Nonlinear fractional differential equations have been intensely studied using fixed point theorems on various different function spaces. Here we combine fixed point theory with complex analysis, considering spaces of analytic functions and the behaviour of complex powers. It is necessary to study carefully the initial value properties of Riemann–Liouville fractional derivatives in order to set up an appropriate initial value problem, since some such problems considered in the literature are not well-posed due to their initial conditions. The problem that emerges turns out to be dimensionally consistent in an unexpected way, and therefore suitable for applications too.

中文翻译：

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用复杂的方法解决一个适定的分数初值问题

已经在各种不同的函数空间上使用不动点定理对非线性分数阶微分方程进行了深入研究。在这里，我们将不动点理论与复分析结合起来，考虑解析函数的空间和复幂的行为。有必要仔细研究 Riemann-Liouville 分数阶导数的初值性质，以建立适当的初值问题，因为文献中考虑的一些此类问题由于其初始条件而不是适定的。出现的问题以意想不到的方式在尺寸上是一致的，因此也适用于应用程序。