Nonlinear Oscillations

, Volume 11, Issue 3, pp 442–460 | Cite as

Invertibility of the nonlinear operator \(\left( {{\mathcal{L}}x} \right)\left( t \right) = H\left( {x\left( t \right),\;\frac{{dx\left( t \right)}} {{dt}}} \right)\) in the space of functions bounded on the axis

  • V. Yu. Slyusarchuk
We obtain conditions for the invertibility of the nonlinear differential operator
$$\left( {{\mathcal{L}}x} \right)\left( t \right) = H\left( {x\left( t \right),\;\frac{{dx\left( t \right)}} {{dt}}} \right)$$
in the space of functions bounded on the axis.


Banach Space Periodic Solution Nonlinear Differential Equation Nonlinear Operator Functional Differential Equation 
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  1. 1.
    Yu. M. Daletskii and M. G. Krein, Stability of Solutions of Differential Equations in a Banach Space [in Russian], Nauka, Moscow (1970).Google Scholar
  2. 2.
    M. A. Krasnosel'skii, V. Sh. Burd, and Yu. S. Kolesov, Nonlinear Almost Periodic Oscillations [in Russian], Nauka, Moscow (1970).Google Scholar
  3. 3.
    P. Hartman, Ordinary Differential Equations, Wiley, New York (1964).MATHGoogle Scholar
  4. 4.
    J. L. Massera and J. J. Schaffer, Linear Differential Equations and Function Spaces, Academic Press, New York (1966).MATHGoogle Scholar
  5. 5.
    Yu. V. Trubnikov and A. I. Perov, Differential Equations with Monotone Nonlinearities [in Russian], Nauka i Tekhnika, Minsk (1986).MATHGoogle Scholar
  6. 6.
    Yu. A. Mitropol'skii, A. M. Samoilenko, and V. L. Kulik, Investigation of Dichotomy of Linear Systems of Differential Equations Using Lyapunov Functions [in Russian], Naukova Dumka, Kiev (1990).Google Scholar
  7. 7.
    É. Mukhamadiev, “On the invertibility of functional operators in the space of functions bounded on the axis,” Mat. Zametki, 11, No. 3, 269–274 (1972).MathSciNetGoogle Scholar
  8. 8.
    V. E. Slyusarchuk, “Nonlocal theorems on bounded solutions of functional differential equations with non-Lipschitz nonlinearities,” in: Investigation of Differential, Difference, and Differential-Difference Equations [in Russian], Naukova Dumka, Kiev (1980), pp. 121–130.Google Scholar
  9. 9.
    V. E. Slyusarchuk, “Invertibility of almost periodic c-continuous functional operators,” Mat. Sb., 116, No. 4, 483–501 (1981).MATHMathSciNetGoogle Scholar
  10. 10.
    V. E. Slyusarchuk, Bounded Solutions of Functional and Functional Differential Equations [in Russian], Doctoral-Degree Thesis (Physics and Mathematics), Rovno (1983).Google Scholar
  11. 11.
    V. E. Slyusarchuk, “Conditions for the existence of bounded solutions of nonlinear differential equations,” Usp. Mat. Nauk, 54, Issue 4, 181–182 (1999).MathSciNetGoogle Scholar
  12. 12.
    V. E. Slyusarchuk, “Necessary and sufficient conditions for the existence and uniqueness of bounded solutions of nonlinear differential equations,” Nonlin. Kolyvannya, 2, No. 4, 523–539 (1999).MathSciNetGoogle Scholar
  13. 13.
    V. E. Slyusarchuk, “Nonlinear differential equations with solutions bounded on ℝ,” Nonlin. Kolyvannya, 11, No. 1, 96–111 (2008).MathSciNetGoogle Scholar
  14. 14.
    G. M. Fikhtengol'ts, A Course in Differential and Integral Calculus [in Russian], Vol. 1, Nauka, Moscow (1966).Google Scholar
  15. 15.
    L. Nirenberg, Topics in Nonlinear Functional Analysis, Courant Institute of Mathematical Sciences, New York University, New York (1974).MATHGoogle Scholar
  16. 16.
    A. D. Aleksandrov and N. Yu. Netsvetaev, Geometry [in Russian], Nauka, Moscow (1990).MATHGoogle Scholar

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© Springer Science+Business Media, Inc. 2008

Authors and Affiliations

  1. 1.National University of Water Management and Nature Resources ApplicationRivneUkraine

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