奇异分数阶微分方程正解的存在性研究
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"Existence of positive solutions for singular higher-order fractional differential equations with infinite-points boundary conditions"
本文研究了一类带有无穷点边值条件的奇异分数阶微分方程正解的存在性。该文中，作者郭丽敏、刘立山和吴永洪利用不动点定理和序列逼近的方法，研究了奇异分数阶微分方程的正解的存在性。
首先，作者给出了格林函数和格林函数的性质，然后利用序列逼近的方法，证明了奇异分数阶微分方程正解的存在性。在该文中，非线性条件里面含有分数阶导数，同时非线性不仅关于时间变量奇异，并且关于空间变量奇异。
该文的主要贡献在于，作者成功地证明了奇异分数阶微分方程正解的存在性，并且该方法可以广泛应用于解决奇异分数阶微分方程问题。
从数学角度来看，该文的研究方法可以分为两个部分：第一部分是格林函数和格林函数的性质的研究，第二部分是序列逼近的方法。通过这两个部分的研究，作者成功地证明了奇异分数阶微分方程正解的存在性。
此外，该文还讨论了奇异分数阶微分方程的边值条件的影响。作者发现，奇异分数阶微分方程的边值条件对其正解的存在性有着重要的影响。因此，该文的研究结果对奇异分数阶微分方程的研究和应用具有重要的指导意义。
该文的研究结果对奇异分数阶微分方程的研究和应用具有重要的意义，为解决奇异分数阶微分方程问题提供了有价值的参考。
知识点：
1. 奇异分数阶微分方程的定义和性质
2. 格林函数和格林函数的性质
3. 序列逼近的方法
4. 奇异分数阶微分方程的边值条件
5. 奇异分数阶微分方程的正解的存在性
相关概念：
1. 分数阶微分方程
2. 奇异问题
3. 无穷点边值条件
4. 序列方法
5. 正规性
中图分类号：O177.91
˖ڍመڙጲ
http://www.paper.edu.cn
singular at x, y, z = 0. The authors obtained the existence and multiplicity of positive solutions
by means of Krasnoselskii’s fixed point theorem. As there do not exists 0 < L < M such that
(3.13) of [15] holds, the results of the multiple solutions are not correct in [15]. In [16], the
authors investigated the fractional differential equation
D
α
0+
u(t) + f(t, u(t), D
µ
0
+
u(t)) = 0, 0 < t < 1,
u(0) = u(1) = 0,
where α, µ ∈ R
1
+
and 1 < α ≤ 2, µ > 0 are real numbers, α − µ ≥ 1, D
α
0+
and D
µ
0
+
are
Riemann-Liouville fractional derivatives, f is a Carath´eodory function and f (t, x, y) is singular
at x = 0. The authors obtained the existence of positive solutions by means of Krasnoselskii’s
fixed point theorem. In [17], the authors investigated the fractional differential equation
D
α
0+
u(t) + f(t, u(t), u
′
(t), D
µ
0
+
u(t)) = 0, 0 < t < 1,
u(0) = 0, u
′
(0) = u
′
(1) = 0,
where α, µ ∈ R
1
+
and 2 < α ≤ 3, 0 < µ ≤ 1 are real numbers, D
α
0+
and D
µ
0
+
are Riemann-
Liouville fractional derivatives, f is a Carath´eodory function and f (t, x, y, z) is singular at
x, y, z = 0. The authors obtained the existence of positive solutions by means of Krasnoselskii’s
fixed point theorem. In [18], the authors investigated the singular problem
D
α
0+
u(t) + q(t)f (t, u(t), u
′
(t), · · · , u
(n−2)
(t)) = 0, 0 < t < 1,
u(0) = u
′
(0) = u
′′
(0) = · · · = u
(n−2)
(0) = u
(n−2)
(1) = 0,
where α ∈ R
1
+
and n − 1 < α ≤ n, n ≥ 2, the nonlinear function f(t, x
1
, x
2
, · · · , x
n−1
) may be
singular at x
i
= 0 (i = 1, 2, · · · , n − 1) and q(t) may be singular at t = 0. The existence results
of positive solutions are obtained by a fixed point theorem for the mixed monotone operator.
Motivated by the results above, in this paper, we utilize the sequential technique and
regularization to investigate the existence of positive solutions of BVP(1.1), where u ∈ C[0, 1]
is said to be a positive solution of BVP (1.1) if and only if u satisfies (1.1) and u(t) > 0
for any t ∈ (0, 1]. By using the sequential technique and regularization on a cone, some new
existence results are obtained for the case where the nonlinearity is allowed to be singular
with respect to both the time variable and the space variable. We should address here that
our work presented in this paper has various new features. Firstly, our study is on singular
nonlinear differential equation boundary value problems, that is, f(t, x
1
, x
2
, · · · , x
n−1
) may have
singularity at t = 0, 1 and x
i
= 0 (i = 1, 2, · · · , n−1), which leads to many difficulties in analysis.
Secondly, compared with [15−17], we complete the proof without the need of imposing the third
condition of the Carath´eodory conditions, that is, the condition |f(t, x
1
, x
2
, · · · , x
n−1
)| ≤ φ
H
(t)
is successfully removed, and at the same time the condition
lim
x→∞
h(x, x, · · · , x)
x
= 0
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