葛 琦,侯成敏

(延邊大學(xué) 理學(xué)院,吉林 延吉 133002)

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一類有序分?jǐn)?shù)階q-差分方程解的存在性

葛 琦,侯成敏

(延邊大學(xué) 理學(xué)院,吉林 延吉 133002)

考慮一類有序分?jǐn)?shù)階q-差分方程解的存在性和唯一性.先利用q-指數(shù)給出該方程解的表達(dá)式,再分別利用Banach壓縮映像原理、Krasnoselskii不動(dòng)點(diǎn)定理、Leray-Schauder選擇定理證明該方程解的存在性和唯一性.

有序分?jǐn)?shù)階q-差分；不動(dòng)點(diǎn)定理；解的存在性

0 引 言

其中:1<α<2;0<λ<1;β>0;f∈C([0,1]×,)表示Caputo型分?jǐn)?shù)階q-導(dǎo)數(shù)(0

1 預(yù)備知識(shí)

定義2[9]函數(shù)f(x)在區(qū)間[0,b]上的q-積分定義為

定義3[9]Riemann-Liouville型分?jǐn)?shù)階q-積分定義為

Caputo型分?jǐn)?shù)階q-導(dǎo)數(shù)定義為

引理1[9]設(shè)α≥0,I是包含原點(diǎn)的實(shí)區(qū)間,且a,b∈I,f(x),g(x)是定義在I到上的函數(shù),則:

2)[a(x-t)](α)=aα(x-t)(α),xDq(x-t)(α)=[α]q(x-t)(α-1)；

3)Dq[fg](x)=Dq[f](x)g(x)+f(qx)Dq[g](x)；

這里iDq表示與變量i有關(guān)的q-導(dǎo)數(shù).

定義4[10]定義標(biāo)準(zhǔn)q-指數(shù)函數(shù)如下：

其中:q>0;z是復(fù)數(shù);

[n]!=[1][2]…[n]; [k]=1+q+q2+…+qk-1;

引理2[9]如果f:[0,1]→是連續(xù)函數(shù),則Iq[f]是連續(xù)函數(shù).

引理3(Banach壓縮映像原理)[11]設(shè)X是實(shí)Banach空間E上的非空閉子集,T:X→X是壓縮算子,則T在X內(nèi)存在唯一的不動(dòng)點(diǎn).

引理4(Krasnoselskii不動(dòng)點(diǎn)定理)[11]設(shè)K是Banach空間E的有界凸閉子集,而T,S:K→E滿足:

1)對(duì)任意x,y∈K有Tx,Tx+Sy∈K;

2)T是壓縮映像;

3)S在K上是全連續(xù)的.

則T+S在K內(nèi)至少存在一個(gè)不動(dòng)點(diǎn).

2)存在一個(gè)x∈?U,對(duì)于λ∈(0,1)有x=λTx.

引理6(Arzela-Ascoli定理)[9]設(shè)D?n是一個(gè)有界域,如果K?)有界,且對(duì)于任意的ε>0,存在δ>0,使得‖x-y‖<δ?|u(x)-u(y)|<ε,??u∈K,則是緊的.

2 主要結(jié)果

引理7方程(1)-(2)與如下積分方程等價(jià)：

其中

(3)

由y(0)=Dq[y](0)=0,得c0=0.由于

所以

又由于

因此

于是

從而有

進(jìn)而由Dq[y](1)=β得

因此

為了證明方程(1)-(2)解的存在性和唯一性,對(duì)Banach空間C([0,1],)賦范數(shù)‖y‖|y(x)|,對(duì)于y∈C([0,1],),定義C([0,1],)上的算子F：

(4)

其中Ky定義見式(3).

定理1假設(shè)存在一個(gè)q-可積的函數(shù)L:[0,1]→,使得對(duì)于?x∈[0,1]及?y1,y2∈,有

|f(x,y1)-f(x,y2)|≤L(x)|y1-y2|,

設(shè)

如果Ω<1,則方程(1)-(2)有唯一解.

證明：先證明由式(4)定義的算子F是一個(gè)壓縮映射.事實(shí)上,對(duì)于?y1,y2∈C([0,1],),有

因此,當(dāng)Ω<1時(shí),算子F是一個(gè)壓縮映射.由引理3知方程(1)-(2)有唯一解.

特別地,當(dāng)定理1中的函數(shù)L是常數(shù)時(shí),即對(duì)?x∈[0,1],L(x)=L,有

又由于

所以,可取

定理2假設(shè):

1)存在一個(gè)q-可積的函數(shù)L:[0,1]→,使得對(duì)于?x∈[0,1]及?y1,y2∈,有|f(x,y1)-f(x,y2)|≤L(x)|y1-y2|；

2)存在一個(gè)連續(xù)函數(shù)G:[0,1]→,使得對(duì)于?x∈[0,1]及?y∈,有|f(x,y)|≤G(x)；

則方程(1)-(2)至少有一個(gè)解.

證明：為應(yīng)用引理4,定義函數(shù)

取正實(shí)數(shù)M1,滿足

其次,類似定理1的證明,易證F2是壓縮映射,即‖F(xiàn)2[y1]+F2[y2]‖≤ψ‖y1-y2‖.

定理3假設(shè):

1)存在連續(xù)函數(shù)G1,G2:[0,1]→和單調(diào)遞增的函數(shù)使得對(duì)于?x∈[0,1]及?y∈,有|f(x,y)|≤G1(x)φ(|y|)+G2(x);

2)存在一個(gè)正常數(shù)N滿足

(5)

其中

則方程(1)-(2)至少有一個(gè)解.

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(責(zé)任編輯：趙立芹)

ExistenceofSolutionsforaClassofSequentialFractionalq-DifferencesEquation

GE Qi,HOU Chengmin

(CollegeofScience,YanbianUniversity,Yanji133002,JilinProvince,China)

We studied the existence and uniqueness of solutions for a class of the sequential fractionalq-differences equation.Firstly,a representation for the solution to this equation was given viaq-exponential.Then the existence and uniqueness of solutions were proven by means of Banach fixed point theorem,Krasnoselskii fixed point theorem and Leray-Schauder alternative theorem.

sequential fractionalq-difference;fixed point theorem;existence of solutions

10.13413/j.cnki.jdxblxb.2015.03.06

2014-07-03.

葛 琦(1975—),女,漢族,碩士,副教授,從事微分方程理論及應(yīng)用的研究,E-mail：geqi9688@163.com.

國家自然科學(xué)基金(批準(zhǔn)號(hào)：11161049)和吉林省教育廳“十二五”科學(xué)技術(shù)研究項(xiàng)目.

O175.6

：A

：1671-5489(2015)03-0377-06