# Monotonicity, Concavity, and Convexity of Fractional Derivative of Functions

Authors: Xian-Feng Zhou; Song Liu; Zhixin Zhang; Wei Jiang
Journal: The Scientific World Journal
 (2013)

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## 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.


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## Body

## 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
(1)Iaαf(t)=1Γ(α)∫at‍(t-s)α-1f(s)ds,pppppppppt∈[a,b],α>0,
where Γ is the Gamma function.Definition 2.
Riemann-Liouville’s derivative of orderα with the lower limit a for a function f∈L1[a,b] can be written as
(2)RLDaαf(t)=1Γ(n-α)dndtn∫at‍(t-s)n-α-1f(s)ds,pppppppt∈[a,b],0<n-1≤α<n.Definition 3.
Suppose that a functionf 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
(3)CDaαf(t)=1Γ(n-α)∫at‍(t-s)n-α-1f(n)(s)ds=Ian-αf(n)(t),t∈[a,b],
where 0<n-1<α≤n.
Particularly, when0<α≤1, it holds that
(4)CDaαf(t)=1Γ(1-α)∫at‍(t-s)-αf˙(s)ds=Ia1-αf˙(t),t∈[a,b].Lemma 4.
There exists a link between Riemann-Liouville and Caputo’s fractional derivative of orderα. Namely,
(5)CDaαf(t)=RLDaαf(t)-∑k=0n-1‍f(k)(a)Γ(k-α+1)(t-a)k-α,ppppppppppppt>a,n-1<Re(α)<n,
where Re(α) denotes the real parts of α.
Particularly, for0<α<1, it holds that
(6)CDaαf(t)=1Γ(1-α)∫at‍(t-s)-αf′(s)ds=DRLaαf(t)-f(a)Γ(1-α)(t-a)-α,t>a.Definition 5.
A functionf:[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
(7)f(θt1+(1-θ)t2)≤θf(t1)+(1-θ)f(t2)
holds.The rest of this paper is organized as follows. Section2 is devoted to monotonicity of solutions of fractional differential equations. In Section 3, we present the monotonicity, the concavity, and the convexity of functions RLDt0αf(t) and CDt0αf(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(8)CDt0αu(t)=g(t,u(t)),t≥t0,
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 that0<α<1. Assume that the solutions of (8) exist.(1)
Ifg(t,u)≥0 on [t0,t1] and u(t0)≥0, then the solutions u(t) of (8) are nonnegative on [t0,t1].(2)
Ifg(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)
Ifg(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
(9)u(t)=u(t0)+1Γ(α)∫t0t‍(t-s)α-1g(s,u(s))ds.
Since g(t,u(t))≥0, it holds that (1/Γ(α))∫t0t‍(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
(10)It01-α(u˙(t))=g(t,u(t)).
Then it follows that
(11)It0α(It01-αu˙(t))=It0α(g(t,u(t))).
That is,
(12)It01(u˙(t))=It0α(g(t,u(t))).
Then we can get that
(13)u˙(t)=ddt(It0αg(t,u(t)))=RLDt01-α(g(t,u(t)))=CDt01-α(g(t,u(t)))+g(t0,u(t0))Γ(1-α)(t-t0)-α=It0α(ddtg(t,u(t)))+g(t0,u(t0))Γ(1-α)(t-t0)-α,ppppppppppppppppppppppppppt∈[t0,t1].
Since (d/dt)g(t,u(t))≥0 on [t0,t1] and g(t0,u(t0))≥0, thus u˙(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 CDt0αu(t)=λu. The solution u(t) of CDt0αu(t)=λu is
(14)u(t)=u(t0)Eα,1(λ(t-t0)α),t≥t0.
If u(t0)≥0, then
(15)g(t,u)=λu=λu(t0)Eα,1(λ(t-t0)α)≥0,pppppppppppppppppppppppt≥t0,ddtg(t,u)=λ2u(t0)(t-t0)α-1Eα,α(λ(t-t0)α)≥0,
which satisfies the conditions of (2) in Theorem 6.Example 8.
Assume that0<α<1. Consider the fractional differential equation
(16)CD0αu(t)=t+sint,t≥0.
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 that0<α<1. Consider the fractional differential equation
(17)CDt0αf(t)=-2,t≥t0.
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))tα-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 that0<α<1 and CDt0αf(t)≥CDt0αg(t) on interval [t0,t1]. Suppose further that f(t0)≥g(t0), then f(t)≥g(t) on [t0,t1].Proof.
SetCDt0αf(t)-CDt0αg(t)=m(t), t∈[t0,t1]. Then
(18)CDt0α(f(t)-g(t))=m(t)≥0,t∈[t0,t1].
Taking It0α on both sides of (18) yields
(19)It0α(CDt0α(f(t)-g(t)))=It0α(m(t)).
That is,
(20)f(t)-g(t)=f(t0)-g(t0)+It0α(m(t)).
Since m(t)≥0, thus It0α(m(t))≥0. Then we have
(21)f(t)-g(t)≥f(t0)-g(t0)≥0,t∈[t0,t1].
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 CD0αv(t)≥CD0αw(t) on ℝ+. If v(0)≥w(0), then v(t)≥w(t) on ℝ+.
Lemma  6.1 in [20] let CD0βx(t)≥CD0βy(t) and x(0)=y(0), where β∈(0,1). Then x(t)≥y(t).


## 3. Monotonicity, Concavity, and Convexity of the FunctionsRLDt0αf(t) and CDt0αf(t)

In this section, we first investigate the monotonicity of the functionsRLDt0αf(t) and CDt0αf(t).Theorem 12.
Assume that0<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 RLDt0αf(t) is nondecreasing on [t0,t1];(2)
f(t0)≥0, f˙(t0)≤0, and f¨(t)≤0 on [t0,t1], then RLDt0αf(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 RLDt0αf(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 RLDt0αf(t) is nondecreasing on [t0,η] and RLDt0αf(t) is not increasing on [η,t1].Proof.
Using formula (6), we have
(22)RLDt0αf(t)=DCt0αf(t)+f(t0)Γ(1-α)(t-t0)-α=1Γ(1-α)∫t0t‍(t-s)-αf˙(s)ds+f(t0)Γ(1-α)(t-t0)-α.
Then we can get that
(23)ddt(RLDt0αf(t))=ddt(CDt0αf(t))-αf(t0)Γ(1-α)(t-t0)-α-1=ddt(It01-αf˙(t))-αf(t0)Γ(1-α)(t-t0)-α-1=RLDt0α(f˙(t))-αf(t0)Γ(1-α)(t-t0)-α-1=CDt0αf˙(t)+f˙(t0)Γ(1-α)(t-t0)-α-αf(t0)Γ(1-α)(t-t0)-α-1=1Γ(1-α)∫t0t‍(t-s)-αf¨(s)ds+f˙(t0)Γ(1-α)(t-t0)-α-αf(t0)Γ(1-α)(t-t0)-α-1.
By assumptions in (1), it follows that (d/dt)(RLDt0αf(t))≥0 on [t0,t1]. Thus RLDt0αf(t) is nondecreasing on [t0,t1]. By assumptions in (2), it follows that RLDt0αf(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),
(24)ddt(RLDt0αf(t))=1Γ(1-α)∫t0t‍(t-s)-αf¨(s)ds+f˙(t0)Γ(1-α)(t-t0)-α-αf(t0)Γ(1-α)(t-t0)-α-1.
Since f(t0)>0 and f˙(t0)>0, then
(25)f˙(t0)Γ(1-α)(t-t0)-α-αf(t0)Γ(1-α)(t-t0)-α-1⟶-∞
as t→t0. By the fact that f¨(t)∈C([t0,t1],(0,+∞)), we have
(26)1Γ(1-α)∫t0t‍(t-s)-αf¨(s)ds⟶0,t⟶t0.
Thus there exists a constant δ1>0 such that (d/dt)(RLDt0αf(t))≤0 on [t0,t0+δ1]. On the other hand, when t≥t0+αf(t0)/f˙(t0),
(27)f˙(t0)Γ(1-α)(t-t0)-α-αf(t0)Γ(1-α)(t-t0)-α-1≥0.
Thus there exists a constant β∈[t0,t1] such that (d/dt)(RLDt0αf(t))≤0 on [t0,β] and (d/dt)(RLDt0αf(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 functionCDt0αf(t).Theorem 13.
Assume that0<α<1. If there exists an interval [t0,t1] such that f¨(t)≥0 on [t0,t1] and f˙(t0)≥0, then CDt0αf(t) is nondecreasing on [t0,t1]. If f¨(t)≤0 on [t0,t1] and f˙(t0)≤0, then CDt0αf(t) is not increasing on [t0,t1].Proof.
Setf˙(t)=g(t). Note that
(28)ddt(CDt0αf(t))=ddt(It01-αf˙(t))=ddt(It01-αg(t))=RLDt0αg(t)=CDt0αg(t)+g(t0)Γ(1-α)(t-t0)-α=1Γ(1-α)∫t0t‍(t-u)-αg˙(u)du+g(t0)Γ(1-α)(t-t0)-α.
If g˙(t)=f¨(t)≥0 on [t0,t1] and g(t0)=f˙(t0)≥0, then (d/dt)(CDt0αf(t))≥0 in t on [t0,t1]. Hence, CDt0αf(t) is nondecreasing on interval [t0,t1]. If g˙(t)=f¨(t)≤0 and g(t0)=f˙(t0)≤0, then (d/dt)(CDt0αf(t))≤0. Hence, CDt0αf(t) is not increasing on [t0,t1]. The proof is completed.The following examples illustrate applications of Theorems12 and 13.Example 14.
Assume that0<α<1. Consider RLDt0αf(t), where f(t)=et, 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 RLDt0α(et) is decreasing on [t0,β] and is increasing on [β,+∞).Example 15.
Assume that0<α<1. Consider CD0.5παsint for t∈[π/2,π]. Since (sint)′′≤0 for t∈[π/2,π] and (sint)′|t=π/2=0, by Theorem 13, CD0.5παsint is decreasing on [π/2,π]. By similar argument, CD1.5παsint is increasing on t∈[3π/2,2π]. Since CDt0αsint = (1/Γ(1-α))∫t0t‍(t-τ)-αcosτd, thus (1/Γ(1-α))∫0.5πt‍(t-τ)-αcosτdτ is decreasing on [π/2,π] and (1/Γ(1-α))∫1.5πt‍(t-τ)-αcosτdτ is increasing on t∈[3π/2,2π].Example 16.
Assume that0<α<1. Consider CDt0αf(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, CDt0αf(t) is not increasing on [1,∞].Next we are to investigate the concavity and the convexity ofRLDt0αf(t) and CDt0αf(t). By formula (23), we have
(29)d2dt2(RLDt0αf(t))=ddt(1Γ(1-α)∫t0t‍(t-s)-αf¨(s)dspppppppp+1Γ(1-α)(t-t0)-αf˙(t0)pppppppp-α1Γ(1-α)(t-t0)-α-1f(t0))=1Γ(1-α)∫t0t‍(t-s)-αf′′′(s)ds+1Γ(1-α)(t-t0)-αf¨(t0)-α1Γ(1-α)(t-t0)-α-1f˙(t0)+α(α+1)1Γ(1-α)(t-t0)-α-2f(t0).
Thus we can obtain the following theorem.Theorem 17.
Assume that0<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 RLDt0αf(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 RLDt0αf(t) is convex on [t0,t1].The next theorem is on the convexity and the concavity ofCDt0αf(t).Theorem 18.
Assume that0<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 CDt0αf(t) is concave on [t0,t1]. If f′′′(t)≤0 on [t0,t1], f¨(t0)≥0 and f˙(t0)≥0, then CDt0αf(t) is convex on [t0,t1].Proof.
By formula (28), we have
(30)d2dt2(CDt0αf(t))=ddt(CDt0αf˙(t)+f˙(t0)Γ(1-α)(t-t0)-α)=ddt(It01-α(f¨(t)))-αf˙(t0)Γ(1-α)(t-t0)-α-1=RLDt0α(f¨(t))-αf˙(t0)Γ(1-α)(t-t0)-α-1=CDt0α(f¨(t))+f¨(t0)Γ(1-α)(t-t0)-α-αf˙(t0)Γ(1-α)(t-t0)-α-1=1Γ(1-α)∫t0t‍(t-s)-αf′′′(s)ds+f¨(t0)Γ(1-α)(t-t0)-α-αf˙(t0)Γ(1-α)(t-t0)-α-1.
If f′′′(t)≥0 on [t0,t1], f¨(t0)≥0, and f˙(t0)≤0, then (d2/dt2)(CDt0αf(t))≥0 on [t0,t1]. Hence, CDt0αf(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)(CDt0αf(t))≤0 on [t0,t1]. Therefore CDt0αf(t) is convex in t on [t0,t1].Example 19.
Assume that0<α<1. Consider the fractional differential equation CDt0αf(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, CDt0α(t-t2) is convex on [t0,0.5].Example 20.
Consider the concavity and convexity of the functionRLDt0αf(t), where f(t)=et, t0∈ℝ. Obviously, Theorems 17 and 18 are useless to the function RLDt0αet. Now we employ the method which is used in the proof of Theorem 17 to investigate it. By formula (29), we have
(31)d2dt2(RLDt0αet)=1Γ(1-α)∫t0t‍(t-s)-αesds+1Γ(1-α)(t-t0)-αet0-α1Γ(1-α)(t-t0)-α-1et0+α(α+1)1Γ(1-α)(t-t0)-α-2et0.
The three terms in the right side of (31) can be reduced to
(32)1Γ(1-α)(t-t0)-αet0-α1Γ(1-α)(t-t0)-α-1et0+α(α+1)1Γ(1-α)(t-t0)-α-2et0=1Γ(1-α)(t-t0)-α-2×et0[(t-t0)2-α(t-t0)+α(α-1)].
It is not difficult to get
(33)(t-t0)2-α(t-t0)+α(α-1)≥0
for t∈[t0+(α+(4α-3α2)0.5)/2,+∞) and
(34)(t-t0)2-α(t-t0)+α(α-1)≤0
for t∈[t0,t0+(α+(4α-3α2)0.5)/2]. Thus (d2/dt2)(RLDt0αet)>0 on [t0+(α+(4α-3α2)0.5)/2,+∞). Consequently, RLDt0αet is concave on [t0+(α+(4α-3α2)0.5)/2,∞). Since (1/Γ(1-α))∫t0t‍(t-s)-αesds→0 as t→t0, and (1/Γ(1-α))∫t0t‍(t-s)-αesds is increasing on [t0,+∞), thus there exists a constant β∈(t0,t0+(α+(4α-3α2)0.5)/2) such that (d2/dt2)(RLDt0αet)<0 on [t0,β] and (d2/dt2)(RLDt0αet)≥0 on [β,∞). Hence RLDt0αet 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 functionsRLDt0αf(t) and CDt0αf(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 RLDt0αf(t) and CDt0αf(t). In the meantime, five examples are given to illustrate the applications of our criteria.



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