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Eventual disconjugacy of selfadjoint fourth order linear differential equations

Author: G. B. Gustafson
Journal: Proc. Amer. Math. Soc. 35 (1972), 187-192
MSC: Primary 34C10
MathSciNet review: 0298126
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Abstract: At the 1969 Differential Equations Conference held at Knoxville, Tennessee, Z. Nehari asked the following question about fourth-order selfadjoint linear differential equations: ``Assume that solutions of the equation have only a finite number of zeros on $ t \geqq A$. Does there exist a half-line on which no solution of the equation has more than three zeros?'' In this paper sufficient conditions are given for the equation $ (p(t)y'') + (q(t)y')' = 0$ to have the property that solutions possess only a finite number of zeros. This theorem is then used to construct an example which answers the above question in the negative. The example also shows that if on each half-line the equation has a solution with two consecutive double zeros, then it need not follow that there is a solution with infinitely many zeros.

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Keywords: Oscillation, nonoscillation, disconjugacy, eventually disconjugate, disconjugate in the sense of Reid, pth conjugate point
Article copyright: © Copyright 1972 American Mathematical Society

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