Two-point boundary value problem for nonlinear differential equation of \(n\)th order (Q1570941)

The authors deal with the solvability and the structure of the set of solutions to twopoint boundary value problems for nonlinear ordinary differential equations of \(n\)th order of the form \[ x^{(n)} = f(t,x,x',...,x^{(n-1)}), \;t \in \;[a,b], \;x^{(i)}(a) = A_{i}, \;0 \leq i \leq k, \;x^{(j)}(b) = B_{j}, 0 \leq j \leq l, \] with \(n \geq 4\), \(f:[a,b] \times \mathbb{R}^{n} \rightarrow \mathbb{R}\) satisfies the Carathéodory conditions, \( k+l+2 \leq n\) and the additional restriction \(\alpha (t) \leq x(t) \leq \beta (t), \forall \;t \in \;[a,b]\), where \(\alpha\) and \(\beta\) are solutions to the given differential equation verifying \(\alpha (t) < \beta (t), \forall \;t \in \;[a,b]\).