OSCILLATION FOR NONAUTONOMOUS NEUTRAL DYNAMIC DELAY EQUATIONS ON TIME SCALES

The article is concerned with oscillation of nonautonomous neutral dynamic delay equations on time scales. Sufficient conditions are established for the existence of bounded positive solutions and for oscillation of all solutions of this equation. Some results extend known results for difference equations when the time scale is the set Z^＋ of positive integers and for differential equations when the time scale is the set IR of real numbers.

Periodic solutions of non-autonomous differential delay equations with superlinear properties

Abstract： The existence of periodic solutions of a class of non- autonomous differential delay equations with the form x′（t）=-∑k=1^n-1f（t,x（t-kr）） is considered, where r 〉 0 is a given constant and f∈C（R×R,R） is odd in x, r-periodic in t and satisfies some superlinear conditions at origin and at infinity. First, the delay system is changed to an equivalent Hamiltonian system. Then the existence of periodic solutions of the Hamiltonian system is studied. Periodic solutions of the Hamiltonian system can be obtained by critical points of a functional defined on a Hilbert space, i.e. , points satisfying φ′（z）=0. By using a linking theorem in critical point theory, the existence of critical points of the functional is obtained. Therefore, the existence of periodic solutions for the Hamiltonian system and its equivalent differential delay equation is established.

Mean Square Stability of the Composite Milstein Method for Nonlinear Stochastic Differential Delay Equations

In this paper, we construct a composite Milstein method for nonlinear stochastic differential delay equations. Then we analyze the mean square stability for this method and obtain the step size condition under which the composite Milstein method is mean square stable. Moreover, we get the step size condition under which the composite Milstein method is global mean square stable. A nonlinear test stochastic differential delay equation is given for numerical tests. The results of numerical tests verify the theoretical results proposed.

THE ASYMPTOTIC BEHAVIOR AND OSCILLATION FOR A CLASS OF THIRD-ORDER NONLINEAR DELAY DYNAMIC EQUATIONS

In this paper,we consider a class of third-order nonlinear delay dynamic equations.First,we establish a Kiguradze-type lemma and some useful estimates.Second,we give a sufficient and necessary condition for the existence of eventually positive solutions having upper bounds and tending to zero.Third,we obtain new oscillation criteria by employing the Potzsche chain rule.Then,using the generalized Riccati transformation technique and averaging method,we establish the Philos-type oscillation criteria.Surprisingly,the integral value of the Philos-type oscillation criteria,which guarantees that all unbounded solutions oscillate,is greater than θ_(4)(t_(1),T).The results of Theorem 3.5 and Remark 3.6 are novel.Finally,we offer four examples to illustrate our results.

Stochastic Maximum Principle for Optimal Advertising Models with Delay and Non-Convex Control Spaces

In this paper we study optimal advertising problems that model the introduction of a new product into the market in the presence of carryover effects of the advertisement and with memory effects in the level of goodwill. In particular, we let the dynamics of the product goodwill to depend on the past, and also on past advertising efforts. We treat the problem by means of the stochastic Pontryagin maximum principle, that here is considered for a class of problems where in the state equation either the state or the control depend on the past. Moreover the control acts on the martingale term and the space of controls U can be chosen to be non-convex but now the space of controls U can be chosen to be non-convex. The maximum principle is thus formulated using a first-order adjoint Backward Stochastic Differential Equations (BSDEs), which can be explicitly computed due to the specific characteristics of the model, and a second-order adjoint relation.

A NOTE ON THE JULIA SETS OF ENTIRE SOLUTIONS TO DELAY DIFFERENTIAL EQUATIONS

Let f be an entire solution of the Tumura-Clunie type non-linear delay differential equation.We mainly investigate the dynamical properties of Julia sets of f,and the lower bound estimates of the measure of related limiting directions is verified.

The Existence of Meromorphic Solutions to Non-Linear Delay Differential Equations

In this paper, we study the existence of the transcendental meromorphic solution of the delay differential equations , where a(z) is a rational function, and are polynomials in w(z) with rational coefficients, k is a positive integer. Under the assumption when above equations own transcendental meromorphic solutions with minimal hyper-type, we derive the concrete conditions on the degree of the right side of them. Specially, when w(z)=0 is a root of , its multiplicity is at most k. Some examples are given here to illustrate that our results are accurate.

Comparison principle and stability criteria for stochastic differential delay equations with Markovian switching

In the present paper we first obtain the comparison principle for the nonlinear stochastic differentialdelay equations with Markovian switching. Later, using this comparison principle, we obtain some stabilitycriteria, including stability in probability, asymptotic stability in probability, stability in the pth mean, asymptoticstability in the pth mean and the pth moment exponential stability of such equations. Finally, an example isgiven to illustrate the effectiveness of our results.

Numerical Algorithm for Solving Multi-Pantograph Delay Equations on the Half-line Using Jacobi Rational Functions with Convergence Analysis

A new spectral Jacobi rational-Gauss collocation （JRC） method is proposed for solving the multi- pantograph delay differential equations on the half-line. The method is based on Jacobi rational functions and Gauss quadrature integration formula. The main idea for obtaining a semi-analytical solution for these equations is essentially developed by reducing the pantograph equations with their initial conditions to systems of algebraic equations in the unknown expansion coefficients. The convergence analysis of the method is analyzed. The method possesses the spectral accuracy. Numerical results indicating the high accuracy and effectiveness of this algorithm are presented. Indeed, the present method is compared favorably with other methods.

Maximal Regularity of Second Order Delay Equations in Banach Spaces

Using known Ca-multiplier result, we give necessary and sufficient conditions for the second order delay equations：u″（t）=Au（t）＋Fut＋Gu′＋f（t）,t∈Rto have maximal regularity in HSlder continuous function spaces C^α （R, X）, where X is a Banach space, A is a closed operator in X, F, G ∈L（C（[-r, 0], X）, X） are delay operators for some fixed r 〉 0.

Strong convergence rate of truncated Euler-Maruyama method for stochastic differential delay equations with Poisson jumps

We study a class of super-linear stochastic differential delay equations with Poisson jumps (SDDEwPJs). The convergence and rate of the convergence of the truncated Euler-Maruyama numerical solutions to SDDEwPJs are investigated under the generalized Khasminskii-type condition.

Moderate deviation and central limit theorem for stochastic differential delay equations with polynomial growth

Employing the weak convergence method, based on a variational representation for expected values of positive functionals of a Brownian motion, we investigate moderate deviation for a class of stochastic differential delay equations with small noises, where the coefficients are allowed to be highly nonlinear growth with respect to the variables. Moreover, we obtain the central limit theorem for stochastic differential delay equations which the coefficients are polynomial growth with respect to the delay variables.

Periodic Solutions of the Differential Delay Equation (?)(t)=-f(x(t-1))

For an odd function f(x)defined only on a finite interval,this paper deals with the existence of periodic solutions and the number of simple periodic solutions of the differential delay equation(DDE)(?)(t)=-f(x(t-1)).By use of the method of qualitative analysis combined with the constructing of special solutions a series of interesting results are obtained on these problems.

Stability of Stochastic Differential Delay Equations with Markovian Switching

The main aim of this paper is to investigate the pth moment exponential stability of stochastic differential delay equations with Markovian switching.A specific Lyapunov function is introduced to obtain the required stability,and the almost sure exponential stability for the delay equations is discussed subsequently.

A NECESSARY AND SUFFICIENT CONDITION FOR ASYMPTOTIC STABILITY OF A FIRST ORDER DELAY EQUATION WITH TWO DELAYS

Consider the following equationwhere 6, c and τ are constants, and τ > 0, bc ≠ 0. In this paper, we establish a necessary and sufficient condition for zero solution of Eq.(*) to be asymptotically stable, which is easy to verify and apply.

Existence of Exactly n+1 Simple 4-Periodic Solutions of the Differential Delay Equation x(t)=-f(x(t-1))

By discussing the zeros of periodic.solutions we give in this paper a criterion for the existence of exactly n+1 simple 4-periodic solutions of the differential delay equation x(T)= -f(x(t-1)).

Global Attractivity of a Kind of Delay Difference Equations

Several new sufficient conditions are given for the global attractivity of solutions of a kind of delay difference equations. They either include or improve some known results and put the study of Ladas' conjecture forward.

Asymptotic Behavior and Nonexistence of Positive Solutions of Delay Difference Equations *L

Aim To obtain new criteria for asymptotic behavior and nonexistence of positive solutions of nonlinear neutral delay difference equations. Methods By means of Hlder inequality and a method of direct analysis, some interesting Lemmas were offered. Results and Conclusion New criteria for asymptotic behavior and nonexistence of positive solutions of nonlinear neutral delay difference equations are established, which extend and improve the results obtained in the literature. Some interesting examples illustrating the importance of our results are also included.

Multiple periodic solutions for non-canonical Hamiltonian systems with application to differential delay equations

We first establish Maslov index for non-canonical Hamiltonian system by using symplectic transformation for Hamiltonian system.Then the existence of multiple periodic solutions for the non-canonical Hamiltonian system is obtained by applying the Maslov index and Morse theory.As an application of the results,we study a class of non-autonomous differential delay equation which can be changed to non-canonical Hamiltonian system and obtain the existence of multiple periodic solutions for the equation by employing variational method.

CONVERGENCE RATE OF THE TRUNCATED EULER-MARUYAMA METHOD FOR NEUTRAL STOCHA STIC DIFFERENTIAL DELAY EQUATIONS WITH MARKOVIAN SWITCHING

The key aim of this paper is to show the strong convergence of the truncated Euler-Maruyama method for neutral stochastic differential delay equations(NSDDEs)with Markovian switching(MS)without the linear growth condition.We present the truncated Euler-Maruyama method of NSDDEs-MS and consider its moment boundedness under the local Lipschitz condition plus Khasminskii-type condition.We also study its strong convergence rates at time T and over a finite interval[0,T].Some numerical examples are given to illustrate the theoretical results.