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

Abstract 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 , $$ Ly : = y''' + \bigl( q ( x,y ) + \lambda \bigr)y = f \in L_{2} ( \mathbb{R} ), \quad \mathbb{R}= ( - \infty,\infty ), \lambda> 0, $$ where q ( x , y ) ≥ 1 $q(x,y) \geq1$ 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 L 2 ( R ) $L_{2} (\mathbb{R})$ where L 2 ( R ) $L_{2} (\mathbb{R})$ is the space of square summable functions. (b) Additional conditions on the third derivative of this solution to belong to the space L 2 ( R ) $L_{2} (\mathbb{R})$