Oscillation criteria for certain difference equations with continuous variables. (English) Zbl 1123.39011
Consider the difference equation with continuous arguments
\[ x(t)-x(t-\tau)+\sum_{i=1}^m p_i(t)x(t-\sigma_i)=0,\quad t\geq 0, \]
where \(0 < \tau < \sigma_1 < \dots < \sigma_m\), \(p_i\in C (\mathbb R, \mathbb R_+)\), \(\lim\inf_{t\to\infty} p_i(t)=\bar p_i\), \(i=1,2,\dots,m\).
The authors give sufficient conditions for the oscillation of all solutions of the above equation in the case when the corresponding limiting equation \[ x(t)-x(t-\tau)+\sum_{i=1}^m \bar p_i x(t-\sigma_i)=0,\quad t\geq 0, \] admits non-oscillatory solutions.
\[ x(t)-x(t-\tau)+\sum_{i=1}^m p_i(t)x(t-\sigma_i)=0,\quad t\geq 0, \]
where \(0 < \tau < \sigma_1 < \dots < \sigma_m\), \(p_i\in C (\mathbb R, \mathbb R_+)\), \(\lim\inf_{t\to\infty} p_i(t)=\bar p_i\), \(i=1,2,\dots,m\).
The authors give sufficient conditions for the oscillation of all solutions of the above equation in the case when the corresponding limiting equation \[ x(t)-x(t-\tau)+\sum_{i=1}^m \bar p_i x(t-\sigma_i)=0,\quad t\geq 0, \] admits non-oscillatory solutions.
Reviewer: B. G. Pachpatte (Aurangabad)
MSC:
| 39A11 | Stability of difference equations (MSC2000) |
