Global Existence for Functional Differential Equations with State-Dependent Delay

Our aim in this work is to study the existence of solutions of a functional differential equation with state-dependent delay. We use Schauder's fixed point theorem to show the existence of solutions. 1. Introduction The theory of functional differential equations has emerged as an important branch of nonlinear analysis. Differential delay equations, and functional differential equations, have been used in modeling scientific phenomena for many years. Often, it has been assumed that the delay is either a fixed constant or is given as an integral in which case it is called a distributed delay [1–5]. In 1806, Poisson [6] published one of the first papers on functional differential equations and studied a geometric problem leading to an example with a state-dependent delay (see also [7]). An extensive theory is developed for evolution equations [8, 9]. Uniqueness and existence results have been established recently for different evolution problems in the papers by Baghli and Benchohra for finite and infinite delay in [10–12]. However, complicated situations in which the delay depends on the unknown functions have been considered in recent years. These equations are frequently called equations with state-dependent delay: see, for instance, [3, 13–15]. Existence results were derived recently for functional differential equations when the solution is depending on the delay for impulsive problems. We refer the reader to the papers by Abada et al. [16], Ait Dads and Ezzinbi [17], Anguraj et al. [18], Hartung et al. [19, 20], Hernández et al. [21], and Li et al. [22]. Over the past several years it has become apparent that equations with state-dependent delay arise also in several areas such as in classical electrodynamics [23], in population models [24–27], in models of commodity price fluctuations [28, 29], in models of blood cell productions [30–33], and in drilling [34]. In this work, we prove the existence of solutions of a class of functional differential equations. Our investigations will be situated in the Banach space of real functions which are defined, continuous, and bounded on the real axis . We will use Schauder's fixed point theorem combined with the semigroup theory to have the existence of solutions of the following functional differential equation with state-dependent delay: where is a given function, is the infinitesimal generator of a strongly continuous semigroup is the phase space to be specified later, , , and is a real Banach space. For any function defined on and any we denote by the element of defined by . Here represents the history of