Nonlinear fractional equations for Caputo differential operators with two fractional orders are studied. One case is a generalization of the Bagley-Torvik equation, another is of Langevin type. These can be confused as being the same but because fractional derivatives do not commute these are different problems. However it is possible to use some common methodology. Some new regularity results for fractional integrals of a certain type are proved. These are used to rigorously prove equivalences between solutions of initial value problems for the fractional derivative equations and solutions of the corresponding integral equations in the space of continuous functions. A novelty is that it is not assumed that the nonlinear term is continuous but that it satisfies the weaker \(L^{p}\)-Carathéodory condition. Existence of solutions on an interval [0, T] in cases where T can be arbitrarily large, so-called global solutions, are proved, obtaining the necessary a priori bounds by using recent fractional Gronwall and fractional Bihari inequalities.

研究了具有两个分数阶的Caputo微分算子的非线性分式方程。一种情况是 Bagley-Torvik 方程的推广，另一种情况是 Langevin 类型。这些可能会被混淆为相同的问题，但因为分数导数不能交换，所以这些是不同的问题。然而，可以使用一些通用的方法。证明了某类分数积分的一些新的正则性结果。这些用于严格证明分数阶导数方程的初值问题的解与连续函数空间中相应积分方程的解之间的等价性。新颖之处在于，不假设非线性项是连续的，而是满足较弱的 \(L^{p}\)-Carathéodory 条件。在 T 可以任意大的情况下，证明了区间 [0, T] 上解的存在性，即所谓的全局解，通过使用最近的分数 Gronwall 和分数 Bihari 不等式获得了必要的先验界限。