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Positive Solutions and Oscillation

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*Agacik Zafer, Yasemin Yalcin and Yeter S.Yilmaz*

**Address.** Department of Mathematics, Middle East Technical University,
06531 Ankara, Turkey

Department of Mathematics, Middle East Technical University, 06531 Ankara,
Turkey

Department of Mathematics, Atilim University, 06830 Ankara, Turkey
**E-mail:** zafer@math.metu.edu.tr

yaseminy@metu.edu.tr

**Abstract.** Sufficient conditions are established for the existence
of positive solutions and oscillation of bounded solutions of $p$-th order
neutral difference equations of the form $$ \Delta^p[x_n+a_nx_{\tau(n)}]+\delta\,
q_nf(x_{\sigma(n)}) = h_n, \quad n\in \N(n_0), $$ where $\delta = \pm 1$,
$\N(n_0)=\{n_0,n_0+1,\ldots \}$, $n_0$ is fixed in $\N=\{1,2,\ldots\}$,
$a, q, h:\;\N(n_0)\rightarrow \R,$ $\tau,\sigma\in \N(n_0)\rightarrow \N$
with $\tau(n)

**AMSclassification.** 39A10, 34A11, 34A99

**Keywords.** Higher order neutral equations, positive solution,
oscillation