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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

$\stackrel{¨}{q}+{V}_{q}\left(t,q\right)=f\left(t\right),\phantom{\rule{2.em}{0ex}}t\in \left({s}_{k-1},{s}_{k}\right),$
${\Delta }\stackrel{˙}{q}\left({s}_{k}\right)={g}_{k}\left(q\left({s}_{k}\right)\right),$

where $k\in ℤ$, $q\in {ℝ}^{n}$, ${\Delta }\stackrel{˙}{q}\left({s}_{k}\right)=\stackrel{˙}{q}\left({s}_{k}^{+}\right)-\stackrel{˙}{q}\left({s}_{k}^{-}\right)$, ${V}_{q}\left(t,q\right)={\text{grad}}_{q}V\left(t,q\right)$, ${g}_{k}\left(q\right)={\text{grad}}_{q}{G}_{k}\left(q\right)$, $f$ is continuous, ${G}_{k}$ is of class ${C}^{1}$ for every $k\in ℤ$, $0={s}_{0}<{s}_{1}<\cdots <{s}_{m}=T$, ${s}_{k+m}={s}_{k}+T$ for certain $m\in ℕ$ 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 ${lim}_{t\to ±\infty }q\left(t\right)=0$ and ${lim}_{t\to ±\infty }\stackrel{˙}{q}\left({t}^{±}\right)=0$. The periodic and homoclinic solutions obtained in the main results are generated by impulses if $f\equiv 0$, 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 Differential equations with impulses 34C25 Periodic solutions of ODE 34C37 Homoclinic and heteroclinic solutions of ODE