Journal of Mathematical Sciences

, Volume 223, Issue 3, pp 210–222 | Cite as

Existence of Solutions of the Boundary-Value Problem for a Nonlinear Differential Equation of Fractional Order

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We find sufficient conditions for the existence of solutions of the boundary-value problem for a nonlinear differential equation containing a mixed Riemann–Liouville derivative of the fractional order.

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References

  1. 1.
    S. G. Samko, A. A. Kilbas, and O. I. Marichev, Integrals and Derivatives of Fractional Orders and Some Their Applications [in Russian], Nauka Tekhnika, Minsk (1987).MATHGoogle Scholar
  2. 2.
    D. Mangeron, “Sopra un problema al contorno per un’equazione differenziale all derivate parziale di qaunt con le caratteristiche reali dopi,” Rend. Accad. Sci. Fis. Mat. Napoli, 2, 28–40 (1932).Google Scholar
  3. 3.
    D. Mangeron, “New methods for determining solutions of mathematical models governing polyvibrating phenomena,” Bull. Inst. Politech. Iasi., 14, No. 1–2, 433–436 (1968).MATHGoogle Scholar
  4. 4.
    G. Birkhoff and W. J. Gordon, “The draftman’s and related equations,” J. Approxim. Theory, 1, 199–208 (1968).CrossRefMATHGoogle Scholar
  5. 5.
    P. Seifert, “Fehlerabschätzungen für differenzenverfahren bei einer hyperbolischen Differentialgleichung,” Beitr. Numer. Math., 2, 193–209 (1974).MATHGoogle Scholar
  6. 6.
    A. N. Vityuk, “On the set of solutions of a boundary-value problem for a hyperbolic differential inclusion,” Differents. Uravn., 35, No. 6, 780–783 (1999).MathSciNetGoogle Scholar
  7. 7.
    A. N. Vityuk and A. V. Mikhailenko, “On the solvability of the boundary-value problem for a differential equation of fractional order,” Differents. Uravn., 47, No. 12, 1803–1807 (2011).MATHGoogle Scholar
  8. 8.
    A. V. Mikhailenko, “On one boundary-value problem for a differential equation of fractional order,” Visn. Odes. Nats. Univ., Ser. Mat. Mekh., 15, Issue 19, 88–93 (2010).Google Scholar
  9. 9.
    B. I. Ptashnik, Ill-Posed Boundary-Value Problems for Partial Differential Equations [in Russian], Naukova Dumka, Kiev (1984).Google Scholar
  10. 10.
    J. Hadamard, “Equations aux derivees partielles, le cas hyperbolique,” L’Enseignement Math., 35, No. 1, 25–29 (1936).MATHGoogle Scholar
  11. 11.
    A. Huber, “Die erste Randwertaufgabe für geschlossene Bereiche bei der Gleichung u xy = f (x, y),” Monatsh. Math. Phys., 39, 79–100 (1932).MathSciNetCrossRefMATHGoogle Scholar
  12. 12.
    A. N. Vityuk and A. V. Mykhailenko, “Darboux problem for differential equation with mixed regularized derivative of fractional order,” Nonlin. Stud., 20, No. 4, 571–580 (2013).MathSciNetMATHGoogle Scholar
  13. 13.
    A. N. Vityuk and A. V. Mikhailenko, “Boundary-value problem for a differential equation of fractional order,” Visn. Odes. Nats. Univ., 19, Issue 2(22), 19–26 (2014).Google Scholar
  14. 14.
    S. Walczak, “Absolutely continuous functions of several variables and their application to differential equations,” Bull. Pol. Acad. Sci. Math., 35, No. 11–12, 733–744 (1987).MathSciNetMATHGoogle Scholar
  15. 15.
    N. O. Virchenko and V. Ya Rybak. Foundations of Fractional Integrodifferentiation [in Ukrainian], Zadruha, Kyiv (2007).Google Scholar
  16. 16.
    A. A. Kilbas, H. M. Srivastava, and J. I. Trujillo Theory and Applications of Fractional Differential Equation, Elsevier, Amsterdam (2006).MATHGoogle Scholar
  17. 17.
    A. A. Chikrii and I. I. Matichin, “On the linear processes with fractional derivatives controlled by conflicts,” Trud. Inst. Mat. Mekh. Ural Otdelen. Ros. Akad. Nauk, 17, No. 2, 256–270 (2001).Google Scholar
  18. 18.
    A. F. Timan, Theory of Functions of Real Variable [in Russian], Fizmatgiz, Moscow (1960).Google Scholar
  19. 19.
    A. N. Vityuk and A. V. Mikhailenko, “On one class of differential equations of fractional order,” Nelin. Kolyv., 11, No. 3, 293–304 (2008); English translation: Nonlin. Oscillat., 11, No. 3, 307–319 (2008).Google Scholar

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© Springer Science+Business Media New York 2017

Authors and Affiliations

  1. 1.Mechnikov Odessa National University, Institute of Mathematics, Economics, and MechanicsOdessaUkraine
  2. 2.Odessa National Economic UniversityOdessaUkraine

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