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Abstract
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.
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 $L^{1}$-Carathéodory function, along with
the periodic boundary conditions
\begin{equation*}
u^{\left( i\right) }\left( 0\right) =u^{\left( i\right) }\left( 1\right) ,
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) , 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 $0 = x_0 < x_1 <...< x_m < x_{m+1}=1.$
The arguments applied make use of the lower and upper solution method
combined with an iterative technique, which is not necessarily monotone,
together with classical results such as Lebesgue Dominated Convergence
Theorem, Ascoli-Arzela Theorem and fixed point theory.
Mathematics Subject Classification: Primary: 34B15, 34A37; Secondary: 34B10, 34K45.
\begin{equation} \\ \end{equation}
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