On the smoothness of solutions of the third order nonlinear differential equation

In 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)Alberto Cabada was partially supported by Ministerio de Economía y Competitividad, Spain, and FEDER, project MTM2013-43014-P, and by the Agencia Estatal de Investigación (AEI) of Spain under grant MTM2016-75140-P, co-financed by the European Community fund FEDERS