河北师范大学学报—

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刊名：河北师范大学学报（自然科学版）Journal of Hebei Normal University (Natural Science)
主办：河北师范大学
ISSN： 1000-5854
CN： 13-1061/N
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非线性Caputo型分数阶微分方程耦合系统边值问题解的存在性和唯一性

薛益民

徐州工程学院数学与物理科学学院, 江苏徐州 221018
DOI： 10.13763/j.cnki.jhebnu.nse.2017.03.003

Existence and Uniqueness of Solutions of the Boundary Value Problem for a Coupled System of Nonlinear Caputo Fractional Differential Equations

XUE Yimin

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摘要/Abstract

摘要：

研究了一类带有积分边界条件非线性Caputo型分数阶微分方程耦合系统^cD^αu（t）+f（t，ν（t））=0，0 < t < 1，
^cD^βν（t）+g（t，u（t））=0，0 < t < 1，
u（0）=u'（0）=…=u^（n-2）（0）=u^（n）（0）=0，u（1）=λ∫⁰₁u（s）ds，
ν（0）=ν'（0）=…=ν^（n-2）（0）=ν^（n）（0）=0，ν（1）=λ∫¹₀ν（s）ds
解的存在性和唯一性问题.利用Schauder不动点定理和Banach压缩映射原理，得到了该耦合系统解的存在性和唯一性的充分条件，并举例说明定理的有效性.

Abstract：

In this paper, we studied the existence and uniqueness of positive solutions for a class of Caputo fractional differential coupled system with integral boundary value conditions below: ^cD^αu(t)+f(t,ν(t))=0,0 < t < 1,
^cD^βν(t)+g(t,u(t))=0,0 < t < 1,
u(0)=u'(0)=…=u^(n-2)(0)=u⁽ⁿ⁾(0)=0,u(1)=λ∫⁰₁u(s)ds,
ν(0)=ν'(0)=…=ν^(n-2)(0)=ν⁽ⁿ⁾(0)=0,ν(1)=λ∫¹₀ν(s)ds
The existence and uniqueness of positive solutions were obtained by applying the Schauder fixed-point theorem and Banach contraction mapping principle, two examples were given to illustrate the advantages of our main results.

关键词

关键词： 积分边值条件 ; 分数阶微分方程 ; 耦合系统 ; Schauder不动点定理

Key words： integral boundary conditions ; fractional differential equations ; coupled system ; Schauder fixed-point theorem

参考文献 16

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