Higher Monotonicity Properties of Certain Sturm-Liouville Functions. III

Abstract

A Sturm-Liouville function is simply a non-trivial solution of the Sturm-Liouville differential equation(1.1)considered, together with everything else in this study, in the real domain. The associated quantities whose higher monotonicity properties are determined here are defined, for fixed λ > –1, to be(1.2)where y(x) is an arbitrary (non-trivial) solution of (1.1) and x1, x2, … is any finite or infinite sequence of consecutive zeros of any non-trivial solution z(x) of (1.1) which may or may not be linearly independent of y(x). The condition λ > –1 is required to assure convergence of the integral defining Mk, and the function W(x) is taken subject to the same restriction.

Monotonicity properties of oscillatory solutions of differential equation $(a(t)\vert y^{\prime }\vert ^{p-1}y^{\prime })^{\prime }+f(t,y,y^{\prime })=0$

“…Our results generalize the ones e.g. of [1] (see also references therein) and [8,11] for (1.3). The obtained results coincide with the known results of [1] for p = 1 and a(t) ≡ 1.…”

“…This problem has a long history which was initiated by P. Hartman, L. Lorch and M. Muldon for Bessel functions and higher monotonicity problem for second order linear equation, see [8,11].…”

On the Composition of Completely Monotonic Functions and Completely Monotonic Sequences and Related Questions

“…However, a similar result of [7], namely Theorem 3.2, cannot be extended in this way. To see this, we note that there exist sequences {xd, {t k }, with {dx k }, {dt k } both completely monotonic, with Xo > to, k = 1,2, ... , and such that the sequence {w(x k )-w(tk)}o need not be monotonic, much less completely monotonic, although W(x) = w'(x) is a completely monotonic function.…”

“…For {x k }, {t k } the zeros of solutions of a certain type of Sturm-Liouville differential equation the sequence {W(Xk) -w(tk)}o would be completely monotonic, according to Theorem 3.2 of [7], whenever w'(x) is completely monotonic.…”

“…Still another result of [7], namely the case when N = 00 of Corollary 3.2, can be freed of dependence on differential equations and thus generalized. The extension is the following.…”