Continuous solutions of systems of nonlinear difference equations (Q726446)

Consider the difference system \[ x(t+1) = A(t)x(t) + F(t,x(t)) \] with \(A(\cdot)\) nonsingular and \(N\)-periodic, \(F\cdot:\mathbb R\times \mathbb R^n\to \mathbb R^n\) being \(C^k\) in \(x\) and \(N\)-periodic in \(t\). For this system it is assumed to be such that the change of variables \[ x = C(t)y \] gives it the form \[ y(t+1) = \Lambda(t)y(t) + \bar{F}(t,y(t)) \] with \(\Lambda(t)\) being continuous, 1-periodic and diagonal. Vector functions of the form \[ \gamma(t,y) = y + \sum_{|i|=2}^k\gamma_i(t)y^i \] are investigated, where \(i=(i_1,\dots,i_n)\), \(|i|=i_1+\dots+i_n\), \(y^i = y_1^{i_1}\dots y_n^{i_n}\), \(\gamma_i(t)\) being \(N\)-periodic, which can be solutions of the transformed system.