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Topological Methods in Nonlinear Analysis

Three solutions for second-order impulsive differential inclusions with Sturm-Liouville boundary conditions via nonsmooth critical point theory
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Three solutions for second-order impulsive differential inclusions with Sturm-Liouville boundary conditions via nonsmooth critical point theory

Authors

  • Yu Tian
  • John R. Graef
  • Lingju Kong
  • Min Wang

DOI:

https://doi.org/10.12775/TMNA.2015.089

Keywords

Differential inclusions, impulsive, Sturm-Liouville boundary conditions, nonsmooth critical point theory

Abstract

A second-order impulsive differential inclusion with Sturm-Liouville boundary conditions is studied. By using a nonsmooth version of a three critical point theorem of Ricceri, the existence of three solutions is obtained.

References

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M. Benchohra, J. Henderson and S. Ntouyas, Impulsive Differential Equations and Inclusions, Hindawi Publishing Corporation, 2006.

V.I. Blagodatskih and A.F. Filippov, Differential inclusions and optimal control, Trudy Mat. Inst. Steklov. 169 (1985), 194-252, 255 (Russian).

A.I. Bulgakov, Integral inclusions with nonconvex images and their applications to boundary value problems for differential inclusions, Mat. Sb. 183 (1992), 63-86 (Russian); transl.: Acad. Sci. Sb. Math. 77 (1994), 193-212.

A.I. Bulgakov and L.I. Tkach, Perturbation of a convex-valued operator by a Hammerstein-type multivalued mapping with nonconvex images, and boundary value problems for functional differential inclusions, Mat. Sb. 189 (1998), no. 6, 3-32 (Russian); transl.: Sb. Math. 189 (1998), 821-848.

F.H. Clarke, Optimization and Nonsmooth Analysis, Wiley, New York, 1983.

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Y.C. Liu, J. Wu and Z.X. Li, Impulsive boundary value problems for Sturm-Liouville type differential inclusion, J. Syst. Sci. Complex. 20 (2007), 370-380.

M.D.P. Monteiro Marques, Differential Inclusions in Nonsmooth Mechanical Problems, Progress in Nonlinear Differential Equations and Their Applications, Birkhauser, Basel, 1993.

D. Motreanu and P.D. Panagiotopoulos, Minimax Theorems and Qualitative Properties of the Solutions of Hemivariational Inequalities, Kluwer, Dordrecht 1999.

N. Ribarska, T. Tsacheva nd M. Krastanov, A note on: "On a critical point theory for multivalued functionals and application to partial differential inclusions", Nonlinear Anal. 43 (2001), 153-158.

B. Ricceri, Existence of three solutions for a class of elliptic eigenvalue problems, Math. Comput. Modelling 32 (2000), 1485-1494.

G.V. Smirnov, Introduction to the Theory of Differential Inclusions, Graduate Studies in Mathematics, vol. 41, Amer. Math. Soc., Providence, 2002.

Y. Tian and W. Ge, Applications of variational methods to boundary value problem for impulsive differential equations, Proc. Edinburgh Math. Soc. 51 (2008), 509-527.

Y. Tian and W. Ge, Variational methods to Sturm-Liouville boundary value problem for impulsive differential equations, Nonlinear Anal. 72 (2010), 277-287.

E. Zeidler, Nonlinear Functional Analysis and its Applications, Vol. III, Springer, New York 1985.

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Published

2016-03-01

How to Cite

1.
TIAN, Yu, GRAEF, John R., KONG, Lingju and WANG, Min. Three solutions for second-order impulsive differential inclusions with Sturm-Liouville boundary conditions via nonsmooth critical point theory. Topological Methods in Nonlinear Analysis. Online. 1 March 2016. Vol. 47, no. 1, pp. 1 - 18. [Accessed 4 July 2025]. DOI 10.12775/TMNA.2015.089.
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