A nonautonomous $C^r-$topological equivalence involving contractions and unbounded nonlinearities

We study the smoothness of the topological equivalence between a linear equation and its nonlinear perturbation, which is regarded as unbounded. To the best of our knowledge, it has not previously been considered such study in the literature. Therefore, the main result of this work copes with this lack, that is, it is shown, on the positive half line, that such topological equivalence is of class $C^r(r\ge 1)$ when the linear part is a uniform contraction and the nonlinearities considered are unbounded.