Fourth Order Impulsive Periodic Boundary Value Problems

Abstract

In this work it is presented an existence result for the impulsive problem composed by the fourth order fully nonlinear equation

$$\begin{aligned} u^{\left( iv\right) }\left( x\right) =f\left( x,u\left( x\right) ,u^{\prime }\left( x\right) ,u^{\prime \prime }\left( x\right) ,u^{\prime \prime \prime }\left( x\right) \right) \end{aligned}$$

for a.e. $x\in \left[ 0,1\right] ~\backslash ~\left\{ x_{1},\ldots ,x_{m}\right\}$ where $f:\left[ 0,1\right] \times \mathbb {R} ^{4}\rightarrow \mathbb {R}$ is a $L^{1}$-Carathéodory function, along with the periodic boundary conditions

$$\begin{aligned} u^{\left( i\right) }\left( 0\right) =u^{\left( i\right) }\left( 1\right) , \quad i=0,1,2,3, \end{aligned}$$

and the impulses

$$\begin{aligned} \begin{array}{c} u\left( x_{j}^{+}\right) =g_{j}\left( u\left( x_{j}\right) \right) , \\ u^{\prime }\left( x_{j}^{+}\right) =h_{j}\left( u^{\prime }\left( x_{j}\right) \right) , \\ u^{\prime \prime }\left( x_{j}^{+}\right) =k_{j}\left( u^{\prime \prime }\left( x_{j}\right) \right) , \\ u^{\prime \prime \prime }\left( x_{j}^{+}\right) =l_{j}\left( u^{\prime \prime \prime }\left( x_{j}\right) \right) , \end{array} \end{aligned}$$

where $x_{j}\in \left( 0,1\right) ,$ for $j=1,\ldots ,m,$ such that $0=x_{0}<x_{1}<\cdots <x_{m}<x_{m+1}=1$, and $g_{j},~h_{j},~k_{j}$ , $l_{j}$ are given real valued functions satisfying some adequate conditions. The arguments used apply lower and upper solutions technique combined with an iterative and non monotone technique.