Monotonicity, Concavity, and Convexity of Fractional Derivative of Functions

Abstract

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 [6], which has been stated in paper [21]. 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.

Now we recall some definitions and lemmas which will be used later. For more detail, see [1–4].

Definition 1. Given an interval [𝑎,𝑏] of ℝ, the fractional order integral of a function 𝑓∈𝐿¹[𝑎,𝑏] of order 𝛼∈ℝ⁺ is defined by 𝐼^𝛼_𝑎𝑓(𝑡)=1Γ(𝛼)^𝑡_𝑎(𝑡−𝑠)^𝛼−1𝑓(𝑠)[]𝑑𝑠,𝑡∈𝑎,𝑏,𝛼>0, (1) where Γ is the Gamma function.

Definition 2. Riemann-Liouville’s derivative of order 𝛼 with the lower limit 𝑎 for a function 𝑓∈𝐿¹[𝑎,𝑏] can be written as ^RL𝐷^𝛼_𝑎𝑓(𝑡)=1Γ()𝑑𝑛−𝛼𝑑𝑡^𝑛𝑛𝑡_𝑎(𝑡−𝑠)^{𝑛−𝛼−1}𝑓(𝑠)[]𝑑𝑠,𝑡∈𝑎,𝑏,0<𝑛−1≤𝛼<𝑛. (2)

Definition 3. Suppose that a function 𝑓 is defined on the interval [𝑎,𝑏] and 𝑓^(𝑛)(𝑡)∈𝐿¹[𝑎,𝑏]. The Caputo’s fractional derivative of order 𝛼 with lower limit 𝑎 for 𝑓 is defined as ^𝐶𝐷^𝛼_𝑎𝑓(𝑡)=1Γ(𝑛−𝛼)^𝑡_𝑎(𝑡−𝑠)^{𝑛−𝛼−1}𝑓^(𝑛)(𝑠)𝑑𝑠=𝐼_𝑎^𝑛−𝛼𝑓^(𝑛)(𝑡),𝑡∈[𝑎,𝑏], (3) where 0<𝑛−1<𝛼≤𝑛.
Particularly, when 0<𝛼≤1, it holds that ^𝐶𝐷^𝛼_𝑎𝑓(𝑡)=1Γ(1−𝛼)^𝑡_𝑎(𝑡−𝑠)^−𝛼𝑓̇(𝑠)𝑑𝑠=𝐼_𝑎^1−𝛼𝑓̇(𝑡),𝑡∈[𝑎,𝑏]. (4)

Lemma 4. There exists a link between Riemann-Liouville and Caputo’s fractional derivative of order 𝛼. Namely, ^𝐶𝐷^𝛼_𝑎𝑓(𝑡)=^𝑅𝐿𝐷^𝛼_𝑎𝑓(𝑡)−^𝑛−1_𝑘=0𝑓^(𝑘)(𝑎)Γ()()𝑘−𝛼+1𝑡−𝑎^𝑘−𝛼,𝑡>𝑎,𝑛−1<Re(𝛼)<𝑛, (5) where Re(𝛼) denotes the real parts of 𝛼.
Particularly, for 0<𝛼<1, it holds that ^𝐶𝐷^𝛼_𝑎𝑓(𝑡)=1Γ(1−𝛼)^𝑡_𝑎(𝑡−𝑠)^−𝛼𝑓^′(𝑠)=𝑑𝑠_𝑅𝐿𝐷^𝛼_𝑎𝑓(𝑡)−𝑓(𝑎)Γ()()1−𝛼𝑡−𝑎^−𝛼,𝑡>𝑎. (6)

Definition 5. A function 𝑓∶[𝑎,𝑏]→ℝ with [𝑎,𝑏]⊂ℝ is said to be convex if whenever 𝑡₁∈[𝑎,𝑏], 𝑡₂∈[𝑎,𝑏], and 𝜃∈[0,1], the inequality 𝑓𝜃𝑡₁+(1−𝜃)𝑡₂≤𝜃𝑓𝑡₁+(1−𝜃)𝑓𝑡₂ (7) 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 ^RL𝐷^𝛼_𝑡0𝑓(𝑡) and ^𝐶𝐷^𝛼_𝑡0𝑓(𝑡). 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 ^𝐶𝐷^𝛼_𝑡0𝑢(𝑡)=𝑔(𝑡,𝑢(𝑡)),𝑡≥𝑡₀, (8) which was discussed in [6, 21], where 0<𝛼<1. The paper [21] gave two examples to show that Lemma  1.7.3 in [6] is invalid. Lemma  1.7.3 in [6] is as follows.

Lemma  1.7.3 in [6] consider (8), where 0<𝛼<1 and 𝑔(𝑡,𝑢)≥0. Then, if the solutions exist, they are nondecreasing in 𝑡.

In [21], the authors gave an improvement of Lemma 1.7.3, which is as follows.

Lemma  2.4 in [21] consider (8). Suppose that 0<𝛼<1, 𝑔(𝑡,𝑢)≥0, and 𝑡₀∈ℝ. If the solutions exists and 𝑢(𝑡₀)≥0, then they are nonnegative. Furthermore, If 𝑔(𝑡,𝑢)=𝜆𝑢 for 𝜆>0, then the solutions are nondecreasing in 𝑡.

Now we will give a more general result for (8), which is an improvement of Lemma  2.4 in [21].

Theorem 6. Assume that 0<𝛼<1. Assume that the solutions of (8) exist.(1)If 𝑔(𝑡,𝑢)≥0 on [𝑡₀,𝑡₁] and 𝑢(𝑡₀)≥0, then the solutions 𝑢(𝑡) of (8) are nonnegative on [𝑡₀,𝑡₁].(2)If 𝑔(𝑡₀,𝑢(𝑡₀))≥0 and (𝑑/𝑑𝑡)𝑔(𝑡,𝑢(𝑡))≥0 on [𝑡₀,𝑡₁], then the solution 𝑢(𝑡) of (8) is nondecreasing on [𝑡₀,𝑡₁].(3)If 𝑔(𝑡₀,𝑢(𝑡₀))≤0 and (𝑑/𝑑𝑡)𝑔(𝑡,𝑢(𝑡))≤0 on [𝑡₀,𝑡₁], then the solution 𝑢(𝑡) of (8) is not increasing on [𝑡₀,𝑡₁].

Proof. The conclusion of (1) is obvious. In fact, (8) is equivalent to 𝑢(𝑡)=𝑢𝑡₀+1Γ(𝛼)^𝑡_𝑡0(𝑡−𝑠)^𝛼−1𝑔(𝑠,𝑢(𝑠))𝑑𝑠. (9) Since 𝑔(𝑡,𝑢(𝑡))≥0, it holds that (1/Γ(𝛼))∫^𝑡_𝑡0(𝑡−𝑠)^𝛼−1𝑔(𝑠,𝑢(𝑠))𝑑𝑠≥0. Noting 𝑢(𝑡₀)≥0, we have 𝑢(𝑡)≥0.
Now we prove the validity of (2) and (3). First, by the definition of the Caputo’s derivative, it holds from (8) that 𝐼_𝑡^1−𝛼₀(𝑢̇(𝑡))=𝑔(𝑡,𝑢(𝑡)). (10) Then it follows that 𝐼^𝛼_𝑡0𝐼_𝑡^1−𝛼₀𝑢̇(𝑡)=𝐼^𝛼_𝑡0𝑔(𝑡,𝑢(𝑡)). (11) That is, 𝐼¹_𝑡0(𝑢̇(𝑡))=𝐼^𝛼_𝑡0𝑔(𝑡,𝑢(𝑡)). (12) Then we can get that 𝑢̇(𝑡)=𝑑𝐼𝑑𝑡^𝛼𝑡0𝑔((𝑡=𝑡,𝑢))_RL𝐷_𝑡^1−𝛼₀𝑔((𝑡=𝑡,𝑢))_𝐶𝐷_𝑡^1−𝛼₀𝑔((𝑡+𝑔𝑡𝑡,𝑢))⁰,𝑢𝑡₀Γ(1−𝛼)𝑡−𝑡₀^−𝛼=𝐼^𝛼_𝑡0𝑑𝑔((𝑡+𝑔𝑡𝑑𝑡𝑡,𝑢))⁰,𝑢𝑡₀Γ(1−𝛼)𝑡−𝑡₀^−𝛼,𝑡∈𝑡₀,𝑡₁. (13) Since (𝑑/𝑑𝑡)𝑔(𝑡,𝑢(𝑡))≥0 on [𝑡₀,𝑡₁] and 𝑔(𝑡₀,𝑢(𝑡₀))≥0, thus 𝑢̇(𝑡)≥0 on [𝑡₀,𝑡₁]. Hence 𝑢(𝑡) is nondecreasing on [𝑡₀,𝑡₁], 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 [21] is a particular case of Theorem 6 of this paper. In fact in Lemma  2.4 in [21], if 𝑔(𝑡,𝑢)=𝜆𝑢, then (8) is ^𝐶𝐷^𝛼_𝑡0𝑢(𝑡)=𝜆𝑢. The solution 𝑢(𝑡) of ^𝐶𝐷^𝛼_𝑡0𝑢(𝑡)=𝜆𝑢 is 𝑢(𝑡)=𝑢𝑡₀𝐸_𝛼,1𝜆𝑡−𝑡₀^𝛼,𝑡≥𝑡₀. (14) If 𝑢(𝑡₀)≥0, then 𝑔(𝑡,𝑢)=𝜆𝑢=𝜆𝑢𝑡₀𝐸_𝛼,1𝜆𝑡−𝑡₀^𝛼≥0,𝑡≥𝑡₀,𝑑𝑔()𝑑𝑡𝑡,𝑢=𝜆²𝑢𝑡₀𝑡−𝑡₀^𝛼−1𝐸_𝛼,𝛼𝜆𝑡−𝑡₀^𝛼≥0, (15) which satisfies the conditions of (2) in Theorem 6.

Example 8. Assume that 0<𝛼<1. Consider the fractional differential equation ^𝐶𝐷^𝛼₀𝑢(𝑡)=𝑡+sin𝑡,𝑡≥0. (16) For 𝑡>0, we have 𝑡+sin𝑡≥0 and (𝑡+sin𝑡)^′=1−cos𝑡≥0. By Theorem 6, we see that 𝑢(𝑡) is nondecreasing in 𝑡 for 𝑡≥0.

Example 9. Assume that 0<𝛼<1. Consider the fractional differential equation ^𝐶𝐷^𝛼_𝑡0𝑓(𝑡)=−2,𝑡≥𝑡₀. (17) Denote 𝑔(𝑡,𝑓(𝑡))=−2, then 𝑔(𝑡₀,𝑓(𝑡₀))<0 and 𝑔(̇𝑡,𝑓(𝑡))=0 for 𝑡≥𝑡₀. By Theorem 6, it follows that 𝑓(𝑡) is not increasing. In fact, by computation we get 𝑓(̇𝑡)=(−2𝛼/Γ(1+𝛼))𝑡^𝛼−1<0 on [𝑡₀,∞), thus 𝑓(𝑡) is decreasing.

The following fractional comparison principle is an improvement of Lemma  6.1 in [20] and Theorem  2.6 in [21]. The method we used here is different from the one used to prove Lemma  6.1 in [20] and the one used to prove Theorem  2.6 in [21].

Theorem 10. Suppose that 0<𝛼<1 and ^𝐶𝐷^𝛼_𝑡0𝑓(𝑡)≥^𝐶𝐷^𝛼_𝑡0𝑔(𝑡) on interval [𝑡₀,𝑡₁]. Suppose further that 𝑓(𝑡₀)≥𝑔(𝑡₀), then 𝑓(𝑡)≥𝑔(𝑡) on [𝑡₀,𝑡₁].

Proof. Set ^𝐶𝐷^𝛼_𝑡0𝑓(𝑡)−^𝐶𝐷^𝛼_𝑡0𝑔(𝑡)=𝑚(𝑡), 𝑡∈[𝑡₀,𝑡₁]. Then ^𝐶𝐷^𝛼_𝑡0𝑓(𝑡)−𝑔(𝑡)=𝑚(𝑡)≥0,𝑡∈𝑡₀,𝑡₁. (18) Taking 𝐼^𝛼_𝑡0 on both sides of (18) yields 𝐼^𝛼_𝑡0^𝐶𝐷^𝛼_𝑡0𝑓(𝑡)−𝑔(𝑡)=𝐼^𝛼_𝑡0(𝑚(𝑡)). (19) That is, 𝑓(𝑡)−𝑔(𝑡)=𝑓𝑡₀−𝑔𝑡₀+𝐼^𝛼_𝑡0(𝑚(𝑡)). (20) Since 𝑚(𝑡)≥0, thus 𝐼^𝛼_𝑡0(𝑚(𝑡))≥0. Then we have 𝑓(𝑡)−𝑔(𝑡)≥𝑓𝑡₀−𝑔𝑡₀≥0,𝑡∈𝑡₀,𝑡₁. (21) Hence 𝑓(𝑡)≥𝑔(𝑡) on [𝑡₀,𝑡₁], and the proof is completed.

Remark 11. The method used to prove Theorem  2.6 in [21] and to prove Lemma  6.1 in [20] is the Laplace transform, which demands 𝑡∈[0,∞). Theorem  2.6 in [21] and Lemma  6.1 in [20] are as follows, respectively.
Theorem  2.6 in [21] suppose that 0<𝛼<1 and ^𝐶𝐷^𝛼₀𝑣(𝑡)≥^𝐶𝐷^𝛼₀𝑤(𝑡) on ℝ₊. If 𝑣(0)≥𝑤(0), then 𝑣(𝑡)≥𝑤(𝑡) on ℝ₊.
Lemma  6.1 in [20] let ^𝐶𝐷^𝛽₀𝑥(𝑡)≥^𝐶𝐷^𝛽₀𝑦(𝑡) and 𝑥(0)=𝑦(0), where 𝛽∈(0,1). Then 𝑥(𝑡)≥𝑦(𝑡).

3. Monotonicity, Concavity, and Convexity of the Functions ^RL𝐷^𝛼_𝑡0𝑓(𝑡) and ^𝐶𝐷^𝛼_𝑡0𝑓(𝑡)

In this section, we first investigate the monotonicity of the functions ^RL𝐷^𝛼_𝑡0𝑓(𝑡) and ^𝐶𝐷^𝛼_𝑡0𝑓(𝑡).

Theorem 12. Assume that 0<𝛼<1. If there exists an interval [𝑡₀,𝑡₁] such that (1)𝑓(𝑡₀)≤0, 𝑓(̇𝑡₀)≥0, and 𝑓(̈𝑡)≥0 on [𝑡₀,𝑡₁], then ^𝑅𝐿𝐷^𝛼_𝑡0𝑓(𝑡) is nondecreasing on [𝑡₀,𝑡₁]; (2)𝑓(𝑡₀)≥0, 𝑓(̇𝑡₀)≤0, and 𝑓(̈𝑡)≤0 on [𝑡₀,𝑡₁], then ^𝑅𝐿𝐷^𝛼_𝑡0𝑓(𝑡) is not increasing on [𝑡₀,𝑡₁]; (3)𝑓(𝑡₀)>0, 𝑓(̇𝑡₀)>0, and 𝑓(̈𝑡)∈𝐶([𝑡₀,𝑡₁],(0,+∞)) (i.e., 𝑓(̈𝑡) is continuous on [𝑡₀,𝑡₁] and 𝑓(̈𝑡)>0), then there exists a constant 𝛽∈[𝑡₀,𝑡₁] such that ^𝑅𝐿𝐷^𝛼_𝑡0𝑓(𝑡) is not increasing on [𝑡₀,𝛽] and is not decreasing on [𝛽,𝑡₁]; (4)𝑓(𝑡₀)<0, 𝑓(̇𝑡₀)<0, and 𝑓(̈𝑡)∈𝐶([𝑡₀,𝑡₁],(−∞,0)), then there exists a constant 𝜂∈[𝑡₀,𝑡₁] such that ^𝑅𝐿𝐷^𝛼_𝑡0𝑓(𝑡) is nondecreasing on [𝑡₀,𝜂] and ^𝑅𝐿𝐷^𝛼_𝑡0𝑓(𝑡) is not increasing on [𝜂,𝑡₁].

Proof. Using formula (6), we have ^RL𝐷^𝛼_𝑡0𝑓(𝑡)=^𝐶𝐷^𝛼_𝑡0𝑓(𝑡)+𝑓𝑡₀Γ(1−𝛼)𝑡−𝑡₀^−𝛼=1Γ(1−𝛼)^𝑡_𝑡0(𝑡−𝑠)^−𝛼𝑓̇(𝑠)+𝑓𝑡𝑑𝑠₀Γ(1−𝛼)𝑡−𝑡₀^−𝛼. (22) Then we can get that 𝑑𝑑𝑡^RL𝐷^𝛼_𝑡0𝑓(𝑡)=𝑑𝑑𝑡^𝐶𝐷^𝛼_𝑡0𝑓(𝑡)−𝛼𝑓𝑡₀Γ(1−𝛼)𝑡−𝑡₀^−𝛼−1=𝑑𝐼𝑑𝑡^{𝑡1−𝛼0}𝑓̇(𝑡)−𝛼𝑓𝑡₀Γ(1−𝛼)𝑡−𝑡₀^−𝛼−1=^RL𝐷^𝛼_𝑡0𝑓̇(𝑡)−𝛼𝑓𝑡₀Γ(1−𝛼)𝑡−𝑡₀^−𝛼−1=^𝐶𝐷^𝛼_𝑡0𝑓̇(𝑡)+𝑓̇𝑡₀Γ(1−𝛼)𝑡−𝑡₀^−𝛼−𝛼𝑓𝑡₀Γ(1−𝛼)𝑡−𝑡₀^−𝛼−1=1Γ(1−𝛼)^𝑡_𝑡0(𝑡−𝑠)^−𝛼𝑓̈(𝑠)𝑓̇𝑡𝑑𝑠+⁰Γ(1−𝛼)𝑡−𝑡₀^−𝛼−𝛼𝑓𝑡₀Γ(1−𝛼)𝑡−𝑡₀^−𝛼−1. (23) By assumptions in (1), it follows that (𝑑/𝑑𝑡)(^RL𝐷^𝛼_𝑡0𝑓(𝑡))≥0 on [𝑡₀,𝑡₁]. Thus ^RL𝐷^𝛼_𝑡0𝑓(𝑡) is nondecreasing on [𝑡₀,𝑡₁]. By assumptions in (2), it follows that ^RL𝐷^𝛼_𝑡0𝑓(𝑡) is not increasing in 𝑡 on [𝑡₀,𝑡₁]. Consequently, the conclusions of (1) and (2) are true.
Let us prove (3). Noting formula (23), 𝑑𝑑𝑡^RL𝐷^𝛼_𝑡0𝑓(𝑡)=1Γ(1−𝛼)^𝑡_𝑡0(𝑡−𝑠)^−𝛼𝑓̈(𝑠)𝑓̇𝑡𝑑𝑠+⁰Γ(1−𝛼)𝑡−𝑡₀^−𝛼−𝛼𝑓𝑡₀Γ(1−𝛼)𝑡−𝑡₀^−𝛼−1. (24) Since 𝑓(𝑡₀)>0 and 𝑓(̇𝑡₀)>0, then 𝑓̇𝑡₀Γ(1−𝛼)𝑡−𝑡₀^−𝛼−𝛼𝑓𝑡₀Γ(1−𝛼)𝑡−𝑡₀^−𝛼−1⟶−∞ (25) as 𝑡→𝑡₀. By the fact that 𝑓(̈𝑡)∈𝐶([𝑡₀,𝑡₁],(0,+∞)), we have 1Γ(1−𝛼)^𝑡_𝑡0(𝑡−𝑠)^−𝛼𝑓̈(𝑠)𝑑𝑠⟶0,𝑡⟶𝑡₀. (26) Thus there exists a constant 𝛿₁>0 such that (𝑑/𝑑𝑡)(^RL𝐷^𝛼_𝑡0𝑓(𝑡))≤0 on [𝑡₀,𝑡₀+𝛿₁]. On the other hand, when 𝑡≥𝑡₀+𝛼𝑓(𝑡₀)/𝑓(̇𝑡₀), 𝑓̇𝑡₀Γ(1−𝛼)𝑡−𝑡₀^−𝛼−𝛼𝑓𝑡₀Γ(1−𝛼)𝑡−𝑡₀^−𝛼−1≥0. (27) Thus there exists a constant 𝛽∈[𝑡₀,𝑡₁] such that (𝑑/𝑑𝑡)(^RL𝐷^𝛼_𝑡0𝑓(𝑡))≤0 on [𝑡₀,𝛽] and (𝑑/𝑑𝑡)(^RL𝐷^𝛼_𝑡0𝑓(𝑡))≥0 on [𝛽,𝑡₁]. 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 ^𝐶𝐷^𝛼_𝑡0𝑓(𝑡).

Theorem 13. Assume that 0<𝛼<1. If there exists an interval [𝑡₀,𝑡₁] such that 𝑓(̈𝑡)≥0 on [𝑡₀,𝑡₁] and 𝑓(̇𝑡₀)≥0, then ^𝐶𝐷^𝛼_𝑡0𝑓(𝑡) is nondecreasing on [𝑡₀,𝑡₁]. If 𝑓(̈𝑡)≤0 on [𝑡₀,𝑡₁] and 𝑓(̇𝑡₀)≤0, then ^𝐶𝐷^𝛼_𝑡0𝑓(𝑡) is not increasing on [𝑡₀,𝑡₁].

Proof. Set 𝑓(̇𝑡)=𝑔(𝑡). Note that 𝑑𝑑𝑡^𝐶𝐷^𝛼_𝑡0𝑓(𝑡)=𝑑𝐼𝑑𝑡^{𝑡1−𝛼0}𝑓̇(𝑡)=𝑑𝐼𝑑𝑡^{𝑡1−𝛼0}𝑔(𝑡)=^RL𝐷^𝛼_𝑡0𝑔(𝑡)=^𝐶𝐷^𝛼_𝑡0𝑔(𝑡)+𝑔𝑡₀Γ(1−𝛼)𝑡−𝑡₀^−𝛼=1Γ(1−𝛼)^𝑡_𝑡0(𝑡−𝑢)^−𝛼𝑔̇(𝑢)+𝑔𝑡𝑑𝑢₀Γ(1−𝛼)𝑡−𝑡₀^−𝛼. (28) If 𝑔(̇𝑡)=𝑓(̈𝑡)≥0 on [𝑡₀,𝑡₁] and 𝑔(𝑡₀)=𝑓(̇𝑡₀)≥0, then (𝑑/𝑑𝑡)(^𝐶𝐷^𝛼_𝑡0𝑓(𝑡))≥0 in 𝑡 on [𝑡₀,𝑡₁]. Hence, ^𝐶𝐷^𝛼_𝑡0𝑓(𝑡) is nondecreasing on interval [𝑡₀,𝑡₁]. If 𝑔(̇𝑡)=𝑓(̈𝑡)≤0 and 𝑔(𝑡₀)=𝑓(̇𝑡₀)≤0, then (𝑑/𝑑𝑡)(^𝐶𝐷^𝛼_𝑡0𝑓(𝑡))≤0. Hence, ^𝐶𝐷^𝛼_𝑡0𝑓(𝑡) is not increasing on [𝑡₀,𝑡₁]. The proof is completed.

The following examples illustrate applications of Theorems 12 and 13.

Example 14. Assume that 0<𝛼<1. Consider ^RL𝐷^𝛼_𝑡0𝑓(𝑡), where 𝑓(𝑡)=𝑒^𝑡, for all 𝑡₀∈ℝ. Since 𝑓(𝑡₀)=𝑓(̇𝑡₀)>0 and 𝑓(̈𝑡)∈𝐶((−∞,+∞),(0,+∞)), by Theorem 12, there exists a constant 𝛽>𝑡₀ such that ^RL𝐷^𝛼_𝑡0(𝑒^𝑡) is decreasing on [𝑡₀,𝛽] and is increasing on [𝛽,+∞).

Example 15. Assume that 0<𝛼<1. Consider ^𝐶𝐷^𝛼_0.5𝜋sin𝑡 for 𝑡∈[𝜋/2,𝜋]. Since (sin𝑡)^′′≤0 for 𝑡∈[𝜋/2,𝜋] and (sin𝑡)^′|_𝑡=𝜋/2=0, by Theorem 13, ^𝐶𝐷^𝛼_0.5𝜋sin𝑡 is decreasing on [𝜋/2,𝜋]. By similar argument, ^𝐶𝐷^𝛼_1.5𝜋sin𝑡 is increasing on 𝑡∈[3𝜋/2,2𝜋]. Since ^𝐶𝐷^𝛼_𝑡0sin𝑡 = (1/Γ(1−𝛼))∫^𝑡_𝑡0(𝑡−𝜏)^−𝛼cos𝜏𝑑, thus (1/Γ(1−𝛼))∫^𝑡_0.5𝜋(𝑡−𝜏)^−𝛼cos𝜏𝑑𝜏 is decreasing on [𝜋/2,𝜋] and (1/Γ(1−𝛼))∫^𝑡_1.5𝜋(𝑡−𝜏)^−𝛼cos𝜏𝑑𝜏 is increasing on 𝑡∈[3𝜋/2,2𝜋].

Example 16. Assume that 0<𝛼<1. Consider ^𝐶𝐷^𝛼_𝑡0𝑓(𝑡); here 𝑡₀=1 and 𝑓(𝑡)=𝑡−𝑡². Obviously, 𝑓(̇𝑡)=1−2𝑡 and 𝑓(̈𝑡)=−2. For 𝑡∈[1,∞], 𝑓(̇1)=1−2<0 and 𝑓(̈𝑡)=−2<0. By Theorem 13, ^𝐶𝐷^𝛼_𝑡0𝑓(𝑡) is not increasing on [1,∞].

Next we are to investigate the concavity and the convexity of ^RL𝐷^𝛼_𝑡0𝑓(𝑡) and ^𝐶𝐷^𝛼_𝑡0𝑓(𝑡). By formula (23), we have 𝑑²𝑑𝑡^2RL𝐷^𝛼_𝑡0𝑓(𝑡)=𝑑1𝑑𝑡Γ(1−𝛼)^𝑡_𝑡0(𝑡−𝑠)^−𝛼𝑓̈(𝑠)+1𝑑𝑠Γ(1−𝛼)𝑡−𝑡₀^−𝛼𝑓̇𝑡₀1−𝛼Γ(1−𝛼)𝑡−𝑡₀^−𝛼−1𝑓𝑡₀=1Γ(1−𝛼)^𝑡_𝑡0(𝑡−𝑠)^−𝛼𝑓^′′′(𝑠)+1𝑑𝑠Γ(1−𝛼)𝑡−𝑡₀^−𝛼𝑓̈𝑡₀1−𝛼Γ(1−𝛼)𝑡−𝑡₀^−𝛼−1𝑓̇𝑡₀()1+𝛼𝛼+1Γ(1−𝛼)𝑡−𝑡₀^−𝛼−2𝑓𝑡₀. (29) Thus we can obtain the following theorem.

Theorem 17. Assume that 0<𝛼<1. If there exists an interval [𝑡₀,𝑡₁] such that 𝑓^′′′(𝑡)≥0 on [𝑡₀,𝑡₁], 𝑓(̈𝑡₀)≥0, 𝑓(̇𝑡₀)≤0 and 𝑓(𝑡₀)>0, then ^𝑅𝐿𝐷^𝛼_𝑡0𝑓(𝑡) is concave on [𝑡₀,𝑡₁]. If 𝑓^′′′(𝑡)≤0 on [𝑡₀,𝑡₁], 𝑓(̈𝑡₀)≤0 and 𝑓(̇𝑡₀)≥0 and 𝑓(𝑡₀)≤0, then ^𝑅𝐿𝐷^𝛼_𝑡0𝑓(𝑡) is convex on [𝑡₀,𝑡₁].

The next theorem is on the convexity and the concavity of ^𝐶𝐷^𝛼_𝑡0𝑓(𝑡).

Theorem 18. Assume that 0<𝛼<1. If there exists an interval [𝑡₀,𝑡₁] such that 𝑓^′′′(𝑡)≥0 on [𝑡₀,𝑡₁], 𝑓(̈𝑡₀)≤0 and 𝑓(̇𝑡₀)≥0, then ^𝐶𝐷^𝛼_𝑡0𝑓(𝑡) is concave on [𝑡₀,𝑡₁]. If 𝑓^′′′(𝑡)≤0 on [𝑡₀,𝑡₁], 𝑓(̈𝑡₀)≥0 and 𝑓(̇𝑡₀)≥0, then ^𝐶𝐷^𝛼_𝑡0𝑓(𝑡) is convex on [𝑡₀,𝑡₁].

Proof. By formula (28), we have 𝑑²𝑑𝑡^2𝐶𝐷^𝛼_𝑡0𝑓(𝑡)=𝑑𝑑𝑡^𝐶𝐷^𝛼_𝑡0𝑓̇(𝑡)+𝑓̇𝑡₀Γ(1−𝛼)𝑡−𝑡₀^−𝛼=𝑑𝐼𝑑𝑡^{𝑡1−𝛼0}𝑓̈(𝑡𝑓̇𝑡)−𝛼⁰Γ(1−𝛼)𝑡−𝑡₀^−𝛼−1=^RL𝐷^𝛼_𝑡0𝑓̈(𝑡)𝑓̇𝑡−𝛼⁰Γ(1−𝛼)𝑡−𝑡₀^−𝛼−1=^𝐶𝐷^𝛼_𝑡0𝑓̈(𝑡)+𝑓̈𝑡₀Γ(1−𝛼)𝑡−𝑡₀^−𝛼𝑓̇𝑡−𝛼⁰Γ(1−𝛼)𝑡−𝑡₀^−𝛼−1=1Γ(1−𝛼)^𝑡_𝑡0(𝑡−𝑠)^−𝛼𝑓^′′′(𝑠)𝑓̈𝑡𝑑𝑠+⁰Γ(1−𝛼)𝑡−𝑡₀^−𝛼𝑓̇𝑡−𝛼⁰Γ(1−𝛼)𝑡−𝑡₀^−𝛼−1. (30) If 𝑓^′′′(𝑡)≥0 on [𝑡₀,𝑡₁], 𝑓(̈𝑡₀)≥0, and 𝑓(̇𝑡₀)≤0, then (𝑑²/𝑑𝑡²)(^𝐶𝐷^𝛼_𝑡0𝑓(𝑡))≥0 on [𝑡₀,𝑡₁]. Hence, ^𝐶𝐷^𝛼_𝑡0𝑓(𝑡) is concave in 𝑡 on [𝑡₀,𝑡₁]. If 𝑓^′′′(𝑡)≤0, on [𝑡₀,𝑡₁], 𝑓(̈𝑡₀)≤0, and 𝑓(̇𝑡₀)≥0, then (𝑑²/𝑑𝑡²)(^𝐶𝐷^𝛼_𝑡0𝑓(𝑡))≤0 on [𝑡₀,𝑡₁]. Therefore ^𝐶𝐷^𝛼_𝑡0𝑓(𝑡) is convex in 𝑡 on [𝑡₀,𝑡₁].

Example 19. Assume that 0<𝛼<1. Consider the fractional differential equation ^𝐶𝐷^𝛼_𝑡0𝑓(𝑡); here 𝑡₀<0 and 𝑓(𝑡)=𝑡−𝑡². Obviously, 𝑓(̇𝑡)=1−2𝑡, 𝑓(̈1)=−2 and 𝑓^′′′(𝑡)=0. For all 𝑡₀<0, it holds that 𝑓(̇𝑡₀)>0, 𝑓(̈𝑡₀)<0, and 𝑓^′′′(𝑡)=0 on [𝑡₀,0.5]. Then by Theorem 18, ^𝐶𝐷^𝛼_𝑡0(𝑡−𝑡²) is convex on [𝑡₀,0.5].

Example 20. Consider the concavity and convexity of the function ^RL𝐷^𝛼_𝑡0𝑓(𝑡), where 𝑓(𝑡)=𝑒^𝑡, 𝑡₀∈ℝ. Obviously, Theorems 17 and 18 are useless to the function ^RL𝐷^𝛼_𝑡0𝑒^𝑡. Now we employ the method which is used in the proof of Theorem 17 to investigate it. By formula (29), we have 𝑑²𝑑𝑡^2RL𝐷^𝛼_𝑡0𝑒^𝑡=1Γ(1−𝛼)^𝑡_𝑡0(𝑡−𝑠)^−𝛼𝑒^𝑠1𝑑𝑠+Γ(1−𝛼)𝑡−𝑡₀^−𝛼𝑒^𝑡01−𝛼Γ(1−𝛼)𝑡−𝑡₀^−𝛼−1𝑒^𝑡0()1+𝛼𝛼+1Γ(1−𝛼)𝑡−𝑡₀^−𝛼−2𝑒^𝑡0. (31) The three terms in the right side of (31) can be reduced to 1Γ(1−𝛼)𝑡−𝑡₀^−𝛼𝑒^𝑡01−𝛼Γ(1−𝛼)𝑡−𝑡₀^−𝛼−1𝑒^𝑡0()1+𝛼𝛼+1Γ(1−𝛼)𝑡−𝑡₀^−𝛼−2𝑒^𝑡0=1Γ(1−𝛼)𝑡−𝑡₀^−𝛼−2×𝑒^𝑡0𝑡−𝑡₀²−𝛼𝑡−𝑡₀+𝛼(𝛼−1). (32) It is not difficult to get 𝑡−𝑡₀²−𝛼𝑡−𝑡₀+𝛼(𝛼−1)≥0 (33) for 𝑡∈[𝑡₀+(𝛼+(4𝛼−3𝛼²)^0.5)/2,+∞) and 𝑡−𝑡₀²−𝛼𝑡−𝑡₀+𝛼(𝛼−1)≤0 (34) for 𝑡∈[𝑡₀,𝑡₀+(𝛼+(4𝛼−3𝛼²)^0.5)/2]. Thus (𝑑²/𝑑𝑡²)(^RL𝐷^𝛼_𝑡0𝑒^𝑡)>0 on [𝑡₀+(𝛼+(4𝛼−3𝛼²)^0.5)/2,+∞). Consequently, ^RL𝐷^𝛼_𝑡0𝑒^𝑡 is concave on [𝑡₀+(𝛼+(4𝛼−3𝛼²)^0.5)/2,∞). Since (1/Γ(1−𝛼))∫^𝑡_𝑡0(𝑡−𝑠)^−𝛼𝑒^𝑠𝑑𝑠→0 as 𝑡→𝑡₀, and (1/Γ(1−𝛼))∫^𝑡_𝑡0(𝑡−𝑠)^−𝛼𝑒^𝑠𝑑𝑠 is increasing on [𝑡₀,+∞), thus there exists a constant 𝛽∈(𝑡₀,𝑡₀+(𝛼+(4𝛼−3𝛼²)^0.5)/2) such that (𝑑²/𝑑𝑡²)(^RL𝐷^𝛼_𝑡0𝑒^𝑡)<0 on [𝑡₀,𝛽] and (𝑑²/𝑑𝑡²)(^RL𝐷^𝛼_𝑡0𝑒^𝑡)≥0 on [𝛽,∞). Hence ^RL𝐷^𝛼_𝑡0𝑒^𝑡 is convex on [𝑡₀,𝛽] and is concave on [𝛽,+∞).

4. Conclusions

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 ^RL𝐷^𝛼_𝑡0𝑓(𝑡) and ^𝐶𝐷^𝛼_𝑡0𝑓(𝑡). 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 ^RL𝐷^𝛼_𝑡0𝑓(𝑡) and ^𝐶𝐷^𝛼_𝑡0𝑓(𝑡). In the meantime, five examples are given to illustrate the applications of our criteria.

Acknowledgments

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.