Properties of solutions ofn-th order linear differential equations

Eventual Disconjugacy of Selfadjoint Fourth Order Linear Differential Equations

“…Put yit)=m1U*it) + m2V*it)-m1Cl-m2C2. Then v-(f) is a nontrivial solution of ip*y")" + iq*y')'=0, y(a)=0, and;;(/)=Wi^i sin kit) + m2h2 cos &(/) on cn^t^drr Because yit) has at least/?-(-3 zeros on the interval a^t<dn, a standard theorem on conjugate points (see [4], [10]) shows that r¡Pia) exists. Furthermore, a constant can be added to yit) so that the solution so obtained has p + 3 double zeros on cn^t^dn.…”

“…Furthermore, given any half-line t^.T, and any integer N>0, there is a solution of the equation with N consecutive double zeros. Therefore, the equation is not disconjugate in the sense of Reid (see [9], [10]). …”

“…+ 3 zeros on a^t^b, counting multiplicities. In [10] it is proved that this minimum exists provided there exists a solution yiO^O that vanishes at t=a and has at leastp+3 zeros on t^a, counting multiplicities.…”

Oscillation Theorems for Systems of Linear Differential Equations

“…The term "disconjugate", as introduced by Wintner [21], refers to the absence of a conjugate point in the sense of Jacobi, and thus originally applied only to selfadjoint equations and systems [3], [4], [14], [15], [19], [20]. However, this concept generalizes in a natural way to general «th order differential equations [1], [8], [9], [10], [13], [17], [18] and thus also to systems which are equivalent to such equations. In all these cases, the right conjugate point r¡(x0) of x0 (■n(x0)>x0) is a continuous function of x0, and the left conjugate point of ^(x0) coincides with x0 [17], [18].…”

“…However, this concept generalizes in a natural way to general «th order differential equations [1], [8], [9], [10], [13], [17], [18] and thus also to systems which are equivalent to such equations. In all these cases, the right conjugate point r¡(x0) of x0 (■n(x0)>x0) is a continuous function of x0, and the left conjugate point of ^(x0) coincides with x0 [17], [18]. In the case of a system which can be reduced to an nth order equation, -n(x0) can be defined in the following way : there exists a solution vector of (1.1) such that every component of y vanishes either at x0 or at r¡(x0), and -n(x0) is the smallest number with this property.…”