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

Published:2011-06-14
Printed: Jun 2012
• Paul W. Eloe,
Department of Mathematics, University of Dayton, Dayton, OH, 45469-2316, USA
• Johnny Henderson,
Department of Mathematics, Baylor University, Waco, TX, 76798-7328, USA
• Rahmat Ali Khan,
Centre for Advanced Mathematics and Physics, National University of Sciences and Technology(NUST), Campus of College of Electrical and Mechanical Engineering, Peshawar Road, Rawalpindi, Pakistan
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## Abstract

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
 MSC Classifications: 34B15 - Nonlinear boundary value problems 34B10 - Nonlocal and multipoint boundary value problems 65D05 - Interpolation

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