A linear functional differential equation with distributions in the input




Tsalyuk, Vadim Z.

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Southwest Texas State University, Department of Mathematics


This paper studies the functional differential equation ẋ(t) = ∫tα dsR(t, s)x(s) + F'(t), t ∈ [α, b], where F' is a generalized derivative, and R(t, ∙) and F are functions of bounded variation. A solution is defined by the difference x - F being absolutely continuous and satisfying the inclusion d/ dt (x(t) - F(t)) ∈ ∫tα dsR(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 x̄ such that x̄(α) = x(α), x̄(b) = x(b), x̄(t) ∈ [x(t - 0), x(t + 0)] for t ∈ (α, b), and d/dt (x(t) - F(t)) = ∫tα dsR(t, s) x̄(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) ∈ C(t, α)x(α) + ∫tα C(t, s) dF(s).



Stieltjes integral, Function of bounded variation, Multivalued integral, Linear functional differential equation


Tsalyuk, V. Z. (2003). A linear functional differential equation with distributions in the input. <i>Electronic Journal of Differential Equations, 2003</i>(104), pp. 1-23.


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