Journal of Inequalities and Applications
Volume 2007 (2007), Article ID 52304, 22 pages
doi:10.1155/2007/52304
Research Article
On the Kneser-Type Solutions for Two-Dimensional Linear
Differential Systems with Deviating Arguments
1Department of Mathematics and Computer Sciences, The Academic College of Judea and Samaria, Ariel 44837, Israel
2Department of Mathematics, University of Tbilisi, University Street 2, Tbilisi 0143, Georgia
Received 7 January 2007; Revised 26 March 2007; Accepted 25 April 2007
Academic Editor: Alberto Cabada
Copyright © 2007 Alexander Domoshnitsky and Roman Koplatadze. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.
Abstract
For the differential system u1'(t)=p(t)u2(τ(t)), u2'(t)=q(t)u1(σ(t)), t∈[0,+∞), where p,q∈Lloc(ℝ+;ℝ+), τ,σ∈C(ℝ+;ℝ+), limt→+∞τ(t)=limt→+∞σ(t)=+∞, we get necessary and sufficient conditions that this system does not have solutions satisfying the condition
u1(t)u2(t)<0 for t∈[t0,+∞). Note one of our results obtained for this system with constant coefficients
and delays (p(t)≡p,q(t)≡q,τ(t)=t−Δ,σ(t)=t−δ, where δ,Δ∈ℝ and Δ+δ>0). The inequality (δ+Δ)pq>2/e is necessary and sufficient for nonexistence of solutions satisfying this condition.