## Variational approach to impulsive differential equations with periodic boundary conditions.(English)Zbl 1186.34089

Summary: The present paper is devoted to studying the second order impulsive differential equations of the following form,
\begin{aligned} & u''(t)= \nabla F(t,u(t))\quad \forall t\in (s_{k-1},s_k),\\ & \Delta u'(s_k)=g(u(s_k)),\\ & u(0)-u(T)=0,\end{aligned}
where $$T>0$$, $$(\{s_k\}$$ is a real sequence such that $$0+s_0<s_1<\cdots<s_m=T$$, $$\Delta u'(s_k)\equiv u'(s^+_k)-u'(s^-_k)=\lim_{s\to s^+_k}u'(s)-\lim_{s\to s^-_k}-u'(s)$$, $$k=1,\dots,m-1$$, $$u'(s^-_0)$$ and $$u'(s^-_m)$$ are defined to be $$u(s^-_m)$$ and $$u'(s^+_0)$$, $$g:\mathbb R\to\mathbb R$$ is continuous and $$F:[0,T]\times \mathbb R\to\mathbb R$$ is continuously differentiable with respect to its second variable.
The main tool that we use is critical point theory. Our results generalize some existing results on periodic solutions for second order ordinary differential equations even when the impulses are absent.

### MSC:

 34K10 Boundary value problems for functional-differential equations 34K45 Functional-differential equations with impulses 58E05 Abstract critical point theory (Morse theory, Lyusternik-Shnirel’man theory, etc.) in infinite-dimensional spaces
Full Text:

### References:

 [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 [2] Akhmetov, M.U.; Zafer, A., Controllability of the vallée-poussin problem for impulsive differential systems, J. optim. theory appl., 102, 263-276, (1999) · Zbl 0941.93004 [3] Akhmetov, M.U.; Zafer, A., Stability of the zero solution of impulsive differential equations by the Lyapunov second method, J. math. anal. appl, 248, 69-82, (2000) · Zbl 0965.34007 [4] Carter, T.E., Optimal impulsive space trajectories based on linear equations, J. optim. theory appl., 70, 277-297, (1991) · Zbl 0732.49025 [5] Carter, T.E., Necessary and sufficient conditions for optional impulsive rendezvous with linear equations of motions, Dynam. control, 10, 219-227, (2000) · Zbl 0980.93058 [6] Chen, J.; Tisdell, C.C.; Yuan, R., On the solvability of periodic boundary value problems with impulse, J. math. anal. appl., 331, 902-912, (2007) · Zbl 1123.34022 [7] Chu, J.; Nieto, J.J., Impulsive periodic solutions of first-order singular differential equations, Bull. London math. soc., 40, 143-150, (2008) · Zbl 1144.34016 [8] Dai, B.; Su, H.; Hu, D., Periodic solution of a delayed ratio-dependent predator – prey model with monotonic functional response and impulse, Nonlinear anal. TMA, 70, 126-134, (2009) · Zbl 1166.34043 [9] 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) [10] Georescu, P.; Morosanu, G., Pest regulation by means of impulsive controls, Appl. math. comput., 190, 790-803, (2007) · Zbl 1117.93006 [11] George, P.K.; Nandakumaran, A.K.; Arapostathis, A., A note on controllability of impulsive systems, J. math. anal. appl, 241, 276-283, (2000) · Zbl 0965.93015 [12] Guan, Z.; Chen, G.; Ueta, T., On impulsive control of a periodically forced chaotic pendulum system, IEEE trans. automat. control, 45, 1724-1727, (2000) · Zbl 0990.93105 [13] Haddad, W.M.; Chellaboina, C.; Nersesov, S.G.; Sergey, G., Stability, dissipativity, and control, (2006), Princeton University Press Princeton, NJ · Zbl 1114.34001 [14] 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 [15] 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 [16] Lakshmikantham, V.; Bainov, D.D.; Simeonov, P.S., Theory of impulsive differential equations, (1989), World Scientific Press Singapore · Zbl 0719.34002 [17] Lenci, S.; Rega, G., Periodic solutions and bifurcations in an impact inverted pendulum under impulsive excitation, Chaos solitons fractals, 11, 2453-2472, (2000) · Zbl 0964.70018 [18] 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 [19] Liu, X.; Willms, A.R., Impulsive controllability of linear dynamical systems with applications to maneuvers of spacecraft, Math. probl. eng., 2, 277-299, (1996) · Zbl 0876.93014 [20] Mawhin, J.; Willem, M., Critical point theory and Hamiltonian systems, (1989), Springer-Verlag New York, Berlin, Heidelberg, London, Paris, Tokyo · Zbl 0676.58017 [21] Nenov, S., Impulsive controllability and optimization problems in populations dynamics, Nonlinear anal. TMA, 36, 881-890, (1999) · Zbl 0941.49021 [22] Nieto, J.J., Basic theory for nonresonance impulsive periodic problems of first order, J. math. anal. appl., 205, 423-433, (1997) · Zbl 0870.34009 [23] Nieto, J.J., Periodic boundary value problems for first-order impulsive ordinary differential equations, Nonlinear anal. TMA, 51, 1223-1232, (2002) · Zbl 1015.34010 [24] J.J. Nieto, D. O’Regan, Variational approach to impulsive differential equations. Nonlinear Anal. RWA, in press (doi:10.1016/j.nonrwa.2007.10.022) [25] Nieto, J.J.; Rodríguez-López, R., Periodic boundary value problem for non-Lipschitzian impulsive functional differential equations, J. math. anal. appl., 318, 593-610, (2006) · Zbl 1101.34051 [26] Nieto, J.J.; Rodríguez-López, R., New comparison results for impulsive integro-differential equations and applications, J. math. anal. appl., 328, 1343-1368, (2007) · Zbl 1113.45007 [27] 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 [28] Prado, A.F.B.A., Bi-impulsive control to build a satellite constellation, Nonlinear dyn. syst. theory, 5, 169-175, (2005) · Zbl 1128.70015 [29] 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 [30] Rabinowitz, P.H., On subharmonic solutions of Hamiltonian systems, Comm. pure. appl. math., 33, 609-633, (1980) · Zbl 0425.34024 [31] Rabinowitz, P.H., Minimax methods in critical point theory with applications to differential equations, () · Zbl 0609.58002 [32] Rogovchenko, Y.V., Impulsive evolution systems: main results and new trends, Dyn. contin. discrete impuls. systems, 3, 57-88, (1997) · Zbl 0879.34014 [33] Samoilenko, A.M.; Perestyuk, N.A., Impulsive differential equations, (1995), World Scientific Publishing Co. Pte. Ltd Singapore · Zbl 0837.34003 [34] Schechter, M., Periodic non-autonomous second-order dynamical systems, J. differential equations, 223, 290-302, (2006) · Zbl 1099.34042 [35] 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 [36] Tang, C., Periodic solutions for nonautonomous second order systems with sublinear nonlinearity, Proc. amer. math. soc., 126, 11, 3263-3270, (1998) · Zbl 0902.34036 [37] Tang, C.; Wu, X., Notes on periodic solutions of subquadratic second order systems, J. math. anal. appl., 185, 8-16, (2003) · Zbl 1054.34075 [38] Y. Tian, W. Ge, Applications of variational methods to boundary value problem for impulsive differential equations, Proc. Edinb. Mat. Soc., (preprint) · Zbl 1163.34015 [39] W. Wang, J. Shen, J.J. Nieto, Permanence periodic solution of predator – prey system with holling type functional response and impulse, Discrete Dynamics in Nature and Society, p. 2007. Article ID 81756, 15 pages, doi:10.1155/2007/81756 [40] Yang, X.; Shen, J., Periodic boundary value problems for second-order impulsive integro-differential equations, J. comput. appl. math., 209, 176-186, (2007) · Zbl 1155.45007 [41] Yao, M.; Zhao, A.; Yan, J., Periodic boundary value problems of second-order impulsive differential equations, Nonlinear anal. TMA, 70, 262-273, (2009) · Zbl 1176.34032 [42] Zavalishchin, S.T.; Sesekin, A.N., Dynamics impulse system, () · Zbl 0880.46031 [43] Zeng, G.; Chen, L.; Sun, L., Existence of periodic solutions of order on of planar impulsive autonomous system, J. comput. appl. math., 186, 466-481, (2006) · Zbl 1088.34040 [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 [45] Zhang, H.; Chen, L.; Nieto, J.J., A delayed epidemic model with stage-structure and pulses for pest management strategy, Nonlinear anal. RWA, 9, 1714-1726, (2008) · Zbl 1154.34394 [46] Zhang, H.; Li, X., Periodic solutions of second-order nonautonomous impulsive differential equations, Int. J. qualitative theory diff. equations appl., 2, 1, 112-124, (2008) · Zbl 1263.34020 [47] Zhang, H.; Xu, W.; Chen, L., A impulsive infective transmission si model for pest control, Math. methods appl. sci., 20, 1169-1184, (2007) · Zbl 1155.34328
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. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.