Stability for impulsive delay differential equations. (English) Zbl 1082.34069

Summary: The stability for the scalar impulsive delay differential equation \[ y'(t)+ a(t)y(t)+F(t,y(\cdot))=0,\quad t\geq 0,\;t\neq\tau_k, \]
\[ y(\tau_k^+)-y(\tau_k)= I_k\bigl(y(\tau_k)\bigr),\;k=1,2,\dots,\lim_{k \to\infty}\tau_k=\infty, \] where delay arguments may be bounded or unbounded, is investigated. Some new stability theorems are established which improve and extend several known results in the literature.


34K45 Functional-differential equations with impulses
34K20 Stability theory of functional-differential equations
Full Text: DOI


[1] Arino, O.; Kimmel, M., Stability analysis of models of cell production system, Math. model., 17, 1269-1300, (1986) · Zbl 0609.92018
[2] Bainov, D.D.; Simeonov, P.S., System with impulse effect, stability, theory and applications, (1989), Wiley New York · Zbl 0676.34035
[3] Balliger, G.; Liu, X., Existence, uniqueness and boundedness results for impulsive delay differential equations, Appl. anal., 74, 71-93, (2000) · Zbl 1031.34081
[4] Berezansky, L.; Braverman, E., Explicit conditions of exponential stability for a linear impulsive delay differential equation, J. math. anal. appl., 214, 439-458, (1997) · Zbl 0893.34069
[5] Gopalsamy, K., Stability and oscillation in delay differential equations of population dynamics, (1992), Kluwer Boston · Zbl 0752.34039
[6] Gyori, I.; Ladas, G., Oscillation theory of delay differential equation with application, (1991), Clarendon Oxford · Zbl 0780.34048
[7] Hale, J.K., Theory of functional differential equations, (1977), Springer New York · Zbl 0425.34048
[8] Kuang, Y., Delay differential equations with applications in population dynamics, (1993), Academic Press London · Zbl 0777.34002
[9] Lakshmikantham, V.; Bainov, D.D.; Simeonov, P.S., Theory of impulsive differential equations, (1989), World Scientific Singapore · Zbl 0719.34002
[10] Liu, X.; Ballinger, G., Existence and continuability of solutions for differential equations with delay and state-dependent impulses, Nonlinear anal., 15, 633-647, (2002) · Zbl 1015.34069
[11] Muroya, Y., On Yoneyama’s 3/2 stability theorems for one-dimensional delay differential equations, J. math. anal. appl., 247, 314-322, (2000) · Zbl 0977.34070
[12] Tang, X.H.; Yu, J.S., Global attractivity of the zero solution of logistic type functional differential equations with impulses, Acta Mathematica sinica, 45, 941-952, (2002) · Zbl 1026.34087
[13] Wazewska-Cztaewska, M.; Lasota, A., Mathematical problems of the dynamics of red blood cells system, annals of the Polish mathematical society, series III, Appl. math., 17, 23-40, (1988)
[14] Yoneyama, T., The 3/2 stability theorem for one-dimensional delay differential equations with unbounded delay, J. math. anal. appl., 165, 133-143, (1992) · Zbl 0755.34074
[15] Yoneyama, T.; Sugie, J., Perturbing uniformly stable nonlinear scalar delay-differential equations, Nonlinear anal., 12, 303-311, (1988) · Zbl 0651.34071
[16] Yorke, J.A., Asymptotic stability for one dimensional differential-delay equations, J. differential equations, 7, 189-202, (1970) · Zbl 0184.12401
[17] Yu, J.S., Explicit conditions for stability of nonlinear scalar delay differential equation with impulses, Nonlinear anal., 46, 3-67, (2001) · Zbl 0986.34063
[18] Yu, J.S.; Zhang, B.G., Stability theorem for delay differential equations with impulses, J. math. anal. appl., 199, 162-175, (1996) · Zbl 0853.34068
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.