楊　帥， 蔡寧寧

(中國(guó)礦業(yè)大學(xué)(北京) 理學(xué)院， 北京 100083)

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一類Caputo分?jǐn)?shù)階微分方程初值問(wèn)題解的存在性

楊帥， 蔡寧寧

(中國(guó)礦業(yè)大學(xué)(北京) 理學(xué)院， 北京 100083)

摘要：將一類Caputo分?jǐn)?shù)階微分方程初值問(wèn)題轉(zhuǎn)化為等價(jià)的Volterra積分方程，通過(guò)構(gòu)造一個(gè)特殊的Banach空間，在此Banach空間上定義算子，將求解Volterra積分方程轉(zhuǎn)化為求算子的不動(dòng)點(diǎn)問(wèn)題，應(yīng)用Schauder 不動(dòng)點(diǎn)定理證明了其解的存在性.

關(guān)鍵詞：Caputo分?jǐn)?shù)階微分方程； 初值問(wèn)題； Volterra積分方程

近年來(lái)，隨著相關(guān)理論的不斷拓展和完善，分?jǐn)?shù)階微分方程已廣泛應(yīng)用于分?jǐn)?shù)物理學(xué)、粘彈性力學(xué)、自動(dòng)控制、混沌與湍流、生物化學(xué)、非牛頓流體力學(xué)、隨機(jī)過(guò)程等諸多科學(xué)領(lǐng)域[1]. 關(guān)于分?jǐn)?shù)階微分方程解的存在性及其求解也取得了豐碩的成果[2-5]. 分?jǐn)?shù)階微分方程初值問(wèn)題是非線性微分方程的一個(gè)重要研究課題，許多學(xué)者都獨(dú)立地探討了各類分?jǐn)?shù)階微分方程初值問(wèn)題[6-9].

本文主要討論如下一類Caputo分?jǐn)?shù)階微分方程邊值問(wèn)題

(1)

1預(yù)備知識(shí)

首先，介紹幾個(gè)基本概念和一些Caputo分?jǐn)?shù)階導(dǎo)數(shù)的性質(zhì)以及相關(guān)引理.

性質(zhì)1常數(shù)的Caputo分?jǐn)?shù)階導(dǎo)數(shù)為0，即

2主要結(jié)果

定理1設(shè)f(x,y(x))在[0,1]×R上連續(xù)，則Caputo分?jǐn)?shù)階微分方程邊值問(wèn)題(1)等價(jià)于下面的第二類非線性Volterra積分方程

(2)

即

等式兩邊積分可得

y(x)=y(0)+

另一方面，設(shè)y(x)是Volterra積分方程(2)的解，即

且在(2)中令x=0可得y(0)=y0，則y(x)是Caputo分?jǐn)?shù)階微分方程邊值問(wèn)題(1)的解.

綜上兩個(gè)方面，得到(1)與(2)等價(jià).

證明由定理1知道，Caputo分?jǐn)?shù)階微分方程邊值問(wèn)題(1)與Volterra積分方程(2)等價(jià)，定義算子A:

Ay(x)=y0+

則方程的解轉(zhuǎn)化為算子A的不動(dòng)點(diǎn)問(wèn)題.

接下來(lái)， 分以下幾步來(lái)證明：

第1步，任取y∈U，可以得到

即Ay(x)∈U，于是算子A:U→U.

?x∈[0,1]

那么

則A:U→U連續(xù).

即A(U)中諸函數(shù)一致有界.

第4步，討論A(U)中諸函數(shù)的等度連續(xù)性.

由于 0<α<1，則

可知A(U)中諸函數(shù)等度連續(xù).

由Ascoli-Arzela定理知A(U)是B相對(duì)緊集. 因此A:U→U全連續(xù). 根據(jù)Schauder 不動(dòng)點(diǎn)定理知A在U中必有不動(dòng)點(diǎn).

綜上，證明了Caputo分?jǐn)?shù)階微分初值問(wèn)題(1)解的存在性，即(1)必有連續(xù)解y∈C[0,1].

參考文獻(xiàn)：

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(編輯：郝秀清)

Existence of solutions of initial value problem for a Caputo fractional differential equation

YANG Shuai， CAI Ning-ning

(College of Science， China University of Mining and Technology， Beijing 100083, China)

Abstract:The initial value problem of a class of Caputo fractional differential equations is transformed into an equivalent Volterra integral equation. By defining a operator on a special Banach space, the solvability of the Volterra integral equation is transformed to a fixed point problem. The existence of its solution is proved by employing Schauder′s fixed point theorem.

Key words:Caputo fractional differential equation; initial value problem; Volterra integral equation

中圖分類號(hào)：0175.14

文獻(xiàn)標(biāo)志碼：A

文章編號(hào)：1672-6197(2016)03-0029-04

作者簡(jiǎn)介：楊帥，男， haotianwuji2@sina.com

收稿日期：2015-07-07