Volterra and Functional Differential Equations

A PARAMETER DEPENDENCE PROBLEM IN FUNCTIONAL DIFFERENTIAL EQUATIONS

Dennis W. Brewer

Department of Mathematics

University of Arkansas

Fayetteville, Arkansas

DOI: 10.1201/9781003420026-13

1. INTROOUCTION

We consider the differentiability with respect to a parameter p of solutions of the linear functional differential equation

$$\begin{array}{ll}\hfill & {\text{x}}^{\prime }\left(\text{t}\right)=\text{L}\left(\text{p}\right){\text{x}}_{\text{t}},\text{ t}\ge 0\hfill \\ \hfill & \text{x}\left(0\right)=\eta ,{\text{ x}}_0=\varphi ,\hfill \end{array}$$ (1.1)

where x_t (s)= x(t+s),t ≥ 0,s ≤ ɸ ∈ L¹ (-∞,0) with values in, ₵ⁿ,ƞ ∈ ₵ⁿ. and, for each purameter p in an open subset P of a normed linear psrameter space P, L(p) is a linear functional from a subset of l¹ (-∞,0) into ₵ⁿ. By a solution of (1,1) we mean a function in L¹ (−∞,t) for each t₁>0 which is absolutely continuous ...