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.

**[1]**F. R. Gantmacher,*Matrizenrechnung. II. Spezielle Fragen und Anwendungen*, Hochschulbücher für Mathematik, Bd. 37, VEB Deutscher Verlag der Wissenschaften, Berlin, 1959 (German). MR**0107647****[2]**Maurice Hanan,*Oscillation criteria for third-order linear differential equations*, Pacific J. Math.**11**(1961), 919–944. MR**145160****[3]**G. H. Hardy, J. E. Littlewood, and G. Pólya,*Inequalities*, Cambridge, at the University Press, 1952. 2d ed. MR**0046395****[4]**Philip Hartman and Aurel Wintner,*On disconjugate differential systems*, Canadian J. Math.**8**(1956), 72–81. MR**74595**, https://doi.org/10.4153/CJM-1956-012-4**[5]**W. J. Kim,*Disconjugacy and disfocality of differential systems*, J. Math. Anal. Appl.**26**(1969), 9–19. MR**236464**, https://doi.org/10.1016/0022-247X(69)90172-3**[6]**A. Lasota and C. Olech,*An optimal solution of Nicoletti’s boundary value problem*, Ann. Polon. Math.**18**(1966), 131–139. MR**204742**, https://doi.org/10.4064/ap-18-2-131-139**[7]**A. Ju. Levin,*Some questions on the oscillation of solutions of linear differential equations*, Dokl. Akad. Nauk SSSR**148**(1963), 512–515 (Russian). MR**0146450****[8]**A. Ju. Levin,*On the distribution of zeros of solutions of a linear differential equation*, Dokl. Akad. Nauk SSSR**156**(1964), 1281–1284 (Russian). MR**0164079****[9]**A. Ju. Levin,*The non-oscillation of solutions of the equation 𝑥⁽ⁿ⁾+𝑝₁(𝑡)𝑥⁽ⁿ⁻¹⁾+\cdots+𝑝_{𝑛}(𝑡)𝑥=0*, Uspehi Mat. Nauk**24**(1969), no. 2 (146), 43–96 (Russian). MR**0254328****[10]**Zeev Nehari,*Oscillation theorems for systems of linear differential equations*, Trans. Amer. Math. Soc.**139**(1969), 339–347. MR**239185**, https://doi.org/10.1090/S0002-9947-1969-0239185-6**[11]**Zeev Nehari,*Nonoscillation and disconjugacy of systems of linear differential equations*, J. Math. Anal. Appl.**42**(1973), 237–254. MR**320437**, https://doi.org/10.1016/0022-247X(73)90136-4**[12]**G. Pólya,*On the mean-value theorem corresponding to a given linear homogeneous differential equation*, Trans. Amer. Math. Soc.**24**(1922), no. 4, 312–324. MR**1501228**, https://doi.org/10.1090/S0002-9947-1922-1501228-5**[13]**Binyamin Schwarz,*Norm conditions for disconjugacy of complex differential systems*, J. Math. Anal. Appl.**28**(1969), 553–568. MR**249722**, https://doi.org/10.1016/0022-247X(69)90008-0**[14]**Thomas L. Sherman,*Properties of solutions of 𝑛𝑡ℎ order linear differential equations*, Pacific J. Math.**15**(1965), 1045–1060. MR**185185****[15]**C. A. Swanson,*Comparison and oscillation theory of linear differential equations*, Academic Press, New York-London, 1968. Mathematics in Science and Engineering, Vol. 48. MR**0463570****[16]**H. W. Turnbull and A. C. Aitken,*An introduction to the theory of canonical matrices*, Dover Publications, Inc., New York, 1961. MR**0123581**

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DOI:
https://doi.org/10.1090/S0002-9947-1974-0350113-7

Article copyright:
© Copyright 1974
American Mathematical Society