Positive Solutions and Oscillation

Agacik Zafer, Yasemin Yalcin and Yeter S.Yilmaz

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