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Bounded solutions and asymptotic stability of nonlinear difference equations in the complex plane. (English) Zbl 1053.39016

The authors deal with the unique solvability in \(l_1\) of a class of nonlinear \(m\)-th order difference equation \[ \begin{aligned} x(n+m)&+\sum _{p=1}^m (\alpha _p+\beta _p(n))x(n+m-p)= \sum _{i=1}^N c_i(n)x(n+q_{i_{1}})x(n+q_{i_{2}})\\ &+\sum _{j=1}^M d_j(n)x(n+q_{j_{3}})x(n+q_{j_{4}})x(n+q_{j_{5}}), \end{aligned} \tag{*} \] where \(m,N,M\) are positive integers, \(q_{i_{1}}, q_{i_{2}}, q_{j_{3}}, q_{j_{4}}, q_{j_{5}}\) are nonnegative integers and \(\alpha _p\in \mathbb C\). Conditions on the sequences \(\beta _p(n),c_i(n),d_j(n)\) and the roots of the polynomial \(r^m+\alpha _1r^{m-1}+\dots +\alpha _m=0\) are given which guarantee that (*) has a unique solution satisfying the initial condition \(x(p)=u_p\), \(p=1,\dots ,m\), and \(\sum _{n=1}^\infty | x(n)| <\infty \).
The main result of the paper is illustrated by a number of examples and several remarks to “insert” this result into the broader context of stability theory of difference equations.

MSC:

39A11 Stability of difference equations (MSC2000)
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