Abstract
By using Leggett and Williams’ multiple positive fixed points theorems, sufficient conditions are obtained for the existence or nonexistence of multiple positive periodic solutions for two kinds of functional difference equations with delays and impulses. An example is employed to illustrate our results.
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References
Cao J. and Ren F., Exponential stability of discrete-time genetic regulatory networks with delays, IEEE Tran. Neur. Net., 19(3), 520–523, (2008)
Yang X., Cui X. and Long Y., Existence and global exponential stability of periodic solution of a cellular neural networks difference equation with delays and impulses, Neural Networks, 22, 970–976, (2009)
Mohamad S., Global exponential stability in continuous-time and discrete-time delayed bidirectional neural networks, Physica D., 159, 233–251, (2001)
Raffoul Y. N., Positive periodic solutions of nonlinear functional difference equations, Electron. J. Differential Equations, 55, 1–8, (2002)
Ma M. and Yu J., Existence of multiple positive periodic solutions for nonlinear functional differential difference equations, J. Math. Anal. Appl., 305, 483–490, (2005)
Liu Y., Periodic solutions of nonlinear functional difference equations at nonresonance case, J. Math. Anal. Appl., 327, 801–815, (2007)
Li Y., Zhu L. and Liu P., Positive periodic solutions of nonlinear functional diffrence equations depending on a parameter, Comput. Math. Appl., 48, 1453–1459, (2004)
Wan A. and Jiang D., Existence of positive periodic solutions for functional differential equations, Kyushu J. Math., 56(1), 193–202, (2002)
Li X., Lin X. and Jiang D., et al, Existence and multiplicity of positive periodic solutions to functional differential equations with impulse effects, Nonlinear Analysis, 62, 683–701, (2005)
Huo H. F., Li W. T. and Liu X. Z., Existence and global attractivity of positive periodic solution of an impulsive delay differential equation, Appl. Anal., 83, 1279–1290, (2004)
Liu Y. and Ge W., Stability theorems and existence results for periodic solutions of nonlinear impulsive delay differential equations with variable coefficients, Nonlinear Analysis, 57, 363–399, (2004)
Song Q. K. and Cao J. D., Dynamical behaviors of discrete-time fuzzy cellular neural networks with variable delays and impulses, Journal of Franklin Institution, 345, 39–59, (2008)
Zhu W., Xu D.Y. and Yang Z. C., Global exponential stability of impulsive delay difference equation, Appl. Math. Comput., 181, 65–72, (2006)
Wang H., Positive periodic solutions of functional differential equations, J. Differential Equations, 202, 354–366, (2004)
Kuang Y. and Smith H. L., Periodic solutions of differential delay equations with thresholdtype delays, oscillations and dynamics in delay equations, Contemp. Math., 129, 153–176, (1992)
Jiang D., Wei J. and Zhang B., Positive periodic solutions of functional differential equations and population models, Electron. J. Differential Equations, 71, 1–13, (2002)
Leggett R. W. and Williams L. R., Multipule positive fixed points of nonlinear operators on ordered Banach spaces, Indiana Univ. Math. J., 28, 673–688, (1979)
Guo D. J., Nonlinear Functional Analysis, Shandong Science and Technology Press, Jinan, 2001 (in Chinese).
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Yang, X., Cao, J. Multiplicity of positive periodic solutions for delayed difference equation with impulses. Differ Equ Dyn Syst 17, 257–267 (2009). https://doi.org/10.1007/s12591-009-0019-5
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DOI: https://doi.org/10.1007/s12591-009-0019-5