On Self-Adjoint Differential Equations of Second Order

Abstract

This note is an abstract of a longer paper which will be published elsewhere.The paper is concerned with the behavior near x = + o of solutions of the self-adjoint differential equation

Relative Oscillation Theory, Weighted Zeros of the Wronskian, and the Spectral Shift Function

“…Only recently it was shown in [5] by Gesztesy, Simon, and one of us, that these limits can be avoided by using a renormalized version of oscillation theory, that is, by counting zeros of Wronskians of solutions instead (see again [30] for a pedagogical discussion respectively [26] for an interesting application to minimal surfaces). That zeros of the Wronskian are related to oscillation theory is indicated by an old paper of Leighton [16], who noted that if two solutions have a non-vanishing Wronskian, then their zeros must intertwine each other.…”

Oscillation criteria for second order self adjoint differential systems

“…Coles' work was subsequently generalized by J. W. Macki and J. S. W. Wong [13] through the use of averaging pairs. In this section we use averaging pairs to establish for equation (1) (12) 2…”

Remarks on Nonoscillation Theorems for a Second Order Nonlinear Differential Equation

“…where B = ¡q sßax(s) ds. An application of Gronwall's inequality to (10) yields that a(t) is bounded on account of (2). Since a(t) is positive, we obtain also from (9) the following estimate:…”

“…We summarize below a number of nonoscillation criteria for equation (1). For the linear equation, we have (I) M. Bôcher [2], y -1, /* ta(t) dt < oo; (II) A. Kneser [9], y = 1, lim sup,^ t2a(t) <\; (III) W. Leighton [10], y = 1, a(t) is nonincreasing, f Vajt) dt < oo.…”