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Abstract This paper mainly deals with the oscillation of nonlinear delay differential equation which is used to describe advertising capital model, analytically and numerically. Firstly, the condition of oscillation of the analytic solution is presented by the technique of the theory of characteristic. Secondly, the asymptotic behavior of non-oscillatory analytic solution is verified. Thirdly, the θ -methods are applied to the mentioned equation, some conditions under which the numerical solution oscillates are obtained. Moreover, it is proved that every non-oscillatory numerical solution tends to the steady state of the model. Finally, some numerical simulations for verifying the theoretical findings are provided. [ABSTRACT FROM AUTHOR]

• We have formulated fractal functional differential equations. • Fractal delay equations are classified and solved. • Fractal linear delay differential equations are solved. In this paper, we give a brief summary of fractal calculus. Fractal functional differential equations are formulated as a framework that provides a mathematical model for the phenomena with fractal time and fractal structure. Fractal retarded, neutral, and renewal delay differential equations with constant coefficients are solved by the method of steps and using Laplace transform. The graphs of solutions are given to show the details. [ABSTRACT FROM AUTHOR]

This paper is concerned with the oscillation of certain third-order nonlinear delay differential equations with damping. We give new characterizations of oscillation of the third-order equation in terms of oscillation of a related, well-studied, second-order linear differential equation without damping. We also establish new oscillation results for the third-order equation by using the integral averaging technique due to Philos. Numerous examples are given throughout. [ABSTRACT FROM AUTHOR]

Recently, there appeared a considerable interest in inverse Sturm–Liouville-type problems with constant delay. However, necessary and sufficient conditions for solvability of such problems were obtained only in one very particular situation. Here we address this gap by obtaining necessary and sufficient conditions in the case of functional-differential pencils possessing a more general form along with a nonlinear dependence on the spectral parameter. For this purpose, we develop the so-called transformation operator approach, which allows reducing the inverse problem to a nonlinear vectorial integral equation. In Appendix A, we obtain as a corollary the analogous result for Sturm–Liouville operators with delay. Remarkably, the present paper is the first work dealing with an inverse problem for functional-differential pencils in any form. Besides generality of the pencils under consideration, an important advantage of studying the inverse problem for them is the possibility of recovering both delayed terms, which is impossible for the Sturm–Liouville operators with two delays. The latter, in turn, is illustrated even for different values of these two delays by a counterexample in Appendix B. We also provide a brief survey on the contemporary state of the inverse spectral theory for operators with delay observing recently answered long-term open questions. [ABSTRACT FROM AUTHOR]

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: 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: 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: This paper investigates the existence and uniqueness theorem of solutions to neutral stochastic differential equations with infinite delay (short for INSFDEs) at a space . Under the uniform Lipschitz condition, linear growth condition is weaken to obtain the moment estimate of the solution for INSFDEs. Furthermore, the existence, uniqueness theorem of the solution for INSFDEs is derived, and the estimate for the error between approximate solution and exact solution is given. On the other hand, under the linear growth condition, the uniform Lipschitz condition is replaced by the local Lipschitz condition, the existence, uniqueness theorem is also valid for INSFDEs on . Moreover, the existence, uniqueness theorem still holds on interval , where is an arbitrary real number. [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 is concerned with an iterative functional differential equationwith the delay depends on the argument of the unknown function and the state derivative. By reducing the equation with the Schröder transformation to another functional differential equation without iteration of the unknown function, we give existence of its local analytic solutions which extend the known results in related literature. [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: 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: By employing Krasnoselskii fixed point theorem, we investigate the existence of multiple positive periodic solutions for a class of state-dependent delay functional differential equations with feedback control. The system considered in this paper is more general and incorporates as special cases various problems which have been studied extensively in the literature. Moreover, some easily verifiable sufficient criteria are established. [Copyright &y& Elsevier]

Abstract: This paper deals with the problem of a type of linear switching systems with discrete and continuous time delay. By constructing Lyapunov functional under a condition on the time delay, we show it stabilizes the system for sufficiently small delays. [Copyright &y& Elsevier]

Abstract: We investigate the behavior of population models in a periodically varying environment. In this paper, we introduce the Fox surplus harvesting model with delay We obtain explicit conditions for existence of global solutions, existence of a positive periodic solution, oscillation of all solutions about positive equilibrium and global attractivity of this equilibrium. Numerical simulations illustrate the effectiveness of the conditions for oscillations and stability of the Fox model. It was also found that oscillations in the intrinsic rate r(t) and the environmental carrying capacity K(t) contributed quantitatively and qualitatively to the system’s dynamics in significant ways, and cannot be neglected in mathematical models of biological systems. [Copyright &y& Elsevier]

Abstract: The segmented formulation of the Tau method is used to approximate the solutions of the parametric nonlinear neutral differential equationwhich represents, for different values of the parameters r, a, b, c and τ, a family of functional differential equations with some of its members arising in areas as different as the number theory, mathematical biology, and population dynamics. For this equation no closed form of analytical solution is available. The numerical results obtained are consistent with the theoretical and practical results reported elsewhere. [Copyright &y& Elsevier]

Abstract: In this paper, we will give a novel criterion for the global stabilization of the zero solutions of a class of neural networks with varying delays and control input described by nonlinear delay differential equations of the neutral type. By using the method of Liapunov–Krasovskii functions, some sufficient conditions for ensuring the global stability of these neutral systems of neural networks are obtained. Some examples are included to illustrate the feasibleness of our results. [Copyright &y& Elsevier]

Abstract: The existence of positive heteroclinic solutions is proven for a class of scalar population models with one discrete delay. Traveling wave solutions for scalar delayed reaction–diffusion equations are also obtained, as perturbations of heteroclinic solutions of the associated equation without diffusion. As an illustration, the results are applied to the Nicholson’s blowflies equation with diffusion in the case of p/δ > e, for which the nonlinearity is non-monotone. [Copyright &y& Elsevier]

Abstract: This paper considers the stability problem of nontrivial solutions of delay differential equations with state-dependent impulses. By using the concepts of reduced system and quasi-stability, the stability problem of a nontrivial solution of the impulsive delay differential equation with state-dependent impulses is converted to that of the trivial solution of the impulsive delay differential equation with fixed impulses. Several stability criteria are then established utilizing the known results for systems with fixed impulses. [Copyright &y& Elsevier]

Abstract: We consider a class of delayed predator–prey system with Holling type III functional response. The uniform persistence and the existence of positive periodic solutions will be studied by using new approaches. Our results improve some in the literature. An example is given to illustrate the obtained results. [Copyright &y& Elsevier]

Abstract: This paper studies a class of stochastic delay differential equations with jumps (SDDEJs). Explicit solutions can hardly be obtained for the SDDEJs. Appropriate numerical approximation schemes such as the Euler scheme are needed to apply SDDEJs in practice or to study their properties. In this paper, it is proved that the Euler approximation solutions converge to the analytic solution for SDDEJs under weaker conditions than the linear growth condition and global Lipschitz condition. An example is given for illustration. [Copyright &y& Elsevier]

Abstract: Here we give simple sufficient conditions in order that the theoretical virus free steady state in models of anti-viral therapies may be Globally Asymptotically Stable, in case of realistic periodical scheduling of the drug. Models with or without distributed delays are considered. [Copyright &y& Elsevier]

A delayed periodic ratio-dependent predator–prey model with prey dispersal and stage structure for predator is investigated. It is assumed that immature individuals and mature individuals of the predator species are divided by a fixed age, and that immature predators don''t have the ability to attack prey, and that predator species is confined to one of the patches while the prey species can disperse between two patches. We first discuss the uniform persistence and impermanence of the model. By using Gaines and Mawhin''s continuation theorem of coincidence degree theory, a set of easily verifiable sufficient conditions are derived for the existence of positive periodic solutions of the proposed model. Numerical simulations are presented to illustrate the feasibility of our main results. [Copyright &y& Elsevier]

This paper deals with H-stability of the linear θ-method with a general variable stepsize for the system of pantograph equations with two delay terms. A sufficient condition such that the system of pantograph equations is asymptotically stable is derived. Furthermore, when the linear θ-method with a general variable stepsize is applied to this system, it is shown that the linear θ-method is H-stable if and only if 1/2<θ⩽1. [Copyright &y& Elsevier]

The sufficient conditions for the stability of the solutions of the second-order linear delay differential equations with variable coefficients is constructed. [Copyright &y& Elsevier]

Delayed duopoly model with gradient dynamics is considered. Linear stability constraints are derived. It is shown that delay increases stability. Hence firms using duopoly model with gradient method in optimized theory with delay has a higher chance of reaching Nash equilibrium. Cycles and chaos regions are determined. [Copyright &y& Elsevier]

In this paper we consider the numerical solution of a delay differential equation undergoing a Hopf bifurcation. We prove that if the delay differential equation has a Hopf bifurcation point at λ=λ*, then the numerical solution of the equation also has a Hopf bifurcation point at λh=λ*+O(h). [Copyright &y& Elsevier]

In this paper we obtain the 3/2 stability theorems for the delay differential equation with impulsesx′(t)+λx(t)+F(t,x(·))=0, t&ges;0, t≠tk,x(tk+)−x(tk)=Ik(x(tk)), k=1,2… [Copyright &y& Elsevier]

In this paper, we mainly consider exponential stability (ES) for a type of neutral stochastic delay differential equations with delay-dependent impulses (NSDDEWDI). By means of the average dwell time (ADT) condition, a new sufficient criterion for ES is given. The infinitesimal operator of Lyapunov functions is controlled by two time-variant coefficients. Furthermore, these new conditions are calculated through an example so as to verify the validity of our results. [ABSTRACT FROM AUTHOR]