On the existence of an invariant torus of a countable system of difference equations defined on an infinite-dimensional torus containing deviation of discrete independent variable (Q2768790)

The author obtains sufficient conditions for the existence of an invariant torus for the system of equations \(\phi_{n+1}=\phi_{n}+a(\phi_{n},\mu)\), \(x_{n+1}=P(\phi_{n+p},\mu,x_{n+k})x_{n}+c(\phi_{n+g+1},\mu)\), where \(\phi=(\phi^1,\phi^2,\phi^3,\ldots)\in M\), \(x=(x^1,x^2,x^3,\ldots)\in M\), \(M\) is the space of bounded number sequences with the norm \(\|x\|=\sup_{i}\{|x^{i}|\}\); the functions \(a(\phi,\mu)=\{a_1(\phi,\mu),a_2(\phi,\mu),\ldots\}\), \(c(\phi,\mu)=\{c_1(\phi,\mu),c_2(\phi,\mu),\ldots\}\) and the infinite-dimensional matrix \(P(\phi,\mu,x)=\{p_{ij}(\phi,\mu,x)\}_{i,j=1}^{\infty}\) are real, periodic on \(\phi^{i}\) \((i=1,2,\ldots)\) with period \(2\pi\); \(p,g,k\) are integer parameters of deviation; \(\mu\in S\subset M\) is parameter; \(S\) is a unit sphere in \(M\).