Limit theorems in the theory of systems of difference equations (Q2754830)

This paper deals with the difference systems in the space \(\mathfrak M\) of bounded sequences. For such systems the authors generalize a number of well known results obtained earlier in finite-dimensional case. The following questions are considered: (a) general results on reducibility of linear system \(x(n+1)=A(n)x(n), n\in \mathbb Z\), with infinite matrix \(A(n)=\{a_{ij}(n)\}_{i,j=1}^\infty \) to a system with constant matrix; (b) Floquet-Lyapunov-like theorems for linear \(N\)-periodic systems; (c) reducibility of linear quasi-periodic systems of the form NEWLINE\[NEWLINEx(n+1)=Ax(n)+P(\varphi(n))x(n),\quad \varphi(n+1)=\varphi(n)+\omega,\tag{1}NEWLINE\]NEWLINE where \(A=\text{diag}(a_1,a_2,\ldots), \varphi \in T^m:=\mathbb R^m/2\pi \mathbb Z^m, \omega \in \mathbb R^m\), and \(P(\varphi)\) is a sufficiently small real analytic infinite matrix on the torus \(T^m\); (d) reducibility of almost periodic system (1) where \(\varphi \in T^\infty =(\mathbb R/2\pi \mathbb Z)^\infty, \omega \in \mathfrak M\); (e) reducibility of almost periodic equation \(x(t+1)=x(t)+\omega +F(x(t),\alpha t)+\lambda,\) on \(T^\infty \), where \(F:T^\infty \times \mathbb R \to \mathfrak M, \omega,\alpha,\lambda \in \mathbb R^\infty \), to the form \(x(t+1)=x(t)+\omega \); (f) existence of periodic solutions to \(N\)-periodic linear inhomogeneous system; (g) construction of approximate \(N\)-periodic solution to quasilinear \(N\)-periodic system \(x(n+1)=x(n)+\epsilon f(n,x(n))\) in resonant case; (h) construction of invariant torus for the system on \(T^\infty \times \mathfrak M\) NEWLINE\[NEWLINE\varphi(n+1)=\varphi(n)+\omega,\quad x(n+1)=P(\varphi(n+p))x(n)+c(\varphi(n+1)).NEWLINE\]NEWLINE The majority of results are obtained on the basis of the same approach: the problem under consideration is first solved for the sequence of truncated finite-dimensional systems of increasing dimension; then the passage to the limit is substantiated.