关键词: 多点, 正解, 边值问题, 不动点定理, p-Laplacian

Abstract:

In this paper, we are concerned with the following multi-point boundary value problem with one-dimensional p-Laplacian
$$
  \left\{
 \begin{array}{rl}
&\disp (\phi_{p}(x'(t)))'+h(t)f(t,x(t),x'(t))=0,\hspace{3mm}0 &\disp x'(0)-\sum_{i=1}^{m-1}\alpha_{i}x(\xi_{i})=0,\ \ \
x'(1)+\sum_{i=1}^{m-1}\beta_{i}x(\eta_{i})=0,
 \end{array}
 \right.
$$
where
$\phi_{p}(s)=|s|^{p-2}s,~p>1,~\alpha_{i}>0,~\beta_{i}>0,~0<\sum\limits_{i=1}^{m-1}\alpha_{i}\xi_{i}\leq1,~
0<\sum\limits_{i=1}^{m-1}\beta_{i}(1-\eta_{i})\leq1,~0=\xi_{0}
<\xi_{1}<\xi_{2}<\cdots<\xi_{m-1}<\eta_{1}<\eta_{2}<\cdots<\eta_{m-1}<\eta_{m}=1,~i=1,2,\cdots,m-1.$
By using a fixed point theorem in a cone, we obtain the existence of three positive solutions at least. The interesting point is that the boundary condition is a new kind of Sturm-Liouville type boundary condition, which has rarely been treated up to now.

Key words: Multi-point, Positive solutions, Boundary value problem, Fixed point theorem,  p-Laplacian