Opuscula Math. 37, no. 3 (2017), 353-379
http://dx.doi.org/10.7494/OpMath.2017.37.3.353

Opuscula Mathematica

# Existence of three solutions for impulsive multi-point boundary value problems

Martin Bohner
Shapour Heidarkhani
Giuseppe Caristi

Abstract. This paper is devoted to the study of the existence of at least three classical solutions for a second-order multi-point boundary value problem with impulsive effects. We use variational methods for smooth functionals defined on reflexive Banach spaces in order to achieve our results. Also by presenting an example, we ensure the applicability of our results.

Keywords: multi-point boundary value problem, impulsive condition, classical solution, variational method, three critical points theorem.

Mathematics Subject Classification: 34B10, 34B15, 34A37.

Full text (pdf)

1. L. Bai, B. Dai, An application of variational method to a class of Dirichlet boundary value problems with impulsive effects, J. Franklin Inst. 348 (2011) 9, 2607-2624.
2. D.D. Baĭnov, P.S. Simeonov, Systems with Impulse Effect. Stability, Theory and Applications, Ellis Horwood Series in Mathematics and Its Applications, New York, 1989.
3. M. Benchohra, J. Henderson, S.K. Ntouyas, Impulsive Differential Equations and Inclusions, Contemporary Mathematics and Its Applications, vol. 2, Hindawi Publishing Corporation, New York, 2006.
4. J.R. Cannon, S. Pérez Esteva, J. van der Hoek, A Galerkin procedure for the diffusion equation subject to the specification of mass, SIAM J. Numer. Anal. 24 (1987) 3, 499-515.
5. J.R. Cannon, The One-dimensional Heat Equation, vol. 23, Encyclopedia of Mathematics and its Applications, Addison-Wesley Publishing Company, Advanced Book Program, Reading, MA, 1984.
6. T.E. Carter, Necessary and sufficient conditions for optimal impulsive rendezvous with linear equations of motion, Dynam. Control 10 (2000) 3, 219-227.
7. L. Chen, J. Sun, Nonlinear boundary value problem of first order impulsive functional differential equations, J. Math. Anal. Appl. 318 (2006) 2, 726-741.
8. J. Chen, C.C. Tisdell, R. Yuan, On the solvability of periodic boundary value problems with impulse, J. Math. Anal. Appl. 331 (2007) 2, 902-912.
9. J. Chu, J.J. Nieto, Impulsive periodic solutions of first-order singular differential equations, Bull. Lond. Math. Soc. 40 (2008) 1, 143-150.
10. B. Dai, H. Su, D. Hu, Periodic solution of a delayed ratio-dependent predator-prey model with monotonic functional response and impulse, Nonlinear Anal. 70 (2009) 1, 126-134.
11. Z. Du, L. Kong, Existence of three solutions for systems of multi-point boundary value problems, Electron. J. Qual. Theory Diff. Equ., Spec. Ed. I, 10 (2009).
12. H.-Y. Feng, W.-G. Ge, Existence of three positive solutions for $$M$$-point boundary-value problem with one-dimensional $$p$$-Laplacian, Taiwanese J. Math. 14 (2010) 2, 647-665.
13. M. Feng, H. Pang, A class of three-point boundary-value problems for second-order impulsive integro-differential equations in Banach spaces, Nonlinear Anal. 70 (2009) 1, 64-82.
14. M. Feng, D. Xie, Multiple positive solutions of multi-point boundary value problem for second-order impulsive differential equations, J. Comput. Appl. Math. 223 (2009) 1, 438-448.
15. W. Feng, J.R.L. Webb, Solvability of $$m$$-point boundary value problems with nonlinear growth, J. Math. Anal. Appl. 212 (1997) 2, 467-480.
16. J.R. Graef, L. Kong, Existence of solutions for nonlinear boundary value problems, Comm. Appl. Nonlinear Anal. 14 (2007) 1, 39-60.
17. J.R. Graef, S. Heidarkhani, L. Kong, A critical points approach to multiplicity results for multi-point boundary value problems, Appl. Anal. 90 (2011) 12, 1909-1925.
18. J.R. Graef, S. Heidarkhani, L. Kong, Infinitely many solutions for systems of multi-point boundary value problems using variational methods, Topol. Methods Nonlinear Anal. 42 (2013) 1, 105-118.
19. J.R. Graef, L. Kong, Q. Kong, Higher order multi-point boundary value problems, Math. Nachr. 284 (2011) 1, 39-52.
20. S. Heidarkhani, Multiple solutions for a class of multipoint boundary value systems driven by a one-dimensional $$(p_1,\dots,p_n)$$-Laplacian operator, Abstr. Appl. Anal. 2012 (2012), Article ID 389530.
21. S. Heidarkhani, M. Ferrara, A. Salari, Infinitely many periodic solutions for a class of perturbed second-order differential equations with impulses, Acta Appl. Math. 139 (2015), 81-94.
22. S. Heidarkhani, A. Salari, Existence of three solutions for impulsive perturbed elastic beam fourth-order equations of Kirchhoff-type, Studia Sci. Math. Hungar. (2016), to appear.
23. J. Henderson, Solutions of multipoint boundary value problems for second order equations, Dynam. Systems Appl. 15 (2006) 1, 111-117.
24. J. Henderson, B. Karna, C.C. Tisdell, Existence of solutions for three-point boundary value problems for second order equations, Proc. Amer. Math. Soc. 133 (2005) 5, 1365-1369.
25. J. Henderson, S.K. Ntouyas, Positive solutions for systems of $$n$$th order three-point nonlocal boundary value problems, Electron. J. Qual. Theory Differ. Equ. 18 (2007).
26. V.A. Il'in, E.I. Moiseev, A nonlocal boundary value problem of the second kind for the Sturm-Liouville operator, Differentsial'nye Uravneniya 23 (1987) 8, 1422-1431.
27. N.I. Ionkin, The solution of a certain boundary value problem of the theory of heat conduction with a nonclassical boundary condition, Differencial'nye Uravnenija 13 (1977) 2, 294-304.
28. G. Jiang, Q. Lu, L. Qian, Complex dynamics of a Holling type II prey-predator system with state feedback control, Chaos Solitons Fractals 31 (2007) 2, 448-461.
29. L.I. Kamynin, A boundary-value problem in the theory of heat conduction with non-classical boundary conditions, Ž. Vyčisl. Mat. i Mat. Fiz. 4 (1964), 1006-1024.
30. V. Lakshmikantham, D.D. Baĭnov, P.S. Simeonov, Theory of Impulsive Differential Equations, vol. 6, Series in Modern Applied Mathematics, World Scientific Publishing Co., Teaneck, NJ, 1989.
31. J. Li, J.J. Nieto, Existence of positive solutions for multipoint boundary value problem on the half-line with impulses, Bound. Value Probl. (2009), 2009:834158.
32. B. Liu, J. Yu, Existence of solution of $$m$$-point boundary value problems of second-order differential systems with impulses, Appl. Math. Comput. 125 (2002) 2-3, 155-175.
33. X. Liu, A.R. Willms, Impulsive controllability of linear dynamical systems with applications to maneuvers of spacecraft, Math. Probl. Eng. 2 (1996) 4, 277-299.
34. R. Ma, Existence of positive solutions for superlinear semipositone $$m$$-point boundary-value problems, Proc. Edinb. Math. Soc. 46 (2003) 2, 279-292.
35. M. Moshinsky, On one-dimensional boundary value problems of a discontinuous nature, Bol. Soc. Mat. Mexicana 4 (1947), 1-25.
36. S.I. Nenov, Impulsive controllability and optimization problems in population dynamics, Nonlinear Anal. 36 (1999), 881-890.
37. D. Qian, X. Li, Periodic solutions for ordinary differential equations with sublinear impulsive effects, J. Math. Anal. Appl. 303 (2005) 1, 288-303.
38. B. Ricceri, A further three critical points theorem, Nonlinear Anal. 71 (2009) 9, 4151-4157.
39. A.M. Samoĭlenko, M.O. Perestyuk, Impulsive Differential Equations, vol. 14, World Scientific Series on Nonlinear Science, Series A: Monographs and Treatises, World Scientific Publishing Co., River Edge, NJ, 1995.
40. J. Simon, Régularité de la solution d'une équation non linéaire dans $$\mathbb{R}^N$$, [in:] Journées d'Analyse Non Linéaire (Proc. Conf., Besançon, 1977), vol. 665, Lecture Notes in Math., Springer, Berlin, 1978, 205-227.
41. J. Sun, H. Chen, J.J. Nieto, M. Otero-Novoa, The multiplicity of solutions for perturbed second-order Hamiltonian systems with impulsive effects, Nonlinear Anal. 72 (2010) 12, 4575-4586.
42. C. Thaiprayoon, D. Samana, J. Tariboon, Multi-point boundary value problem for first order impulsive integro-differential equations with multi-point jump conditions, Bound. Value Probl. 2012, 2012:38.
43. Y. Tian, W. Ge, Applications of variational methods to boundary-value problem for impulsive differential equations, Proc. Edinb. Math. Soc. 51 (2008) 2, 509-527.
44. Y. Tian, W. Ge, Variational methods to Sturm-Liouville boundary value problem for impulsive differential equations, Nonlinear Anal. 72 (2010) 1, 277-287.
45. S.P. Timoshenko, Theory of Elastic Stability, 2nd ed., McGraw-Hill Book Co., New York, 1961.
46. J. Xiao, J.J. Nieto, Z. Luo, Multiplicity of solutions for nonlinear second order impulsive differential equations with linear derivative dependence via variational methods, Commun. Nonlinear Sci. Numer. Simul. 17 (2012) 1, 426-432.
47. E. Zeidler, Nonlinear functional analysis and its applications. II/A: Linear monotone operators, Springer-Verlag, New York, 1990.
48. H. Zhang, Z. Li, Variational approach to impulsive differential equations with periodic boundary conditions, Nonlinear Anal. Real World Appl. 11 (2010) 1, 67-78.
• Martin Bohner
• Department of Mathematics and Statistics, Missouri S&T, Rolla, MO 65409-0020, USA
• Shapour Heidarkhani
• Razi University, Faculty of Sciences, Department of Mathematics, 67149 Kermanshah, Iran
• Razi University, Faculty of Sciences, Department of Mathematics, 67149 Kermanshah, Iran
• Giuseppe Caristi
• University of Messina, Department of Economics, Messina, Italy
• Communicated by Marek Galewski.
• Revised: 2016-12-07.
• Accepted: 2016-12-08.
• Published online: 2017-01-30.