In this paper, we study a nonlinear implicit differential equation with initial conditions. The considered problem involves the fractional Caputo derivatives under some conditions on the order. We prove an existence and uniqueness analytic result by application of Banach principle. Then, another result that deals with the existence of at least one solution is delivered and some sufficient conditions for this result are established by means of the fixed point theorem of Schaefer. At the end, we discuss two examples to illustrate the applicability of the main results.

中文翻译：

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具有Caputo导数的非线性隐式分数阶方程

在本文中，我们研究了具有初始条件的非线性隐式微分方程。所考虑的问题涉及阶上某些条件下的分数阶Caputo衍生物。通过应用Banach原理证明了存在性和唯一性分析结果。然后，传递解决至少一个解的存在的另一个结果，并通过Schaefer不动点定理为该结果建立一些充分的条件。最后，我们讨论两个例子来说明主要结果的适用性。