Oscillation and Asymptotic Properties of Second Order Half-Linear Differential Equations with Mixed Deviating Arguments

In this paper, we study oscillation and asymptotic properties for half-linear second order differential equations with mixed argument of the form r(t)(y′(t))α′=p(t)yα(τ(t)). Such differential equation may possesses two types of nonoscillatory solutions either from the class N0 (positive decreasing solutions) or N2 (positive increasing solutions). We establish new criteria for N0=∅ and N2=∅ provided that delayed and advanced parts of deviating argument are large enough. As a consequence of these results, we provide new oscillatory criteria. The presented results essentially improve existing ones even for a linear case of considered equations.

A fixed-point approach for decaying solutions of difference equations

A boundary value problem associated with the difference equation with advanced argument * Δ ( a n Φ ( Δ x n ) ) + b n Φ ( x n + p ) = 0 , n ≥ 1 is presented, where Φ ( u ) = | u | α sgn u , α > 0, p is a positive integer and the sequences a , b , are positive. We deal with a particular type of decaying solution of (*), that is the so-called intermediate solution (see below for the definition). In particular, we prove the existence of this type of solution for (*) by reducing it to a suitable boundary value problem associated with a difference equation without deviating argument. Our approach is based on a fixed-point result for difference equations, which originates from existing ones stated in the continuous case. Some examples and suggestions for future research complete the paper. This article is part of the theme issue ‘Topological degree and fixed point theories in differential and difference equations’.

Matrix Measure Approach for Stability and Synchronization of Complex-Valued Neural Networks with Deviating Argument

This paper concentrates on global exponential stability and synchronization for complex-valued neural networks (CVNNs) with deviating argument by matrix measure approach. The Lyapunov function is no longer required, and some sufficient conditions are firstly obtained to ascertain the addressed system to be exponentially stable under different activation functions. Moreover, after designing a suitable controller, the synchronization of two complex-valued coupled neural networks is realized, and the derived condition is easy to be confirmed. Finally, some numerical examples are given to demonstrate the superiority and feasibility of the presented theoretical analysis and results.

Robust Exponential Stability of Recurrent Neural Networks with Deviating Argument and Stochastic Disturbance

It is well known that deviating argument and stochastic disturbance may derail the stability of recurrent neural networks (RNNs). This paper discusses the robustness of global exponential stability (GES) of RNNs accompanied with deviating argument and stochastic disturbance. For a given global exponentially stable RNNs, it is interesting to know how much the length of the interval of piecewise function and the interference intensity so that the disturbed system may still be exponentially stable. The available upper boundary of the range of piecewise variables and the interference intensity in the disturbed RNNs to keep GES are the solutions of some transcendental equations. Finally, some examples are provided to demonstrate the efficacy of the inferential results.

Law of universal gravitation with finite velocity of gravity and mathematical model of motion of a finite number of material points

The law of universal gravitation is intro- duced taking into account the finiteness of the gravi- tational velocity. Based on this law, a mathematical model of the motion of a finite number of material points is constructed, a separate case of which is the classical model of the motion of points, which is de- scribed by a system of ordinary differential equations. The constructed model is a system of nonlinear dif- ferential equations with deviating argument and func- tional equations. It more accurately describes the dy- namics of a finite number of material points than the corresponding classical model. A mathematical model of the motion of two material points is also considered.

Introduction

This introductory chapter provides an overview of time-delay systems. Time-delay systems, also called systems with after-effect or dead-time, hereditary systems, equations with deviating argument, or differential-difference equations, are ubiquitous in practice. Some representative examples are found in chemical industry, electrical and mechanical engineering, biomedical engineering, and management and traffic science. The most common forms of time delay in dynamic phenomena that arise in engineering practice are actuator and sensor delays. Due to the time it takes to receive the information needed for decision-making, to compute control decisions, and to execute these decisions, feedback systems often operate in the presence of delays. The chapter then illustrates the possible methods in control of time-delay systems. This book develops adaptive and robust predictor feedback laws for the compensation of the five uncertainties for general linear time-invariant (LTI) systems with input delays.