Matucci, Serena The \(l^p\) trichotomy for difference systems and applications. (English) Zbl 1090.39501 Arch. Math., Brno 36, Suppl., 519-529 (2000). 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. Reviewer: Ladislav Adamec (Brno) Cited in 2 Documents MSC: 39A10 Additive difference equations 39A11 Stability of difference equations (MSC2000) 39A12 Discrete version of topics in analysis Keywords:nonlinear difference systems; asymptotic behavior of solutions; \(l^p\) trichotomy PDF BibTeX XML Cite \textit{S. Matucci}, Arch. Math., Brno 36, 519--529 (2000; Zbl 1090.39501) Full Text: EuDML EMIS OpenURL