帶積分邊值條件下分數(shù)階脈沖微分方程解的存在性*

蔣 偉, 周宗福

(安徽大學 數(shù)學科學學院，合肥 230601)

帶積分邊值條件下分數(shù)階脈沖微分方程解的存在性*

蔣 偉, 周宗福**

(安徽大學 數(shù)學科學學院，合肥 230601)

針對分數(shù)階脈沖微分方程解的存在性研究, 提出一類帶積分邊值條件的分數(shù)階脈沖微分方程邊值問題; 通過上下解方法, 利用Schauder 不動點定理得到此邊值問題解的存在性結(jié)果; 最后給出了一個例子來說明所得結(jié)果的應(yīng)用性.

積分邊值條件;分數(shù)階脈沖微分方程;Schauder不動點定理

0 引 言

近年來, 分數(shù)階微分方程應(yīng)用面十分廣泛, 除了在數(shù)學各方面的應(yīng)用外, 還廣泛應(yīng)用于生物系統(tǒng)的電傳導(dǎo)、流體力學、粘彈性力學、電化學分析、分數(shù)控制系統(tǒng)與分數(shù)控制器等.分數(shù)階微分方程理論吸引了很多學者研究[1-15].本文研究分數(shù)階脈沖微分方程邊值問題解的存在性. 近幾年, 關(guān)于這類問題的研究可以參考文獻[15-18].

在本文中，研究一類積分邊值條件下的分數(shù)階脈沖微分方程:

(1)

積分邊值條件:

(2)

1

0=t0

J′=J{t1,t2,…tm},J0=[0,t1]

Jk=(tk,tk+1],k=1,2,…,m

w(t)∈C(J,J+),t-r≤w(t)≤t(t∈J),r>0

文獻[19]研究了無窮區(qū)間上的分數(shù)階微分方程積分邊值問題:

文獻[20]研究了如下的分數(shù)階脈沖微分方程邊值問題:

非線性邊值條件:

g0(u(0),u(T))=0,g1(u′(0),u′(T))=0

在以上文獻的基礎(chǔ)上,利用Schauder不動點定理討論邊值問題(1)(2)解的存在性.

1 預(yù)備知識

定義1 函數(shù)f的q>0階Riemann-Liouville積分定義為

定義2 函數(shù)f的q>0階Caputo積分定義為

n-1

② 常值函數(shù)的Caputo分數(shù)階導(dǎo)數(shù)為0.

根據(jù)引理1,可以得到如下的引理.

引理3 令f(t)=L[J,R], 邊值問題

(3)

的解:

t∈Jk,k=0,1,2,…,m

其中，

證明 令u是式(3)的解, 由引理2,有

其中C1,C2∈R, 因此可得

如果t∈J1, 則

其中d1,d2∈R.因此有

因此

依次類推可得

t∈Jk,k=0,1,2,…,m

于是得到

證畢.

2 主要結(jié)果

定義算子T:PC(J+)→PC(J+),

其中，

顯然,T的不動點就是邊值問題(1)(2)的解.

引理4 算子T是全連續(xù)的.

因此有

因此

也就意味著

所以TD一致有界.

另一方面, 對任意的u∈D,t∈Jk,0≤k≤m,有

因此, 對任意t1,t2∈Jk且t1

所以TD是全連續(xù)的，因此TD在PC(J+)上是相對緊的，所以T是全連續(xù)的.證畢.

下控制函數(shù):

易知Ga(u),gb(u),Ya(u),yb(u),Ha(t,u,w),hb(t,u,w)關(guān)于u,w是單調(diào)不減的, 且

gb(u)≤Ik(u)≤Ga(u)

hb(t,u,w)≤f(t,u,w)≤Ha(t,u,w)

(t,u,w)∈[0,1]×[a,b]×[a,b]

其中，

Ha(s,u(s),u(w(s)))dsdμ(τ)+

Ha(s,u(s),u(w(s)))dsdμ(τ)+

Ha(s,u(s),u(w(s)))dsdμ(τ)+

hb(s,u(s),u(w(s)))dsdμ(τ)+

hb(s,u(s),u(w(s)))dsdμ(τ)+

下證T(S)?S:?v(t)∈S, 有

3 實例分析

下面給出一個例子說明主要結(jié)果的應(yīng)用.

例1 考慮下面的邊值問題:

邊值條件為

由方程(5)可知其中

f(t,u,v)=2t2+cost+sinu+arctanv

(6)

Ha(t,u,v)=4+t+u+v

hb(t,u,v)=t2+sinu+arctanv

4 結(jié) 論

近年來,分數(shù)階微分方程成為研究的熱點,由于其初值條件的復(fù)雜性,以及它們某些物理意義還沒有得到普遍認可,所以還有大量的工作要做. 本文是研究帶積分邊值條件的分數(shù)階脈沖微分方程邊值問題解的存在性. 首先介紹一些定義、引理,然后定義了一個全連續(xù)算子,于是由Schauder不動點定理可知,此算子存在一個不動點,且這個不動點是邊值問題的一個解.

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責任編輯：李翠薇

The Existence of Solution for Impulsive Fractional Differential Equations with Integral Boundary Value Condition

JIANG Wei, ZHOU Zong-fu

(School of Mathematical Science, Anhui University, Hefei 230601, China)

For the existence of solution to impulsive fractional differential equations, this paper proposes boundary value problems of impulsive fractional differential equations with integral boundary value condition, and the existence of solution for boundary value problems is obtained by upper and lower solutions together with Schauder fixed point theorem. Finally, an example is given to illustrate the application of the obtained results.

integral boundary value condition; impulsive fractional differential equation; Schauder fixed point theorem

2016-11-23；

2017-01-17.

國家自然科學基金(11371027)；安徽省自然科學基金(1608085MA12).

蔣偉(1992-)，女，安徽滁州人，從事泛函微分方程的研究.

**通訊作者：周宗福(1964-)，男，安徽合肥人，教授，從事泛函微分方程的研究.E-mail:zhouzf12@126.com.

10.16055/j.issn.1672-058X.2017.0004.005

O175

A

1672-058X(2017)04-0024-08