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Inclusion theorems for nonlinear difference equations with applications. (English) Zbl 1056.39003
The paper concerns real nonlinear difference equations of order $$m\geq 1$$ $F(x_n, x_{n+1},\dots, x_{n+m})= 0,\tag{1}$ $$F: \mathbb{R}^{m+1}\to \mathbb{R}$$, $$n\in\mathbb{N}_0$$. It is assumed that $$F$$ depends on $$n$$ explicitly, but this is not indicated. The objective is to prove an inclusion theorem of the following kind: given a suitable pair of sequences $$(\varphi_n)$$, $$(\psi_n)$$ with $$\psi_n> 0$$, then for arbitrary $$\varepsilon> 0$$ there exists a solution $$(x_n)$$ of (1) and an $$n_0(\varepsilon)$$ such that $\psi_n- \varepsilon\psi_n\leq x_n\leq \varphi_n+ \varepsilon\psi_n,\tag{2}$ for $$n\geq n_0(\varepsilon)$$. It is always assumed that $$\phi_n= O(\varphi_n)$$ as $$n\to\infty$$ so that (2) implies $$x_n= \varphi_n+ O(\psi_n)$$. The set of all sequences $$(x_n)$$ with (2) is called an asymptotic stripe $$X(\varepsilon)$$, i.e. $$(y_n)\in X(\varepsilon)$$ implies the existence of a real sequence $$(C_n)$$ with $$y_n= \varphi_n+ C_n\psi_n$$ and $$| C_n|\leq\varepsilon$$ for $$n\geq n_0(\varepsilon)$$.

##### MSC:
 39A10 Additive difference equations 39A11 Stability of difference equations (MSC2000) 39A12 Discrete version of topics in analysis
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##### References:
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