[go: up one dir, main page]
More Web Proxy on the site http://driver.im/
Skip to main content
Springer Nature Link
Log in

On Kneser Solutions of the n-th Order Nonlinear Differential Inclusions

  • Published:
Czechoslovak Mathematical Journal Aims and scope Submit manuscript

Abstract

The paper deals with the existence of a Kneser solution of the n-th order nonlinear differential inclusion

$${x^{(n)}}(t) \in - {A_1}(t,x,(t),...,{x^{(n - 1)}}(t)){x^{(n - 1)}}(t) - ... - {A_n}(t,x(t),...,{x^{(n - 1)}}(t))x(t)\;\text{for}\;\text{a.a.}\;t\; \in [a,\infty ),$$

where a ∈ (0,∞), and Ai: [a,∞) × ℝn → ℝ, i = 1,..., n, are upper-Carathéodory mappings. The derived result is finally illustrated by the third order Kneser problem.

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
£29.99 /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 includes VAT (United Kingdom)

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

References

  1. R. P. Agarwal, D. O’Regan: Infinite Interval Problems for Differential, Difference and Integral Equations. Kluwer Academic Publishers, Dordrecht, 2001.

    Book  MATH  Google Scholar 

  2. J. Andres, G. Gabor, L. Górniewicz: Topological structure of solution sets to multivalued asymptotic problems, Z. Anal. Anwend. 19 (2000), 35–60.

    Article  MATH  Google Scholar 

  3. J. Andres, G. Gabor, L. Górniewicz: Acyclicity of solution sets to functional inclusions, Nonlinear Anal., Theory Methods Appl., Ser. A, Theory Methods. 49 (2002), 671–688.

    Article  MathSciNet  MATH  Google Scholar 

  4. J. Andres, L. Górniewicz: Topological Fixed Point Principles for Boundary Value Problems. Topological Fixed Point Theory and Its Applications 1, Kluwer Academic Publishers, Dordrecht, 2003.

    Book  MATH  Google Scholar 

  5. J. Andres, M. Pavlačková: Asymptotic boundary value problems for second-order differential systems, Nonlinear Anal., Theory Methods Appl., Ser. A, Theory Methods. 71 (2009), 1462–1473.

    Article  MathSciNet  MATH  Google Scholar 

  6. J. Andres, M. Pavlačková: Boundary value problems on noncompact intervals for the n-th order vector differential inclusions, Electron. J. Qual. Theory Differ. Equ. 2016 (2016), 19 pages.

    Article  MathSciNet  MATH  Google Scholar 

  7. J. Appell, E. De Pascale, N. H. Thái, P. P. Zabreˇıko: Multi-valued superpositions, Diss. Math. 345 (1995), 97 pages.

    MathSciNet  MATH  Google Scholar 

  8. J.-P. Aubin, A. Cellina: Differential Inclusions. Set-Valued Maps and Viability Theory. Grundlehren der Mathematischen Wissenschaften 264, Springer, Berlin, 1984.

    Book  MATH  Google Scholar 

  9. M. Bartušek, M. Cecchi, M. Marini: On Kneser solutions of nonlinear third order differential equations, J. Math. Anal. Appl. 261 (2001), 72–84.

    Article  MathSciNet  MATH  Google Scholar 

  10. M. Bartušek, Z. Došlá: Oscillation of third order differential equation with damping term, Czech. Math. J. 65 (2015), 301–316.

    Article  MathSciNet  MATH  Google Scholar 

  11. K. Borsuk: Theory of Retracts. Monografie Matematyczne 44, PWN, Warszawa, 1967.

    MATH  Google Scholar 

  12. M. Cecchi, M. Furi, M. Marini: About the solvability of ordinary differential equations with asymptotic boundary conditions, Boll. Unione Mat. Ital., VI. Ser., C, Anal. Funz. Appl. 4 (1985), 329–345.

    MathSciNet  MATH  Google Scholar 

  13. E. Fermi: Un metodo statistico per la determinazione di alcune prioriet á dell’atomo, Rend. R. Accad. Nat. Lincei. 6 (1927), 602–607. (In Italian.)

    Google Scholar 

  14. A. F. Filippov: Differential Equations with Discontinuous Right-Hand Sides. Mathematics and Its Applications (Soviet Series) 18, Kluwer Academic Publishers, Dordrecht, 1988.

    Book  Google Scholar 

  15. G. Gabor: On the acyclicity of fixed point sets of multivalued maps, Topol. Methods Nonlinear Anal. 14 (1999), 327–343.

    Article  MathSciNet  MATH  Google Scholar 

  16. L. Górniewicz: Topological Fixed Point Theory of Multivalued Mappings. Mathematics and Its Applications 495, Kluwer Academic Publishers, Dordrecht, 1999.

    Book  MATH  Google Scholar 

  17. J. R. Graef, J. Henderson, A. Ouahab: Impulsive Differential Inclusions. A Fixed Point Approach. De Gruyter Series in Nonlinear Analysis and Applications 20, De Gruyter, Berlin, 2013.

    Book  MATH  Google Scholar 

  18. P. Hartman, A. Wintner: On the non-increasing solutions of y′′ = f(x, y, y′), Am. J. Math. 73 (1951), 390–404.

    Article  MathSciNet  MATH  Google Scholar 

  19. L. V. Kantorovich, G. P. Akilov: Functional Analysis in Normed Spaces. International Series of Monographs in Pure and Applied Mathematics 46, Pergamon Press, Oxford, 1964.

    MATH  Google Scholar 

  20. I. T. Kiguradze, T. A. Chanturia: Asymptotic Properties of Solutions of Nonautonomous Ordinary Differential Equations. Mathematics and Its Applications (Soviet Series) 89, Kluwer Academic Publishers, Dordrecht, 1993.

    Book  Google Scholar 

  21. I. T. Kiguradze, B. L. Shekhter: Singular boundary value problems for second-order ordinary differential equations, J. Soviet Math. 43 (1988), 2340–2417 (In English. Russian original.); translation from Itogi Nauki Tekh., Ser. Sovrem. Probl. Mat. 30 (1987), 105–201.

    Article  MATH  Google Scholar 

  22. A. Kneser: Untersuchung und asymptotische Darstellung der Integrale gewisser Differentialgleichungen bei grossen reellen Werten des Arguments I, II, J. für Math. 116 (1896), 178–212; 117 (1896), 72–103. (In German.) zbl

    MATH  Google Scholar 

  23. V. A. Kozlov: On Kneser solutions of higher order nonlinear ordinary differential equations, Ark. Mat. 37 (1999), 305–322.

    Article  MathSciNet  MATH  Google Scholar 

  24. J. Kurzweil: Ordinary Differential Equations. Introduction to the Theory of Ordinary Differential Equations in the Real Domain. Studies in Applied Mechanics 13, Elsevier Scientific Publishing, Amsterdam; SNTL Publishers of Technical Literature, Praha, 1986.

    MATH  Google Scholar 

  25. D. O’Regan, A. Petruşel: Leray-Schauder, Lefschetz and Krasnoselskii fixed point theory in Fréchet spaces for general classes of Volterra operators, Fixed Point Theory. 9 (2008), 497–513.

    MathSciNet  MATH  Google Scholar 

  26. S. Padhi, S. Pati: Theory of Third-Order Differential Equations. Springer, New Delhi, 2014.

    Book  MATH  Google Scholar 

  27. N. Partsvania, Z. Sokhadze: Oscillatory and monotone solutions of first-order nonlinear delay differential equations, Georgian Math. J. 23 (2016), 269–277.

    Article  MathSciNet  MATH  Google Scholar 

  28. L. H. Thomas: The calculation of atomic fields, Proceedings Cambridge. 23 (1927), 542–548. zbl doi

    Article  MATH  Google Scholar 

  29. I. I. Vrabie: Compactness Methods for Nonlinear Evolutions. Pitman Monographs and Surveys in Pure and Applied Mathematics 75, Longman Scientific &Technical, Harlow; John Wiley & Sons, New York, 1995.

    Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Department of Mathematical Analysis and Applications of Mathematics, Faculty of Science, Palacký University, 17. listopadu 12, 771 46, Olomouc, Czech Republic

    Martina Pavlačková

Authors
  1. Martina Pavlačková
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Martina Pavlačková.

Additional information

The research has been supported by the grant No. 14-06958S “Singularities and impulses in boundary value problems for nonlinear ordinary differential equations” of the Grant Agency of the Czech Republic.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Pavlačková, M. On Kneser Solutions of the n-th Order Nonlinear Differential Inclusions. Czech Math J 69, 99–116 (2019). https://doi.org/10.21136/CMJ.2018.0191-17

Download citation

  • Received:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.21136/CMJ.2018.0191-17

Keywords

MSC 2010

Search

Navigation