On higher order fully periodic boundary value problems

In this paper we present sufficient conditions for the existence of periodic solutions of the higher order fully differential equation u ( n ) ( x ) = f ( x , u ( x ) , u ′ ( x ) , … , u ( n − 1 ) ( x ) ) , with n ≥ 3 , x ∈ [ a , b ] and f : [ a , b ] × R n → R a continuous function verifying a Nagumo-type growth condition. type of lower and upper solutions, eventually non-ordered, allows us to obtain, not only the existence, but also some qualitative properties on the solution. The last section contains two examples to stress the application to both cases of n odd and n even.