Asymptotic behavior of oscillatory solutions of first order delay differential equations of unstable type (Q2868891)

The author studies the following first order delay differential equation of unstable type NEWLINE\[NEWLINEx'(t)=q(t)x(\sigma(t)), \quad t\geq t_0, \tag{*}NEWLINE\]NEWLINE where \(q: [t_0, \infty)\) is a piecewise continuous function, \(\sigma: [t_0, \infty)\to (-\infty, \infty)\) is a continuous, monotone increasing function such that \(\sigma(t)\leq t\) and \(\sigma(t)\to \infty\) as \(t\in \infty\). The first result of this paper is that if NEWLINE\[NEWLINE\limsup\limits_{t\to \infty}\int_{\sigma(t)}^tq(s)ds<\frac{5}{2},NEWLINE\]NEWLINE then every oscillatory solution of (*) tend to zero as \(t\to \infty\). This result is an improvement of Györi's result in a special case. The author also shows that for the nonhomogenous equation NEWLINE\[NEWLINEx'(t)=q(t)x(\sigma(t))+f(t), \quad t\geq t_0,\tag{**}NEWLINE\]NEWLINE the above inequality is also a sufficient condition for every oscillatory solution of (**) tend to zero as \(t\to \infty\) when \(f\) is a continuous function of certain type. For a special case of (**), NEWLINE\[NEWLINEx'(t)=qx(t-\tau)+ke^{-\lambda t},NEWLINE\]NEWLINE the author proves that \(q\tau<3\pi/2\) is a necessary and sufficient condition for every oscillatory solution to tend to zero as \(t\to \infty\). Finally, by an example the author shows that NEWLINE\[NEWLINE\limsup\limits_{t\to \infty}\int_{\sigma(t)}^tq(s)ds<\frac{15}{4}NEWLINE\]NEWLINE is a necessary condition for every oscillatory solution of (*) to tend to zero as \(t\to \infty\).