Existence of the Local Solution of a Non Homogeneous Schrödinger Type Equation

Abstract

In this article, we prove that initial value problem associated to the Schrödinger type non homogeneous equation in periodic Sobolev spaces has a local solution in [0, T] with T > 0, and the solution has continuous dependence with respect to the initial data and the non homogeneous part of the problem. We do this in a intuitive way using Fourier theory and introducing a Co- group inspired by the work of Iorio [1] and Santiago [7]. Also, we prove the uniqueness solution of the Schrödinger type homogeneous equation, using its conservative property, inspired by the work of Iorio [1] and Santiago [6]. Finally, we study its generalization to n-th order equation.