Monotone and Concave Positive Solutions to Three-Point Boundary Value Problems of Higher-Order Fractional Differential Equations

and Applied Analysis 3 Furthermore, using the assumptions imposed on the function g(t, u) and integrating by parts, we obtain


Introduction
Applications of fractional differential equations can be found in various areas, including engineering, physics, and chemistry [1][2][3][4]. In recent years, the interest in the study of fractional differential equations has been growing rapidly.
As one of the focal topics in the research of fractional differential equations, the study of the boundary value problems (BVPs for short) recently has attracted a great deal of attention from many researchers. A series of works have been presented to discuss the existence of (positive) solutions in the BVPs for fractional differential equations [5][6][7][8][9][10][11][12][13][14][15].
However, there are few results in the literature to discuss the positive, monotone, and concave solutions to the BVPs of fractional differential equations; it is difficult to establish the relation between the monotonicity and concavity of a function and its fractional derivatives. It is worth pointing out that Wang et al. [7] obtained the existence and multiplicity results of the positive, monotone, and concave solutions to the following problem: where 0+ is the Caputo fractional derivative of order . The multiplicity results of solutions are obtained by using the Legget-Williams fixed point theorem. However, the question of how to establish the connection between the monotonicity and concavity of a function and its fractional derivatives is far from being solved; and the concavity of a function is also not used sufficiently.
Motivated by the aforementioned results, we then turn to investigating the existence of monotone and concave positive solutions for the following boundary value problem (BVP for short): where 0+ is the Caputo fractional derivative of order . The case = 2 was discussed in [16] by virtue of the Avery-Henderson and Legget-Williams fixed point theorems. While in the setting of the fractional-order derivatives, as far as we know, the existence of positive solutions for BVP (2) has not been discussed in the literature.
We now make the following assumptions to be used later: (A1) the function The rest of paper is organized as follows. Section 2 preliminarily provides some definitions and lemmas which are crucial to the following discussion, and the connection between the monotonicity and concavity of a function and its Caputo derivatives is established in this section. Section 3 gives some sufficient conditions for the existence of at least two positive solutions of BVP (2) by means of the Avery-Henderson fixed point theorem. Section 4 gives some sufficient conditions for the existence of at least three positive solutions by virtue of the five-functional fixed point theorem. In addition, the sufficient conditions also guarantee that the positive solutions obtained are monotone and concave. Finally, Section 5 provides an example to illustrate a possible application of the obtained results.

Preliminaries
In this section, we preliminarily provide some definitions and lemmas to be used in the following discussion.
The following two lemmas are fundamental in finding an integral representation of solutions of BVP (2).
Abstract and Applied Analysis 3 Furthermore, using the assumptions imposed on the function ( , ) and integrating by parts, we obtain This yields that, for every in (0, 1], Since the second term of the right-hand side of the above equality is continuous on the interval [0, 1], ( ) ∈ (0, 1] ∩ (0, 1]. Consequently, direct computations produce The proof is completed. By Lemma 6, we next present an integral representation of the solution of the linearized problem corresponding to BVP (2).
Here = { : ⩽ }, and denotes the characteristic function of the set .
Proof. Lemma 6 implies Differentiating (19) with respect to up to the order − 1 and using the boundary conditions that From the above equation and the condition that (1) + (1) = ( ), it follows that Substituting 0 into (20), we have where ( , ) is defined by (16). The proof is completed.
We now give some properties of the functions ( , ).

Lemma 9.
If condition (A2) holds, then Proof. It follows from the definition of 1 ( , ) that, for ⩾ , On the other hand, for ⩽ , the assertion for 1 ( , ) is obvious.
As for the assertion for 2 ( , ), it is sufficient to verify that 2 ( , ) ⩾ 0 for each in [0, 1]. In fact, the definition of The proof is completed.
The following results establish the connection between the monotonicity and concavity of a function and its Caputo fractional derivatives under some conditions. Proof. Set ℎ( ) = 0+ ( ). Then, as in the proof of Lemma 8, the assumptions made on 0+ ( ) and (0) yield This implies for = 1, 2, . . . − 1. Thus the desired results follow from the nonpositivty of ℎ( ). The proof is completed.
Lemmas 8-10 yield the following important properties of the solution of BVP (14), which is easy to check.
Also, for a given positive real number , define a function set P by Naturally, we denote that P = { ∈ P | ‖ ‖ ⩽ } and that P = { ∈ P | ‖ ‖ = }. Next, define the operator A : P → E by for any ∈ P. We now show some important properties on this map.

Lemma 13. Assume that hypotheses (A1)-(A2) are all fulfilled.
Then A(P) ⊂ P and A : P → P is completely continuous.
Proof. It is easy to check that A(P) ⊂ P. Moreover, analysis similar to that in [6] shows that A : P → P is completely continuous. The proof is completed. Proof. If is a solution of BVP (2), then Lemma 11 implies ∈ P. Furthermore, replacing ℎ( ) in Lemma 8 by ( , ( )), we get A = . Hence is a fixed point of A in P.
On the other hand, if ∈ P and A = , then The above equation and Lemma 7 imply Moreover, it is easy to check that all the boundary conditions in BVP (2) are satisfied. Therefore is a positive solution of BVP (2). We consequently complete the proof.

Two Positive Solutions in Boundary Value Problems
In this section, we aim to adopt the well-known Avery-Henderson fixed point theorem to prove the existence of at least two positive solutions in BVP (2). For the sake of selfcontainment, we first state the Avery-Henderson fixed point theorem as follows.

Theorem 16. Assume that hypotheses (A1)-(A2) all hold and that there exist positive real numbers , , and such that
Furthermore, assume that satisfies the following conditions: Then BVP (2) has at least two positive solutions 1 and 2 such that Proof. Let the cone P and the operator A be defined by (28) and (30), respectively. Furthermore, define the increasing, nonnegative, and continuous functionals , , and on P, respectively, by Evidently, ( ) = ( ) ⩽ ( ) for each in P. Moreover, for each in P, Lemma 12 implies that for each in P. Also, notice that ( ) = ( ) for each in [0, 1] and in P( , ). In addition, Lemma 13 guarantees that the operator A : P( , ) → P is completely continuous.
Next, we are to verify that all the conditions of Theorem 15 are satisfied with respect to the operator A.

Three Positive Solutions in Boundary Value Problems
In this section, we are to prove the existence of at least three positive solutions in BVP (2) by using the five-functional fixed point theorem which is attributed to Avery [19]. Let , , be nonnegative continuous convex functionals on P.
and are supposed to be nonnegative continuous concave functionals on P. Thus, for nonnegative real numbers ℎ, , , , and , define five convex sets, respectively, by Theorem 17 (see [19]). Let P be a cone in a real Banach space E. Suppose that and are nonnegative continuous concave functionals on P and that , , and are nonnegative continuous convex functionals on P such that, for some positive numbers and , for all ∈ P( , ). In addition, suppose that A : P( , ) → P( , ) is a completely continuous operator and that there exist nonnegative real numbers ℎ, , , with 0 < < such that Then the operator A admits at least three fixed points 1 , 2 , and 3 ∈ P( , ) satisfying ( 1 ) < , < ( 2 ), and < ( 3 ) with ( 3 ) < , respectively.
With this theorem, we are now in a position to establish the following result on the existence of at least three positive solutions in BVP (2). Then BVP (2) admits at least three positive solutions 1 ( ), for each in P.
Next, we intend to verify that all the conditions in Theorem 17 hold with respect to the operator A. We first claim that the operator A : P( , ) → P( , ) is completely continuous. By Lemma 13, we only need to show that A ⊂ P( , ) for each in P( , ). To this end, let ∈ P( , ). (52) Hence we obtain the desired result. Now, it remains to verify that conditions (i)-(iv) in Theorem 17 are satisfied.