TY - JOUR

T1 - Solution estimates for linear differential equations with delay

AU - Berezansky, Leonid

AU - Braverman, Elena

N1 - Funding Information:
The second author acknowledges the support of NSERC , the grant RGPIN-2015-05976 . Both authors are grateful to the reviewer for valuable comments and suggestions.
Publisher Copyright:
© 2019 Elsevier Inc.

PY - 2020/5/1

Y1 - 2020/5/1

N2 - In this paper, we give explicit exponential estimates |x(t)|≤Me−γ(t−t0), where t ≥ t0, M > 0, for solutions of a linear scalar delay differential equation x˙(t)+∑k=1mbk(t)x(hk(t))=f(t),t≥t0,x(t)=ϕ(t),t≤t0.We consider two different cases: when γ > 0 (corresponding to exponential stability) and the case of γ < 0 when the solution is, generally, growing. In the first case, together with the exponential estimate, we also obtain an exponential stability test, in the second case we get estimation for solution growth. Here both the coefficients and the delays are measurable, not necessarily continuous.

AB - In this paper, we give explicit exponential estimates |x(t)|≤Me−γ(t−t0), where t ≥ t0, M > 0, for solutions of a linear scalar delay differential equation x˙(t)+∑k=1mbk(t)x(hk(t))=f(t),t≥t0,x(t)=ϕ(t),t≤t0.We consider two different cases: when γ > 0 (corresponding to exponential stability) and the case of γ < 0 when the solution is, generally, growing. In the first case, together with the exponential estimate, we also obtain an exponential stability test, in the second case we get estimation for solution growth. Here both the coefficients and the delays are measurable, not necessarily continuous.

KW - Explicit solution estimates

KW - Exponential stability

KW - Linear delay differential equations

KW - Variable delays and coefficients

UR - http://www.scopus.com/inward/record.url?scp=85077073616&partnerID=8YFLogxK

U2 - 10.1016/j.amc.2019.124962

DO - 10.1016/j.amc.2019.124962

M3 - Article

AN - SCOPUS:85077073616

SN - 0096-3003

VL - 372

JO - Applied Mathematics and Computation

JF - Applied Mathematics and Computation

M1 - 124962

ER -