A necessary and sufficient condition for the oscillation of a delay equation with continuous and piecewise constant arguments
The equation \[ x'(t)+px(x-l)+qx([t-k])=0 \tag{1} \] is considered with \(p,q\in \mathbb{R}\), \(l,k\in \mathbb{N}=\{0,1,2,\dots\}\) and \([\cdot ]\) denotes the greatest-integer function. An open problem posed by \textit{I. Győri} and \textit{G. Ladas} [Oscillation theory of delay differential equations: With applications. Clarendon Press, Oxford (1991; Zbl 0780.34048)] is partly answered: When \(l\in [0,\infty)\), obtain a ``characteristic equation'' for (1). The following is proved: A necessary and sufficient condition for the oscillation of (1) is that the characteristic equation \[ \lambda \exp (p\lambda ^{-l})-1+q\lambda ^{-k}\int_0^1\exp (p\lambda ^{-l}\eta) d\eta =0 \tag{2} \] has no real roots in \((0,\infty)\). The improved \(Z\)-transform is used in the proof. For the equation \[ x'(t)+px(x-\tau)+qx([t-k])=0 \tag{3} \] with \(p,q,\tau \in (0,\infty)\), \(k\in \mathbb{N}\) the following holds: Let \([\tau ]=l\). Then if \((2)\) has no real roots in \((0,\infty)\), (3) is oscillatory.
scientific article; zbMATH DE number 1225607