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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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##### References:
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