分數(shù)階脈沖微分方程邊值問題解的存在性

江衛(wèi)華,李慶敏,周彩蓮

(河北科技大學理學院，河北石家莊 050018)

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分數(shù)階脈沖微分方程邊值問題解的存在性

江衛(wèi)華,李慶敏,周彩蓮

(河北科技大學理學院，河北石家莊 050018)

為了解決對半無窮區(qū)間上具有可數(shù)個脈沖點且?guī)в蟹e分邊界條件的分數(shù)階脈沖微分方程邊值問題，具體研究此類微分方程邊值問題解的存在性。通過定義合適的Banach空間、范數(shù)以及算子，合理運用分數(shù)階微積分的性質(zhì)，分別應(yīng)用壓縮映像原理和Krasnoselskii不動點定理證明了分數(shù)階脈沖微分方程邊值問題解的存在性，最后通過實例驗證了此類方程邊值問題解的存在性。

常微分方程解析理論；脈沖；壓縮映像原理；Krasnoselskii不動點定理；邊值問題；半無窮區(qū)間

1 問題提出

分數(shù)階微積分是對整數(shù)階微積分理論的拓展，它可以更好地描述某些客觀事物或規(guī)律，應(yīng)用廣泛，比如在處理光學和熱學系統(tǒng)、流變學及材料和力學系統(tǒng)、信號處理和系統(tǒng)辨識、控制等問題的過程中，經(jīng)常會用到分數(shù)階微積分的理論。所以分數(shù)階微積分理論受到了人們越來越多的關(guān)注[1-12]。此外，脈沖微分方程也有廣泛的應(yīng)用，許多學者對脈沖微分方程的理論及其應(yīng)用[13-24]進行了深入的研究。

文獻[2]中GUO應(yīng)用Banach空間中的錐拉伸與壓縮不動點定理研究了半無窮區(qū)間上具有可數(shù)個脈沖點的二階奇異脈沖微分方程邊值問題:

解的存在性。

文獻[4]中AHMAD等根據(jù)壓縮映像原理和Krasnoselskii不動點定理研究了有限區(qū)間上具有有限個脈沖點的非線性分數(shù)階脈沖微分方程邊值問題：

解的存在性。

受上述文獻的啟發(fā)，本文將應(yīng)用壓縮映像原理和Krasnoselskii不動點定理研究半無窮區(qū)間上具有可數(shù)個脈沖點的分數(shù)階脈沖微分方程邊值問題：

2 預(yù)備知識

定義1u:J→R是連續(xù)函數(shù)，u的α階Riemann-Liuville積分的定義式為

定義2u:J→R是連續(xù)函數(shù)，u的α階Riemann-Liuville導數(shù)的定義式為

定理1 (壓縮映像原理)

設(shè)X是完備的度量空間，T是X上的壓縮映像，那么T有且僅有1個不動點。

定理2 (Krasnoselskii不動點定理)

設(shè)M是Banach空間X中的一個非空凸閉子集。假設(shè)A,B是2個算子，滿足：

a) 對任意的x,y∈M，有Ax+By∈M；

b)A是全連續(xù)映射；

c)B是一個壓縮映射，

則至少存在一個z∈M，使得z=Az+Bz。

引理4 對于給定的函數(shù)y∈C(Jk)，k=1,2,…,u(t)是分數(shù)階脈沖微分方程邊值問題:

的解當且僅當u(t)滿足

證明 設(shè)u(t)是分數(shù)階脈沖微分方程邊值問題(2)的解，由引理2可得當t∈[0,t1]時，

同理由u(t)的連續(xù)性可知b=0，所以t∈Jk=(tk-1,tk]時,

因此，對?t∈J有

(4)

3 主要結(jié)果

H2)存在常數(shù)γk∈J，使得對?t∈J,u,v∈R，有

|Ik(u)-Ik(v)|≤γk‖u-v‖，

H4)存在函數(shù)F∈C[R,J]，常數(shù)ηk∈J，使得對?t∈J,u∈R，有

|Ik(u)|≤ηkF(u)，

證明 定義算子T:PC1[J,R]→PC1[J,R]如下：

所以

對?u,v∈PC1[J,R]，?t∈J有

所以‖Tu-Tv‖S≤ρ‖u-v‖。

證明 定義算子如下：

所以

由條件H3)—條件H4)可知：對?r>0，

|Ik(u)|≤ηkF(u)≤Nηk,k=1,2,…,

對?u,v∈Br，?t∈J,

所以‖Au+Bv‖S≤r。

下證Bu為壓縮算子。對?u,v∈Br，?t∈J有

對?t∈J，

下證算子A的緊性。取un∈Br={u∈PC1[J,R]:‖u‖≤r}。對?t∈J，

定義函數(shù)

對?t1,t2∈Jk，當t2>t1時，有

由積分的絕對連續(xù)性可知：存在δ3，使得|t2-t1|<δ3時，

由一元連續(xù)函數(shù)的一致連續(xù)性可知：存在δ4，使得|t2-t1|<δ4時，

令

對?ξ

對?ξi0時，

4 舉 例

例1 考慮半無窮區(qū)間上分數(shù)階脈沖微分方程邊值問題

例2 考慮半無窮區(qū)間上分數(shù)階脈沖微分方程邊值問題

因此，根據(jù)定理4可得該分數(shù)階脈沖微分方程邊值問題至少有1個解。

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Existence of solutions to boundary value problem of fractional differential equations with impulsive

JIANG Weihua, LI Qingmin, ZHOU Cailian

(School of Science, Hebei University of Science and Technology, Shijiazhuang, Hebei 050018, China)

In order to solve the boundary value problem of fractional impulsive differential equations with countable impulses and integral boundary conditions on the half line, the existence of solutions to the boundary problem is specifically studied. By defining suitable Banach spaces, norms and operators, using the properties of fractional calculus and applying the contraction mapping principle and Krasnoselskii's fixed point theorem, the existence of solutions for the boundary value problem of fractional impulsive differential equations with countable impulses and integral boundary conditions on the half line is proved, and examples are given to illustrate the existence of solutions to this kind of equation boundary value problems.

analytic theory of ordinary differential equation; impulse; contraction mapping theorem; Krasnoselskii’s fixed point theorem; boundary value problem; the half line

1008-1542(2016)06-0562-13

10.7535/hbkd.2016yx06007

2016-03-24；

2016-09-10；責任編輯：張 軍

河北省自然科學基金(A2013208108)

江衛(wèi)華(1964—)，女，河北邯鄲人，教授，博士，主要從事應(yīng)用泛函分析、常微分方程邊值問題方面的研究。

E-mail:jianghua64@163.com

O175.8 MSC(2010)主題分類：34B18

A

江衛(wèi)華，李慶敏，周彩蓮.分數(shù)階脈沖微分方程邊值問題解的存在性[J].河北科技大學學報,2016,37(6):562-574. JIANG Weihua, LI Qingmin, ZHOU Cailian.Existence of solutions to boundary value problem of fractional differential equations with impulsive[J].Journal of Hebei University of Science and Technology,2016,37(6):562-574.