Existence of positive periodic solutions for two types of secondorder nonlinear neutral differential equations with variable delay
DOI:
https://doi.org/10.4067/S071609172013000400006Keywords:
Positive periodic solutions, Nonlinear neutral differential equations, Fixed point theorem.Abstract
In this article we study the existence of positive periodic solutions for two types of secondorder nonlinear neutral differential equation with variable delay. The main tool employed here is the Krasnoselskii's fixedpoint theoremdealing withasum of twomappings, one is a contraction and the other is completely continuous. The results obtained here generalize the work of Cheung, Ren and Han 7.References
[1] A. Ardjouni and A. Djoudi, Existence of periodic solutions for nonlinear neutral dynamic equations with variable delay on a time scale. Commun Nonlinear Sci Numer Simulat 17, pp. 3061—3069, (2012).
[2] A. Ardjouni and A. Djoudi, Periodic solutions for a secondorder nonlinear neutral differential equation with variable delay, Electronic Journal of Differential Equations, Vol. No. 128, pp. 17, (2011).
[3] A. Ardjouni and A. Djoudi, Periodic solutions in totally nonlinear dynamic equations with functional delay on a time scale, Rend. Sem. Mat. Univ. Politec. Torino Vol. 68, 4, pp. 349359, (2010).
[4] T. A. Burton, Liapunov functionals, fixed points and stability by Krasnoselskii’s theorem. Nonlinear Stud. 9(2002), No. 2, 181190.
[5] T. A. Burton, Stability by Fixed Point Theory for Functional Differential Equations, Dover Publications, New York, 2006.
[6] F. D. Chen, Positive periodic solutions of neutral LotkaVolterra system with feedback control, Appl. Math. Comput. 162, no. 3, pp. 1279 1302, (2005).
[7] W. S. Cheung, J. Ren and W. Han, Positive periodic solutions of second order neutral functional differential equations, Nonlinear Analysis 71, pp. 39483955, (2009).
[8] H. Deham and A. Djoudi, Periodic solutions for nonlinear differential equation with functional delay, Georgian Mathematical Journal 15, No. 4, pp. 635642, (2008).
[9] H. Deham and A. Djoudi, Existence of periodic solutions for neutral nonlinear differential equations with variable delay, Electronic Journal of Differential Equations, Vol. No. 127, pp. 18, (2010).
[10] Y. M. Dib, M. R. Maroun and Y. N. Raffoul, Periodicity and stability in neutral nonlinear differential equations with functional delay, Electronic Journal of Differential Equations, Vol. No. 142, pp. 111, (2005).
[11] E. R. Kaufmann, A nonlinear neutral periodic differential equation, Electron. J. Differential Equations, No. 88, pp. 1—8, (2010).
[12] Y. Liu and W. Ge, Positive periodic solutions of nonlinear duffing equations with delay and variable coefficients, Tamsui Oxf. J. Math. Sci. 20, pp. 235255, (2004).
[13] Y. Luo, W. Wang and J. Shen, Existence of positive periodic solutions for two kinds of neutral functional differential equations, Applied Mathematics Letters 21, pp. 581587, (2008).
[14] Y. N. Raffoul, Periodic solutions for neutral nonlinear differential equations with functional delay, Electron. J. Differential Equations, No. 102, pp. 1—7, (2003).
[15] Y. N. Raffoul, Stability in neutral nonlinear differential equations with functional delays using fixedpoint theory, Math. Comput. Modelling 40, No. 78, pp. 691—700, (2004).
[16] Y. N. Raffoul, Positive periodic solutions in neutral nonlinear differential equations, E. J. Qualitative Theory of Diff. Equ., No. 16, pp. 1—10, (2007).
[17] D. S. Smart, Fixed point theorems; Cambridge Tracts in Mathematics, No. 66. Cambridge University Press, LondonNew York, (1974).
[18] Q. Wang, Positive periodic solutions of neutral delay equations (in Chinese), Acta Math. Sinica (N.S.) 6, pp. 789795, (1996).
[19] E. Yankson, Positive periodic solutions for secondorder neutral differential equations with functional delay, Electron. J. Differential Equations, No. 14, pp. 1—6, (2012).
[20] W. Zeng, Almost periodic solutions for nonlinear Duffing equations, Acta Math. Sinica (N. S.) 13, pp. 373380, (1997).
[2] A. Ardjouni and A. Djoudi, Periodic solutions for a secondorder nonlinear neutral differential equation with variable delay, Electronic Journal of Differential Equations, Vol. No. 128, pp. 17, (2011).
[3] A. Ardjouni and A. Djoudi, Periodic solutions in totally nonlinear dynamic equations with functional delay on a time scale, Rend. Sem. Mat. Univ. Politec. Torino Vol. 68, 4, pp. 349359, (2010).
[4] T. A. Burton, Liapunov functionals, fixed points and stability by Krasnoselskii’s theorem. Nonlinear Stud. 9(2002), No. 2, 181190.
[5] T. A. Burton, Stability by Fixed Point Theory for Functional Differential Equations, Dover Publications, New York, 2006.
[6] F. D. Chen, Positive periodic solutions of neutral LotkaVolterra system with feedback control, Appl. Math. Comput. 162, no. 3, pp. 1279 1302, (2005).
[7] W. S. Cheung, J. Ren and W. Han, Positive periodic solutions of second order neutral functional differential equations, Nonlinear Analysis 71, pp. 39483955, (2009).
[8] H. Deham and A. Djoudi, Periodic solutions for nonlinear differential equation with functional delay, Georgian Mathematical Journal 15, No. 4, pp. 635642, (2008).
[9] H. Deham and A. Djoudi, Existence of periodic solutions for neutral nonlinear differential equations with variable delay, Electronic Journal of Differential Equations, Vol. No. 127, pp. 18, (2010).
[10] Y. M. Dib, M. R. Maroun and Y. N. Raffoul, Periodicity and stability in neutral nonlinear differential equations with functional delay, Electronic Journal of Differential Equations, Vol. No. 142, pp. 111, (2005).
[11] E. R. Kaufmann, A nonlinear neutral periodic differential equation, Electron. J. Differential Equations, No. 88, pp. 1—8, (2010).
[12] Y. Liu and W. Ge, Positive periodic solutions of nonlinear duffing equations with delay and variable coefficients, Tamsui Oxf. J. Math. Sci. 20, pp. 235255, (2004).
[13] Y. Luo, W. Wang and J. Shen, Existence of positive periodic solutions for two kinds of neutral functional differential equations, Applied Mathematics Letters 21, pp. 581587, (2008).
[14] Y. N. Raffoul, Periodic solutions for neutral nonlinear differential equations with functional delay, Electron. J. Differential Equations, No. 102, pp. 1—7, (2003).
[15] Y. N. Raffoul, Stability in neutral nonlinear differential equations with functional delays using fixedpoint theory, Math. Comput. Modelling 40, No. 78, pp. 691—700, (2004).
[16] Y. N. Raffoul, Positive periodic solutions in neutral nonlinear differential equations, E. J. Qualitative Theory of Diff. Equ., No. 16, pp. 1—10, (2007).
[17] D. S. Smart, Fixed point theorems; Cambridge Tracts in Mathematics, No. 66. Cambridge University Press, LondonNew York, (1974).
[18] Q. Wang, Positive periodic solutions of neutral delay equations (in Chinese), Acta Math. Sinica (N.S.) 6, pp. 789795, (1996).
[19] E. Yankson, Positive periodic solutions for secondorder neutral differential equations with functional delay, Electron. J. Differential Equations, No. 14, pp. 1—6, (2012).
[20] W. Zeng, Almost periodic solutions for nonlinear Duffing equations, Acta Math. Sinica (N. S.) 13, pp. 373380, (1997).
How to Cite
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A. Ardjouni and A. Djoudi, “Existence of positive periodic solutions for two types of secondorder nonlinear neutral differential equations with variable delay”, Proyecciones (Antofagasta, On line), vol. 32, no. 4, pp. 377391, 1.
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