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Hemivariational Inequality Approach to Evolutionary Constrained Problems on Star-Shaped Sets

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Abstract

In this paper, we consider a nonconvex evolutionary constrained problem for a star-shaped set. The problem is a generalization of the classical evolution variational inequality of parabolic type. We provide an existence result; the proof is based on the hemivariational inequality approach, a surjectivity theorem for multivalued pseudomonotone operators in reflexive Banach spaces, and a penalization method. The admissible set of constraints is closed and star-shaped with respect to a certain ball; this allows one to use a discontinuity property of the generalized Clarke subdifferential of the distance function. An application of our results to a heat conduction problem with nonconvex constraints is provided.

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References

  1. Hartman, P., Stampacchia, G.: On some non linear elliptic differential-functional equations. Acta Math. 115, 271–310 (1966)

    Article  MATH  MathSciNet  Google Scholar 

  2. Stampacchia, G.: Formes bilinaires sur les ensemble convexes. C.R. Acad.Sci., Paris (1964)

    Google Scholar 

  3. Lions, J.L., Stampacchia, G.: Variational inequalities. Commun. Pure Appl. Math. 20, 493–519 (1967)

    Article  MATH  MathSciNet  Google Scholar 

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

    Book  MATH  Google Scholar 

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

    Book  MATH  Google Scholar 

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

    Article  MATH  MathSciNet  Google Scholar 

  7. Migórski, S.: Evolution hemivariational inequalities with applications. In: Gao, D.Y., Motreanu, D. (eds.) Handbook of Nonconvex Analysis and Applications, Chap. 8, pp. 409–473. International Press, Boston (2010)

    Google Scholar 

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

    Article  MATH  MathSciNet  Google Scholar 

  9. Naniewicz, Z., Panagiotopoulos, P.D.: Mathematical Theory of Hemivariational Inequalities and Applications, Monographs and Textbooks in Pure and Applied Mathematics, vol. 188. Marcel Dekker, New York (1995)

    Google Scholar 

  10. Migórski, S., Ochal, A., Sofonea, M.: Nonlinear Inclusions and Hemivariational Inequalities. Models and Analysis of Contact Problems, Advances in Mechanics and Mathematics, vol. 26. Springer, New York (2013)

    Google Scholar 

  11. Naniewicz, Z.: Hemivariational inequality approach to constrained problems for star-shaped admissible sets. J. Optim. Theory Appl. 83, 97–112 (1994)

    Article  MATH  MathSciNet  Google Scholar 

  12. Goeleven, D.: On the hemivariational inequality approach to nonconvex constrained problems in the theory of von Kárman plates. Z. Angew. Math. Mech. (ZAMM) 75, 861–866 (1995)

    Article  MATH  MathSciNet  Google Scholar 

  13. Browder, F.E., Hess, P.: Nonlinear mappings of monotone type in Banach spaces. J. Funct. Anal. 11, 251–294 (1972)

    Article  MATH  MathSciNet  Google Scholar 

  14. Denkowski, Z., Migórski, S., Papageorgiou, N.S.: An Introduction to Nonlinear Analysis: Applications. Kluwer/Plenum, New York (2003)

    Book  Google Scholar 

  15. Zeidler, E.: Nonlinear Functional Analysis and Applications II A/B. Springer, New York (1990)

    Book  Google Scholar 

  16. Denkowski, Z., Migórski, S.: A system of evolution hemivariational inequalities modeling thermoviscoelastic frictional contact. Nonlinear Anal. 60, 1415–1441 (2005)

    Article  MATH  MathSciNet  Google Scholar 

  17. Papageorgiou, N.S., Papalini, F., Renzacci, F.: Existence of solutions and periodic solutions for nonlinear evolution inclusions. Rend. Circ. Mat. Palermo 48, 341–364 (1999)

    Article  MATH  MathSciNet  Google Scholar 

  18. Denkowski, Z., Migórski, S., Papageorgiou, N.S.: An Introduction to Nonlinear Analysis: Theory. Kluwer Academic/Plenum Publishers, Boston, Dordrecht, London, New York (2003)

    Book  Google Scholar 

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

    MATH  Google Scholar 

  20. Preiss, D.: Differentiability of Lipschitz functions on Banach spaces. J. Funct. Anal. 91, 312–345 (1990)

    Article  MATH  MathSciNet  Google Scholar 

  21. Berkovits, J., Mustonen, V.: Monotonicity methods for nonlinear evolution equations. Nonlinear Anal. 27, 1397–1405 (1996)

    Article  MATH  MathSciNet  Google Scholar 

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Acknowledgments

The authors are gratefully indebted to the anonymous referee for his/her insightful comments that improved the paper. The research was supported by the Marie Curie International Research Staff Exchange Scheme Fellowship within the 7th European Community Framework Programme under Grant Agreement No. 295118, the National Science Center of Poland under Maestro Advanced Project No. DEC-2012/06/A/ST1/00262, the National Science Center of Poland under Grant No. N N201 604640, and the International Project co-financed by the Ministry of Science and Higher Education of Republic of Poland under Grant No. W111/7.PR/2012. The research was supported by the Ministry of Science and Higher Education of Republic of Poland and the Ministry of Science and Technology of the People’s Republic of China under the 35th Session Project No. 35-7 within the Bilateral S&T Cooperation Program between Poland and China for the years 2012–2014. The second author was supported by NNSF of China Grants Nos. 11271087 and 61263006, and the last author was also supported by the Guangxi Education Department Grant No. 2013YB067.

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Authors and Affiliations

  1. Faculty of Mathematics and Computer Science, Jagiellonian University, ul. Lojasiewicza 6, 30348,  Krakow, Poland

    Leszek Gasiński, Stanisław Migórski & Anna Ochal

  2. College of Sciences, Guangxi University for Nationalities, Nanning,  530006, Guangxi, People’s Republic of China

    Zhenhai Liu & Zijia Peng

Authors
  1. Leszek Gasiński
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  2. Zhenhai Liu
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  3. Stanisław Migórski
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  4. Anna Ochal
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  5. Zijia Peng
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Correspondence to Zijia Peng.

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Gasiński, L., Liu, Z., Migórski, S. et al. Hemivariational Inequality Approach to Evolutionary Constrained Problems on Star-Shaped Sets. J Optim Theory Appl 164, 514–533 (2015). https://doi.org/10.1007/s10957-014-0587-6

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  • DOI: https://doi.org/10.1007/s10957-014-0587-6

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