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A partly sharp oscillation criterion for first-order delay differential equations. (English) Zbl 1510.34145

Summary: In this paper, a partly sharp oscillation criterion is established for the first-order delay differential equation \[x^\prime ( t ) + p ( t ) x ( \tau ( t ) ) = 0 , \quad t \geq T_0 ,\] where \(p ( t )\) is continuous and \(p ( t ) \geq 0\) for all \(t \geq T_0 ; \tau ( t )\) is continuous, non-decreasing, \( \tau ( t ) < t\) for all \(t \geq T_0 ,\) and \(\lim_{t \to \infty} \tau ( t ) = + \infty \). By improving techniques from previous researches, we show that when \(\alpha = \liminf_{t \to \infty} \int_{\tau ( t )}^t p ( s ) d s \in ( 0 , \frac{1}{e} ] ,\) all solutions of the equation are oscillatory if \[\limsup_{t \to \infty} \int_{\tau (t)}^t p (s) d s > 2 \alpha + \frac{2}{\lambda_1} - 1\] under no additional constraints, where \(\lambda_1\) is the smaller root of the equation \(\lambda = e^{\alpha \lambda} .\) This result is proved to be weaker than previous ones, and sharp when \(\alpha \in ( 0 , \frac{ \ln 2}{2} )\) by constructing a specific example.

MSC:

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