Hostname: page-component-7479d7b7d-8zxtt Total loading time: 0 Render date: 2024-07-12T20:13:27.194Z Has data issue: false hasContentIssue false
  • English
  • Français

On the Monotone Nature of Boundary Value Functions for nth-Order Differential Equations

Published online by Cambridge University Press:  20 November 2018

A. C. Peterson
A. C. Peterson*
Affiliation:
University of Nebraska, Lincoln, Nebraska
Rights & Permissions [Opens in a new window]

Extract

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.

We are concerned with the nth (n≥3) order linear differential equation

1

where the coefficients are continuous on (∞, ∞). Our main result is to give conditions under which the two-point boundary value function rij(t) (see Definition 2) are strictly increasing continuously differentiable functions of t. Levin [1] states without proof a similar theorem concerning just the monotone nature of the rij(t) but assumes that the coefficients in (1) satisfy the standard differentiability conditions when one works with the formal adjoint of (1). Bogar [2] looks at the same problem for an nth-order quasi-differential equation where he makes no assumption concerning the differentiability of the coefficients in the quasi differential equation that he considers. Bogar gives conditions under which the rij(t) are strictly increasing and continuous. The different approach of the author to this problem also enables the author to establish the continuous differentiability of the rij(t) and to express the derivatives in terms of the principal solutions Uj(x, t), j=0, 1,…,n-1 (see Definition 4).

Type
Research Article
Information
Canadian Mathematical Bulletin , Volume 15 , Issue 2 , June 1972 , pp. 253 - 258
Copyright
Copyright © Canadian Mathematical Society 1972

References

1. Levin, A. Ju., Distribution of the zeros of solutions of a linear differential equation, Soviet Math. Dokl. (1964), 818-821.Google Scholar
2. Bogar, G. A., Properties of two-point boundary value functions, Proc. Amer. Math. Soc. 23 (1969), 335-339.Google Scholar
3. Dolan, J. M.,Oscillating behavior of solutions of linear differential equations of third order, Unpublished doctoral dissertation, Univ. of Tennessee, Knoxville, Tenn., 1967.Google Scholar
4. Barrett, J. H., Oscillatory theory of ordinary linear differential equations, Advances in Math. 3 (1969), 415-509.Google Scholar
5. Peterson, A. C., Distribution of zeros of solutions of a fourth order differential equation, Pac. J. Math. 30 (1969), 751-764.Google Scholar
6. Jutkin, G. A., On the question of the distribution of zeros of solutions of linear differential equations, (Russian) Diff. Equations, 15 (1969), 1821-1829.Google Scholar
7. Hartman, P.,Ordinary Differential Equations, Wiley, New York, 1964.Google Scholar