Uniqueness Of Periodic Solutions Research Articles

The existence of pseudo-almost periodic solutions for some nonlinear differential equations in Banach spaces

Let ( E , ‖ . ‖ ) be a real Banach, where [ x , h ] − denotes the limit of the quotient ( ‖ x ‖ − ‖ x − t h ‖ ) / t as 0 < t → 0 . Consider the differential equation d x d t = A ( t , x ) + f ( t ) where we assume, amongst other properties, that A is strongly dissipative in the following sense: [ x − y , A ( t , x ) − A ( t , y ) ] − ≤ p ( t ) ‖ x − y ‖ θ ( ‖ x − y ‖ ) where θ is a function of the type u α ( α ≥ 0 ) and p is a function majorized by an almost periodic function with negative mean value. First, we show that this equation has a unique pseudo-almost periodic solution. Then we apply this result to the class of the following equations: d x d t + q ( t ) ‖ x ‖ α x = f ( t ) ( α ≥ 0 ) , where q is a pseudo-almost periodic function with positive mean value.

Existence and globally exponential stability of periodic solution of BAM neural networks with impulses and recent-history distributed delays

In this paper, a class of generalized bi-directional associative memory (BAM) neural networks with recent history distributed delays and impulses are considered. By using Lyapunov functional method and fixed point theorem, we derive several sufficient conditions for the existence and globally exponential stability of a unique periodic solution of the networks. The results of this paper are new and different from previously known results.

Rate of decay of stable periodic solutions of Duffing equations

In this paper, we consider the second-order equations of Duffing type. Bounds for the derivative of the restoring force are given that ensure the existence and uniqueness of a periodic solution. Furthermore, the stability of the unique periodic solution is analyzed; the sharp rate of exponential decay is determined for a solution that is near to the unique periodic solution.

EXISTENCE AND GLOBAL EXPONENTIAL STABILITY OF PERIODIC SOLUTION OF A CLASS OF IMPULSIVE NETWORKS WITH INFINITE DELAYS

Sufficient conditions are obtained for the existence and global exponential stability of a unique periodic solution of a class of impulsive tow-neuron networks with variable and unbounded delays. The approaches are based on Mawhin's continuation theorem of coincidence degree theory and Lyapunov functions.

Existence and global attractivity of unique positive periodic solution for a model of hematopoiesis

In this paper, we consider the generalized model of hematopoiesis x ′ ( t ) = − a ( t ) x ( t ) + ∑ i = 1 m b i ( t ) 1 + x n ( t − τ i ( t ) ) . By using a fixed point theorem, some criteria are established for the existence of the unique positive ω-periodic solution x ˜ of the above equation. In particular, we not only give the conclusion of convergence of x k to x ˜ , where { x k } is a successive sequence, but also show that x ˜ is a global attractor of all other positive solutions.

Existence, uniqueness, and stability of periodic solutions of an equation of duffing type

We consider a second-order equation of Duffing type. Bounds for the derivative of the restoring force are given which ensure the existence and uniqueness of a periodic solution. Furthermore, the unique periodic solution is asymptotically stable with sharp rate of exponential decay. In particular, for a restoring term independent of the variable $t$, a necessary and sufficient condition is obtained which guarantees the existence and uniqueness of a periodic solution that is stable.

Permanence and Periodic Solution of Predator-Prey System with Holling Type Functional Response and Impulses

We considered a nonautonomous two dimensional predator-prey system with impulsive effect. Conditions for the permanence of the system and for the existence of a unique stable periodic solution are obtained.

Existence and Uniqueness of Positive Periodic Solutions for a Class of Differential Delay Equations

The existence of a unique positive periodic solution for the differential delay equation $x'(t) = -a(t)x(t) + b(t)f (x(t - c(t)))$ is established. Our method is based on Hilbert’s projective metric and the contraction mapping principle.

GLOBAL EXPONENTIAL STABILITY AND PERIODIC OSCILLATIONS OF REACTION–DIFFUSION BAM NEURAL NETWORKS WITH PERIODIC COEFFICIENTS AND GENERAL DELAYS

For a large class of reaction–diffusion bidirectional associative memory (RDBAM) neural networks with periodic coefficients and general delays, several new delay-dependent or delay-independent sufficient conditions ensuring the existence and global exponential stability of a unique periodic solution are given, by constructing suitable Lyapunov functionals and employing some analytic techniques such as Poincaré mapping. The presented conditions are easily verifiable and useful in the design and applications of RDBAM neural networks. Moreover, the employed analytic techniques do not require the symmetry of the bidirectional connection weight matrix, the boundedness, monotonicity and differentiability of activation functions of the network. In several ways, the results generalize and improve those established in the current literature.

Global stability of the Armstrong-Frederick model with periodic biaxial inputs

The paper is concerned with the study of plasticity models described by differential equations with stop and play operators. We suggest sufficient conditions for the global stability of a unique periodic solution for the scalar models and for the vector models with biaxial inputs of a particular form, namely the sum of a uniaxial function and a constant term. For another class of simple biaxial inputs, we present an example of the existence of unstable periodic solutions.

The existence and exponential attractivity of κ-almost periodic sequence solution of discrete time neural networks

In the present paper, several sufficient conditions are obtained for the existence and exponential attractivity of a unique κ-almost periodic sequence solution of discrete time neural network. Our results generalize the corresponding results about almost periodic sequence solution in common sense. It is shown that discretization step κ affects the dynamical characteristics of discrete-time analogues of continuous time neural networks and exponential convergence is dependent on small discretization step size. Our results on exponential attractivity of κ-almost periodic sequence solution can provide us with relevant estimates on how precise such networks can perform during real-time computations. Finally, computer simulations are performed in the end to show the feasibility of our results.

Periodic solutions of neutral nonlinear system of differential equations with functional delay

We study the existence of periodic solutions of the nonlinear neutral system of differential equations of the form d d t x ( t ) = A ( t ) x ( t ) + d d t Q ( t , x ( t − g ( t ) ) ) + G ( t , x ( t ) , x ( t − g ( t ) ) ) . In the process we use the fundamental matrix solution of y ′ = A ( t ) y and convert the given neutral differential equation into an equivalent integral equation. Then we construct appropriate mappings and employ Krasnoselskii's fixed point theorem to show the existence of a periodic solution of this neutral differential equation. We also use the contraction mapping principle to show the existence of a unique periodic solution of the equation.

Existence of periodic solution for a harvested system with impulses at variable times

In this Letter, a mathematical model for the impulsive exploitation of a single species is proposed and analyzed. An impulsive harvested population model at variable times is investigated, and the sufficient and necessary conditions for the existence of periodic solution are established. We find that the system has a unique nontrivial periodic solution if and only if the initial value and the impulsive times satisfy some conditions specifically, which assure that density of the single species is a periodic surge. A numerical example is presented to illustrate the efficiency of the proposed results.

Dynamics for a class of general hematopoiesis model with periodic coefficients

Sufficient conditions are obtained for the existence and global attractivity of a unique positive periodic solution x ˜ ( t ) of (∗) x ′ ( t ) = - a ( t ) x ( t ) + b ( t ) 1 + x n ( t - τ ( t ) ) , t > 0 , where n > 1, a and b are continuous positive periodic function. Also, some sufficient conditions are established for oscillation of all positive solutions of (∗) about x ˜ ( t ) . For the proof of existence and uniqueness of x ˜ ( t ) , the method used here is better than contraction mapping principle.

Existence and global exponential stability of periodic solution of CNNs with impulses

Abstract Sufficient conditions are obtained for the existence and global exponential stability of a unique periodic solution of cellular neural networks with variable time delays and impulses by using Mawhin’s continuation theorem of coincidence degree and by means of a method based on delay differential inequality.

Stability for delayed reaction–diffusion neural networks

We consider a Hopfield neural network model with diffusive terms, non-decreasing and discontinuous neural activation functions, time-dependent delays and time-periodic coefficients. We provide conditions on interconnection matrices and delays which guarantee that for each periodic input the model has a unique periodic solution that is globally exponentially stable. Even in the case without diffusion, such conditions improve recent results on classical delayed Hopfield neural networks with discontinuous activation functions. Numerical examples illustrate the results.

Global attractivity of a periodic ecological model with m-predators and n-preys by “Pure-delay type” system

In this paper, based on the comparison theorem and Lyapunov method, we study the following periodic Lotka-Volterra model with m-predators and n-preys by “pure-delay type”: x ˙ i ( t ) = x i ( t ) b i ( t ) − ∑ k = 1 n a i k ( t ) x k ( t − T i k ) − ∑ l = 1 m c i l ( t ) y l ( t − σ i l ) , i = 1,2 , … , n , y ˙ j ( t ) = y j ( t ) − r j ( t ) + ∑ k = 1 n d j k ( t ) x k ( t − ξ j k ) − ∑ l = 1 m e j l ( t ) y l ( t − η j l ) , j = 1,2 , … , m . By proposing a new concept of generalized uniform M-matrix, a set of easily verifiable sufficient conditions are obtained for the existence and global attractivity of a unique positive periodic solution of the above model. The obtained results improve and generalize some known results.

Existence and uniqueness of periodic solution for a class of differential systems

In this paper, sufficient conditions are obtained for the existence of a unique periodic solution for a class of differential systems.

Authors' reply [to comments on 'Asymptotic state tracking in a class of nonlinear systems via learning-based inversion'

Jian-Xin Xu and Jing Xu (see ibid., p.720, 2006) made several remarks on the results by the authors (Young-Hoon Kim and In-Joon Ha) (see ibid., vol.45, no11, p.2011-27, 2000) and have pointed out an error therein as summarized here. 1) Possibly long waiting time could make the proposed learning controller less efficient than other learning controllers. 2) There is a derivation error in the proof of Theorem 6, which may lead to the revision of the theorem. 3) The uniqueness of solution was assumed implicitly in the proof of Yoshizawa's result (Yoshizawa, 1975). In this context, the theoretical analysis conducted by Young-Hoon Kim and In-Joon Ha (2000) may be valid only under the assumption of unique periodic solution. The authors reply to all three points.

Existence and global attractivity of positive periodic solutions for the impulsive delay Nicholson's blowflies model

In this paper we shall consider the following nonlinear impulsive delay population model: (0.1) x ′ ( t ) = - δ ( t ) x ( t ) + p ( t ) x ( t - m ω ) e - α ( t ) x ( t - m ω ) a.e. t > 0 , t ≠ t k , x ( t k + ) = ( 1 + b k ) x ( t k ) , k = 1 , 2 , … , where m is a positive integer, δ ( t ) , α ( t ) and p ( t ) are positive periodic continuous functions with period ω > 0 . In the nondelay case ( m = 0 ), we show that (0.1) has a unique positive periodic solution x * ( t ) which is globally asymptotically stable. In the delay case, we present sufficient conditions for the global attractivity of x * ( t ) . Our results imply that under the appropriate linear periodic impulsive perturbations, the impulsive delay equation (0.1) preserves the original periodic property of the nonimpulsive delay equation. In particular, our work extends and improves some known results.