The paper deals with the mappings of Banach space E given in a form of quasilinear difference equation x_(n+1) = Ax_n + F_n(x_n), n ≥0 (1) where A is linear continuous operator, {F_n : E → E} are nonlinear bounded operators satisfying identity F_n(0) ≡ 0. Side by side with the above equation we consider an equation of the first approximation, that is, the linear difference equation y_(n+1)= Ay_n, n ≥0 (2) We will discuss the assertions which guarantee local stability or instability for the trivial solution of (1) if (2) to be of this specificity. The proposal paper not only generalizes well known finite dimensional stability analysis results for quasilinear difference equations. Using spectral properties of operator A as a basis, our research shows that the infinite dimension of the space E not only strongle complicates computations and proofs of relevant theorems on stability analysis by the first approximation but also can have significant influence to statement of these results.