Eventual disconjugacy of selfadjoint fourth order linear differential equations

Gustafson, G. B.

doi:10.1090/S0002-9939-1972-0298126-1

Eventual disconjugacy of selfadjoint fourth order linear differential equations
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by G. B. Gustafson

Proc. Amer. Math. Soc. 35 (1972), 187-192

DOI: https://doi.org/10.1090/S0002-9939-1972-0298126-1

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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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