Convergent solutions of certain nonlinear differential equations with maxima

We study nonlinear differential equations with maxima of the form {x′(t)=f(t,x(t),maxu∈Itx(u)),witht∈Ix(t)=χ(t),witht∈[t0−h,t0].We prove that this equation has the property of asymptotic equilibrium, and we give an asymptotic representation for the solutions of this differential equation. Moreover, we prove the stability of the convergent solutions. In the given examples we can apreciate that the asymptotic representation of the solutions is satisfied for sufficiently large values of tt. However, the existence of the asymptotic solution can be obtained for all t∈[−h,∞)t∈[−h,∞) by assuming some more strict hypotheses, i.e., for initial conditions small enough.