Electron. J. Diff. Eqns., Vol. 2003(2003), No. 104, pp. 1-23.

A linear functional differential equation with distributions in the input

Vadim Z. Tsalyuk

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 [11] (in this article we review and extend the results in [11]). 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.

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Vadim Z. Tsalyuk
Mathematics Department
Kuban State University
Stavropol'skaya 149, Krasnodar 350040, Russia
email: vts@math.kubsu.ru
http://public.kubsu.ru/vts (in Russian)

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