Skip to main content
SpringerLink
Log in

Existence of variational quasi-hemivariational inequalities involving a set-valued operatorand a nonlinear term

  • Original Paper
  • Published:
Optimization Letters Aims and scope Submit manuscript

Abstract

This paper is devoted to the existence of solutions for a class of variational quasi-hemivariational inequalities involving a lower hemicontinuous set-valued operator and a nonlinear term in reflexive Banach spaces. In the case when the constraint set is bounded, under certain generalized monotonicity conditions, we prove an existence result of solutions for the problem by means of F-KKM theorem. In the case when the constraint set is unbounded, under certain coercivity conditions, we construct an existence theorem of solutions and a boundedness theorem of the solution set for the problem, respectively. Moreover, a necessary and sufficient condition to the existence of solutions is 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  MathSciNet  MATH  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  MathSciNet  MATH  Google Scholar 

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

    Book  MATH  Google Scholar 

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

    MATH  Google Scholar 

  5. Costea, N., Rădulescu, V.: Existence results for hemivariational inequalities involving relaxed \(\eta -\alpha \) monotone mappings. Commun. Appl. Anal. 13, 293–304 (2009)

    MathSciNet  MATH  Google Scholar 

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

    Article  MathSciNet  MATH  Google Scholar 

  7. Costea, N., Ion, D.A., Lupu, C.: Variational-like inequality problems involving set-valued maps and generalized monotonicity. J. Optim. Theory Appl. 155, 79–99 (2012)

    Article  MathSciNet  MATH  Google Scholar 

  8. Costea, N., Rădulescu, V.: Inequality problems of quasi-hemivariational type involving set-valued operators and a nonlinear term. J. Glob. Optim. 52, 743–756 (2012)

    Article  MATH  Google Scholar 

  9. Facchinei, F., Pang, J.S.: Finite-Dimensional Variational Inequalities and Complementary Problems. Springer, New York (2003)

    Google Scholar 

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

    Article  MathSciNet  MATH  Google Scholar 

  11. Farajzadeh, A., Karamian, A., Plubtieng, S.: On the translations of quasimonotone maps and monotonicity, J. Inequalities Appl. (2012) http://www.journalofinequalitiesandapplications.com/content/2012/1/192

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

    Article  MathSciNet  MATH  Google Scholar 

  13. Hadjisavvas, N.: Translations of quasimonotone maps and monotonicity. Appl. Math. Lett. 19, 913–915 (2006)

    Article  MathSciNet  MATH  Google Scholar 

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

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

    Article  MathSciNet  MATH  Google Scholar 

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

    Article  MathSciNet  MATH  Google Scholar 

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

    MATH  Google Scholar 

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

    Article  MATH  Google Scholar 

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

    Article  MathSciNet  MATH  Google Scholar 

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

    Article  MathSciNet  MATH  Google Scholar 

  21. Motreanu, D., Rădulescu, V.: Variational and Non-Variational Methods in Nonlinear Analysis and Boundary Value Problems. Kluwer Academic Publishers, Boston, Dordrecht, London (2003)

    Book  Google Scholar 

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

    Google Scholar 

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

    MathSciNet  Google Scholar 

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

    Book  MATH  Google Scholar 

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

    Article  MathSciNet  MATH  Google Scholar 

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

    Book  MATH  Google Scholar 

  27. 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  MATH  Google Scholar 

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

    Article  MathSciNet  MATH  Google Scholar 

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

    Article  MathSciNet  MATH  Google Scholar 

  30. Tang, G.J., Huang, N.J.: Existence theorems of the variational-hemivariational inequalities. J. Glob. Optim. 56, 605–622 (2013)

    Article  MathSciNet  MATH  Google Scholar 

  31. Xiao, Y.B., Huang, N.J.: Generalized quasi-variational-like hemivariational inequalities. Nonliear Anal. 69, 637–646 (2008)

    Article  MathSciNet  MATH  Google Scholar 

  32. 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  MathSciNet  MATH  Google Scholar 

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

    Article  MathSciNet  MATH  Google Scholar 

Download references

Acknowledgments

The authors are grateful to the editor for the valuable comments and suggestions. This work was supported by Guangxi Natural Science Foundation (2013GXNSFBA019015), Scientific Research Foundation of Guangxi University for Nationalities (2012QD015) and Open fund of Guangxi key laboratory of hybrid computation and IC design analysis (HCIC201308).

Author information

Authors and Affiliations

  1. School of Science, Guangxi University for Nationalities, Nanning, 530006, Guangxi, People’s Republic of China

    Guo-ji Tang

  2. School of Information Technology, Jiangxi University of Finance and Economics, Nanchang, 330013, People’s Republic of China

    Xing Wang

  3. Department of Mathematics, Southwest Jiaotong University, Chengdu, 610031, People’s Republic of China

    Zhong-bao Wang

Authors
  1. Guo-ji Tang
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Xing Wang
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. Zhong-bao Wang
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Guo-ji Tang.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Tang, Gj., Wang, X. & Wang, Zb. Existence of variational quasi-hemivariational inequalities involving a set-valued operatorand a nonlinear term. Optim Lett 9, 75–90 (2015). https://doi.org/10.1007/s11590-014-0739-5

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s11590-014-0739-5

Keywords

Search

Navigation