FOCAL POINTS FOR A LINEAR-DIFFERENTIAL EQUATION WHOSE COEFFICIENTS ARE OF CONSTANT SIGNS

The differential equation considered is (FORMULA PRESENTED), where (FORMULA PRESENTED). The focal point ζ (a) is defined as the least value of s, s > a, such that there exists a nontrivial solution y which satisfies (FORMULA PRESENTED) Our method is based on a characterization of f (a) by solutions which satisfy > 0, 1 « 0,…» n - 1, on [a, b], b > ζ(a). We study the behavior of the function ζ and the dependence of ζ (a) on …, p0- 1 when at least a certain pi(x) does not vanish identically near a or near £(a). As an application we prove the existence of an eigenvalue of a related boundary value problem. © 1979 American Mathematical Society.