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

Alexander Domoshnitsky1 and Roman Koplatadze2

 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

Recommended by Alberto Cabada

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.