具有微分算子的分?jǐn)?shù)階微分方程邊值問題解的存在性與唯一性

吳貴云, 劉錫平, 楊 浩

(上海理工大學(xué)理學(xué)院,上海 200093)

具有微分算子的分?jǐn)?shù)階微分方程邊值問題解的存在性與唯一性

吳貴云, 劉錫平, 楊 浩

(上海理工大學(xué)理學(xué)院,上海 200093)

研究一類具有分?jǐn)?shù)階線性微分算子的Riemann-Liouville型分?jǐn)?shù)階非線性微分方程兩點邊值問題解的存在性和唯一性.通過求出相應(yīng)邊值問題的Green函數(shù)并證明其性質(zhì),建立積分算子方程,應(yīng)用壓縮映射原理證明了這類邊值問題解的存在性與唯一性定理.運用Krasnoselskii’s不動點理論建立并證明了該邊值問題解的存在性與唯一性定理.最后給出了兩個應(yīng)用實例,用以說明本文所得結(jié)論的有效性.

Riemann-Liouville分?jǐn)?shù)階導(dǎo)數(shù);分?jǐn)?shù)階微分方程;微分算子;邊值問題;存在性與唯一性

由于分?jǐn)?shù)階微分方程在工程技術(shù)和科學(xué)研究中具有廣泛的應(yīng)用背景,因此,分?jǐn)?shù)階微分方程理論研究受到了廣泛關(guān)注.近年來,有大量關(guān)于分?jǐn)?shù)階微分方程邊值及初值問題的研究成果出現(xiàn),參見文獻(xiàn)[1-10]以及其中的參考文獻(xiàn).

對于一些較復(fù)雜的問題,常常用微分算子方程來描述.文獻(xiàn)[1-2]討論了具有線性微分算子的分?jǐn)?shù)階微分方程初值問題正解的存在性.本文研究一類具有線性微分算子的分?jǐn)?shù)階微分方程兩點邊值問題解的存在性和唯一性.

1 準(zhǔn)備工作

本文研究具有線性微分算子的分?jǐn)?shù)階微分方程兩點邊值問題

解的存在性和唯一性,其中線性微分算子L(D)= Dα-rtnDβ,n是非負(fù)整數(shù),r∈瓗,0＜β＜1＜α＜2,且Dα,Dβ是標(biāo)準(zhǔn)的Riemann-Liouville分?jǐn)?shù)階導(dǎo)數(shù).f:[0,1]×瓗→瓗是非線性連續(xù)函數(shù).

有關(guān)Riemann-Liouville型分?jǐn)?shù)階積分與分?jǐn)?shù)階導(dǎo)數(shù)的定義請參考文獻(xiàn)[3].

引理1[4]若x在[0,+∞)上是連續(xù)的,且0＜β＜1,β≤α,那么

引理2 設(shè)r∈瓗,n是非負(fù)整數(shù),0＜β＜1＜α＜2,則邊值問題(1)與積分方程

證明 運用分?jǐn)?shù)階微積分的相關(guān)引理可以得出方程L(D)x(t)+f(t,x(t))=0的等價方程為

式中,c1,c2為任意常數(shù).

根據(jù)邊界條件可求得

引理3 設(shè)r∈瓗,0＜β＜1＜α＜2,n為非負(fù)整數(shù),則函數(shù)G(t,s),Hk(t,s)具有以下性質(zhì):

a.G(t,s)連續(xù),并且0≤G(t,s)≤G(s,s),

證明 a.由函數(shù)的定義式(2),易得G(t,s)連續(xù),并且G(t,s)≥0.

當(dāng)s＜t時,有

2 主要結(jié)論

下面應(yīng)用Krasnoselskii’s不動點定理研究邊值問題(1)解的存在性.

根據(jù)Krasnoselskii’s不動點定理[11],可得邊值問題(1)至少有一個解.

3 應(yīng)用舉例

以下給出兩個實例,用于說明所得到的主要結(jié)論.

例1 考慮具有線性微分算子的分?jǐn)?shù)階微分方程邊值問題

綜上所述,邊值問題(4)滿足了定理1的條件,根據(jù)定理1,邊值問題(4)存在唯一解.容易看出x(t)≡0不是邊值問題(4)的解,因此,邊值問題(4)存在唯一非平凡解.

例2 考慮分?jǐn)?shù)階微分方程

容易驗證,邊值問題(5)滿足定理2的條件,根據(jù)定理2,邊值問題(5)至少存在一個解,并且該解為非平凡解.

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(編輯:丁紅藝)

Existence and Uniqueness of Solutions for Boundary Value Problems of Fractional Differential Equations with Differential Operator

WUGuiyun, LIU Xiping, YANGHao
(College of Science,University of Shanghai for Science and Technology,Shanghai 200093,China)

The existence and uniqueness of solutions for a class of two points boundary value problems of Riemann-Liouville nonlinear fractional differential equation with linear fractional differential operator were considered.Through finding the Green function of the boundary value problem and determining its properties,an integral operator equation was established and some sufficient conditions for the existence and uniqueness of solutions were derived by applying the contraction principle and the fixed point theorem. Some examples were given to illustrate the results.

Riemann-Liouville fractional derivative;fractional differential equation; differential operator;boundary value problem;existence and uniqueness

O 175.8

A

1007-6735(2015)03-0205-05

10.13255/j.cnki.jusst.2015.03.001

2014-03-03

國家自然科學(xué)基金資助項目(11171220)

吳貴云(1990-),女,碩士研究生.研究方向:應(yīng)用微分方程.E-mail:763924848@qq.com

劉錫平(1962-),男,教授.研究方向:應(yīng)用微分方程.E-mail:xipingliu@usst.edu.cn