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Variational approach to impulsive differential equations with periodic boundary conditions. (English) Zbl 1186.34089

Summary: The present paper is devoted to studying the second order impulsive differential equations of the following form,
\[ \begin{aligned} & u''(t)= \nabla F(t,u(t))\quad \forall t\in (s_{k-1},s_k),\\ & \Delta u'(s_k)=g(u(s_k)),\\ & u(0)-u(T)=0,\end{aligned} \]
where \(T>0\), \((\{s_k\}\) is a real sequence such that \(0+s_0<s_1<\cdots<s_m=T\), \(\Delta u'(s_k)\equiv u'(s^+_k)-u'(s^-_k)=\lim_{s\to s^+_k}u'(s)-\lim_{s\to s^-_k}-u'(s)\), \(k=1,\dots,m-1\), \(u'(s^-_0)\) and \(u'(s^-_m)\) are defined to be \(u(s^-_m)\) and \(u'(s^+_0)\), \(g:\mathbb R\to\mathbb R\) is continuous and \(F:[0,T]\times \mathbb R\to\mathbb R\) is continuously differentiable with respect to its second variable.
The main tool that we use is critical point theory. Our results generalize some existing results on periodic solutions for second order ordinary differential equations even when the impulses are absent.

MSC:

34K10 Boundary value problems for functional-differential equations
34K45 Functional-differential equations with impulses
58E05 Abstract critical point theory (Morse theory, Lyusternik-Shnirel’man theory, etc.) in infinite-dimensional spaces
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