【作者】 李镇；

【导师】 李雪梅；

【作者基本信息】 郑州大学 ， 基础数学， 2007， 硕士

【摘要】 本文研究下述2×2常微分方程的两种边值问题，其中u(x)，v(x)，w(x)是实连续函数。第一种边界条件如下：第二种边界条件是所谓的周期边界条件：上述方程分别和两种边界条件构成两个特征值问题。它们的基本问题是特征值的存在性，个数与分布，以及相应的特征函数系的完备性。本文将它们分别归纳为两个常微分算子，通过Grean函数的构造，证明了预解式的存在性，从而证明了它们都是自伴算子。然后将这两个常微分算子转化为积分算子，证明了积分算子是自伴的全连续算子，利用自伴全连续算子的谱理论的一般结果，证明了它们的特征函数系在L~2(0，π)上的完备性。从而证明了前述两个微分算子的特征函数系在L~2(0，π)上的完备性。最后，应用标准的方法又获得了函数在较严格的限制下按特征函数系展开为一致收敛的广义Fourier级数的展开定理。更多还原

【Abstract】 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.更多还原