Hostname: page-component-586b7cd67f-gb8f7 Total loading time: 0 Render date: 2024-11-24T06:25:03.592Z Has data issue: false hasContentIssue false
  • English
  • Français

On Transformation and Oscillation of Linear Differential System

Published online by Cambridge University Press:  20 November 2018

Donald F. St. Mary
Donald F. St. Mary*
Affiliation:
University of Massachusetts, Amherst, Massachusetts; University of Oklahoma, Norman, Oklahoma
Rights & Permissions [Opens in a new window]

Abstract

Core share and HTML view are not available for this content. However, as you have access to this content, a full PDF is available via the ‘Save PDF’ action button.

In this paper we study second order linear differential systems. We examine the relationship between oscillation of n-dimensional systems and certain associated n-dimensional systems, where m ≧ n. Several theorems are presented which unify and encompass in the linear case a number of results from the literature. In particular, we present a transformation which extends an oscillation theorem due to Allegretto and Erbe [1], and a comparison theorem due to Kreith [9], and explains some work of Howard [7].

Type
Research Article
Information
Canadian Journal of Mathematics , Volume 29 , Issue 2 , 01 April 1977 , pp. 392 - 399
Copyright
Copyright © Canadian Mathematical Society 1977

References

1. Allegretto, W. and Erbe, L., Oscillation criteria for matrix differential inequalities, Can. Math. Bull. 16 (1973), 5–10.Google Scholar
2. Barrett, J. H., Matrix systems of second order differential equations, Portugaliae Mathematica U(1955), 79–89.Google Scholar
3. Coppel, W. A., Disconjugacy, Lecture Notes in Mathematics 220 (Springer-Verlag, Berlin, Heidelberg, New York, 1971).Google Scholar
4. Eliason, S. B., Results on oscillation for a second order linear matrix system of differential equations, submitted for publication.Google Scholar
5. Eliason, S. B. and D. F. St., Mary, On oscillation of linear differential systems, to appear.Google Scholar
6. Hartman, P., Ordinary differential equations (John Wiley & Sons, New York, 1964).Google Scholar
7. Howard, H. C., Oscillation criteria for matrix differential equations, Can. J. Math. 19 (1967), 184–199.Google Scholar
8. Kartsatos, A. G., Oscillation of nonlinear systems of matrix differential equations, Proc. Amer. Math. Soc. 30 (1971), 97–101.Google Scholar
9. Kreith, K., Oscillation criteria for nonlinear matrix differential equations, Proc. Amer. Math. Soc. 26 (1970), 270–272.Google Scholar
10. Leigh ton, W., On self-adjoint differential equations of second order, J. London Math. Soc. 27 (1952), 37–47.Google Scholar
11. Noussair, E. S. and Swanson, C. A., Oscillation criteria for differential systems, J. Math. Anal. Appl. 36 (1971), 575–580.Google Scholar
12. Reid, W. T., Ordinary differential equations (John Wiley & Sons, New York, 1971).Google Scholar
13. Simons, W., Disconjugacy criteria for systems of self-adjoint differential equations, J. London Math. Soc. 6 (1973), 373–381.Google Scholar
14. Swanson, C. A., Oscillation criteria for nonlinear matrix differential inequalities, Proc. Amer. Math. Soc. 24 (1970), 824–827.Google Scholar
15. Willet, D., On the oscillatory behavior of the solutions of second order linear differential equations, Ann. Polon. Math. 21 (1969), 175–194.Google Scholar