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We consider a linear scalar delay differential equation (DDE), consisting of two arbitrary distributed time delays. We formulate necessary conditions for stability of the trivial solution which are independent of the distributions. For the case of one discrete and one gamma distributed delay, we give an explicit description of the region of stability of the trivial solution and discuss how this depends on the model parameters. [ABSTRACT FROM AUTHOR]

In this paper, we study the asymptotic behavior of solutions of a class of non-autonomous delay differential equations. These equations have important practical applications and generalize those on which Bernfeld and Haddock conjectured that each solution of the equations tends to a constant. It is shown that every solution of the equations is bounded and tends to a constant as t → + ∞ , which improves and extends some existing ones. [ABSTRACT FROM AUTHOR]

Abstract: The -Laplace equations have some applications in continuum mechanics. On the basis of this background detail, we study oscillation and asymptotic behavior of solutions to two classes of higher-order delay damped differential equations with -Laplacian like operators. Some new criteria are presented that improve the related contributions to the subject. Several examples are provided to illustrate the relevance of new theorems. [Copyright &y& Elsevier]

Abstract: We consider a network of three identical neurons incorporating distributed and discrete signal transmission delays. The model for such a network is a system of coupled nonlinear delay differential equations. It is established that two cases of a single Hopf bifurcation may occur at the trivial equilibrium of the system, as a consequence of the symmetry of the network. These single Hopf bifurcations are the simple and the double root. The present paper looks at the simple root case, and addresses the issue of absolute stability of the trivial equilibrium and stability switching, leading up to calculation of the critical delay and formulation of a Hopf bifurcation theorem. [Copyright &y& Elsevier]

Abstract: In this paper, we study the existence and asymptotic stability in pth moment of mild solutions to nonlinear impulsive stochastic partial differential equations with infinite delay. By employing a fixed point approach, sufficient conditions are derived for achieving the required result. These conditions do not require the monotone decreasing behaviour of the delays. [Copyright &y& Elsevier]

Abstract: In this paper, essentially strongly order-preserving and conditionally set-condensing semiflows are considered. Obtained is a new type of generic quasi-convergence principles implying the existence of an open and dense set of stable quasi-convergent points when the state space is order bounded. The generic quasi-convergence principles are then applied to essentially cooperative and irreducible systems in the forms of ordinary differential equations and delay differential equations, giving some results of theoretical and practical significance. [Copyright &y& Elsevier]

Abstract: Recently P. Palumbo, S. Panunzi and A. De Gaetano analyzed a delay model of the glucose–insulin system. They proved its persistence, the existence of a unique positive equilibrium point, as well as the local stability of this point. In this paper we consider further the uniform persistence of such equilibrium solutions and their global stability. Using the omega limit set of a persistent solution and constructing a full time solution, we also investigate the effect of delays in connection with the behavior of oscillating solutions to the system. The model is shown to admit global stability under certain conditions of the parameters. It is also shown that the model admits slowly oscillating behavior, which demonstrates that the model is physiologically consistent and actually applicable to diabetological research. [Copyright &y& Elsevier]

Abstract: We consider a switching system with time delay composed of a finite number of linear delay differential equations (DDEs). Each DDE consists of a sum of a linear ODE part and a linear DDE part. We study two particular cases: (a) all the ODE parts are stable and (b) all the ODE parts are unstable and determine conditions for delay independent stability. For case (a), we extend a standard result of linear DDEs via the multiple Lyapunov function and functional methods. For case (b) the standard DDE result is not directly applicable, however, we are able to obtain uniform asymptotic stability using the single Lyapunov function and functional methods. [Copyright &y& Elsevier]

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 study the asymptotic stability of the zero solution of even order linear delay differential equations of the form where and are certain constants and . Here is a constant delay. In proving our results we make use of Pontryagin''s theory for quasi-polynomials. [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: A mathematical model for HIV/AIDS with explicit incubation period is presented as a system of discrete time delay differential equations and its important mathematical features are analysed. The disease-free and endemic equilibria are found and their local stability investigated. We use the Lyapunov functional approach to show the global stability of the endemic equilibrium. Qualitative analysis of the model including positivity and boundedness of solutions, and persistence are also presented. The HIV/AIDS model is numerically analysed to asses the effects of incubation period on the dynamics of HIV/AIDS and the demographic impact of the epidemic using the demographic and epidemiological parameters for Zimbabwe. [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: In a previous paper we gave sufficient conditions for a system of delay differential equations to present Bautin-type bifurcation. In the present work we present an example of delay equation that satisfies these conditions. [Copyright &y& Elsevier]

Abstract: This paper concerns the regularity of a functional differential equation in the form: , , where A is the generator of an analytic semigroup on a Banach space X, and are -bounded linear operator for . By spectral analysis, it is shown that the associated solution semigroup of this equation is eventually differentiable. [Copyright &y& Elsevier]

Abstract: In this paper, we employ a well-known fixed-point index theorem to study the existence and non-existence of positive periodic solutions for the periodic impulsive functional differential equations with two parameters. Several existence and non-existence results are established. [Copyright &y& Elsevier]

Abstract: We consider the boundary value problems: , , , , subject to some boundary conditions. By using a generalization of the Leggett–Williams fixed-point theorem due to Avery and Peterson, we provide sufficient conditions for the existence of at least three positive solutions to the above problems. [Copyright &y& Elsevier]

Abstract: The paper discusses the pth moment exponential stability for a general class of neutral stochastic functional differential equations of the Ito type. This investigation can be very complicated, even in many special cases, by using usual methods based on Lyapunov functionals. In this paper we present criteria which are relatively easy to verify the pth moment exponential stability of the solutions of such equations. [Copyright &y& Elsevier]

Abstract: This paper considers the stability problem for a class of impulsive systems of functional differential equations. By using Lyapunov functions and the Razumikhin technique, several stability criteria in terms of two measures are established. Some examples are worked out to illustrate the theorems. [Copyright &y& Elsevier]

Abstract: We consider general nonlinear dynamical systems in a Banach space with dependence on parameters in a second Banach space. An abstract theoretical framework for sensitivity equations is developed. An application to measure dependent delay differential systems arising in a class of HIV models is presented. [Copyright &y& Elsevier]

Abstract: The purpose of this contribution is to give sufficient conditions for the existence of global solutions or left semi-global solutions for some classes of delayed functional differential equations. The principle of monotone sequences is used as a main tool of investigation. Inequalities for coordinates of global solutions are derived as a consequence of used method. Examples illustrate the results. [Copyright &y& Elsevier]

Abstract: In this article we study asymptotic properties of solutions to first order linear neutral differential equations with variable coefficients and constant delays. Results are stated in terms of the solution to a characteristic equation. By doing this, we extend some of the results obtained for delay equations in [J.G. Dix, Ch.G. Philos, I.K. Purnaras, An asymptotic property of solutions to linear non-autonomous delay differential equations, Electron. J. Differential Equations 2005 (2005) 1–9] to neutral equations. [Copyright &y& Elsevier]

Abstract: We consider the synchronized periodic oscillation in a ring neural network model with two different delays in self-connection and nearest neighbor coupling. Employing the center manifold theorem and normal form method introduced by Hassard et al., we give an algorithm for determining the Hopf bifurcation properties. Using the global Hopf bifurcation theorem for FDE due to Wu and Bendixson''s criterion for high-dimensional ODE due to Li and Muldowney, we obtain several groups of conditions that guarantee the model have multiple synchronized periodic solutions when the transfer coefficient or time delay is sufficiently large. [Copyright &y& Elsevier]

Abstract: First, we systematize earlier results on the global stability of the model of population growth. Second, we investigate the effect of delay on the asymptotic behavior when the nonlinearity f is a unimodal function. Our results can be applied to several population models [Elements of Mathematical Ecology, 2001 ; Appl. Anal. 43 (1992) 109–124; Math. Comput. Modelling, in press; Funkt. Biol. Med. 256 (1982) 156–164; Math. Comput. Modelling 35 (2002) 719–731; Mat. Stos. 6 (1976) 25–40] because the function f does not need to be monotone or differentiable. Specifically, our results generalize earlier result of [Delay Differential Equations with Applications in Population Dynamics, 1993], since our function f may not be differentiable. [Copyright &y& Elsevier]

In this paper, by means of the method of upper and lower solutions and monotone iterative technique, the existence of maximal and minimal solutions of the boundary value problems for first order impulsive delay differential equations is established. [Copyright &y& Elsevier]