EXISTENCE OF THREE POSITIVE SOLUTIONS FOR M-POINT BOUNDARY-VALUE PROBLEM WITH ONE-DIMENSIONAL P-LAPLACIAN

Abstract

In this paper, we consider the multipoint boundary value problem for the one-dimensional $p$-Laplacian $$\left(\phi_p(u'(t))\right)' + q(t) f\left(t,u(t),u'(t)\right) = 0,~~~~\ \ \ \ \ \ t \in (0,1), $$ subject to the boundary value conditions: $$ u(0) = \sum_{i=1}^{m-2} a_{i} u(\xi_{i}),\ \ \ \ \ \ u(1) = \sum_{i=1}^{m-2} b_{i} u(\xi_{i}), $$ where $\phi_{p}(s) = |s|^{p-2}s$, $p \gt 1$, $\xi_{i} \in (0,1)$ with $0 \lt \xi_{1} \lt \xi_{2} \lt \cdots \lt \xi_{m-2} \lt 1$ and $a_{i}, b_{i} \in [0,1)$, $0 \leq \sum\limits_{i=1}^{m-2} a_{i} \lt 1$, $0 \leq \sum\limits_{i=1}^{m-2} b_{i} \lt 1$. Using a fixed point theorem due to Avery and Peterson, we study the existence of at least three positive solutions to the above boundary value problem. The interesting point is the nonlinear term $f$ is involved with the first-order derivative explicitly.