Oscillation of deviating differential equations. (English) Zbl 1499.34344

Summary: Consider the first-order linear delay (advanced) differential equation \[x'(t)+p(t)x(\tau(t))=0\quad(x'(t)-q(t)x(\sigma(t))=0),\quad t\geq t_{0},\] where \(p\) \((q)\) is a continuous function of nonnegative real numbers and the argument \(\tau(t)\) \((\sigma(t))\) is not necessarily monotone. Based on an iterative technique, a new oscillation criterion is established when the well-known conditions \[\limsup\limits_{t\rightarrow\infty}\int_{\tau(t)}^{t}p(s)ds>1\quad\biggl(\limsup\limits_{t\rightarrow\infty}\int_{t}^{\sigma(t)}q(s)ds>1\bigg)\] and \[\liminf_{t\rightarrow\infty}\int_{\tau(t)}^{t}p(s)ds>\frac{1}{\mathrm{e}}\quad\biggl(\liminf_{t\rightarrow\infty}\int_{t}^{\sigma(t)}q(s)ds>\frac{1}{\mathrm{e}}\bigg)\] are not satisfied. An example, numerically solved in MATLAB, is also given to illustrate the applicability and strength of the obtained condition over known ones.


34K11 Oscillation theory of functional-differential equations
34K06 Linear functional-differential equations


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