In this paper, we study some existence results for fractional differential equations subject to some kind of initial conditions. First, we focus on the linear problem and we give an explicit form of solutions by reduction to an integral problem. We analyze some properties of the solutions to the linear problem in terms of its coefficients. Then we provide examples of application of the mentioned properties. Secondly, with the help of this theory, we study the nonlinear problem subject to initial value conditions. By using the upper and lower solutions method and the monotone iterative algorithm, we show the existence and localization of solutions to the nonlinear fractional differential equation.

中文翻译：

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存在上下解的分数阶微分方程的存在性结果

在本文中，我们研究了在某种初始条件下分数阶微分方程的一些存在性结果。首先，我们关注线性问题，并通过简化为整数问题给出明确的解决方案形式。我们根据其系数分析线性问题解的某些性质。然后，我们提供上述属性的应用示例。其次，借助这一理论，我们研究了具有初始值条件的非线性问题。通过使用上下解方法和单调迭代算法，我们证明了非线性分数阶微分方程解的存在性和局限性。