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Abstract: Some new oscillation results are presented that improve those reported in [C. Zhang, T. Li, B. Sun, and E. Thandapani, On the oscillation of higher-order half-linear delay differential equations, Appl. Math. Lett. 24 (2011) 1618–1621]. [Copyright &y& Elsevier]

Abstract: This paper discusses the controllability of fractional control systems with control delay. We firstly give the solution expression for fractional control systems with control delay, then give the necessary and sufficient conditions for the controllability of fractional control systems with control delay. [Copyright &y& Elsevier]

Abstract: We provide new results on the existence of extremal solutions for discontinuous differential equations with a deviated argument which can be either delayed or advanced. The boundary condition is allowed to be discontinuous and to depend functionally on the unknown solution. [Copyright &y& Elsevier]

Abstract: This work is concerned with a class of semilinear stochastic functional parabolic differential equations of retarded type. We first establish conditions to ensure the existence of a unique non-negative solution of the stochastic delay partial differential equation under investigation. Subsequently, the problem of explosive or blow-up solutions in mean -norm sense, , in a finite time to the stochastic system is considered. Several examples are given to illustrate the theory established in the work. [Copyright &y& Elsevier]

Abstract: We establish a coupled fixed point theorem for a meaningful class of mixed monotone multivalued operators, and then we use it to derive some results on the existence of quasisolutions and unique solutions to first-order functional differential equations with state-dependent deviating arguments. Our results are very general and can be applied to functional equations featuring discontinuities with respect to all of their arguments, but we emphasize that they are new even for differential equations with continuously state-dependent delays. [Copyright &y& Elsevier]

Abstract: In this work, we study the oscillatory behaviour of the differential equation of mixed type with delays and advances , both differentiable. Some analytical and numerical criteria are obtained in order to guarantee that all solutions are oscillatory. [Copyright &y& Elsevier]

Abstract: In this paper we are concerned with oscillatory functional differential equations (that is, those equations where all the solutions oscillate) under a numerical approximation. Our interest is in the preservation of qualitative properties of solutions under a numerical discretisation. We give conditions under which an equation is oscillatory, and consider whether the discrete schemes derived using linear -methods will also be oscillatory. We conclude with some general theory. [Copyright &y& Elsevier]

Abstract: In this paper the existence of mild solutions for a class of abstract neutral integro–differential equations with state-dependent delay is studied. [Copyright &y& Elsevier]

Abstract: In this paper, we study a class of partial functional differential equations with finite delay, whose linear part is not necessarily densely defined but satisfies the Hille–Yosida condition. Using the classical theory about global attractors in infinite dimensional dynamical systems, we establish some sufficient conditions for guaranteeing the existence of a global attractor under small delays. [Copyright &y& Elsevier]

Abstract: New explicit exponential stability conditions are obtained for the nonautonomous linear equation where and is an oscillating function. We apply the comparison method based on the Bohl–Perron type theorem. Coefficients and delays are not assumed to be continuous. Some real-world applications and several examples are also discussed. [Copyright &y& Elsevier]

Abstract: We consider the symmetric schemes in Boundary Value Methods (BVMs) applied to delay differential equations with real coefficients a and b. If the numerical solution tends to zero whenever the exact solution does, the symmetric scheme with -boundary conditions is called -stable. Three families of symmetric schemes, namely the Extended Trapezoidal Rules of first (ETRs) and second (ETR2s) kind, and the Top Order Methods (TOMs), are considered in this paper. By using the boundary locus technology, the delay-dependent stability region of the symmetric schemes are analyzed and their boundaries are found. Then by using a necessary and sufficient condition, the considered symmetric schemes are proved to be -stable. [Copyright &y& Elsevier]

Abstract: We present a sufficient condition for a zero of a function that arises typically as the characteristic equation of a linear functional differential equations of neutral type, to be simple and dominant. This knowledge is useful in order to derive the asymptotic behaviour of solutions of such equations. A simple characteristic equation, arisen from the study of delay equations with small delay, is analyzed in greater detail. [Copyright &y& Elsevier]

Abstract: This paper deals with the stability of a class of impulsive infinite delay differential equations. By using the Razumikhin technique and Lyapunov functions, we obtain a new criterion on the uniform asymptotic stability and global stability for such differential equations. The result is less restrictive and conservative than that given in some earlier references. Also, our result shows that impulses can make unstable systems stable. An example is given to illustrate the feasibility and advantage of the result. [Copyright &y& Elsevier]

Abstract: This paper is concerned with the uniqueness and globally exponential stability of traveling wave solutions in a lattice delayed differential equation for a single species with two age classes and a fixed maturation period living in a two-dimensional spatial unbounded environment. Under bistable assumption and realistic assumption on the birth function, we construct various pairs of super- and subsolutions and employ the squeezing technique to prove that the equation has a uniqueness traveling-wave solution. Moreover, the wave front is globally asymptotic stable with phase shift. Comparing with the previous results, we consider the symmetry of the traveling wave solutions. Our results can be expended to the case with multi-dimension lattice. [Copyright &y& Elsevier]

Abstract: In this paper, we consider the discretization of parameter-dependent delay differential equation of the form It is shown that if the delay differential equation undergoes a Hopf bifurcation at , then the discrete scheme undergoes a Hopf bifurcation at for sufficiently small step size h, where is the order of the strictly stable linear multistep method. The direction of numerical Hopf bifurcation and stability of bifurcating invariant curve are the same as that of the corresponding delay differential equation. [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]

Abstract: The main objective of this paper is to establish some new delay integral inequalities, which provide explicit bounds on unknown functions. The inequalities given here can be used as tools in the qualitative theory of certain delay differential equations and delay integral equations. [Copyright &y& Elsevier]

Abstract: In this note, we study the exponential stability of impulsive functional differential systems with infinite delays by using the Razumikhin technique and Lyapunov functions. Several Razumikhin-type theorems on exponential stability are obtained, which shows that certain impulsive perturbations may make unstable systems exponentially stable. Some examples are discussed to illustrate our results. [Copyright &y& Elsevier]

Abstract: We present new oscillation criteria for the second-order neutral delay differential equations with distributed deviating argumentwhere . New oscillation criteria are established, which are based on a class of new functions defined in the sequel. Our results are sharper than some previous results which can be seen by the example at the end of this paper. [Copyright &y& Elsevier]

Abstract: Stability of IMEX (implicit–explicit) Runge–Kutta methods applied to delay differential equations (DDEs) is studied on the basis of the scalar test equation , where is a constant delay and are complex parameters. More specifically, P-stability regions of the methods are defined and analyzed in the same way as in the case of the standard Runge–Kutta methods. A new IMEX method which possesses a superior stability property for DDEs is proposed. Some numerical examples which confirm the results of our analysis are presented. [Copyright &y& Elsevier]

Abstract: This manuscript presents many detailed studies of the numerical solution (by the restart method) of a general class of functional differential equations with state dependent lags; that is, where the delays depend upon the (unknown) solution. It emphasizes the sharpness of earlier published results, including those that relate the multiplicity of various shifted zeros of the lag function to the rates of convergence of methods for locating these zeros and to the rates of convergence of the global solution of the delay equation. The studies were selected to clarify the unusual aspects of numerical methods for delay equations, particularly those with state dependent lags. Although the results given here are mainly analytical in nature, they are ones that every numerical analyst and software designer interested in delay equations should master since they form the technological basis for modern software for state dependent delay differential equations. Indeed, almost all modern delay equation software, in one form or another, is based on these results. [Copyright &y& Elsevier]