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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.

##### MSC:
 34A37 Ordinary differential equations with impulses 34C25 Periodic solutions to ordinary differential equations 34C37 Homoclinic and heteroclinic solutions to ordinary differential equations
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