Existence of Solution for a p-Laplacian Multi-point Boundary Value Problem at Resonance

Abstract

In this paper, we are concerned with the p-Laplacian multi-point boundary value problem

$$\begin{aligned} (\phi _{p}(x''(t)))'= & {} f(t,x(t),x'(t),x''(t)),\,t\in (0, 1),\\ \phi _{p}(x''(0))= & {} \sum _{i=1}^{m}\alpha _{i}\phi _{p}(x''(\xi _{i})),\\ ~~x'(1)= & {} \sum _{j=1}^{n}\beta _{j}x'(\eta _{j}), ~~x''(1)=0, \end{aligned}$$

where $\phi _p(s)=|s|^{p-2}s,~p>1, \phi _{q}=\phi _{p}^{-1}, \frac{1}{p}+\frac{1}{q}=1, f: [0, 1]\times R^3\rightarrow R$ is a continuous function, $0<\xi _{1}<\xi _{2}<\cdots<\xi _{m}<1, \alpha _{i}\in R, i=1,2,\ldots , m, m\ge 2$ and $0<\eta _{1}<\cdots<\eta _{n}<1, \beta _{j}\in R, j=1,\ldots , n, n\ge 1$. Based on the extension of Mawhin’s continuation theorem, a new general existence result of the p-Laplacian problem is established in the resonance case.