The Completeness Of Eigenfuction System For Two Regular 2×2 Sturm-Liouville Problems

In this paper, two eigenvalue problems of 2×2 Sturm-Liouville are discussed. The 2×2 Sturm-Liouville equations system is of the formWhere u(x),v(x) and w(x) are real continues functions.The most important boundary-value conditions for the equations system areandAbove equations system with two system of boundary conditions constitute two eigenvalue problems respectively. The existence, number, distribution and completeness of eigenfunctions are essential problem for them.At first, we can induce the two ordinary differential operators from the two eigenvalue problems respectively. The existence of resolvent is proved through Green function, thereby it is proved that the two ordinary differential operators are selfadjoint operators.Secondly, We transform the two ordinary differential operators into integral operators, and prove that the integral operators are selfadjoint completely continous operators. By using the general fact of the spectral theory for selfadjoint completely continous oprator, the completeness of their eigenfunction system inL~2(0,π) is proved. Therefore the completeness of the eigenfunction system for the two ordinany differential operators in L~2 (0,π) is proved.Finally, madding use of the standard method, the expansion theorem is obtained, which says that the function can be expanded into the uniformly convergent generalized Fourier series in eigenfunction system under the relatively strict conditions.