Existence of Positive Periodic Solutions of First Order Neutral Differential Equations

Abstract

In this paper, we consider two classes of first order neutral nonlinear differential equations and we give some new sufficient conditions for the existence of positive periodic solutions of these equations by using the Krasnoselskii's fixed point theorem. An illustrative example is provided to support the theory developed in this study.

Keywords

• Fixed point
• Neutral equations
• Positive periodic solution
• First-order

References

1. [1] R. P. Agarwal, M. Bohner and W.-T. li, Nonoscillation and Oscillation: Theory for Functional Differential Equations, Marcel Dekker, 2004.
2. [2] T. Candan, Existence of positive periodic solutions of first order neutral differential equations with variable coefficients, Appl. Math. Lett., 52 (2016), 142-148.
3. [3] T. Candan, Existence of positive periodic solutions of first order neutral differential equations, Math. Methods Appl. Sci., 40(1) (2017), 205-209.
4. [4] T. Candan, Existence of positive periodic solution of second-order neutral differential equations, Turkish J. Math., 42(3) (2018), 797–806.
5. [5] J. Dˇzurina, Said R. Grace, Irena Jadlovsk´a and Tongxing Li, Oscillation criteria for second-order Emden-Fowler delay differential equations with a sublinear neutral term, Math. Nachr., 293(5) (2020), 910-922.
6. [6] J. R. Graef and L. Kong, Periodic solutions of first order functional differential equations, Appl. Math. Lett., 24 (2011), 1981-1985.
7. [7] Tongxing Li and Yuriy V. Rogovchenko, Oscillation criteria for second-order superlinear Emden-Fowler neutral differential equations, Monatsh. Math., 184(3) (2017), 489-500.
8. [8] Tongxing Li and Yuriy V. Rogovchenko, On the asymptotic behavior of solutions to a class of third-order nonlinear neutral differential equations, Appl. Math. Lett., 105 (2020), Art. 106293.

9. [9] Z. Li and X. Wang, Existence of positive periodic solutions for neutral functional differential equations, Electron. J. Differential Equations., 34 (2006), 8 pp.
10. [10] Z. Liu, X. Li, S. M. Kang and Y. C. Kwun, Positive periodic solutions for first-order neutral functional differential equations with periodic delays, Abstr. Appl. Anal., (2012), 185692, 12 pp.
11. [11] Y. Luo, W. Wang and J. Shen, Existence of positive periodic solutions for two kinds of neutral functional differential equations, Appl. Math. Lett., 21 (2008), 581-587.
12. [12] M. B. Mesmouli, A. Ardjouni and A. Djoudi, Positive periodic solutions for first-order nonlinear neutral functional differential equations with periodic delay, Transylv. J. Math. Mech., 6 (2014), 151-162.
13. [13] L. Yao, Global Exponential Convergence of Neutral Type Shunting Inhibitory Cellular Neural Networks with D Operator, Neural Processing Letters, 45 (2017), 401–409.