Generalized Ordinary Differential Equations in Abstract Spaces and Applications. Группа авторов

of a regulated function

and a time

to a regulated function. The main reference to this chapter is [209]. Measure FDEs in the integral form described above feature in Chapter 4 as supporting actors, because now their solutions can be related to solutions of the generalized ODEs, whose right-hand sides involve functions which look like

      This characteristic of generalized ODEs plays an important role in the entire manuscript, since it allows one to translate results from generalized ODEs to measure FDEs.

      Chapter 5, based on [78], brings together the foundations of the theory of generalized ODEs. Section 5.1 concerns local existence and uniqueness of a solution of a nonautonomous generalized ODE with applications to measure FDEs and functional dynamic equations on time scales. Second 5.2 is devoted to results on prolongation of solutions of generalized ODEs, measure differential equations, and dynamic equations on time scales.

      Chapter 6 deals with a very important class of differential equations, the class of linear generalized ODEs. The origins of linear generalized ODEs goes back to the papers [209–211]. Here, we recall the notion of fundamental operator associated with a linear generalized ODE for Banach space-valued functions and we travel on the same road as the authors of [45] to obtain a variation-of-constants formula for a linear perturbed generalized ODE. Concerning applications, we extend the class of equations to include linear measure FDEs.

      After linear generalized ODEs are investigated, we move to results on continuous dependence of solutions on parameters. This is the core of Chapter 7 which is based on [4, 95, 96, 177]. Given a family of generalized ODEs, we present sufficient conditions so that the family of their corresponding solutions converge uniformly, on compact sets, to the solution of the limiting generalized ODE. We also prove that given a generalized ODE and its solution

, where

is a Banach space, one can obtain a family of generalized ODEs whose solutions converge uniformly to

on

.

As we mentioned before, many types of differential equations can be regarded as generalized ODEs. This fact allows us to derive stability results for these equations through the relations between the solutions of a certain equations and the solutions of a generalized ODEs. At the present time, the stability theory for generalized ODEs is undergoing a remarkable development. Recent results in this respect are contained in [3, 7, 80, 89, 90] and are gathered in Chapter 8. We also show the effectiveness of Lyapunov's Direct Method to obtain several stability results, in addition to proving converse Lyapunov theorems for some types of stability. The types of stability explore here are variational stability, Lyapunov stability, regular stability, and many relations permeating these concepts.

      The existence of periodic solutions to any kind of equation is also of great interest, especially in applications. Chapter 9is devoted to this matter in the framework of generalized ODEs, whose results are also specified to measure differential equations and impulsive differential equations. Section 9.1 brings together a result which provides conditions for the solutions of a linear generalized ODE taking values in

to be periodic and a result which relates periodic solutions of linear nonhomogeneous generalized ODEs to periodic solutions of linear homogeneous generalized ODEs. Still in this section, a characterization of the fundamental matrix of periodic linear generalized ODEs is established. This is the analogue of the Floquet theorem for generalized ODEs involving finite dimensional space-valued functions. In Section 9.2, inspired in an approach by Jean Mawhin to treat periodic boundary value problem (we write periodic BVP for short), we introduce the concept of a

-periodic solution for a nonlinear homogeneous generalized ODE in Banach spaces, where

and

. A result that ensures a correspondence between solutions of a

-periodic BVP and

-periodic solutions of a nonlinear homogeneous generalized ODE is the spotlight here. Then, the existence of a

-periodic solution is guaranteed.

      Averaging methods are used to investigate the solutions of a nonautonomous differential equations by means of the solutions of an “averaged ” autonomous equation. In Chapter 10, we present a periodic averaging principle as well as a nonperiodic one for generalized ODEs. The main references to this chapter are [83, 178].

      Chapter 11 is designed to provide the reader with a systematic account of recent developments in the boundedness theory for generalized ODEs. The results of this chapter were borrowed from the articles [2, 79].

      Chapter 12 is devoted to the control theory in the setting of abstract generalized ODEs. In its first section, we introduce concepts of observability, exact controllability, and approximate controllability, and we give necessary and sufficient conditions for a system of generalized ODEs to be exactly controllable, approximately controllable, or observable. In Section 12.2, we apply the results to classical ODEs.

      The study of exponential dichotomy for linear generalized ODEs of type

      is the heartwood of Chapter 13, where sufficient conditions for the existence of exponential dichotomies are obtained, as well as conditions for the existence of bounded solutions for the nonhomogeneous equation