Uniqueness Implies Existence and Uniqueness Conditions for a Class of (k + j)-Point Boundary Value Problems for n-th Order Differential Equations
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- Journal:
- Canadian Mathematical Bulletin / Volume 55 / Issue 2 / 01 June 2012
- Published online by Cambridge University Press:
- 20 November 2018, pp. 285-296
- Print publication:
- 01 June 2012
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- Article
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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.
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