On distance between zeros of solutions of third-order differential equations (Q2759182)

Here, the oscillatory solutions to the third-order differential equation NEWLINE\[NEWLINEy'''+q(t)y'+p(t)y=0 \tag{E}NEWLINE\]NEWLINE \(p\in C^1([0,\infty))\), \(p,q'\in L_{\text{loc}}([0,\infty))\) are studied. The author presents sufficient conditions for \(t_{n+1}-t_n\to\infty\) or \(t_{n-2}-t_n\to\infty\) as \(n\to\infty\), where \(\{t_n\}\) is a sequence of zeros of an oscillatory solution to equation (E). One of the main results is the following theorem : Let \(0<\sigma<\infty\). Let \(q(t)\geq 0\), \(q'(t)\leq 0\) for \(t\in[0,\infty)\) and \(q(t)\to 0\) as \(t\to\infty\). Suppose that \(|q'(t)-p(t)|^\sigma\) is locally integrable on \([0,\infty)\) and \(\lim_{t\to\infty}\int_t^{t+\delta_0}|q'(s)-p(s)|^\sigma\text ds=0\) for some \(\delta_0>0\). If \(\{t_n\}\) is an increasing sequence of zeros of an oscillatory solution \(y(t)\) to \((E)\), then \(t_{n-2}-t_n\to\infty\) as \(n\to\infty\). NEWLINENEWLINENEWLINEThis result extends the recent work of the authors [J. Math. Anal. Appl. 233, No. 2, 445-460 (1999; Zbl 0932.34030)]. In the last section, a lower bound for the spacing \(d-a\) is derived for the solution \(y(t)\) to equation (E) with \(y (a)=y'(a)=0\), \(y'(c)=0\) and \(y''(d)=0\), \(a<d<c\). Examples are given to illustrate the fact that the authors significantly extend the earlier results.