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Periodic-Type Solutions for Differential Equations with Positively Homogeneous Functionals

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We establish efficient conditions guaranteeing the existence of a solution of the periodic-type boundary-value problem for a two-dimensional system of nonlinear functional-differential equations in the case where the right-hand side of the system is the sum of positively homogeneous terms of degrees λ and 1/λ and other terms with relatively slow growth at infinity. The general results are reformulated in a special case of differential equations with maxima.

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References

  1. D. D. Bainov, S. Hristova, Differential Equations with Maxima, Chapman & Hall/CRC Pure and Applied Mathematics (2011).

  2. A. D. Myshkis, “On some problems in the theory of differential equations with deviating argument,” Rus. Math. Surv., 32, 181–213 (1977).

    Article  MathSciNet  MATH  Google Scholar 

  3. V. Petukhov, “Questions about qualitative investigations of differential equations with ‘maxima’,” Izv. Vyssh. Uchebn. Zaved., Matematika, 3, 116–119 (1964).

    Google Scholar 

  4. N. Bantsur and E. Trofimchuk, “On the existence of T -periodic solutions of essentially nonlinear scalar differential equations with maxima,” Nelin. Kolyv., 1, 1–5 (1998).

    MathSciNet  MATH  Google Scholar 

  5. S. Dashkovskiy, S. Hristova, and K. Sapozhnikova, “Stability properties of systems with maximum,” J. Nonlin. Evolut. Equat. Appl., 2019, No. 3, 35–58 (2020).

    MathSciNet  MATH  Google Scholar 

  6. P. González and M. Pinto, “Convergent solutions of certain nonlinear differential equations with maxima,” Math. Comput. Model., 45, 1–10 (2007).

  7. R. Hakl, “On periodic-type boundary value problems for functional differential equations with a positively homogeneous operator,” Miskolc Math. Notes, 5, No. 1, 33–55 (2004).

    Article  MathSciNet  MATH  Google Scholar 

  8. R. Hakl, On a Boundary-Value Problem for Nonlinear Functional Differential Equations, Bound. Value Probl., Article number: 964950 (2005).

  9. R. Hakl, “On a periodic type boundary-value problem for functional differential equations with a positively homogeneous operator,” Mem. Different. Equat. Math. Phys., 40, 17–54 (2007).

    MathSciNet  MATH  Google Scholar 

  10. R. Hakl and M. Zamora, “Fredholm-type theorem for boundary value problems for systems of nonlinear functional differential equations,” Bound. Value Probl., 2014, No. 113, 9 p. (2014).

  11. A. Ivanov, E. Liz, and S. Trofimchuk, “Halanay inequality, Yorke 3/2 stability criterion, and differential equations with maxima,” Tohoku Math. J., 54, No. 2, 277–295 (2002).

    Article  MathSciNet  MATH  Google Scholar 

  12. M. Pinto and S. Trofimchuk, “Stability and existence of multiple periodic solutions for a quasilinear differential equation with maxima,” Proc. Roy. Soc. Edinburgh, Sect. A., 130, No. 5, 1103–1118 (2000).

    Article  MathSciNet  MATH  Google Scholar 

  13. A. Samoilenko, E. Trofimchuk, and N. Bantsur, “Periodic and almost periodic solutions of the systems of differential equations with maxima,” Dopov. Nats. Akad. Nauk Ukr., Mat. Prirodoznav. Tekh. Nauki, No. 1, 53–57 (1998).

  14. G. H. Sarafova and D. D. Bainov, “Application of A. M. Samoĭlenko’s numerical-analytic method to the investigation of periodic linear differential equations with maxima,” Rev. Roumaine Sci. Tech. Sér. Méc. Appl., 26, 595–603 (1981).

  15. E. Trofimchuk, E. Liz, and S. Trofimchuk, The Peak-and-End Rule and Differential Equations with Maxima: a View on the Unpredictability of Happiness, arXiv:2106.10843.

  16. A. R. Magomedov, Ordinary Differential Equations with Maxima, Baku, Elm (1991).

    Google Scholar 

  17. N. Bantsur and E. Trofimchuk, “Existence and stability of the periodic and almost periodic solutions of quasilinear systems with maxima,” Ukr. Math. J., 50, 847–856 (1998).

    Article  Google Scholar 

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Authors and Affiliations

  1. Institute of Mathematics, Czech Academy of Sciences, Zitná, 25, Praha1, 11567, Czech Republic

    R. Hakl

  2. “I. Sikorsky Kyiv Polytechnic Institute” Ukrainian National Technical University, Peremoha Ave., 37, Kyiv, 03056, Ukraine

    E. Trofimchuk

  3. Institute of Mathematics, University of Talca, Poniente str., 1, 1141, Talca, Chile

    S. Trofimchuk

Authors
  1. R. Hakl
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  2. E. Trofimchuk
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  3. S. Trofimchuk
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Correspondence to S. Trofimchuk.

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Published in Neliniini Kolyvannya, Vol. 25, No. 1, pp. 119–132, January–March, 2022.

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Hakl, R., Trofimchuk, E. & Trofimchuk, S. Periodic-Type Solutions for Differential Equations with Positively Homogeneous Functionals. J Math Sci 274, 126–141 (2023). https://doi.org/10.1007/s10958-023-06575-y

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  • DOI: https://doi.org/10.1007/s10958-023-06575-y

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