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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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