# zbMATH — the first resource for mathematics

Existence of solutions for a class of second-order Hamiltonian systems with impulsive effects. (English) Zbl 1193.34057
The authors give sufficient conditions the existence of a solution to the following boundary value problem
$\begin{cases} \ddot{u}(t)=\nabla F(t,u(t))\quad &\text{a.e. }t\in [0,T];\\ u(0)-u(T)=\dot{u}(0)-\dot{u}(T)=0,\\ \triangle \dot{u}^j(t_j)=\dot{u}^j(t_j^+)-\dot{u}^j(t_j^-)=I_{ij}(u^i(t_j)), & i=1,2,\dots,N;\quad j=1,2,\dots,p. \end{cases}$ Here, $$t_0=0<t_1<t_2<\cdots<t_p<t_{p+1}=T, u(t)=(u^1(t),u^2(t),\dots,u^N(t)), I_{ij}:\mathbb{R}\to \mathbb{R}$$ $$(i=1,2,\dots,N$$, $$j=1,2,\dots,p)$$ are continuous and $$F:[0,T]\times \mathbb{R}^N \to \mathbb{R}$$ satisfies the following assumption:
(A) $$F(t,x)$$ is measurable in $$t$$ for every $$x\in \mathbb{R}^N$$ and continuously differentiable in $$x$$ for a.e. $$t \in [0,T]$$ and there exist $$a\in C(\mathbb{R}^+,\mathbb{R}^+), b\in L^1(0,T;\mathbb{R}^+)$$ such that
$|F(t,x)|\leq a(|x|)b(x),\quad |\nabla F(t,x)|\leq a(|x|)b(x)$
for all $$x\in \mathbb{R}^N$$ and a.e. $$t\in [0,T]$$.
Two illustrative examples are given.

##### MSC:
 34B37 Boundary value problems with impulses for ordinary differential equations 37J99 Dynamical aspects of finite-dimensional Hamiltonian and Lagrangian systems
##### Keywords:
Hamiltonian systems; impulse; critical points
Full Text:
##### References:
 [1] Mawhin, J.; Willem, M., Critical point theory and Hamiltonian systems, (1989), Springer-Verlag Berlin · Zbl 0676.58017 [2] Berger, M.S.; Schechter, M., On the solvability of semilinear gradient operator equations, Adv. math., 25, 97-132, (1977) · Zbl 0354.47025 [3] Long, Y.M., Nonlinear oscillations for classical Hamiltonian systems with bi-even subquadratic potentials, Nonlinear anal., 24, 1665-1671, (1995) · Zbl 0824.34042 [4] Rabinowitz, P.H., On subharmonic solutions of Hamiltonian systems, Comm. pure appl. math., 33, 609-633, (1980) · Zbl 0425.34024 [5] Tang, C.L., Periodic solutions for nonautonomous second systems with sublinear nonlinearity, Proc. amer. math. soc., 126, 3263-3270, (1998) · Zbl 0902.34036 [6] Nieto, J.J., Impulsive resonance periodic problems of first order, Appl. math. lett., 15, 489-493, (2002) · Zbl 1022.34025 [7] 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 [8] 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 [9] Chang, Y.K.; Nieto, J.J.; Li, W.S., On impulsive hyperbolic differential inclusions with nonlocal initial conditions, J. optim. theory appl., 140, 431-442, (2009) · Zbl 1159.49042 [10] 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 [11] Li, Y.K., Positive periodic solutions of nonlinear differential systems with impulses, Nonlinear anal., 68, 2389-2405, (2008) · Zbl 1162.34064 [12] Zeng, G.Z.; Wang, F.Y.; 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 [13] Nieto, J.J.; Rodriguez-Lopez, R., Periodic boundary value problem for non-Lipschitzian impulsive functional differential equations, J. math. anal. appl., 318, 593-610, (2006) · Zbl 1101.34051 [14] Luo, Z.; Nieto, J.J., New results of periodic boundary value problem for impulsive integro-differential equations, Nonlinear anal., 70, 2248-2260, (2009) · Zbl 1166.45002 [15] 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 [16] 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., 60, 3291-3298, (2008) · Zbl 1158.34049 [17] Hernandez, E.; Henriquez, H.R.; McKibben, M.A., Existence results for abstract impulsive second-order neutral functional differential equations, Nonlinear anal., 70, 2736-2751, (2009) · Zbl 1173.34049 [18] 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. modelling, 40, 509-518, (2004) · Zbl 1112.34052 [19] Zhang, H.; Chen, L.S.; Nieto, J.J., A delayed epidemic model with stage-structure and pulses for management strategy, Nonlinear anal. real world appl., 9, 1714-1726, (2008) · Zbl 1154.34394 [20] Nieto, J.J.; O’Regan, D., Variational approach to impulsive differential equations, Nonlinear anal. real world appl., 10, 680-690, (2009) · Zbl 1167.34318 [21] Pasquero, S., On the simultaneous presence of unilateral and kinetic constraints in time-dependent impulsive mechanics, J. math. phys., 47, 082903.19, (2006) [22] 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 [23] Lee, E.K.; Lee, Y.H., Multiple positive solutions of singular two point boundary value problems for second order impulsive differential equation, Appl. math. comput., 158, 745-759, (2004) · Zbl 1069.34035 [24] Lin, X.N.; Jiang, D.Q., Multiple positive solutions of Dirichlet boundary value problems for second order impulsive differential equations, J. math. anal. appl., 321, 501-514, (2006) · Zbl 1103.34015 [25] Zhou, J.W.; Li, Y.K., Existence and multiplicity of solutions for some Dirichlet problems with impulsive effects, Nonlinear anal., 71, 2856-2865, (2009) · Zbl 1175.34035
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.