In this paper, we investigate the existence of the extremal solutions for a class of impulsive fractional integro-differential equation with order $$1< \alpha < 2$$, combine upper and lower solution method with the theory of $$(\alpha ,\beta )$$-resolvent family, a series of results about the solutions to such differential equations is obtained. The results improve and extend some relevant results in ordinary differential equations and partial differential equations. Finally, an example is given to illustrate the feasibility of our results.

中文翻译：

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脉冲分数阶演化方程的上下解法

本文研究一类阶次$$1<\alpha<2$$的脉冲分数阶积分微分方程极值解的存在性，结合上下解法和$$(\alpha, β )$$-resolvent 族，得到了关于此类微分方程解的一系列结果。结果改进和扩展了常微分方程和偏微分方程中的一些相关结果。最后，给出一个例子来说明我们的结果的可行性。