A linear functional differential equation with distributions in the input
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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  (in this article we review and extend the results in ). 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).
CitationTsalyuk, V. Z. (2003). A linear functional differential equation with distributions in the input. Electronic Journal of Differential Equations, 2003(104), pp. 1-23.
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