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Green’s Function for Periodic Solutions in Alternately Advanced and Delayed Differential Systems

Acta Mathematicae Applicatae Sinica, English Series volume 36, pages 936–951 (2020)Cite this article

Abstract

In this paper we investigate the existence of the periodic solutions of a nonlinear differential equation with a general piecewise constant argument, in short DEPCAG, that is, the argument is a general step function. We consider the critical case, when associated linear homogeneous system admits nontrivial periodic solutions. Criteria of existence of periodic solutions of such equations are obtained. In the process we use the Green’s function for periodic solutions and convert the given DEPCAG into an equivalent integral equation. Then we construct appropriate mappings and employ Krasnoselskii’s fixed point theorem to show the existence of a periodic solution of this type of nonlinear differential equations. We also use the contraction mapping principle to show the existence of a unique periodic solution. Appropriate examples are given to show the feasibility of our results.

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References

  1. Aftabizadeh, A.R., Wiener, J. Oscillatory and periodic solutions of an equation alternately of retarded and advanced types. Appl. Anal., 23: 219–231 (1986)

    MathSciNet  MATH  Article  Google Scholar 

  2. A.R. Aftabizadeh, J. Wiener, J.M. Xu. Oscillatory and periodic solutions of delay differential equations with piecewise constant argument. Proc. Amer. Math. Soc., 99: 673–679 (1987)

    MathSciNet  MATH  Article  Google Scholar 

  3. Akhmet, M.U. Integral manifolds of differential equations with piecewise constant argument of generalized type. Nonlinear Anal. TMA., 66: 367–383 (2007)

    MathSciNet  MATH  Article  Google Scholar 

  4. Akhmet, M.U. On the reduction principle for differential equations with piecewise constant argument of generalized type. J. Math. Anal. Appl., 336: 646–663 (2007)

    MathSciNet  MATH  Article  Google Scholar 

  5. Akhmet, M.U., Buyukadali, C., Ergenc, T. Periodic solutions of the hybrid system with small parameter. Nonlinear Anal. Hybrid Syst., 2: 532–543 (2008)

    MathSciNet  MATH  Article  Google Scholar 

  6. Alonso, A., Hong, J., Obaya, R. Almost periodic type solutions of differential equations with piecewise constant argument via almost periodic type sequences. Appl. Math. Lett., 13: 131–137 (2000)

    MathSciNet  MATH  Article  Google Scholar 

  7. Burton, T.A. Stability and Periodic Solutions of Ordinary and Functional Differential Equations. Academic Press Inc., New York, 1985

    MATH  Google Scholar 

  8. Burton, T.A. Krasnoselskii’s inversion principle and fixed points. Nonlinear Anal. TMA., 30: 3975–3986 (1997)

    MathSciNet  MATH  Article  Google Scholar 

  9. Burton, T.A. A fixed-point theorem of Krasnoselskii. Appl. Math. Lett., 11: 85–88 (1998)

    MathSciNet  MATH  Article  Google Scholar 

  10. Burton, T.A., Furumochi, T. Periodic and asymptotically periodic solutions of neutral integral equations. E.J. Qualitative Theory of Diff. Equ., 10: 1–19 (2000)

    MATH  Google Scholar 

  11. Burton, T.A., Furumochi, T. Existence theorems and periodic solutions of neutral integral equations. Nonlinear Anal. TMA., 43: 527–546 (2001)

    MathSciNet  MATH  Article  Google Scholar 

  12. Cabada, A., Ferreiro, J.B., Nieto, J.J. Green’s function and comparison principles for first order periodic differential equations with piecewise constant arguments. J. Math. Anal. Appl., 291: 690–697 (2004)

    MathSciNet  MATH  Article  Google Scholar 

  13. Cooke, K.L., Wiener, J. Retarded differential equations with piecewise constant delays. J. Math. Anal. Appl., 99: 265–297 (1984)

    MathSciNet  MATH  Article  Google Scholar 

  14. Cooke, K.L., Wiener, J. An equation alternately of retarded and advanced type. Proc. Amer. Math. Soc., 99: 726–732 (1987)

    MathSciNet  MATH  Article  Google Scholar 

  15. Chiu, K.S., Pinto, M. Oscillatory and periodic solutions in alternately advanced and delayed differential equations. Carpathian J. Math., 29(2): 149–158 (2013)

    MathSciNet  MATH  Google Scholar 

  16. Chiu, K.S. Stability of oscillatory solutions of differential equations with a general piecewise constant argument. E.J. Qualitative Theory of Diff. Equ., 88: 1–15 (2011)

    Google Scholar 

  17. Chiu, K.S., Pinto, M. Variation of parameters formula and Gronwall inequality for differential equations with a general piecewise constant argument. Acta Math. Appl. Sin. Engl. Ser, 27(4): 561–568 (2011)

    MathSciNet  MATH  Article  Google Scholar 

  18. Chiu, K.S., Pinto, M. Periodic solutions of differential equations with a general piecewise constant argument and applications. E.J. Qualitative Theory of Diff. Equ., 46: 1–19 (2010)

    MATH  Google Scholar 

  19. Chiu, K.S., Pinto, M., Jeng, J.Ch. Existence and global convergence of periodic solutions in recurrent neural network models with a general piecewise alternately advanced and retarded argument. Acta Appl. Math., 133: 133–152 (2014)

    MathSciNet  MATH  Article  Google Scholar 

  20. Chiu, K.S. On generalized impulsive piecewise constant delay differential equations. Science China Mathematics., 58: 1981–2002 (2015)

    MathSciNet  MATH  Article  Google Scholar 

  21. Chiu, K.S., Jeng, J.Ch. Stability of oscillatory solutions of differential equations with general piecewise constant arguments of mixed type. Math. Nachr., 288: 1085–1097 (2015)

    MathSciNet  MATH  Article  Google Scholar 

  22. Chiu, K.S. Exponential stability and periodic solutions of impulsive neural network models with piecewise constant argument. Acta Appl. Math., 151: 199–226 (2017)

    MathSciNet  MATH  Article  Google Scholar 

  23. Chiu, K.S. Asymptotic equivalence of alternately advanced and delayed differential systems with piecewise constant generalized arguments. Acta Math. Sci., 38: 220–236 (2018)

    MathSciNet  MATH  Article  Google Scholar 

  24. Chiu, K.S., Li, T. Oscillatory and periodic solutions of differential equations with piecewise constant generalized mixed arguments. Math. Nachr., 292: 2153–2164 (2019)

    MathSciNet  MATH  Article  Google Scholar 

  25. Dai, L.M. Nonlinear Dynamics of Piecewise of Constant Systems and Implememtation of Piecewise Constants Arguments. World Scientific, Singapore, 2008

    Book  Google Scholar 

  26. Karakoc, F., Bereketoglu, H., Seyhan, G. Oscillatory and periodic solutions of impulsive differential equations with piecewise constant argument. Acta Appl. Math., 110: 499–510 (2009)

    MathSciNet  MATH  Article  Google Scholar 

  27. Liu, Y., Yang, P., Ge, W. Periodic solutions of higher-order delay differential equations. Nonlinear Anal. TMA., 63: 136–152 (2005)

    MathSciNet  MATH  Article  Google Scholar 

  28. Liu, X.L., Li, W.T. Periodic solutions for dynamic equations on time scales. Nonlinear Anal. TMA., 67: 1457–1463 (2007)

    MathSciNet  MATH  Article  Google Scholar 

  29. Myshkis, A.D. On certain problems in the theory of differential equations with deviating arguments. Uspekhi Mat. Nauk., 32: 173–202 (1977)

    Google Scholar 

  30. Nieto, J.J., Rodríguez-López, R. Green’s function for second-order periodic boundary value problems with piecewise constant arguments. J. Math. Anal. Appl., 304: 33–57 (2005)

    MathSciNet  MATH  Article  Google Scholar 

  31. Pinto, M. Asymptotic equivalence of nonlinear and quasilinear differential equations with piecewise constant arguments. Math. Comput. Model., 49: 1750–1758 (2009)

    MATH  Article  Google Scholar 

  32. Pinto, M. Dichotomy and existence of periodic solutions of quasilinear functional differential equations. Nonlinear Anal., TMA., 72: 1227–1234 (2010)

    MathSciNet  MATH  Article  Google Scholar 

  33. Pinto, M. Cauchy and Green matrices type and stability in alternately advanced and delayed differential systems. J. Difference Equ. Appl., 17(2): 235–254 (2011)

    MathSciNet  MATH  Article  Google Scholar 

  34. Pinto, M., Sepúlveda, D. h-asymptotic stability by fixed point in neutral nonlinear differential equations with delay. Nonlinear Anal. TMA., 74: 3926–3933 (2011)

    MathSciNet  MATH  Article  Google Scholar 

  35. Raffoul, Y.N. Periodic solutions for neutral nonlinear differential equations with functional delay. Electron. J. Differential Equations, 2003(102): 1–7 (2003)

    MathSciNet  MATH  Google Scholar 

  36. Shah, S.M., Wiener, J. Advanced differential equations with piecewise constant argument deviations. Internat. J. Math. and Math. Sci., 6: 671–703 (1983)

    MathSciNet  MATH  Article  Google Scholar 

  37. Smart, D.R. Fixed Points Theorems. Cambridge University Press, Cambridge, 1980

    MATH  Google Scholar 

  38. Wiener, J. Generalized Solutions of Functional Differential Equations. World Scientific, Singapore, 1993

    MATH  Book  Google Scholar 

  39. Wang, G.Q. Existence theorem of periodic solutions for a delay nonlinear differential equation with piecewise constant arguments. J. Math. Anal. Appl., 298: 298–307 (2004)

    MathSciNet  MATH  Article  Google Scholar 

  40. Wang, G.Q., Cheng, S.S. Periodic solutions of discrete Rayleigh equations with deviating arguments. Taiwanese J. Math., 13(6B): 2051–2067 (2009)

    MathSciNet  MATH  Article  Google Scholar 

  41. Wang, G.Q., Cheng, S.S. Existence and uniqueness of periodic solutions for a second-order nonlinear differential equation with piecewise constant argument. Int. J. Math. Math. Sci., 2009, Art. ID 950797, 14 pp

  42. Xia, Y.H., Huang, Z., Han, M. Existence of almost periodic solutions for forced perturbed systems with piecewise constant argument. J. Math. Anal. Appl., 333: 798–816 (2007)

    MathSciNet  MATH  Article  Google Scholar 

  43. Yang, P., Liu, Y., Ge, W. Green’s function for second order differential equations with piecewise constant argument. Nonlinear Anal. TMA., 64: 1812–1830 (2006)

    MathSciNet  MATH  Article  Google Scholar 

  44. Yuan, R. The existence of almost periodic solutions of retarded differential equations with piecewise constant argument. Nonlinear Anal. TMA., 48: 1013–1032 (2002)

    MathSciNet  MATH  Article  Google Scholar 

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Acknowledgments

The author thanks the referees very much for their valuable suggestions which made this paper much improved. This research was in part supported by FGI 05-16 DIUMCE.

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Authors and Affiliations

  1. Departamento de Matemática, Facultad de Ciencias Básicas, Universidad Metropolitana de Ciencias de la Educación, Santiago, Chile

    Kuo-Shou Chiu

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  1. Kuo-Shou Chiu
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Correspondence to Kuo-Shou Chiu.

Additional information

This research was in part supported by FGI 05-16 DIUMCE.

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Chiu, KS. Green’s Function for Periodic Solutions in Alternately Advanced and Delayed Differential Systems. Acta Math. Appl. Sin. Engl. Ser. 36, 936–951 (2020). https://doi.org/10.1007/s10255-020-0975-7

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  • DOI: https://doi.org/10.1007/s10255-020-0975-7

Keywords

  • piecewise constant arguments
  • Green’s function
  • periodic solutions
  • hybrid equations
  • fixed point theorems

2000 MR Subject Classification

  • 34A36
  • 34B27
  • 34K13
  • 37C25