Uniqueness Implies Existence and Uniqueness Conditions for a Class of $(k+j)$-Point Boundary Value Problems for $n$-th Order Differential Equations

http://dx.doi.org/10.4153/CMB-2011-117-0
Canad. Math. Bull. 55(2012), 285-296

  Published:2011-06-14
 Printed: Jun 2012

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 $2\le k\le n-1$.