Abstract
A criterion is given which is necessary and sufficient that certain homogeneous linear boundary value problems have only the trivial solution. The condition includes results of Pólya [Traps. Amer. Math. Soc. 24 (1922), pp. 312–324] on disconjugacy and Muldowney [Proc. Amer. Math. Soc., 74 (1979), pp. 49–55] on disfocality. A mean value theorem and a positivity result are also obtained. Analogues of the Sturm comparison principle are established including a generalization of a theorem of Hartman [Amer. J. Math., 91 (1969), pp. 306–362; 93 (1971), pp. 439–451] and Levin [Soviet Math. Dokl. 4 (1963), pp. 121–124] which gives a necessary and sufficient condition for invertibility in terms of the existence of solutions to a family of differential inequalities.
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