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In this paper we introduce new effective technique for investigation of oscillation for the second-order trinomial differential equation with retarded and advanced arguments y ′ ′ (t) = p (t) y (τ (t)) + q (t) y (σ (t)). Our criteria improve the existing ones and the progress is illustrated via several examples. [ABSTRACT FROM AUTHOR]

In this paper, we consider the global asymptotical stability of stochastic functional differential equations with impulsive effects. First, by constructing the Lyapunov function, some stability criteria of impulsive stochastic functional differential equations are established. Second, we propose an application to investigate the effectiveness of the obtained results. [ABSTRACT FROM AUTHOR]

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]

This paper is concerned with the existence and uniqueness of mean random attractors of a class non-autonomous stochastic delay p -Laplacian lattice systems defined on a high-dimensional integer set Z d driven by a family infinite-dimensional superlinear noise. We first establish the global-in-time existence and uniqueness of the solutions in C ([ τ , ∞) , L 2 k (Ω , ℓ 2 (Z d))) ∩ L q (Ω , L l o c q ((τ , ∞) , ℓ q (Z d))) for any k ⩾ 1 when the draft term has an arbitrary polynomial growth rate q > 2 and the coefficient of the noise admits a superlinear growth order q ̃ ∈ [ 2 , q). We then show that the mean random dynamical system generated by the solution operators has a unique weakly compact and weakly attracting mean random attractor in the high-order product Bochner space L 2 k (Ω , F ; ℓ 2 (Z d)) × L 2 k (Ω , F ; L 2 k ((− ϱ , 0) , ℓ 2 (Z d))) , where ϱ is the time delay parameter. The dissipative property of the draft term is employed to carefully controlling the superlinear growth diffusion term. When k = 1 , our results are new even in the product Hilbert space L 2 (Ω , F ; ℓ 2 (Z d)) × L 2 (Ω , F ; L 2 ((− ϱ , 0) , ℓ 2 (Z d))). This work can be regard as a further study of mean attractors of stochastic p -Laplacian lattice systems in the works of Wang and Wang (2020) and Chen et al. (2023). [ABSTRACT FROM AUTHOR]

This paper addresses the dynamic behaviors of a diffusive Nicholson's blowflies equation with two distinct distributed delays, i.e., diapause and developmental delays. By applying some differential inequality techniques and the fluctuation lemma, two novel criteria to ensure the global exponential stability (in the L 2 norm) and characterize long time dynamic behavior are established. The obtained theoretical results encompass some existing ones. In particular, two explanatory examples are performed to substantiate our analytical findings. [ABSTRACT FROM AUTHOR]

In this paper, we propose a novel algorithm for simultaneously identifying the time-delay parameter and fractional order in nonlinear fractional delay differential equations. Building upon the optimal perturbation method, we introduce an important perturbation function and develop a modified optimal perturbation method to identify the unknown parameters. Our proposed algorithm overcomes the issue of the optimal perturbation method being prone to local optimal values and requires relatively fewer iterations. Numerical examples are presented to confirm the validity of our algorithm. [ABSTRACT FROM AUTHOR]

In this paper, we firstly develop an inequality on the rising function, which help us to deal with the fractional integral (sum) in terms of Hölder inequality. Secondly, a finite-time stability criterion of a class of nonlinear fractional delay difference systems with order 1 < α < 2 is addressed by virtue of a new discrete Gronwall inequality and Jensen inequality. Finally, two numerical examples are presented to demonstrate our theoretical result. [ABSTRACT FROM AUTHOR]

Consider a stochastic differential delay equation driven by G -Brownian motion (G -SDDE) d x (t) = f (x (t) , x (t − τ)) d t + g (x (t) , x (t − τ)) d B (t) + h (x (t) , x (t − τ)) d 〈 B 〉 (t). Under the global Lipschitz condition for the G -SDDE, we show that the G -SDDE is exponentially stable in mean square if and only if for sufficiently small step size, the Euler–Maruyama (EM) method is exponentially stable in mean square. Thus, we can carry out careful numerical simulations to investigate the exponential stability of the underlying G -SDDE in practice, in the absence of an appropriate Lyapunov function. A numerical example is provided to illustrate our results. [ABSTRACT FROM AUTHOR]

Abstract In this paper, we investigate the global existence of almost surely positive solution to a stochastic Nicholson's blowflies delay differential equation with regime switching, and give the estimation of the path. The results presented in this paper extend some corresponding results in Wang et al. Stochastic Nicholson's blowflies delayed differential equations, Appl. Math. Lett. 87 (2019) 20–26. [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]

We have recently introduced a very high order Discontinuous Galerkin (DG) method for the solution of stiff ordinary differential equations (ODEs) and delay differential equations (DDEs). In this communication we focus on the detection of breaking points frequently arising in the solution of stiff DDEs. Breaking points are discontinuities appearing in the solution of DDEs and/or in some of its derivatives. Breaking points affect the efficiency of the numerical method by forcing very small time steps in their vicinity. They also locally affect the solution accuracy. We therefore introduce a very simple breaking point detection algorithm in order to further improve the efficiency our DG method. [ABSTRACT FROM AUTHOR]

In this paper, we prove a small time large deviation principle for a type of stochastic differential delay equations. The key point is to translate small time asymptotic behavior problems into Freidlin–Wentzell type large deviation problems. The weak convergence method plays an important role in obtaining the large deviation principle. [ABSTRACT FROM AUTHOR]

This paper is concerned with the numerical solutions of nonlinear second-order boundary value problems with time-variable delay. By adapting Runge–Kutta–Nyström (RKN) methods and combining Lagrange interpolation, a class of modified RKN (MRKN) methods are suggested for solving the problems. Under some suitable conditions, MRKN methods are proved to be convergent of order min { p , q } , where p , q are the local orders of MRKN methods and Lagrange interpolation, respectively. Numerical experiments further confirm the computational effectiveness and accuracy of MRKN methods. [ABSTRACT FROM AUTHOR]

The COVID-19 pandemic has brought a serious threat to human life safety worldwide. SARS-CoV-2 virus mainly binds to the target cell surface receptor ACE2 (Angiotensin-converting enzyme 2) through the S protein expressed on the surface of the virus, resulting in infection of target cells. During this infection process, the target cell ACE2 receptor plays a very important mediating role. In this paper, a delay differential equation model containing the mediated effect of target cell receptor is established according to the mechanism of SARS-CoV-2 virus invasion of target cells, and the global stability of the infection-free equilibrium and the infected equilibrium of the model is obtained by using the basic reproduction number ℛ 0 and constructing the appropriate Lyapunov functional. The expression of the basic reproduction number ℛ 0 intuitively gives the dependence on the expression ratio of the target cell surface ACE2 receptor, which is helpful for the understanding of the mechanism of SARS-CoV-2 virus infection. [ABSTRACT FROM AUTHOR]

Abstract It is well-known that for the oscillation of all solutions of the linear delay differential equation x ′ (t) + p (t) x (t − τ) = 0 , t ≥ t 0 , with p ∈ C ([ t 0 , ∞) , R +) and τ > 0 it is necessary that B ≔ lim sup t → ∞ A (t) ≥ 1 e , where A (t) ≔ ∫ t − τ t p (s) d s. Our main result shows that if the function A is slowly varying at infinity (in additive form), then under mild additional assumptions B > 1 e implies the oscillation of all solutions of the above linear delay differential equation. The applicability of the obtained results and the importance of the slowly varying assumption on A are illustrated by examples. [ABSTRACT FROM AUTHOR]

Abstract In this paper, we consider an impulsive competitive system with infinite delays and investigate the extinction and stability of the system. For the corresponding impulsive logistic model, our stability conditions are weaker than those of Yang et al. (2011). [ABSTRACT FROM AUTHOR]

Abstract This article is devoted to the boundedness problem for a class of non-autonomous stochastic delay differential systems driven by G -Brownian motion. Using the G -Lyapunov function method, some sufficient conditions are derived for the pth moment asymptotical boundedness and stability. [ABSTRACT FROM AUTHOR]

Abstract The purpose of this paper is to study the second-order nonlinear noncanonical differential equation r (t) y ′ (t) ′ + p (t) f (y (τ (t))) = 0 under the condition ∫ ∞ 1 r (t) d t < ∞. Contrary to most existing results, oscillation of the studied equation is attained via only one condition. We consider both delay and advanced differential equations. A particular example of Euler type equation is provided in order to illustrate the significance of our main results. [ABSTRACT FROM AUTHOR]

Abstract The long term behavior of solutions of stochastic delay differential equations with a fading stochastic perturbations is investigated. It is shown that if the level of stochastic perturbations fades on the infinity, for instance, if it is given by square integrable function, then an asymptotically stable deterministic system remains to be an asymptotically stable (in mean square). [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]

This paper is devoted to the investigation of the ultimate boundedness of impulsive stochastic delay differential equations. The p th moment global exponential ultimate boundedness criteria are obtained based on the Itô formula and the successful construction of suitable Lyapunov functionals, and the estimated exponential convergence rate and the ultimate bound are provided as well. It is shown that unbounded stochastic delay differential equations can be turned into bounded ones by impulses. [ABSTRACT FROM AUTHOR]

This paper is concerned with the delay dependent dissipativity for a class of nonlinear neutral delay differential equations. By taking into account the influence of the delay, a new dissipativity criterion is derived by using a generalized Halanay’s inequality, which is less conservative than those in the existing literature in some cases. At last, two examples are provided to validate our theoretical results. [ABSTRACT FROM AUTHOR]

A coupled Gompertz-like system of delay differential equations is considered. We prove the existence of T -periodic solutions under resonance assuming a Lazer–Leach type condition. [ABSTRACT FROM AUTHOR]

In this paper, a new algorithm is presented to solve the impulsive delay initial value problems. This is the first time to propose the simplified reproducing kernel method (SRKM for short) to solve the impulsive delay differential equations. Then the uniform convergence of the numerical solution is proved, and the time consuming Schmidt orthogonalization process is avoided. The proposed method is proved to be stable and is not less than second order convergence. The algorithm is employed successfully in some numerical examples. [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]

This paper concerns with the ultimate boundedness problem for impulsive fractional delay differential equations. Based on the impulsive fractional differential inequality, the boundedness of Mittag-Leffler functions, and the successful construction of suitable Lyapunov functionals, some algebraic criteria are derived for testing the global ultimate boundedness of the equations, and the estimations of the global attractive sets are provided as well. One example is also given to show the effectiveness of the obtained theoretical results. [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]

In this paper, the interval qualitative properties of a class of mixed nonlinear impulsive differential equations are studied. Under the hypothesis of delay σ ( t ) being variable, the ratio of functions x ( t − σ ( t ) ) and x ( t ) on each considered interval is estimated, then Riccati transformation and \omega functions are applied to obtain interval oscillation criteria. The results gained by Özbekler and Zafer (2011) for delay σ ( t ) = 0 and by Guo et al. (2012) for delay σ ( t ) being nonnegative constant are developed. [ABSTRACT FROM AUTHOR]

In this note, we shall consider the norm continuity of a class of solution semigroups associated with linear functional differential equations of neutral type with time lag r > 0 in Hilbert spaces. The norm continuity plays an important role in the analysis of asymptotic stability of the system under consideration by means of spectrum approaches. We shall show that for a square integrable neutral delay term and an unbounded infinitesimal generator A multiplied by a square integrable weight function in the distributed delay term, the associated solution semigroup of the system is norm continuous at every t > r . [ABSTRACT FROM AUTHOR]

The purpose of this paper is to study the second-order half-linear delay differential equation r ( t ) y ′ ( t ) α ′ + q ( t ) y α ( τ ( t ) ) = 0 under the condition ∫ ∞ r − 1 ∕ α ( t ) d t < ∞ . Contrary to most existing results, oscillation of the studied equation is attained via only one condition. A particular example of Euler type equation is provided in order to illustrate the significance of our main results. [ABSTRACT FROM AUTHOR]

We study asymptotic behavior of solutions to a class of higher-order sublinear Emden–Fowler delay differential equations. Our theorems improve several results reported recently in the literature. Two examples are provided to illustrate the importance and advantages of new criteria. [ABSTRACT FROM AUTHOR]

In this paper, we firstly introduce a concept of delayed Mittag-Leffler type matrix function, an extension of Mittag-Leffler matrix function for linear fractional ODEs, which help us to seek explicit formula of solutions to fractional delay differential equations by using the variation of constants method. Secondly, we present the finite time stability results by virtue of delayed Mittag-Leffler type matrix. Finally, an example is given to illustrate our theoretical results. [ABSTRACT FROM AUTHOR]

In this paper, the distance between adjacent zeros of oscillatory solutions for second order nonlinear neutral delay differential equations ( a ( t ) z ′ ( t ) ) ′ + q ( t ) f ( x ( t − σ ) ) = 0 , t ≥ t 0 , where z ( t ) = x ( t ) + p ( t ) x ( t − τ ) is investigated. By means of inequality techniques, specific function sequences and nonincreasing solutions for corresponding first order differential inequality, some new estimates for the distribution of zeros of oscillatory solutions have been presented, which have extended and improved some known results. [ABSTRACT FROM AUTHOR]

We study oscillatory behavior of solutions to a class of even-order neutral differential equations relating oscillation of higher-order equations to that of a pair of associated first-order delay differential equations. As illustrated with two examples in the final part of the paper, our criteria improve a number of related results reported in the literature. [ABSTRACT FROM AUTHOR]

The oscillatory behavior of the solutions to a differential equation with several non-monotone delay arguments and non-negative coefficients is studied. A new sufficient oscillation condition, involving lim sup , is obtained. An example illustrating the significance of the result is also given. [ABSTRACT FROM AUTHOR]

Exponential stability of the second order linear delay differential equation in x and u -control x ̈ (t) + a 1 (t) x ̇ (h 1 (t)) + a 2 (t) x (h 2 (t)) + a 3 (t) u (h 3 (t)) = 0 is studied, where indirect feedback control u ̇ (t) + b 1 (t) u (g 1 (t)) + b 2 (t) x (g 2 (t)) = 0 connects u with the solution. Explicit sufficient conditions guarantee that both x and u decay exponentially. [ABSTRACT FROM AUTHOR]

To discuss the influence of cytokine-enhanced rate on viral infection, we propose a delay differential equation model with nonlinear incidence rate. By constructing suitable Lyapunov functionals and using LaSalle invariance principle, the global asymptotic stability of the equilibria of the proposed model is investigated. [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]

In this article, a Shifted-Legendre orthonormal method for solving delay heat conduction equations is considered. This method, based on Legendre orthogonal polynomials, gives the relevant properties of ɛ -optimal approximation solution, and proves the convergence and effectiveness of the constructed numerical algorithm. Numerical examples are given to verify the feasibility of the proposed algorithm. [ABSTRACT FROM AUTHOR]

A substitution is introduced which exploits the parameters of the high-order delayed linear non-autonomous models and consequently allows the use of the M -matrix methods for the stability analysis. To demonstrate the efficacy of the proposed algorithm we obtain explicit and practical stability conditions for the second and third order non-autonomous delayed models. [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]

The objective of this paper is to study asymptotic properties of the fourth-order linear trinomial delay differential equation ( E ) y ( 4 ) ( t ) + p ( t ) y ′ ( t ) + q ( t ) y ( τ ( t ) ) = 0 . Applying suitable comparison principles, we present new criteria for all solutions of ( E ) to be Kneser ones. Obtained results essentially simplify the examination of the studied equations. [ABSTRACT FROM AUTHOR]

We study the asymptotic integration problem for second-order nonlinear delay differential equations of the form ( p ( t ) x ′ ( t ) ) ′ + q ( t ) x ( t ) = f ( t , x ( g ( t ) ) ) . It is shown that if u and v are principal and nonprincipal solutions of equation ( p ( t ) x ′ ) ′ + q ( t ) x = 0 , then there are solutions x 1 ( t ) and x 2 ( t ) of the above nonlinear equation such that x 1 ( t ) = a u ( t ) + o ( u ( t ) ) , t → ∞ and x 2 ( t ) = b v ( t ) + o ( v ( t ) ) , t → ∞ . [ABSTRACT FROM AUTHOR]

The paper deals with nonlinear delay reaction–diffusion equations of the form u t = a u x x + F ( u , u ̄ ) , where u = u ( x , t ) and u ̄ = u ( x , t − τ ) , with τ denoting the delay time. We present a number of traveling-wave solutions of the form u = w ( z ) , z = k x + λ t , that can be represented in terms of elementary functions. We consider equations with quadratic, power-law, exponential and logarithmic nonlinearities as well as more complex equations with the kinetic function dependent on one to four arbitrary functions of a single argument. All of the solutions obtained involve free parameters and so may be suitable for solving certain model problems as well as testing numerical and approximate analytical methods for delay reaction–diffusion equations and more complex nonlinear delay PDEs. [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]

In this paper we use variational methods to investigate the existence of periodic solutions for some second order delay differential systems with impulsive effects. Some new methods and results are obtained. [ABSTRACT FROM AUTHOR]

A nonsymmetric discontinuous Galerkin FEM with interior penalties has been applied to one-dimensional singularly perturbed problem with a constant negative shift. Using higher order polynomials on Shishkin-type layer-adapted meshes, a robust convergence has been proved in the corresponding energy norm. Numerical experiments support theoretical findings. [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]