Aleroev Mukhamed, Aleroev Temirkhan
Department of Higher Mathematics, Moscow Automobile and Road Construction State Technical University (STU-MADI), Leningradsky Ave. 64, 125319 Moscow, Russia.
Department of Applied Mathematics, The National Research Moscow State University of Civil Engineering (NRU MGSU), Yaroslavskoe Highway 26, 129337 Moscow, Russia.
Entropy (Basel). 2022 Apr 6;24(4):515. doi: 10.3390/e24040515.
In the spectral analysis of operators associated with Sturm-Liouville-type boundary value problems for fractional differential equations, the problem of positive definiteness or the problem of Hermitian nonnegativity of the corresponding kernels plays an important role. The present paper is mainly devoted to this problem. It should be noted that the operators under study are non-self-adjoint, their spectral structure is not well investigated. In this paper we use various methods to prove the Hermitian non-negativity of the studied kernels; in particular, a study of matrices that approximate the Green's function of the boundary value problem for a differential equation of fractional order is carried out. Using the well-known Livshits theorem, it is shown that the system of eigenfunctions of considered operator is complete in the space L2(0,1). Generally speaking, it should be noted that this very important problem turned out to be very difficult.
在与分数阶微分方程的Sturm-Liouville型边值问题相关的算子的谱分析中，相应核的正定问题或埃尔米特非负性问题起着重要作用。本文主要致力于此问题。应当指出，所研究的算子是非自伴的，其谱结构尚未得到充分研究。在本文中，我们使用各种方法来证明所研究核的埃尔米特非负性；特别是，对逼近分数阶微分方程边值问题格林函数的矩阵进行了研究。利用著名的利夫希茨定理，证明了所考虑算子的特征函数系在空间L2(0,1)中是完备的。一般来说，应当指出，这个非常重要的问题结果非常困难。
