Sign-changing solutions on a kind of fourth-order Neumann boundary value problem

In this paper, we consider the fourth-order Neumann boundary value problem u(4)(t)−2u″(t)+u(t)=f(t,u(t)) for all t∈[0,1] and subject to u′(0)=u′(1)=u‴(0)=u‴(1)=0. Using the fixed point index and the critical group, we establish the existence theorem of solutions that guarantees the problem has at least one positive solution and two sign-changing solutions under certain conditions.