Berdyshev, Abdumauvlen S.Birgebaev, Ahtai B.Cabada Fernández, Alberto2018-10-222018-10-222017Berdyshev, A., Birgebaev, A. & Cabada, A. (2017). On the smoothness of solutions of the third order nonlinear differential equation. Boundary Value Problems, 2017:69. doi: https://doi.org/10.1186/s13661-017-0799-41687-2762http://hdl.handle.net/10347/17605In this work we study the following third order differential equation: Ly := y + (q(x, y) + λ)y = f ∈ L2(R), R = (–∞,∞),λ > 0, where q(x, y) ≥ 1 is a continuous function in all its variables. We will deal with the following questions: (a) The existence of a solution to equation (1) in the space L2(R) where L2(R) is the space of square summable functions. (b) Additional conditions on the third derivative of this solution to belong to the space L2(R)eng© The Author(s) 2017. This article is distributed under the terms of the Creative Commons Attribution 4.0 International License (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted use, distribution, and reproduction in any medium, provided you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license, and indicate if changes were madehttp://creativecommons.org/licenses/by/4.0/Nonlinear equationSolvabilitySeparabilitySmoothnessjournal article10.1186/s13661-017-0799-41687-2770open access