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

Abstract

We study the three-point boundary value problem of higher-order fractional differential equations of the form , , , , , where is the Caputo fractional derivative of order , and the function is continuously differentiable. Here, , , . By virtue of some fixed point theorems, some sufficient criteria for the existence and multiplicity results of positive solutions are established and the obtained results also guarantee that the positive solutions discussed are monotone and concave.

1. Introduction

Applications of fractional differential equations can be found in various areas, including engineering, physics, and chemistry [1–4]. 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 [5–15].

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 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 is the Caputo fractional derivative of order . The case 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 is continuously differentiable;(A2), , .

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.

2. Preliminaries

In this section, we preliminarily provide some definitions and lemmas to be used in the following discussion.

Definition 1 (see [3]). The fractional integral of order of a function is given by provided the right side is pointwise defined on .

Definition 2 (see [3]). The Riemann-Liouville fractional derivative of order of a continuous function is given by where , provided the right side is pointwise defined on .

Remark 3. Consider , where , .

Definition 4 (see [4]). For a function given on the interval , the Caputo fractional derivative of order of is defined by where , denotes the integer part of .

Lemma 5 (see [3]). Let and be positive numbers. If , then , and the equations and are satisfied for each in .

Lemma 6 (see [3]). Let . If or , then for some in , ,  .

The following two lemmas are fundamental in finding an integral representation of solutions of BVP (2).

Lemma 7. Let be a continuously differentiable function. If a function in is a solution of the equation , then and , and the relation holds for each in .

Proof. Let be a solution of the equation .
Since is continuous on , Lemma 5 implies that and that The above equation, together with Remark 3 and Lemma 5, yields Hence is continuously differentiable on and .
Generally, noticing , we have which implies for .
Furthermore, using the assumptions imposed on the function and integrating by parts, we obtain This yields that, for every in , Since the second term of the right-hand side of the above equality is continuous on the interval ,  . 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).

Lemma 8. Let ; if (A1)-(A2) hold, then BVP has a unique solution where Here , and denotes the characteristic function of the set .

Proof. Lemma 6 implies Differentiating (19) with respect to up to the order and using the boundary conditions that , we obtain From the above equation and the condition that , it follows that Substituting 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 for all in and , where .

Proof. It follows from the definition of that, for , On the other hand, for , the assertion for is obvious.
As for the assertion for , it is sufficient to verify that for each in . In fact, the definition of and condition (A2) directly imply 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.

Lemma 10. Let be a function defined on . Assume for . Suppose that is continuously differentiable on ; if on , then on for .

Proof. Set . Then, as in the proof of Lemma 8, the assumptions made on and yield This implies for . Thus the desired results follow from the nonpositivty of . The proof is completed.

Lemmas 8–10 yield the following important properties of the solution of BVP (14), which is easy to check.

Lemma 11. Let . If condition (A2) holds, then the solution of BVP (14) is nonnegative, monotone, and concave on .

Lemma 12 (see [17]). If a function is nonnegative and concave on and , then(i) for each in , where ;(ii) for all in with .

Now, denote by the classical Banach space with the norm , where . Furthermore, define a cone, denoted by , through Also, for a given positive real number , define a function set by Naturally, we denote that and that .

Next, define the operator by for any . We now show some important properties on this map.

Lemma 13. Assume that hypotheses (A1)-(A2) are all fulfilled. Then and is completely continuous.

Proof. It is easy to check that . Moreover, analysis similar to that in [6] shows that is completely continuous. The proof is completed.

Lemma 14. If (A1)-(A2) hold, then a function in is a solution of BVP (2) if and only if it is a fixed point of in .

Proof. If is a solution of BVP (2), then Lemma 11 implies . Furthermore, replacing in Lemma 8 by , we get . Hence is a fixed point of in .
On the other hand, if and , 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.

3. 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 self-containment, we first state the Avery-Henderson fixed point theorem as follows.

Theorem 15 (see [18]). Let be a cone in a real Banach space . For each , set . Let and be increasing, nonnegative continuous functional on , and let be a nonnegative continuous functional on with such that, for some and , for all . Suppose that there exist a completely continuous operator and three positive numbers such that and (i) for all ; (ii) for all ; (iii) and for all . Then, the operator has at least two fixed points, denoted by and , belonging to and satisfying with and with .

Now, select and such that . Let We are now in a position to obtain the following result.

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:(C1) for in ;(C2) for in ;(C3) for in .Then BVP (2) has at least two positive solutions and such that

Proof. Let the cone and the operator be defined by (28) and (30), respectively. Furthermore, define the increasing, nonnegative, and continuous functionals , , and on , respectively, by Evidently, for each in .
Moreover, for each in , Lemma 12 implies that . Observing , we have for each in . Also, notice that for each in and in . In addition, Lemma 13 guarantees that the operator is completely continuous.
Next, we are to verify that all the conditions of Theorem 15 are satisfied with respect to the operator .
Let . Then . This implies that for each in , which, combined with (39), yields that for each in . This inequality and assumption () imply for each in . Now, from the definition of the operator and Lemmas 8 and 12, we obtain that Thus condition (i) in Theorem 15 is satisfied.
We now claim that condition (ii) in Theorem 15 is satisfied. To this end, let . Then, , from which we have for each in . Analogously, it follows from inequality (39) that, for each in , which implies for each in . This, combined with assumption (), yields for each in . Thus we have which consequently implies the validity of condition (ii) in Theorem 15.
Finally, notice that the constant function so that . Letting , we get . This with assumption (C3) implies that and for each in . Similarly, we have Thus condition (iii) in Theorem 15 is satisfied.
Consequently, an application of Theorem 15 implies that BVP (2) has at least two positive solutions, denoted by and , satisfying with and with , respectively.

4. 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 . and are supposed to be nonnegative continuous concave functionals on . Thus, for nonnegative real numbers , , , , and , define five convex sets, respectively, by

Theorem 17 (see [19]). Let be a cone in a real Banach space . Suppose that and are nonnegative continuous concave functionals on and that , , and are nonnegative continuous convex functionals on such that, for some positive numbers and , for all . In addition, suppose that is a completely continuous operator and that there exist nonnegative real numbers , , , with such that(i) and for ;(ii) and for ;(iii) for with ;(iv) for with .Then the operator admits at least three fixed points , , and satisfying , , and with , 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).

Theorem 18. Suppose that hypotheses (A1)-(A2) are all fulfilled. Assume that there exist positive real numbers , , and such that Furthermore, assume that satisfies the following conditions:(H1) for in ;(H2) for in ;(H3) for in .Then BVP (2) admits at least three positive solutions , , and , defined on , satisfying, respectively,

Proof. Let the cone and the operator be defined by (28) and (30), respectively. Define, respectively, the nonnegative continuous concave functionals on the as follows: It is obvious that for in . Moreover, from Lemma 12, it follows that for each in .
Next, we intend to verify that all the conditions in Theorem 17 hold with respect to the operator . We first claim that the operator is completely continuous. By Lemma 13, we only need to show that for each in . To this end, let . Then, , which, combined with (51), implies that for in and in . Thus, it follows from assumption (H1) that, for , , from which we have the following estimations: Hence we obtain the desired result. Now, it remains to verify that conditions (i)–(iv) in Theorem 17 are satisfied.
Since the constant function belongs to the set the set is not empty. Analogously, since the set is nonemptyIn addition, for in , inequality (51) implies for each in . From assumption (H2), we thus obtain Hence, it follows from (58) and Lemma 12 that Therefore condition (i) in Theorem 17 is satisfied.
We next claim that condition (ii) in Theorem 17 is satisfied. To see this, letting , then we get for each in . Thus assumption (H3) yields . Furthermore, we have Accordingly, the validity of condition (ii) in Theorem 17 is verified.
Aside from conditions (i) and (ii), we are finally to verify the validity of conditions (iii) and (iv). For this purpose, on the one hand, consider with . Thus we have On the other hand, consider with . In such a case, we obtain Therefore both conditions (iii) and (iv) in Theorem 17 are satisfied. Consequently, by virtue of Theorem 17, BVP (2) has at least three positive solutions defined on satisfying , , and with .

5. An Illustrative Example

Consider BVP where for in . Here, , and .

We claim that the above BVP has at least three positive solutions. To see this, letting , , we have Selecting , , and , then it is easy to check that these parameters satisfy Now, we can verify that conditions (H1)–(H3) in Theorem 17 are satisfied. Indeed, direct computations produce the following estimations: Thus conditions (H1)–(H3) in Theorem 17 are satisfied for the above specified functions and parameters. Therefore, in light of Theorem 17, the assertion made above is verified.

Conflict of Interests

The authors declare that there is no conflict of interests regarding the publication of this paper.

Acknowledgment

This work was supported by the Natural Science Foundation of Hunan Province of China (Grant no. 11JJ3007).