Analysis of Fractional Dynamic SystemsView this Special Issue
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Monotonicity, Concavity, and Convexity of Fractional Derivative of Functions
The monotonicity of the solutions of a class of nonlinear fractional differential equations is studied first, and the existing results were extended. Then we discuss monotonicity, concavity, and convexity of fractional derivative of some functions and derive corresponding criteria. Several examples are provided to illustrate the applications of our results.
1. Introduction and Preliminaries
Fractional calculus is a generalization of the traditional integer order calculus. Recently, fractional differential equations have received increasing attention since behavior of many physical systems can be properly described as fractional differential systems. Most of the present works focused on the existence, uniqueness, and stability of solutions for fractional differential equations, controllability and observability for fractional differential systems, numerical methods for fractional dynamical systems, and so on see the monographs [1–4] and the papers [5–24]. However, there existed a flaw in paper , which has been stated in paper . The main reason that the flaw arose is that one is unknown of monotonicity, concavity, and convexity of fractional derivative of a function.
It is well known that the monotonicity, the concavity, and the convexity of a function play an important role in studying the sensitivity analysis for variational inequalities, variational inclusions, and complementarity. Since fractional derivative of a function is usually not an elementary function, its properties are more complicated than those of integer order derivative of the function. The focal point of this paper is to investigate the monotonicity, the concavity, and the convexity of fractional derivative of some functions.
Definition 1. Given an interval of , the fractional order integral of a function of order is defined by where is the Gamma function.
Definition 2. Riemann-Liouville’s derivative of order with the lower limit for a function can be written as
Definition 3. Suppose that a function is defined on the interval and . The Caputo’s fractional derivative of order with lower limit for is defined as
Particularly, when , it holds that
Lemma 4. There exists a link between Riemann-Liouville and Caputo’s fractional derivative of order . Namely,
where denotes the real parts of .
Particularly, for , it holds that
Definition 5. A function with is said to be convex if whenever , , and , the inequality holds.
The rest of this paper is organized as follows. Section 2 is devoted to monotonicity of solutions of fractional differential equations. In Section 3, we present the monotonicity, the concavity, and the convexity of functions and . Summarizing this paper forms the content of Section 4.
2. Monotonicity of Solutions of Nonlinear Fractional Differential Equations
In this section, we mainly investigate the monotonicity of the solution of nonlinear fractional differential equation with Caputo’s derivative which was discussed in [6, 21], where . The paper  gave two examples to show that Lemma 1.7.3 in  is invalid. Lemma 1.7.3 in  is as follows.
In , the authors gave an improvement of Lemma , which is as follows.
Theorem 6. Assume that . Assume that the solutions of (8) exist.(1)If on and , then the solutions of (8) are nonnegative on .(2)If and on , then the solution of (8) is nondecreasing on .(3)If and on , then the solution of (8) is not increasing on .
Proof. The conclusion of (1) is obvious. In fact, (8) is equivalent to
Since , it holds that . Noting , we have .
Now we prove the validity of (2) and (3). First, by the definition of the Caputo’s derivative, it holds from (8) that Then it follows that That is, Then we can get that Since on and , thus on . Hence is nondecreasing on , and (2) holds.
Similar to the proof of (2), we can prove that (3) holds. This completes the proof.
Remark 7. Lemma 2.4 in  is a particular case of Theorem 6 of this paper. In fact in Lemma 2.4 in , if , then (8) is . The solution of is If , then which satisfies the conditions of (2) in Theorem 6.
Example 8. Assume that . Consider the fractional differential equation For , we have and . By Theorem 6, we see that is nondecreasing in for .
Example 9. Assume that . Consider the fractional differential equation Denote , then and for . By Theorem 6, it follows that is not increasing. In fact, by computation we get on , thus is decreasing.
The following fractional comparison principle is an improvement of Lemma 6.1 in  and Theorem 2.6 in . The method we used here is different from the one used to prove Lemma 6.1 in  and the one used to prove Theorem 2.6 in .
Theorem 10. Suppose that and on interval . Suppose further that , then on .
Proof. Set , . Then Taking on both sides of (18) yields That is, Since , thus . Then we have Hence on , and the proof is completed.
Remark 11. The method used to prove Theorem 2.6 in  and to prove Lemma 6.1 in  is the Laplace transform, which demands . Theorem 2.6 in  and Lemma 6.1 in  are as follows, respectively.
Theorem 2.6 in  suppose that and on . If , then on .
Lemma 6.1 in  let and , where . Then .
3. Monotonicity, Concavity, and Convexity of the Functions and
In this section, we first investigate the monotonicity of the functions and .
Theorem 12. Assume that . If there exists an interval such that (1), , and on , then is nondecreasing on ; (2), , and on , then is not increasing on ; (3), , and (i.e., is continuous on and ), then there exists a constant such that is not increasing on and is not decreasing on ; (4), , and , then there exists a constant such that is nondecreasing on and is not increasing on .
Proof. Using formula (6), we have
Then we can get that
By assumptions in (1), it follows that on . Thus is nondecreasing on . By assumptions in (2), it follows that is not increasing in on . Consequently, the conclusions of (1) and (2) are true.
Let us prove (3). Noting formula (23), Since and , then as . By the fact that , we have Thus there exists a constant such that on . On the other hand, when , Thus there exists a constant such that on and on . Therefore, the conclusion of (3) is valid.
The proof of (4) is similar to that of (3). This completes the proof.
Now we are to investigate the monotonicity of the function .
Theorem 13. Assume that . If there exists an interval such that on and , then is nondecreasing on . If on and , then is not increasing on .
Proof. Set . Note that If on and , then in on . Hence, is nondecreasing on interval . If and , then . Hence, is not increasing on . The proof is completed.
Example 14. Assume that . Consider , where , for all . Since and , by Theorem 12, there exists a constant such that is decreasing on and is increasing on .
Example 15. Assume that . Consider for . Since for and , by Theorem 13, is decreasing on . By similar argument, is increasing on . Since = , thus is decreasing on and is increasing on .
Example 16. Assume that . Consider ; here and . Obviously, and . For , and . By Theorem 13, is not increasing on .
Next we are to investigate the concavity and the convexity of and . By formula (23), we have Thus we can obtain the following theorem.
Theorem 17. Assume that . If there exists an interval such that on , , and , then is concave on . If on , and and , then is convex on .
The next theorem is on the convexity and the concavity of .
Theorem 18. Assume that . If there exists an interval such that on , and , then is concave on . If on , and , then is convex on .
Proof. By formula (28), we have If on , , and , then on . Hence, is concave in on . If , on , , and , then on . Therefore is convex in on .
Example 19. Assume that . Consider the fractional differential equation ; here and . Obviously, , and . For all , it holds that , , and on . Then by Theorem 18, is convex on .
Example 20. Consider the concavity and convexity of the function , where , . Obviously, Theorems 17 and 18 are useless to the function . Now we employ the method which is used in the proof of Theorem 17 to investigate it. By formula (29), we have The three terms in the right side of (31) can be reduced to It is not difficult to get for and for . Thus on . Consequently, is concave on . Since as , and is increasing on , thus there exists a constant such that on and on . Hence is convex on and is concave on .
In this paper, we first investigate the monotonicity of solutions of nonlinear fractional differential equations with the Caputo’s derivative. The results we derive are an improvement of the existing results. Meanwhile, several examples are provided to illustrate the applicability of our results.
The main part of this paper is to study the monotonicity, the concavity, and the convexity of the functions and . Based on the relation between the Riemann-Liouville fractional derivative and the Caputo’s derivative, we obtain the criteria on the monotonicity, the concavity, and the convexity of the functions and . In the meantime, five examples are given to illustrate the applications of our criteria.
This paper was supported by the Natural Science Foundation of China (11371027, 11071001, and 11201248), Program of Natural Science Research in Anhui Universities (KJ2011A020, KJ2013A032), the Research Fund for Doctoral Program of Higher Education of China (20123401120001), Anhui Provincial Natural Science Foundation (1208085MA13), Scientific Research Starting Fund for Dr. of Anhui University (023033190001, and 023033190181), and the 211 Project of Anhui University (KJQN1001, 023033050055). The authors would like to thank the editors and the reviewers for their valuable comments and suggestions, which helped to improve the quality of this paper.
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