Approximation of the Solutions to Quasilinear Parabolic Problems with Perturbed VMOx Coefficients

We consider the Cauchy-Dirichlet problem for second-order quasilinear operators of parabolic type in non-divergence form. The data are Cara\-thé\-o\-dory functions, and the principal part is of $VMO_x$-type with respect to the variables $ (x,t).$ Assuming the existence of a strong solution $u_0,$ we apply the Implicit Function Theorem in a neighbourhood of this solution to show that small bounded perturbations of the data lead to small perturbations of the solution $u_0$ itself. Furthermore, we employ the Newton Iteration Procedure to construct an approximating sequence that converges to $u_0$ in the corresponding Sobolev space.