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 PDF

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

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.

References

John H. Barrett, Systems-disconjugacy of a fourth-order differential equations, Proc. Amer. Math. Soc. 12 (1961), 205–213. MR 130443, DOI 10.1090/S0002-9939-1961-0130443-0

Oscillation theory of ordinary differential equations

J. M. Dolan, On the relationship between the oscillatory behavior of a linear third-order differential equation and its adjoint, J. Differential Equations 7 (1970), 367–388. MR 255908, DOI 10.1016/0022-0396(70)90116-6

Conjugate point properties for nth order linear differential equations

Philip Hartman, Ordinary differential equations, John Wiley & Sons, Inc., New York-London-Sydney, 1964. MR 0171038
Walter Leighton and Zeev Nehari, On the oscillation of solutions of self-adjoint linear differential equations of the fourth order, Trans. Amer. Math. Soc. 89 (1958), 325–377. MR 102639, DOI 10.1090/S0002-9947-1958-0102639-X
Zeev Nehari, Non-oscillation criteria for $n-th$ order linear differential equations, Duke Math. J. 32 (1965), 607–615. MR 186883
Zeev Nehari, Disconjugate linear differential operators, Trans. Amer. Math. Soc. 129 (1967), 500–516. MR 219781, DOI 10.1090/S0002-9947-1967-0219781-0
William T. Reid, Oscillation criteria for self-adjoint differential systems, Trans. Amer. Math. Soc. 101 (1961), 91–106. MR 133518, DOI 10.1090/S0002-9947-1961-0133518-X
Thomas L. Sherman, Properties of solutions of $n\textrm {th}$ order linear differential equations, Pacific J. Math. 15 (1965), 1045–1060. MR 185185
Marko Švec, Sur une propriété des intégrales de l’équation $y^{(n)}+Q(x)y=0,$, $n=3,\,4$, Czechoslovak Math. J. 7(82) (1957), 450–462 (French, with Russian summary). MR 95313