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.     

Ground States of K-component Coupled Nonlinear Schrödinger Equations with Inverse-square Potential

In this paper, the authors study ground states for a class of K-component coupled nonlinear Schr¨odinger equations with a sign-changing potential which is periodic or asymptotically periodic. The resulting problem engages three major difficulties: One is that the associated functional is strongly indefinite, the second is that, due to the asymptotically periodic assumption, the associated functional loses the Z^N -translation invariance,many effective methods for periodic problems cannot be applied to asymptotically periodic ones. The third difficulty is singular potential μ_i/|x|² , which does not belong to the Kato’s class. These enable them to develop a direct approach and new tricks to overcome the difficulties caused by singularity and the dropping of periodicity of potential.     

Existence of periodic and subharmonic solutions for second order functional difference equations

In this paper, by using the critical point theory, the existence of periodic and subharmonic solutions to a class of second order functional difference equations is obtained. The main approach used in our paper is variational technique and the Saddle Point Theorem. The problem is to solve the existence of periodic and subharmonic solutions of second order forward and backward difference equations.     

POSITIVE PERIODIC SOLUTIONS OF NONLINEAR FUNCTIONAL DIFFERENCE EQUATION

Using the Krasnoselskii's fixed point theorem, the existence of positive periodic solutions to a class of nonlinear functional difference equations is studied in this paper. Some sufficient conditions for the existence of positive periodic solutions are presented.     

MULTIPLE POSITIVE PERIODIC SOLUTIONS IN A IMPULSIVE PREDATOR-PREY SYSTEM WITH TYPEⅣFUNCTIONAL RESPONSES

The main purpose of this paper is to explore the existence of positive periodic solutions to impulsive predator-prey systems with typeⅣfunctional responses. Sufficient criteria are obtained for the existence of strictly positive periodic solutions.The approach is based on a continuation theorem in the coincidence degree theory as well as some prior estimates.This is also the first time that multiple positive periodic solutions are obtained using coincidence degree theory in impulsive ecological systems.     

PSEUDO-ALMOST PERIODIC SOLUTIONS TO SOME FUNCTIONAL DIFFERENTIAL EQUATIONS

This paper is concerned with the pseudo-almost periodic solutions to some functional differential equations. By the exponential dichotomy theory and Schauder’s fixed-point theorem, some results on the existence and uniqueness of pseudo-almost periodic solutions to the system are obtained.     

MULTIPLE PERIODIC SOLUTIONS TO SECOND ORDER FUNCTIONAL DIFFERENTIAL EQUATION WITH INFINITE DELAY

This paper is concerned with the existence of positive periodic solutions to a second order functional differential equation with infinite delay.Under the appropriate conditions,some existence and multiplicity of positive periodic solutions are derived by an abstract fixed-point theorem.     

Square-mean almost periodic solutions to some stochastic evolution equations

This paper concerns the square-mean almost periodic mild solutions to a class of abstract nonautonomous functional integro-differential stochastic evolution equations in a real separable Hilbert space. By using the so-called "Acquistapace–Terreni" conditions and the Banach fixed point theorem, we establish the existence, uniqueness and the asymptotical stability of square-mean almost periodic solutions to such nonautonomous stochastic differential equations. As an application, almost periodic solution to a concrete nonautonomous stochastic integro-differential equation is considered to illustrate the applicability of our abstract results.     

Periodic Solutions for p-Laplacian Neutral Functional Differential Equation with a Deviating Argument

In this paper, by using the continuation theorem of coincidence degree theory and some analysis methods, we study a kind of periodic solutions to p-Laplacian neutral functional differential equation with a deviating argument,some new results on the existence of periodic solutions is 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.     

On Monotone Method for Second Order Periodic Boundary Value Problems and Periodic Solutions of Functional Difference Equations

In this paper, we show that the method of monotone iterative technique is valid to obtain two monotone sequences that converge uniformly to extremal solutions to second order periodic boundary value problems and periodic solutions of functional difference equations. We obtain some new results under the lower solution a and upper solution βwith α≤β.     

带有变时滞延迟和分布延迟的竞争型神经网络周期解在时间尺度上的的全局指数稳定性

In this paper, competitive neural networks with time-varying and distributed delays are investigated. By utilizing Lyapunov functional methods, the global exponential stability of periodic solutions of the neural networks is discussed on time scales. In addition, an example is given to illustrate the effectiveness of the theoretical results.     

EXISTENCE OF PERIODIC SOLUTIONS TO A SYSTEM WITH FUNCTIONAL RESPONSE ON TIME SCALES

This paper investigates the existence of periodic solutions of a three-species food-chain diffusive system with Beddington-DeAngelis functional responses and time delays in a two-patch environment on time scales. By using a continuation theorem based on coincidence degree theory, we obtain sufficient criteria for the existence of periodic solutions for the system. Moreover, when the time scale T is chosen as R or Z, the existence of the periodic solutions of the corresponding continuous and discrete models follows. Therefore, the method is unified to provide the existence of the desired solutions for continuous differential equations and discrete difference equations.     

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.     

A COMPETITIVE AND PREY-PREDATOR MODEL WITH STAGE-STRUCTURE

The uniform persistence is proved for a non-autonomous competitive and prey-predator model with ratio-dependent functional response and stage-structure. By constructing a Liapunov functional, we establish the conditions of existence and uniqueness for the positive periodic solution, which is globally asymptotically stable. We get a unique almost periodic solution for an almost periodic system as well under corresponding conditions .by means of the Razumikhin function method.     

A PRIORI BOUNDS FOR PERIODIC SOLUTIONS TO A FUNCTIONAL DIFFERENTIAL EQUATION WITH DELAY

A priori bounds are established for periodic solutions to a functional differential equation with delay.By means of these bounds,an existence theorem for periodic solutions is obtained using Mawhin’s continuation theorem.Our work generalizes the known result.     

Associative Cones and Integrable Systems

We identify R7 as the pure imaginary part of octonions. Then the multiplication in octonions gives a natural almost complex structure for the unit sphere S6. It is known that a cone over a surface M in S6 is an associative submanifold of R7 if and only if M is almost complex in S6. In this paper, we show that the Gauss-Codazzi equation for almost complex curves in S6 are the equation for primitive maps associated to the 6-symmetric space G2/T2, and use this to explain some of the known results. Moreover, the equation for S1-symmetric almost complex curves in S6 is the periodic Toda lattice, and a discussion of periodic solutions is given.     

MULTIPLE PERIODICITY TO A CLASS OF DELAYED STATE-DEPENDENT PREDATOR-PREY SYSTEM

The existence of the multiple positive periodic solutions to a delayed state- dependent predator-prey system with non-monotonic functional response is con- sidered.Using the continuation theorem based on Mawhin's coincidence degree, some more generalized results are obtained.     

HORN-TYPE THEOREM IN FRéCHET SPACES AND APPLICATION~(**)

This paper establishes a Horn-type fixed point theorem in Fréchet spaces which is ananswer to Burton's problem in these spaces.As an application of the result,the authorsobtain an existence theorem of periodic solutions for functional differential equations withinfinite delay.