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| The theory of impulsive problem is experiencing a rapid development in the
last few years. Mainly because they have been used to describe some
phenomena, arising from different disciplines like physics or biology,
subject to instantaneous change at some time instants called moments. Second
order periodic impulsive problems were studied to some extent, however very few papers were dedicated to the study of third
and higher order impulsive problems. One can refer for instance and the references therein.
The high order impulsive problem considered is composed by
the fully nonlinear equation%
\begin{equation}
u^{\left( n\right) }\left( x\right) =f\left( x,u\left( x\right) ,u^{\prime
}\left( x\right) ,...,u^{\left( n-1\right) }\left( x\right) \right)
\end{equation}%
for a. e. $x\in I:=\left[ 0,1\right] ~\backslash ~\left\{
x_{1},...,x_{m}\right\} $ where $f:\left[ 0,1\right] \times \mathbb{R}%
^{n}\rightarrow \mathbb{R}$ is $L^{1}$-Carath\'{e}odory function, along with
the periodic boundary conditions%
\begin{equation}
u^{\left( i\right) }\left( 0\right) =u^{\left( i\right) }\left( 1\right) ,%
\text{ \ \ }i=0,...,n-1,
\end{equation}%
and the impulsive conditions%
\begin{equation}
\begin{array}{c}
u^{\left( i\right) }\left( x_{j}^{+}\right) =g_{j}^{i}\left( u\left(
x_{j}\right) \right) ,\text{ \ \ }i=0,...,n-1,%
\end{array}
\end{equation}%
where $g_{j}^{i},$ for $j=1,...,m,$are given real valued functions
satisfying some adequate conditions, and $x_{j}\in \left( 0,1\right) ,$ such
that $0=x_{0} |
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