In this paper, we considered the spectral problem and initial value problem of a nonlocal Sturm-Liouville equation with fractional integrals and fractional derivatives. We proved that the fractional operator associated to the nonlocal Sturm-Liouville equation is self-adjoint in Hilbert space. And then, we derived the corresponding spectral problem consists of countable number of real eigenvalues, and the algebraic multiplicity of each eigenvalue is simple. We also discussed the orthogonal completeness of the corresponding eigenfunction system in the Hilbert spaces. Furthermore, we obtained asymptotic properties of eigenvalues and the number of zeros of eigenfunctions by using the perturbation theory for linear operators. Finally, we studied the uniqueness of solutions for the nonlocal Sturm-Liouville equation under some initial value conditions.

在本文中，我们考虑了带有分数积分和分数导数的非局部Sturm-Liouville方程的频谱问题和初值问题。我们证明了与非局部Sturm-Liouville方程相关的分数算子在希尔伯特空间中是自伴的。然后，我们推导了对应的频谱问题，该问题由可数的真实特征值组成，并且每个特征值的代数多重性很简单。我们还讨论了希尔伯特空间中相应特征函数系统的正交完整性。此外，通过使用线性算子的扰动理论，我们获得了特征值的渐近性质和特征函数的零个数。最后，我们研究了在某些初始值条件下非局部Sturm-Liouville方程解的唯一性。