Hostname: page-component-78c5997874-j824f Total loading time: 0 Render date: 2024-11-04T21:06:23.640Z Has data issue: false hasContentIssue false
  • English
  • Français

Principal solutions of positive linear Hamiltonian systems

Published online by Cambridge University Press:  09 April 2009

Don Hinton
Don Hinton
Affiliation:
Mathematics Department, The University of Tennessee, Knoxville, 37916, Tennessee, U.S.A.
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.

The Hamiltonian system Y′ = BY + CZ, Z′ = – AY – B*Z is considered where the coefficients are continuous on I = [a, ∞, C = C* ≧ 0, and A = A* ≦ 0. A solution (Y, Z) satisfying Y*Z = Z*Y is defined to be principal (coprincipal) provided that (i) Y−1 exists on I (Z−1 exists on I) and (ii) as t→∞ ( as t → ∞). Three conditions are given which are separtely equivalent to the condition that a solution is principal iff it is coprincipal. For a self-adjoint scalar operator L of order 2n, this problem is related to the deficiency index problem and to a problem of Anderson and Lazer (1970) which concerns the number of lnearly independent solutions of L (y) =0 satisfying y(k) ∈ (a, ∞) (k = 0, …, n).

Type
Research Article
Information
Journal of the Australian Mathematical Society , Volume 22 , Issue 4 , December 1976 , pp. 411 - 420
Copyright
Copyright © Australian Mathematical Society 1976

References

Ahlbrandt, C. D. (1972), ‘Principal and antiprincipal solutions of self-adjoint differential systems and their reciprocals’, Rocky Mountain J. Math. 2, 169–182.CrossRefGoogle Scholar
Ahlbrandt, C. D. (1976), ‘The question of equivalence of principal nd coprincipal solutions of self-adjoint differential systems’, Illinois J. Math. 20, 72–81.CrossRefGoogle Scholar
Anderson, L. R. and Lazer, A. C. (1970), ‘On the solutions of linear self-adjoint differntial equations’, Trans. Amer. Math. Soc. 152, 519–530.CrossRefGoogle Scholar
Coppel, W. A. (1971), Disconjugacy (Springer-Verlag Lecture Notes in Mathematics, vol. 220, Berlin).CrossRefGoogle Scholar
Glazman, I. M. (1965), Direct methods of qualitative spectral analysis of singular differntial operators (Israel program for scientific translation, Jerusalem).Google Scholar
Hinton, D. B. (1972), ‘Limit point criteria for differential equations’, Canad. J. Math. 24, 293–305.CrossRefGoogle Scholar
Hinton, D. B. (1973), ‘Limit point criteria for positive definite fourth-order differential operators’, Quart. J. Math. Oxford (2) 24, 367–376.CrossRefGoogle Scholar
Reid, W. T. (1971), Ordinary differential equations (Wiley, New York).Google Scholar