Periodic and homoclinic solutions generated by impulses. (English) Zbl 1225.34019
The topic of interest is the following class of second order differential equations with impulses
\[ \ddot{q}+V_q(t,q)=f(t),\qquad t \in (s_{k-1},s_k), \]
\[ \Delta \dot{q}(s_k)= g_{k}(q(s_k)), \]
where \(k \in \mathbb{Z}\), \(q \in \mathbb{R}^n\), \(\Delta \dot{q}(s_k)= \dot{q}(s_k^+)- \dot{q}(s_k^-)\), \(V_q(t,q)=\text{grad}_q V(t,q)\), \(g_k(q)=\text{grad}_q G_k(q)\), \(f\) is continuous, \(G_k\) is of class \(C^1\) for every \(k \in \mathbb{Z}\), \(0=s_0<s_1<\dots < s_m=T\), \(s_{k+m}=s_k+T\) for certain \(m\in \mathbb{N}\) and \(T>0\), \(V\) is continuously differentiable and \(T\)-periodic, and \(g_k\) is \(m\)-periodic in \(k\).
The existence of periodic and homoclinic solutions to this problem is studied via variational methods. In particular, sufficient conditions are given for the existence of at least one non-trivial periodic solution, which is generated by impulses if \(f\equiv 0\). An estimate (lower bound) of the number of periodic solutions generated by impulses is also given, showing that this lower bound depends on the number of impulses in a period of the solution. Moreover, under appropriate conditions, the existence of at least a non-trivial homoclinic solution is obtained, that is, a solution satisfying that \(\lim_{t \to \pm \infty}q(t)=0\) and \(\lim_{t \to \pm \infty}\dot{q}(t^{\pm})=0\). The periodic and homoclinic solutions obtained in the main results are generated by impulses if \(f\equiv0\), due to the non-existence of non-trivial periodic and homoclinic solutions of the problem when \(f\) and \(g_k\) vanish identically.
The main tools for the proofs of the main theorems are the mountain pass theorem and a result on the existence of pairs of critical points by D. C. Clark [Math. J., Indiana Univ. 22, 65–74 (1972; Zbl 0228.58006)] (see also [P. H. Rabinowitz [Reg. Conf. Ser. Math. 65 (1986; Zbl 0609.58002)]), as well as the theory of Sobolev spaces.
\[ \ddot{q}+V_q(t,q)=f(t),\qquad t \in (s_{k-1},s_k), \]
\[ \Delta \dot{q}(s_k)= g_{k}(q(s_k)), \]
where \(k \in \mathbb{Z}\), \(q \in \mathbb{R}^n\), \(\Delta \dot{q}(s_k)= \dot{q}(s_k^+)- \dot{q}(s_k^-)\), \(V_q(t,q)=\text{grad}_q V(t,q)\), \(g_k(q)=\text{grad}_q G_k(q)\), \(f\) is continuous, \(G_k\) is of class \(C^1\) for every \(k \in \mathbb{Z}\), \(0=s_0<s_1<\dots < s_m=T\), \(s_{k+m}=s_k+T\) for certain \(m\in \mathbb{N}\) and \(T>0\), \(V\) is continuously differentiable and \(T\)-periodic, and \(g_k\) is \(m\)-periodic in \(k\).
The existence of periodic and homoclinic solutions to this problem is studied via variational methods. In particular, sufficient conditions are given for the existence of at least one non-trivial periodic solution, which is generated by impulses if \(f\equiv 0\). An estimate (lower bound) of the number of periodic solutions generated by impulses is also given, showing that this lower bound depends on the number of impulses in a period of the solution. Moreover, under appropriate conditions, the existence of at least a non-trivial homoclinic solution is obtained, that is, a solution satisfying that \(\lim_{t \to \pm \infty}q(t)=0\) and \(\lim_{t \to \pm \infty}\dot{q}(t^{\pm})=0\). The periodic and homoclinic solutions obtained in the main results are generated by impulses if \(f\equiv0\), due to the non-existence of non-trivial periodic and homoclinic solutions of the problem when \(f\) and \(g_k\) vanish identically.
The main tools for the proofs of the main theorems are the mountain pass theorem and a result on the existence of pairs of critical points by D. C. Clark [Math. J., Indiana Univ. 22, 65–74 (1972; Zbl 0228.58006)] (see also [P. H. Rabinowitz [Reg. Conf. Ser. Math. 65 (1986; Zbl 0609.58002)]), as well as the theory of Sobolev spaces.
MSC:
| 34A37 | Ordinary differential equations with impulses |
| 34C25 | Periodic solutions to ordinary differential equations |
| 34C37 | Homoclinic and heteroclinic solutions to ordinary differential equations |
References:
| [1] | Benchohra, M.; Henderson, J.; Ntouyas, S. K., Impulsive Differential Equations and Inclusions (2006), Hindawi Publishing Corporation: Hindawi Publishing Corporation New York · Zbl 1130.34003 |
| [2] | Chu, J.; Nieto, J. J., Impulsive periodic solutions of first-order singular differential equations, Bull. Lond. Math. Soc., 40, 143-150 (2008) · Zbl 1144.34016 |
| [3] | Haddad, W. M.; Chellaboina, C.; Nersesov, S. G.; Sergey, G., Impulsive and Hybrid Dynamical Systems. Stability, Dissipativity, and Control (2006), Princeton University Press: Princeton University Press Princeton, NJ · Zbl 1114.34001 |
| [4] | Lakshmikantham, V.; Bainov, D. D.; Simeonov, P. S., Theory of Impulsive Differential Equations (1989), World Scientific Press: World Scientific Press Singapore · Zbl 0719.34002 |
| [5] | Nieto, J. J., Basic theory for nonresonance impulsive periodic problems of first order, J. Math. Anal. Appl., 205, 423-433 (1997) · Zbl 0870.34009 |
| [6] | Nieto, J. J., Periodic boundary value problems for first-order impulsive ordinary differential equations, Nonlinear Anal. TMA, 51, 1223-1232 (2002) · Zbl 1015.34010 |
| [7] | Nieto, J. J.; O’Regan, D., Variational approach to impulsive differential equations, Nonlinear Anal. RWA, 10, 2, 680-690 (2009) · Zbl 1167.34318 |
| [8] | 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 |
| [9] | Samoilenko, A. M.; Perestyuk, N. A., Impulsive Differential Equations (1995), World Scientific Publishing Co. Pte. Ltd.: World Scientific Publishing Co. Pte. Ltd. Singapore · Zbl 0837.34003 |
| [10] | 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 |
| [11] | Zavalishchin, S. T.; Sesekin, A. N., Dynamic Impulse System. Theory and Applications (1997), Kluwer Academic Publishers Group: Kluwer Academic Publishers Group Dordrecht · Zbl 0880.46031 |
| [12] | 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 |
| [13] | 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 |
| [14] | Rogovchenko, Y. V., Impulsive evolution systems: main results and new trends, Dyn. Contin. Discrete Impuls. Syst., 3, 57-88 (1997) · Zbl 0879.34014 |
| [15] | Tian, Y.; Ge, W. G., Applications of variational methods to boundary-value problem for impulsive differential equations, Proc. Edinb. Math. Soc., 51, 509-527 (2008) · Zbl 1163.34015 |
| [16] | Zhang, H.; Li, Z. X., Periodic solutions of second-order nonautonomous impulsive differential equations, Int. J. Qual. Theory Differ. Equ. Appl., 2, 1, 112-124 (2008) · Zbl 1263.34020 |
| [17] | Zhang, H.; Li, Z. X., Variational approach to impulsive differential equation with periodic boundary conditions, Nonlinear Anal. RWA, 11, 1, 67-78 (2010) · Zbl 1186.34089 |
| [18] | Zhang, Z.; Yuan, R., An application of variational methods to Dirichlet boundary value problem with impulses, Nonlinear Anal. RWA, 11, 1, 155-162 (2010) · Zbl 1191.34039 |
| [19] | Izydorek, Mare; Janczewska, Joanna, Homoclinic solutions for nonautonomous second order Hamiltonian systems with a coercive potential, J. Math. Anal. Appl., 335, 1119-1127 (2007) · Zbl 1118.37032 |
| [20] | Izydorek, Marek; Janczewska, Joanna, Homoclinic solutions for a class of the second order Hamiltonian systems, J. Differential Equations, 219, 375-398 (2005) · Zbl 1080.37067 |
| [21] | Kirchgraber, U.; Stoffer, U., Chaotic behavior in simple dynamical systems, SIAM Rev., 32, 424-452 (1990) · Zbl 0715.58024 |
| [22] | Rabinowitz, Paul H., Homoclinic orbits for a class of Hamiltonian systems, Proc. Roy. Soc. Edinburgh Sect. A, 114, 33-38 (1990) · Zbl 0705.34054 |
| [23] | Moser, J., Stable and Random Motions in Dynamical Systems (1973), Princeton University Press: Princeton University Press Princeton · Zbl 0271.70009 |
| [24] | Séré, Eric, Existence of infinitely many homoclinic orbits in Hamiltonian system, Math. Z., 209, 27-42 (1992) · Zbl 0725.58017 |
| [25] | Smale, S., Diffeomorphisms with many periodic points, (Cairns, S., Differential and Combinatorial Topology (1965), Princeton University Press: Princeton University Press Princeton) · Zbl 0142.41103 |
| [26] | Zelati, Vittorio Coti; Ekeland, I.; Séré, Eric, A variational approach to homoclinic orbits in Hamiltonian system, Math. Ann., 288, 133-160 (1990) · Zbl 0731.34050 |
| [27] | Zelati, Vittorio Coti; Rabinowitz, Paul H., Homoclinic orbits for second order Hamiltonian system possessing superquadratic potentials, J. Amer. Math. Soc., 4, 4, 693-727 (1991) · Zbl 0744.34045 |
| [28] | Clark, D. C., A variant of Lusternik-Schnirelman theory, Indiana Univ. Math. J., 22, 65-74 (1972) · Zbl 0228.58006 |
| [29] | Rabinowitz, P. H., Minimax methods in critical point theory with applications to differential equations, (C.B.M.S Reg. Conf. Ser., vol. 56 (1986), Amer. Math. Soc.: Amer. Math. Soc. Providence, RI) · Zbl 0609.58002 |
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