LAUSR

In this paper, we are concerned with the p-Laplacian multi-point boundary value problem $$\begin{aligned} (\phi _{p}(x''(t)))'= & {} f(t,x(t),x'(t),x''(t)),\,t\in (0, 1),\\ \phi _{p}(x''(0))= & {} \sum _{i=1}^{m}\alpha _{i}\phi _{p}(x''(\xi _{i})),\\ ~~x'(1)= & {} \sum _{j=1}^{n}\beta _{j}x'(\eta _{j}), ~~x''(1)=0, \end{aligned}$$(ϕp(x′′(t)))′=f(t,x(t),x′(t),x′′(t)),t∈(0,1),ϕp(x′′(0))=∑i=1mαiϕp(x′′(ξi)),x′(1)=∑j=1nβjx′(ηj),x′′(1)=0,where $$\phi _p(s)=|s|^{p-2}s,~p>1, \phi _{q}=\phi _{p}^{-1}, \frac{1}{p}+\frac{1}{q}=1, f: [0, 1]\times R^3\rightarrow R$$ϕp(s)=|s|p-2s,p>1,ϕq=ϕp-1,1p+1q=1,f:[0,1]×R3→R is a continuous function, $$0<\xi _{1}<\xi _{2}<\cdots<\xi _{m}<1, \alpha _{i}\in R, i=1,2,\ldots , m, m\ge 2$$0<ξ1<ξ2<⋯<ξm<1,αi∈R,i=1,2,…,m,m≥2 and $$0<\eta _{1}<\cdots<\eta _{n}<1, \beta _{j}\in R, j=1,\ldots , n, n\ge 1$$0<η1<⋯<ηn<1,βj∈R,j=1,…,n,n≥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.