Hostname: page-component-f554764f5-sl7kg Total loading time: 0 Render date: 2025-04-10T20:18:50.090Z Has data issue: false hasContentIssue false
  • English
  • Français

On the First Conjugate Point Function for Nonlinear Differential Equations

Published online by Cambridge University Press:  20 November 2018

Allan C. Peterson  and
Dwight V. Sukup
Allan C. Peterson
Affiliation:
University of Nebraska-Lincoln Lincoln, Nebraska68508
Dwight V. Sukup
Affiliation:
University of South Dakota Vermillion, South Dakota57069
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.

We are concerned with the nth order differential equation y(n) = (x, y, y′, …,y(n-1)), where it is assumed throughout that f is continuous on [α,β) × Rn, α < β≤∞, and that solutions of initial value problems are unique and exist on [α, β). The definition of the first conjugate point function η1(t) for linear homogeneous equations is extended to this nonlinear case. Our main concern is what properties of this conjugacy function are valid in the nonlinear case.

Type
Research Article
Information
Canadian Mathematical Bulletin , Volume 18 , Issue 4 , October 1975 , pp. 577 - 585
Copyright
Copyright © Canadian Mathematical Society 1975

References

1. Beesack, P., On the Green’s function of an N-point boundary value problem, Pac. J. of Math. 12 (1962), 801–812.Google Scholar
2. Hartman, P., Unrestricted n-parameter families, Rend. Cire. Mat. Palermo 7 (1958), 123–142.Google Scholar
3. Hartman, P., Ordinary Differential Equations, John Wiley, New York, 1964.Google Scholar
4. Jackson, L., Subfunctions and second-order ordinary differential inequalities, Advances in Math. 2 (1968), 307–363.Google Scholar
5. Jackson, L., Lecture notes given at the Univ. of Utah, 1971–2.Google Scholar
6. Jackson, L., Uniqueness of solutions of boundary value problems for ordinary differential equations, SIAM J. on Applied Math., 24 (1973), 535–538.Google Scholar
7. Jackson, L., Uniqueness and existence of solutions of boundary value problems for third order ordinary differential equations, J. of Diff. Eqs. 13 (1973), 432–437.Google Scholar
8. Lasota, A. and Opial, Z., On the existence and uniqueness of solutions of a boundary value problem for an ordinary second-order differential equation, Colloquium Math. 18 (1967), 1–5.Google Scholar
9. Peterson, D., Uniqueness, existence, and comparison theorems for ordinary differentialequations; Ph.D. thesis, Univ. of Nebraska, 1973.Google Scholar
10. Sherman, T., Properties of solutions of Nth order linear differential equations, Pac. J. of Math. 15 (1965), 1045–1060.Google Scholar
11. Sherman, T., Conjugate points and simple zeros for ordinary linear differential equations, Trans. AMS 146 (1969), 397–411.Google Scholar
12. Spencer, J., Relations between boundary value functions for a nonlinear differential equation and its variational equations, Canad. Math. Bull., to appear.Google Scholar
13. Spencer, J., Boundary value functions for nonlinear differential equations, J. of Diff. Eqs., to appear.Google Scholar
14. Sukup, D., Boundary value problems for nonlinear differential equations, Ph.D. thesis, Univ. of Nebraska, 1974.Google Scholar
15. Sukup, D., On the existence of solutions of multipoint boundary value problems, R. M. J. of Math, to appear.Google Scholar