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Positive Periodic Solutions for a Kind of Second-Order Neutral Differential Equations with Variable Coefficient and Delay

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Abstract

In this article, we discuss a type of second-order neutral differential equations with variable coefficient and delay:

$$\begin{aligned} (x(t)-c(t)x(t-\tau (t)))''+a(t)x(t)=f(t,x(t-\delta (t))), \end{aligned}$$

where \(c(t)\in C({\mathbb {R}},{\mathbb {R}})\) and \(|c(t)|\ne 1\). By employing Krasnoselskii’s fixed-point theorem and properties of the neutral operator \((Ax)(t):=x(t)-c(t)x(t-\tau (t))\), some sufficient conditions for the existence of periodic solutions are established.

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References

  1. Ardjouni, A., Djoudi, A.: Existence of positive periodic solutions for two types of second-order nonlinear neutral differential equations with variable delay. Proyecc. J. Math. 32, 377–391 (2013)

    MathSciNet  MATH  Google Scholar 

  2. Agarwal, R.P., Grace, S.R., O’Regan, D.: Oscillation theory for difference and functional differential equations. Kluwer Academic, New York (2000)

    Book  MATH  Google Scholar 

  3. Bai, M., Xu, S.: On a two-phase size-structured population model with infinite states-at-birth and distributed delay in birth process. J. Biol. Dyn. 8, 42–56 (2014)

    Article  MathSciNet  Google Scholar 

  4. Candan, T.: Existence of positive periodic solutions of first order neutral differential equations with variable coefficients. Appl. Math. Lett. 52, 142–148 (2016)

    Article  MathSciNet  MATH  Google Scholar 

  5. Cheung, W.S., Ren, J.L., Han, W.W.: Positive periodic solution of second-order neutral functional differential equations. Nonlinear Anal. 71, 3948–3955 (2009)

    Article  MathSciNet  MATH  Google Scholar 

  6. Han, W.W., Ren, J.L.: Some results on second-order neutral functional differential equations with infinite distributed delay. Nonlinear Anal. 70, 1393–1406 (2009)

    Article  MathSciNet  MATH  Google Scholar 

  7. Kuang, Y.: Delay differential equations with applications in population dynamics. Academic Press, New York (1993)

    MATH  Google Scholar 

  8. Lu, S.P., Ge, W.G.: Periodic solutions for a kind of second order differential equation with multiple deviating arguments. Appl. Math. Compu. 2003(146), 195–209 (2003)

    MathSciNet  MATH  Google Scholar 

  9. Luo, Y., Wei, W.B., Shen, J.H.: Existence of positive periodic solutions for two kinds of neutral functional differential equations. Appl. Math. Lett. 21, 1257–1262 (2008)

    Article  MathSciNet  Google Scholar 

  10. Liu, Z.Q., Li, X., Kang, S.M., Kwun, Y.C.: Positive periodic solutions for first-order neutral functional differential equations with periodic delays. Abstr. Appl. Anal. 185692, 1–12 (2012)

    MathSciNet  MATH  Google Scholar 

  11. Ren, J.L., Cheng, Z.B., Siegmund, S.: Neutral operator and neutral differential equation. Abstr. Appl. Anal. 2011(969276), 1–29 (2011)

    Article  MathSciNet  MATH  Google Scholar 

  12. Torres, P.: Existence of one-signed periodic solutions of some second-order differential equations via a Krasnoselskii fixed theorem. J. Diff. Equ. 190, 643–662 (2003)

    Article  MathSciNet  MATH  Google Scholar 

  13. Wu, J., Wang, Z.C.: Two periodic solutions of second-order neutral functional differential equations. J. Math. Anal. Appl. 329, 677–689 (2007)

    Article  MathSciNet  MATH  Google Scholar 

  14. Wang, H.Y.: Positive periodic solutions of functional differential equations. J. Differential Equations 202, 354–366 (2004)

    Article  MathSciNet  MATH  Google Scholar 

  15. Wan, A.Y., Jiang, D.Q., Xu, X.J.: A new existence theory for positive periodic solutions to functional differential equations. Comput. Math. Appl. 47, 1257–1262 (2004)

    Article  MathSciNet  MATH  Google Scholar 

  16. Xin, Y., Cheng, Z.B.: Neutral operator with variable parameter and third-order neutral differential equation. Adv. Diff. Equ. 2014(273), 1–18 (2014)

    MathSciNet  Google Scholar 

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Authors and Affiliations

  1. School of Mathematics and Information Science, Henan Polytechnic University, Jiaozuo, 454000, China

    Zhibo Cheng & Feifan Li

  2. Department of Mathematics, Sichuan University, Chengdu, 610064, China

    Zhibo Cheng

Authors
  1. Zhibo Cheng
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  2. Feifan Li
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Correspondence to Zhibo Cheng.

Additional information

Research is supported by the National Natural Science Foundation of China (no. 11501170), China Postdoctoral Science Foundation funded project (no. 2016M590886), Fundamental Research Funds for the Universities of Henan Province (NSFRF170302), Education Department of Henan Province Project (no. 14A110002), Henan Polytechnic University Outstanding Youth Fund (J2015-02) and Henan Polytechnic University Doctor Fund (B2013-055).

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Cheng, Z., Li, F. Positive Periodic Solutions for a Kind of Second-Order Neutral Differential Equations with Variable Coefficient and Delay. Mediterr. J. Math. 15, 134 (2018). https://doi.org/10.1007/s00009-018-1184-y

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  • DOI: https://doi.org/10.1007/s00009-018-1184-y

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