Dependence of eigenvalues of regular Sturm- Liouville operators on the boundary condition

In this paper,we study the continuous dependence of eigenvalue of Sturm-Liouville differential operators on the boundary condition by using of implicit function theorem.The work not only provides a new and elementary proof of the above results,but also explicitly presents the expressions for derivatives of the n-th eigenvalue with respect to given parameters.Further-more,we obtain the new results of the position and number of the generated double eigenvalues under the real coupled boundary condition.