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Name:
Miloslav Znojil
Reviewer number:
9689
Email:
znojil@ujf.cas.cz
Item's zbl-Number:
DE 017 567 559
Author(s):
Clark, Steve; Hinton, Don:
Shorttitle:
Positive eigenvalues of second order boundary value problems and a theorem of M. G. Krein.
Source:
Proc. Am. Math. Soc. 130, No. 10, 3005-3015 (2002)
Classification:
34C10Oscillation theory
Primary Classification:
34L15Estimation of eigenvalues, upper and lower bounds
Secondary Classification:
34B24Sturm Liouville theory
34D10Perturbations
Keywords:
systems of ordinary linear differential equations; stable boundedness of solutions; positivity of eigenvalues; Opial-like inequalities
Review:


The authors found that M. G. Krein never published a proof of one
of his theorems on stability of solutions of equations with
periodic coefficients. The paper fills the gap. More than that:
conditions are derived which guarantee the positivity of
eigenvalues of an auxiliary n-dimensional system (with both the
Dirichlet and antiperiodic boundary conditions) while, in the next
step, the (required) stable boundedness of solutions of the
original system, which means the property of solutions being
bounded on the full line, is proved. The property is true for all
periodic perturbations of a sufficiently small p=1 norm. A neat
and nice study, with the main technical emphasis on the above
mentioned positivity of eigenvalues which, in its turn, relies
heavily on several (and, in a preliminary section, separately
proved) Opial-like inequalities (for pairs of functions) and on
the ample use of a specific index which ''measures" oscillations.
Remarks to the editors:

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