By using coincidence degree theory and Lyapunov functions, we study the existence and global exponential stability of antiperiodic solutions for a class of generalized neural networks with impulses and arbitrary delays on time scales. Some completely new sufficient conditions are established. Finally, an example is given to illustrate our results. These results are of great significance in designs and applications of globally stable anti-periodic Cohen-Grossberg neural networks with delays and impulses .
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:
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