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)),u(n)(x)=f(x,u(x),u′(x),…,u(n−1)(x)), with n≥3n≥3,x∈[a,b]x∈[a,b] and f:[a,b]×Rn→Rf:[a,b]×Rn→R a continuous function verifying a Nagumo-type growth condition.A new 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 nn odd and nn even.