Principal Solutions of Disconjugate n-th Order Linear Differential Equations

Factorization and disconjugacy of third order differential equations

“…We show thaty^f) > 0 for t > a. Proceeding as in the proof of Corollary 2 we show that equation (2) has no nontrivial solution satisfying boundary condition (7). Similarly, using the third and fourth inequalities from the hypothesis, and arguing with respect to the adjoint equation (4) we show that (4) has no nontrivial solution satisfying boundary condition (7) for any c, d in / with c < d. The proof is then completed just as in Theorem 3.…”

“…The first two inequalities in the hypothesis imply, by Corollary 2, that no nontrivial solution of equation (2) satisfies a boundary condition of type (7). Similarly the second pair of inequalities imply that no nontrivial solution of equation (4) satisfies a boundary condition of type (7).…”

Difference equations: disconjugacy, principal solutions, Green’s functions, complete monotonicity

“…The object of this paper is to carry analogous results on linear «th order differential equations over to (1.1). These include the formulation of the concept of disconjugacy in §1, the analogues of Pólya's [17] criterion for disconjugacy in §5, the generalized Sturm comparison theorems of [6], [7], [13] in §6, discussion of the sign of Green's function of [11], [12], [16] (cf. [2, pp.…”

“…[2, pp. 105-109]) in §7, existence and properties of principal solutions of [6], [7] in §8, inequalities for certain solutions in §9, and completely monotone families of solutions of equations depending on a parameter [8] in §10. We illustrate the result of §10 with an application to Bessel functions.…”

“…Many of the proofs for (1.1) are similar to those for differential equations and, for the sake of brevity, will be omitted or only indicated. The principal tools in [6], [7], [13], [8] for differential equations are identities for Wronskian determinants, Rolle's theorem, and results of [9]. We enumerate equivalent identities for determinants in §2, an analogue of Rolle's theorem is Proposition 5.1 below, while results of [9] are stated for both differential and difference equations.…”