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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]

The objective of this study was to get some sufficient conditions which guarantee the asymptotic stability and uniform boundedness of the null solution of some differential equations of third order with the variable delay. The most efficient tool for the study of the stability and boundedness of solutions of a given nonlinear differential equation is provided by Lyapunov theory. However the construction of such functions which are positive definite with corresponding negative definite derivatives is in general a difficult task, especially for higher-order differential equations with delay. Such functions and their time derivatives along the system under consideration must satisfy some fundamental inequalities. Here the Lyapunov second method or direct method is used as a basic tool. By defining an appropriate Lyapunov functional, we prove two new theorems on the asymptotic stability and uniform boundedness of the null solution of the considered equation. Our results obtained in this work improve and extend some existing well-known related results in the relevant literature which were obtained for nonlinear differential equations of third order with a constant delay. We also give an example to illustrate the importance of the theoretical analysis in this work and to test the effectiveness of the method employed. [ABSTRACT FROM AUTHOR]

This paper is concerned with certain non-linear unperturbed and perturbed systems of integro-delay differential equations (IDDEs). We investigate fundamental properties of solutions such as uniformly stability (US), uniformly asymptotically stability (UAS), integrability and instability of the un-perturbed system of the IDDEs as well as the boundedness of the perturbed system of IDDEs. In this paper, five new and improved fundamental qualitative results, which have less conservative conditions, are obtained on the mentioned fundamental properties of solutions. The technique used in the proofs depends on Lyapunov-Krasovski functionals (LKFs). In particular cases, three examples and their numerical simulations are provided as numerical applications of this paper. This paper provides new, extensive and improved contributions to the theory of IDDEs. [ABSTRACT FROM AUTHOR]

In this paper we study the oscillatory and global asymptotic stability of a single neuron model with two delays and a general activation function. New sufficient conditions for the oscillation and nonoscillation of the model are given. We obtain both delay-dependent and delay-independent global asymptotic stability criteria. Some of our results are new even for models with one delay. [ABSTRACT FROM AUTHOR]

Abstract: In this paper, we introduce a class of smooth essentially strongly order-preserving semiflows and improve the limit set dichotomy for essentially strongly order-preserving semiflows. Generic convergence and stability properties of this class of smooth essentially strongly order-preserving semiflows are then developed. We also establish the generalized Krein–Rutman Theorem for a compact and eventually essentially strongly positive linear operator. By applying the main results of this paper to essentially cooperative and irreducible systems of delay differential equations, we obtain some results on generic convergence and stability, the linearized stability of an equilibrium and the existence of the most unstable manifold in these systems. The obtained results improve some corresponding ones already known. [Copyright &y& Elsevier]

Abstract: In this paper we consider a linear scalar neutral stochastic differential equation with variable delays and give conditions to ensure that the zero solution is asymptotically mean square stable by means of fixed point theory. These conditions do not require the boundedness of delays, nor do they ask for a fixed sign on the coefficient functions. An asymptotic mean square stability theorem with a necessary and sufficient condition is proved. Some well-known results are improved and generalized. [Copyright &y& Elsevier]

Abstract: In this paper we are concerned with the exponential asymptotic stability of the solution of a class of differential equations with state dependent delays. Our approach is based on the Crandall–Liggett approximation and the properties of semigroups. [Copyright &y& Elsevier]

Abstract: This paper is concerned with a class of essentially strongly order-preserving semiflows, which are defined on an ordered metric space and are generalizations of strongly order-preserving semiflows. For essentially strongly order-preserving semiflows, we prove several principles, which are analogues of the nonordering principle for limit sets, the limit set dichtomy and the sequential limit set trichotomy for strongly order-preserving semiflows. Then, under certain compactness hypotheses, we obtain some results on convergence, quasiconvergence and stability in essentially strongly order-preserving semiflows. Finally, some applications are made to quasimonotone systems of delay differential equations and reaction–diffusion equations with delay, and the main advantages of our results over the classical ones are that we do not require the delicate choice of state space and the technical ignition assumption. [Copyright &y& Elsevier]

Abstract: The mathematical modeling of many problems in biology (the model for myelinated axons) has been formulated as nonlinear delay differential equations. Some basic properties of the myelinated axons are discussed. The numerical scheme for solving this model is described. The convergence and stability of numerical treatments that are discussed analytically are tested. The dependence of both the solution and the internodal delay τ on the various physical parameters are investigated and its numerical results are presented and interpreted. [Copyright &y& Elsevier]

Abstract: This paper deals with the stability of Runge-Kutta methods of collocation type adapted to the numerical solution of initial value problems for delay differential equations. In order to obtain the adaptation of these Runge-Kutta methods to delay equations, the interpolation procedure is considered that uses the relevant local collocation polynomials from the past. Our analysis concerns a well-known stability condition that forms a natural generalization of the classical concept of A-stability to the case of delay differential equations. Up to now, the problem has been open in the literature of whether the stability condition above is fulfilled in the interesting case of the Radau IIA methods if the number of stages v ≥ 2. In this paper, we prove that if v = 2 or v = 3, then the answer is negative. Next, by strong numerical evidence, we arrive at a conjecture stating that the stability condition is not satisfied in the case of the Radau IIA methods whenever v ≥ 2. We finally establish further insight by numerically examining the relevant stability regions, in ℂ × ℂ, for v = 2,3,…, 10. [Copyright &y& Elsevier]

We study scalar first order linear autonomous neutral delay differential equations with distributed type delays. This article presents some new results on the asymptotic behavior, the nonoscillation and the stability. These results are obtained via a real root (with an appropriate property) of the characteristic equation. Applications to the special cases such as (non-neutral) delay differential equations are also presented. [ABSTRACT FROM AUTHOR]

This paper studies stability problems of impulsive control systems with time delay. Several criteria on asymptotic stability are established using the method of Lyapunov functions and Lyapunov functionals. It is shown that by using impulsive feedback control, a linear delay differential system can be stabilized even if it contains no stable matrix. Some examples are also discussed to illustrate the theorems. [Copyright &y& Elsevier]

In the paper, employing the variational Lyapunov method, stability and instability properties in terms of two measures for impulsive delay differential equations with fixed moments of impulsive effects are discussed. Some stability and instability criteria are obtained. These results much generalize the known ones. Some examples are given to illustrate the advantages of them as well. [Copyright &y& Elsevier]

Transition from excitability to asymptotic periodicity in an excitable system, modeled by the FitzHugh–Nagumo equations, with multiple time-delays, is analyzed. It is demonstrated that, for intermediate time-lags, the system has two coexisting attractors, a hyperbolic stable fixed point and a stable limit cycle. The fixed point is destabilized via subcritical Hopf bifurcation for much larger values of the time-lags. [ABSTRACT FROM AUTHOR]

The paper studies equation (1.1) in two cases: