Skip to main content
SpringerLink
Log in

On the behaviors of solutions of systems of non-linear differential equations with multiple constant delays

  • Original Paper
  • Published:
Revista de la Real Academia de Ciencias Exactas, Físicas y Naturales. Serie A. Matemáticas Aims and scope Submit manuscript

Abstract

In this paper, non-perturbed and perturbed systems of non-linear differential equations with multiple constant delays are considered. Five new theorems on the qualitative properties of solutions, uniform asymptotic stability (UAS) and instability of trivial solution, boundedness and integrability of solutions, are obtained. The technique of the proofs is based on the construction of two new Lyapunov–Krasovskiĭ functionals (LKFs). An advantage of the new LKFs used here is that they allow to eliminate the Gronwall’s inequality and to obtain more convenient conditions. When we compare our results with the related results in the literature, the established conditions here are new, more convenient and general, less conservative, and they are more suitable for applications. We provide three examples to show the applications of the results of this paper.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Subscribe and save

Springer+ Basic
$34.99 /Month
  • Get 10 units per month
  • Download Article/Chapter or eBook
  • 1 Unit = 1 Article or 1 Chapter
  • Cancel anytime
Subscribe now

Buy Now

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Fig. 1
Fig. 2
Fig. 3
Fig. 4
Fig. 5
Fig. 6

Similar content being viewed by others

References

  1. Akbulut, I., Tunç, C.: On the stability of solutions of neutral differential equations of first order. Int. J. Math. Comput. Sci. 14(4), 849–866 (2019)

    MathSciNet  MATH  Google Scholar 

  2. Arino,O., Hbid, M.L., Ait Dads, E.: Delay differential equations and applications. In: Proceedings of the NATO Advanced Study Institute held at the Cadi Ayyad University, Marrakech, September 9–21, 2002. NATO Science Series II: Mathematics, Physics and Chemistry, vol. 205. Springer, Dordrecht (2006)

  3. Azbelev,N., Maksimov, V., Rakhmatullina, L.: Introduction to the theory of linear functional-differential equations. In: Advanced Series in Mathematical Science and Engineering, vol. 3. World Federation Publishers Company, Atlanta (1995)

  4. Berezansky, L., Braverman, E.: On stability of some linear and nonlinear delay differential equations. J. Math. Anal. Appl. 314(2), 391–411 (2006)

    Article  MathSciNet  MATH  Google Scholar 

  5. Berezansky, L., Braverman, E.: Stability conditions for scalar delay differential equations with a non-delay term. Appl. Math. Comput. 250, 157–164 (2015)

    Article  MathSciNet  MATH  Google Scholar 

  6. Berezansky, L., Braverman, E.: On stability of delay equations with positive and negative coefficients with applications. Z. Anal. Anwend. 38(2), 157–189 (2019)

    Article  MathSciNet  MATH  Google Scholar 

  7. Berezansky, L., Braverman, E.: On stability of linear neutral differential equations with variable delays. Czechoslov. Math. J. 69(144), 863–891 (2019)

    Article  MathSciNet  MATH  Google Scholar 

  8. Berezansky, L., Braverman, E.: Solution estimates for linear differential equations with delay. Appl. Math. Comput. 372, 124962 (2020). (10 pp)

    Article  MathSciNet  MATH  Google Scholar 

  9. Bohner, M., Stamova, I.M.: Asymptotic stability criteria for a class of impulsive functional differential systems. Appl. Math. Inf. Sci. 8(4), 1475–1483 (2014)

    Article  MathSciNet  Google Scholar 

  10. Bojor, F.: Florin note on the stability of first order linear differential equations. Opusc. Math. 32(1), 67–74 (2012)

    Article  MathSciNet  MATH  Google Scholar 

  11. Burton,T.A.: Stability and periodic solutions of ordinary and functional differential equations. Corrected version of the 1985 original. Dover Publications, Inc., Mineola (2005)

  12. Du, X.T.: Some kinds of Liapunov functional in stability theory of RFDE. Acta Math. Appl. Sin. (English Ser.) 11(2), 214–224 (1995)

    Article  MathSciNet  MATH  Google Scholar 

  13. Gil, M.I.: Stability of delay differential equations with oscillating coefficients. Electron. J. Differ. Equ. 99, 5 (2010)

    MathSciNet  MATH  Google Scholar 

  14. Graef, J.R., Tunç, C.: Continuability and boundedness of multi-delay functional integro-differential equations of the second order. Rev. R. Acad. Cienc. Exactas Fís. Nat. Ser. A Mat. RACSAM 109(1), 169–173 (2015)

    Article  MathSciNet  MATH  Google Scholar 

  15. Hale,J.: Theory of Functional Differential Equations, 2nd edn. Applied Mathematical Sciences, vol. 3. Springer, New York (1977)

  16. Hale,J.K., Verduyn Lunel, S.M.: Introduction to Functional-Differential Equations. Applied Mathematical Sciences, vol. 99. Springer, New York (1993)

  17. Kiri,Y., Ueda, Y.: Stability Criteria for Some System of Delay Differential Equations. Theory, Numerics and Applications of Hyperbolic Problems. II. Springer Proc. Math. Stat., vol. 237, pp. 137–144. Springer, Cham (2018)

  18. Kolmanovskii,V., Myshkis, A.: Applied Theory of Functional-Differential Equations. Mathematics and its Applications (Soviet Series), vol. 85. Kluwer Academic Publishers Group, Dordrecht (1992)

  19. Kolmanovskii,V.B., Nosov, V.R.: Stability of Functional-Differential Equations. Mathematics in Science and Engineering, vol. 180. Academic Press, Inc. [Harcourt Brace Jovanovich, Publishers], London (1986)

  20. Krasovskiĭ,N.N.: Stability of Motion. Applications of Lyapunov's second method to differential systems and equations with delay. Translated by J. L. Brenner Stanford University Press, Stanford (1963)

  21. Kuang,Y.: Delay Differential Equations with Applications in Population Dynamics. Mathematics in Science and Engineering, vol. 191. Academic Press, Inc., Boston (1993)

  22. Lakshmikantham,V., Wen, L.Z., Zhang, B.G.: Theory of Differential Equations with Unbounded Delay. Mathematics and its Applications, vol. 298. Kluwer Academic Publishers Group, Dordrecht (1994)

  23. Slyn’ko, V.I., Tunç, C.: Global asymptotic stability of nonlinear periodic impulsive equations. Miskolc Math. Notes 19(1), 595–610 (2018)

    Article  MathSciNet  MATH  Google Scholar 

  24. Slyn’ko, V.I., Tunç, C.: Instability of set differential equations. J. Math. Anal. Appl. 467(2), 935–947 (2018)

    Article  MathSciNet  MATH  Google Scholar 

  25. Slyn’ko, V.I., Tunç, C.: Sufficient stability conditions for linear periodic impulsive systems with delay(Russian). Avtomat. i Telemekh. 11, 47–66 (2018). (translation in Autom. Remote Control 79 (2018), no. 11, 1989–2004)

    Article  MATH  Google Scholar 

  26. Slyn’ko, V.I., Tunç, C.: Stability of abstract linear switched impulsive differential equations. Autom. J. IFAC 107, 433–441 (2019)

    Article  MathSciNet  MATH  Google Scholar 

  27. Smith,H.: An Introduction to Delay Differential Equations with Applications to the Life Sciences. Texts in Applied Mathematics, vol. 57. Springer, New York (2011)

  28. Tunç, C.: A note on boundedness of solutions to a class of non-autonomous differential equations of second order. Appl. Anal. Discrete Math. 4(2), 361–372 (2010)

    Article  MathSciNet  MATH  Google Scholar 

  29. Tunç, C., Tunç, O.: A note on certain qualitative properties of a second order linear differential system. Appl. Math. Inf. Sci. 9(2), 953–956 (2015)

    MathSciNet  MATH  Google Scholar 

  30. Tunç, C., Tunç, O.: On the boundedness and integration of non-oscillatory solutions of certain linear differential equations of second order. J. Adv. Res. 7(1), 165–168 (2016)

    Article  Google Scholar 

  31. Tunç, C., Tunç, O.: A note on the stability and boundedness of solutions to non-linear differential systems of second order. J. Assoc. Arab Univ. Basic Appl. Sci. 24, 169–175 (2017)

    Google Scholar 

  32. Tunç, C., Tunç, O.: Qualitative analysis for a variable delay system of differential equations of second order. J. Taibah Univ. Sci. 13(1), 468–477 (2019)

    Article  MATH  Google Scholar 

  33. Tunç, C., Tunç, O.: On the stability, integrability and boundedness analyses of systems of integro-differential equations with time-delay retardation. Rev. R. Acad. Cienc. Exactas Fís. Nat. Ser. A Mat. RACSAM 115(3), 115 (2021)

    Article  MathSciNet  MATH  Google Scholar 

  34. Tunç, O., Tunç, C., Wang, Y.: Delay-dependent stability, integrability and boundedeness criteria for delay differential systems. Axioms 10(3), 138 (2021). https://doi.org/10.3390/axioms10030138

    Article  Google Scholar 

  35. Benchohra, M., Bouriah, S., Nieto, J.J.: Existence of periodic solutions for nonlinear implicit Hadamard’s fractional differential equations. Rev. R. Acad. Cienc. Exactas Fís. Nat. Ser. A Mat. RACSAM 12(1), 25–35 (2018)

    Article  MathSciNet  MATH  Google Scholar 

  36. Saker, S.H., Sudha, B., Arahet, M.A., Thandapani, E.: Distribution of zeros of second order superlinear and sublinear neutral delay differential equations. Rev. R. Acad. Cienc. Exactas Fís. Nat. Ser. A Mat. RACSAM 113(3), 1907–1915 (2019)

    Article  MathSciNet  MATH  Google Scholar 

  37. Zhang, S.: The uniqueness result of solutions to initial value problems of differential equations of variable-order. Rev. R. Acad. Cienc. Exactas Fís. Nat. Ser. A Mat. RACSAM 112(2), 407–423 (2018)

    Article  MathSciNet  MATH  Google Scholar 

  38. Alzabut, J.O.: Existence of periodic solutions for a type of linear difference equations with distributed delay. Adv. Differ. Equ. 53, 14 (2012)

    MathSciNet  MATH  Google Scholar 

  39. Alzabut, J.O., Obaidat, S.: Almost periodic solutions for Fox production harvesting model with delay. Electron. J. Qual. Theory Differ. Equ. 34, 12 (2012)

    MathSciNet  MATH  Google Scholar 

  40. Alzabut, J.O., Tunç, C.: Existence of periodic solutions for Rayleigh equations with state-dependent delay. Electron. J. Differ. Equ. 77, 8 (2012)

    MathSciNet  MATH  Google Scholar 

  41. Saker, S.H., Alzabut, J.O.: On the impulsive delay hematopoiesis model with periodic coefficients. Rocky Mt. J. Math. 39(5), 1657–1688 (2009)

    Article  MathSciNet  MATH  Google Scholar 

  42. Tunç, C., Tunç, O., Wang, Y., Yao, J.C.: Qualitative analyses of differential systems with time-varying delays via Lyapunov–Krasovskiĭ approach. Mathematics 9(11), 1196 (2021)

    Article  Google Scholar 

  43. Džurina, J., Grace, S.R., Jadlovská, I., Li, T.: Oscillation criteria for second-order Emden-Fowler delay differential equations with a sublinear neutral term. Math. Nachr. 293(5), 910–922 (2020)

    Article  MathSciNet  MATH  Google Scholar 

  44. Li, T., Rogovchenko, Y.V.: On the asymptotic behavior of solutions to a class of third-order nonlinear neutral differential equations. Appl. Math. Lett. 105, 106293 (2020). (7 pp)

    Article  MathSciNet  MATH  Google Scholar 

  45. Li, T., Pintus, N., Viglialoro, G.: Properties of solutions to porous medium problems with different sources and boundary conditions. Z. Angew. Math. Phys. 70(3), 18 (2019). (Paper No. 86)

    Article  MathSciNet  MATH  Google Scholar 

  46. Li, T., Viglialoro, G.: Boundedness for a nonlocal reaction chemotaxis model even in the attraction-dominated regime. Differ. Integral Equ. 34(5–6), 315–336 (2021)

    MathSciNet  MATH  Google Scholar 

  47. Viglialoro, G., Woolley, T.E.: Solvability of a Keller–Segel system with signal-dependent sensitivity and essentially sublinear production. Appl. Anal. 99(14), 2507–2525 (2020)

    Article  MathSciNet  MATH  Google Scholar 

  48. Tunç, O., Atan, Ö., Tunç, C., Yao, J.C.: Qualitative analyses of integro-fractional differential equations with caputo derivatives and retardations via the lyapunov-razumikhin method. Axioms 10(2), 58 (2021). https://doi.org/10.3390/axioms10020058

    Article  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Department of Computer Programing, Baskale Vocational School, Van Yuzuncu Yil University, Campus, 65080, Van, Turkey

    Osman Tunç

Authors
  1. Osman Tunç
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Osman Tunç.

Additional information

Publisher's Note

Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Tunç, O. On the behaviors of solutions of systems of non-linear differential equations with multiple constant delays. Rev. Real Acad. Cienc. Exactas Fis. Nat. Ser. A-Mat. 115, 164 (2021). https://doi.org/10.1007/s13398-021-01104-5

Download citation

  • Received:

  • Accepted:

  • Published:

  • DOI: https://doi.org/10.1007/s13398-021-01104-5

Keywords

Mathematics Subject Classification

Search

Navigation