Skip to main content
SpringerLink
Log in

On the Approximate Controllability of Second-Order Evolution Hemivariational Inequalities

  • Published:
Results in Mathematics Aims and scope Submit manuscript

Abstract

In our manuscript, we organize a group of sufficient conditions for the approximate controllability of second-order evolution hemivariational inequalities. By applying a suitable fixed-point theorem for multivalued maps, we prove our results. Lastly, we present an example to illustrate the obtained theory.

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. Ahmed, H.M., El-Owaidy, H.M., AL-Nahhas, M.A.: Neutral fractional stochastic partial differential equations with Clarke subdifferential. Appl. Anal. 66, 1–13 (2019). https://doi.org/10.1080/00036811.2020.1714035

    Article  Google Scholar 

  2. Arthi, G., Balachandran, K.: Controllability of second-order impulsive evolution systems with infinite delay. Nonlinear Anal. Hybrid Syst. 11, 139–153 (2014)

    Article  MathSciNet  MATH  Google Scholar 

  3. Bashirov, A.E., Mahmudov, N.I.: On concepts of controllability for deterministic and stochastic systems. SIAM J. Control Optim. 37(6), 1808–1821 (1999)

    Article  MathSciNet  MATH  Google Scholar 

  4. Batty, C.J.K., Chill, R., Srivastava, S.: Maximal regularity for second-order non-autonomous Cauchy problems. Studia Math. 189, 205–223 (2008)

    Article  MathSciNet  MATH  Google Scholar 

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

    Article  MathSciNet  MATH  Google Scholar 

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

    Article  MathSciNet  MATH  Google Scholar 

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

    MATH  Google Scholar 

  8. Dhage, B.C.: Multi-valued mappings and fixed points II. Tamkang J. Math. 37(1), 27–46 (2006)

    Article  MathSciNet  MATH  Google Scholar 

  9. Haslinger, J., Panagiotopoulos, P.D.: Optimal control of systems governed by hemivariational inequalities. Exist. Approx. Results Nonlinear Anal. 24(1), 105–119 (1995)

    Article  MATH  Google Scholar 

  10. Henríquez, H.R.: Existence of solutions of non-autonomous second-order functional differential equations with infinite delay. Nonlinear Anal. Theory Methods Appl. 74, 3333–3352 (2011)

    Article  MathSciNet  MATH  Google Scholar 

  11. Henríquez, H.R., Hernández, E.: Existence of solutions of a second-order abstract functional Cauchy problem with nonlocal conditions. Ann. Pol. Math. 88(2), 141–159 (2006)

    Article  MathSciNet  MATH  Google Scholar 

  12. Hernández, E., Henriquez, H.R., McKibben, M.A.: Existence results for abstract impulsive second-order neutral functional differential equations. Nonlinear Anal. Theory Methods Appl. 70, 2736–2751 (2009)

    Article  MathSciNet  MATH  Google Scholar 

  13. Huang, Y., Liu, Z.H., Migórski, S.: Elliptic hemivariational inequalities with nonhomogeneous Neumann boundary conditions and their applications to static frictional contact problems. Acta Appl. Math. 138, 153–170 (2015)

    Article  MathSciNet  MATH  Google Scholar 

  14. Kisyński, J.: On cosine operator functions and one parameter group of operators. Studia Math. 49, 93–105 (1972)

    Article  MathSciNet  MATH  Google Scholar 

  15. Kozak, M.: A fundamental solution of a second-order differential equation in a Banach space. Univ. Iagel. Acta Math. 32, 275–289 (1995)

    MathSciNet  MATH  Google Scholar 

  16. Li, X., Liu, Z.H., Migórski, S.: Approximate controllability for second-order nonlinear evolution hemivariational inequalities. Electron. J. Qualit. Theory Differ. Equ. 100, 1–16 (2015)

    MathSciNet  MATH  Google Scholar 

  17. Lightbourne, J.H., Rankin, S.: A partial functional differential equation of Sobolev type. J. Math. Anal. Appl. 93, 328–337 (1983)

    Article  MathSciNet  MATH  Google Scholar 

  18. Liu, Z.H., Li, X.: Approximate controllability for a class of hemivariational inequalities. Nonlinear Anal. Real World Appl. 22, 581–591 (2015)

    Article  MathSciNet  MATH  Google Scholar 

  19. Lu, L., Liu, Z.H.: Existence and controllability results for stochastic fractional evolution hemivariational inequalities. Appl. Math. Comput. 268, 1164–1176 (2015)

    MathSciNet  MATH  Google Scholar 

  20. Mahmudov, N.I.: Existence and approximate controllability of Sobolev type fractional stochastic evolution equations. Bull. Pol. Acad. Sci. Tech. Sci. 62(2), 205–215 (2014)

    Google Scholar 

  21. Mahmudov, N. I.: Approximate controllability of fractional Sobolev-type evolution equations in Banach spaces. Abstr. Appl. Anal. 1–9. Article ID 502839 (2013)

  22. Mahmudov, N.I., Denker, A.: On controllability of linear stochastic systems. Int. J. Control 73, 144–151 (2000)

    Article  MathSciNet  MATH  Google Scholar 

  23. Mahmudov, N.I., Murugesu, R., Ravichandran, C., Vijayakumar, V.: Approximate controllability results for fractional semilinear integro-differential inclusions in Hilbert spaces. Results Math. 71(1), 45–61 (2017)

    Article  MathSciNet  MATH  Google Scholar 

  24. Mahmudov, N.I., Vijayakumar, V., Murugesu, R.: Approximate controllability of second-order evolution differential inclusions in Hilbert spaces. Mediterr. J. Math. 13(5), 3433–3454 (2016)

    Article  MathSciNet  MATH  Google Scholar 

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

    Article  MathSciNet  MATH  Google Scholar 

  26. Migórski, S., Ochal, A.: Quasi-static hemivariational inequality via vanishing acceleration approach. SIAM J. Math. Anal. 41, 1415–1435 (2009)

    Article  MathSciNet  MATH  Google Scholar 

  27. 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)

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

    MATH  Google Scholar 

  29. Panagiotopoulos, P.D.: Hemivariational inequality and fan-variational inequality, new applications and results. Atti del Seminario Matematico e Fisico dell’ Universita di Modena, XLIII, pp. 159–191 (1995)

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

    Book  MATH  Google Scholar 

  31. Panagiotopoulos, P.D., Pop, G.: On a type of hyperbolic variational-hemivariational inequalities. J. Appl. Anal. 5(1), 95–112 (1999)

    Article  MathSciNet  MATH  Google Scholar 

  32. Serizawa, H., Watanabe, M.: Time-dependent perturbation for cosine families in Banach spaces. Houst. J. Math. 12, 579–586 (1986)

    MathSciNet  MATH  Google Scholar 

  33. Sivasankaran, S., Mallika Arjunan, M., Vijayakumar, V.: Existence of global solutions for second order impulsive abstract partial differential equations. Nonlinear Anal. Theory Methods Appl. 74(17), 6747–6757 (2011)

    Article  MathSciNet  MATH  Google Scholar 

  34. Travis, C.C., Webb, G.F.: Cosine families and abstract nonlinear second-order differential equations. Acta Math. Acad. Sci. Hung. 32, 76–96 (1978)

    Article  MathSciNet  MATH  Google Scholar 

  35. Travis, C.C., Webb, G.F.: Compactness, regularity, and uniform continuity properties of strongly continuous cosine families. Houst. J. Math. 3(4), 555–567 (1977)

    MathSciNet  MATH  Google Scholar 

  36. Vijayakumar, V.: Approximate controllability for a class of second-order stochastic evolution inclusions of Clarke’s subdifferential type. Results Math. 73(42), 1–23 (2018)

    MathSciNet  MATH  Google Scholar 

  37. Vijayakumar, V.: Approximate controllability results for abstract neutral integro-differential inclusions with infinite delay in Hilbert spaces. IMA J. Math. Control Inf. 35(1), 297–314 (2018)

    MathSciNet  MATH  Google Scholar 

  38. Vijayakumar, V.: Approximate controllability results for analytic resolvent integro-differential inclusions in Hilbert spaces. Int. J. Control 91(1), 204–214 (2018)

    Article  MathSciNet  MATH  Google Scholar 

  39. Vijayakumar, V.: Approximate controllability results for impulsive neutral differential inclusions of Sobolev-type with infinite delay. Int. J. Control 91(10), 2366–2386 (2018)

    Article  MathSciNet  MATH  Google Scholar 

  40. Vijayakumar, V., Henríquez, H.R.: Existence of global solutions for a class of abstract second order nonlocal Cauchy problem with impulsive conditions in Banach spaces. Numer. Funct. Anal. Optim. 39(6), 704–736 (2018)

    Article  MathSciNet  MATH  Google Scholar 

  41. Vijayakumar, V., Murugesu, R., Poongodi, R., Dhanalakshmi, S.: Controllability of second-order impulsive nonlocal Cauchy problem via measure of noncompactness. Mediterr. J. Math. 14(1), 29–51 (2017)

    Article  MathSciNet  MATH  Google Scholar 

  42. Vijayakumar, V., Sivasankaran, S., Mallika Arjunan, M.: Existence of solutions for second-order impulsive neutral functional integro-differential equations with infinite delay. Nonlinear Stud. 19(2), 327–343 (2012)

    MathSciNet  MATH  Google Scholar 

  43. Vijayakumar, V., Udhayakumar, R., Dineshkumar, C.: Approximate controllability of second-order nonlocal neutral differential evolution inclusions. IMA J. Math. Control Inf. (2020). https://doi.org/10.1093/imamci/dnaa001

    Article  Google Scholar 

Download references

Acknowledgements

The authors are immensely grateful to the anonymous referees for their careful reading of this paper and helpful comments, which have been very useful for improving the quality of this paper.

Author information

Authors and Affiliations

  1. Department of Mathematics, Eastern Mediterranean University, T.R. North Cyprus, Mersin 10, Gazimagusa, Turkey

    N. I. Mahmudov

  2. Department of Mathematics, School of Advanced Sciences, Vellore Institute of Technology, Vellore, Tamil Nadu, 632 014, India

    R. Udhayakumar & V. Vijayakumar

Authors
  1. N. I. Mahmudov
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. R. Udhayakumar
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. V. Vijayakumar
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to N. I. Mahmudov.

Additional information

Publisher's Note

Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Mahmudov, N.I., Udhayakumar, R. & Vijayakumar, V. On the Approximate Controllability of Second-Order Evolution Hemivariational Inequalities. Results Math 75, 160 (2020). https://doi.org/10.1007/s00025-020-01293-2

Download citation

  • Received:

  • Accepted:

  • Published:

  • DOI: https://doi.org/10.1007/s00025-020-01293-2

Mathematics Subject Classification

Keywords

Search

Navigation