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Asymptotic behavior of solutions of nonlinear higher-order neutral type difference equations. (English) Zbl 1098.39006
The authors study the asymptotic behavior of the solutions from the \(m\)th-order neutral type difference equation \[ \Delta^m(y_m+py_{n-k})=f(n,y_{n-1},\Delta y_{n-1}, \Delta^2 y_{n-1},\dots \Delta^{m-1} y_{n-1}), \] where \(f\) is a function of \(m+1\) variables and \(\Delta^m\) is the \(m\)th-order forward difference operator defined by: \[ \Delta y_n=y_{n+1}-y_n,\quad \Delta^iy_n=\Delta(\Delta^{i-1}y_n),\quad 1\leq i\leq m, \quad m\geq 3. \] A solution \(\{y_n\}_n\) of the above equation is called nonoscillatory if and only if it is either eventually positive (\(y_n>0\) for sufficiently large \(n\)) or eventually negative (\(y_n<0\) for sufficiently large \(n\)). The main results of the paper establish sufficient conditions under which the nonoscillatory solutions of the equation written above behave asymptotically like a sequence \(a_1n^{m-1}+a_2n^{m-2}+\dots+a_{m-1}n+a_m\) with \(a_i\in\mathbb R\) for any \(i\in\{1,2,\dots,m\}\). Some applications of the proved theorems are also given. This work extends some results proved for second order difference equation by E. Thandapani, R. Arul and P. S. Raja [Math. Comput. Modelling 39, No. 13, 1457–1465 (2004; Zbl 1067.39018)].

MSC:
39A11 Stability of difference equations (MSC2000)
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