带有Riemann-Stieltjes积分边界条件的非线性奇异分数阶微分方程边值问题正解的存在性

Existence and Multiplicity of Positive Solutions for Nonlinear Singular Fractional Differential Equation with Riemann-Stieltjes Integral Boundary Conditions

本文主要研究下列带有Riemann-Stieltjes积分边值条件的奇异分数阶微分方程问题正解的存在性和多重性:{D_0~α+u(t)+βω(t)f(t,u(t))=0,0<t<1,u(0)=u'(0)=u''(0)=···=u^(n=2)(0)=0,u(1)=λ∫_0~ηg(s)u(s)dA(s),其中β>0是参数,α>2,n-1<α≤n,0<η≤1,0≤(λη~α)/α<1,函数A(s)是有界变差函数,g∈L^1[0,1],D_(0+)~α是Riemann-Liouville分数阶微分;ω:(0,1)→(0,+∞)连续,ω∈L^1(0,1)且ω(t)在t=0和t=1处奇异,非线性项f:[0,1]×(0,+∞)→(0,+∞)连续且f(t,x)在x=0处奇异.本文首先给出了该问题的Green函数及其性质,然后在一些条件下,运用Green函数的性质和不动点指数理论,并利用相关线性算子的第一特征值,得到了问题正解的存在性和多重性.接下来,以注的形式,说明了一些相关的边值问题.最后,我们给出了相关的例子,来说明我们主要结果的实用性.

In this paper, we consider the existence and multiplicity of positive solutions to the nonlinear singular fractional differential equation with Riemann-Stieltjes integral bound- ary conditions： {D0a＋u（t）＋βw（t）f（t,u（t））=0,0〈t〈1,u（0）=u＇（0）=u＇＇（0）=…=u（n-2）（0）=0,u（1）=λ∫0ηg（s）u（s）dA（s）, where β〉0 is a parameter, α〉2,n-1〈α≤n,0〈η≤1,0≤ληa/a〈1 denotes a bounded variation function, g（s）∈ L1[0, 1], D0＋a is the standard Riemann- Liouville derivative, w ： （0, 1）→ [0, ＋∞） is continuous and w（t） may be singular at t = 0 and t = 1, f： [0, 1] × （0, ＋∞）→[0, ＋∞） is continuous and f（t, x） may have singularity at x = 0. In this paper, we deduce the Green function and its properties. The existence and multiplicity of positive solutions are obtained by means of the fixed point index theory in cones under suitable conditions on f concerning the properties of the Green function and the first eigenvalue corresponding to the relevant linear operator. Next, some related boundary value problems are also introduced in the form of annotation. Finally, an interesting example is presented to illustrate the usefulness of our main results.