Department of Mathematics, Yarmouk University, Irbid, Jordan.
Chaos. 2019 Jan;29(1):013135. doi: 10.1063/1.5083202.
In this paper, we study linear and nonlinear fractional eigenvalue problems involving the Atangana-Baleanu fractional derivative of the order 1<δ<2. We first estimate the fractional derivative of a function at its extreme points and apply it to obtain a maximum principle for the linear fractional boundary value problem. We then estimate the eigenvalues of the nonlinear eigenvalue problem and obtain necessary conditions to guarantee the existence of eigenfunctions. We also obtain a uniqueness result and a norm estimate of solutions of the linear problem. The obtained maximum principle and results are based on a condition that connects the boundary conditions, the order of the fractional derivative, and the Mittag-Leffler kernel. This condition is different from the ones obtained in previous results with different types of fractional derivatives.
在本文中，我们研究了涉及阶数 1<δ<2 的 Atangana-Baleanu 分数导数的线性和非线性分数特征值问题。我们首先估计了函数在其极值点处的分数导数，并将其应用于获得线性分数边值问题的最大原理。然后，我们估计了非线性特征值问题的特征值，并获得了保证特征函数存在的必要条件。我们还获得了线性问题解的唯一性结果和范数估计。所得到的最大原理和结果基于一个条件，该条件连接了边界条件、分数导数的阶数和 Mittag-Leffler 核。这个条件与使用不同类型分数导数获得的之前的结果中的条件不同。
