非线性分数阶微分方程边值问题正解的存在性

数学物理学报 ›› 2011, Vol. 31 ›› Issue (2): 401-409.

非线性分数阶微分方程边值问题正解的存在性

许晓婕^1,2, 孙新国¹, 吕炜¹

1.中国石油大学(华东) 数学与计算科学学院山东东营 257061|2.东北师范大学数学与统计学院长春 130024

收稿日期:2009-03-05 修回日期:2010-04-20 出版日期:2011-04-25 发布日期:2011-04-25
基金资助:
中国石油大学(华东)基础研究基金(y070815)资助

Existence of Positive Solutions for |Boundary Value Problems with Nonlinear Fractional Differential Equations

XU Xiao-Jie, SUN Xin-Guo, LV Wei

1.School of Mathematics and Computational Science, China University of Petroleum (East China), Dongying 257061|2.School of Mathematics and Statistics, Northeast Normal University, Changchun 130024

Received:2009-03-05 Revised:2010-04-20 Online:2011-04-25 Published:2011-04-25
Supported by:
中国石油大学(华东)基础研究基金(y070815)资助

摘要/Abstract

摘要：

该文研究了下面分数阶微分方程边值问题格林函数的相关性质
D^α₀₊u(t)=f(t, u(t)), 0<t<1,
u(0)=u(1)=u'(0)=u'(1)=0,
其中3<α≤4 是实数, D^α₀₊是标准的Riemann-Liouville微分, f: [0,1]×[0, ∞)→[0, ∞) 连续. 应用格林函数的性质构造了锥, 从而应用一些不动点定理得到了正解的存在性.

关键词: 分数阶微分方程, 边值问题, 正解, 分数阶格林函数, 不动点定理

Abstract:

In this paper, the authors consider the properties of Green's function for the nonlinear fractional differential equation boundary-value problem
D^α₀₊u(t)=f(t, u(t)), 0<t<1,
u(0)=u(1)=u'(0)=u'(1)=0,
where 3<α≤4 is a real number, and D^α₀₊ is the standard Riemann-Liouville differentiation, and f: [0,1]×[0, ∞)→[0, ∞) is continuous.
As an application of Green's function, the authors give some multiple positive solutions for nonlinear by means of some fixed-point theorem on cones.

Key words: Fractional differential equation, Boundary-value problem, Positive solution, Fractional Green’s function, Fixed-point theorem

中图分类号:

34B18

许晓婕, 孙新国, 吕炜. 非线性分数阶微分方程边值问题正解的存在性[J]. 数学物理学报, 2011, 31(2): 401-409.

XU Xiao-Jie, SUN Xin-Guo, LV Wei. Existence of Positive Solutions for |Boundary Value Problems with Nonlinear Fractional Differential Equations[J]. Acta mathematica scientia,Series A, 2011, 31(2): 401-409.

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