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We study the Tricomi problem for equations of mixed type with multiple functional delay and advance. A general solution of the equations is obtained. The problem is uniquely solvable. [ABSTRACT FROM AUTHOR]

In this paper, by using the fixed point method, we prove two new results on the Hyers-Ulam-Rassias and the Hyers-Ulam stability for the first order delay differential equation of the form y'(t) = F(t, y(t), y(t -t )). Our results improve some related results in the literature. [ABSTRACT FROM AUTHOR]

By using the standard Liapunov-Krasovskii functional approach, in this paper, new stability, boundedness and ultimately boundedness criteria are established for a class of vector functional differential equations of third order with retarded argument. [ABSTRACT FROM AUTHOR]

A numerical method is presented for solving the singularly perturbed multipantograph delay equations with a boundary layer at one end point. The original problem is reduced to boundary layer and regular domain problems. The regular domain problem is solved by combining the asymptotic expansion and the reproducing kernel method (RKM). The boundary layer problem is treated by the method of scaling and the RKM. Two numerical examples are provided to illustrate the effectiveness of the present method. The results from the numerical example show that the present method can provide very accurate analytical approximate solutions. [ABSTRACT FROM AUTHOR]

Monotonicity of solutions is an important property in the investigation of oscillatory behaviors of differential equations. A number of papers provide some existence criteria for eventually positive increasing solutions. However, relatively little attention is paid to eventually positive solutions that are also eventually decreasing solutions. For this reason, we establish several new and sharp oscillatory criteria for impulsive functional differential equations from this viewpoint. [ABSTRACT FROM AUTHOR]

Abstract: In this paper, some sufficient conditions are obtained to guarantee the existence of multiple periodic solutions with minimal period 4 of the delay differential equation by using relative Morse index and some considerate analysis techniques. [Copyright &y& Elsevier]

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]

In this article, by using a fixed point theorem, we study the existence and regularity of mild solutions for a class of abstract neutral functional differential equations with infinite delay. The fraction power theory and α-norm is used to discuss the problem so that the obtained results can be applied to equations with terms involving spatial derivatives. A stability result for the autonomous case is also established. We conclude with an example that illustrates the applications of the results obtained. [ABSTRACT FROM AUTHOR]

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]

This paper deals with the existence and multiplicity of periodic solutions to delay differential equations of the form where $${z\in {\bf R}^N, f\in C({\bf R}^{N}, {\bf R}^N)}$$. By using the S pseudo geometrical index theory in the critical point theory, some known results for Kaplan-Yorke type differential delay equations are generalized to higher dimensional case. As a result, the Kaplan-Yorke's conjecture is proved to be true in the case of higher dimensional systems. [ABSTRACT FROM AUTHOR]

Abstract: We consider retarded functional differential equations in the setting of Kurzweil–Henstock integrable functions and we state an averaging result for these equations. Our result generalizes previous ones. [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]

In this paper, the stability of impulsive functional differential equations with infinite delays are investigated. By using Lyapunov functions and the Razumikhin technique, a new theorem on the uniform asymptotic stability and global asymptotic stability for such differential equations is obtained. An example is given to illustrate the feasibility of the result. [ABSTRACT FROM AUTHOR]

In this paper, we give some sufficient conditions on the instability of the zero solution of a kind of eighth order nonlinear differential equations of retarded type by using the Lyapunov direct method. The obtained sufficient conditions improve an existing result in the literature. [ABSTRACT FROM AUTHOR]

This article is concerned with the stability of θ-methods for linear delay-integro-differential systems. The linear θ-methods for the modified linear delay-integro-differential systems are established. It is shown that the linear θ-methods are GP-stable if θ∈[1/2, 1]. Finally, some numerical experiments are given to demonstrate the conclusions. [ABSTRACT FROM AUTHOR]

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]

This paper establishes the existence-and-uniqueness theorems of the global solution to a class of stochastic delay differential equations only with the local Lipschitz condition but does not consider the linear growth condition and studies stochastic ultimate boundedness and the moment average in time. Here we allow the coefficients of these equations to be polynomial or their growths are controlled by polynomial speed. To illustrate our idea clearly, we consider a scalar equation and a special n-dimensional equation carefully. [ABSTRACT FROM AUTHOR]

In this paper, we are concerned with the existence of periodic solutions to nonlinear third order delay differential equation: x‴(t)+ψ(x′(t))x″(t)+g(x′(t-r))+f(x(t))= p(t,x(t),x(t-r),x′(t),x′(t-r),x″(t)) when p(t,x(t),x(t-r),x′(t),x′(t-r),x″(t)) is a periodic function of period T, T > 0. With use of Lyapunov's functional approach, we establish some new sufficient conditions which guarantee that there exists a periodic solution of this equation of period T, T > 0. For illustrations, an example is also given on the existence of periodic solutions. [ABSTRACT FROM AUTHOR]

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: 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: The stability and boundedness of the solution for stochastic functional differential equation with finite delay have been studied by several authors, but there is almost no work on the stability of the solutions for stochastic functional differential equations with infinite delay. The main aim of this paper is to close this gap. We establish criteria of pth moment -bounded for neutral stochastic functional differential equations with infinite delay and exponentially stable criteria for stochastic functional differential equations with infinite delay, and we also illustrate the result with an example. [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: 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: 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: 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]

We establish certain sufficient conditions guaranteeing the boundedness of all solutions of some third-order delay differential equations. [ABSTRACT FROM AUTHOR]

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]

We apply multi-scale analysis to stochastic delay-differential equations with multiplicative or parametric noise, deriving approximate stochastic equations for the amplitudes of oscillatory solutions near critical delays. Reduced equations for the envelope of the oscillations provides an efficient analysis of the dynamics by separating the influence of the noise from the intrinsic oscillations over long time scales. We show how this analysis can be used to compute Lyapunov exponents and extended to nonlinear models where the noise has additional resonances. [ABSTRACT FROM AUTHOR]

Certain linear autonomous delay as well as neutral delay difference equations are considered. A class of linear autonomous delay difference equations with continuous variable is also considered. Some results on the behavior of the solutions are established via two distinct positive roots of the corresponding characteristic equation. [ABSTRACT FROM AUTHOR]

We consider scalar delay differential equations x'(t) = -∂x(t)+f (t, xt) (*)with non- linear f satisfying a sort of negative feedback condition combined with a boundedness condition. The well-known Mackey --Glass-type equations,equations satisfying the Yorke condition, and equations with maxima all fall within our considerations. Here, we establish a criterion for the global asymptotical stabilityof a unique steadystate to (*). As an example, we study Nicholson's blowflies equation, where our computations support the Smith conjecture about the equivalence between global and local asymptotical stabilities in this population model. [ABSTRACT FROM AUTHOR]