A Map-type Gronwall Inequality on Functional Differential Equations with State-dependence

In this paper, in order to investigate a Gronwall inequality with statedependence, another auxiliary map-type Gronwall inequality is discussed by modifying the technique of sequential monotonization on functions into the one on maps. Then we employ the state-dependent Gronwall inequality to give the estimate and boundedness of solutions for a functional differential equation with state-dependence. Finally, we exhibit a concrete example of bounded solutions as application.