IJPAM: Volume 116, No. 3 (2017)

Title

EVENTUAL STABILITY WITH TWO MEASURES
FOR NONLINEAR DIFFERENTIAL EQUATIONS
WITH NON-INSTANTANEOUS IMPULSES

Authors

Todor Kostadinov$^1$, Krasimira Ivanova$^2$
$^1$Technical University of Sofia, branch Plovdiv
Plovdiv, 4000, ul. Canko Dyustabanov 25, BULGARIA
$^2$Department of Applied Mathematics
University of Plovdiv Paisii Hilendarski
Plovdiv, 4000, ul. Tzar Asen 24, BULGARIA

Abstract

The integral stability of the solutions of a nonlinear differential equation with non-instantaneous impulses is studied using Lyapunov like functions. In these differential equation we have impulses, which start abruptly at some points and their action continue on given finite intervals. Sufficient conditions for integral stability are established.

History

Received: 2017-07-08
Revised: 2017-09-06
Published: October 26, 2017

AMS Classification, Key Words

AMS Subject Classification: 34A34, 34A08, 34D20
Key Words and Phrases: non-instantaneous impulses, integral stability, Lyapunov functions

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Bibliography

1
R. P. Agarwal, S.Hristova, Strict stability in terms of two measures for impulsive differential equations with ‘supremum’ , Appl. Anal., 91, 7, (2012), 1379-1392.

2
R. Agarwal, D. O'Regan, S. Hristova, Stability of Caputo fractional differential equations with non-instantaneous impulses, Commun. Appl. Anal.. (accepted)

3
R Agarwal, D O'Regan, S Hristova, Stability by Lyapunov like functions of nonlinear differential equations with non-instantaneous impulses, J. Appl. Math. Comput. 53 (1-2), (2015), 147-168.

4
M. U. Akhmetov and A. Zafer, Stability of the zero solution of impulsive differential equations by the Lyapunov second method, J. Math. Anal. Appl., 248, (2000), 69-82.

5
M. Feckan, J.R. Wang, Y. Zhou, Periodic solutions for nonlinear evolution equations with non-instantaneous impulses, Nonauton. Dyn. Syst., 1, (2014), 93-101.

6
J. Henderson, S. Hristova, Eventual practical stability and cone valued Lyapunov functions for differential equations with" maxima", Commun. Appl. Anal., 14, 3, (2010), 515-526.

7
E. Hernandez, D. O'Regan, On a new class of abstract impulsive differential equations, Proc. Amer. Math. Soc., 141, (2013), 1641-1649.

8
S. Hristova, Qualitative Investigations and Approximate Methods for Impulsive Equations, Nova Sci. Publ. Inc., New York, 2009.

9
S. Hristova, Stability on a cone in terms of two measures for impulsive differential equations with “supremum”, Appl. Math. Lett., 23, 5, (2010), 508-511.

10
S.G. Hristova, Razumikhin method and cone valued Lyapunov functions for impulsive differential equations with'supremum', Inter. J. Dynam. Syst. Diff. Eq. 2 , 3, (2009) DOI: 10.1504/IJDSDE.2009.031103

11
S. Hristova, Integral stability in terms of two measures for impulsive differential equations with 'supremum", Commun. Appl. Nonl. Anal., 16, 3, (2009),37-49.

12
S Hristova, Lipschitz stability for impulsive differential equations with ‘supremum’, Int. Electron. J. Pure Appl. Math , 1, (4), (2010), 345-358.

13
S. G. Hristova, A. Georgieva, Practical Stability in terms of two measures for impulsive differential equations with “supremum”, Inter. J. Diff. Eq., 2011, Article ID 703189, (2011).

14
S. Hristova, R. Terzieva, Lipschitz stability of differential equations with non-instantaneous impulses, Adv. Diff. Eq., 2016, 1, (2016), 322

15
V. Lakshmikantham, D.D. Bainov, P.S. Simeonov, Theory of Impulsive Differential Equations, World Scientific, Singapore, 1989.

16
V. Lakshmikantham, S. Leela, Differential and Integral Inequalities , vol I, Academic Press, New York, 1969.

17
Y. M. Liao, J. R. Wang, A note on stability of impulsive differential equations, Boundary Value Probl., 2014, 2014:67.

18
Z. Lin, W. Wei, J. R. Wang, Existence and stability results for impulsive integro-differential equations, Facta Univer. (Nis), ser. Math. Inform., 29, No 2 (2014), 119-130.

19
D.N. Pandey, S. Das, N. Sukavanam, Existence of solutions for a second order neutral differential equation with state dependent delay and not instantaneous impulses, Intern. J. Nonlinear Sci., 18, 2, (2014), 145-155.

20
M. Pierri, D. O’Regan, V. Rolnik, Existence of solutions for semi-linear abstract differential equations with not instantaneous impulses, Appl. Math. Comput., 219, (2013), 6743-6749.

21
A.A. Soliman, M.H. Abdalla, Integral stability criteria of nonlinear differential systems, Math. Comput. Model.  48 (2008), 258-267.

22
A. Sood, S. K. Srivastava, On stability of differential systems with noninstantaneous impulses, Math. Probl. Eng., 2015, 2015, Article ID 691687, 5 pages.

23
J. R. Wang, X. Li, Periodic BVP for integer/fractional order nonlinear differential equations with non-instantaneous impulses, J. Appl. Math. Comput., 46, 1-2, (2014), 321-334.

How to Cite?

DOI: 10.12732/ijpam.v116i3.21 How to cite this paper?

Source:
International Journal of Pure and Applied Mathematics
ISSN printed version: 1311-8080
ISSN on-line version: 1314-3395
Year: 2017
Volume: 116
Issue: 3
Pages: 751 - 764


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