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Oscillation and asymptotic behavior of neutral differential equations with deviating arguments. (English) Zbl 0566.34057

Consider the neutral differential equation \[ (1)\quad (d/dt)[y(t)+py(t- \tau)]+qy(t-\sigma)=0,\quad t\geq t_ 0, \] where \(q\neq 0,p,\tau\), and \(\sigma\) are real numbers. Let y(t) be a nonoscillatory solution of (1). Then \(\lim_{t\to \infty}y(t)\) is determined for all cases, except: i) \(p\leq -1\), \(q<0\), and \(\tau >0\); ii) \(-1\leq p<0\), \(q>0\), and \(\tau <0\); iii) \(p>0\), \(q<0\), and \(\tau\) \(\neq 0\). Two conjectures (as well as evidence indicating their possible validity) are given to cover the missing cases i), ii), and iii). It is also shown that if \(q\tau\geq 0\), or if \(q\tau <0\) and \(p\geq 0\), then each of the following conditions implies that every solution of (1) is oscillatory: a) \(p=-1\); b) \(p\neq - 1\) and \(q/(1+p)(\sigma -\tau)>1/e\); c) \(p\neq -1\) and \(q/(1+p)\sigma >1/e\); d) \(1+pq(\sigma +\tau)\leq 0\); e) \(p\neq 0\) and \(1+(q/p^ 2)(\sigma -2\tau)\leq 0.\)

MSC:

34K99 Functional-differential equations (including equations with delayed, advanced or state-dependent argument)
34K25 Asymptotic theory of functional-differential equations
34C10 Oscillation theory, zeros, disconjugacy and comparison theory for ordinary differential equations
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