In this paper, we study the constructive existence of solutions of multi-point boundary value problem for Hilfer fractional differential equation at resonance. The constructive existence of solutions is very important in practice, because it gives approximation method to solve equation as well as the existence. First, we obtain reversibility of some fractional integral operator which is useful for description and get analytic expressions of the inverse operator with Mittag–Leffler function. Next, we have the solution expressions of initial value problem and boundary-value problem for linear Hilfer fractional differential equations. And we get the sufficient conditions for the existence and nonexistence of solutions of multi-point value problem for nonlinear Hilfer fractional differential equation at resonance and establish the approximation method to solve it by the method of upper and lower solutions. Finally, we illustrate our results with an example.

中文翻译：

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Hilfer分数阶微分方程共振点多点边值问题解的构造性存在

在本文中，我们研究了Hilfer分数阶微分方程共振时多点边值问题解的构造性存在。解的建设性存在在实践中非常重要，因为它提供了求解方程和存在的近似方法。首先，我们获得了一些分数积分算子的可逆性，这对于描述非常有用，并获得了具有Mittag–Leffler函数的逆算子的解析表达式。接下来，我们得到了线性希尔弗分数阶微分方程的初值问题和边值问题的解表达式。并为存在共振条件的非线性希尔弗分数阶微分方程多点值问题的解的存在和不存在提供了充分的条件，并建立了上下求解的近似方法。最后，我们通过一个例子说明我们的结果。