Dichotomy Of Linear Differential Equations Research Articles

Piecewise pseudo almost periodic solutions of interval general BAM neural networks with mixed time-varying delays and impulsive perturbations

<abstract><p>This paper is concerned with piecewise pseudo almost periodic solutions of a class of interval general BAM neural networks with mixed time-varying delays and impulsive perturbations. By adopting the exponential dichotomy of linear differential equations and the fixed point theory of contraction mapping. The sufficient conditions for the existence of piecewise pseudo almost periodic solutions of the interval general BAM neural networks with mixed time-varying delays and impulsive perturbations are obtained. By adopting differential inequality techniques and mathematical methods of induction, the global exponential stability for the piecewise pseudo almost periodic solutions of the interval general BAM neural networks with mixed time-varying delays and impulsive perturbations is discussed. An example is given to illustrate the effectiveness of the results obtained in the paper.</p></abstract>

Almost Automorphic Solutions for Quaternion-Valued Hopfield Neural Networks with Mixed Time-Varying Delays and Leakage Delays

This paper considers a class of quaternion-valued Hopfield neural networks with mixed time-varying delays and leakage delays. By utilizing the exponential dichotomy of linear differential equations, Banach’s fixed point theorem and differential inequality techniques, the authors obtain some sufficient conditions to ensure the existence and global exponential stability of almost automorphic solutions for this class of quaternion-valued neural networks. The results are completely new. Finally, the authors give an example to illustrate the feasibility of the results.

Almost Periodic Synchronization of Fuzzy Cellular Neural Networks with Time–Varying Delays via State–Feedback and Impulsive Control

Abstract In this paper, we are concerned with drive-response synchronization for a class of fuzzy cellular neural networks with time varying delays. Based on the exponential dichotomy of linear differential equations, the Banach fixed point theorem and the differential inequality technique, we obtain the existence of almost periodic solutions of this class of networks. Then, we design a state feedback and an impulsive controller, and construct a suitable Lyapunov function to study the problem of global exponential almost periodic synchronization for the drive-response systems considered. At the end of the paper, we provide an example to verify the effectiveness of the theoretical results.

Dynamics of impulsive neutral-type BAM neural networks

In this letter, the existence and the global exponential stability of piecewise pseudo almost periodic solutions (PAPT) for bidirectional associative memory neural networks (BAMNNs) with time-varying delay in leakage (or forgetting) terms and impulsive are investigated by applying contraction mapping fixed point theorem, the exponential dichotomy of linear differential equations and differential inequality techniques. Furthermore, we give an explanatory example to illustrate the efficiency of the theoretical predictions.

Influence of Leakage Delay on Almost Periodic Solutions for BAM Neural Networks

In this paper, we deal with a class of BAM neural networks with time-varying leakage delays. By applying the exponential dichotomy of linear differential equations, fixed point theorems and differential inequality techniques, we obtain some sufficient conditions which guarantee the existence and exponential stability of almost periodic solutions for such class of BAM neural networks. An example is provided to illustrate the effectiveness of the theoretical predictions. The results obtained in this paper are completely new and complement the previously known publications.

Almost Periodic Synchronization for Quaternion-Valued Neural Networks with Time-Varying Delays

This paper focuses on the global exponential almost periodic synchronization of quaternion-valued neural networks with time-varying delays. By virtue of the exponential dichotomy of linear differential equations, Banach’s fixed point theorem, Lyapunov functional method, and differential inequality technique, some sufficient conditions are established for assuring the existence and global exponential synchronization of almost periodic solutions of the delayed quaternion-valued neural networks, which are completely new. Finally, we give one example with simulation to show the applicability and effectiveness of our main results.

Existence and Global Exponential Stability of Almost Periodic Solutions for Shunting Inhibitory Cellular Neural Networks with Distributed Leakage Delays on Time Scales

In this paper, we deal with a class of shunting inhibitory cellular neural networks (SICNNs) with distributed leakage delays on time scales. Some sufficient conditions which ensure the existence and exponential stability of almost periodic solutions for such class of SICNNs are obtained by applying the exponential dichotomy of linear differential equations, Lapunov functional method and contraction mapping principle. An example is given to illustrate the effectiveness of the theoretical results. The results obtained in this paper are completely new and complement the previously known studies.

Existence and exponential stability of almost periodic solutions for neutral‐type BAM neural networks with distributed leakage delays

This paper is concerned with a class of neutral‐type BAM neural networks with distributed leakage delays. By applying the exponential dichotomy of linear differential equations, Lyapunov functional method and contraction mapping principle, we establish some sufficient conditions which ensure the existence and exponential stability of almost periodic solutions for such BAM neural networks. An example is given to illustrate the effectiveness of the theoretical findings. The results obtained in this article are completely new and complement the previously known studies. Copyright © 2016 John Wiley & Sons, Ltd.

Existence and exponential stability of almost periodic solution for neutral delay BAM neural networks with time-varying delays in leakage terms

This paper is concerned with a class of neutral delay BAM neural networks with time-varying delays in leakage terms. Some sufficient conditions are established to ensure the existence and exponential stability for such class of neural networks by employing the exponential dichotomy of linear differential equations, fixed point theorems and differential inequality techniques. An example is provided to show the effectiveness of the theoretical results. The results of this paper are completely new and complementary to the previously known results.

Lp-Dichotomy of Linear Differential Equations in an Arbitrary Banach Space

The notion of Lp-dichotomy for linear differential equations with possibly unbounded operator is introduced. By help of Banach fixed point theorem sufficient conditions for the existence of bounded solutions of nonlinear differential equations with an Lp-dichotomous linear part are obtained.

Exponential dichotomy of linear impulsive differential equations in a Banach space

A number of processes in physics, chemistry, biology, control theory, and robotics during their evolutionary development are subject to the action of short-time forces in the form of impulses. In most cases the duration of the action of these forces is negligibly small, as a result of which one can assume that the forces act only at certain moments of time. The impulsive differential equations represent a mathematical model of such processes. The work of Mil 'man and Myshkis (1960) marked the beginning of the mathematical theory of these equations, and the work of Bainov et al. (1988, 1989) marked the beginning of the mathematical theory of the same equations in abstract spaces. Samoilenko and Perestyuk (1987) published the first monograph dedicated to this subject. In the present paper the exponential dichotomy of linear impulsive equations in a Banach space is investigated. The exponential dichotomy of linear differential equations in the finite-dimensional case without impulses was investigated by Palmer (1979a, b; 1984a, b) and Eiaydi and Hajek (1988).

Exponential dichotomies for linear differential equations with delays: Periodic and autonomous equations

We consider a homogeneous linear differential equation with delays u.+Mu=0,where u takes values in a Banach space and M is a memory with bounded recall under natural Caratheodory conditions, i.e., a linear mapping from the continuous to the locally Bochner-integrable functions such that Mu on [a, b] depends only on u on [a−l, b] for a suitable recall bound l, and M satisfies a reasonable boundedness condition. The presence of the conditional exponential stability behaviour called an exponential dichotomy—which is known to be related to the existence of bounded solutions of the inhomogeneous equation for right-hand sides in a suitable function space — is examined when the equation is periodic; it is shown to be equivalent to a splitting of the spectrum of the transition operators corresponding to a period. More detailed results are obtained when the equation is autonomous.

Dichotomies for linear differential equations with delays: the Carathéodory case

This work is concerned with differential equations with delays, on [0, ∞[, of the form\(\dot u\)+Mu=r, and the corresponding homogeneous equation\(\dot u\)+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.