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

Rath, R. N.; Misra, N.; Rath, S. K.

Acta Math. Acad. Paedagog. Nyházi. (N.S.) 25, No. 1, 55-63 (2009).

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.

MSC:

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

Keywords:

oscillation; asymptotic behavior; neutral functional difference equation