Linear Functional Differential Equations as Semigroups on Product Spaces

Abstract

In this paper we consider the well-posedness of linear functional differential equations on product spaces. Let L and D be linear $\mathbb{R}^n $-valued functions with domains $\mathfrak{D}(L)$ and $\mathfrak{D}(D)$ subspaces of the Lebesgue measurable $\mathbb{R}^n $-valued functions on $[ - r,0]$ and such that $W^{1,p} ([ - r,0];\mathbb{R}^n ) \subseteq \mathfrak{D}(L) \cap \mathfrak{D}(D)$. Under weak conditions on D and L we establish the equivalence between generalized solutions to the functional differential equation \[ \frac{d}{{dt}}Dx_t = Lx_t + f(t)\] and mild solutions to the Cauchy problem in $\mathbb{R}^n \times L_p ([ - r,0];\mathbb{R}^n )$\[ \dot z(t) = \mathfrak{a} z(t) + (f(t),0), \] where $\mathfrak{a}$ is the operator defined on \[ \mathfrak{D}(\mathfrak{a}) = \left\{ {(\eta ,\varphi ) \in \mathbb{R}^n \times L_p {{\left( {[ - r,0];\mathbb{R}^n } \right)} / \varphi } \in W^{1,p} \left( {[ - r,0];\mathbb{R}^n } \right),D\varphi = \eta } \right\},\] by \[ \mathfrak{a}(\eta ,\varphi ) = (L\varphi ,\dot \varphi ). \] The results are applicable to neutral functional differential equations and certain singular integral equations.