Disconjugacy of Second-order Linear Differential Equations with Non- Negative Coefficients

His basic criterion for disconjugacy turned out to be 1 0 is continuous and fal/P= °°, then the substitution t — fll/p transforms the first equation into one where p = l and the new f-interval is infinite. Thus this more general case reduces to Nehari's and again the criterion is 1 then the well-known Leighton-Wintner Oscillation Theorem [4; 8] states that the equation (py')'+fy = 0 is oscillatory.2 Also, if both integrals are finite, the equation is nonoscillatory.3 These conclusions may be strengthened to strongly oscillatory and strongly nonoscillatory, respectively, as is evident from the definitions [5]. The cases of interest in this paper are those where one integral is infinite and one is finite. As will be seen in §5, if /"/= oo and (py')'+fy = 0 is disconjugate, then X(o)—>0 (as the reciprocal of flf) and another disconjugacy criterion is needed for this case. However, by noting that the necessary conditions for disconjugacy are related to quadratic functionals, which Reid [6] has connected to the nonexistence of focal points, Nehari's and other necessary conditions are obtained for more general conditions on the coefficients. Also, the case of /"/= °°, together with disconjugacy of (1), is discussed.