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FURTHER RESULTS ON POSITIVE PERIODIC SOLUTIONS OF IMPULSIVE FUNCTIONAL DIFFERENTIAL EQUATIONS AND APPLICATIONS

Part of: Qualitative theory Functional-differential and differential-difference equations Boundary value problems

Published online by Cambridge University Press:  27 July 2009

YUJI LIU
YUJI LIU*
Affiliation:
Department of Mathematics, Guangdong University of Business Studies, Guangzhou 510320, People’s Republic of China (email: [email protected])
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Abstract

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A class of first-order impulsive functional differential equations with forcing terms is considered. It is shown that, under certain assumptions, there exist positive T-periodic solutions, and under some other assumptions, there exists no positive T-periodic solution. Applications and examples are given to illustrate the main results.

Keywords

impulsefirst-order functional differential equationpositiveT-periodic solutionfixed point theorem

MSC classification

Secondary: 34B10: Nonlocal and multipoint boundary value problems 34K10: Boundary value problems 34C25: Periodic solutions 92D25: Population dynamics (general)
Type
Research Article
Information
The ANZIAM Journal , Volume 50 , Issue 4 , April 2009 , pp. 513 - 533
Copyright
Copyright © Australian Mathematical Society 2009

References

[1]Bartha, M., “Periodic solutions for differential equations with state-dependent delay and positive feedback”, Nonlinear Anal. 53 (2003) 839–857.CrossRefGoogle Scholar
[2]Chen, L. and Sun, J., “Nonlinear boundary value problem of first order impulsive functional differential equations”, J. Math. Anal. Appl. 318 (2006) 726–741.CrossRefGoogle Scholar
[3]Cheng, S. and Zhang, G., “Existence of positive periodic solutions for non-autonomous functional differential equations”, Electron. J. Differential Equations 59 (2001) 1–8.Google Scholar
[4]Choisy, M., Guegan, J. and Rohani, P., “Dynamics of infectious diseases and pulse vaccination: teasing apart the embedded resonance effects”, Physica D 22 (2006) 26–35.CrossRefGoogle Scholar
[5]Ding, W., Mi, J. and Han, M., “Periodic boundary value problems for the first order impulsive functional differential equations”, Appl. Math. Comput. 165 (2005) 433–446.Google Scholar
[6]d’Onofrio, A., “On pulse vaccination strategy in the SIR epidemic model with vertical transmission”, Appl. Math. Lett. 18 (2005) 729–732.CrossRefGoogle Scholar
[7]d’Onofrio, A., “A general framework for modeling tumor-immune system competition and immunotherapy: mathematical analysis and biomedical inferences”, Physica D 208 (2005) 220–235.CrossRefGoogle Scholar
[8]Gao, S., Chen, L., Nieto, J. and Torres, A., “Analysis of a delayed epidemic model with pulse vaccination and saturation incidence”, Vaccine 24 (2006) 6037–6045.CrossRefGoogle ScholarPubMed
[9]Gurney, W., Blythe, S. and Nisbet, R., “Nicholson blowflies revised”, Nature 287 (1980) 17–21.CrossRefGoogle Scholar
[10]Hale, J. and Lonel, S., Introduction to functional differential equations (Springer, Berlin, 1993).CrossRefGoogle Scholar
[11]Jiang, D., Nieto, J. and Zuo, W., “On monotone method for first and second order periodic boundary value problems and periodic solutions of functional differential equations”, J. Math. Anal. Appl. 289 (2004) 691–699.CrossRefGoogle Scholar
[12]Kang, S. and Zhang, G., “Existence of nontrivial periodic solutions for first order functional differential equations”, Appl. Math. Lett. 18 (2005) 101–107.CrossRefGoogle Scholar
[13]Kuang, Y. and Smith, H., “Slowly oscillating periodic solutions of nonautonomous state-dependent delay equations”, Nonlinear Anal. 19 (1992) 855–872.CrossRefGoogle Scholar
[14]Lakshmikantham, V., Bainov, D. and Simenov, P., Theory of impulsive differential equations (World Scientific, Singapore, 1989).CrossRefGoogle Scholar
[15]Li, W. and Huo, H., “Existence and global attractivity of positive periodic solutions of functional differential equations with impulses”, Nonlinear Anal. 59 (2004) 857–877.CrossRefGoogle Scholar
[16]Li, W. and Huo, H., “Global attractivity of positive periodic solutions for an impulsive delay periodic model of respiratory dynamics”, J. Comput. Appl. Math. 174 (2005) 227–238.CrossRefGoogle Scholar
[17]Li, X., Lin, X., Jiang, D. and Zhang, X., “Existence and multiplicity of positive periodic solutions to functional differential equations with impulse effects”, Nonlinear Anal. 62 (2005) 683–701.CrossRefGoogle Scholar
[18]Li, J., Nieto, J. and Shen, J., “Impulsive periodic boundary value problems of first order differential equations”, J. Math. Anal. Appl. 325 (2007) 226–236.CrossRefGoogle Scholar
[19]Li, J. and Shen, J., “Existence of positive periodic solutions to a class of functional differential equations with impulses”, Math. Appl. (Wuhan) 17 (2004) 456–463.Google Scholar
[20]Li, J. and Shen, J., “Periodic boundary value problems for delay differential equations with impulses”, J. Comput. Appl. Math. 193 (2006) 563–573.CrossRefGoogle Scholar
[21]Li, W. and Wang, Z., “Existence and global attractivity of positive periodic solutions for the impulsive delay Nicholson’s blowflies model”, J. Comput. Appl. Math. 50 (2005) 41–47.CrossRefGoogle Scholar
[22]Li, X., Zhang, X. and Jiang, D., “A new existence theory for positive periodic solutions to functional differential equations with impulse effects”, Comput. Math. Appl. 51 (2006) 1761–1772.CrossRefGoogle Scholar
[23]Li, Y. and Zhu, L., “Positive periodic solutions for a class of higher-dimensional state-dependent delay functional differential equations with feedback control”, Appl. Math. Comput. 159 (2004) 783–795.Google Scholar
[24]Liang, R. and Shen, J., “Periodic boundary value problem for the first order impulsive functional differential equations”, J. Comput. Appl. Math. 202 (2007) 498–510.CrossRefGoogle Scholar
[25]Liu, Y., “Further results on periodic boundary value problems for nonlinear first order impulsive functional differential equations”, J. Math. Anal. Appl. 327 (2007) 435–452.CrossRefGoogle Scholar
[26]Liu, Y., Bai, Z. and Ge, W., “Positive periodic solutions of impulsive delay differential equations with sign-changing coefficients”, Port. Math. 61 (2004) 177–192.Google Scholar
[27]Liu, Y. and Ge, W., “Asymptotic behavior of certain delay differential equations with forcing term”, J. Math. Anal. Appl. 280 (2003) 350–363.CrossRefGoogle Scholar
[28]Liu, Y. and Ge, W., “Global attractivity in delay “food-limited” models with exponential impulses”, J. Math. Anal. Appl. 287 (2003) 200–216.CrossRefGoogle Scholar
[29]Liu, Y. and Ge, W., “Positive periodic solutions of state-dependent functional differential equations”, Appl. Anal. 84 (2005) 1079–1094.CrossRefGoogle Scholar
[30]Liu, P. and Li, Y., “Positive periodic solutions of infinite delay functional differential equations depending on a parameter”, Appl. Math. Comput. 150 (2004) 159–168.Google Scholar
[31]Liu, Y., Xia, J. and Ge, W., “Positive periodic solutions of impulsive functional differential equations”, J. Appl. Math. Comput. 19 (2005) 261–280.CrossRefGoogle Scholar
[32]Liu, G., Yan, J. and Zhang, F., “Existence and global attractivity of unique positive periodic solution for a model of hematopoiesis”, J. Math. Anal. Appl. 334 (2007) 157–171.CrossRefGoogle Scholar
[33]Liu, Y. and Zhang, B., “Global attractivity of a class of delay differential equations with impulses”, ANZIAM J. 45 (2003) 271–284.CrossRefGoogle Scholar
[34]Mackey, M. and Glass, I., “Oscillations and chaos in physiological control systems”, Science 197 (1977) 287–289.CrossRefGoogle ScholarPubMed
[35]Nieto, J. and Rodriguez-Lopez, R., “Remarks on periodic boundary value problems for functional differential equations”, J. Comput. Appl. Math. 158 (2003) 339–353.CrossRefGoogle Scholar
[36]Nieto, J. and Rodriguez-Lopez, R., “Monotone method for first-order functional differential equations”, Comput. Math. Appl. 52 (2006) 471–484.CrossRefGoogle Scholar
[37]Nieto, J. and Rodriguez-Lopez, R., “Periodic boundary value problem for non-Lipschitzian impulsive functional differential equations”, J. Math. Anal. Appl. 318 (2006) 593–610.CrossRefGoogle Scholar
[38]Nieto, J. and Rodriguez-Lopez, R., “New comparison results for impulsive integro-differential equations and applications”, J. Math. Anal. Appl. 328 (2007) 1343–1368.CrossRefGoogle Scholar
[39]Pielou, E., Mathematical ecology (Wiley Interscience, New York, 1977).Google Scholar
[40]Qian, D. and Li, X., “Periodic solutions for ordinary differential equations with sublinear impulsive effects”, J. Math. Anal. Appl. 303 (2004) 288–303.CrossRefGoogle Scholar
[41]Rost, G., “On the global attractivity controversy for a delay model of hematopoiesis”, Appl. Math. Comput. 190 (2007) 846–850.Google Scholar
[42]Saker, S. and Agarwal, S., “Oscillatory and global attractivity in a periodic Nicholson’s blowflies model”, Math. Comput. Modelling 35 (2002) 719–731.CrossRefGoogle Scholar
[43]Tang, S. and Chen, L., “Density-dependent birth rate, birth pulses and their population dynamic consequences”, J. Math. Biol. 44 (2002) 185–199.CrossRefGoogle ScholarPubMed
[44]Tang, S. and Chen, L., “Global attractivity in a “food-limited” population model with impulsive effects”, J. Math. Anal. Appl. 292 (2004) 211–221.CrossRefGoogle Scholar
[45]Wang, H., “Positive periodic solutions of functional differential equations”, J. Differential Equations 202 (2004) 354–366.CrossRefGoogle Scholar
[46]Wang, W., Wang, H. and Li, Z., “The dynamic complexity of a three-species Beddington-type food chain with impulsive control strategy”, Chaos Solitons Fractals 32 (2007) 1772–1785.CrossRefGoogle Scholar
[47]Wazewska-Czyzewska, M. and Lasota, A., “Mathematical models of red cells system”, Mat. Stosow. 6 (1976) 25–40.Google Scholar
[48]Weng, P. and Liang, M., “The existence and behavior of periodic solutions of a Hematopoiesis model”, Math. Appl. 8 (1995) 434–439.Google Scholar
[49]Yan, J., “Existence and global attractivity of positive periodic solution for an impulsive Lasota–Wazewska model”, J. Math. Anal. Appl. 279 (2003) 111–120.CrossRefGoogle Scholar
[50]Yan, J., “Existence of positive periodic solutions of impulsive functional differential equations with two parameters”, J. Math. Anal. Appl. 327 (2007) 854–868.CrossRefGoogle Scholar
[51]Yan, J., Zhao, A. and Nieto, J., “Existence and global attractivity of positive periodic solution of periodic single species impulsive Lotka–Volterra systems”, Math. Comput. Modelling 40 (2004) 509–518.CrossRefGoogle Scholar
[52]Zhang, G. and Cheng, S., “Positive periodic solutions of nonautonomous functional differential equations depend on a parameter”, Abstr. Anal. Appl. 7 (2002) 279–286.CrossRefGoogle Scholar
[53]Zhang, W. and Fan, M., “Periodicity in a generalized ecological competition system governed by impulsive differential equations with delays”, Math. Comput. Modelling 39 (2004) 479–493.CrossRefGoogle Scholar
[54]Zhang, X., Shuai, Z. and Wang, K., “Optimal impulsive harvesting policy for single population”, Nonlinear Anal. Real World Appl. 4 (2003) 639–651.CrossRefGoogle Scholar