Positive solutions for a system of third-order differential equation with multi-point and integral conditions

Jebari, Rochdi ; Boukricha, Abderrahman

Commentationes Mathematicae Universitatis Carolinae, Tome 56 (2015) no. 2, pp. 187-207.

Résumé

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, 01, 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.

Détail

DOI : 10.14712/1213-7243.2015.117

Classification : 34B10, 34B15, 34B18, 34B27
Keywords: third-order differential equation; multi-point and integral boundary conditions; Guo-Krasnosel'skii fixed point theorem in cone; positive solutions