Existence of solutions for 2mth-order periodic boundary value problems

In this paper, the existence results of solutions are obtained for the 2mth-order differential equation periodic boundary value problem: (-1)mu(2m)(t)+∑i=1m(-1)m-iaiu(2(m-i))(t)=f(t,u(t)) for all t ∈ [0, 1] with u(i)(0) = u(i)(1), i = 0, 1, … , 2m − 1, where f∈C1([0,1]×R1,R1),ai∈R1,i=1,2,…,m. By using the Morse theory, we impose certain conditions on f which are able to guarantee that the problem has at least one nontrivial solution, two nontrivial solutions and infinitely many solutions, separately.