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Transactions of the American Mathematical Society

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Conjugate points of vector-matrix differential equations


Author: Roger T. Lewis
Journal: Trans. Amer. Math. Soc. 231 (1977), 167-178
MSC: Primary 34C10
DOI: https://doi.org/10.1090/S0002-9947-1977-0442364-0
MathSciNet review: 0442364
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Abstract | References | Similar Articles | Additional Information

Abstract: The system of equations

$\displaystyle \sum\limits_{k = 0}^n {{{( - 1)}^{n - k}}{{\left( {{P_k}(x){y^{(n - k)}}(x)} \right)}^{(n - k)}}} = 0\quad (0 \leqslant x < \infty )$

is considered where the coefficients are real, continuous, symmetric matrices, y is a vector, and $ {P_0}(x)$ is positive definite.

It is shown that the well-known quadratic functional criterion for existence of conjugate points for this system can be further utilized to extend results of the associated scalar equation to the vector-matrix case, and in some cases the scalar results are also improved. The existence and nonexistence criteria for conjugate points of this system are stated in terms of integral conditions on the eigenvalues or norms of the coefficient matrices.


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  • [1] C. D. Ahlbrandt, Disconjugacy criteria for self-adjoint differential systems, J. Differential Equations 6 (1969), 271-295. MR 39 #5855. MR 0244541 (39:5855)
  • [2] G. J. Etgen, Oscillation criteria for nonlinear second order matrix differential equations, Proc. Amer. Math. Soc. 27 (1971), 259-267. MR 43 #617. MR 0274859 (43:617)
  • [3] -, Oscillatory properties of certain nonlinear matrix differential systems of second order, Trans. Amer. Math. Soc. 122 (1966), 289-310. MR 32 #7834. MR 0190421 (32:7834)
  • [4] I. M. Gel'fand and S. V. Fomin, Calculus of variations, Fizmatgiz, Moscow, 1961; English transl., Prentice-Hall, Englewood Cliffs, N.J., 1963. MR 28 #3352; 28 #3353. MR 0160139 (28:3353)
  • [5] I. M. Glazman, Direct methods of qualitative spectral analysis of singular differential operators, Fizmatgiz, Moscow, 1963; English transl., Israel Program for Scientific Translations, Jerusalem, 1965; Davey, New York, 1966. MR 32 #2938; 32 #8210. MR 0190800 (32:8210)
  • [6] H. C. Howard, Oscillation criteria for matrix differential equations, Canad. J. Math. 19 (1967), 184-199. MR 35 #3126. MR 0212252 (35:3126)
  • [7] H. Kaufman and R. L. Sternberg, A two-point boundary problem for ordinary self-adjoint differential equations of even order, Duke Math. J. 20 (1953), 527-531. MR 15, 530. MR 0059443 (15:530c)
  • [8] K. Kreith, Oscillation criteria for nonlinear matrix differential equations, Proc. Amer. Math. Soc. 26 (1970), 270-272. MR 41 #8759. MR 0264163 (41:8759)
  • [9] R. T. Lewis, Oscillation and nonoscillation criteria for some self-adjoint even order linear differential operators, Pacific J. Math. 51 (1974), 221-234. MR 50 #2605. MR 0350112 (50:2605)
  • [10] -, The existence of conjugate points for self-adjoint differential equations of even order, Proc. Amer. Math. Soc. 56 (1976), 162-166. MR 0399576 (53:3419)
  • [11] V. V. Martynov, The conditions for discreteness and continuity of the spectrum in case of a self-adjoint system of even order differential equations, Differencial'nye Uravnenija 1 (1965), 1578-1591. MR 32 #6263. MR 0188831 (32:6263)
  • [12] Z. Nehari, Conjugate points, triangular matrices, and Riccati equations, Trans. Amer. Math. Soc. 199 (1974), 181-198. MR 50 #2606. MR 0350113 (50:2606)
  • [13] B. Noble, Applied linear algebra, Prentice-Hall, Englewood Cliffs, N.J., 1969. MR 40 #153. MR 0246884 (40:153)
  • [14] E. S. Noussair and C. A. Swanson, Oscillation criteria for differential systems, J. Math. Anal. Appl. 36 (1971), 575-580. MR 45 #5477. MR 0296417 (45:5477)
  • [15] W. T. Reid, Ordinary differential equations, Wiley, New York, 1971. MR 42 #7963. MR 0273082 (42:7963)
  • [16] C. A. Swanson, Oscillation criteria for nonlinear matrix differential inequalities, Proc. Amer. Math. Soc. 24 (1970), 824-827. MR 41 #3890. MR 0259248 (41:3890)
  • [17] E. C. Tomastik, Oscillation of nonlinear matrix differential equations of second order, Proc. Amer. Math. Soc. 19 (1968), 1427-1431. MR 38 #372. MR 0232046 (38:372)
  • [18] -, Singular quadratic functionals of n dependent variables, Trans. Amer. Math. Soc. 124 (1966), 60-76. MR 33 #4743. MR 0196556 (33:4743)
  • [19] -, Principal quadratic functionals, Trans. Amer. Math. Soc. 218 (1976), 297-309. MR 0405208 (53:9002)

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DOI: https://doi.org/10.1090/S0002-9947-1977-0442364-0
Article copyright: © Copyright 1977 American Mathematical Society

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