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A linear system of delayed differential equations with multiple delays x ̇ (t) = − ∑ i = 1 s c i (t) A i (t) x (t − τ i (t)) , t ∈ [ t 0 , ∞) , is considered where x is an n -dimensional column vector, t 0 ∈ R , s is a fixed integer, delays τ i are positive and bounded, entries of n by n matrices A i as well as functions c i are nonnegative, and the sums of columns of the matrix A i (t) are identical and equal to a function α i (t). It is proved that, on [ t 0 , ∞) , the system has a solution with positive coordinates if and only if the scalar equation y ̇ (t) = − ∑ i = 1 s c i (t) α i (t) y (t − τ i (t)) , t ∈ [ t 0 , ∞) , has a positive solution. Some asymptotic properties of solutions related to both equations are also discussed. Illustrative examples are considered and some open problems formulated. • System of delayed differential equations. • Scalar delayed differential equations. • Multiple delays. • Novel positivity criterion. • Topological principle. [ABSTRACT FROM AUTHOR]

In this paper, we study the second-order linear delay differential equation of the form (E) y ′ ′ (t) + p (t) y (τ (t)) = 0. We establish new oscillation criteria for (1.1), which improve a number of related ones in the literature. Our approach essentially involves establishing stronger monotonicity properties for the positive solutions of (1.1) than those presented in known works. Two main approaches for the investigation of (1.1) will be used, namely a comparison principle with first-order delay differential inequalities and generalization of very effective Koplatadze's technique. We illustrate the improvement over the known results by applying and comparing our method with the other known results for (1.1). [ABSTRACT FROM AUTHOR]

The paper studies a class of linear differential equations with several delayed arguments formed by iterates of a given function. The main result of this paper improves the existing stability criteria and formulates an effective necessary and sufficient condition relating stability of the studied differential equations to stability of some auxiliary difference equations. In the case of a two-delay equation, this condition is presented explicitly in terms of the equation's parameters. As an accompanying result, the asymptotic decay rate of the solutions is described as well. • A class of delay differential equations with commensurate delayed arguments is considered. • An effective (optimal) asymptotic stability criterion for this class is derived. • An asymptotic estimate of the solutions is described as well. • Comparisons with the existing results are provided. [ABSTRACT FROM AUTHOR]