应用泛函分析学报
應用汎函分析學報
응용범함분석학보
ACTA ANALYSIS FUNCTIONALIS APPLICATA
2011年
3期
274-284
,共11页
初值问题%周期边值问题%分数阶微分方程%序列Riemann-Liouville分数阶导数%Caputo 分数阶导数%上下解
初值問題%週期邊值問題%分數階微分方程%序列Riemann-Liouville分數階導數%Caputo 分數階導數%上下解
초치문제%주기변치문제%분수계미분방정%서렬Riemann-Liouville분수계도수%Caputo 분수계도수%상하해
initial value problem%periodic boundary value problem%fractional differential equation%sequential Riemann-liouville fractional derivatives%Caputo fractional derivative%upper solution and lower solution
第一部分,介绍分数阶导数的定义和著名的Mittag-Leffler函数的性质.第二部分,利用单调迭代方法给出了具有2序列Riemann-Liouville分数阶导数微分方程初值问题解的存在性和唯—性.第三部分,利用上下解方法和Schauder不动点定理给出了具有2序列Riemann-Liouville分数阶导数微分方程周期边值问题解的存在性.第四部分,利用Leray-Schauder不动点定理和Banach压缩映像原理建立了具有n序列Riemann-Liouville分数阶导数微分方程初值问题解的存在性、唯—性和解对初值的连续依赖性.第五部分,利用锥上的不动点定理给出了具有Caputo分数阶导数微分方程边值问题,在超线性(次线性)条件下C3[0,1]正解存在的充分必要条件.最后一部分,通过建立比较定理和利用单调迭代方法给出了具有Caputo分数阶导数脉冲微分方程周期边值问题最大解和最小解的存在性.
第一部分,介紹分數階導數的定義和著名的Mittag-Leffler函數的性質.第二部分,利用單調迭代方法給齣瞭具有2序列Riemann-Liouville分數階導數微分方程初值問題解的存在性和唯—性.第三部分,利用上下解方法和Schauder不動點定理給齣瞭具有2序列Riemann-Liouville分數階導數微分方程週期邊值問題解的存在性.第四部分,利用Leray-Schauder不動點定理和Banach壓縮映像原理建立瞭具有n序列Riemann-Liouville分數階導數微分方程初值問題解的存在性、唯—性和解對初值的連續依賴性.第五部分,利用錐上的不動點定理給齣瞭具有Caputo分數階導數微分方程邊值問題,在超線性(次線性)條件下C3[0,1]正解存在的充分必要條件.最後一部分,通過建立比較定理和利用單調迭代方法給齣瞭具有Caputo分數階導數脈遲微分方程週期邊值問題最大解和最小解的存在性.
제일부분,개소분수계도수적정의화저명적Mittag-Leffler함수적성질.제이부분,이용단조질대방법급출료구유2서렬Riemann-Liouville분수계도수미분방정초치문제해적존재성화유—성.제삼부분,이용상하해방법화Schauder불동점정리급출료구유2서렬Riemann-Liouville분수계도수미분방정주기변치문제해적존재성.제사부분,이용Leray-Schauder불동점정리화Banach압축영상원리건립료구유n서렬Riemann-Liouville분수계도수미분방정초치문제해적존재성、유—성화해대초치적련속의뢰성.제오부분,이용추상적불동점정리급출료구유Caputo분수계도수미분방정변치문제,재초선성(차선성)조건하C3[0,1]정해존재적충분필요조건.최후일부분,통과건립비교정리화이용단조질대방법급출료구유Caputo분수계도수맥충미분방정주기변치문제최대해화최소해적존재성.
In this report,there are six parts.In the first part,we introduce some definitions of fractional derivative and discuss the properties of the well-known Mittag-Leffier function.In the second part,we consider the existence and uniqueness of solution of the initial value problem for fractional differential equation involving 2 sequential Riemann-Liouville fractional derivative by using monotone iterative method.In the third part,we consider the existence of solution of the periodic boundary value problem for fractional differential equation involving 2 sequential RiemannLiouville fractional derivative by means of the method of upper and lower solutions and Schauder fixed point theorem.In the fourth part,we have established the existence,uniqueness and continuous dependence results of solutions for the initial value problem of fractional differential equation involving n sequential Riemann-Liouville fractional derivative by means of the Leray-Schauder type fixed point theorem and Banach contraction principle.In the fifth part,we investigate the existence of positive solutions of singular superlinear (or sublinear) boundary value problems for fractional differential equation involving Caputo fractional derivative,a necessary and sufficient condition for the existence of C3[0,1] positive solutions is given by means of the fixed point theorems on cones.In the last part,we consider the existence of minimal and maximal solutions for the periodic boundary value problem of impulsive fractional differential equation involving Caputo fractional derivative by using a comparison result and the monotone iterative method.
