×

The existence and multiplicity of solutions for an impulsive differential equation with two parameters via a variational method. (English) Zbl 1198.34037

The authors consider the impulsive boundary value problem for an ordinary differential equation of second order
\[ \begin{aligned} &-u''(t) +cu(t) = \lambda g(t,u(t)) \quad \text{a.e.}\;t \in[0,T], t\neq t_k,\\ &\triangle u'(t_k) = I_k(u(t_k)), \quad k = 1,\ldots, p-1,\\ &u(0) = u(T), \quad u'(0) = u'(T),\end{aligned} \]
where \(0 < t_1 < \ldots < t_{p-1} < T\); \(c, \lambda \in {\mathbb R}\), \(\lambda \neq 0\); \(g : [0,T] \times {\mathbb R} \to {\mathbb R}\) is a continuous function; \(I_k : {\mathbb R} \to {\mathbb R}\) are continuous. Some new criteria to guarantee the existence of at least one solution, two solutions and infinitely many solutions according to the values of the pair \((c,\lambda)\) are given. The results are obtained by using variational methods and critical point theory.

MSC:

34B37 Boundary value problems with impulses for ordinary differential equations
34B08 Parameter dependent boundary value problems for ordinary differential equations
58E05 Abstract critical point theory (Morse theory, Lyusternik-Shnirel’man theory, etc.) in infinite-dimensional spaces
PDFBibTeX XMLCite
Full Text: DOI

References:

[1] Lakshmikantham, V.; Bainov, D. D.; Simeonov, P. S., Theory of Impulsive Differential Equations (1989), World Scientific: World Scientific Singapore · Zbl 0719.34002
[2] Agarwal, R. P.; Franco, D.; O’Regan, D., Singular boundary value problems for first and second order impulsive differential equations, Aequationes Math., 69, 83-96 (2005) · Zbl 1073.34025
[3] Nieto, J. J.; Rodriguez-Lopez, R., Boundary value problems for a class of impulsive functional equations, Comput. Math. Appl., 55, 2715-2731 (2008) · Zbl 1142.34362
[4] Li, J.; Nieto, J. J.; Shen, J., Impulsive periodic boundary value problems of first-order differential equations, J. Math. Anal. Appl., 325, 226-299 (2007) · Zbl 1110.34019
[5] Nieto, J. J.; Rodriguez-Lopez, R., New comparison results for impulsive integro-differential equations and applications, J. Math. Anal. Appl., 328, 1343-1368 (2007) · Zbl 1113.45007
[6] Samoilenko, A. M.; Perestyuk, N. A., Impulsive Differential Equations (1995), World Scientific: World Scientific Singapore · Zbl 0837.34003
[7] Zhang, H.; Chen, L.; Nieto, J. J., A delayed epidemic model with stage structure and pulses for management strategy, Nonlinear Anal. RWA, 9, 1714-1726 (2008) · Zbl 1154.34394
[8] Benchohra, M.; Henderson, J.; Ntouyas, S. K., Impulsive Differential Equations and Inclusions, vol. 2 (2006), Hindawi Publishing Corporation: Hindawi Publishing Corporation New York · Zbl 1130.34003
[9] Zeng, G.; Wang, F.; Nieto, J. J., Complexity of a delayed predator-prey model with impulsive harvest and Holling-type II functional response, Adv. Complex Syst., 11, 77-97 (2008) · Zbl 1168.34052
[10] Meng, X.; Li, Z.; Nieto, J. J., Dynamic analysis of Michaelis-Menten chemostat-type competition models with time delay and pulse in a polluted environment, J. Math. Chem., 47, 123-144 (2009) · Zbl 1194.92075
[11] Qian, D.; Li, X., Periodic solutions for ordinary differential equations with sublinear impulsive effects, J. Math. Anal. Appl., 303, 288-303 (2005) · Zbl 1071.34005
[12] Chen, L.; Sun, J., Nonlinear boundary value problem for first order impulsive functional differential equations, J. Math. Anal. Appl., 318, 726-741 (2006) · Zbl 1102.34052
[13] Chen, L.; Tisdel, C. C.; Yuan, R., On the solvability of periodic boundary value problems with impulse, J. Math. Anal. Appl., 331, 233-244 (2007)
[14] Chu, J.; Nieto, J. J., Impulsive periodic solution of first-order singular differential equations, Bull. London. Math. Soc., 40, 143-150 (2008) · Zbl 1144.34016
[15] Bonanno, G.; Di Bella, B., A boundary value problem for fourth-order elastic beam equations, J. Math. Anal. Appl., 343, 1166-1176 (2008) · Zbl 1145.34005
[16] Li, F.; Zhang, Z.; Zhang, Q., Existence of solutions to a class of nonlinear second order two-point boundary value problems, J. Math. Anal. Appl., 312, 357-373 (2005) · Zbl 1088.34012
[17] Gyulov, T.; Morosanu, G.; Tersian, S., Existence for a semilinear sixth-order ODE, J. Math. Anal. Appl., 321, 86-98 (2006) · Zbl 1106.34007
[18] Liu, X.; Li, W., Existence and multiplicity of solutions for fourth-order boundary value problems with three parameters, Math. Comput. Modelling., 46, 525-534 (2007) · Zbl 1135.34007
[19] Yang, Y.; Zhang, J., Existence of solutions for some fourth-order boundary value problems with parameters, Nonlinear Anal., 69, 1364-1375 (2008) · Zbl 1166.34012
[20] Zhang, X.; Zhou, Y., Periodic solutions of non-autonomous second order Hamiltonian systems, J. Math. Anal. Appl., 345, 929-933 (2008) · Zbl 1173.34329
[21] Rabinowitz, P. H., Minimax methods in critical point theory with applications to differential equations, (CBMS Regional Conf. Ser. in. Math., vol. 65 (1986), American Mathematical Society: American Mathematical Society Providence, RI) · Zbl 0152.10003
[22] Mawhin, J.; Willem, M., Critical Point Theory and Hamiltonian Systems (1989), Springer-Verlag: Springer-Verlag Berlin · Zbl 0676.58017
[23] Tian, Y.; Ge, W., Applications of variational methods to boundary-value problem for impulsive differential equations, Proc. Edinb. Math. Soc., 51, 509-527 (2008) · Zbl 1163.34015
[24] Nieto, J. J.; O’Regan, D., Variational approach to impulsive differential equations, Nonlinear Anal. RWA, 10, 680-690 (2009) · Zbl 1167.34318
[25] Zhang, Z.; Yuan, R., An application of variational methods to Dirichlet boundary value problem with impulses, Nonlinear Anal. RWA, 11, 155-162 (2010) · Zbl 1191.34039
[26] Zhou, J.; Li, Y., Existence and multiplicity of solutions for some Dirichlet problems with impulse effects, Nonlinear Anal., 71, 2856-2865 (2009) · Zbl 1175.34035
[27] Zhang, H.; Li, Z., Variational approach to impulsive differential equations with periodic boundary conditions, Nonlinear Anal. RWA, 11, 67-78 (2010) · Zbl 1186.34089
[28] Sun, J.; Chen, H., Variational method to the impulsive equation with Neumann boundary conditions, Bound. Value Probl. (2009), Article ID 316812, 17 pages · Zbl 1184.34039
[29] Brezis, H.; Nirenberg, L., Remarks on finding critical points, Comm. Pure Appl. Math., XLIV, 939-963 (1991) · Zbl 0751.58006
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.