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PROC

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.

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.

PROC

In this the authors consider the nonlinear fully equation

\begin{equation*} u^{(iv)} (x) + f( x,u(x) ,u^{\prime}(x) ,u^{\prime \prime}(x) ,u^{\prime \prime \prime}(x) ) = 0 \end{equation*} for $x\in [ 0,1] ,$ where $f:[ 0,1] \times \mathbb{R} ^{4} \to \mathbb{R}$ is a continuous functions, coupled with the Lidstone boundary conditions, \begin{equation*} u(0) = u(1) = u^{\prime \prime}(0) = u^{\prime \prime }(1) = 0. \end{equation*}

They discuss how different definitions of lower and upper solutions can generalize existence and location results for boundary value problems with Lidstone boundary data. In addition, they replace the usual bilateral Nagumo condition by a one-sided condition, allowing the nonlinearity to be unbounded$.$ An example will show that this unilateral condition generalizes the usual one and stress the potentialities of the new definitions.

\begin{equation*} u^{(iv)} (x) + f( x,u(x) ,u^{\prime}(x) ,u^{\prime \prime}(x) ,u^{\prime \prime \prime}(x) ) = 0 \end{equation*} for $x\in [ 0,1] ,$ where $f:[ 0,1] \times \mathbb{R} ^{4} \to \mathbb{R}$ is a continuous functions, coupled with the Lidstone boundary conditions, \begin{equation*} u(0) = u(1) = u^{\prime \prime}(0) = u^{\prime \prime }(1) = 0. \end{equation*}

They discuss how different definitions of lower and upper solutions can generalize existence and location results for boundary value problems with Lidstone boundary data. In addition, they replace the usual bilateral Nagumo condition by a one-sided condition, allowing the nonlinearity to be unbounded$.$ An example will show that this unilateral condition generalizes the usual one and stress the potentialities of the new definitions.

PROC

In this paper it is considered a fourth order problem composed of a fully
nonlinear differential equation and functional boundary conditions
satisfying some monotone conditions.This functional dependence on $u,u^' $ and $u^{''}$and generalizes several types of boundary
conditions such as Sturm-Liouville, multipoint, maximum and/or minimum
arguments, or nonlocal. The main theorem is an existence and location result
as it provides not only the existence, but also some qualitative information
about the solution.

PROC

In this work the authors present some existence, non-existence and location
results of the problem composed of the fourth order fully nonlinear equation
\begin{equation*}
u^{\left( 4\right) }\left( x\right) +f( x,u\left( x\right) ,u^{\prime
}\left( x\right) ,u^{\prime \prime }\left( x\right) ,u^{\prime \prime \prime
}\left( x\right) ) =s\text{ }p(x)
\end{equation*}
for $x\in \left[ a,b\right] ,$ where $f:\left[ a,b\right] \times \mathbb{R}
^{4}\rightarrow \mathbb{R},$ $p:\left[ a,b\right] \rightarrow \mathbb{R}^{+}$
are continuous functions and $s$ a real parameter, with the boundary
conditions
\begin{equation*}
u\left( a\right) =A,\text{ }u^{\prime }\left( a\right) =B,\text{ }u^{\prime
\prime \prime }\left( a\right) =C,\text{ }u^{\prime \prime \prime }\left(
b\right) =D,\text{ }
\end{equation*}
for $A,B,C,D\in \mathbb{R}.$ In this work they use an
Ambrosetti-Prodi type approach, with some new features: the existence
part is obtained in presence of nonlinearities not necessarily bounded, and
in the multiplicity result it is not assumed a speed growth condition or an
asymptotic condition, as it is usual in the literature for these type of
higher order problems.

The arguments used apply lower and upper solutions technique and topological degree theory.

An application is made to a continuous model of the human spine, used in aircraft ejections, vehicle crash situations, and some forms of scoliosis.

The arguments used apply lower and upper solutions technique and topological degree theory.

An application is made to a continuous model of the human spine, used in aircraft ejections, vehicle crash situations, and some forms of scoliosis.

PROC

In this paper, given $f : I \times (C(I))^2 \times \mathbb{R}^2 \leftarrow \mathbb{R}$ a $L^1$ Carathéodory
function, it is considered the functional fourth order equation
$u^(iv) (x) = f(x, u, u', u'' (x), u''' (x))$
together with the nonlinear functional boundary conditions
$L_0(u, u', u'', u (a)) = 0 = L_1(u, u', u'', u' (a))$
$L_2(u, u', u'', u'' (a), u''' (a)) = 0 = L_3(u, u', u'', u'' (b}, u''' (b)):$
Here $L_i, i$ = 0; 1; 2; 3, are continuous functions satisfying some adequate
monotonicity assumptions.
It will be proved an existence and location result in presence of non ordered
lower and upper solutions and without monotone assumptions on the right
hand side of the equation.

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