On quotient-convergence factors (Q1359574)

The problem \[ y_{n+1}= \lambda\cdot y_n,\qquad y_0=x, \] and the recurrence \[ x_{n+1}= \lambda\cdot x_n+\Omega x_n, \qquad x_0=x, \] \((n=0,1,2,\dots)\) are considered where \(x_n\), \(y_n\) belong to a certain linear space \(X\) over the field of real numbers \(\mathbb{R}\), \(\Omega\) is a linear operator on \(X\) and \(\lambda\in \mathbb{R}\) is fixed. It is expected then that the behaviour of the perturbed sequence \(\{x_n\}\) resembles that of \(\{y_n\}\).