Boundary Value Problems for Generalized ODEs

Bonotto, Everaldo M.; Federson, M.; Mesquita, Camila Campos

doi:10.1007/s12220-022-01090-z

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“…More recently, the stability theory and the qualitative property of GODEs are developed by many scholars. For example, the variation of constants formula for the GODEs was proposed by Collegari et al [17], the boundedness of solutions for GODEs were derived by Afonso et al [18] and Federson et al [19], the concepts of exponential dichotomy and its robustness results were given by Bonotto et al [20,21], the results on topological properties of flows for nonnegative time in the framework of GODEs were also presented by Bonotto et al [22], the existence of periodic solutions for autonomous GODEs can be seen Federson et al [23], the converse Lyapunov theorem for GODEs were described by Andrade da Silva et al [24] and the boundary value problems for GODEs was obtained from Bonotto et al [25]. Other concepts on the GODEs were considered in the monograph of Bonotto et al [26].…”

Section: Introduction 1history Of the Generalized Ordinary Differenti...mentioning

confidence: 99%

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

Lü¹,

Xia²

2023

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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.

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