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

Abstract

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∈ℝ 1,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. © 2011 Elsevier Inc. All rights reserved.