一類(lèi)反周期分?jǐn)?shù)階q-差分邊值問(wèn)題解的存在性

孫明哲, 侯成敏

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

(f)(x)=(x-qt)(α-1)f(t)dqt, α>0, x∈[0,1],

研究簡(jiǎn)報(bào)

一類(lèi)反周期分?jǐn)?shù)階q-差分邊值問(wèn)題解的存在性

孫明哲, 侯成敏

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

利用基本的不動(dòng)點(diǎn)定理研究一類(lèi)帶有反周期非線(xiàn)性分?jǐn)?shù)階q-差分方程邊值問(wèn)題, 得到了邊值問(wèn)題解的存在與唯一的充分條件, 并通過(guò)具體方程驗(yàn)證了所得結(jié)論.

分?jǐn)?shù)階q-差分; 邊值問(wèn)題; 不動(dòng)點(diǎn)定理

q-微積分在量子力學(xué)、核及高能物理等研究領(lǐng)域應(yīng)用廣泛.文獻(xiàn)[1-3]給出了分?jǐn)?shù)階q-微積分的基本概念和性質(zhì); 文獻(xiàn)[4-9]給出了分?jǐn)?shù)階q-差分邊值問(wèn)題的研究成果.但對(duì)于含有分?jǐn)?shù)階邊值條件的邊值問(wèn)題目前報(bào)道很少, 本文討論帶有分?jǐn)?shù)階邊值條件的q-差分邊值問(wèn)題:

解的存在性和唯一性, 其中: 1<α<2, 0<ν<1為常數(shù);f(t,u(t)): [0,1]×[0,+∞)→為連續(xù)函數(shù)是Caputo分?jǐn)?shù)階q-導(dǎo)數(shù).

定義1[8]分?jǐn)?shù)階q-積分定義為

引理1[10]設(shè)X是一個(gè)Banach空間,T:X→X是一個(gè)完全連續(xù)算子, 集合V={u∈X|u=μTu, 0<μ<1}有界, 則算子T在X上有不動(dòng)點(diǎn).

定理1設(shè)1<α<2, 0<ν<1, 則u(t)是問(wèn)題(1)解的充要條件是u(t)有如下形式:

其中

證明: 假設(shè)u(t)是問(wèn)題(1)的解, 則有

利用邊值條件(2)解得

將C1,C2代入式(4)得式(3).反之, 如果u(t)滿(mǎn)足式(3), 則易推出u(t)是問(wèn)題(1)-(2)的解.證畢.

記Banach空間C=C([0,1],), 賦范數(shù)‖u‖|u(t)|.定義算子T:C→C, 且

由定理1知問(wèn)題(5)有解當(dāng)且僅當(dāng)算子T有不動(dòng)點(diǎn).

定理2假設(shè)對(duì)于t∈[0,1],u∈C, 存在常數(shù)M>0, 滿(mǎn)足|f(t,u)|≤M, 則問(wèn)題(1)-(2)至少存在一個(gè)解.

證明: 首先, 考慮算子T是完全連續(xù)的.由f的連續(xù)性易知T是連續(xù)的.令Ω?C有界, 則由u∈Ω, |f(t,u)|≤M, 得

表明‖Tu(t)‖≤M2, 此外

因此對(duì)于t1,t2∈[0,1],t1

故T在[0,1]上是等度連續(xù)的, 由Arzela-Ascoli定理知, 算子T:C→C是完全連續(xù)的.

其次, 考慮集合V={u∈C|u=μTu, 0<μ<1}有界.若u∈V,u=μTu, 0<μ<1, 對(duì)于任意的t∈[0,1], 有|u(t)|≤μ|Tu(t)|≤|Tu(t)|≤M2, 從而對(duì)于任意的t∈[0,1], 有‖u(t)‖≤M2, 即V有界.

由引理1知,T至少有一個(gè)不動(dòng)點(diǎn), 即問(wèn)題(1)-(2)至少存在一個(gè)解.證畢.

定義Ω={u∈C|‖u‖

由引理2知,T至少有一個(gè)不動(dòng)點(diǎn), 即問(wèn)題(1)-(2)至少存在一個(gè)解.

定理4假設(shè)f: [0,1]×→為連續(xù)函數(shù), 滿(mǎn)足|f(t,u)-f(t,v)|≤L|u-v|, ?t∈[0,1],u,v∈且L<, 則問(wèn)題(1)-(2)存在唯一解.

因此‖(Tu)(t)‖≤r.

對(duì)于u,v∈C,t∈[0,1], 有

例1考慮邊值問(wèn)題

易知

滿(mǎn)足定理2的假設(shè)條件, 因此問(wèn)題(6)至少有一個(gè)解.

例2考慮邊值問(wèn)題

例3考慮邊值問(wèn)題

滿(mǎn)足定理4的假設(shè)條件, 故問(wèn)題(8)有唯一解.

[1]Jackson F H.q-Difference Equations [J].Amer J Math, 1910, 32(4)： 305-314.

[2]Al-Salam W A.Some Fractionalq-Integrals andq-Derivatives [J].Proc Edinb Math Soc, 1966/1967, 15(2): 135-140.

[3]Agarwal R P.Certain Fractionalq-Integrals andq-Derivatives [J].Proc Cambridge Philos Soc, 1969, 66: 365-370.

[4]YANG Wengui.Positive Solutions for Boundary Value Problems Involving Nonlinear Frationalq-Difference Equations [J].Differ Equ Appl, 2013, 5(2): 205-219.

[5]ZHAO Yulin, CHEN Haibo, ZHANG Qiming.Existence and Multiplicity of Positive Solutions for Nonhomogeneous Boundary Value Problems with Fractionalq-Derivative [J].Bound Value Probl, 2013, 2013: 103.

[6]ZHAO Yulin, YE Guobing, CHEN Haibo.Multiple Positive Solutions of a Singular Semipositone Integral Boundary Value Problem for Fractionalq-Derivatives Equation [J/OL].Abstr Appl Anal, 2013.http: dx.doi.org/10.1155/2013/643571.

[7]Ferreira R A C.Nontrivial Solutions for Fractionalq-Difference Boundary Value Problems [J].Electron J Qual Theory Differ Equ, 2010, 70(10): 1-10.

[8]Ahmad B, Nieto J J.Anti-periodic Fractional Boundary Value Problems with Nonlinear Term Depending on Low Order Derivative [J].Fractional Calculus and Applied Analysis, 2012, 15(3): 451-462.

[9]WANG Fang, LIU Zhenhai.Anti-periodic Fractional Boundary Value Problems for Nonlinear Differential Equations of Fractional Order [J].Adv Differ Equ, 2012, 2012: 116.

[10]Smart D R.Fixed Point Theorems [M].Cambridge: Cambridge University Press, 1980.

ExistenceofSolutionsforaClassofAnti-periodicBoundaryValueProblemswithFractionalq-DifferenceEquations

SUN Mingzhe, HOU Chengmin
(DepartmentofMathematics,CollegeofScience,YanbianUniversity,Yanji133002,JilinProvince,China)

We studied a class of the fractionalq-differences boundary value problem with the fractionalq-differences boundary conditions with the aid of some standard fixed point theorems, obtaining sufficient conditions for the existence and uniqueness results of solutions.As the application, some equations were presented to illustrate the main results.

fractionalq-difference; boundary value problem; fixed point theorem

2014-01-23.

孫明哲(1979—), 女, 漢族， 碩士, 講師, 從事微分方程理論及應(yīng)用的研究, E-mail: mzsun@ybu.edu.cn.通信作者： 侯成敏(1964—), 女, 漢族， 碩士, 教授, 從事微分方程理論及應(yīng)用的研究, E-mail: cmhou@foxmail.com.

國(guó)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(批準(zhǔn)號(hào): 吉教科合字[2014]第20號(hào)).

O175.8

A

1671-5489(2014)06-1215-04

10.13413/j.cnki.jdxblxb.2014.06.21

趙立芹)