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Oscillatory and asymptotic behaviour of perturbed
quasilinear second order difference equations

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*
E. Thandapani and L. Ramuppillai
*

** Address.**
Department of Mathematics, Madras University
P.\,G.\,Centre, Salem-636 011, Tamil Nadu, INDIA

** E-mail:**

**Abstract.**
This paper deals with oscillatory and asymptotic behaviour of
solutions of second order quasilinear difference equation of the
form
$$
\Delta (a_{n-1}| \Delta y_{n-1}|^{\alpha-1} \Delta y_{n-1})+
F(n, y_n)= G(n, y_n, \Delta y_n),
\quad n\in N(n_0)
\tag{E}
$$
where $\alpha > 0$. Some sufficient conditions for all solutions of (E)
to be oscillatory are obtained. Asymptotic behaviour of
nonoscillatory solutions of (E) are also considered.

**AMSclassification.**
39A10

**Keywords.**
Perturbed quasilinear difference equation, oscillatory
solution