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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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