# Article

Keywords:
third-order differential equation; multi-point and integral boundary conditions; Guo-Krasnosel'skii fixed point theorem in cone; positive solutions
Summary:
This paper concerns the following system of nonlinear third-order boundary value problem: \begin{equation*} u_{i}'''(t)+f_{i}(t,u_{1}(t),\dots ,u_{n}(t),u'_{1}(t),\dots ,u'_{n}(t))= 0, 0<t<1, i\in \{1,\dots ,n\} \end{equation*} with the following multi-point and integral boundary conditions: $$\begin{cases} u_{i}(0)=0 u_{i}'(0)=0 u_{i}'(1)= \sum^{p}_{j=1}\beta_{j,i}u_{i}'(\eta_{j,i}) + \int^{1}_{0}h_{i}(u_{1}(s),\dots ,u_{n}(s))\,ds \end{cases}$$ where $\beta_{j,i}>0$, $0< \eta_{1,i}<\dots <\eta_{p,i}<\frac{1}{2}$, $f_{i}:[0,1]\times \mathbb{R}^{n}\times \mathbb{R}^{n}\rightarrow \mathbb{R}$ and $h_{i}:[0,1]\times \mathbb{R}^{n}\rightarrow \mathbb{R}$ are continuous functions for all $i\in \{1,\dots ,n\}$ and $j\in \{1,\dots ,p\}$. Using Guo-Krasnosel'skii fixed point theorem in cone, we discuss the existence of positive solutions of this problem. We also prove nonexistence of positive solutions and we give some examples to illustrate our results.
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