We study the long-time behavior of solutions for a general class of nonlinear fractional differential equations. These equations involve Hadamard fractional derivatives of different orders. We determine sufficient conditions on the nonlinear terms which guarantee that solutions exist globally and decay to zero as a logarithmic function. For this purpose, we combine and generalize some versions of Gronwall–Bellman inequality, appropriate regularization techniques and several properties of the Hadamard fractional derivative. Our findings are illustrated by examples.

中文翻译：

---

Hadamard分数微分问题的对数型的稳定性。

我们研究了一类非线性分数阶微分方程解的长时间行为。这些方程式涉及不同阶数的Hadamard分数导数。我们在非线性项上确定了充分的条件，这些条件保证了解在全局范围内存在并作为对数函数衰减为零。为此，我们结合并推广了Gronwall-Bellman不等式的某些版本，适当的正则化技术以及Hadamard分数导数的若干性质。实例说明了我们的发现。