Open Access
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

© Wuhan University 2022

Licence Creative CommonsThis 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

(1)

(2)

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 [1-7]. 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[8-23] and nonlocal boundary conditions[24-35] 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 four-point boundary value problems for fractional differential equations. For examples, in Ref.[28], the authors considered a class of four-point 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 four-point nonlinear boundary value problem

was considered. The existence of solutions of the problem were established.

Ji and Ge [30] studied the following four-point 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 Avery-Peterson 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 Riemann-Liouville fractional integral of order of a function u(t) is given by

provided the right side is point-wise 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 point-wise 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

(3)

(4)

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 R1>0 such that . Denote

Then for , we have

By means of the Arzela-Ascoli 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 (1-2) has at least three positive solutions satisfying

Proof   Problem (1-2) 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 (S1) of Lemma 3 is satisfied. Secondly, for all,

Thus, condition (S2) of Lemma 3 holds. Finally we show that (S3) also holds. We see that and . Suppose that Then by assumption

Thus, all conditions of Lemma 3 are satisfied. Hence problem (1-2) has at least three positive solutions satisfying

3 Example

Here we present an example to illustrate the main theorem. Consider the boundary value problem

(5)

(6)

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 (5-6) 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 (5-6) explicitly. The early results for positive solutions to this kind of fractional differential equations are not applicable to this problem.

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