Open Journal of Applied Sciences, 2015, 5, 651-660 Published Online October 2015 in SciRes. http://www.scirp.org/journal/ojapps http://dx.doi.org/10.4236/ojapps.2015.510064 How to cite this paper: Sood, A. and Srivastava, S.K. (2015) Integral Φ 0 -Stability of Impulsive Differential Equations. Open Journal of Applied Sciences, 5, 651-660. http://dx.doi.org/10.4236/ojapps.2015.510064 Integral Φ 0 -Stability of Impulsive Differential Equations Anju Sood 1 , Sanjay K. Srivastava 2 1 Applied Sciences Department (Research Scholar-1113002), Punjab Technical University, Kapurthala, India 2 Applied Sciences Department (Mathematics), Beant College of Engineering and Technology, Gurdaspur, India Email: anjusood36@yahoo.com Received 24 September 2015; accepted 27 October 2015; published 30 October 2015 Copyright © 2015 by authors and Scientific Research Publishing Inc. This work is licensed under the Creative Commons Attribution International License (CC BY). http://creativecommons.org/licenses/by/4.0/ Abstract In this paper, the notions of integral 0 φ -stability of ordinary impulsive differential equations are introduced. The definition of integral 0 φ -stability depends significantly on the fixed time impulses. Sufficient conditions for integral 0 φ -stability are obtained by using comparison principle and piecewise continuous cone valued Lyapunov functions. A new comparison lemma, connecting the solutions of given impulsive differential system to the solution of a vector valued impulsive diffe- rential system is also established. Keywords Integral 0 φ -Stability, Cone Valued Lyapunov Functions, Impulsive Differential Equations, Fixed Time Impulses 1. Introduction Impulsive differential equations have been developed in modeling impulsive problems in physics, population dynamics, ecology, biological systems, industrial robotics, optimal control, bio-technology and so forth. In view of the vast applications, the fundamental and qualitative properties i.e. stability, boundedness etc. of such equa- tions are studied extensively in past decades. Several types of stability have been defined and established in lite- rature by academicians for impulsive ordinary differential equations. Various techniques such as scalar valued piecewise continuous Lyapunov functions, vector valued piecewise continuous Lyapunov functions, Rajumikhin method, comparison principle etc. have been employed to establish stability results. To the best of our knowledge, the concept of integral stability and 0 φ -stability were introduced for ordinary differential equations by Lakshmikantham in 1969 [1] and by Akpan in 1992 [2] respectively. Later, these sta-