王翠菁

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分?jǐn)?shù)階微分方程非零邊值問(wèn)題解的存在性

王翠菁1, 2

(1. 中國(guó)礦業(yè)大學(xué) 理學(xué)院, 江蘇 徐州, 221008; 2. 徐州工業(yè)職業(yè)技術(shù)學(xué)院 信息管理技術(shù)學(xué)院, 江蘇 徐州, 221140)

利用壓縮映射原理和Krasnoselskii’s不動(dòng)點(diǎn)定理, 在Banach空間下討論非線性分?jǐn)?shù)階微分方程非零邊值問(wèn)題()=(,()), 0<<1;(0)=(0),(1)=β()解的存在性, 其中1<α≤2是一個(gè)實(shí)數(shù),是Caputo型微分.

壓縮映射原理; Caputo型微分; Krasnoselskii’s不動(dòng)點(diǎn)定理

分?jǐn)?shù)計(jì)算[1-3]始于17世紀(jì), 由Leibniz、Euler、Lagrange、Able以及其他人的先驅(qū)工作, 才逐步發(fā)展起來(lái). 對(duì)于分?jǐn)?shù)階微分方程, 它在不同的科學(xué)領(lǐng)域也起著越來(lái)越重要的作用[4-6]. 有許多學(xué)者在微分方程系統(tǒng)方面已獲得不少的研究成果[7-10].

例如, 張淑琴[8]利用錐不動(dòng)點(diǎn)理論、Leggett-Williams定理討論了Caputo型分?jǐn)?shù)階微分方程邊值問(wèn)題:

正解及多個(gè)正解的存在性.

本文在文獻(xiàn)[9-10]的啟發(fā)下, 在Banach空間下討論非線性分?jǐn)?shù)階微分方程非零邊值問(wèn)題()=(,()), 0<<1;(0)=￠(0),(1)=β()解的存在性和唯一性, 其中1<≤2是一個(gè)實(shí)數(shù),是Caputo型微分, 并且假設(shè)是連續(xù)的,是Banach空間. 令, 記為在全體連續(xù)函數(shù)構(gòu)成的Banach空間賦予其范數(shù)(拓?fù)湟恢率諗康?.

1 預(yù)備知識(shí)

.

由引理1、引理2可得引理3.

2 主要結(jié)論

為了后面的分析, 我們做如下假設(shè):

例如，在班級(jí)當(dāng)中，我們要時(shí)刻嚴(yán)格要求幼兒遵守班級(jí)中的規(guī)章制度，使幼兒懂得制度的不可破壞性和必要性。如對(duì)于班級(jí)的值日制度，我們要對(duì)幼兒進(jìn)行明確的分工，將每一件事情的責(zé)任分配到位，讓幼兒在進(jìn)行簡(jiǎn)單的值日過(guò)程中，明白自己該做什么事，要做什么事等，這種制度化的管理使教師和幼兒都能夠輕松完成自己在班級(jí)中的責(zé)任，從而使幼兒園的班級(jí)管理更有秩序，更加簡(jiǎn)單，使幼兒園的管理質(zhì)量和水平得到有效的提升和增強(qiáng)。

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[2] Miller K S, Ross B. An introduction to the Fractional calculus and fractional differential equations[M]. New York: Wiley, 1993.

[3] Podlubny I. Fractional differential equations mathematics in science and engineering[M]. New York: Academic Press, 1999.

[4] Oldham K B, Spanier J. The fractional calculus[M]. New York: Academic Press, 1974.

[5] Tatom F B. The relationship between fractional calculus and fractals[J]. Fractals, 1995, 3: 217-229.

[6] Nonnenmacher T F, Metzle R. On the Riemann-Liouville fractional calculus and some recent applications[J]. Fractals, 1995, 3: 557-566.

[7] Bai Z, Lu H. Positive solutions for boundary value problem of nonlinear fractional differential equation[J]. J Math Anal Appl, 2005, 311: 495–505.

[8] Shuqin Zhang. Positive solutions for boundary-value problems of nonlinear fractional differential equations[J]. Electronic Journal of Differential Equations, 2006, 68: 1-12.

[9] Bashir Ahmad. Existence of solutions for fractional differential equations of order q∈(2,3) with anti-periodic boundary conditions[J]. J Appl Math Comput, 2010, 34: 385-391.

[10] Bashir Ahmad, Juan J, Nieto. Existence of solutions for nonlocal boundary value problems of higher-order nonlinear fractional differential equations[J]. Abstract and Applied Analysis, 2009, Article ID 494720: 1-9.

The existence of solutions for nonzero boundary value problem of fractional differential equations

WANG Cui-jing1,2

(1 College of Sciences, China University of Mining & Technology, Xuzhou 221008, China; 2 Information Management Technology Institute, Xuzhou College of Industrial Technology, Xuzhou 221140, China)

By using the contraction mapping principle and Krasnoselskii’s fixed point theorem,we obtain the existence results in a Banach space for nonzero boundary value problem of fractional differential equation()=(,()), 0<<1;(0)=(0),(1)=β(),Where 1<α≤2 is a real number, andis a Caputo′s differentiation.

contraction mapping principle; Caputo′s differentiation; Krasnoselskii’s fixed point theorem

O 175.8

1672-6146(2012)01-0011-03

10.3969/j.issn.1672-6146.2012.01.004

2011-12-15

王翠菁(1982-), 女, 講師, 研究方向?yàn)槲⒎址匠踢呏祮?wèn)題. E-mail: 76124768@qq.com

(責(zé)任編校: 劉曉霞)