The \(l^p\) trichotomy for difference systems and applications. (English) Zbl 1090.39501

The paper deals with a quasi-linear system \[ x(n+1) = A(n)x(n)+f(n,x(n)),\qquad n\in {\mathbb Z},\tag{1} \] where \(A(n)\) in an \(m\times m\) matrix and \(f:{\mathbb Y}\times {\mathbb R}^m\to {\mathbb R}^m\) is a continuous function.
The aim is to study existence of bounded solutions of (1) having zero limit as \(n\to \pm \infty \), under the assumption that solutions of the associated linear homogeneous system \[ x(n+1)=A(n)x(n),\qquad n\in {\mathbb Z}, \tag{2} \] are not all bounded on \(\mathbb Z\).
The main result discuss existence of at least one solution of the boundary value problem \[ \begin{matrix} y(n+1)=A(n)y(n)+f(n,y(n))\\ y(-\infty )=y(+\infty )=0\\ y(0)=\xi \end{matrix} \] under the assumption that the associated linear system (2) possesses an \(l^p\) trichotomy. The \(l^p\) trichotomy for a linear difference system is considered as an extension of exponential trichotomy and \(l^p\) dichotomy.


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