Existence of solutions for high order ordinary differential equations with some periodic-type boundary condition

We consider the following high order periodic-type boundary value problem \[ \lefteqn{(PBVP)} \left\{\begin{array}{lll} (E)~u^{(n)}(t)= f(t,u(t),u^{(1)}(t), \cdots, u^{(n-2)}(t), u^{(n-1)}(t))~\mbox{for}~t\in (0,T) \\ (PBC)~\left\{\begin{array}{lll} u^{(i)}(0)=0,~0\leq i\leq n-3,\\ u^{(n-2)}(0)= u^{(n-2)}(T),\\ u^{(n-1)}(0)= u^{(n-1)}(T), \end{array}\right. \end{array}\right. \] where $f\in C([0,T]\times\mathbb{R}^n,\mathbb{R})$, $n\geq 2$ and satisfies the so-called Nagumo's condition. In this article, we will use a general upper and lower solution method to establish an existence theorem for solutions of $(PBVP)$.