HOMOCLINIC SOLUTIONS FOR A CLASS OF SECOND ORDER NEUTRAL FUNCTIONAL DIFFERENTIAL SYSTEMS

By means of an extension of Mawhin＇s continuation theorem and some analysis methods, the existence of a set with 2kT-periodic solutions for a class of second order neutral functional differential systems is studied, and then the homoclinic solutions are obtained as the limit points of a certain subsequence of the above set.

EXISTENCE OF TWO PERIODIC SOLUTIONS TO n-DIMENSIONAL NEUTRAL FUNCTIONAL DIFFERENTIAL SYSTEM

In this paper,some suffcient criteria for the existence and the multiplicity of positive periodic solutions to a class of first-order n-dimensional neutral functional differential system are established. Our main results generalize related studies in the previous literatures. The fixed point theorem in cones is required to prove our results.

EXISTENCE OF POSITIVE PERIODIC SOLUTIONS FOR NEUTRAL FUNCTIONAL DIFFERENTIAL EQUATIONS WITH DEVIATING ARGUMENTS

By means of an abstract continuation theorem, the existence criteria are established for the positive periodic solutions of a neutral functional differential equation d N d t=N(t)[a(t)-β(t)N(t)-b(t)N(t-σ(t))-c(t)N′(t-τ(t))].

RAZUMIKHIN-TYPE THEOREM FOR NEUTRAL STOCHASTIC FUNCTIONAL DIFFERENTIAL EQUATIONS WITH UNBOUNDED DELAY

This paper establishes the Razumikhin-type theorem on stability for neutral stochastic functional differential equations with unbounded delay. To overcome difficulties from unbounded delay, we develop several different techniques to investigate stability. To show our idea clearly, we examine neutral stochastic delay differential equations with unbounded delay and linear neutral stochastic Volterra unbounded-delay-integro-differential equations.

RAZUMIKHIN-TYPE THEOREMS OF NEUTRAL STOCHASTIC FUNCTIONAL DIFFERENTIAL EQUATIONS

The stability of stochastic functional differential equation with Markovian switching was studied by several authors,but there was almost no work on the stability of the neutral stochastic functional differential equations with Markovian switching.The aim of this article is to close this gap.The authors establish Razumikhin-type theorem of the neutral stochastic functional differential equations with Markovian switching,and those without Markovian switching.

Periodic Solutions in UMD Spaces for Some Neutral Partial Functional Differential Equations

The aim of this work is to study the existence of a periodic solution for some neutral partial functional differential equations. Our approach is based on the R-boundedness of linear operators Lp-multipliers and UMD-spaces.

PROBLEM OF PERIODIC SOLUTIONS FOR NEUTRAL FUNCTIONAL DIFFERENTIAL EQUATION

The problem of periodic solutions for a kind of kth-order linear neutral functional differential equation is studied. By using the theory of Fourier expansions, a sufficient and necessary condition to guarantee the existence and uniqueness of periodic solution is obtained. Further, by applying this result and Schauder's fixed point principle, a kind of kth-order nonlinear neutral functional differential equation is investigated, and some new results on existence of the periodic solutions are given as well. These results improve and extend some known results in recent literature.

POSITIVE SOLUTIONS OF NEUTRAL FUNCTIONAL DIFFERENTIAL EQUATIONS

This paper is to obtain sufficient conditions under which the neutral functional differential equation d/dx[x(t) + integral(c)(t) x(s)d(s) mu(t, s)] + integral(c)(t) f(t, x(s))d(s) eta(t, s) = 0, t greater than or equal to t(0) greater than or equal to c (1) has a positive solution on [c, +infinity). Some results in [1] are generalized. Then we apply our results to functional differential equations of special form and obtain sufficient conditions for those equations to have a positive solution.

Stability of Neutral Stochastic Differential Equations with Multiple Variable Delays

This paper discusses the pth moment stability of neutral stochastic differential equations with multiple variable delays. The equation has a much more general form than the neutral stochastic differential equations with delay. A new kind of φ-function is introduced to address the stability, which is more general than the exponential stability and polynomial stability. Using a specific Lyapunov function, a stability criteria for the neutral stochastic differential equations with multiple variable delays is established, by which it is relatively easy to verify the stability of such equations. Finally, the proposed theories are illustrated by two examples.

Approximate controllability of impulsive neutral stochastic functional differential system with state-dependent delay in Hilbert spaces

Many practical systems in physical and technical sciences have impulsive dynamical behaviors during the evolution process which can be modeled by impulsive differential equations. In this paper, we prove the approximate controllability of control systems governed by a class of impulsive neutral stochastic functional differential system with state-dependent delay in Hilbert spaces. Sufficient conditions for approximate controllability of the control systems are established under the natural assumption that the corresponding linear system is approximately controllable. The results are obtained by using semigroup theory, stochastic analysis techniques, fixed point approach and abstract phase space axioms. An example is provided to illustrate the application of the obtained results.

STABILITY AND ALMOST PERIODIC SOLUTIONS OF FUNCTIONAL DIFFERENTIAL EQUATIONS OF NEUTRAL TYPE

Based on the inherence theorem to the operator D(t) in [9], we investigate the stability with respect to the hull and the existence on almost periodic solutions for neutral functional differential equations (NFDE). For periodic Eq. (1), we prove that if Eq. (1) possesses the solution ξ(t) that is uniformly asymptotically stable with respect to H.f(ξ),then Eq. (1 ) has an mω -- periodic solution p(t), for some integer m ≥1. Furthermore, we prove that if almost periodic Eq. (1 ) possesses the solution ξ(t) that is stable under disturbance from H+ (ξ,D,f), then Eq. (1) has an almost periodic solution.

On the Existence of Positive Periodic Solutions for Neutral Functional Differential Equation with Multiple Deviating Arguments

By means of an abstract continuation theory for k-set contraction and continuation theorem of coincidence degree principle, some criteria are established for the existence of positive periodic solutions of following neutral functional differential equation

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.

A Priori Bounds for Periodic Solutions of a Kind of Second Order Neutral Functional Differential Equation with Multiple Deviating Arguments

Thc main aim of this paper is to use the continuation theorem of coincidence degree theory for studying the existence of periodic solutions to a kind of neutral functional differential equation as follows：（x（t）-^n∑i=1cix（t-ri））″=f（x（t））x′＋g（x（t-τ））＋p（t）.In order to do so, we analyze the structure of the linear difference operator A ： C2π→C2π, [Ax]（t） =x（t）-∑^ni=1cix（t-ri）to determine some flmdamental properties first, which we are going to use throughout this paper. Meanwhile, we also prove some new inequalities which are useful for estimating a priori bounds of periodie solutions.

PERIODIC SOLUTION TO A CLASS OF SECOND ORDER NEUTRAL FUNCTIONAL DIFFERENTIAL EQUATION WITH A STATEMENT-DEPENDENT DEVIATION VARIABLE

We obtain sufficient conditions for class of second order neutral functional theorem of the coincidence degree. the existence of periodic solution to a differential equation, using continuation

Existence of Periodic Solutions for Neutral Functional Differential Equations with Nonlinear Difference Operator

In this paper, the authors consider the problem of existence of periodic solutions for a second order neutral functional differential system with nonlinear difference D-operator. For such a system, since the possible periodic solutions may not be differentiable, our method is based on topological degree theory of condensing field, not based on Leray Schauder topological degree theory associated to completely continuous field.

Existence of Homoclinic Solutions for a Class of Neutral Functional Differential Equations

By means of Mawhin＇s continuation theorem and some analysis methods, the existence of 2kT-periodic solutions is studied for a class of neutral functional differential equations, and then a homoclinic solution is obtained as a limit of a certain subsequence of the above periodic solutions set.

Contractivity and Exponential Stability of Solutions to Nonlinear Neutral Functional Differential Equations in Banach Spaces

A series of eontractivity and exponential stability results for the solutions to nonlinear neutral functional differential equations （NFDEs） in Banach spaces are obtained, which provide unified theoretical foundation for the contractivity analysis of solutions to nonlinear problems in functional differential equations （FDEs）, neutral delay differential equations （NDDEs） and NFDEs of other types which appear in practice.

EXISTENCE OF PERIODIC SOLUTIONS TO A p-LAPLACIAN NEUTRAL FUNCTIONAL DIFFERENTIAL EQUATION WITH VARIABLE PARAMETER

By the generalized Borsuk theorem in coincidence degree theory, a p-Laplacian neutral functional differential equation is studied. A new result on the existence of periodic solution is obtained. The interest is that some coeffcient in it is not a constant function and its sign can be changeable, which is different from that in the known literatures.

Periodic Solutions to a Kind of Second Order Neutral Functional Differential Equation in the Critical Case

In this paper, the authors consider the existence of periodic solutions for a kind of second neutral functional differential equation as follows：（x（t） - cx（t -τ）＂ = g（t, x（t - μ（t））） ＋ e（t）,in the critical case ｜c｜ = 1. By employing Mawhin＇s continuation theorem and some analysis techniques, some new results are obtained.