Positive Solutions of the Equation ẋ(t)=−c(t)x(t−τ) in the Critical Case

Abstract

Ž .Ž . Ž . The equation x t y c t x t y is considered in the critical case. For it, thė asymptotic behavior of dominant and subdominant solutions is studied. A generalization is made and connections with known results are discussed. ᮊ 2000 Academic Press

Existence of positive solutions of delay differential equations

“…In this section we shall investigate the existence of positive solutions for equation (1). The main result is in the following theorem.…”

“…It is due mainly to the technical difficulties arising in analysis of the existence problem. We refer the reader to [1]- [10] and the references cited therein. However, mostly in the papers the investigation of the existence of nonoscillatory solutions which are bounded by the positive functions, absents.…”

Sufficient conditions for the existence of global solutions of delayed differential equations

“…Many recent papers are devoted to the investigation of asymptotic properties of solutions of delayed functional differential equations for t → +∞ (let us cite, at least the papers [1][2][3][4][5][6][7][8]11] and the book [10]). Often such investigations are in a sense local since the corresponding properties of the solutions are stated "only" on a neighborhood of the point +∞.…”

Subdominant positive solutions of the discrete equation Δu(k + n) = −p(k)u(k)

“…In [6], it was investigated that if (1.2) admits a positive solutionx on an interval I, then it admits on I two positive solutions x 1 and x 2 , satisfying Moreover, every solution x of (1.2) on I is represented by the formula x(t) = Kx 1 (t) + O x 2 (t) , (1.4) where K ∈ R depends on x and O is the Landau order symbol. In this formula, the solutions x 1 , x 2 can be changed to any couple of positive on I solutionsx 1 ,x 2 of (1.2) satisfying the property lim t→∞x 2 (t) x 1 (t) = 0 (1.5) (see [6, pages 638-639]).…”