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Periodic Solutions of a Second-Order Functional Differential Equation with State-Dependent Argument

  • Hou Yu ZhaoEmail author
  • Jia Liu
Article
  • 43 Downloads

Abstract

In this paper, we use Schauder and Banach fixed point theorem to study the existence, uniqueness and stability of periodic solutions of a class of iterative differential equation
$$\begin{aligned} c_0x''(t)+c_1x'(t)+c_2x(t)=x(p(t)+bx(t))+h(t). \end{aligned}$$

Keywords

Iterative differential equation periodic solutions fixed point theorem 

Mathematics Subject Classification

39B12 39B82 

References

  1. 1.
    Bellman, R., Cooke, K.: Differential-Difference Equations. Academic Press, Cambridge (1963)zbMATHGoogle Scholar
  2. 2.
    Cooke, K. L.: Functional differential systems: some models and perturbation problems. In: Proceedings of the international symposium on differential equations and dynamical systems, Puerto Rico, 1965. Academic Press, New York (1967)Google Scholar
  3. 3.
    Eder, E.: The functional differential equation \(x^{\prime }(t)=x(x(t))\). J. Differ. Equ. 54, 390–400 (1984)CrossRefGoogle Scholar
  4. 4.
    Fečkan, M.: On a certain type of functional differential equations. Math. Slovaca 43, 39–43 (1993)MathSciNetzbMATHGoogle Scholar
  5. 5.
    Hale, J.: Theory of functional differential equations. Springer, Berlin (1977)CrossRefGoogle Scholar
  6. 6.
    Liu, H.Z., Li, W.R.: Discussion on the analytic solutions of the second-order iterated differential equation. Bull. Korean Math. Soc. 43(4), 791–804 (2006)MathSciNetCrossRefGoogle Scholar
  7. 7.
    Shi, B., Li, Z.X.: Existence of solutions and bifurcation of a class of first-order functional differential equations. Acta Math. Appl. Sin. 18, 83–89 (1995)MathSciNetzbMATHGoogle Scholar
  8. 8.
    Si, J.G., Cheng, S.S.: Analytic solutions of a functional differential equation with state dependent argument. Taiwan. J. Math. 1(4), 471–480 (1997)MathSciNetCrossRefGoogle Scholar
  9. 9.
    Si, J.G., Ma, M.H.: Local invertible analytic solution of a functional differential equation with deviating arguments depending on the state derivative. J. Math. Anal. Appl. 327, 723–734 (2007)MathSciNetCrossRefGoogle Scholar
  10. 10.
    Si, J.G., Wang, X.P., Cheng, S.S.: Analytic solutions of a functional differential equation with a state derivative dependent delay. Aequ. Math. 57, 75–86 (1999)MathSciNetCrossRefGoogle Scholar
  11. 11.
    Si, J.G., Zhang, W.N., Kim, G.H.: Analytic solutions of an iterative functional differential equation. Appl. Math. Comput. 150, 647–659 (2004)MathSciNetzbMATHGoogle Scholar
  12. 12.
    Stephen, B.H.: On the existence of periodic solutions of \(z^{\prime }(t)=-az(t-r+\mu k(t, z(t)))+F(t)\). J. Differ. Equ. 6, 408–419 (1969)MathSciNetCrossRefGoogle Scholar
  13. 13.
    Wu, H.Z.: A class of functional differential equations with deviating arguments depending on the state. Acta Math. Sin. 38(6), 803–809 (1995)zbMATHGoogle Scholar
  14. 14.
    Zhao, H.Y., Fečkan, M.: Periodic solutions for a class of differential equations with delays depending on state. Math. Commun. 23, 29–42 (2018)MathSciNetzbMATHGoogle Scholar

Copyright information

© Springer Nature Switzerland AG 2018

Authors and Affiliations

  1. 1.School of MathematicsChongqing Normal UniversityChongqingPeople’s Republic of China
  2. 2.Department of Architectural Economic ManagementShandong Urban Construction Vocational CollegeJinanPeople’s Republic of China

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