Remote Access Proceedings of the American Mathematical Society
Green Open Access

Proceedings of the American Mathematical Society

ISSN 1088-6826(online) ISSN 0002-9939(print)



On a conjecture for oscillation of second-order ordinary differential systems

Author: Angelo B. Mingarelli
Journal: Proc. Amer. Math. Soc. 82 (1981), 593-598
MSC: Primary 34C10; Secondary 34A30
MathSciNet review: 614884
Full-text PDF Free Access

Abstract | References | Similar Articles | Additional Information

Abstract: We present here some results pertaining to the oscillatory behavior at infinity of the vector differential equation

$\displaystyle y'' + Q(t)y = 0,\quad t \in [0,\infty )$

, where $ Q(t)$ is a real continuous $ n \times n$ symmetric matrix function. It has been conjectured (cf., e.g. [6]) that the criterion

$\displaystyle \mathop {\lim }\limits_{t \to \infty } {\lambda _1}\left\{ {\int_0^t {Q(s)\;ds} } \right\} = \infty $

where $ {\lambda _1}( \cdot )$ denotes the maximum eigenvalue of the matrix concerned, implies oscillation. We show that this is so under the tacit assumption

$\displaystyle \mathop {\lim \inf }\limits_{t \to \infty } {t^{ - 1}}{\text{tr}}\left\{ {\int_0^t {Q(s)\;ds} } \right\} > - \infty $

where $ {\text{tr}}( \cdot )$ represents the trace of the matrix under consideration.

References [Enhancements On Off] (What's this?)

  • [1] R. Bellman, Introduction to matrix analysis, 2nd ed., McGraw-Hill, New York, 1970. MR 0258847 (41:3493)
  • [2] W. A. Coppell, Disconjugacy, Lecture Notes in Math., vol. 220, Springer-Verlag, New York, 1971. MR 0460785 (57:778)
  • [3] G. J. Etgen and J. F. Pawlowski, Oscillation criteria for second order self-adjoint differential systems, Pacific J. Math. 66 (1976), 99-110. MR 0440147 (55:13027)
  • [4] P. Hartman, Ordinary differential equations, Hartman, Baltimore, Md., 1973. MR 0344555 (49:9294)
  • [5] -, Oscillation criteria for self-adjoint second order differential systems and "principal sectional curvatures", J. Differential Equations 34 (1979), 326-338. MR 550049 (81a:34034)
  • [6] D. Hinton and R. T. Lewis, Oscillation theory for generalized second order differential equations, Rocky Mountain J. Math. 10 (1980), 751-756. MR 595103 (82c:34039)
  • [7] W. T. Reid, Ordinary differential equations, Wiley, New York, 1971. MR 0273082 (42:7963)

Similar Articles

Retrieve articles in Proceedings of the American Mathematical Society with MSC: 34C10, 34A30

Retrieve articles in all journals with MSC: 34C10, 34A30

Additional Information

Keywords: Conjugate points, disconjugacy, oscillation at infinity, differential systems
Article copyright: © Copyright 1981 American Mathematical Society

American Mathematical Society