## Boundedness for impulsive delay differential equations and applications to population growth models.(English)Zbl 1037.34061

Here, the authors apply Lyapunov functions and Razumikhin techniques for the boundedness of the following impulsive functional-differential equation $\dot{x}(t)=f(t,x_t),\quad t\neq \tau_k, \qquad \Delta x(t)=I(t,x_t-), \quad t=\tau_k,\;\lim \tau_k=\infty.$ Applications to some population growth models with delays are also given.

### MSC:

 34K12 Growth, boundedness, comparison of solutions to functional-differential equations 34K45 Functional-differential equations with impulses 34K60 Qualitative investigation and simulation of models involving functional-differential equations 92D25 Population dynamics (general) 34A37 Ordinary differential equations with impulses 34D40 Ultimate boundedness (MSC2000)
Full Text:

### References:

 [1] Ballinger, G.; Liu, X., On boundedness of solutions of impulsive systems, Nonlinear stud., 4, 1, 121-131, (1997) · Zbl 0879.34015 [2] Ballinger, G.; Liu, X., Permanence of population growth models with impulsive effects, Math. comput. modelling, 26, 59-72, (1997) · Zbl 1185.34014 [3] Ballinger, G.; Liu, X., Existence and uniqueness results for impulsive delay differential equations, Dcdis, 5, 579-591, (1999) · Zbl 0955.34068 [4] Ballinger, G.; Liu, X., Existence, uniqueness and boundedness results for impulsive delay differential equations, Appl. anal., 74, 71-93, (2000) · Zbl 1031.34081 [5] Gopalsamy, K., Stability and oscillations in delay differential equations of population dynamics, (1992), Kluwer Academic Publishers Dordrecht, Netherlands · Zbl 0752.34039 [6] Hale, J.K.; Lunel, S.M.V., Introduction to functional differential equations, (1993), Springer New York [7] Hofbauer, J.; Sigmund, K., The theory of evolution and dynamical systems, (1988), Cambridge University Press Cambridge, MA [8] Lakshmikantham, V.; Bainov, D.D.; Simeonov, P.S., Theory of impulsive differential equations, (1989), World Scientific Singapore · Zbl 0719.34002 [9] Lakshmikantham, V.; Liu, X., Stability criteria for impulsive differential equations in terms of two measures, J. math. anal. appl., 137, 591-604, (1989) · Zbl 0688.34031 [10] Liu, X., Stability results for impulsive differential systems with applications to population growth models, Dyn. stability systems, 9, 163-174, (1994) · Zbl 0808.34056 [11] Liu, X.; Ballinger, G., Uniform asymptotic stability of impulsive differential equations, Comput. math. appl., 41, 903-915, (2001) · Zbl 0989.34061 [12] Liu, X.; Ballinger, G., Existence and continuability of solutions for differential equations with delays and state-dependent impulses, Nonlinear anal. TMA, 51, 633-647, (2002) · Zbl 1015.34069 [13] Liu, X.; Shen, J.H., Razumikhin-type theorems on boundedness for impulsive functional differential equations, Dyn. systems appl., 9, 389-404, (2000) · Zbl 0971.34059
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.