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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]

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]

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]

Abstract: In this work, we present an implicit compact difference scheme for solving a class of neutral delay parabolic differential equations (NDPDEs). The unique solvability and unconditional stability of the scheme are proved. The temporal accuracy of the scheme is improved by using different Richardson extrapolation techniques for linear and nonlinear problems, and fourth-order accuracy in both temporal and spatial dimensions is obtained. Finally, numerical experiments are conducted to verify the accuracy and efficiency of the algorithms. [Copyright &y& Elsevier]

Abstract: In this paper, the global stability problem for a general discrete Cohen–Grossberg neural network with finite and infinite delays is investigated. A simple criterion ensuring the global asymptotical stability is established, by applying the Lyapunov method and graph theory. Finally, an example showing the effectiveness of the provided criterion is given. [Copyright &y& Elsevier]

Abstract: A sufficient condition of stability of exponential Runge–Kutta methods for delay differential equations is obtained. Furthermore, a relationship between P-stability and GP-stability is established. It is proved that the numerical methods can preserve the analytical stability for a class of test problems. [Copyright &y& Elsevier]

Abstract: We describe a surface in which is called the stability cone. We prove necessary and sufficient stability conditions for the delay differential matrix equation . These conditions are formulated in terms of the location with respect to the stability cone of some points determined by the eigenvalues of matrices and the delay value. We require that matrices admit a simultaneous triangularization. [Copyright &y& Elsevier]

Abstract: Several thermomechanical models have been proposed from a heuristic point of view. A mathematical analysis should help to clarify the applicability of these models, among those recent thermal or viscoelastic models. Single-phase-lag and dual-phase-lag heat conduction models can be interpreted as formal expansions of delay equations. The delay equations are shown to be ill-posed, as are the formal expansions of higher order — in contrast to lower-order expansions leading to Fourier’s or Cattaneo’s law. The ill-posedness is proved, showing the lack of continuous dependence on the data, and thus showing that these models (delay or higher-order expansion ones) are highly explosive. In this note we shall present conditions for when this happens. [Copyright &y& Elsevier]

Abstract: In this work, we investigate the stability of a class of impulsive functional differential equations. Some general stability theorems are obtained. Our results can be applied to finite delay impulsive systems or infinite delay impulsive systems or impulsive systems involving both finite and infinite delays, in a unified way. Examples are also given to illustrate that applying our theorems yields better conclusions than the results in the literature. [Copyright &y& Elsevier]

The paper studies equation (1.1) in two cases: