A linear functional differential equation with distributions in the input
| dc.contributor.author | Tsalyuk, Vadim Z. | |
| dc.date.accessioned | 2021-01-27T19:46:27Z | |
| dc.date.available | 2021-01-27T19:46:27Z | |
| dc.date.issued | 2003-10-13 | |
| dc.description.abstract | 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). | |
| dc.description.department | Mathematics | |
| dc.format | Text | |
| dc.format.extent | 23 pages | |
| dc.format.medium | 1 file (.pdf) | |
| dc.identifier.citation | 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. | |
| dc.identifier.issn | 1072-6691 | |
| dc.identifier.uri | https://hdl.handle.net/10877/13155 | |
| dc.language.iso | en | |
| dc.publisher | Southwest Texas State University, Department of Mathematics | |
| dc.rights | Attribution 4.0 International | |
| dc.rights.holder | This work is licensed under a Creative Commons Attribution 4.0 International License. | |
| dc.rights.uri | https://creativecommons.org/licenses/by/4.0/ | |
| dc.source | Electronic Journal of Differential Equations, 2003, San Marcos, Texas: Southwest Texas State University and University of North Texas. | |
| dc.subject | Stieltjes integral | |
| dc.subject | Function of bounded variation | |
| dc.subject | Multivalued integral | |
| dc.subject | Linear functional differential equation | |
| dc.title | A linear functional differential equation with distributions in the input | |
| dc.type | Article |