Existenceofpositivesolutionsforsingularhigher-orderfractionaldifferentialequationswithinfinite-pointsboundaryconditions资源-CSDN文库

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根据提供的文件信息，我们可以总结出以下知识点： 1. 高阶分数阶微分方程的正解存在性：本文所研究的问题涉及到在数学领域内具有重要地位的高阶分数阶微分方程。这类方程在自然科学和工程技术的许多领域中都有广泛的应用，如物理、化学、生物科学以及在力学中的振动和扩散问题。 2. 奇异问题的处理：文章中提到的“奇异”，指的是微分方程中包含的非线性项可能在某些点（比如时间变量的端点或空间变量的特定点）变得无法定义或者无限大，处理这类问题通常更为复杂。 3. 无穷点边值条件：在分数阶微分方程的理论中，边值条件是用来指定在定义域的边界上解应满足的条件。文中提到的“无穷点”是指边值条件中包含了无穷多的点，这是一种比较特殊和复杂的边值问题。 4. 不动点定理的应用：不动点定理是数学中一个非常重要的定理，常用于证明数学模型中各种函数或映射存在不动点，即在某种映射下保持不变的点。在本研究中，利用不动点定理来证明正解的存在性。 5. 序列逼近方法：序列逼近方法是求解数学问题的一种有效手段，它涉及到构造一系列逼近的序列，并利用极限理论来分析原问题的解的性质。在本文中，通过序列逼近方法来研究正解的存在性。 6. 格林函数及其性质：格林函数是偏微分方程理论中的一个概念，它与线性微分方程的解有着密切联系。通过研究格林函数，我们可以获得关于原微分方程解的重要信息，包括解的性质等。 7. 分数微分方程：分数微分方程是指含有分数导数的微分方程。与传统的整数阶微分方程不同，分数导数可以提供非局部和历史依赖性的信息，这些特性使得分数微分方程在描述具有记忆和遗传性质的过程方面具有优势。 8. 正解：在微分方程的研究中，“正解”指的是满足一定条件的非平凡解，其在某种意义上大于零（或小于零）。正解在研究物理、生物和工程技术中的问题时非常重要，因为它们可以描述稳定的状态或者某种偏好方向的过程。 9. 正规性：正规性在数学中涉及函数的连续性、光滑性等属性。在本研究中，通过正规化技术来研究分数微分方程正解的存在性，可能涉及到对非线性项进行正规化处理，使其在全定义域内有良好定义。上述内容涉及了关于带有无穷点的高阶分数阶微分方程的正解存在性研究的基础理论和方法，展现了作者在非线性分析和分数微分方程领域所进行的深入探讨。研究的成果不仅为理论数学提供了新的视角和工具，同时也为解决相关科学和工程问题提供了新的思路。

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˖ڍመ᝶஠ڙጲ

http://www.paper.edu.cn

带有无穷点的高阶分数阶微分方程正解的存

在性

郭丽敏

，刘立山

1,2

，吴永洪

曲阜师范大学数学科学学院，山东省曲阜市　邮编 273165

Department of Mathematics and Statistics, Curtin University, Perth,

WA 6845, Australia

摘要：本文利用不动点定理和序列逼近的方法，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。此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。首先，给出格林函数和格林函数的性质，然后利用序列逼近的

方法得到方正正解的存在性。

关键词：分数微分方程; 奇异问题; 无穷点边值条件; 正解; 序列方法和正规性.

中图分类号： O177.91

Existence of positive solutions for singular

higher-order fractional diﬀerential equations with

inﬁnite-points boundary conditions

LIMIN GUO

, LISHAN LIU

1,2

, YONGHONG WU

School of Mathematical Science, Qufu Normal University, Qufu 273165,

Shandong, People’s Republic of China

Department of Mathematics and Statistics, Curtin University, Perth,

WA 6845, Australia

Abstract: In this paper, a class of singular fractional diﬀerential equations with

inﬁnite-points boundary conditions is considered. The fractional orders are involved in the

nonlinearity of the boundary value problem and the nonlinearity is allowed to be singular

with respect to not only the time variable but also the space variable. Firstly, we give Green’s

function and establish its properties. Then, we utilize the sequential technique and

regularization to investigate the existence of positive solutions.

Key words: Fractional diﬀerential equation; Singular problem; Inﬁnite-points boundary

conditions; Positive solution; Sequential techniques and regularization

Foundations: The authors were supported ﬁnancially by the National Natural Science Foundation of China (11371221,

11571296).

Author Introduction: Correspondence author: Lishan Liu, male, professor, major research direction: nonlinear analysis

and application.

- 1 -

˖ڍመ᝶஠ڙጲ

http://www.paper.edu.cn

1. Introduction

In this paper, we consider the following class of nonlinear singular fractional diﬀerential

equations











u(t) + f(t, u(t), D

u(t), D

u(t), · · · , D

n−2

u(t)) = 0, 0 < t < 1,

u(0) = u

′

(0) = · · · = u

(n−2)

(0) = 0,

(n−2)

(1) =

∞



j=1

u(ξ

(1.1)

where α, µ

∈ R

= [0, +∞), n ∈ N (natural number set) and n − 1 < α ≤ n, n ≥ 4,

i − 1 < µ

≤ i (i = 1, 2, · · · , n − 2), f(t, x

, x

, · · · , x

n−1

) may have singularity at t = 0, 1

and x

= 0 (i = 1, 2, · · · , n − 1), 0 < ξ

< ξ

< . . . < ξ

< 1, η

> 0,



∞

j=1

α−1

< Θ,

Θ = (α − 1)(α − 2) · · · (α − n + 2) and D

u,D

u (i = 1, 2, · · · , n) are the Riemann-Liouville

derivative.

Boundary value problems for nonlinear fractional diﬀerential equations arise from the s-

tudies of complex problems in many disciplinary areas such as ﬂuid ﬂows, electrical networks,

rheology, biology chemical physics, and so on. Fractional-order models have been shown to be

more accurate than integer order models, and in the application of these models, it is important

to theoretically establish the conditions for the existence of positive solutions. In recent years,

many authors investigated the existence of positive solutions for fractional equation boundary

value problems (see [1-24] and the references therein), and a great deal of results have been

developed for diﬀerential and integral boundary value problems. In [14], the author considered

the following fractional diﬀerential equation











u(t) + g(t)f (t, u(t)) = 0, 0 < t < 1,

u(0) = u

′

(0) = . . . = u

(n−2)

(0) = 0,

(i)

(1) =

∞



i=1

u(ξ

), 1 ≤ i ≤ n − 2,

where α ∈ R

, n ∈ N and n − 1 < α ≤ n, n ≥ 3, α

≥ 0, 0 < ξ

< ξ

< . . . < ξ

j−1

< . . . < 1 (j = 1, 2, · · · ), D

are the standard Riemann-Liouville derivative and f ∈

C((0, 1) × (0, +∞), R

)) permits singularities with respect to both the time (t = 0, 1) and the

space variables (u = 0). The author the established the existence and multiplicity of positive

solutions. In [15], the authors investigated the fractional diﬀerential equation







u(t) + f(t, u(t), D

u(t), D

u(t)) = 0, 0 < t < 1,

u(0) = u

′

(0) = u

′′

(0) = u

′′

(1) = 0,

where α, v, µ ∈ R

and 3 < α ≤ 4, 0 < v ≤ 1, 0 < µ ≤ 1 are real numbers, D

, D

and

are Riemann-Liouville fractional derivatives, f(t, x, y, z) is a Carath´eodory function and is

- 2 -

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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







u(t) + f(t, u(t), D

u(t)) = 0, 0 < t < 1,

u(0) = u(1) = 0,

where α, µ ∈ R

and 1 < α ≤ 2, µ > 0 are real numbers, α − µ ≥ 1, D

and D

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







u(t) + f(t, u(t), u

′

(t), D

u(t)) = 0, 0 < t < 1,

u(0) = 0, u

′

(0) = u

′

(1) = 0,

where α, µ ∈ R

and 2 < α ≤ 3, 0 < µ ≤ 1 are real numbers, D

and D

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







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

and n − 1 < α ≤ n, n ≥ 2, the nonlinear function f(t, x

, x

, · · · , x

n−1

) may be

singular at x

= 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

, x

, · · · , x

n−1

) may have

singularity at t = 0, 1 and x

= 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

, x

, · · · , x

n−1

)| ≤ φ

(t)

is successfully removed, and at the same time the condition

lim

x→∞

h(x, x, · · · , x)

= 0

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