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Evolution hemivariational inequality problems with doubly nonlinear operators

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Abstract

This paper deals with a class of doubly nonlinear hemivariational inequality problems. We establish the existence results and investigate the periodic and symmetric solutions under suitable conditions.

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References

  1. Akagi G.: Doubly nonlinear evolution equations governed by time-dependent subdifferentials in reflexive Banach spaces. J. Differ. Equ. 231, 32–56 (2006)

    Article  Google Scholar 

  2. Carl S.: Existence and comparision results for noncoercive and nonmonotone multivalued elliptic problems. Nonlinear Anal. 65, 1532–1546 (2006)

    Article  Google Scholar 

  3. Carl S., Motreanu D.: General comparison principle for quasilinear elliptic inclusions. Nonlinear Anal. 70, 1105–1112 (2009)

    Article  Google Scholar 

  4. Carl S., Motreanu D.: Extremal solutions of quasilinear parabolic inclusions with generalized Clarke’s gradient. J. Differ. Equ. 191, 206–233 (2003)

    Article  Google Scholar 

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

    Google Scholar 

  6. Ekeland I., Téman R.: Convex Analysis and Variational Problems. SIAM, Philadelphia (1999)

    Google Scholar 

  7. Kenmochi N., Pawlow I.: A class of nonlinear elliptic-parabolic equations with time-dependent constrains. Nonlinear Anal. Theory. Methods Appl. 10, 1181–1202 (1986)

    Article  Google Scholar 

  8. Kubo M., Yamazaki N.: Elliptic-parabolic variational inequalities with time-depedent contrains. Discret. Continuous Dyn. Syst. 19, 335–359 (2007)

    Article  Google Scholar 

  9. Liu Z.: A class of evolution hemivariational inequalities. Nonlinear Anal. Theory Methods Appl. 36, 91–100 (1999)

    Article  Google Scholar 

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

    Article  Google Scholar 

  11. Liu Z.: Periodic solutions for double degenerate quasilinear parabolic equations. Nonlinear Anal. Theory Methods Appl. 51(7), 1245–1257 (2002)

    Article  Google Scholar 

  12. Liu Z., Zou J.Z.: Strong convergence results for hemivariational inequalities. Sci. China Ser. A Math. 49(7), 893–901 (2006)

    Article  Google Scholar 

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

    Article  Google Scholar 

  14. Liu Z.: On boundary variational-hemivariational inequalities of elliptic type. Proc. R. Soc. Edinburgh Sect. A Math. 140(2), 419–434 (2010)

    Article  Google Scholar 

  15. Gilbert R.P., Panagiotopoulos P.D., Pardalos P.M.: From Convexity to Nonconvexity, Series: Nonconvex Optimization and its Applications, vol. 55. Springer, Berlin (2001)

    Google Scholar 

  16. Maitre E., Witomski P.: A pseudo-monotonicity adapted to doubly nonlinear elliptic-parabolic equations. Nonlinear Anal. 50, 223–250 (2002)

    Article  Google Scholar 

  17. Giannessi F., Maugeri A., Pardalos P.M.: Equilibrium Problems: Nonsmooth Optimization and Variational Inequality Models, Series: Nonconvex Optimization and its Applications, vol. 58. Springer, Berlin (2002)

    Google Scholar 

  18. Miettinen M., Panagiotopoulous P.D.: On parabolic hemivariational inequalities and applications. Nonlinear Anal. Theory Methods Appl. 35, 885–915 (1999)

    Article  Google Scholar 

  19. Migórski S.: on existence of solutions for parabolic hemivariational inequalities. J. Comput. Appl. Math. 129, 77–87 (2001)

    Article  Google Scholar 

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

    Article  Google Scholar 

  21. Migórski S., Ochal A.: Existence of solutions for second order evolution inclusions with application to mechanical contact problems. Optimization 55, 101–120 (2006)

    Article  Google Scholar 

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

    Google Scholar 

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

    Google Scholar 

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

    Google Scholar 

  25. Rockafellar R.T.: Integral functionals, normal integrands and mesurable selections. In: Waelbroeck, L. (eds) Nonlinear Operators and the Calculus of Variations. Lecture Notes in Mathematics, vol. 543, pp. 157–207. Springer, Berlin (1976)

    Chapter  Google Scholar 

  26. Simon J.: Compact sets in the space L p[0, T; H]. Annali di Matematica Pura ed Applicata 146, 65–96 (1986)

    Article  Google Scholar 

  27. Visintin A.: Homogenization of doubly nonlinear stefan-type problem. SIAM J. Math. Anal. 39, 987–1017 (2007)

    Article  Google Scholar 

  28. Zeidler E.: Nonlinear Functional Analysis and its Application, vol. II A & B. Springer, Berlin (1990)

    Book  Google Scholar 

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Correspondence to Zhenhai Liu.

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Project supported by NNSF of China Grant No. 10971019, the Foundation (2010) of Guangxi Education Department and Hunan Provincial Innovation Foundation For Postgraduate No. CX2010B117.

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Peng, Z., Liu, Z. Evolution hemivariational inequality problems with doubly nonlinear operators. J Glob Optim 51, 413–427 (2011). https://doi.org/10.1007/s10898-010-9634-5

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  • DOI: https://doi.org/10.1007/s10898-010-9634-5

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