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 ) m u ( 2 m ) ( t ) + ∑ i = 1 m ( - 1 ) m - i a i u ( 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 ∈ C 1 ( [ 0 , 1 ] × R 1 , R 1 ) , a i ∈ 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.