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Zbl 1035.39008
Saker, S. H.
Oscillation of second order nonlinear delay difference equations.
(English)
[J] Bull. Korean Math. Soc. 40, No. 3, 489-501 (2003). ISSN 1015-8634

Consider the nonlinear delay difference equation $$\Delta (p_n\Delta x_n)+q_nf(x_{n-\sigma })=0,\quad n=0,1,2,\dots \tag*$$ where $\Delta u_n=u_{n+1}-u_n$ for any sequence $\{u_n\}$ of real numbers, $\sigma$ is a nonnegative integer, $\{p_n\}_{n=0}^\infty$ and $\{q_n\}_{n=0}^\infty$ are sequences of real numbers such that $p_n>0,$ $\sum^\infty \frac 1{p_n}<\infty,$ $q_n\geq 0$ and $q_n$ has a positive subsequence, and $f$ is a continuous nondecreasing real valued function which satisfies $uf(u)>0$ for $u\neq 0$ and $f(u)/u\geq \gamma >0.$ The author establishes some sufficient conditions which guarantee that every solution of (*) is oscillatory or converges to zero.
[Fozi Dannan (Doha)]
MSC 2000:
*39A11 Stability of difference equations

Keywords: oscillation; nonlinear delay difference equations

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