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Transactions of the American Mathematical Society

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Conjugate points, triangular matrices, and Riccati equations


Author: Zeev Nehari
Journal: Trans. Amer. Math. Soc. 199 (1974), 181-198
MSC: Primary 34C10
DOI: https://doi.org/10.1090/S0002-9947-1974-0350113-7
MathSciNet review: 0350113
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Abstract: Let $ A$ be a real continuous $ n \times n$ matrix on an interval $ \Gamma $, and let the $ n$-vector $ x$ be a solution of the differential equation $ x' = Ax$ on $ \Gamma $. If $ [\alpha ,\beta ] \in \Gamma ,\beta $ is called a conjugate point of $ \alpha $ if the equation has a nontrivial solution vector $ x = ({x_1},{\kern 1pt} \ldots ,{x_n})$ such that $ {x_1}(\alpha ) = \ldots = {x_k}(\alpha ) = {x_{k + 1}}(\beta ) = \ldots = {x_n}(\beta ) = 0$ for some $ k \in [1,n - 1]$.

It is shown that the absence on $ ({t_1},{t_2})$ of a point conjugate to $ {t_1}$ with respect to the equation $ x' = Ax$ is equivalent to the existence on $ ({t_1},{t_2})$ of a continuous matrix solution $ L$ of the nonlinear differential equation $ L' = {[L{A^ \ast }{L^{ - 1}}]_{{\tau _0}}}L$ with the initial condition $ L({t_1}) = I$, where $ {[B]_{{\tau _0}}}$ denotes the matrix obtained from the $ n \times n$ matrix $ B$ by replacing the elements on and above the main diagonal by zeros. This nonlinear equation--which may be regarded as a generalization of the Riccati equation, to which it reduces for $ n = 2$--can be used to derive criteria for the presence or absence of conjugate points on a given interval.


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DOI: https://doi.org/10.1090/S0002-9947-1974-0350113-7
Article copyright: © Copyright 1974 American Mathematical Society

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