Sufficient conditions for oscillatory behaviour of a first order neutral difference equation with oscillating coefficients.

*(English)*Zbl 1224.39018Summary: We obtain sufficient conditions so that every solution of neutral functional difference equation

$${\Delta}({y}_{n}-{p}_{n}{y}_{\tau \left(n\right)})+{q}_{n}G\left({y}_{\sigma \left(n\right)}\right)={f}_{n}$$

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; $\left\{{\tau}_{n}\right\}$ and $\left\{{\sigma}_{n}\right\}$ 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 (difference equations) |

39A10 | Additive difference equations |

39A12 | Discrete version of topics in analysis |

39A22 | Growth, boundedness, comparison of solutions (difference equations) |

34K40 | Neutral functional-differential equations |

34K11 | Oscillation theory of functional-differential equations |