Sufficient conditions for oscillatory behaviour of a first order neutral difference equation with oscillating coefficients. (English) Zbl 1224.39018

Summary: We obtain sufficient conditions so that every solution of neutral functional difference equation \[ \Delta(y_n - p_n y_{\tau(n)}) + q_n G(y_{\sigma(n)}) = f_n \] oscillates or tends to zero as \(n\to \infty\). Here, \(\Delta\) is the forward difference operator given by \(\Delta x_n = x_{n+1}-x_n\), and \(p_n\), \(q_n\), \(f_n\) are the terms of oscillating infinite sequences; \(\{\tau_n\}\) and \(\{\sigma_n\}\) are non-decreasing sequences, which are less than \(n\) and approaches \(\infty\) as \(n\) approaches \(\infty\). This paper generalizes and improves some recent results.


39A21 Oscillation theory for difference equations
39A10 Additive difference equations
39A12 Discrete version of topics in analysis
39A22 Growth, boundedness, comparison of solutions to difference equations
34K40 Neutral functional-differential equations
34K11 Oscillation theory of functional-differential equations
Full Text: EuDML