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

Abstract

In this paper, we study the topological conjugacy between the linear generalized 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 Kurzweil integrable. GODEs are completely different from the classical ODEs. Note that the GODEs in Banach space are defined via its solution. $\frac{dx}{d\tau}$ is only a notation and %$\frac{dx}{d\tau}$ it does not indicate that the solution has a derivative. The solution of the GODEs can be discontinuous and even the number of discontinuous points is countable, so that many classical theorems and tools are no longer applicable to the GODEs.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 valid for the GODEs. In this paper, we study the linearization and its H\"{o}lder continuity of the GODEs. Firstly, we construct the formula for bounded solutions of the nonlinear GODEs in the Kurzweil integral sense. Afterwards, we establish a Hartman-Grobman type linearization theorem which is a bridge connecting the linear GODEs with their nonlinear perturbations. Further, we show that the conjugacies are both H\"{o}lder continuous by using the Gronwall-type inequality (in the Perron-Stieltjes integral sense) and other nontrivial estimate techniques. %The GODEs include measure differential equations, impulsive differential equations, functional differential equations and the classical ordinary differential equations as special cases. Finally, applications to the measure differential equations and impulsive differential equations, our results are very effective.