# Existence of positive periodic solutions for two types of second-order nonlinear neutral differential equations with variable delay

## DOI:

https://doi.org/10.4067/S0716-09172013000400006## Keywords:

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 second-order nonlinear neutral differential equation with variable delay. The main tool employed here is the Krasnosel-skii'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 second-order nonlinear neutral differential equation with variable delay, Electronic Journal of Differential Equations, Vol. No. 128, pp. 1-7, (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. 349-359, (2010).

[4] T. A. Burton, Liapunov functionals, fixed points and stability by Krasnoselskii’s theorem. Nonlinear Stud. 9(2002), No. 2, 181-190.

[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 Lotka-Volterra 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. 3948-3955, (2009).

[8] H. Deham and A. Djoudi, Periodic solutions for nonlinear differential equation with functional delay, Georgian Mathematical Journal 15, No. 4, pp. 635-642, (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. 1-8, (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. 1-11, (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. 235-255, (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. 581-587, (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 fixed-point theory, Math. Comput. Modelling 40, No. 7-8, 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, London-New York, (1974).

[18] Q. Wang, Positive periodic solutions of neutral delay equations (in Chinese), Acta Math. Sinica (N.S.) 6, pp. 789-795, (1996).

[19] E. Yankson, Positive periodic solutions for second-order 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. 373-380, (1997).

[2] A. Ardjouni and A. Djoudi, Periodic solutions for a second-order nonlinear neutral differential equation with variable delay, Electronic Journal of Differential Equations, Vol. No. 128, pp. 1-7, (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. 349-359, (2010).

[4] T. A. Burton, Liapunov functionals, fixed points and stability by Krasnoselskii’s theorem. Nonlinear Stud. 9(2002), No. 2, 181-190.

[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 Lotka-Volterra 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. 3948-3955, (2009).

[8] H. Deham and A. Djoudi, Periodic solutions for nonlinear differential equation with functional delay, Georgian Mathematical Journal 15, No. 4, pp. 635-642, (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. 1-8, (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. 1-11, (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. 235-255, (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. 581-587, (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 fixed-point theory, Math. Comput. Modelling 40, No. 7-8, 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, London-New York, (1974).

[18] Q. Wang, Positive periodic solutions of neutral delay equations (in Chinese), Acta Math. Sinica (N.S.) 6, pp. 789-795, (1996).

[19] E. Yankson, Positive periodic solutions for second-order 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. 373-380, (1997).

## How to Cite

[1]

A. Ardjouni and A. Djoudi, “Existence of positive periodic solutions for two types of second-order nonlinear neutral differential equations with variable delay”,

*Proyecciones (Antofagasta, On line)*, vol. 32, no. 4, pp. 377-391, 1.## Issue

## Section

Artículos