Asymptotic behavior of solutions of nonlinear neutral delay difference equations with positive and negative coefficients (Q2896295)

The object of this paper is the neutral delay difference equation NEWLINE\[NEWLINE \triangle[x(n)+R(n)x(n-m)]+p(n)f(x(n-k))-q(n)f(x(n-\ell))=0\;,\;n\geq n_0>0 NEWLINE\]NEWLINE where \(\triangle x(n) = x(n+1)-x(n)\); \(\{p(n)\}_n\), \(\{q(n)\}_n\), \(\{R(n)\}_n\) are sequences of reals and \(k\), \(\ell\), \(m\) are positive integers.NEWLINENEWLINEThe main result of the paper concerns boundedness of all solutions of the equation under the following assumptionsNEWLINENEWLINE(A1) \(k>\ell\); there exists \(M>0\) such that NEWLINE\[NEWLINE |x|<|f(x)|<M\;,\;x\in R;\;xf(x)>0 NEWLINE\]NEWLINENEWLINENEWLINE(A2) \(\limsup_{n\rightarrow\infty}|R(n)|=\mu<1\);NEWLINENEWLINE(A3) \(H(n)=p(n)-q(n+\ell-k)>0\;,\;n\geq n_1=n_0+k-\ell\);NEWLINENEWLINE(A4) \(\limsup_{n\rightarrow\infty}\sum_{i=n-k}^{n-\ell-1}q(i+\ell)<1/M\);NEWLINENEWLINE(A5) NEWLINE\[NEWLINE\begin{multlined} \limsup_{n\rightarrow\infty}\left[\sum_{i=n-k}^{n+k}H(i+k) + {{q(n+\ell)}\over{H(n+k)}}\sum_{i=n-k+1}^nH(i+2k)\right. \\ \left. + \mu\left(1+ {{H(n+m+k)}\over{H(n+k)}}\right) + \sum_{i=n-k}^{n-\ell-1}q(i+\ell)\right]<2/M\end{multlined} NEWLINE\]NEWLINE Other sufficient conditions in order that every solution tends to some constants as \(n\rightarrow\infty\) are then obtained.