Examples



mdbootstrap.com



 
Статья
2013

Periodic boundary-value problem for a fourth-order differential equation


A. R. AbdullaevA. R. Abdullaev, E. A. SkachkovaE. A. Skachkova
Русская математика
https://doi.org/10.3103/S1066369X13120013
Abstract / Full Text

In the present paper we obtain sufficient conditions for solvability of a periodic boundary-value problem for a fourth-order ordinary differential equation. The research technique is based on a solvability theorem for a quasi-linear operator equation in the resonance case. We formulate sufficient conditions for existence of periodic solutions in terms of the initial equation. The main result of the paper clarifies the existence theorem established by B. Mehry and D. Shadman in Sci. Iran. 15 (2), 182–185 (2008).

Author information
  • Perm National Research Polytechnic University, Komsomol’skii pr. 29, Perm, 614990, RussiaA. R. Abdullaev
  • Perm State National Research University, ul. Bukireva 15, Perm, 614990, RussiaE. A. Skachkova
References
  1. N. V. Azbelev, V. P. Maksimov, and L. F. Rakhmatullina, Introduction to the Theory of Functional-Differential Equations (Nauka, Moscow, 1991).
  2. C. De Coster, C. Fabry, and F. Munyamarere, “Nonresonance Conditions for Fourth-Order Nonlinear Boundary Value Problems,” Internat. J. Math. Sci. 17(4), 725–740 (1994).
  3. L. Pinda, “On a Fourth Order Periodic Boundary Value Problem,” Arch. Math. (Brno) 30(1), 1–8 (1994).
  4. C. Bereanu, “Periodic Solutions of Some Fourth-Order Nonlinear Differential Equations,” Nonlinear Anal.: Theory, Methods & Appl. 71(1–2), 53–57 (2009).
  5. Q. Yao, “Existence, Multiplicity and Infinite Solvability of Positive Solutions to a Nonlinear Fourth-Order Periodic Boundary Value Problem,” Nonlinear Anal.: Theory, Methods & Appl. 63(2), 237–246 (2005).
  6. P. Pietramala, “ANote on a BeamEquation with Nonlinear Boundary Conditions,” BoundaryValue Problems 2011, Art. ID 376782, 1–14 (2011).
  7. B. Mehri and D. Shadman, “On the Existence of Periodic Solutions for Nonlinear Ordinary Differential Equations,” Sci. Iran. 15(2), 182–185 (2008).
  8. A. R. Abdullaev and A. B. Burmistrova, Elements of Theory of Topological Noetherian Operators (Chelyabinsk State Univ., Chelyabinsk, 1994).
  9. V. A. Trenogin, Functional Analysis (Fizmatlit, Moscow, 2002).
  10. A. R. Abdullaev, “Solvability of Quasi-Linear Boundary-Value Problems for Functional-Differential Equations,” in Functional-Differential Equations (Perm, 1992), pp. 80–87 [in Russian].
  11. S. Fučik and A. Kufner, Nonlinear Differential Equations (Nakladatelstvi Technicke Literatury, Praha, 1978; Nauka, Moscow, 1988).