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Keywords:
nonlinear difference equtions; oscillation; eventually positive solutions; characteristic equation
Summary:
In this paper we consider the nonlinear difference equation with several delays \[ (ax_{n+1}+bx_{n})^k-(cx_{n})^k+\sum \limits _{i=1}^{m} p_{i}(n)x^k_{n-\sigma _{i}}=0 \] where $a,b,c\in (0,\infty )$, $k=q/r, q, r$ are positive odd integers, $m$, $\sigma _{i}$ are positive integers, $\lbrace p_{i}(n)\rbrace $, $i=1,2,\dots ,m, $ is a real sequence with $p_{i}(n)\ge 0$ for all large $n$, and $\liminf _{n\rightarrow \infty }p_{i}(n)=p_{i}<\infty $, $i=1,2,\dots ,m$. Some sufficient conditions for the oscillation of all solutions of the above equation are obtained.
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