Fixed points and stability of neutral stochastic delay differential equations

In this paper we consider a linear scalar neutral stochastic differential equation with variable delays and give conditions to ensure that the zero solution is asymptotically mean square stable by means of fixed point theory. These conditions do not require the boundedness of delays, nor do they ask for a fixed sign on the coefficient functions. An asymptotic mean square stability theorem with a necessary and sufficient condition is proved. Some well-known results are improved and generalized.     

Robustness of the controllability for the heat equation under the influence of multiple impulses and delays

For many control systems in real life, impulses and delays are intrinsic properties that do not modify their behavior. Thus, we conjecture that under certain conditions the abrupt changes and delays as perturbations of a system, that could model a real situation, do not modify properties such as controllability. In this regard, we prove the approximate controllability of the semilinear heat equation under the influence of multiple impulses and delays, this is done by using new techniques, avoiding fixed point theorems, employed by A.E. Bashirov et al.     

Asymptotic stability of impulsive stochastic partial differential equations with infinite delays

In this paper, we study the existence and asymptotic stability in pth moment of mild solutions to nonlinear impulsive stochastic partial differential equations with infinite delay. By employing a fixed point approach, sufficient conditions are derived for achieving the required result. These conditions do not require the monotone decreasing behaviour of the delays.     

Stability results for stochastic delayed recurrent neural networks with discrete and distributed delays

We present new conditions for asymptotic stability and exponential stability of a class of stochastic recurrent neural networks with discrete and distributed time varying delays. Our approach is based on the method using fixed point theory, which do not resort to any Liapunov function or Liapunov functional. Our results neither require the boundedness, monotonicity and differentiability of the activation functions nor differentiability of the time varying delays. In particular, a class of neural networks without stochastic perturbations is also considered. Examples are given to illustrate our main results.     

Permanence and periodicity of a delayed ratio-dependent predator-prey model with stage structure

A periodic ratio-dependent predator-prey model with time delays and stage structure for both prey and predator is investigated. It is assumed that immature individuals and mature individuals of each species are divided by a fixed age, and that immature predators do not have the ability to attack prey. Sufficient conditions are derived for the permanence and existence of positive periodic solutions of the model. Numerical simulations are presented to illustrate the feasibility of our main results.     

Stability in functional differential equations established using fixed point theory

In this paper we consider a nonlinear scalar delay differential equation with variable delays and give some new conditions for the boundedness and stability by means of Krasnoselskii’s fixed point theory. A stability theorem with a necessary and sufficient condition is proved. The results in [T.A. Burton, Stability by fixed point theory or Liapunov’s theory: A comparison, Fixed Point Theory 4 (2003) 15–32; T.A. Burton, T. Furumochi, Asymptotic behavior of solutions of functional differential equations by fixed point theorems, Dynamic Systems and Applications 11 (2002) 499–519; B. Zhang, Fixed points and stability in differential equations with variable delays, Nonlinear Analysis 63 (2005) e233–e242] are improved and generalized. Some examples are given to illustrate our theory.     

Global exponential stability for interval general bidirectional associative memory (BAM) neural networks with proportional delays

This paper is concerned with interval general bidirectional associative memory (BAM) neural networks with proportional delays. Using appropriate nonlinear variable transformations, the interval general BAM neural networks with proportional delays can be equivalently transformed into the interval general BAM neural networks with constant delays. The sufficient condition for the existence and uniqueness of equilibrium point of the model is established by applying Brouwer's fixed point theorem. By constructing suitable delay differential inequalities, some sufficient conditions for the global exponential stability of the model are obtained. Two examples are given to illustrate the effectiveness of the obtained results. This paper ends with a brief conclusion. Copyright © 2016 John Wiley & Sons, Ltd.     

Existence and uniqueness of mild solutions for semilinear integro-differential equations of fractional order with nonlocal initial conditions and delays

In this paper, we establish the existence and uniqueness of mild solutions for a class of semilinear integro-differential equations of fractional order with nonlocal initial conditions and delays in α-norm. And the main tool is the fixed point theorem due to Sadovskii. Some known results are generalized.     

Exponential stability of positive neural networks in bidirectional associative memory model with delays

This paper is concerned with the problem of exponential stability of positive neural networks in bidirectional associative memory (BAM) model with multiple time‐varying delays and nonlinear self‐excitation rates. On the basis of a systematic approach involving extended comparison techniques via differential inequalities, we first prove the positivity of state trajectories initializing from a positive cone called the admissible set of initial conditions. In combination with the use of Brouwer's fixed point theorem and M‐matrix theory, we then derive conditions for the existence and global exponential stability of a unique positive equilibrium of the model. An extension to the case of BAM neural networks with proportional delays is also presented. The effectiveness of the obtained results is illustrated by a numerical example with simulations.     

Relative controllability of fractional dynamical systems with delays in control

This paper is concerned with the controllability of nonlinear fractional dynamical systems with time varying multiple delays and distributed delays in control defined in finite dimensional spaces. Sufficient conditions for controllability results are obtained using the Schauder fixed point theorem and the controllability Grammian matrix which is defined by Mittag–Leffler matrix function. Examples are provided to illustrate the theory.     

Delayed dynamics in heterogeneous competition with product differentiation

Heterogeneous duopolies with product differentiation and isoelastic price functions are examined, in which one firm is quantity setter and the other is price setter. The reaction functions and the Cournot–Bertrand (CB) equilibrium are first determined. It is shown that the best response dynamics with continuous time scales and without time delays, is always locally asymptotically stable. This stability can be, however, lost in the presence of time delays. Both fixed and continuously distributed time delays are examined, stability conditions derived and the stability regions determined and illustrated. The results are compared to Cournot–Cournot (CC) and Bertrand–Bertrand (BB) dynamics. It turns out that continuously distributed lags have a smaller destabilizing effect on the equilibria than fixed lags, and both homogeneous (CC and BB) competitions are more stable than the heterogeneous competitions.     

Dynamics of a class of Cohen–Grossberg neural networks with time-varying delays

Without assuming the boundedness, monotonicity, and differentiability of activation functions and any symmetry of interconnections, we establish some sufficient conditions for the global exponential stability of a unique equilibrium and the existence of periodic solution for the Cohen–Grossberg neural network with time-varying delays. Brouwer's fixed point theorem, matrix theory, a continuation theorem of the coincidence degree and inequality analysis are employed. Our results are all independent of the delays and maybe more convenient to design a circuit network.     

Fixed points and exponential stability of mild solutions of stochastic partial differential equations with delays

The fixed-point theory is first used to consider the stability for stochastic partial differential equations with delays. Some conditions for the exponential stability in pth mean as well as in sample path of mild solutions are given. These conditions do not require the monotone decreasing behavior of the delays, which is necessary in [T. Caraballo, K. Liu, Exponential stability of mild solutions of stochastic partial differential equations with delays, Stoch. Anal. Appl. 17 (1999) 743-763; Ruhollan Jahanipur, Stability of stochastic delay evolution equations with monotone nonlinearity, Stoch. Anal. Appl. 21 (2003) 161-181]. Even in this special case, our results also improve the results in [T. Caraballo, K. Liu, Exponential stability of mild solutions of stochastic partial differential equations with delays, Stoch. Anal. Appl. 17 (1999) 743-763].     

Stability results for nonlinear functional differential equations using fixed point methods

We present new conditions for stability of the zero solution for three distinct classes of scalar nonlinear delay differential equations. Our approach is based on fixed point methods and has the advantage that our conditions neither require boundedness of delays nor fixed sign conditions on the coefficient functions. Our work extends and improves a number of recent stability results for nonlinear functional differential equations in a unified framework. A number of examples are given to illustrate our main results.     

The maximum principle in optimal control problems with concentrated and distributed delays in controls

In the present work there has been posed and studied a general nonlinear optimal problem and a quasi-linear optimal problem with fixed time and free right end. It contains absolutely continuous monotone delays in phase coordinates and absolutely continuous monotone and distributed delays in controls. For these problems the necessary and, respectively, sufficient conditions of optimality in the form of the maximum principle have been proved.     

Neural networks with discrete and distributed time-varying delays: A general stability analysis

In this paper, the global asymptotic and exponential stability are investigated for a class of neural networks with both the discrete and distributed time-varying delays. By using appropriate Lyapunov–Krasovskii functional and linear matrix inequality (LMI) technique, several delay-dependent sufficient conditions are obtained to guarantee the global asymptotic and exponential stability of the addressed neural networks. These conditions are expressed in terms of LMIs, and are dependent on both the discrete and distributed time delays. Therefore, the stability of the neural networks can be checked readily by resorting to the Matlab LMI toolbox. In addition, the proposed stability criteria do not require the monotonicity of the activation functions and the differentiability of the discrete and distributed time-varying delays, which means that our results generalize and further improve those in the earlier publications. A simulation example is given to show the effectiveness and less conservatism of the obtained conditions.     

Existence of multiple positive periodic solutions for differential equation with state-dependent delays

By utilizing a fixed point theorem in cones, we present some sufficient conditions which guarantee the existence of multiple positive periodic solutions for a class of differential equations with state-dependent delays. Our results extend and improve some previous results.     

Exponential stability of discrete-time systems with slowly time-varying delays

Slowly time-varying delays are seldom, but do need to be, considered in the context of discrete-time systems. This paper addresses the exponential stability issue of discrete-time systems with slowly time-varying delays. The basic idea is to transform, by utilizing the switching transformation approach, the original system with slowly time-varying delays into an equivalent switched system with special switching signal. Different types of delays correspond to different types of switching signals, and the stability issue of the original system is converted into that of a switched system. It is the first time that the method of switched homogeneous polynomial Lyapunov function is applied to general delayed systems. Some sufficient exponential stability conditions for the original system are proposed in several situations. It is numerically shown that the conservativeness of the proposed conditions reduces as the degree of the switched homogeneous polynomial Lyapunov function increases.     

中立多变时滞Volterra型随机动力系统的稳定性

探讨了一类非线性随机积分微分动力系统,并通过Banach不动点方法,给出了该系统零解均方渐近稳定的充要条件,形成了中立多变时滞Volterra型随机积分微分动力系统零解均方渐近稳定性定理。与前人的研究方法不同,该文根据多变时滞随机动力系统各时滞的特点,灵活构造算子,相比以往文献的方法更加灵活实用。文章的结论一定程度上改进和发展了相关研究论文的结果。另外,文章所得结论补充并推广了不动点方法在研究非线性中立多变时滞Volterra型随机积分微分动力系统零解稳定性方面的成果。     

State constrained optimal control problems with time delays

In a recent, related, paper, necessary conditions in the form of a Maximum Principle were derived for optimal control problems with time delays in both state and control variables. Different versions of the necessary conditions covered fixed end-time problems and, under additional hypotheses, free end-time problems. These conditions improved on previous conditions in the following respects. They provided the first fully non-smooth Pontryagin Maximum Principle for problems involving delays in both state and control variables, only special cases of which were previously available. They provide a strong version of the Weierstrass condition for general problems with possibly non-commensurate control delays, whereas the earlier literature does so only under structural assumptions about the dynamic constraint. They also provided a new ‘two-sided’ generalized transversality condition, associated with the optimal end-time. This paper provides an extension of the Pontryagin Maximum Principle of the earlier paper for time delay systems, to allow for the presence of a unilateral state constraint. The new results fully recover the necessary conditions of the earlier paper when the state constraint is absent, and therefore retain all their advantages but in a setting of greater generality.