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In this paper, we study the topological conjugacy between the lineargeneralized ODEs (for short, GODEs) \[  \frac{dx}{d\tau}=D[A(t)x] \] and their nonlinear perturbation \[  \frac{dx}{d\tau}=D[A(t)x+F(x,t)] \] on Banach space $\mathscr{X}$, where$A:\mathbb{R}\to\mathscr{B}(\mathscr{X})$ is a bounded linear operator on$\mathscr{X}$ and $F:\mathscr{X}\times \mathbb{R}\to \mathscr{X}$ is Kurzweilintegrable. GODEs are completely different from the classical ODEs. Note thatthe GODEs in Banach space are defined via its solution. $\frac{dx}{d\tau}$ isonly a notation and %$\frac{dx}{d\tau}$ it does not indicate that the solutionhas a derivative. The solution of the GODEs can be discontinuous and even thenumber of discontinuous points is countable, so that many classical theoremsand tools are no longer applicable to the GODEs.  For instances, the chain rule and the multiplication rule of derivatives,differential mean value theorem, and integral mean value theorem are not validfor the GODEs. In this paper, we study the linearization and its H\"{o}ldercontinuity of the GODEs.  Firstly, we construct the formula for bounded solutions of the nonlinearGODEs in the Kurzweil integral sense.  Afterwards, we establish a Hartman-Grobman type linearization theorem whichis a bridge connecting the linear GODEs with their nonlinear perturbations.Further, we show that the conjugacies are both H\"{o}lder continuous by usingthe Gronwall-type inequality (in the Perron-Stieltjes integral sense) and othernontrivial estimate techniques. %The GODEs include measure differentialequations, impulsive differential equations, functional differential equationsand the classical ordinary differential equations as special cases.  Finally, applications to the measure differential equations and impulsivedifferential equations, our results are very effective.

Linearization and H\"{o}lder continuity of generalized ODEs with application to measure differential equations