Nonlinear boundary value problems for discontinuous delayed differential equations

Abstract

In this paper nonlinear boundary value problems for discontinuous delayed differential equations are considered. Some existence and boundedness results of solutions are obtained via the method of upper and lower solutions, which may be discontinuous. Our analysis can be applied to those phenomena from physics and control theory which have been successfully described by delayed differential equations with discontinuous functions, such as electric, pneumatic, and hydraulic networks. It is also an important step to precede the design of control signals when finite-time or practical stability control are concerned. §1 IntroductionIn this paper we deal with the following nonlinear boundary value problem for a discontinuous delayed differential equationwhere f satisfies the well-known Carathéodory conditions and B : C × BV ([σ, σ + A]) → R is continuous with respect to both of its variables. We denote by BV ([σ, σ + A]) the space of functions of bounded variation on [σ, σ + A].We first give in this section the appropriate generalization to delayed differential equations of the well-known Carathéodory conditions of ordinary differential equations(ODEs for short), which was first proposed by Carathéodory. Consider the initial value problem of scalar delayed differential equationThe system (2) is said to be a Carathéodory system if the function f : [σ, σ + A] × C → R satisfies the Carathéodory conditions :