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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]

Abstract A new criterion for the existence of positive solutions of the second-order delayed differential equation y ̈ (t) = f (t , y t , y ̇ t) , t ∈ [ t 0 , ∞) is given with applications to linear equations. Open problems for future research are formulated. [ABSTRACT FROM AUTHOR]

Abstract There exist a well-developed stability theory for neutral differential equations of the first order and only a few results on functional differential equations of the second order. One of the aims of this paper is to fill this gap. Explicit tests for stability of linear neutral delay differential equations of the second order are obtained. [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]

In this paper, we establish some new criteria for oscillation and asymptotic behavior of solutions of a certain class of third-order neutral differential equations with continuously distributed delay. We study the case of noncanonical equations subject to various conditions. An example is given to illustrate the main results. [ABSTRACT FROM AUTHOR]

An n -th order delayed differential equation y (n) (t) = f (t , y t , y t ′ , ... , y t (n − 1) ) is considered, where y t (θ) = y (t + θ) , θ ∈ [ − τ , 0 ] , τ > 0 , if t → ∞. A criterion is formulated guaranteeing the existence of a solution y = y (t) in a cone 0 < (− 1) i − 1 y (i − 1) (t) < (− 1) i − 1 φ (i − 1) (t) , i = 1 , ... , n where φ is an n -times continuously differentiable function such that 0 < (− 1) i φ (i) (t) , i = 0 , ... , n. The proof is based on a similar result proved first for a system of delayed differential equations equivalent in a sense. Particular linear cases are considered and an open problem is formulated as well. [ABSTRACT FROM AUTHOR]

We investigate the oscillation of a first order delay differential equation with non-negative coefficient and non-monotone arguments. New oscillation criterion of lim sup type is obtained. An example is given to show the applicability and strength of the obtained condition over known ones. [ABSTRACT FROM AUTHOR]

In this article, we study the existence of nonoscillatory solutions of higher order differential and delay differential equations with forcing term. Some new sufficient conditions are given. We use the Schauder’s fixed point theorem to prove our results. [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]

This paper establishes the basic structure of the exponential Euler difference form for Caputo–Fabrizio fractional-order differential equations (CF-FODEs) with multiple lags. The research shows that the acquired difference form (i.e., discrete-time CF-FODEs) belongs to the scope of implicit Euler differences and then the fractional PECE algorithm is proposed to solve this implicit difference. At last, global dynamics of the discrete-time CF-FODEs is discussed from the viewpoint of control theory. [ABSTRACT FROM AUTHOR]

Abstract: This work deals with the existence of positive -periodic solutions for the delay differential equations. The main results are illustrated with several examples. [Copyright &y& Elsevier]

Abstract: In this paper, the delay-dependent dissipativity of nonlinear delay differential equations is studied. A new dissipativity criterion is derived, which is less conservative than those in the existing literature in some cases, especially for equations with small delays. [Copyright &y& Elsevier]

Abstract: In this paper a single species dynamics governed by the logistic type delayed equation with an external influence (more precisely, suppression) on the population size is studied. Conditions guaranteeing global stability of the zero steady state are proved. These conditions are necessary and sufficient for a periodic or quasi-periodic suppression rate. Moreover, if the external influence is periodic and the zero steady state is repulsive, the existence of a periodic solution is shown. From the applications point of view, the logistic type equations are often used in the description of underlying tumour dynamics. In this case the external influence reflects the tumour treatment and the global stability of the zero steady state can be interpreted as the cure. [Copyright &y& Elsevier]

Abstract: This paper is concerned with an impulsive neutral differential equation of Euler form with unbounded delays Sufficient conditions are obtained for every solution of (∗) to tend to a constant as . [Copyright &y& Elsevier]

Abstract: The asymptotic behavior of the bounded solutions of the delayed system is investigated, where is a given constant, , , is strictly increasing on , and either for all or for all . It is shown that every bounded solution of the system tends to a constant vector as . The result obtained extends the existing ones in the literature. [Copyright &y& Elsevier]

Abstract: This work is concerned with the existence of positive solutions for the neutral delay differential equation with positive and negative coefficients where . [Copyright &y& Elsevier]

The representation and behavior is investigated of the solutions to an equation with a single constant delay x ̇ (t) = − c (t) x (t − τ) , t ≥ t 0 ≥ 0 with c a positive function provided that it has a positive solution. A scheme is suggested for determining the representation of solutions defined by arbitrary initial function if dominant and subdominant solutions are known. Applications are given to equations with constant and perturbed coefficients with open problems included for future investigation. • Delayed differential equation. • Adjoint advanced differential equation. • Positive solution. • Duality. [ABSTRACT FROM AUTHOR]

Abstract: The numerical approximation of neutral dissipative initial value problems with piecewise constant delay by fixed time stepping Runge–Kutta methods is considered and two sufficient conditions for the dissipativity of the system are given. The concept of (weak) -dissipativity is introduced. It is shown that under some assumptions a -irreducible, algebraically stable Runge–Kutta method is (weakly) -dissipative. [Copyright &y& Elsevier]

Abstract: For a scalar delay differential equation , we obtain new explicit conditions for the existence of a positive solution. [Copyright &y& Elsevier]

Abstract: Consider the neutral delay difference equation with positive and negative coefficients, where and and . Some sufficient conditions for the existence of a nonoscillatory solution of the above equation expressed in terms of are obtained. [Copyright &y& Elsevier]

Abstract: The second order nonlinear delay differential equation with periodic coefficients is considered in this work. By using Krasnoselskii’s fixed point theorem and the contraction mapping principle, we establish some criteria for the existence and uniqueness of periodic solutions to the delay differential equation. [Copyright &y& Elsevier]

Abstract: In this letter we consider a linear differential system with multiple delays which has nonisolated equilibria. In order to study the asymptotic behavior of linear delay differential equations, characteristic equations are generally used. But it is hard to establish the properties of zeros of the characteristic equations, especially if there are multiple time delays. So we use the invariance principle combined with two functionals to show whether any solutions converge. One of the functionals plays the role of a Lyapunov functional, and the other is a conserved quantity. Furthermore we give explicit expressions for the limits of the solutions by using the conserved quantity. [Copyright &y& Elsevier]

Abstract: In this work, we consider the stability of impulsive infinite delay differential equations. By using Lyapunov functions and the Razumikhin technique, we get some results that are more general than ones given before. And in using the Razumikhin technique, we use a new technique that has been given by Shunian Zhang; we extend this technique to study impulsive systems. An example is also discussed in this work to illustrate the advantage of the results obtained. [Copyright &y& Elsevier]

This paper studies the property of continuous dependence of solutions with respect to initial values for impulsive delay differential equations. Criteria on continuous dependence of solutions with respect to initial functions are established. It is shown, however, that due to the discontinuities of the function space and the time delays in the system, continuous dependence with respect to the initial time, or more generally with respect to the entire initial data, is unfortunately not generally satisfied. [Copyright &y& Elsevier]

The paper studies equation (1.1) in two cases:

In this paper, we deal with the asymptotic and oscillatory behavior of the solutions of higher-order differential equations with middle term of the form L w ′ + p ν f w m − 1 ν + q ν g w σ ν = 0 , where L w ≔ r ν w m − 1 ν α . By using generalized Riccati transformations we study the asymptotic behavior and derive a new oscillation criterion. The results obtained here extend and improve some well-known results which have been published recently in the literature. An example is given to illustrate the applicability of the obtained results. [ABSTRACT FROM AUTHOR]

The paper discusses the problem of necessary and sufficient stability conditions for a test fractional delay differential equation and its discretization. First, we recall the existing condition for asymptotic stability of the exact equation and consider an appropriate fractional delay difference equation as its discrete counterpart. Then, using the Laplace transform method combined with the boundary locus technique, we derive asymptotic stability conditions in the discrete case as well. Since the studied fractional delay difference equation serves as a backward Euler discretization of the underlying differential equation, we discuss a related problem of numerical stability (with a negative conclusion). Also, as a by-product of our observations, a fractional analogue of the classical Levin–May stability condition is presented. [ABSTRACT FROM AUTHOR]

In this paper, we introduce new oscillatory criteria for the second order noncanonical differential equation with delay argument (r (t) (y ′ (t)) α) ′ + p (t) y β (τ (t)) = 0. Our oscillatory results essentially extend the earlier ones. [ABSTRACT FROM AUTHOR]

Abstract: A solution of a linear delay differential equation can have an infinite number of isolated zeros on a finite interval. As is well known, for ordinary differential equations this is impossible. [Copyright &y& Elsevier]