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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 {\it D. C. Clark} [Math. J., Indiana Univ. 22, 65--74 (1972; Zbl 0228.58006)] (see also [{\it P. H. Rabinowitz} [Reg. Conf. Ser. Math. 65 (1986; Zbl 0609.58002)]), as well as the theory of Sobolev spaces.

34A37Differential equations with impulses
34C25Periodic solutions of ODE
34C37Homoclinic and heteroclinic solutions of ODE
Full Text: DOI
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