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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 {\it J. Lopez Fenner} and {\it 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:
34C20Transformation and reduction of ODE and systems, normal forms
34A37Differential equations with impulses
34D09Dichotomy, trichotomy
34C41Equivalence, asymptotic equivalence
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