Existence of solutions of third-order p-Laplacian resonant multi-point boundary value problems

A class of third-order ordinary differential equation resonant multi-point boundary value problem with p-Laplacian operator is studied. By integrating on both sides of the equation once, its equivalent integral equation is obtained. By defining the linear operator L and using the Mawhin continuation theorem, it is shown that the integral equation has at least one solution when f(t,u,p) is allowed to be negative. The existence of the solution of the third-order resonant multi-point boundary value problem with p-Laplacian operator is thus proved.