Some New Existence and Uniqueness Results for an Integral Boundary Value Problem of Caputo Fractional Differential Equations

Abstract

In this work, we consider an integral boundary value problem of Caputo fractional differential equations. Based on a fixed-point theorem of generalized concave operators, we obtain the existence and uniqueness of positive solutions. As applications of main results, we give two examples in the end.

1. Introduction

The fractional differential equations appear naturally in fields of physics, chemistry, electrodynamics of complex medium, control of dynamical systems, and so on. So differential equations with fractional order have been studied by many researchers, and then the theory of fractional boundary value problems has been noticed more and more in recent years; see [1–26], for example.

In [25], the author investigated the existence of positive solutions for an integral boundary value problem of fractional differential equations by using a fixed-point theorem on cones. In [21, 22], Yang gave the existence and nonexistence results for fractional differential equation integral boundary value problems. In these papers, we can find that there are no uniqueness results on positive solutions. Moreover, there are still very few results on the uniqueness of positive solutions for an integral boundary value problem of Caputo fractional differential equations. To fill this gap, we study the existence and uniqueness of positive solutions for the following integral boundary value problem for fractional differential equations where

In [19], the authors studied the existence of positive solutions for (1) by the fixed-point theorems on cones. We also do not find the uniqueness results on positive solutions in [19]. In this paper, we will show the existence and uniqueness of positive solutions for problem (1). Our mail tool is a fixed-point theorem of generalized concave operators in ordered Banach spaces.

The paper is organized as follows: in Section 2, we will present some useful definitions, preliminaries, and lemmas. The existence and uniqueness results are proved in Section 3. In Section 4, we finish the paper with two examples.

2. Preliminaries

Now we list some conditions for convenience. is increasing in for each .For , there exists such that . are right continuous on , left continuous at , and increasing on , with are the Riemann-Stieltjes integrals of with respect to ; moreover,

is continuous with existing. is decreasing in for each and for .For any , there exists such that In the following, we present some definitions and lemmas.

Definition 1 (see [12]). The Riemann-Liouville fractional integral of order for function is provided that the right side is point-wise defined on .

Definition 2 (see [12]). The Riemann-Liouville fractional derivative of order for function is where , provided that the right side is point-wise defined on .

Definition 3 (see [23]). If , then the Caputo fractional derivatives of order exist almost everywhere on and are represented by

Next we recall some concepts. Let be a real Banach space with norm , and let be the zero element of . is a cone in . For all , the notation means that there exist such that . Clearly, is an equivalence relation. Given , we denote by the set Clearly, is convex and for all

Definition 4. A cone is said to be normal if and only if there exists a constant such that where is called the normality constant of .

Definition 5. An operator is increasing (decreasing) if implies

Lemma 6 (see [19]). Suppose that hold and ; then the integral boundary value problem has a solution where

Lemma 7 (see [19]). Let Then Green’s function has the following properties:(i) (ii)  where

Lemma 8. Let ; then and , where .

We now present a fixed-point theorem of generalized concave operators which will be used in the later proofs.

Theorem 9 (see [27]). Let and be a normal cone. Assume that is increasing and ;for any and , there exists with respect to such that Thenthere are and such that operator equation has a unique solution in .

Remark 10. An operator is said to be generalized concave if satisfies condition .

3. Existence and Uniqueness of Positive Solutions for Problem (1)

In this section, we use Theorem 9 to study problem (1) and we obtain some new results on the existence and uniqueness of positive solutions. This is also the main motivation for the study of (1) in the present work.

Set , . Evidently, is a Banach space with the norm , , the standard cone, and it is normal. Our main result is summarized in the following theorem.

Theorem 11. Assume that hold and . Thenthere exist such that the integral boundary value problem (1) has a unique positive solution in , where

Proof. Define an operator by We know that is a solution of problem (1) if and only if is a fixed point of the operator .
Firstly, we show that is increasing, generalized concave. From , we know that , and from (10) and  (13) we have , so we have for It follows from that is increasing.
Now we prove that is generalized concave. For any and , from we have Thus, we have
Secondly, we prove that Note that , and we have From (10), it is easy to obtain that From (14), we have We know by (16) that is decreasing, so when obtains the minimum, and , so we have and from (17), we have So from (25), (27), and (28), (22) has the following form: After simple calculation, we get , where . So we need to discuss several cases:
When , we have Let ; then From (28) and (29), we obtain that When , we have Let ; then From (28) and (29), we obtain that When , we have Let ; then From (28) and (29), we obtain that Let Then from (31), (33), and (35), we have On the other hand, from (25), (27), and (28), we have Also, we need to discuss the following cases:
When , we have We can see that From (28),  (38), and Lemma 8, we obtain that (ii) When , we have Let , and then . From (28), (38), and Lemma 8, we obtain that (iii) When , we have Let , and then From (28), (38), and Lemma 8, we obtain that Let Then from (40), (42), and (44), we have So, from (37) and (46) we have that is, Finally, an application of Theorem 9 implies that
(i) there exist such that , that is, (ii) operator equation has a unique solution in That is, is the unique positive solution for problem (1) in .

Theorem 12. Assume that hold and for . Then
(i) there exist such that where (ii) the integral boundary value problem (1) has a unique positive solution in , where

Proof. Similar to the proof of Theorem 11, we consider the operator by From , we know that It follows from that is decreasing. For any and from , we have Thus, we have Further, for , we have So we obtain . Consequently, is increasing and for , Let Then and So the operator is generalized concave.
Next we prove that Note that , and we can obtain Because , where , we need to discuss several cases:
(i) When , we have and we can see that From (28) and (29), we obtain that (ii) When , we have Note that ; then From (28) and (29), we obtain that (iii) When , we have Note that ; then From (28) and (29), we obtain that Let Then from (57), (59), and (61), we have Further, we need to discuss the following cases:
(i) When , we have Let , and then From (28), (38), and Lemma 8, we obtain that (ii) When , we have Note that and from (28), (38), and Lemma 8, we obtain that (iii) When , we have Note that and from (28), (38), and Lemma 8, we obtain that Let Then from (65), (67), and (69), we have So, from (63) and (71) we have Hence, we have Because , where , we need to discuss the following cases:
(i) When , we have and we can see that From (28) and (73), we obtain that (ii) When , we have Let ; then From (28) and (73), we obtain that (iii) When , we have Let ; then From (28) and (73), we obtain that Let Then from (76), (78), and (80), we have Similarly, we need to discuss the following cases:
(i) When , we have Let , and then From (28), (74), and Lemma 8, we obtain that (ii) When , we have Let , and then From (28), (74), and Lemma 8, we obtain that (iii) When , we have Note that and from (28), (74), and Lemma 8, we obtain that Let Then from (84), (86), and (88), we have From (82) and (90), we can get that is, Finally, an application of Theorem 9 implies the following: (i) there exist such that ; (ii) has a unique solution in . Let , and then That is, Moreover, Next, we show that is the unique fixed point of in In view of , by the uniqueness of solutions for the operator equation , we have that Suppose that is another fixed point of in , so we have Hence, we have So problem (1) has a unique positive solution in

4. Examples

Example 1. Consider the following fractional boundary value problem of form (1):

Conclusion 1. Problem (93) has a unique positive solution in , where

Proof. From (93), we know , and it is not difficult to find the following:(1) and is continuous, increasing in .(2)Let , then , and we have (3)From (93), we know  so we can get (4) exists.So we claim that the conditions hold. Hence, by Theorem 11, we can obtain that (93) has a unique positive solution in , where

Example 2. Consider the following fractional boundary value problem of form (1):

Conclusion 2. Problem (98) has a unique positive solution in , where

Proof. From (98), we know , and it is not difficult to find the following:(1) and is decreasing in .(2)Let , and then , so we have (3)From (98), we know  so we can get (4) exists.So the conditions hold. Hence, by Theorem 12, we can obtain that (98) has a unique positive solution in , where

Conflicts of Interest

The authors declare that they have no conflicts of interest regarding the publication of this paper.

Authors’ Contributions

The authors declare that they share equal responsibility. All authors read and approved the final manuscript.

Acknowledgments

This paper was supported financially by the Youth Science Foundation of China (11201272), Shanxi Province Science Foundation (2015011005), and the Youth Science Foundation of Shanxi University of Finance and Economic (2014026).