具有Hilfer分數階脈沖微分方程邊值問題解的存在性

郭春靜 孟凡猛 陳坤 江衛華

摘 要：為了拓展邊值問題的基本理論，研究一類具有有限個脈沖點的Hilfer分數階脈沖微分方程邊值問題解的存在性。首先，求出微分方程等價的積分方程；其次，定義恰當的Banach空間和范數，構造合適的算子，在非線性項滿足不同條件的情況下，運用Krasnoselskii不動點定理，分別得到此類邊值問題存在解的充分條件；最后，通過2個實例驗證研究結果的普適性。結果表明，含有Hilfer分數階導數的脈沖微分方程邊值問題的解具有存在性。運用Krasnoselskii不動點定理能夠有效解決具有Hilfer分數階脈沖微分方程邊值問題解的存在性問題，豐富了分數階微分方程理論，為解決其他類型的脈沖分數階微分方程邊值問題提供了借鑒與參考。

關鍵詞：解析理論；脈沖；邊值問題；Krasnoselskii不動點定理；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

近幾十年來，分數階微分方程受到研究者的廣泛關注。人們之所以對分數階微分方程產生興趣，主要是因為分數階導數對于科學技術領域的不同過程、材料記憶以及遺傳特性描述發揮著重要作用。脈沖現象實際上是一種間斷、突然的變化，經常伴隨一些物理系統的出現。脈沖微分方程廣泛應用于力學、醫學、生態學等諸多領域。為了更加精確地描述這類演化過程，許多研究人員對具有脈沖條件的微分方程展開討論，參見文獻［1］—文獻［9］。在文獻［10］中，FENG等運用不動點定理研究了下列整數階脈沖微分方程邊值問題：

4 結 語

本文運用Krasnoselskii不動點定理，研究了一類具有Hilfer分數階導數的脈沖微分方程邊值問題，得到了這類邊值問題解的存在性，通過2個具體實例說明了結論的正確性。研究結果表明，Krasnoselskii不動點定理能夠有效解決具有Hilfer分數階脈沖微分方程邊值問題解的存在性問題，推廣了具有Riemann-Liouville分數階導數或者Caputo分數階導數脈沖微分方程邊值問題的結果，豐富了分數階微分方程理論，為解決其他類型的脈沖分數階微分方程邊值問題提供了借鑒與參考。

但是本文是在非線性項連續的條件下考慮的脈沖邊值問題，限制條件較強。在今后的研究中，將會探索削弱非線性項滿足的條件，進一步研究此類問題存在解的更一般化的結果。

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