Linearization and Hölder continuity of generalized ODEs with application to measure differential equations

Abstract

The generalized ordinary differential equations (for short, GODEs) in Banach space were defined via its solution, which include measure differential equations, impulsive differential equations, functional differential equations and the classical ordinary differential equations as special cases. It should be mentioned that even the symbol dx dτ does not indicate that the solution has a derivative. In this paper, we study the linearization and its Hölder continuity of a class of GODEs. Firstly, we construct the formulas for bounded solutions of the nonlinear GODEs in the Kurzweil integral sense under the linear GODEs have an exponential dichotomy. Afterwards, we establish a topological conjugacy between the linear and nonlinear GODEs. Further, we show that the conjugacies are both Hölder continuous by using the Gronwall-type inequality (in the Perron-Stieltjes integral sense) and other nontrivial estimate techniques. Finally, applications to the measure differential equations and impulsive differential equations, our results are very effective.