×

On the zeros of oscillatory solutions of linear second-order differential equations. (English) Zbl 0756.34035

Let \(y(x)\) be a real non-trival solution of the differential equation \((f)y''+f(x)y=0\) and let \(\{x_ n\}_ N\) denote the sequence of consecutive zeros of \(y\). The relationship between the asymptotic behaviour of \(f(x)\) and that of \(\{x_ n\}_ N\) is studied. Certain sequences related to \(\{x_ n\}_ N\) as well as the comparison of \(\{x_ n\}_ N\) with corresponding sequences belonging to \((F)\), where \(f(x)\) and \(F(x)\) differ only slightly, are studied too. The above mentioned relationship is based on the so called Mammana identity \(v''v- {1\over 2}v^{'2}+2fv^ 2=2w^ 2\), where \(v=y^ 2_ 1+y^ 2_ 2\), \(y_ 1,y_ 2\) being solutions of \((f)\), and \(w\) is their (constant) Wronskian. Under certain conditions \(fv^ 2\) is asymptotically constant [see e.g. the reviewer, Differ. Uravn. 15, 2115-2124 (1979; Zbl 0423.34045)]. The authors study in fact the above mentioned conditions and derive interesting theorems.

MSC:

34C10 Oscillation theory, zeros, disconjugacy and comparison theory for ordinary differential equations
60F05 Central limit and other weak theorems
34A30 Linear ordinary differential equations and systems

Citations:

Zbl 0423.34045
PDFBibTeX XMLCite