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One method for investigating the solvability of boundary value problems for an implicit differential equation

Annotation

The article concernes a boundary value problem with linear boundary conditions of general form for the scalar differential equation f(t,x(t),x ̇(t))=y ̂(t), not resolved with respect to the derivative x ̇ of the required function. It is assumed that the function f satisfies the Caratheodory conditions, and the function y ̂ is measurable. The method proposed for studying such a boundary value problem is based on the results about operator equation with a mapping acting from a metric space to a set with distance (this distance satisfies only one axiom of a metric: it is equal to zero if and only if the elements coincide). In terms of the covering set of the function f(t,x_1,•): R→R and the Lipschitz set of the function f(t,•,x_2): R →R, conditions for the existence of solutions and their stability to perturbations of the function f generating the differential equation, as well as to perturbations of the right-hand sides of the boundary value problem: the function y ̂ and the value of the boundary condition, are obtained.

Keywords

implicit differential equation, linear boundary conditions, existence of solutions to a boundary value problem, covering mapping of metric spaces

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DOI

10.20310/2686-9667-2021-26-136-404-413

UDC

517.988.6, 517.922

Pages

404-413

References

[1] E.S. Zhukovskiy, W. Merchela, “On covering mappings in generalized metric spaces in studying implicit differential equations”, Ufa Mathematical Journal, 12:4 (2020), 41–54. [2] S. Benarab, E.S. Zhukovskii, W. Merchela, “Theorems on perturbations of covering mappings in spaces with a distance and in spaces with a binary relation”, Trudy Inst. Mat. i Mekh. UrO RAN, 25, 2019, 52–63 (In Russian). [3] W. Merchela, “On stability of solutions of integral equations in the class of measurable functions”, Russian Universities Reports. Mathematics, 26:133 (2021), 44–54 (In Russian). [4] E.R. Avakov, A.V. Arutyunov, E.S. Zhukovskii, “Covering mappings and their applications to differential equations not solved with respect to the derivative”, Differential Equations, 45:5 (2009), 627–649. [5] A.V. Arutyunov, E.S. Zhukovskii, S.E. Zhukovskii, “On the well-posedness of differential equations unsolved for the derivative”, Differential Equations, 47:11 (2011), 1541–1555. [6] E.S. Zhukovskii, E.A. Pluzhnikova, “Covering Mappings in a Product of Metric Spaces and Boundary Value Problems for Differential Equations Unsolved for the Derivative”, Differential Equations, 49:4 (2013), 420–436. [7] E.S. Zhukovskii, E.A. Pluzhnikova, “A theorem on operator covering in the product of metric spaces”, Tambov University Reports. Series: Natural and Technical Sciences, 16:1 (2011), 70–72 (In Russian). [8] D. Burago, Yu. Burago, S. Ivanov, A Course in Metric Geometry, Graduate Studies in Mathematics, 33, American Mathematical Society, Providence, Rhode Island, 2001. [9] A.V. Arutyunov, A.V. Greshnov, “(q_1; q_2) -quasimetric spaces. Covering mappings and coincidence points”, Izv. Math., 82:2 (2018), 245–272. [10] E.S. Zhukovskiy, “The fixed points of contractions of f -quasimetric spaces”, Siberian Mathematical Journal, 59:6 (2018), 1063-1072. [11] A.V. Arutyunov, “Covering mappings in metric spaces and fixed points”, Proceedings of the Russian Academy of Sciences, 416:2 (2007), 151–155 (In Russian). [12] I.V. Shragin, “Superpositional measurability under generalized Caratheodory conditions”, Tambov University Reports. Series: Natural and Technical Sciences, 19:2 (2014), 476–478 (In Russian). [13] I.D. Serova, “Superpositional measurability of a multivalued function under generalized Caratheodory conditions”, Russian Universities Reports. Mathematics, 26:135 (2021), 305–314 (In Russian). [14] N.V. Azbelev, V.P. Maksimov, L.F. Rakhmatullina, Introduction to the theory of functional differential equations, Nauka Publ., Moscow, 1991 (In Russian).

Received

2021-09-29

Section of issue

Scientific articles

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