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 { (phi p(x '))'=f(t,x,x '),t is an element of[0,1] x '(0) = 0,x(1) =& sum;(m-2)i=1aix(xi i) where phi p(s)=|s|(p-2)s,p>1,f: [0,1]xR(2)-> Ris a continuous function, ai>0(i=1,2,...,m-2)with & sum;m-2i=1ai=1,0<xi 1<xi 2<<middle dot><middle dot><middle dot><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. Mean while some examples are also given to illustrate our main results