Exponentially dichotomous linear systems of differential equations with piecewise constant argument

Date

2012-01-01

Author

Akhmet, Marat

Metadata

Show full item record

This work is licensed under a Creative Commons Attribution-NonCommercial-NoDerivatives 4.0 International License.

Item Usage Stats

143
views

0
downloads

© 2012 L & H Scientific Publishing, LLC.We consider differential equations with piecewise constant argument of generalized type. It is the first time, an attention is given to the exponential dichotomy of linear systems. Bounded, almost periodic and periodic solutions and their stability are discussed. The study is made in such a way that further construction of the theory will follow for ordinary differential equations. The results are illustrated by examples.

Subject Keywords

Almost periodic solutions, Exponential dichotomy, Periodic solutions, Piecewise constant argument

URI

https://www.scopus.com/inward/record.uri?partnerID=HzOxMe3b&scp=84888172227&origin=inward
https://hdl.handle.net/11511/99047

Journal

Discontinuity, Nonlinearity, and Complexity

DOI

https://doi.org/10.5890/dnc.2012.09.001

Collections

Department of Mathematics, Article

Suggestions

OpenMETU
Core

Almost periodic solutions of second order neutral differential equations with functional response on piecewise constant argument Akhmet, Marat (2013-01-01) © 2013 L & H Scientific Publishing, LLC.We consider second order functional differential equations with generalized piecewise constant argument. Conditions for existence, uniqueness and stability of Bohr almost periodic solutions are defined. Appropriate examples which illustrate the results are provided.
Exponential stability of periodic solutions of recurrent neural networks with functional dependence on piecewise constant argument Akhmet, Marat; Cengiz, Nur (null; 2015-08-25) Akhmet [1] generalized differential equations with piecewise constant argument by taking any piecewise constant functions as arguments, and recently he introduced functional dependence on piecewise constant argument [2]. These equations play an important role in applications such as neural networks [3]. In this study, we develope a model of recurrent neural network with functional dependence on piecewise constant argument of generalized type given by x 0 (t) = −Ax (t) + Ex (γ (t)) + Bh (xt) + Cg xγ(t) + D...
Global exponential stability of neural networks with non-smooth and impact activations Akhmet, Marat (2012-10-01) In this paper, we consider a model of impulsive recurrent neural networks with piecewise constant argument. The dynamics are presented by differential equations with discontinuities such as impulses at fixed moments and piecewise constant argument of generalized type. Sufficient conditions ensuring the existence, uniqueness and global exponential stability of the equilibrium point are obtained. By employing Green's function we derive new result of existence of the periodic solution. The global exponential s...
Impulsive Hopfield-type neural network system with piecewise constant argument Akhmet, Marat; Yılmaz, Elanur (2010-08-01) In this paper we introduce an impulsive Hopfield-type neural network system with piecewise constant argument of generalized type. Sufficient conditions for the existence of the unique equilibrium are obtained. Existence and uniqueness of solutions of such systems are established. Stability criterion based on linear approximation is proposed. Some sufficient conditions for the existence and stability of periodic solutions are derived. An example with numerical simulations is given to illustrate our results.
Almost periodic solutions of the linear differential equation with piecewise constant argument Akhmet, Marat (2009-10-01) The paper is concerned with the existence and stability of almost periodic solutions of linear systems with piecewise constant argument where t∈R, x ∈ Rn [·] is the greatest integer function. The Wexler inequality [1]-[4] for the Cauchy's matrix is used. The results can be easily extended for the quasilinear case. A new technique of investigation of equations with piecewise argument, based on an integral representation formula, is proposed. Copyright © 2009 Watam Press.

Citation Formats

M. Akhmet, “Exponentially dichotomous linear systems of differential equations with piecewise constant argument,” Discontinuity, Nonlinearity, and Complexity, vol. 1, no. 4, pp. 337–352, 2012, Accessed: 00, 2022. [Online]. Available: https://www.scopus.com/inward/record.uri?partnerID=HzOxMe3b&scp=84888172227&origin=inward.