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An application of variational methods to Dirichlet boundary value problem with impulses. (English) Zbl 1191.34039
The authors consider the impulsive Dirichlet boundary value problem $$\aligned &-u''(t) + \lambda u(t) = f(t,u(t)) + p(t), \quad t \in [0,T], \\ &\triangle u'(t_j) = I_j(u(t_j)),\ j = 1,\dots,k, \\ &u(0) = u(T) = 0, \endaligned$$ where $0 < t_1 < \dots < t_k < T$ are impulse instants, the impulsive functions $I_j : {\Bbb R} \to {\Bbb R}$ and the right-hand side $f$ are continuous, $p \in L^2[0,T]$, $\lambda > -\pi^2/T^2$. Sufficient conditions for the existence of at least one and infinitely many weak solutions are found. The proofs are based on the critical points theory.

34B37Boundary value problems for ODE with impulses
58E05Abstract critical point theory
47J30Variational methods (nonlinear operator equations)
Full Text: DOI
[1] 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 · doi:10.1007/s00010-004-2735-9
[2] Ahmad, B.; Nieto, J. J.: Existence and approximation of solutions for a class of nonlinear impulsive functional differential equations with anti-periodic boundary conditions, Nonlinear anal. TMA 69, No. 10, 3291-3298 (2008) · Zbl 1158.34049 · doi:10.1016/j.na.2007.09.018
[3] Ambrosetti, A.; Rabinowitz, P. H.: Dual variational methods in critical point theory and applications, J. funct. Anal. 14, 349-381 (1973) · Zbl 0273.49063 · doi:10.1016/0022-1236(73)90051-7
[4] Braverman, E.; Zhukovskiy, S.: The problem of a lazy tester, or exponential dichotomy for impulsive differential equations revisited, Nonlinear anal. Hybrid syst. 2, 971-979 (2008) · Zbl 1217.34091 · doi:10.1016/j.nahs.2008.04.002
[5] Carter, T. E.: Optimal impulsive space trajectories based on linear equations, J. optim. Theory appl. 70, 277-297 (1991) · Zbl 0732.49025 · doi:10.1007/BF00940627
[6] Carter, T. E.: Necessary and sufficient conditions for optimal impulsive rendezvous with linear equations of motion, Dynam. control. 10, 219-227 (2000) · Zbl 0980.93058 · doi:10.1023/A:1008376427023
[7] Chen, J.; Tisdel, C. C.; Yuan, R.: On the solvability of periodic boundary value problems with impulse, J. math. Anal. appl. 331, No. 2, 902-912 (2007) · Zbl 1123.34022 · doi:10.1016/j.jmaa.2006.09.021
[8] 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 · doi:10.1016/j.jmaa.2005.08.012
[9] Choisy, M.; Guégan, J. F.; Rohani, P.: Dynamics of infectious diseases and pulse vaccination: teasing apart the embedded resonance effects, Physica D 223, 26-35 (2006) · Zbl 1110.34031 · doi:10.1016/j.physd.2006.08.006
[10] Chu, J.; Nieto, J. J.: Impulsive periodic solutions of first-order singular differential equations, Bull. London math. Soc. 40, No. 1, 143-150 (2008) · Zbl 1144.34016 · doi:10.1112/blms/bdm110
[11] Dai, B.; Su, H.; Hu, D.: Periodic solution of a delayed ratio-dependent predator--prey model with monotonic functional response and impulse, Nonlinear anal. 70, 126-134 (2009) · Zbl 1166.34043 · doi:10.1016/j.na.2007.11.036
[12] De Coster, C.; Habets, P.: Upper and lower solutions in the theory of ODE boundary value problems: classical and recent results, Nonlinear analysis and boundary value problems for ordinary differential equations CISM-ICMS 371, 1-78 (1996) · Zbl 0889.34018
[13] D’onofrio, A.: On pulse vaccination strategy in the SIR epidemic model with vertical transmission, Appl. math. Lett. 18, 729-732 (2005) · Zbl 1064.92041 · doi:10.1016/j.aml.2004.05.012
[14] Gao, S.; Chen, L.; Nieto, J. J.; Torres, A.: Analysis of a delayed epidemic model with pulse vaccination and saturation incidence, Vaccine 24, 6037-6045 (2006)
[15] George, R. K.; Nandakumaran, A. K.; Arapostathis, A.: A note on controllability of impulsive systems, J. math. Anal. appl. 241, 276-283 (2000) · Zbl 0965.93015 · doi:10.1006/jmaa.1999.6632
[16] H. Guo, L. Chen, Time-limited pest control of a Lotka--Volterra model with impulsive harvest, Nonlinear Anal. RWA, doi:10.1016/j.nonrwa.2007.11.007 (in press) · Zbl 1167.34306
[17] E. Hernandez, H.R. Henriquez, M.A. McKibben, Existence results for abstract impulsive second-order neutral functional differential equations, Nonlinear Anal., doi:10.1016/j.na.2008.03.062 (in press)
[18] Izydorek, M.; Janczewska, J.: Homoclinic solutions for a class of the second order Hamiltonian systems, J. differential equations. 219, 375-389 (2005) · Zbl 1080.37067 · doi:10.1016/j.jde.2005.06.029
[19] Jiang, G.; Lu, Q.: Impulsive state feedback control of a predator--prey model, J. comput. Appl. math. 200, 193-207 (2007) · Zbl 1134.49024 · doi:10.1016/j.cam.2005.12.013
[20] Jiang, G.; Lu, Q.; Qian, L.: Chaos and its control in an impulsive differential system, Chaos solitons fractals 34, 1135-1147 (2007) · Zbl 1142.93424 · doi:10.1016/j.chaos.2006.04.024
[21] Jiang, G.; Lu, Q.; Qian, L.: Complex dynamics of a Holling type II prey--predator system with state feedback control, Chaos, solitons fractals 31, 448-461 (2007) · Zbl 1203.34071 · doi:10.1016/j.chaos.2005.09.077
[22] Lakshmikantham, V.; Bainov, D. D.; Simeonov, P. S.: Theory of impulsive differential equations, (1989) · Zbl 0718.34011
[23] Li, J.; Nieto, J. J.; Shen, J.: Impulsive periodic boundary value problems of first-order differential equations, J. math. Anal. appl. 325, 226-236 (2007) · Zbl 1110.34019 · doi:10.1016/j.jmaa.2005.04.005
[24] Liu, X.; Willms, A. R.: Impulsive controllability of linear dynamical systems with applications to maneuvers of spacecraft, Math. problems engineer. 2, 277-299 (1996) · Zbl 0876.93014 · doi:10.1155/S1024123X9600035X
[25] Z. Luo, J.J. Nieto, New results for the periodic boundary value problem for impulsive integro-differential equations, Nonlinear Anal., doi:10.1016/j.na.2008.03.004 (in press) · Zbl 1166.45002
[26] Mawhin, J.: Topological degree and boundary value problems for nonlinear differential equations, Lecture notes in mathematics 1537, 74-142 (1993) · Zbl 0798.34025
[27] Mawhin, J.; Willem, M.: Critical point theory and Hamiltonian systems, (1989) · Zbl 0676.58017
[28] Mohamad, S.; Gopalsamy, K.; Akca, H.: Exponential stability of artificial neural networks with distributed delays and large impulses, Nonlinear anal. RWA 9, 872-888 (2008) · Zbl 1154.34042 · doi:10.1016/j.nonrwa.2007.01.011
[29] Nenov, S.: Impulsive controllability and optimization problems in population dynamics, Nonlinear anal. 36, 881-890 (1999) · Zbl 0941.49021 · doi:10.1016/S0362-546X(99)00627-6
[30] Nieto, Juan J.: Basic theory for nonresonance impulsive periodic problems of first order, J. math. Anal. appl. 205, No. 2, 423-433 (1997) · Zbl 0870.34009 · doi:10.1006/jmaa.1997.5207
[31] 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 · doi:10.1016/j.jmaa.2006.06.029
[32] Nieto, J. J.; O’regan, D.: Variational approach to impulsive differential equations, Nonlinear anal. RWA (2007)
[33] Pei, Y.; Li, C.; Chen, L.; Wang, C.: Complex dynamics of one-prey multi-predator system with defensive ability of prey and impulsive biological control on predators, Adv. complex syst. 8, 483-495 (2005) · Zbl 1082.92046 · doi:10.1142/S0219525905000579
[34] Prado, A. F. B.A.: Bi-impulsive control to build a satellite constellation, Nonlinear dyn. Syst. theory. 5, 169-175 (2005) · Zbl 1128.70015 · http://www.e-ndst.kiev.ua/v5n2.htm
[35] 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 · doi:10.1016/j.jmaa.2004.08.034
[36] Rabinowitz, P. H.: Minimax methods in critical point theory with applications to differential equations, CBMS regional conf. Ser. in. Math. 65 (1986) · Zbl 0609.58002
[37] Samoilenko, A. M.; Perestyuk, N. A.: Impulsive differential equations, (1995) · Zbl 0837.34003
[38] Shen, J.; Li, J.: Existence and global attractivity of positive periodic solutions for impulsive predator--prey model with dispersion and time delays, Nonlinear anal. RWA 10, 227-243 (2009) · Zbl 1154.34372 · doi:10.1016/j.nonrwa.2007.08.026
[39] 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 · doi:10.1017/S0013091506001532
[40] Wang, W.; Shen, J.; Nieto, J. J.: Permanence and periodic solution of predator prey system with Holling type functional response and impulses, Discrete dyn. Nat. soc. (2007) · Zbl 1146.37370 · doi:10.1155/2007/81756
[41] Wei, C.; Chen, L.: A delayed epidemic model with pulse vaccination, Discrete dyn. Nat. soc. (2008) · Zbl 1149.92329 · doi:10.1155/2008/746951
[42] Xia, Y.: Positive periodic solutions for a neutral impulsive delayed Lotka--Volterra competition system with the effect of toxic substance, Nonlinear anal. RWA 8, 204-221 (2007) · Zbl 1121.34075 · doi:10.1016/j.nonrwa.2005.07.002
[43] Yan, J.; Zhao, A.; Nieto, J. J.: Existence and global attractivity of positive periodic solution of periodic single-species impulsive Lotka--Volterra systems, Math. comput. Modeling 40, 509-518 (2004) · Zbl 1112.34052 · doi:10.1016/j.mcm.2003.12.011
[44] Zeng, G.; Wang, F.; Nieto, J. J.: Complexity of delayed predator--prey model with impulsive harvest and Holling type-II functional response, Adv. complex syst. 11, 77-97 (2008) · Zbl 1168.34052 · doi:10.1142/S0219525908001519
[45] 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 · doi:10.1016/j.nonrwa.2007.05.004
[46] Zhang, H.; Xu, W.; Chen, L.: A impulsive infective transmission SI model for pest control, Math. methods appl. Sci. 30, 1169-1184 (2007) · Zbl 1155.34328 · doi:10.1002/mma.834
[47] Zhou, J.; Xiang, L.; Liu, Z.: Synchronization in complex delayed dynamical networks with impulsive effects, Physica A 384, 684-692 (2007)