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 be a real continuous matrix on an interval , and let the -vector be a solution of the differential equation on . If is called a conjugate point of if the equation has a nontrivial solution vector such that for some .

It is shown that the absence on of a point conjugate to with respect to the equation is equivalent to the existence on of a continuous matrix solution of the nonlinear differential equation with the initial condition , where denotes the matrix obtained from the matrix 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 --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