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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 online by Cambridge University Press:  20 November 2018

Paul W. Eloe ,
Johnny Henderson  and
Rahmat Ali Khan
Paul W. Eloe
Affiliation:
Department of Mathematics, University of Dayton, Dayton, OH, 45469-2316, USAe-mail: Paul.Eloe@notes.udayton.edu
Johnny Henderson
Affiliation:
Department of Mathematics, Baylor University, Waco, TX, 76798-7328, USAe-mail: Johnny Henderson@baylor.edu
Rahmat Ali Khan
Affiliation:
Centre for Advanced Mathematics and Physics, National University of Sciences and Technology(NUST), Campus of College of Electrical and Mechanical Engineering, Peshawar Road, Rawalpindi, Pakistane-mail: rahmat alipk@yahoo.com
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Abstract

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

Keywords

34B1534B1065D05boundary value problemuniquenessexistenceunique solvabilitynonlinear interpolation
Type
Research Article
Information
Canadian Mathematical Bulletin , Volume 55 , Issue 2 , 01 June 2012 , pp. 285 - 296
Copyright
Copyright © Canadian Mathematical Society 2012

References

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