Jiří Šremr

Some Remarks on Functional Differential Equations in Abstract Spaces

abstract:
The aim of this paper is to present some remarks concerning the functional differential equation
$$ v'(t)=G(v)(t) $$
in a~Banach space $\mathbb{X}$, where $G:\cabx\to\babx$ is a continuous operator and $\cabx$, resp. $\babx$, denotes the Banach space of continuous, resp. Bochner integrable, abstract functions.

It is proved, in particular, that both initial value problems (Darboux and Cauchy) for the hyperbolic functional differential equation
$$ \frac{\partial^2 u(t,x)}{\partial t\,\partial x}=F(u)(t,x) $$
with a Carathéodory right-hand side on the rectangle $[a,b]\times[c,d]$ can be rewritten as initial value problems for abstract functional differential equation with a suitable operator $G$ and $\mathbb{X}=\ccdr$.

Mathematics Subject Classification: 34Gxx, 34Kxx, 35Lxx

Key words and phrases: Functional differential equation in a Banach space, hyperbolic functional differential equation, initial value problem