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.