φ0-STABILITY OF IMPULSIVE HYBRID SETVALUED DIFFERENTIAL EQUATIONS WITH DELAY BY PERTURBING LYAPUNOV FUNCTIONS

Titleφ0-STABILITY OF IMPULSIVE HYBRID SETVALUED DIFFERENTIAL EQUATIONS WITH DELAY BY PERTURBING LYAPUNOV FUNCTIONS
Publication TypeJournal Article
Year of Publication2008
AuthorsAHMAD, BASHIR, SIVASUNDARAM, S
Volume12
Issue2
Start Page137
Pagination9
Date Published2008
ISSN1083-2564
AMS34D20, 34K45
Abstract

We study ${\varphi_0}$-stability of the null solution of impulsive set differential system with delay by means of the perturbing Lyapunov function method. Sufficient conditions for the ${\varphi_0}$- stability of the null solution of impulsive set differential equations with delay are presented.

URLhttp://www.acadsol.eu/en/articles/12/2/3.pdf
Refereed DesignationRefereed
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REFERENCES
[1] Akpan, E.P. and Akinyele, O., On the φ0-stability of comparison differential systems, J. Math.
Anal. Appl. 164(1992) 307–324.
[2] Bainov, D.D. and Simeonov, P.S., Systems with Impulse Effect, Ellis Horwood, Chichester, 1989.
[3] Brandao Lopes Pinto, A.J., De Blasi, F.S. and Iervolino, F., Uniqueness and existence theorems
for differential equations with compact convex valued solutions, Boll. Unione. Mat. Italy 3(1970) 47–54.
[4] Gnana Bhaskar, T. and Lakshmikantham, V., Set differential equations and flow invariance,
Appl. Anal. 82(2003) 357–368.
[5] Gnana Bhaskar, T. and Lakshmikantham, V., Lyapunov stability for set differential equations,
Dynam. Systems Appl. 13(2004) 1–10.
[6] Guo, D and Lakshmikantham, V., Nonlinear Problems in Abstract Cones, Academic Press, New York, 1988.
[7] Hale, J., Theory of Functional Differential Equations, Springer-Verlag, New York, 1977.
[8] Lakshmikantham, V., Bainov, D.D. and Simeonov, P.S., Theory of Impulsive Differential Equations,
World Scientific, Singapore, 1989.
[9] Lakshmikantham, V. and Leela, S., On perturbing Lyapunov functions, Math. Systems Theory
10(1976) 85–90.
[10] Lakshmikantham, V., Leela, S. and Vatsala, A.S., Setvalued hybrid differential equations and
stability in terms of two measures, J. Hybrid Systems 2(2002) 169–187.
[11] Lakshmikantham, V. and Leela, S., Vatsala, A.S., Interconnection between set and fuzzy
differential equations, Nonlinear Anal. 54(2003) 351–360.
[12] Lakshmikantham, V., Leela, S. and Vatsala, A.S., Stability theory for set differential equations,
Dyn. Contin. Discrete impuls. syst. 11(2004) 181–189.
[13] Lakshmikantham, V., Matrosov, V.M. and Sivasundaram, S., Vector Lyapunov Functions and
Stability Analysis of Nonlinear Systems, Kluwer Academic Publishers, Dordrecht, 1991.
[14] Lakshmikantham, V. and Tolstonogov, A., Existence and interrelation between set and fuzzy
differential equations, Nonlinear Anal. 55(2003) 255–268.
[15] Leela, S., Mcrae, F.A. and Sivasundaram, S., controllability of impulsive differential equations,
J. Math. Anal. Appl. 177(1993) 24–30.
[16] Mcrae, F.A. and Devi, J.V., Impulsive set differential equations with delay, Applicable Anal.
84(2005) 329–341.
[17] Samoilenko, A.M., and Perestyuk, N.A.,Impulsive Differential Equations, World Scientific,
Singapore, 1995.
[18] Soliman, A.A., On cone perturbing Lyapunov function for impulsive differential systems, App.
Math. Comput. 163(2005) 1069–1079.
[19] Teran, P. An embedding theorem for convex fuzzy sets, Fuzzy Sets an Systems 152(2005)
191–208.
[20] Tolstonogov, A., Differential Inclusions In A Banach Space, Kluwer Academic Publishers,
Dordrecht, 2000.