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

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.

Show me the PDF file (285K), TEX file, and other files for this article.

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)

Return to the EJDE web page