Permanence of equilibrium points in the basin of attraction and existence of periodic solutions for autonomous measure differential equations and dynamic equations on time scales via generalized ODEs

Abstract

It is well-known that generalized ODEs encompass several types of differential equations as, for instance, functional differential equations, measure differential equations, dynamic equations on time scales, impulsive differential equations and any combinations among them, not to mention integrals equations, among others. The aim of this paper is to establish a theory of autonomous equations in the setting of generalized ODEs. Thus, we introduce the notion of autonomous generalized ODEs as well as new classes of right-hand sides for nonautonomous generalized ODEs. Amongst the main results, we prove that one of these new classes coincide with the original class of right-hand sides introduced by Kurzweil in 1957. We also prove that autonomous generalized ODEs do not enlarge the class of autonomous ODEs with uniformly continuous right-hand sides. Motivated by this fact, we then enlarge the class of autonomous generalized ODEs so that discontinuities can be taken into account. We then introduce a more general class of autonomous generalized ODEs, in whose integral form, a Stieltjes-type integral appears. A correspondence between these equations and autonomous measure differential equations is established and several results are obtained. We mention local existence and uniqueness of solutions, continuous dependence of solutions on initial values, existence of periodic solutions and permanence of asymptotically stable equilibrium point in the basin of attraction. All these results are, then, specified not only for autonomous generalized ODEs, but also for autonomous measure differential equations and dynamic equations on time scales.