Analysis of global exponential stability and periodic solutions of FCNNs with constant delays and time-varying delays

In this paper, we investigate a class of fuzzy cellular neural networks with constant delays and time-varying delays. By constructing suitable Lyapunov functional and employing Young inequality, we find sufficient conditions for the existence, uniqueness, global exponential stability of equilibrium, and the existence of periodic solutions of fuzzy cellular neural networks with time-varying delays. The results of this paper are new and they extend previously known results.     

Stability in Lagrange sense for Cohen–Grossberg neural networks with time-varying delays and finite distributed delays

In this paper, we study the global exponential stability in Lagrange sense for a class of Cohen–Grossberg neural networks with time-varying delays and finite distributed delays. Based on the Lyapunov stability theory, several global exponential attractive sets in which all trajectories converge are obtained. We analyze three different types of activation functions which include both bounded and unbounded activation functions. These results can also be applied to analyze monostable as well as multistable and more extensive neural networks due to making no assumptions on the number of equilibria. Meanwhile, the results obtained in this paper are more general and challenging than that of the existing references. Finally, one example is given and analyzed to verify our results.     

Interactional models for adults of two populations with maturation delays

This paper is concerned with interactional models for adults of two species delayed by their mature periods. The existence and local stability of equilibria are discussed thoroughly for competitive systems, cooperative systems and predator-prey systems, respectively. For systems with interaction of competition and cooperation, it is found that the two populations are uniformly persistent if the positive equilibrium is stable. For predator-prey interaction, however, some further conditions are needed to guarantee the persistence of the systems.     

Almost periodic solutions for Lotka–Volterra systems with delays

This paper studies a general class of delayed almost periodic Lotka–Volterra system with time-varying delays and distributed delays. By using the definition of almost periodic function, the sufficient conditions for the existence and uniqueness of globally exponentially stable almost periodic solution are obtained. The conditions can be easily reduced to special cases of cooperative systems and competitive systems.     

Permanence for nonautonomous Lotka–Volterra cooperative systems with delays

In this paper, we consider a class of nonautonomous two species Lotka–Volterra cooperative population systems with time delays, and establish sufficient conditions which ensure the system to be permanent. We improve and extend the known condition of the permanence in [G. Lu and Z. Lu, Permanence for two species Lotka–Volterra cooperative systems with delays, Math. Biosci. Eng. 5 (2008) 477–484] to nonautonomous two-species Lotka–Volterra cooperative systems. Moreover, our conditions need no restriction on the size of time delays.     

Stability of linear impulsive Itô differential equations with bounded delays

We use the method of “model” equations to study the exponential p-stability (2 ≤ p < ∞) of the trivial solution with respect to the initial function for a linear impulsive system of Itô differential equations with bounded delays. The specific form of the equation and the method used permit one to analyze the stability of solutions starting from an arbitrary point of the half-line [0,∞) and obtain constructive sufficient conditions in terms of the parameters of the equations to be studied.     

Asymptotic behavior of stochastic two-dimensional Navier–Stokes equations with delays

The paper proves the L ²-exponential stability of weak solutions of two-dimensional stochastic Navier?CStokes equations in the presence of delays. The results extend some of the existing results.     

Cucker–Smale flocking under rooted leadership and time-varying heterogeneous delays

In this paper, a Cucker–Smale model with time-varying heterogeneous delays in the communication process is investigated, where the delay on each communication link is time-varying and independent of other communication links. A general update rule based on the distance between agents is used to determine the edge weights of the communication network. The method of constructing augmented system is used so that the system stability is transformed into a product convergence problem of time-varying sub-stochastic matrices equivalently. By analyzing this convergence problem, a sufficient condition is established to achieve flocking, which relates to the initial states, the structure of communication network and the upper bound of time delays.     

Bistable traveling waves for a competitive–cooperative system with nonlocal delays

This paper is devoted to the study of bistable traveling waves for a competitive–cooperative reaction and diffusion system with nonlocal time delays. The existence of bistable waves is established by appealing to the theory of monotone semiflows and the finite-delay approximations. Then the global stability of such traveling waves is obtained via a squeezing technique and a dynamical systems approach.     

Travelling wavefronts of scalar reaction–diffusion equations with and without delays

This paper deals with the existence of travelling wavefronts for scalar nonlinear reaction–diffusion equations with and without delays in one-dimensional space. New iterative techniques for a class of integral operators of Hammerstein type are established and applied to tackle the existence of travelling wavefronts in a unified way. Our results without delays only require the functions involved to be continuous and satisfy a suitable monotonicity condition. Our results with multiple delays employ the usual C¹-assumption but generalize the well-known results.     

Robust reliable H∞ control for neural networks with mixed time delays

This paper considers the problem of robust reliable H_∞ control for neural networks. The system has time-varying delays, parametric uncertainties and faulty actuators. The faulty actuators are considered as a disturbance signal to the system which is augmented with system disturbance input. Based on the LMI technique and the Lyapunov stability theory, a new set of sufficient conditions is obtained for the existence of the robust reliable H_∞ controller. An example is also presented illustrate the results.     

Dichotomies for linear differential equations with delays: the Carathéodory case

Summary This work is concerned with differential equations with delays, on [0, ∞[, of the form +Mu=r, and the corresponding homogeneous equation +Mu=0, where u and r take values in a Banach space E, and M is a ? memory ?. The main results relate the existence of dichotomies or exponential dichotomies of the solutions of the homogeneous equation (a kind of conditional stability) to the admissibility of certain pairs of function spaces for the inhomogeneous equation. The ? memory ? M is quite general: it is a linear mapping that takes continuous E-valued functions on [−1, ∞[ into locally integrable functions on [0, ∞[ in such a way that Mu on [a, b] ⊂ [0, ∞[ depends only on u on [a −1, b], and satisfies a reasonable boundedness condition. Entrata in Redazione il 10 settembre 1973. This work was partially supported by NSF Grant GP-33364X.     

Periodicity in a nonlinear discrete predator–prey system with state dependent delays

With the help of a continuation theorem based on Gaines and Mawhin's coincidence degree, easily verifiable criteria are established for the global existence of positive periodic solutions of the following nonlinear discrete state dependent delays predator–prey systemwhere $a_i,c_j,d_i:Z\to R^{+}$ are positive $\omega$-periodic, $\omega$ is a fixed positive integer. $b_1,b_2:Z\to R^{+}$ are $\omega$-periodic and ${\sum }_{k=0}^{\omega -1}{b}_i(k)>0$. ${\tau }_i,{\sigma }_j,{\rho }_i:Z\times R\times R\to R(i=1,2,\dots ,n,j=1,2,\dots ,m)$ are $\omega$-periodic with respect to their first arguments, respectively. ${\alpha }_i,{\beta }_j,{\gamma }_i(i=1,2,\dots ,n,j=1,2,\dots ,m)$ are positive constants.     

Global asymptotic behavior in a Lotka–Volterra competition system with spatio-temporal delays

This paper is concerned with a Lotka–Volterra competition system with spatio-temporal delays. By using the linearization method, we show the local asymptotic behavior of the nonnegative steady-state solutions. Especially, the global asymptotic stability of the positive steady-state solution is investigated by the method of upper and lower solutions. The result of global asymptotic stability implies that the system has no nonconstant positive steady-state solution.