Given any sequence z = (z n) n≥1 of positive real numbers and any set E of complex sequences, we write Ez for the set of all sequences y = (y n) n≥1 such that y/z = (y n /z n) n≥1 ∈ E; in particular, c z = s (c) z denotes the set of all sequences y such that y/z converges. Starting with the equation F x = F b we deal with some perturbed equation of the form ε + F x = F b , where ε is a linear space of sequences. In this way we solve the previous equation where ε = (Ea)T and (E; F) ∈ {(l∞, c) ; (c0, l∞) ; (c0, c) ; (l^p, c) ; (l^p, l∞) ; (w0, l∞)} with p ≥1, and T is a triangle.