Skip to main content
SpringerLink
Log in

Necessary and sufficient conditions for the existence and ɛ-uniqueness of bounded solutions of the equation x′ = f(x) − h(t)

  • Published:
Mathematical Notes Aims and scope Submit manuscript

Abstract

We introduce the notion of ɛ-unique bounded solution to the nonlinear differential equation x′ = f(x) − h(t), where f: ℝ → ℝ is a continuous function and h(t) is an arbitrary continuous function bounded on ℝ. We derive necessary and sufficient conditions for the existence and ɛ-uniqueness of bounded solutions to this equation.

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

Access this article

Log in via an institution

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

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

References

  1. V. E. Slyusarchuk, “Conditions for the existence of bounded solutions to nonlinear differential equations,” Uspekhi Mat. Nauk 54(4), 181–182 (1999) [Russian Math. Surveys 54 (4), 852–853 (1999)].

    MathSciNet  Google Scholar 

  2. V. E. Slyusarchuk, “Necessary and sufficient conditions for the existence and uniqueness of bounded solutions to nonlinear differential equations,” Neliniini Kolivannya 2(4), 523–539 (1999) [in Ukrainian].

    MathSciNet  Google Scholar 

  3. V. E. Slyusarchuk, “Necessary and sufficient conditions for the existence and uniqueness of bounded and almost-periodic solutions of nonlinear differential equations,” Acta Appl.Math. 65(1–3), 333–341 (2001).

    Article  MathSciNet  Google Scholar 

  4. V. Yu. Slyusarchuk, “Necessary and sufficient conditions of Lipschitz reversibility of nonlinear differential operator d/dt−f in the space of bounded functions on axis,” Mat. Stud. 15(1), 77–86 (2001) [in Ukrainian].

    MathSciNet  Google Scholar 

  5. V. E. Slyusarchuk, “Necessary and sufficient conditions for the invertibility of the nonlinear differential operator d/dt + f in the space of bounded functions on the real axis,” in Differential Equations and Nonlinear Oscillation, Proc. Ukrainian Mathematical Congress — 2001, Sec. 3 (Inst. Math., NAS of Ukraine, Kiev, 2002), pp. 138–151 [in Russian].

    Google Scholar 

  6. V. E. Slyusarchuk, “Necessary and sufficient conditions for the Lipschitz invertibility of the nonlinear differential mapping d/dt + f in the spaces L p(ℝ,ℝ), 1 ⩽ p⩽∞,” Mat. Zametki 73(6), 891–903 (2003) [Math. Notes 73 (6), 843–854 (2003)].

    MathSciNet  Google Scholar 

  7. N. N. Bogolyubov, Yu. A. Mitropol’skii, and A. M. Samoilenko, Accelerated Convergence Method in Nonlinear Mechanics (Naukova Dumka, Kiev, 1969) [in Russian].

    Google Scholar 

  8. R. Reissig, G. Sansone and R. Conti, Qualitative Theory of Nonlinear Differential Equations (Edizioni cremonese, Roma, 1963; Nauka, Moscow, 1974).

    Google Scholar 

  9. E. F. Mishchenko and N. Kh. Rozov, Differential Equations with a Small Parameter, and Relaxation Oscillations (Nauka, Moscow, 1975) [in Russian].

    Google Scholar 

  10. V. A. Pliss, Integral Sets of Periodic Systems of Differential Equations (Nauka, Moscow, 1977) [in Russian].

    MATH  Google Scholar 

  11. Yu. V.TrubnikovandA. I. Perov, Differential Equations with Monotone Nonlinearities (Nauka i Tekhnika, Minsk, 1986) [in Russian].

    Google Scholar 

  12. A. I. Perov, “On bounded solutions of nonlinear systems of ordinary differential equations,” Vestn. VGU. Ser. Fiz., Matem., No. 1, 165–168 (2003).

  13. V. E. Slyusarchuk, “Nonlinear differential equations with solutions bounded on R,” Neliniini Kolivannya 11(1), 96–111 (2008) [in Ukrainian].

    MathSciNet  Google Scholar 

  14. G. M. Fikhtengol’ts, Course of Differential and Integral Calculus, in 3 volumes (Nauka, Moscow, 1966), Vol. 1 [in Russian].

    Google Scholar 

  15. A. N. Kolmogorov and S. V. Fomin, Elements of Theory of Functions and Functional Analysis (Nauka, Moscow, 1968) [in Russian].

    Google Scholar 

  16. L. Nirenberg, Topics in Nonlinear Functional Analysis (Courant Inst, of Math. Sci., New York, 1973; Mir, Moscow, 1977).

    Google Scholar 

Download references

Author information

Authors and Affiliations

  1. National University for the Management of Water and Natural Resources, Rivne, Ukraine

    V. E. Slyusarchuk

Authors
  1. V. E. Slyusarchuk
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to V. E. Slyusarchuk.

Additional information

Original Russian Text © V. E. Slyusarchuk, 2011, published in Matematicheskie Zametki, 2011, Vol. 90, No. 1, pp. 137–142.

Rights and permissions

Reprints and permissions

About this article

Cite this article

Slyusarchuk, V.E. Necessary and sufficient conditions for the existence and ɛ-uniqueness of bounded solutions of the equation x′ = f(x) − h(t). Math Notes 90, 136 (2011). https://doi.org/10.1134/S0001434611070133

Download citation

  • Received:

  • Published:

  • DOI: https://doi.org/10.1134/S0001434611070133

Keywords

Search

Navigation