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

Eloe, Paul W. ; Henderson, Johnny ; Khan, Rahmat Ali

Canadian mathematical bulletin, Tome 55 (2012) no. 2, pp. 285-296

Résumé

For the $n$ -th order nonlinear differential equation, ${{y}^{(n)}}\,=\,f(x,\,y,\,y\prime ,\ldots ,\,{{y}^{(n-1)}})$ , 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 $1\,\le \,k\,\le \,n\,-\,j$ . 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 $1\,\le \,k\,\le \,n\,-\,j$ . This result is analogous to $n$ -point disconjugacy implies $k$ -point disconjugacy for $2\,\le \,k\,\le \,n\,-\,1$ .

Détail

DOI : 10.4153/CMB-2011-117-0

Mots-clés : 34B15, 34B10, 65D05, boundary value problem, uniqueness, existence, unique solvability, nonlinear interpolation

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