Electronic Journal of Differential Equations,
Vol. 2003(2003), No. 104, pp. 1-23.
Title: A linear functional differential equation
with distributions in the input
Author: Vadim Z. Tsalyuk (Kuban State University, Russia)
Abstract:
This paper studies the functional differential equation
$$
\dot x(t) = \int_a^t {d_s R(t,s)\, x(s)} + F'(t), \quad t \in [a,b],
$$
where $F'$ is a generalized derivative, and
$R(t,\cdot)$ and $F$ are functions of bounded variation.
A solution is defined by the difference $x - F$
being absolutely continuous and satisfying the inclusion
$$
\frac{d}{dt} (x(t) - F(t)) \in \int_a^t {d_s R(t,s)\,x(s)}.
$$
Here, the integral in the right is the multivalued Stieltjes integral
presented in \cite{VTs1} (in this article we review and extend the results
in \cite{VTs1}).
We show that the solution set for the initial-value problem is
nonempty, compact, and convex.
A solution $x$ is said to have memory if there exists the
function $\bar x$ such that $\bar x(a) = x(a)$, $\bar x(b) = x(b)$,
$\bar x(t) \in [x(t-0),x(t+0)]$ for $t \in (a,b)$, and
$\frac{d}{dt} (x(t) - F(t)) = \int_a^t {d_s R(t,s)\, \bar{x}(s)}$,
where Lebesgue-Stieltjes integral is used.
We show that such solutions form a nonempty, compact, and convex
set. It is shown that solutions with memory obey the Cauchy-type
formula
$$
x(t) \in C(t,a)x(a) + \int_a^t C(t,s)\, dF(s).
$$
Submitted June 14, 2003. Published October 13, 2003.
Math Subject Classifications: 26A42, 28B20, 34A60.
Key Words: Stieltjes integral; function of bounded variation;
multivalued integral; linear functional differential equation.