A General Initial Value Problem

Abstract

Suppose r ≥ O is a given real number, R = (−∞,∞), Rⁿ is a real or complex n-dimensional linear vector space with norm |•|, C([a,b],Rⁿ) is the Banach space of continuous functions mapping the interval [a,b] into Rⁿ with the topology of uniform convergence. If [a,b] = [−r,0] we let C = C([−r,0],Rⁿ) and designate the norm of an element ϕ in C by $\left| {\rm{\phi }} \right|{\rm{ = sup}}_{{\rm{ - r}} \le \theta \le {\rm{0}}} \left| {{\rm{\phi }}\left( \theta \right)} \right|$. Even though single bars are used for norms in different spaces, no confusion should arise. If σ ∈ R, A ≥ 0 and x ∈ C([σ−r,σ+A],Rⁿ), then for any t ∈ [σ,σ+A], we let x_t ∈ C be defined by X_t(θ) = x(t+θ), −r ≤ θ ≤ 0. If · = d/dt and f: R × C→Rⁿ is a given function, we say that the relation

$${\rm{\dot x}}\left( {\rm{t}} \right) = {\rm{f}}\left( {{\rm{t,x}}_{\rm{t}} } \right)$$

((2.1))

is a functional differential equation of retarded type or simply a functional differential equation. A function x is said to be a solution of (2.1) if there are σ ∈ R, A > 0 such that x ∈ C([σ−r,σ+A),Rⁿ) and x(t) satisfies (2.1) for t ∈ (σ,σ + A). In such a case, we say x is a solution of (2.1) on [σ−r,σ+ A). For a given σ ∈ R and a given ϕ ∈ C we say x = x(cr,cp) is a solution of (2.1) with initial value ϕ at σ or simply a solution of (2.1) through (σ,ϕ) if there is an A >0 such that x(σ,ϕ) is a solution of (2.1) on [σ−r,σ+A) and x_σ(σ,ϕ) = ϕ.