一类Caputo分数阶微分方程边值问题多解的存在性

Existence of Multiple Solutions for a Caputo Fractional Difference Equation Boundary Value Problem

关键词:  分数阶微分方程  边值问题  Leggett-Williams不动点定理

Key words:  fractional difference equation  boundary value problem  Leggett-Williams fixed point theorems

摘要:

研究一类Caputo分数阶微分方程边值问题:
⎧D₀^α+u(t)+f(t,u(t))=0,t∈(0,1)
⎨
⎩u'(0)=u(1)=0
多解的存在性,其中1 < α ≤ 2,f:[0,+∞)×R→[0,+∞)是连续的,D₀₊^α是标准的Caputo微分.先将微分方程边值问题转化为积分方程,再转化为积分算子不动点问题,最后利用Leggett-Williams不动点定理得出Caputo分数阶微分方程边值问题至少有3个正解存在,其中格林函数的性质和非线性项的条件至关重要.

Abstract:

We investigate the existence and multiplicity of positive solutions for nonlinear Caputo fractional differential equation boundary value problem
⎧D₀^α+u(t)+f(t,u(t))=0,t∈(0,1)
⎨
⎩u'(0)=u(1)=0,
Where 1 < α ≤ 2,f:[0,+∞)×R→[0,+∞) is continuous,and D₀₊^α is the standard Caputo differentiation.In the process of proof,we first transform it into integral equation,then differential equation boundary value problem is further converted to discuss the problem of integral operator fixed point. Finally,by means of Leggett-Williams fixed point theorems on cone,existence results of at least three positive solutions are obtained.The properties of the Green function and the conditions of the nonlinear term is very important.