Global Solutions and Exponential Stability of Stochastic Functional Differential Equations with Infinite Delay

This paper proves that, under the local Lipschitz condition, the stochastic functional differential equations with infinite delay have global solutions without the linear growth condition. Furthermore, the pth moment exponential stability conditions are given. Finally, one example is presented to illustrate our theory.

ON A PERIODIC PREY-PREDATOR SYSTEM WITH INFINITE DELAYS

Based on the theory of coincidence degree,the existence of positive periodic solutions is established for a periodic prey predator system with infinite delays (t)=x(t)[α(t)-γ(t)y(t)-γ(t)∫ ∞ 0K 1(t,s)y(t-s) d s- ∫ ∞ 0∫ ∞ 0R 1(t,s,θ)y(t-s)y(t-θ) d θ d s], (t)=y(t)[-β(t)+μ(t)x(t)+μ(t)∫ ∞ 0K 2(t,s)x(t-s) d s+ ∫ ∞ 0∫ ∞ 0R 2(t,s,θ)x(t-θ)x(t-s) d θ d s],where α,γ,β,μ are positive continuous ω periodic functions, K i∈C (R×[0,∞),(0,∞))( i =1,2) are ω periodic with respect to their first arguments,.respectively, R i∈C (R×[0,∞)×[0,∞),(0,∞))( i =1,2) are ω periodic with respect to their first arguments,respectively.

THE RAZUMIKHIN TYPE THEOREMS OF STABILITY AND BOUNDEDNESS FOR FUNCTIONAL DIFFERENTIAL EQUATIONS WITH INFINITE DELAY

In this paper, we obtain some new Razumikhin type theorems of stability andboundedness for functional differential equations with infinite delay. Under thecondition of V((ξ)) V(t), we substitute the requirement satisfying V 0 insome sets of points {(t, x)|V(t, x) = αi or βi, i = 1, 2,...} for the requirementV 0 in classical theorems of stability and boundedness (for reference, see[1]-[3]).

The Existence and Uniqueness of the Solution for Neutral Stochastic Functional Differential Equations with Infinite Delay

The main aim of this paper is to establish the existence-and-uniqueness theorem for neutral stochastic functional differential equations with infinite delay at phase space BC（（-∞, 0]; R^n） An example is given for illustration.

THE EXISTENCE AND UNIQUENESS AND STABILITY OF ALMOST PERIODIC SOLUTIONS FOR FUNCTIONAL DIFFERENTIAL EQUATIONS WITH INFINITE DELAYS

This paper deals with the problems on the existence and uniqueness and stability of almost periodic solutions for functional differential equations with infinite delays.The author obtains some sufficient conditions which ganrantee the existence and uniqueness and stability of almost periodic solutions with module containment.The results extend all the results of the paper and solve the two open problems proposed in under much weaker conditions than that proposed in.

Dynamic Behavior of a Logistic Type Equation with Infinite Delay

A non-autonomous Logistic type equation with infinite delay is investigated. For general nonautonomous case, sufficient conditions which guarantee the uniform persistence and globally attractivity of the system arc obtained; For almost periodic case, by means of a suitable Lyapunov functional, sufficient conditions are derived for the existence and uniqueness of almost periodic solution of the system. Some new results are obtained.

Stability for Impulsive Functional Differential Equations with Infinite Delays

In this paper, some theorems of uniform stability and uniform asymptotic stability for impulsive functional differential equations with infinite delay are proved by using Lyapunov functionals and Razumikhin techniques. An example is also proved at the end to illustrate the application of the obtained results.

PERIODIC SOLUTIONS OF SCALAR NEUTRAL VOLTERRA INTEGRO-DIFFERENTIAL EQUATIONS WITH INFINITE DELAY

This paper deals with the existence and uniqueness of periodic solutions of the following scalar neutral Volterra integro-differential equation with infinite delaywhere a, C, D, f are continuous functions, also a(t + T) = a(t), C(t + T,s + T) = C(t, s), D(t + T,s + T) = D(t, s), f(t + T) = f(t). Sufficient conditions on the existence and uniqueness of periodic solution to this equation are obtained by the contraction mapping theorem.

Existence Results for Functional Differential Inclusions with Infinite Delay

The aim of the present paper is to investigate the existence of solutions to functional differential inclusions with infinite delay in Banach spaces. A relevant set of phase space axioms is proposed. The main tools used in this paper are certain fixed point theorems based on the setcontraction theory.

Stability Criteria of Nonlinear Impulsive Differential Equations with Infinite Delays

In this paper, we consider the uniform stability and uniformly asymptotical stability of nonlinear impulsive infinite delay differential equations. Instead of putting all components of the state variable x in one Lyapunov function, several Lyapunov-Razumikhin functions of partial components of the state variable x are used so that the conditions ensuring that stability are simpler and less restrictive; moreover, examples are discussed to illustrate the advantage of the results obtained.

THE POSITIVE PERIODIC SOLUTIONS OF PERIODIC RFDEs WITH INFINITE DELAY

This paper is concerned with the periodic retarded functional differential equati-ons(RFDEs) with infinite delay. The sufficient conditions for the existence of noncon-stant positive periodic solutions are established by combining the theory of monotone semiflows generated by RFDEs with infinite delay and the fixed point theorems of solution operators. A nontrivial application of the results obtained here to a well-known nonautonomous Lotka-Volterra system with infinite delay is also presented.

EXISTENCE OF POSITIVE PERIODIC SOLUTIONS FOR INFINITE DELAY FUNCTIONAL DIFFERENTIAL EQUATIONS

In this paper, the existence of positive periodic solutions for infinite delay functional differential equations(FDEs for short) is discussed by utilizing a fixed point theorem on a cone in Banach space. Some results on the existence of positive periodic solutions are derived.

THE STABILITY OF A CLASS OF FUNCTIONAL DIFFERENTIAL EQUATIONS WITH INFINITE DELAYS

This paper deals with the problem on the stability for zero solution to a class of functional differential equations with infinite delays. We give up the usual confine to the boundedness of the coefficient matrix of the equations and obtain some new results which guarantee the stability and asymptotic stability for zero solution of the equations. The results are of simple forms, easily checked and applicable, and extend the relative results of [1].

EXPONENTIAL ESTIMATE OF SOLUTION TO STOCHASTIC FUNCTIONAL DIFFERENTIAL EQUATIONS WITH INFINITE DELAY

In this paper, by the Burkholder-Davis-Gundy inequality and It? formula, the exponential estimate of the solution to stochastic functional differential equations with infinite delay is established in the phase space BC((-∞,0];Rd). Furthermore, the sample Lyapunov exponent of the solution is obtained, which is less than a positive constant 2√K + 65K. Moreover, a pth moment of the solution is studied.

PERSISTENCE OF NONAUTONOMOUS PREDATOR-PREY SYSTEMS WITH INFINITE DELAY

Nonautonomous predator-prey systems with infinite delay is considered at phase space Cg in this paper. Some suitable conditions of persistence of the populations are obtained. The results are different from ones of Wang Ke which was considered in phase space Ch.

STABILITY OF THE SOLUTIONS TO STOCHASTIC FUNCTIONAL DIFFERENTIAL EQUATIONS WITH INFINITE DELAY

In this paper,we obtain the stability of solutions to stochastic functional differential equations with infinite delay at phase space BC((-∞,0];Rd),under non-Lipschitz condition with Lipschitz condition being considered as a special case and a weakened linear growth condition by means of the corollary of Bihari inequality.

The Existence and Uniqueness for the Solution of Neutral Stochastic Functional Differential Equations with Infinite Delay

In this paper, we will make use of a new method to study the existence and uniqueness for the solution of neutral stochastic functional differential equations with infinite delay （INSFDEs for short） in the phase space BC（（？∞,0];Rd）. By constructing a new iterative scheme, the existence and uniqueness for the solution of INSFDEs can be directly obtained only under uniform Lipschitz condition, linear grown condition and contractive condition. Meanwhile, the moment estimate of the solution and the estimate for the error between the approximate solution and the accurate solution can be both given. Compared with the previous results, our method is partially different from the Picard iterative method and our results can complement the earlier publications in the existing literatures.

PERIODIC SOLUTIONS TO NEUTRAL FUNCTIONAL DIFFERENTIAL EQUATIONS WITH INFINITE DELAY

A new and convenient method is used to study the existence of periodic solutions to neutral functional differential equations with infinite delay. A new criterion for the existence of periodic solutions is obtained in this paper.

Approximate controllability results of semilinear integrodifferential equations with infinite delays

We study the approximate controllability of control systems governed by a class of semilinear integrod- ifferential equations with infinite delays. Sufficient conditions for approximate controllability of semilinear control systems are established provided the approximate controllability of the corresponding linear control systems. The results are obtained by using fixed-point theorems and semigroup theory. As an illustration of the application of the approximate controllability results, a simple example is provided.

ON PERIODIC SOLUTIONS OF FUNCTIONAL DIFFERENTIAL EQUATIONS WITH INFINITE DELAY

In this paper,using Mawhin's continuation theorem in the theory of coincidence degree,we first prove the general existence theorem of periodic solutions for F.D.Es with infinite delay:dx(t)/dt=f(t,x_t),x(t)∈R^n,which is an extension of Mawhin's existence theorem of periodic solutions of F.D.Es with finite delay.Second,as an application of it,we obtain the existence theorem of positive periodic solutions of the Lotka-Volterra equations:dx(t)/dt=x(t)(a-kx(t)-by(t)),dy(t)/dt=-cy(t)+d integral from n=0 to +∞ x(t-s)y(t-s)dμ(s)+p(t).