Vol. 89, No. 2, 1980

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A note on disconjugacy for second order systems

Hal Leslie Smith

Vol. 89 (1980), No. 2, 447–452

It is well-known that the equation

x′′ + A(t)x = 0

is disconjugate on [a,b] if and only if there exists a solution which is positive on [a,b], in the case that A(t) is scalar-valued. In this note we generalize this simple result to the case where A(t) = (aij(t)) is an n × n matrix-valued function which satisfies certain generalized sign conditions. These results apply, for instance, if the off diagonal elements are nonnegative. Simple necessary and sufficient conditions are given for disconjugacy if A(t) A and these are used to construct examples showing the necessity of sign conditions on A(t) for the above mentioned results and other results of Sturm type for systems to be valid.

Mathematical Subject Classification 2000
Primary: 34C10
Received: 15 July 1977
Revised: 21 April 1980
Published: 1 August 1980
Hal Leslie Smith