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Conditions for the existence of bounded solutions of one class of nonlinear differential equations

Abstract

For systems of nonlinear differential equations (dx/dt) = A(x)x + f(t) in a Banach space, we establish sufficient conditions for the existence of their solutions bounded on the entire real axis R.

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References

  1. 1.

    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).

  2. 2.

    Yu. M. Daletskii and M. G. Krein, Stability of Solutions of Differential Equations in a Banach Space [in Russian], Nauka, Moscow (1970).

  3. 3.

    J. L. Massera and J. J. Schaffer, Linear Differential Equations and Function Spaces, Academic Press, New York (1966).

  4. 4.

    M. A. Krasnosel’skii, V. Sh. Burd, and Yu. S. Kolesov, Nonlinear Almost Periodic Oscillations [in Russian], Nauka, Moscow (1970).

  5. 5.

    L. Nirenberg, Topics in Nonlinear Functional Analysis, Courant Institute of Mathematical Sciences, New York University, New York (1974).

  6. 6.

    V. E. Slyusarchuk, “Invertibility of almost periodic c-continuous functional operators,” Mat. Sb., 116(158), no. 4 (12), 483–501 (1981).

  7. 7.

    É. Mukhamadiev, Investigation in the Theory of Periodic and Bounded Solutions of Differential Equations [in Russian], Doctoral-Degree Thesis (Physics and Mathematics), Dushanbe (1978).

  8. 8.

    V. E. Slyusarchuk, “Invertibility of nonautonomous differential-functional operators,” Mat. Sb., 130(172), no. 1 (5), 86–104 (1986).

  9. 9.

    V. E. Slyusarchuk, “Weakly nonlinear perturbations of normally solvable functional-differential and discrete equations,” Ukr. Mat. Zh., 39, No. 5, 660–662 (1987).

  10. 10.

    V. E. Slyusarchuk, “Necessary and sufficient conditions for the invertibility of nonautonomous functional-differential operators,” Mat. Zametki, 42, No. 2, 262–267 (1987).

  11. 11.

    V. E. Slyusarchuk, “Necessary and sufficient conditions for the invertibility of uniformly c-continuous functional-differential operators,” Ukr. Mat. Zh., 41, No. 2, 201–205 (1989).

  12. 12.

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

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Translated from Ukrains’kyi Matematychnyi Zhurnal, Vol. 58, No. 3, pp. 317–325, March, 2006.

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Hrod, I.M. Conditions for the existence of bounded solutions of one class of nonlinear differential equations. Ukr Math J 58, 357–367 (2006). https://doi.org/10.1007/s11253-006-0071-z

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Keywords

  • Banach Space
  • Nonlinear Differential Equation
  • Arbitrary Element
  • Closed Ball
  • Arbitrary Sequence