We study an even order system boundary value problem with periodic boundary conditions. By establishing the existence of a positive eigenvalue of an associated linear system Sturm-Liouville problem, we obtain new conditions for the boundary value problem to have a positive solution. Our major tools are the Krein-Rutman theorem for linear spectra and the fixed point index theory for compact operators.

For the $n$-th order nonlinear differential equation, $y^{(n)} = f(x, y, y', \dots, y^{(n-1)})$, we consider uniqueness implies uniqueness and existence results for solutions satisfying certain $(k+j)$-point boundary conditions for $1\le j \le n-1$ and $1\leq k \leq n-j$. We define $(k;j)$-point unique solvability in analogy to $k$-point disconjugacy and we show that $(n-j_{0};j_{0})$-point unique solvability implies $(k;j)$-point unique solvability for $1\le j \le j_{0}$, and $1\leq k \leq n-j$. This result is analogous to $n$-point disconjugacy implies $k$-point disconjugacy for $2\le k\le n-1$.