Lyapunov method for boundedness of solutions of nonlinear impulsive functional differential equations Ivanka M. Stamova Department of Mathematics, Bourgas Free University, 8000 Bourgas, Bulgaria Abstract Sufficient conditions are investigated for boundedness of solutions of nonlinear impulsive functional differential equa- tions with impulses at fixed moments. The main tools are Lyapunov functions and Gronwall type of inequality. Ó 2005 Elsevier Inc. All rights reserved. Keywords: Impulsive functional differential equation; Lyapunov functions; Gronwall type of inequality; Boundedness 1. Introduction The impulsive differential equations can be successfully used for mathematical simulation of processes and phenomena which are subject to short-term perturbations during their evolution. The duration of the perturbations is negligible in comparison with the duration of the process considered, and they can be thought of as momentary. The theory of impulsive ordinary differential equations started with the pio- neer paper of Mil’man and Myshkis [13] and it is a subject of intensive investigations during the last three decades [1–5,8,9,14]. A natural generalization of impulsive ordinary differential equations are the impulsive functional differential equations. Such a generalization of the notion of an impulsive differential equation provide natural framework for mathematical simulation of many processes and phenomena in theory of optimal control, biology, population dynamics, bio-technologies, medicine, electronics, radio engineering, and economics. At the present time the theory of impulsive functional differential equations undergoes rapid development [1,4,5,9,15,16]. In this paper we investigate sufficient conditions for boundedness of solutions of nonlinear impulsive functional differential equations involving fixed moments of impulsive effect. Lyapunov functions and Gronwall type inequality are employed to obtain an estimate on the solutions in terms of delays and impulses. 0096-3003/$ - see front matter Ó 2005 Elsevier Inc. All rights reserved. doi:10.1016/j.amc.2005.09.107 E-mail address: stamova@bfu.bg Applied Mathematics and Computation 177 (2006) 714–719 www.elsevier.com/locate/amc