On Some Directions in Researching the Oscillation of Solutions to First Order DDE
摘要
We investigate the oscillation of the first-order delay differential equation \(\begin{aligned} x^{\prime }(t)+p(t)x(\tau (t))=0,t\ge t_{0} \in \mathbb {R}, \end{aligned}\) where \(p :\mathbb {R} \rightarrow [0,+\infty )\) , \(\tau :\mathbb {R} \rightarrow \mathbb {R}, \tau (t)\le t\) , \(\lim _{t\rightarrow \infty }\tau (t)=\infty \) . We trace the evolution of the relevant literature, from the seminal results of \(\begin{aligned} \liminf _{t\rightarrow \infty }\int _{\tau (t)}^{t}p(s)ds>\frac{1}{e}, \end{aligned}\) and \(\begin{aligned} \limsup _{t\rightarrow \infty }\int _{\tau (t)}^{t}p(s)ds>1, \end{aligned}\) to the recent condition (for nondecreasing delayed argument \(\tau (t)\) ) \(\begin{aligned} \limsup _{t\rightarrow \infty }\int _{\tau (t)}^{t}p(s)ds>\kappa (\liminf _{t\rightarrow \infty }\int _{\tau (t)}^{t}p(s)ds), \end{aligned}\) where \(\kappa :[0,\frac{1}{e}]\rightarrow [\frac{1}{e},1]\) is strictly decreasing, continuous. This condition is sharp in the sense that, for any given values of \(\liminf _{t\rightarrow \infty }\int _{\tau (t)}^{t}p(s)ds \in [0,\frac{1}{e}]\) and \(\varepsilon >0\) , one cannot improve the condition to \(\begin{aligned} \limsup _{t\rightarrow \infty }\int _{\tau (t)}^{t}p(s)ds>\kappa (\liminf _{t\rightarrow \infty }\int _{\tau (t)}^{t}p(s)ds)-\varepsilon . \end{aligned}\) In particular, \(\kappa (0)=1,\kappa (\frac{1}{e})=\frac{1}{e}\) . Our focus is two-fold. Firstly, we describe the intuitions and methods applied to oscillation. Secondly, we examine the overall advancements and improvements upon the criteria.