Three Positive Periodic Solutions for Dynamic Equations With Piecewise Constant Argument and Impulse on Time Scales

Abstract

Abstract. In this paper, by using the Leggett-Williams fixed point theorem, the existence of three positive periodic solutions for differential equations with piecewise constant argument and impulse on time scales is investigated. Some easily verifiable sufficient criteria are established. Finally, an example is given to illustrate the results.2010 Mathematics Subject Classification. 34N05, 34K45, 34K13.1. Introduction. Impulsive differential equations, which arise in physics, population dynamics, economics, etc., are important mathematical tools for a better understanding of many real-world models, we refer the reader to [1][2][3][4][5] and the references therein. The study of differential equations on time scales, which has been created in order to unify the study of differential and difference equations, is an area of mathematics that has recently gained a lot of attention, moreover, many results on this issue have been well documented in the monographs [6][7][8]. The study of differential equations with piecewise constant arguments (EPCA) was initiated by Aftabizadeh and Wiener [9]. They observed that the change of sign in the argument deviation leads not only to interesting periodic properties but also to complications in the asymptotic and oscillatory behaviour of solutions. Various qualitative behaviours of solutions for EPCA have been investigated by many authors (see e.g. Refs. [9][10][11][12][13][14][15][16][17]).To the best of the authors' knowledge, there have been no results about the existence of multiple solutions of impulsive differential equations with piecewise constant arguments and parameters. In this paper, by using the Leggett-Williams multiple fixed point theorem, we shall consider the following equation on time scales:

Existence and global exponential stability of periodic solutions for n-dimensional neutral dynamic equations on time scales

“…Since then, the study on dynamic equations on time scales has received much attention of many scholars. For example, in DaCunha ( 2005 ), the author studied the stability of the following linear dynamic equation on time scales: In Du and Tien ( 2007 ), the authors obtained some conditions ensuring the stability of the trivial solution for the following dynamic equation on time scales: For other studies on dynamic equations on time scales, we refer the reader to Bohner and Peterson ( 2001 ), Graef and Hill ( 2015 ), Li and Sun ( 2013 ), Li and Xu ( 2011 ), Lupulescu and Younus ( 2011 ), Su and Feng ( 2014 ), Wang et al. ( 2010 ), Zhang et al.…”