On the linearization theorem of Fenner and Pinto

This paper reports an improvement of the linearization theorem of Fenner and Pinto (1999) [22]. Fenner and Pinto presented a version of Hartman’s result. They showed that there exists a one-to-one correspondence between solutions of the linear system and the nonlinear system. Moreover, if H(t,x)H(t,x) denotes the transformation, then H(t,x)−xH(t,x)−x is uniformly bounded. However, no proof of the Hölder regularity of the transformation H(t,x)H(t,x) appears in Fenner and Pinto (1999) [22]. The main objective in this paper is precisely to give a proof of the Hölder regularity of the transformation H(t,x)H(t,x). Namely, we show that the conjugating function H(t,x)H(t,x) in the Hartman–Grobman theorem, is always Hölder continuous (and has Hölder continuous inverse). Moreover, we weakened an important assumption in Fenner and Pinto (1999) [22]. Fenner and Pinto obtained the linearization theorem by setting that the whole linear system should satisfy IS condition. In this paper, this assumption is reduced. In fact, it is enough to assume that the linear system partially satisfies IS condition. Therefore, we improve the linearization theorem of Fenner and Pinto.