Rochdi Jebari, Abderrahman Boukricha
Positive solutions for a system of third-order differential equation with multi-point and integral conditions

Abstract: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, \ 00$, $0< \eta_{1,i}<\dots <\eta_{p,i}<\frac{1}{2}$, $f_{i}:[0,1]\times \mathbb{R}^{n}\times \mathbb{R}^{n}\rightarrow \mathbb{R}$ 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.

Keywords: third-order differential equation; multi-point and integral boundary conditions; Guo-Krasnosel'skii fixed point theorem in cone; positive solutions

DOI: DOI 10.14712/1213-7243.2015.117
AMS Subject Classification: 34B15 34B18 34B27