Positive solutions for a system of third-order differential equation with multi-point and integral conditions
Positive solutions for a system of third-order differential equation with multi-point and integral conditions
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 $\beta_{j,i}>0$, $0< \eta_{1,i}<\dots <\eta_{p,i}<\frac{1}{2}$, $f_{i}:[0,1]\times \mathbb{R}^{n}\times \mathbb{R}^{n}\rightarrow \mathbb{R}$ …