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THREE POSITIVE PERIODIC SOLUTIONS FOR DYNAMIC EQUATIONS WITH PIECEWISE CONSTANT ARGUMENT AND IMPULSE ON TIME SCALES*

Published online by Cambridge University Press:  08 December 2010

YONGKUN LI  and
ERLIANG XU
YONGKUN LI
Affiliation:
Department of Mathematics, Yunnan University, Kunming, Yunnan 650091, P.R. China
ERLIANG XU
Affiliation:
Department of Mathematics, Yunnan University, Kunming, Yunnan 650091, P.R. China
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Abstract

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In this paper, by using the Leggett–Williams fixed point theorem, the existence of three positive periodic solutions for differential equations with piecewise constant argument and impulse on time scales is investigated. Some easily verifiable sufficient criteria are established. Finally, an example is given to illustrate the results.

Keywords

34N0534K4534K13
Type
Research Article
Information
Glasgow Mathematical Journal , Volume 53 , Issue 2 , May 2011 , pp. 369 - 377
Copyright
Copyright © Glasgow Mathematical Journal Trust 2010

References

REFERENCES

1.Lakshmikantham, V., Bainov, D. D. and Simeonov, P. S., Theorey of Impulsive Differential Equations (World Scientific, Singapore, 1989).Google Scholar
2.Samoilenko, A. M. and Perestyuk, N. A., Impulsive Differential Equations (World Scientific, Singapore, 1995).CrossRefGoogle Scholar
3.Zavalishchin, S. T. and Sesekin, A. N., Dynamic Impulse Systems. Theory and applications (Kluwer Academic Publishers Group, Dordrecht, 1997).CrossRefGoogle Scholar
4.Zhang, W. and Fan, M., Periodic in a generalized ecological competition system governed by impulsive differential equations with delays, J. Math. Comput. Model. 39 (2004), 479–493.Google Scholar
5.Bainov, D. D. and Simeonov, P. S., Impulsive differential equations: Periodic solutions and applications, in: Pitman Monographs and Surveys in Pure and Applied Mathematics, vol. 66 (Longman Scientific, New York, 1993).Google Scholar
6.Lakshmikantham, V., Sivasundaram, S. and Kaymarkcalan, B., Dynamic systems on measure chains (Kluwer Academic Publishers, Dordrecht, 1996).Google Scholar
7.Bohner, M. and Peterson, A., Dynamic equations on time scales, An introduction with applications (Birkhauser, Boston, 2001).CrossRefGoogle Scholar
8.Bohner, M. and Peterson, A., Advances in dynamic equations on time scales (Birkhauser, Boston, 2003).CrossRefGoogle Scholar
9.Aftabizadeh, A. R. and Wiener, J., Oscillatory and periodic solutions of an equation alternately of retarded and advanced type, Appl. Anal. 23 (1986), 219–231.CrossRefGoogle Scholar
10.Cooke, K. L. and Wiener, J., An equation alternately of retarded and advanced type, Proc. Amer. Math. Soc. 99 (1987), 726–732.CrossRefGoogle Scholar
11.Akhmet, M. U., Almost periodic solutions of differential equations with piecewise constant argument of generalized type, Nonlinear Anal. 2 (2008), 456–467.Google Scholar
12.Li, H. X., Almost periodic solutions of second-order neutral equations with piecewise constant arguments, Nonlinear Anal. 65 (2006), 1512–1520.Google Scholar
13.Li, H. X., Almost periodic weak solutions neutral delay-differential equations with piecewise constant argument, Nonlinear Anal. 64 (2006), 530–545.CrossRefGoogle Scholar
14.Akhmet, M. U., Stability of differential equations with piecewise constant arguments of generalized type, Nonlinear Anal. 68 (2008), 794–803.CrossRefGoogle Scholar
15.Pinto, M., Asymptotic equivalence of nonlinear and quasi linear differential equations with piecewise constant arguments, Math. Comput. Model. 49 (2009), 1750–1758.CrossRefGoogle Scholar
16.Akhmet, M. U. and Yilmaz, E., Impulsive Hopfield-type neural networks system with piecewise constant argument, Nonlinear Anal. Real World Appl. 11 (2010), 2584–2593.CrossRefGoogle Scholar
17.Liang, H., Liu, M. Z. and Lv, W. J., Stability of θ-schemes in the numerical solution of a partial dirrerential equation with piecewise continuous arguments, Appl. Math. Lett. 23 (2010), 198–206.CrossRefGoogle Scholar
18.Kaufmann, E. R. and Raffoul, Y. N., Periodic solutions for a neutral nonlinear dynamical equation on a time scale, J. Math. Anal. Appl. 319 (2006), 315–325.Google Scholar
19.Leggett, R. W. and Williams, L. R., Multiple positive fixed points of nonlinear operator on ordered Banach spaces, Indiana Univ. Math. J. 28 (1979), 673–688.CrossRefGoogle Scholar
20.Nieto, J. J., Basic theory for nonresonance impulsive periodic problems of first order, J. Math. Anal. Appl. 205 (1997), 423–433.Google Scholar