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1. CMB 2011 (vol 56 pp. 102)
Eigenvalue Approach to Even Order System Periodic Boundary Value Problems 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.
Keywords:Green's function, high order system boundary value problems, positive solutions, Sturm-Liouville problem Categories:34B18, 34B24 |
2. CMB 2011 (vol 55 pp. 285)
Uniqueness Implies Existence and Uniqueness Conditions for a Class of $(k+j)$-Point Boundary Value Problems for $n$-th Order Differential Equations |
Uniqueness Implies Existence and Uniqueness Conditions for a Class of $(k+j)$-Point Boundary Value Problems for $n$-th Order Differential Equations 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 $2\le k\le
n-1$.
Keywords:boundary value problem, uniqueness, existence, unique solvability, nonlinear interpolation Categories:34B15, 34B10, 65D05 |