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

In this work we study the following third order differential equation: 1 L y : = y ‴ + ( q ( x , y ) + λ ) y = f ∈ L 2 ( R ) , R = ( − ∞ , ∞ ) , λ > 0 , where q ( x , y ) ≥ 1 is a continuous function in all its variables. We will deal with the following questions: The existence of a solution to equation ( 1 ) in the space L 2 ( R ) where L 2 ( R ) is the space of square summable functions. Additional conditions on the third derivative of this solution to belong to the space L 2 ( R ) .