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**On the linearization theorem of Fenner and Pinto.**
*(English)*
Zbl 1272.34048

Summary: This paper reports an improvement of the linearization theorem of J. Lopez Fenner and M. Pinto [Nonlinear Anal., Theory Methods Appl. 38, No.3, A, 307–325 (1999; Zbl 0931.34007)]. 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 their paper. 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 mentioned paper by Fenner and Pinto. They obtained the linearization theorem by setting that the whole linear system should satisfy the integrable summable condition (IS condition). In this paper, this assumption is reduced. In fact, it is enough to assume that the linear system partially satisfies the IS condition. Therefore, we improve the linearization theorem of Fenner and Pinto.

### MSC:

34C20 | Transformation and reduction of ordinary differential equations and systems, normal forms |

34A37 | Ordinary differential equations with impulses |

34D09 | Dichotomy, trichotomy of solutions to ordinary differential equations |

34C41 | Equivalence and asymptotic equivalence of ordinary differential equations |