郭春静 孟凡猛 陈坤 江卫华

摘 要：为了拓展边值问题的基本理论，研究一类具有有限个脉冲点的Hilfer分数阶脉冲微分方程边值问题解的存在性。首先，求出微分方程等价的积分方程；其次，定义恰当的Banach空间和范数，构造合适的算子，在非线性项满足不同条件的情况下，运用Krasnoselskii不动点定理，分别得到此类边值问题存在解的充分条件；最后，通过2个实例验证研究结果的普适性。结果表明，含有Hilfer分数阶导数的脉冲微分方程边值问题的解具有存在性。运用Krasnoselskii不动点定理能够有效解决具有Hilfer分数阶脉冲微分方程边值问题解的存在性问题，丰富了分数阶微分方程理论，为解决其他类型的脉冲分数阶微分方程边值问题提供了借鉴与参考。

关键词：解析理论；脉冲；边值问题；Krasnoselskii不动点定理；解的存在性

Existence of solutions for boundary value problems of fractional impulsive differential equations with Hilfer

GUO Chunjing¹，MENG Fanmeng¹，CHEN Kun²，JIANG Weihua¹

（1.School of Sciences， Hebei University of Science and Technology， Shijiazhuang， Hebei 050018，China；2.Office of Academic Affairs，Shijiazhuang Peoples Medical College，Shijiazhuang，Hebei 050091，China）

Abstract：In order to extend the basic theory of boundary value problems， the existence of solutions for a class of Hilfer fractional impulsive differential equations with finite impulsive points was studied. Firstly， the integral equation equivalent to the differential equation was obtained; Secondly， appropriate Banach spaces and norms were defined， and appropriate operators were constructed. When the nonlinear term satisfies different conditions， sufficient conditions for the existence of solutions of such boundary value problems were obtained by using Krasnoselskii fixed point theorem; Finally， two examples were used to illustrate the universality of the research results. It is shown that the solution of the boundary value problem of impulsive differential equations with Hilfer fractional derivative exists. By using the Krasnoselskii fixed-point theorem， the existence of solutions for impulsive differential equation boundary value problems with Hilfer fractional order can be effectively solved， which provides some reference for solving other types of impulsive fractional differential equation boundary value problems.

Keywords：analytic theory; impulse; boundary value problem; Krasnoselskii fixed point theorem; existence of solutions

近幾十年来，分数阶微分方程受到研究者的广泛关注。人们之所以对分数阶微分方程产生兴趣，主要是因为分数阶导数对于科学技术领域的不同过程、材料记忆以及遗传特性描述发挥着重要作用。脉冲现象实际上是一种间断、突然的变化，经常伴随一些物理系统的出现。脉冲微分方程广泛应用于力学、医学、生态学等诸多领域。为了更加精确地描述这类演化过程，许多研究人员对具有脉冲条件的微分方程展开讨论，参见文献［1］—文献［9］。在文献［10］中，FENG等运用不动点定理研究了下列整数阶脉冲微分方程边值问题：

4 结 语

本文运用Krasnoselskii不动点定理，研究了一类具有Hilfer分数阶导数的脉冲微分方程边值问题，得到了这类边值问题解的存在性，通过2个具体实例说明了结论的正确性。研究结果表明，Krasnoselskii不动点定理能够有效解决具有Hilfer分数阶脉冲微分方程边值问题解的存在性问题，推广了具有Riemann-Liouville分数阶导数或者Caputo分数阶导数脉冲微分方程边值问题的结果，丰富了分数阶微分方程理论，为解决其他类型的脉冲分数阶微分方程边值问题提供了借鉴与参考。

但是本文是在非线性项连续的条件下考虑的脉冲边值问题，限制条件较强。在今后的研究中，将会探索削弱非线性项满足的条件，进一步研究此类问题存在解的更一般化的结果。

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