Vadim Z. Tsalyuk

This paper studies the functional differential equation

where is a generalized derivative, and and are functions of bounded variation. A solution is defined by the difference being absolutely continuous and satisfying the inclusion

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 is said to have memory if there exists the function such that , , for , and , 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

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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