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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.
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Abstract

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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

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