Chapter 8 - Delay differential equations

Abstract

In this chapter, we introduce the concept of delay differential equations. We focus on the existence and uniqueness of solutions and introduce the step method to solve certain delay differential equations on bounded intervals. Then we adjust the methods of the previous chapter concerning Lyapunov functions to suit delay differential equations. Once we do that, we make use of Lyapunov functions to study the stability and existence of solutions. Toward the end of the chapter, we use the fixed point theory on Banach spaces developed in Chapter 2 to deduce the boundedness, existence of periodic and positive periodic solutions, and stability for neutral delay differential equations. We end the chapter by utilizing Lyapunov functionals and obtain the exponential stability for totally nonlinear finite delay integro-differential equations.