Issue 
Wuhan Univ. J. Nat. Sci.
Volume 27, Number 4, August 2022



Page(s)  287  295  
DOI  https://doi.org/10.1051/wujns/2022274287  
Published online  26 September 2022 
Mathematics
CLC number: O 175.8
Existence of Triple Positive Solutions to a FourPoint Boundary Value Problem of a Fractional Differential Equations
College of Mathematics and Statistics, Hefei Normal University, Hefei 230061, Anhui, China
Received:
5
January
2022
In this paper, the existence result of at least triple positive solutions to a boundary value problem of a fractional differential equations is achieved by means of the AveryPeterson fixed point theorem and the careful analysis of the associate Green's function in which the derivative of unknown function is involved in the nonlinear term explicitly. An example illustrating our main result is given. Our results complement the previous work in the area of the positive solutions of fractional differential equations.
Key words: fixed point / positive solution / cone / AveryPeterson fixed point theorem
Biography: YANG Liu, male, Professor, research direction: differential equations. Email:yliu722@163.com
Fundation item: Supported by the National Natural Science Foundation of China (12001152), the Natural Science Foundation of Anhui Province (A2008085QA08, KJ2020A0089, KJ2021A0927)
© Wuhan University 2022
This is an Open Access article distributed under the terms of the Creative Commons Attribution License (https://creativecommons.org/licenses/by/4.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.
0 Introduction
In this paper we consider the boundary value problem of nonlinear fractional differential equations
where and and denotes the Caputo's fractional derivative of order .
Due to the development of the theory of fractional calculus and its applications, such as Bode's analysis of feedback amplifiers, aerodynamics and polymer rheology in the fields of physics, etc, many works on the basic theory of fractional calculus and fractional order differential equations have been done^{ [17]}. Recently, there have been many papers dealing with the solutions or positive solutions to boundary value problems for nonlinear fractional differential equations (FBVPs) with local boundary conditions^{[823]} and nonlocal boundary conditions^{[2435]} and references along this line.
Zhang^{[13]} proved the existence of positive solution to the boundary value problem of fractional order differential equation
where and
By using the fixed point theorem, Goodrich^{[22]} considered the following class of nonlinear fractional differential equations with the given boundary conditions for multiplicity of positive solutions as
Specially, there are a few researches concerning fourpoint boundary value problems for fractional differential equations. For examples, in Ref.[28], the authors considered a class of fourpoint fractional boundary value problem of the form
where u' denotes the first order derivative of function u and is continuous and is the Caputo's fractional derivative of order .
In Ref.[29], the following fourpoint nonlinear boundary value problem
was considered. The existence of solutions of the problem were established.
Ji and Ge ^{[30] }studied the following fourpoint nonlocal boundary value problems of fractional order
where and is Caputo's fractional derivative. By using the fixed point theorem, multiplicity results of positive solutions are obtained.
We noticed that in these work the existence results of positive solutions were all established under the assumption that the derivative of the unknown function was not involved in the nonlinear term explicitly. The main reason is that one can not derive the concavity or convexity of the function by the sign of its fractional derivative. On account of the practical meaning of u'(t), it is interesting to consider the boundary value problem of fractional differential equations in which the derivative of the unknown function is involved in the nonlinear term explicitly.
In this paper, by using the careful analysis of the associated Green's function and defining the special cone in a suitable Banach space together with the AveryPeterson fixed point theorem, we overcome the difficulties bringing by the lack of the concavity or convexity of the unknown function and show the existence of multiple positive solutions of problem (1)(2).The results complete and extend the previous work on boundary value problem of fractional differential equations.
1 Preliminaries
Definition 1 The RiemannLiouville fractional integral of order of a function u(t) is given by
provided the right side is pointwise defined on
Definition 2 The Caputo's fractional derivative of order of a continuous function u(t) is given by
where , provided that the right side is pointwise defined on
Lemma 1 Let . The fractional differential equation has solution
Lemma 2 Assume that u(t) is differentiable with a fractional derivative of order . Then
where n is the smallest integer greater than or equal to .
Definition 3 The map is said to be a nonnegative continuous convex functional on cone P of a real Banach space E provided that is continuous and
Definition 4 The map is said to be a nonnegative continuous concave functional on cone P of a real Banach space E provided that is continuous and
Let be nonnegative continuous convex functionals on P, be a nonnegative continuous concave functional on P and be a nonnegative continuous functional on P. Then for positive numbers a, b, c and d, we define the following convex sets:
and a closed set
Lemma 3 ^{[36]}Let P be a cone in Banach space E. Let be nonnegative continuous convex functionals on P, be a nonnegative continuous concave functional on P, and be a nonnegative continuous functional on P satisfying
where is the closure of the set . Suppose is completely continuous and there exist positive numbers a, b, c with a<b such that
Then T has at least three fixed points such that:
2 Main Results
Lemma 4^{[30]}Given , then boundary value problem
is equivalent to
where
Furthermore, the function G(t, s) satisfies that
Lemma 5 The function G(t, s) satisfies the following properties:
1) G(t, s) is decreasing with respect to t;
2) where
Proof 1) To prove that 1) is true, we begin with
It is easy to find that G(t, s) is decreasing with respect to t.
2) From the expression and monotonicity of function G(t, s) with respect to t , we have
Thus, for
For
For Then, we conclude that
Lemma 6 Assume that y(t)>0 and u(t) is a solution to problem (3)(4). Then
Proof The fact that and
ensure that
Let the space endowed with the norm
It is well known that X is a Banach space . Define the cone by
Lemma 7 Let be the operator defined by Then is completely continuous.
Proof First, we will show that the operator is continuous. For any with , we have
From the continuity of function f, we obtain
Thus
Therefore,
which implies that These ensure that T is continuous. Second, we will show that T is completely continuous.
Let be bounded. Then there exists a positive constant R_{1}>0 such that . Denote
Then for , we have
By means of the ArzelaAscoli theorem, we claim that T is completely continuous. Finally, we see that
Considering the definition of the operator T together with Lemma 6, one can find that
Thus, we conclude that is a completely continuous operator.
Let the nonnegative continuous concave functional , the nonnegative continuous convex functionals and the nonnegative continuous functional be defined on the cone by
By Lemmas 5 and 6, the functionals defined above satisfy that
Therefore condition of Lemma 3 is satisfied.
Assume that there exist constants and
such that
Theorem 1 Under assumptions , problem (12) has at least three positive solutions satisfying
Proof Problem (12) has a solution if and only if u solves the operator equation
For , we have Then
Thus
Hence,
The fact that the constant function and implies that . For , we have . From assumption, we see
Thus
which means These ensure that condition (S_{1}) of Lemma 3 is satisfied. Secondly, for all,
Thus, condition (S_{2}) of Lemma 3 holds. Finally we show that (S_{3}) also holds. We see that and . Suppose that Then by assumption
Thus, all conditions of Lemma 3 are satisfied. Hence problem (12) has at least three positive solutions satisfying
3 Example
Here we present an example to illustrate the main theorem. Consider the boundary value problem
where and
By a straightforward calculation, we see that
We choose positive constants a=1, b=5, d=1 000 and check that the nonlinear term f(t, u,v) satisfies
Then all assumptions of Theorem 1 are satisfied. Thus problem (56) has at least three positive solutions satisfying
Remark We see that the first order derivative of function u(t) is involved in the nonlinear term of the problem (56) explicitly. The early results for positive solutions to this kind of fractional differential equations are not applicable to this problem.
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