Solvability of a nonlinear second order m-point boundary value problem with p-Laplacian at resonance

We study the existence of solutions of the nonlinear second order m-point boundary value problem with p-Laplacian at resonance $\{\begin{array}{c}{({\varphi }_p(x^{\prime }))}^{\prime }=f(t,x,x^{\prime }),t\in [0,1],\hfill \\ x^{\prime }(0)=0,x(1)={\sum }_{i=1}^{m-2}{a}_{i}x({\xi }_i),\hfill \end{array}$ \( \textstyle\begin{cases} (\phi _{p}(x'))'=f(t,x,x'),\quad t\in [0,1],\\ x'(0)=0, \qquad x(1)=\sum_{i=1}^{m-2}a_{i}x(\xi _{i}), \end{cases} \) where ${\varphi }_p(s)={|s|}^{p-2}s$ $\phi _{p}(s)=|s|^{p-2}s$ , $p>1$ $p>1$ , $f:[0,1]\times {\mathbb{R}}^2\to \mathbb{R}$ $f:[0,1]\times \mathbb{R}^{2}\to \mathbb{R}$ is a continuous function, $a_i>0$ $a_{i}>0$ ( $i=1,2,\dots ,m-2$ $i=1,2,\ldots ,m-2$ ) with ${\sum }_{i=1}^{m-2}{a}_i=1$ $\sum_{i=1}^{m-2}a_{i}=1$ , $0<{\xi }_1<{\xi }_2<\cdots <{\xi }_{m-2}<1$ $0<\xi _{1}<\xi _{2}<\cdots <\xi _{m-2}<1$ . Based on the topological transversality method together with the barrier strip technique and the cut-off technique, we obtain new existence results of solutions of the above problem. Meanwhile some examples are also given to illustrate our main results.