiperiodic solutions for hemivariational inequalities. Nonlinear Anal. 68, 747–767 (2008)

    Article  Google Scholar 

  35. Park J.Y., Ha T.G.: Existence of anti-periodic solutions for quasilinear parabolic hemivariational inequalities. Nonlinear Anal. 71, 3203–3217 (2009)

    Article  Google Scholar 

  36. Peng Z.J., Liu Z.H.: Evolution hemivariational inequality problems with doubly nonlinear operators. J. Glob. Optim. 51, 413–427 (2011)

    Article  Google Scholar 

  37. Xiao Y.B., Huang N.J.: Sub-super-solution method for a class of higher order evolution hemivariational inequalities. Nonliear Anal. 71, 558–570 (2009)

    Article  Google Scholar 

  38. Xiao Y.B., Huang N.J.: Well-posedness for a class of variational-hemivariational inequalities with perturbations. J. Optim. Theory Appl. 151, 33–51 (2011)

    Article  Google Scholar 

  39. Yin H.Y., Xu C.X., Zhang Z.X.: The F-complementarity problem and its equivalence with the least element problem. Acta Math. Sin. 44, 679–686 (2001)

    Google Scholar 

  40. Zhang Y.L., He Y.R.: On stably quasimonotone hemivariational inequalities. Nonlinear Anal. 74, 3324–3332 (2011)

    Article  Google Scholar 

  41. Zhong R.Y., Huang N.J.: Stability analysis for Minty mixed variational inequality in reflexive Banach spaces. J. Optim. Theory Appl. 147, 454–472 (2010)

    Article  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Department of Mathematics, Sichuan University, Chengdu, 610064, Sichuan, People’s Republic of China

    Guo-ji Tang & Nan-jing Huang

Authors
  1. Guo-ji Tang
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Nan-jing Huang
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Nan-jing Huang.

Rights and permissions

Reprints and permissions

About this article

Cite this article

Tang, Gj., Huang, Nj. Existence theorems of the variational-hemivariational inequalities. J Glob Optim 56, 605–622 (2013). https://doi.org/10.1007/s10898-012-9884-5

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s10898-012-9884-5

Keywords

Search

Navigation