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Existence and multiplicity of positive periodic solutions to functional differential equations with impulse effects. (English) Zbl 1084.34071
Existence and global attractivity of positive periodic solutions are studied for the following nonlinear delay equation with impulses $$\dot{y}(t)=-a(t)y(t)+g(t, y(t-\tau(t)),\quad t\neq t_j\ y(t_j^{+})=y(t_j^-)+I_j(y(t_j)).$$ As corollaries the authors obtain explicit conditions for many popular equations of mathematical biology: logistic equation, Mackey-Glass equation, blood cell production equation, and Nicholson’s blowflies equation.

##### MSC:
 34K45 Functional-differential equations with impulses 34K13 Periodic solutions of functional differential equations
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##### References:
 [1] Akhmetov, M. U.; Zafer, A.: Successive approximation method for quasilinear impulsive differential equations with control. Appl. math. Lett. 13, No. 5, 99-105 (2000) · Zbl 1125.93349 [2] Anokhin, A. V.; Berezansky, L.; Braverman, E.: Exponential stability of linear delay impulsive differential equations. J. math. Anal. appl. 193, 923-941 (1995) · Zbl 0837.34076 [3] Bainov, D. D.; Covachev, V.; Stamova, I.: Stability under persistent disturbances of impulsive differential-difference equations of neutral type. J. math. Anal. appl. 187, 790-808 (1994) · Zbl 0811.34057 [4] Bainov, D. D.; Simeonov, P. S.: Impulsive differential equations: periodic solutions and applications. (1993) · Zbl 0815.34001 [5] Cabada, A.; Nieto, J. J.; Franco, D.; Trofimchuk, S. I.: A generalization of the monotone method for second order periodic boundary value problem with impulses at fixed points. Dynamics continuous discrete impulsive systems 7, 145-158 (2000) · Zbl 0953.34020 [6] Deimling, K.: Nonlinear functional analysis. (1985) · Zbl 0559.47040 [7] Franco, D.; Liz, E.; Nieto, J. J.; Rogovchenko, Y. V.: A contribution to the study of functional differential equations with impulses. Math. nachr. 218, 49-60 (2000) · Zbl 0966.34073 [8] Gopalsamy, K.; Weng, P.: Global attractivity and level crossing in model of hoematopoiesis. Bull. inst. Math. acad. Sinica 22, 341-360 (1994) · Zbl 0829.34067 [9] Gurney, W. S. C.; Blythe, S. P.; Nisbet, R. M.: Nicholson’s blowfies revisited. Nature 287, 17-20 (1980) [10] Gyori, I.; Trofimchuk, S. I.: On the existence of rapidly oscillatory solutions in the Nicholson blowflies equation. Nonlinear anal. 48, 1033-1042 (2002) · Zbl 1007.34063 [11] He, Z.; Yu, J.: Periodic boundary value problem for first-order impulsive functional differential equations. J. comput. Appl: math. 138, 205-217 (2002) · Zbl 1004.34052 [12] Z. Jin, The study for ecological and epidemical models influenced by impulses, Doctoral Thesis, Xian Jiaotong University, 2001 (in Chinese). [13] Joseph, W.; So, H.; Yu, J.: Global attractivity and uniformly persistence in Nicholson’s blowfies. Differential equation dynamics systems 2, 11-18 (1994) · Zbl 0869.34056 [14] Krasnoselskii, M. A.: Positive solutions of operator equations. (1964) [15] Lakshmikantham, V.; Bainov, D. D.; Simeonov, P. S.: Theory of impulsive differential equations. (1989) · Zbl 0719.34002 [16] Lan, K.; Webb, J. R. L.: Positive solutions of semilinear differential equations with singularities. J. differential equations 148, 407-421 (1998) · Zbl 0909.34013 [17] Li, W.; Huo, H.: Existence and global attractivity of positive periodic solutions of functional differential with impulsives. Nonlinear anal. 59, 857-877 (2004) · Zbl 1061.34059 [18] Li, W.; Huo, H.: Global attractivity of positive periodic solutions for an impulsive delay periodic model of respiratory dynamics. J. comput. Appl: math. 174, 227-238 (2005) · Zbl 1070.34089 [19] Liu, X.; Shen, J.: Asymptotic behavior of solutions of impulsive neutral differential equations. Appl. math. Lett. 12, No. 7, 51-58 (1999) · Zbl 0943.34065 [20] E. Liz, Boundary value problems for new types of differential equations, Ph.D. Thesis, University of Vigo-Spain, 1994 (in Spanish). · Zbl 0812.45002 [21] Luo, J.; Yu, J.: Global asymptotic stability of nonautonomous mathematical ecological equations with distributed deviating arguments. Acta math. Sinica 41, 1273-1282 (1998) · Zbl 1027.34088 [22] Mackey, M. C.; Glass, L.: Oscillations and chaos in physiological control systems. Science 197, 287-289 (1987) [23] Mei, M.: Asymptotic stability of travelling waves for Nicholson’s blowflies equation with diffusion. Proc. roy. Soc. Edinburgh section 134, 579-594 (2004) · Zbl 1059.34019 [24] Nieto, J. J.: Basic theory for nonresonance impulsive periodic problems of first order. J. math. Anal. appl. 205, 423-433 (1997) · Zbl 0870.34009 [25] Nieto, J. J.: Impulsive resonance periodic problems of first order. Appl. math. Lett. 15, 489-493 (2002) · Zbl 1022.34025 [26] Qian, D.; Li, X.: Periodic solutions for ordinary differential equations with sublinear impulsive effects. J. math. Anal. appl. 303, 288-303 (2005) · Zbl 1071.34005 [27] Rogovchenko, Y. V.: Impulsive evolution systems: Main results and new trends. Dynamics continuous discrete impulsive systems 3, 57-88 (1997) · Zbl 0879.34014 [28] Samoilenko, A. M.; Perestyuk, N. A.: Impulsive differential equations. (1995) · Zbl 0837.34003 [29] Shen, J. H.: On some asymptotic stability results of impulsive integro-differential equations. Chinese math. Ann. 17A, 759-765 (1996) · Zbl 0877.34051 [30] Shen, J. H.: The existence of non-oscillatory solutions of delay differential equations with impulses. Appl. math. Comput. 77, 156-165 (1996) · Zbl 0861.34044 [31] Shen, J. H.; Yan, J. R.: Razumikhin type stability theorems for impulsive functional differential equations. Nonlinear anal. TMA 33, 519-531 (1998) · Zbl 0933.34083 [32] Wan, A.; Jiang, D.: Existence of positive periodic solutions for functional differential equations. Kyushu J. Math. 56, No. 1, 193-202 (2002) · Zbl 1012.34068 [33] Wan, A.; Jiang, D.; Xu, X.: A new existence theory for positive periodic solutions to functional differential equations. Comput. math. Appl. 47, 1257-1262 (2004) · Zbl 1073.34082 [34] Yan, J.: Oscillation of first-order impulsive differential equations with advanced argument. Comput. math. Appl. 42, 1353-1363 (2001) · Zbl 1005.34061 [35] Yan, J.; Zhao, A.; Nieto, J. J.: Existence and global attractivity of positive periodic solution of periodic single-species impulsive Lotka -- Volterra systems. Math. comput. Model. 40, 509-518 (2004) · Zbl 1112.34052 [36] Yu, J. S.; Zhang, B. G.: Stability theorems for delay differential equations with impulses. J. math. Anal. appl. 199, 162-175 (1996) · Zbl 0853.34068 [37] Zhang, W.; Fan, M.: Periodicity in a generalized ecological competition system governed by impulsive differential equations with delays. Math. comput. Model. 39, 479-493 (2004) · Zbl 1065.92066