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In this paper, under appropriate hypotheses, we have the existence of a solution semigroup of partial differential equations with delay operator. These equations are used to describe time–age-structured cell cycle model. We also prove that the solution semigroup is a frequently hypercyclic semigroup. [ABSTRACT FROM AUTHOR]

This paper proposes a new fitted operator strategy for solving singularly perturbed parabolic partial differential equation with delay on the spatial variable. We decomposed the problem into three piecewise equations. The delay term in the equation is expanded by Taylor series, the time variable is discretized by implicit Euler method, and the space variable is discretized by central difference methods. After developing the fitting operator method, we accelerate the order of convergence of the time direction using Richardson extrapolation scheme and obtained O h 2 + k 2 uniform order of convergence. Finally, three examples are given to illustrate the effectiveness of the method. The result shows the proposed method is more accurate than some of the methods that exist in the literature. [ABSTRACT FROM AUTHOR]

In this paper, the authors studied oscillatory behavior of solutions of fourth-order delay difference equation Δ c 3 n Δ c 2 n Δ c 1 n Δ u n + p n f u n − k = 0 under the conditions ∑ n = n 0 ∞ c i n < ∞ , i = 1 , 2 , 3. New oscillation criteria have been obtained which greatly reduce the number of conditions required for the studied equation. Some examples are presented to show the strength and applicability of the main results. [ABSTRACT FROM AUTHOR]

In this paper, a mathematical model of pneumococcal pneumonia with time delays is proposed. The stability theory of delay differential equations is used to analyze the model. The results show that the disease-free equilibrium is asymptotically stable if the control reproduction ratio R0 is less than unity and unstable otherwise. The stability of equilibria with delays shows that the endemic equilibrium is locally stable without delays and stable if the delays are under conditions. The existence of Hopf-bifurcation is investigated and transversality conditions are proved. The model results suggest that, as the respective delays exceed some critical value past the endemic equilibrium, the system loses stability through the process of local birth or death of oscillations. Further, a decrease or an increase in the delays leads to asymptotic stability or instability of the endemic equilibrium, respectively. The analytical results are supported by numerical simulations. [ABSTRACT FROM AUTHOR]

In this paper, some new results are obtained for the even order neutral delay difference equation ΔanΔm-1xn+pnxn-kα+qnxn-lβ=0, where m≥2 is an even integer, which ensure that all solutions of the studied equation are oscillatory. Our results extend, include, and correct some of the existing results. Examples are provided to illustrate the importance of the main results. [ABSTRACT FROM AUTHOR]

We consider applications of the g-Drazin inverse to some classes of abstract Cauchy problems, namely, the heat equation with operator coefficient and delay differential equations in Banach space. [ABSTRACT FROM AUTHOR]

The existence, multiplicity, and properties of positive solutions for a third order nonlinear neutral delay difference equation are discussed. Six examples are given to illustrate the results presented in this paper. [ABSTRACT FROM AUTHOR]

This paper deals with stability and Hopf bifurcation analyses of a mathematical model of HIV infection of CD4+ T-cells. The model is based on a system of delay differential equations with logistic growth term and antiretroviral treatment with a discrete time delay, which plays a main role in changing the stability of each steady state. By fixing the time delay as a bifurcation parameter, we get a limit cycle bifurcation about the infected steady state. We study the effect of the time delay on the stability of the endemically infected equilibrium. We derive explicit formulae to determine the stability and direction of the limit cycles by using center manifold theory and normal form method. Numerical simulations are presented to illustrate the results. [ABSTRACT FROM AUTHOR]

We study the second-order neutral delay differential equation [r(t)Φγ(z'(t))]' + q(t)Φβ(x(σ(t))) = 0, where Φα(t) = |t|α-1t, α ≥ 1 and z(t) = x(t) + p(t)x(τ(t)). Based on the conversion into a certain first-order delay differential equation we provide sufficient conditions for nonexistence of eventually positive solutions of two different types. We cover both cases of convergent and divergent integral ∫∞ r-1/γ (t)dt. A suitable combination of our results yields new oscillation criteria for this equation. Examples are shown to exhibit that our results improve related results published recently by several authors. The results are new even in the linear case. [ABSTRACT FROM AUTHOR]

A linearized compact difference scheme is provided for a class of variable coefficient parabolic systems with delay. The unique solvability, unconditional stability, and convergence of the difference scheme are proved, where the convergence order is four in space and two in time. A numerical test is presented to illustrate the theoretical results. [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]

We discuss the dynamical behavior of the stochastic delay three-specie mutualism system. We develop the technique for stochastic differential equations to deal with the asymptotic property. Using it we obtain the existence of the unique positive solution, the asymptotic properties, and the nonpersistence. Finally, we give the numerical examinations to illustrate our results. [ABSTRACT FROM AUTHOR]

We drive a scalar delay differential system to model the congestion of a wireless access network setting. The Hopf bifurcation of this system is investigated using the control and bifurcation theory; it is proved that there exists a critical value of delay for the stability. When the delay value passes through the critical value, the system loses its stability and a Hopf bifurcation occurs. Furthermore, the direction and stability of the bifurcating periodic solutions are derived by applying the normal form theory and the center manifold theorem. Finally, some examples and numerical simulations are presented to show the feasibility of the theoretical results. [ABSTRACT FROM AUTHOR]

We establish some oscillation criteria for the following certain even order neutral delay differential equations with mixed nonlinearities: (r(t)|z(n-1)(t)|α-1z(n-1)(t))' +q0(t)|(x)(τ0(t))|α-1x(τ0 (t))+q1(t)|x(τ1(t))|β-1x (τ1(t))+q2(t)|(x(τ2(t))|γ-1 x(τ2(t)) = 0, t ≥ t0, where z(t) = x(t) + p(t)x(σ(t)), n is even integer, and γ > α > β > 0. Our results generalize and improve some known results for oscillation of certain even order neutral delay differential equations with mixed nonlinearities. [ABSTRACT FROM AUTHOR]

We are concerned with oscillation of the first order neutral delay differential equation [x(t) - px(t -τ)]' + qx(t - σ) = 0 with constant coefficients, and we obtain some necessary and sufficient conditions of oscillation for all the solutions in respective cases 0 < p < 1 and p > 1. [ABSTRACT FROM AUTHOR]

Almost sure exponential stability of the split-step backward Euler (SSBE) method applied to an Itô-type stochastic differential equation with time-varying delay is discussed by the techniques based on Doob-Mayer decomposition and semimartingale convergence theorem. Numerical experiments confirm the theoretical analysis. [ABSTRACT FROM AUTHOR]

We study the oscillatory behaviour of all solutions of first-order neutral equations with variable coefficients. The obtained results extend and improve some of the well-known results in the literature. Some examples are given to show the evidence of our new results. [ABSTRACT FROM AUTHOR]

This paper investigates the problem of robust nonfragile fuzzy H∞ filtering for uncertain Takagi-Sugeno (T-S) fuzzy systems with interval time-varying delays. Attention is focused on the design of a filter such that the filtering error system preserves a prescribed H∞ performance, where the filter to be designed is assumed to have gain perturbations. By developing a delay decomposition approach, both lower and upper bound information of the delayed plant states can be taken into full consideration; the proposed delay-fractional-dependent stability condition for the filter error systems is obtained based on the direct Lyapunov method allied with an appropriate and variable Lyapunov-Krasovskii functional choice and with tighter upper bound of some integral terms in the derivation process. Then, a new robust nonfragile fuzzy H∞ filter scheme is proposed, and a sufficient condition for the existence of such a filter is established in terms of linear matrix inequalities (LMIs). Finally, some numerical examples are utilized to demonstrate the effectiveness and reduced conservatism of the proposed approach. [ABSTRACT FROM AUTHOR]

In 2010, Agarwal et al. studied the existence of a one-dimensional fractional neutral functional differential equation. In this paper, we study an initial value problem for a class of k-dimensional systems of fractional neutral functional differential equations by using Krasnoselskii's fixed point theorem. In fact, our main result generalizes their main result in a sense. [ABSTRACT FROM AUTHOR]

A comparison theorem on oscillation behavior is firstly established for a class of even-order nonlinear neutral delay difference equations. By using the obtained comparison theorem, two oscillation criteria are derived for the class of even-order nonlinear neutral delay difference equations. Two examples are given to show the effectiveness of the obtained results. [ABSTRACT FROM AUTHOR]

This paper focuses on the stability issue of discrete-time networked control systems with random Markovian delays and uncertain transition probabilities, wherein the random time delays exist in the sensor-to-controller and controller-to-actuator. The resulting closed-loop system is modeled as a discrete-time Markovian delays system governed by two Markov chains. Using Lyapunov stability theory, a result is established on the Markovian structure and ensured that the closed-loop system is stochastically stable. A simulation example illustrates the validity and feasibility of the results. [ABSTRACT FROM AUTHOR]

We investigate the stability of stochastic delay differential systems with delayed impulses by Razumikhin methods. Some criteria on the pth moment and almost sure exponential stability are obtained. It is shown that an unstable stochastic delay system can be successfully stabilized by delayed impulses. Moreover, it is also shown that if a continuous dynamic system is stable, then, under some conditions, the delayed impulses do not destroy the stability of the systems. The effectiveness of the proposed results is illustrated by two examples. [ABSTRACT FROM AUTHOR]

An averaging principle for a class of stochastic differential delay equations (SDDEs) driven by fractional Brownian motion (fBm) with Hurst parameter in (1/2, 1) is considered, where stochastic integration is convolved as the path integrals. The solutions to the original SDDEs can be approximated by solutions to the corresponding averaged SDDEs in the sense of both convergence in mean square and in probability, respectively. Two examples are carried out to illustrate the proposed averaging principle. [ABSTRACT FROM AUTHOR]

A delay differential system is investigated based on a previously proposed nutrient-phytoplankton model. The time delay is regarded as a bifurcation parameter. Our aim is to determine how the time delay affects the system. First, we study the existence and local stability of two equilibria using the characteristic equation and identify the condition where a Hopf bifurcation can occur. Second, the formulae that determine the direction of the Hopf bifurcation and the stability of periodic solutions are obtained using the normal form and the center manifold theory. Furthermore, our main results are illustrated using numerical simulations. [ABSTRACT FROM AUTHOR]

A class of fractional-order BAM neural networks with delays in the leakage terms is considered. By using inequality technique and analysis method, several delay-dependent sufficient conditions are established to ensure the uniform stability of such networks. Moreover, the sufficient conditions guaranteeing the existence, uniqueness, and stability of the equilibrium point are also obtained. In addition, three simulation examples are given to demonstrate the effectiveness of the obtained results. [ABSTRACT FROM AUTHOR]

In the paper by Mocanu (1980), Mocanu has obtained sufficient conditions for a function in the classes C¹ (U), respectively, and C²(U) to be univalent and to map U onto a domain which is starlike (with respect to origin), respectively, and convex. Those conditions are similar to those in the analytic case. In the paper by Mocanu (1981), Mocanu has obtained sufficient conditions of univalency for complex functions in the class C¹ which are also similar to those in the analytic case. Having those papers as inspiration, we try to introduce the notion of subordination for nonanalytic functions of classes C¹ and C² following the classical theory of differential subordination for analytic functions introduced by Miller and Mocanu in their papers (1978 and 1981) and developed in their book (2000). Let Ω be any set in the complex plane ℂ, let p be a nonanalytic function in the unit disc U, p ∈ C ² (U), and let ψ(r, s, t; z) : ℂ³ x U → ℂ. In this paper, we consider the problem of determining properties of the function p, nonanalytic in the unit disc U, such that p satisfies the differential subordination ψ (p(z), Dp(z), D²p(z) - Dp(z); z) ⊂ Ω ⇒ p(U) ⊂ Δ. [ABSTRACT FROM AUTHOR]

This investigation aims at developing a methodology to establish stability and bifurcation dynamics generated by a class of delayed economic model, whose state variable is described by the scalar delay differential equation of the form d² p(t)/dt² = -μδ(p(t))(dp(t)/dt)-μbp(t-τ1)-μ(a0p(t-τ2)/(a1 +p(t-τ2))) + μ(d0 -g0). At appropriate parameter values, linear stability and Hopf bifurcation including its direction and stability of the economic model are obtained. The main tools to obtain our results are the normal form method and the center manifold theory introduced by Hassard. Simulations show that the theoretically predicted values are in excellent agreement with the numerically observed behavior. Our results extend and complement some earlier publications. [ABSTRACT FROM AUTHOR]

We prove the existence and uniqueness of solutions for two classes of infinite delay nonlinear fractional order differential equations involving Riemann-Liouville fractional derivatives. The analysis is based on the alternative of the Leray-Schauder fixed-point theorem, the Banach fixed-point theorem, and the Arzela-Ascoli theorem in Ω = {y : (-∞,b] → R : y| (-∞,0] ∊ B} such that y| [0,b] is continuous and B is a phase space. [ABSTRACT FROM AUTHOR]

We study the half-linear delay differential equation ... We establish a new a priori bound for the nonoscillatory solution of this equation and utilize this bound to derive new oscillation criteria for this equation in terms of oscillation criteria for an ordinary half-linear differential equation. The presented results extend and improve previous results of other authors. An extension to neutral equations is also provided. [ABSTRACT FROM AUTHOR]

A nonlinear generalized difference equation with both delays and the maximum value of the unknown function over a discrete past time interval are studied. A nonlinear boundary value problem of antiperiodic type for the given difference equation is set up. One of the main characteristics of the considered difference equation is the presence of the unknown function in both sides of the equation. It leads to impossibility for using the step method for explicit solving of the nonlinear difference equation. In this paper, an approximate method, namely, the monotone iterative technique, is applied to solve the problem. An important feature of the given algorithm is that each successive approximation of the unknown solution is equal to the unique solution of an appropriately constructed initial value problem for a linear difference equation with "maxima," and an algorithm for its explicit solving is given. Also, each approximation is a lower/upper solution of the given nonlinear boundary value problem. The suggested scheme for approximate solving is computer realized, and it is appliedtoa particular example, which is a generalizationofamodel inpopulation dynamics. [ABSTRACT FROM AUTHOR]

An efficient computational method is given in order to solve the systems of nonlinear infinite-delay-differential equations (IDDEs) with proportional delays. Representation of the solution and an iterative method are established in the reproducing kernel space. Some examples are displayed to demonstrate the computation efficiency of the method. [ABSTRACT FROM AUTHOR]

By the critical point theory, infinitely many 4σ-periodic solutions are obtained for the system of delay differential equations x(t) = -f(x(t - σ)), where σ ∊ (0,+8) and f ∊ C(R", R"). It is shown that all the periodic solutions derived here are brought about by the time delay. [ABSTRACT FROM AUTHOR]

This paper is concerned with asymptotical behavior for a class of impulsive delay differential equations. The new criteria for determining attracting sets and attracting basin of the impulsive systemare obtained by developing the properties of quasi-invariant sets. Examples and numerical simulations are given to illustrate the effectiveness of our results. In addition, we show that the impulsive effects may play a key role to these asymptotical properties even though the solutions of corresponding nonimpulsive systems are unbounded. [ABSTRACT FROM AUTHOR]

This paper is concerned with a semiratio-dependent predator-prey systemwith nonmonotonic functional response and two delays. It is shown that the positive equilibriumof the system is locally asymptotically stablewhen the time delay is small enough. Change of stability of the positive equilibrium will cause bifurcating periodic solutions as the time delay passes through a sequence of critical values. The properties of Hopf bifurcation such as direction and stability are determined by using the normal form method and center manifold theorem. Numerical simulations confirm our theoretical findings. [ABSTRACT FROM AUTHOR]

Some new oscillation criteria are given for first order neutral delay differential equations with variable coefficients. Our results generalize and extend some of the well-known results in the literature. Some examples are considered to illustrate the main results. [ABSTRACT FROM AUTHOR]

We study the global asymptotic stability of the following difference equation: x n+1 = f(xn-k1, xn-k2,...,xn-ks;xn-m1,xn-m2,...,xn-mt),n = 0, 1,..., where 0 ≤ k1 < k2< ...< ks and 0 ≤ m1 < m2 < ... < mt with {k1,k2,...,ks}∩{m1,m2,...,mt} = Φ, the initial values are positive, and f ∊ C(Es+t, (0,+∞)) with E ∊ {(0,+∞), [0,+∞)}. We give sufficient conditions under which the unique positive equilibrium x of that equation is globally asymptotically stable. [ABSTRACT FROM AUTHOR]

By applying the method of coincidence degree, some criteria are established for the existence of antiperiodic solutions for a generalized high-order (p, q)-Laplacian neutral differential system with delays ... in the critical case |c| = |d| = 1. The results of this paper are completely new. Finally, an example is employed to illustrate our results. [ABSTRACT FROM AUTHOR]

Using a method based on the concept of the Kuratowski measure of the noncompactness of a bounded set as well as some new estimates of the equicontinuity of the solutions, we prove the existence of a unique pullback attractor in higher regularity space for the multivalued process associated with the nonautonomous 2D-Navier-Stokes model with delays and without the uniqueness of solutions. [ABSTRACT FROM AUTHOR]

A coupled system of nine identical cells with delays and D3 × D3-symmetry is considered. The individual cells are modelled by a scalar delay differential equation which includes linear decay and nonlinear delayed feedback. By analyzing the corresponding characteristic equations, the linear stability of the equilibrium is given. We also investigate the simultaneous occurrence of multiple periodic solutions and spatiotemporal patterns of the bifurcating periodic oscillations by using the equivariant bifurcation theory of delay differential equations combined with representation theory of Lie groups. Numerical simulations are carried out to illustrate our theoretical results. [ABSTRACT FROM AUTHOR]

This paper deals with the oscillation of third-order nonlinear impulsive equations with delay. The results in this paper improve and extend some results for the equations without impulses. Some examples are given to illustrate the main results. [ABSTRACT FROM AUTHOR]

This paper investigates global synchronization in an array of coupled neural networks with time-varying delays and unbounded distributed delays. In the coupled neural networks, limited transmission efficiency between coupled nodes, whichmakes themodel more practical, is considered. Based on a novel integral inequality and the Lyapunov functionalmethod, sufficient synchronization criteria are derived. The derived synchronization criteria are formulated by linear matrix inequalities (LMIs) and can be easily verified by using Matlab LMI Toolbox. It is displayed that, when some of the transmission efficiencies are limited, the dynamics of the synchronized state are different from those of the isolated node. Furthermore, the transmission efficiency and inner coupling matrices between nodes play important roles in the final synchronized state. The derivative of the time-varying delay can be any given value, and the time-varying delay can be unbounded. The outer-coupling matrices can be symmetric or asymmetric. Numerical simulations are finally given to demonstrate the effectiveness of the theoretical results. [ABSTRACT FROM AUTHOR]

Some flaws on impulsive fractional differential equations (systems) have been found. This paper is concerned with the complete controllability of impulsive fractional linear time-invariant dynamical systems with delay. The criteria on the controllability of the system, which is sufficient and necessary, are established by constructing suitable control inputs. Two examples are provided to illustrate the obtained results. [ABSTRACT FROM AUTHOR]

We introduce impulsive cellular neural network models with piecewise alternately advanced and retarded argument (in short IDEPCA).Themodel with the advanced argument is systemwith strong anticipation. Some sufficient conditions are established for the existence and global exponential stability of a unique equilibrium. The approaches are based on employing Banach's fixed point theorem and a new IDEPCA integral inequality of Gronwall type. The criteria given are easily verifiable, possess many adjustable parameters, and depend on impulses and piecewise constant argument deviations, which provides exibility for the design and analysis of cellular neural network models. Several numerical examples and simulations are also given to show the feasibility and effectiveness of our results. [ABSTRACT FROM AUTHOR]

We investigate global dynamics for a system of delay differential equations which describes a virus-immune interaction in vivo. The model has two time delays describing time needed for infection of cell and CTLs generation. Our model admits three possible equilibria: infection-free equilibrium, CTL-absent infection equilibrium, and CTL-present infection equilibrium. The effect of time delay on stability of the equilibria of the CTL immune response model has been studied. [ABSTRACT FROM AUTHOR]

The asymptotic behaviour of a real two-dimensional differential system... with unbounded nonconstant delays t-Θk(t) = 0 satisfying limt?∞?k(t) = 8 is studied under the assumption of instability. Here, A, Bk, and h are supposed to be matrix functions and a vector function. The conditions for the instable properties of solutions and the conditions for the existence of bounded solutions are given. The methods are based on the transformation of the considered real system to one equation with complex-valued coefficients. Asymptotic properties are studied by means of a Lyapunov-Krasovskii functional and the suitableWa?zewski topological principle. The results generalize some previous ones, where the asymptotic properties for two-dimensional systems with one constant or nonconstant delay were studied. [ABSTRACT FROM AUTHOR]

The existence results of positive ω-periodic solutions are obtained for the second-order functional differential equation with multiple delays u''(t) + a(t)u(t) = f(t,u(t),...,u(t - Tn(t))) ε where a(t) ε C(R) is a positive ω-periodic function, f : R x [0, +∞)n+1 ➝ is a continuous function which is ω-periodic in t, and τ T1(t),...,Tn(t) ε C(R, [0, +∞)) are ω-periodic functions. The existence conditions concern the first eigenvalue of the associated linear periodic boundary problem. Our discussion is based on the fixed-point index theory in cones. [ABSTRACT FROM AUTHOR]

The existence results of positive ω-periodic solutions are obtained for the second-order differential equation with delays -u'' + a(t) = f(t,u(t - T1), ..., u(t - Tn)), where a Έ C(R, (0, ∞)) is a ω-periodic function, f : R x [0, ∞)n → [0, ∞) is a continuous function, which is ω-periodic in t, and T1, T 2, ..., Tn are positive constants. Our discussion is based on the fixed point index theory in cones. [ABSTRACT FROM AUTHOR]

Our purpose is to prove the existence of periodic solutions for a competition Lotka-Volterra system on time scales, and one example is given to illustrate our results. [ABSTRACT FROM AUTHOR]

This paper discusses dissipativity problem for system of linear state/input delay equations. Motivated by dissipativity theory of control systems, we choose a new quadratic supply rate. Using the concept of dissipativity, necessary and sufficient conditions for the linear state/input delay systems to be dissipative and exponentially dissipative are derived. The connection of dissipativity with stability is also considered. Finally, passivity and finite gain are explored, correspondingly. The positive-real and bounded-real lemmas are derived. [ABSTRACT FROM AUTHOR]

This paper studies the following third order neutral delay discrete equation Δ(anΔ2(xn+pnxn-τ))+ f(n, xn-d1n, . . ., xn-dln) = gn, n ≥ n0, where τ, l ∈ N, n0 ∈ N ∪ {0}, {an} n∈Nn0, {pn} n∈Nn0, {gn} n∈Nn0 are real sequences with an≠0 for n ≥ n0, {din} n∈Nn0 ⊆ Z with limn→∞(n - din= +∞ for i ∈ {1, 2, . . ., l} and f ∈ C(Nn0 × Rl,R. By using a nonlinear alternative theorem of Leray-Schauder type, we get sufficient conditions which ensure the existence of bounded positive solutions for the equation. Three examples are given to illustrate the results obtained in this paper. [ABSTRACT FROM AUTHOR]