Integral Φ0-Stability of Impulsive Differential Equations


In this paper, the notions of integral Φ0-stability of ordinary impulsive differential equations are introduced. The definition of integral Φ0-stability depends significantly on the fixed time impulses. Sufficient conditions for integral Φ0-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 differential system is also established.

Share and Cite:

Sood, A. and Srivastava, S. (2015) Integral Φ0-Stability of Impulsive Differential Equations. Open Journal of Applied Sciences, 5, 651-660. doi: 10.4236/ojapps.2015.510064.

Conflicts of Interest

The authors declare no conflicts of interest.


[1] Lakshmikantham, V. and Leela, S. (1969) Differential and Integral Inequalities—Theory and Applications. Academic Press, New York, 131-190.
[2] Akpan, E.P. and Akinyele, O. (1992) On the -Stability of Nonlinear Systems of Comparison Differential Systems. Journal of Mathematical Analysis and Applications, 164, 307-324.
[3] Akpan, E.P. (1993) On the -Stability of Perturbed Nonlinear Differential Systems. International Centre for Theoretical Physics, 1-13.
[4] Soliman, A.A. and Abdalla, M.H. (2010) Integral Stability Criteria of Nonlinear Differential Systems. Mathematical and Computer Modelling, 48, 258-267.
[5] Hristova, S.G. and Russinov, I. (2009) Stability in Terms of Two Measures for Initial Time Differences for Differential Equations by Perturbing Lyapunov Functions. International Journal of Pure and Applied Mathematics, 51, 19-32.
[6] Lakshmikantham, V., Bainov, D. and Simenov, P.S. (1989) Theory of Impulsive Differential Equations. World Scientific Publishing Co. Pvt. Ltd., Singapore, USA, England.
[7] Hristova, S.G. (2010) Integral Stability in Terms of Two Measures for Impulsive Differential Equations. Mathematical and Computer Modelling, 51, 100-108.

Copyright © 2021 by authors and Scientific Research Publishing Inc.

Creative Commons License

This work and the related PDF file are licensed under a Creative Commons Attribution 4.0 International License.