Search

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]

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]

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]