一類混合分?jǐn)?shù)階q-差分邊值問題解的存在性

楊瀟， 白俊杰， 葛琦

( 延邊大學(xué)理學(xué)院 數(shù)學(xué)系， 吉林 延吉 133002 )

一類混合分?jǐn)?shù)階q-差分邊值問題解的存在性

楊瀟， 白俊杰， 葛琦*

( 延邊大學(xué)理學(xué)院 數(shù)學(xué)系， 吉林 延吉 133002 )

研究了一類帶有分?jǐn)?shù)階q-差分邊值條件的混合分?jǐn)?shù)階q-差分方程解的存在性.首先分析了格林函數(shù)的性質(zhì),然后借助Lipschitz條件，在Banach代數(shù)中利用不動點(diǎn)定理研究了該方程解的存在性，最后通過實(shí)例驗(yàn)證了所得結(jié)論的合理性.

混合分?jǐn)?shù)階q-差分； 不動點(diǎn)定理； 解的存在性

(1)

其中2<α≤3, 0

1 預(yù)備知識

定義2[6]函數(shù)f(x)在區(qū)間[0,b]上的q-積分定義為:

定義3[6]Riemann-Liouville型分?jǐn)?shù)階q-積分定義為:

Riemann-Liouville型分?jǐn)?shù)階q-導(dǎo)數(shù)定義為:

引理1[6]設(shè)α>0, p是正整數(shù),則

引理2[6]設(shè)α,β≥0, f(x)是定義在[0,1]上的函數(shù),則

引理4[6]若α>0, a≤b≤t, 則(t-a)(α)≥(t-b)(α).

引理5[8]設(shè)S是Banach代數(shù)X上的一個非空的、有界閉凸子集,設(shè)A∶X→X和B∶S→X是兩個算子，且滿足：

(i) A滿足Lipschitz條件,其中L為Lipschitz常數(shù);

(ii) B是完全連續(xù)的;

(iii) 對于?y∈S, 有x=A xBy x∈S；

(iv) L M<1, 其中M=‖B(S)‖=sup{‖B(x)‖:x∈S}.

則算子方程A xB x=x在S中有一個解.

2 Green函數(shù)的性質(zhì)

定理1 設(shè)h∶[0,1]→[0,+∞)是連續(xù)的，且2<α<3, 1<ν<2, 邊值問題

(2)

于是有

引理6[6]Green函數(shù)G(t,q s)具有以下性質(zhì)：

(i) G(t,q s)≥0, t,s∈[0,1];

3 解的存在性

證明 設(shè)X=C([0,1],R), 定義X的子集W={x∈X:‖x‖≤N}, 其中

顯然W是X上的一個有界閉凸子集.由定理1知,邊值問題(1)與如下方程等價:

(3)

定義兩個算子A∶X→X和B∶W→X:

A x(t)=f(t,x(t)), t∈[0,1],

首先A是X上的Lipschitz算子，其中L是Lipschitz常數(shù).事實(shí)上，設(shè)?x,y∈X， 由條件(H1)有

因此,對于?x,y∈X, 有‖A x-Ay‖≤L‖x-y‖.

即B是W上的連續(xù)算子.其次證明B是W上的緊算子,即B(W)是一致有界的,且在X上是等度連續(xù)的.事實(shí)上,設(shè)x∈W, 由引理6和條件(H2)知，對于?t∈[0,1]有

下面證明引理5中的條件(iii)成立.事實(shí)上,設(shè)x∈X滿足x=A xBy (?y∈W), 由條件(H1)有

綜合以上結(jié)果可知，引理5中的條件均被滿足,于是算子方程A xBx=x在W中有一個解,即邊值問題(1)在W中存在一個解.

注1 當(dāng)f(t,x)=1時,由文獻(xiàn)[6]可知邊值問題(1)的解存在.

例1 考慮混合分?jǐn)?shù)階q-差分邊值問題

(4)

[1] Ferreira R A C. Nontrivial solutions for fractionalq-difference boundary value problems[J]. Theory of Differential Equations, 2010,70:1-10.

[2] Ferreira R A C. Positive solutions for a class of boundary value problems with fractionalq-differences[J]. Computers and Mathematics with Applications, 2011,61(2):367-373.

[3] Ricardo Almeida, Natália Martins. Existence results for fractionalq-difference equations of orderα∈]2,3[ with three-point boundary conditions[J]. Communications in Nonlinear Science and Numerical Simulation, 2014,19(6):1675-1685.

[4] Zhao Yulin, Chen Haibo, Zhang Qiming. Existence results for fractionalq-difference equations with nonlocalq-integral boundary conditions[J]. Advances in Difference Equations, 2013,2013:1-15.

[5] Zhao Yulin, Ye Guobing, Chen Haibo. Multiple positive solutions of a singular semipositione integral boundary value problem for fractionalq-derivatives equation[J]. Abstract and Applied Analysis, 2013,2013:1-12.

[6] 孫明哲，韓筱爽.一類分?jǐn)?shù)階q-差分邊值問題的正解[J].延邊大學(xué)學(xué)報:自然科學(xué)版，2013,39(4):252-255.

[7] Bashir Ahmad, Sotiris K Ntouyas. Fractionalq-difference hybrid equations and inclusions with Dirichlet boundary conditions[J]. Advances in Difference Equations, 2014,2014:1-14.

[8] Sun Shurong, Zhao Yige, Han Zhenlai. The existence of solutions for boundary value problems of fractional hybrid differential equations[J]. Communications in Nonlinear Science and Numerical Simulation, 2012,17(12):4961-4967.

Existence of solutions for a class of boundary value problems with hybrid fractional q-differences

YANG Xiao, BAI Junjie, GE Qi*

(DepartmentofMathematics，CollegeofScience,YanbianUniversity，Yanji133002，China)

We study the existence of solutions for a class of the fractional q-differences hybrid equation with the fractionalq-differences boundary conditions. Firstly, some characteristics of the Green function were analyzed. The second, we obtained sufficient conditions for the existence of positive solutions to this equation under Lipschitz condition using fixed point theorems in Banach algebra. The end, the main results were illustrated with the aid of examples.

hybrid fractionalq-differences; fixed point theorem; existence of solutions

2014-11-16 *通信作者： 葛琦(1975—)，女，副教授，研究方向?yàn)槌Ｎ⒎址匠汤碚摷捌鋺?yīng)用.

國家自然科學(xué)基金資助項(xiàng)目(11161049);吉林省教育廳“十二五”科學(xué)技術(shù)研究項(xiàng)目(2015—2016)

1004-4353(2015)01-0021-04

O175.6

A