New Gronwall-Bellman Type Inequalities and Applications in the Analysis for Solutions to Fractional Differential Equations

Abstract

Some new Gronwall-Bellman type inequalities are presented in this paper. Based on these inequalities, new explicit bounds for the related unknown functions are derived. The inequalities established can also be used as a handy tool in the research of qualitative as well as quantitative analysis for solutions to some fractional differential equations defined in the sense of the modified Riemann-Liouville fractional derivative. For illustrating the validity of the results established, we present some applications for them, in which the boundedness, uniqueness, and continuous dependence on the initial value for the solutions to some certain fractional differential and integral equations are investigated.

1. Introduction

It is well known that the Gronwall-Bellman inequality [1, 2] and their generalizations can provide explicit bounds for solutions to differential and integral equations as well as difference equations. Recently, many authors have researched various generalizations of the Gronwall-Bellman inequality (e.g., see [3–26] and the references therein). These Gronwall-Bellman type inequalities established have proved to be useful in the research of boundedness, global existence, uniqueness, stability, and continuous dependence of solutions to differential and integral equations as well as difference equations. However, in the research for the properties of solutions to some fractional differential and integral equations, the earlier inequalities established are inadequate to fulfill such analysis, and it is necessary to establish new Gronwall-Bellman type inequalities so as to obtain the desired result.

On the other hand, recently, Jumarie presented a new definition for the fractional derivative named the modified Riemann-Liouville fractional derivative (see [27, 28]). The modified Riemann-Liouville fractional derivative is defined by the following expression.

Definition 1. The modified Riemann-Liouville derivative of order 𝛼 is defined by the following expression: 𝐷^𝛼_𝑡𝑓(𝑡)=1Γ()𝑑1−𝛼𝑑𝑡^𝑡₀(𝑡−𝜉)^−𝛼⋅𝑓(𝜉)(0)𝑓−𝑓𝑑𝜉,0<𝛼<1,_(𝑛)(𝑡)^{(𝛼−𝑛)},𝑛≤𝛼<𝑛+1,𝑛≥1. (1)

Definition 2. The Riemann-Liouville fractional integral of order 𝛼 on the interval [0,𝑡] is defined by𝐼^𝛼𝑓(𝑡)=1Γ(1+𝛼)^𝑡₀𝑓(𝑠)(𝑑𝑠)^𝛼=1Γ(𝛼)^𝑡₀(𝑡−𝑠)^𝛼−1𝑓(𝑠)𝑑𝑠. (2)
Some important properties for the modified Riemann-Liouville derivative and fractional integral are listed as follows (the interval concerned below is always defined by [0,𝑡]):(a)𝐷^𝛼_𝑡𝑡^𝑟=(Γ(1+𝑟)/Γ(1+𝑟−𝛼))𝑡^𝑟−𝛼. (b)𝐷^𝛼_𝑡(𝑓(𝑡)𝑔(𝑡))=𝑔(𝑡)𝐷^𝛼_𝑡𝑓(𝑡)+𝑓(𝑡)𝐷^𝛼_𝑡𝑔(𝑡). (c)𝐷^𝛼_𝑡𝑓[𝑔(𝑡)]=𝑓^′_𝑔[𝑔(𝑡)]𝐷^𝛼_𝑡𝑔(𝑡)=𝐷^𝛼_𝑔𝑓[𝑔(𝑡)](𝑔^′(𝑡))^𝛼. (d)𝐼^𝛼(𝐷^𝛼_𝑡𝑓(𝑡))=𝑓(𝑡)−𝑓(0). (e)𝐼^𝛼(𝑔(𝑡)𝐷^𝛼_𝑡𝑓(𝑡))=𝑓(𝑡)𝑔(𝑡)−𝑓(0)𝑔(0)−𝐼^𝛼(𝑓(𝑡)𝐷^𝛼_𝑡𝑔(𝑡)). (f)The modified Riemann-Liouville derivative for a constant is zero.

The modified Riemann-Liouville derivative has many excellent characters in handling many fractional calculus problems. Many authors have investigated various applications of the modified Riemann-Liouville fractional derivative. For example, in [29–31], the authors seeked exact solutions for some types of fractional differential equations based on the modified Riemann-Liouville fractional derivative, and in [32], the modified Riemann-Liouville fractional derivative was used in fractional calculus of variations, where the authors considered the fractional basic problem with free boundary conditions as well as problems with isoperimetric and holonomic constraints in the calculus of variations. In [33], Khan et al. presented a fractional homotopy perturbation method (FHPM) for solving fractional differential equations of any fractional order based on the modified Riemann-Liouville fractional derivative. In [34–36], fractional variational iteration method based on the modified Riemann-Liouville fractional derivative was concerned. In [37], a fractional variational homotopy perturbation iteration method was proposed.

Motivated by the wide applications of the modified Riemann-Liouville fractional derivative, in this paper, we use this type of fractional derivative to establish some fractional Gronwall-Bellman type inequalities. Based on these inequalities and some basic properties of the modified Riemann-Liouville fractional derivative, we derive explicit bounds for unknown functions concerned in these inequalities. As for applications, we apply these inequalities to research qualitative properties such as the boundedness, uniqueness, and continuous dependence on initial data for solutions to some certain fractional differential and integral equations.

We organize the rest of this paper as follows. In Section 2, we present the main inequalities, and based on them derive explicit bounds for unknown functions in these inequalities. Then in Section 3, we apply the results established in Section 2 to research boundedness, uniqueness, and continuous dependence on initial data for the solution to some certain fractional differential and integral equations.

2. Main Results

Lemma 3 (see [38]). Assume that 𝑎≥0,𝑝≥𝑞≥0; and 𝑝≠0, then, for any 𝐾>0, one has 𝑎^𝑞/𝑝≤𝑞𝑝𝐾^{(𝑞−𝑝)/𝑝}𝑎+𝑝−𝑞𝑝𝐾^𝑞/𝑝. (3)

Lemma 4. Let 𝛼>0, 𝑎,𝑏,𝑢 be continuous functions defined on 𝑡≥0. Then for 𝑡≥0, 𝐷^𝛼_𝑡𝑢(𝑡)≤𝑎(𝑡)+𝑏(𝑡)𝑢(𝑡) (4) implies 𝑢(𝑡)≤𝑢(0)exp^𝑡𝛼₀^/Γ(1+𝛼)𝑏(𝑠Γ(1+𝛼))^1/𝛼+1𝑑𝑠Γ(𝛼)^𝑡₀(𝑡−𝜏)^𝛼−1𝑎(𝜏)×exp−^𝑡𝛼_𝜏^/Γ(1+𝛼)_{𝛼/Γ(1+𝛼)}𝑏(𝑠Γ(1+𝛼))^1/𝛼𝑑𝑠𝑑𝜏. (5)

Proof. By the properties (a),(b), and (c) we have the following observation: 𝐷^𝛼_𝑡𝑢(𝑡)exp−^𝑡𝛼₀^/Γ(1+𝛼)𝑏(𝑠Γ(1+𝛼))^1/𝛼−𝑑𝑠=exp^𝑡𝛼₀^/Γ(1+𝛼)𝑏(𝑠Γ(1+𝛼))^1/𝛼𝑑𝑠×𝐷^𝛼_𝑡𝑢(𝑡)+𝑢(𝑡)𝐷^𝛼_𝑡×exp−^𝑡𝛼₀^/Γ(1+𝛼)𝑏(𝑠Γ(1+𝛼))^1/𝛼−𝑑𝑠=exp^𝑡𝛼₀^/Γ(1+𝛼)𝑏(𝑠Γ(1+𝛼))^1/𝛼𝑑𝑠×𝐷^𝛼_𝑡𝑢(𝑡)(𝑡)𝑢(𝑡)−−𝑏×exp^𝑡𝛼₀^/Γ(1+𝛼)𝑏(𝑠Γ(1+𝛼))^1/𝛼𝑑𝑠×𝐷^𝛼_𝑡𝑡^𝛼Γ()−1+𝛼=exp^𝑡𝛼₀^/Γ(1+𝛼)𝑏(𝑠Γ(1+𝛼))^1/𝛼×𝐷𝑑𝑠_𝛼𝑡𝑢(𝑡)(𝑡)𝑢(𝑡(𝑡)−−𝑏)≤𝑎exp^t𝛼₀^/Γ(1+𝛼)𝑏(𝑠Γ(1+𝛼))^1/𝛼𝑑𝑠. (6)
Substituting 𝑡 with 𝜏, fulfilling fractional integral of order 𝛼 for (6) with respect to 𝜏 from 0 to 𝑡, we deduce that 𝑢(𝑡)exp−^𝑡𝛼₀^/Γ(1+𝛼)𝑏(𝑠Γ(1+𝛼))^1/𝛼(0)+1𝑑𝑠≤𝑢Γ(𝛼)^𝑡₀(𝑡−𝜏)^𝛼−1𝑎(𝜏)×exp−^𝜏𝛼₀^/Γ(1+𝛼)𝑏(𝑠Γ(1+𝛼))^1/𝛼𝑑𝑠𝑑𝜏, (7) which implies 𝑢(𝑡)≤exp^𝑡𝛼₀^{/(Γ(1+𝛼))}𝑏(𝑠Γ(1+𝛼))^1/𝛼×𝑢(0)+1𝑑𝑠Γ(𝛼)×^𝑡₀(𝑡−𝜏)^𝛼−1𝑎(𝜏)×exp−^𝜏𝛼₀^{/(Γ(1+𝛼))}𝑏(𝑠Γ(1+𝛼))^1/𝛼𝑑𝑠𝑑𝜏. (8)
The desired result can be obtained subsequently.

Lemma 5. Suppose 𝛼>0, the functions 𝑢, 𝑎, 𝑏, 𝑔, and ℎ are nonnegative continuous functions defined on 𝑡≥0, and 𝑎,𝑏 are nondecreasing, 𝑝,𝑞, and 𝑟 are constants with 𝑝≥𝑞>0,𝑝≥𝑟>0. If the following inequality holds: 𝑢^𝑝(𝑡)(𝑡)+𝑏(𝑡)≤𝑎Γ(𝛼)×^𝑡₀(𝑡−𝑠)^𝛼−1×𝑔(𝑠)𝑢^𝑞(𝑠)+^𝑠₀ℎ(𝜉)𝑢^𝑟(𝜉)𝑑𝜉𝑑𝑠,𝑡≥0, (9) then one has the following explicit estimate for u(t): 𝑢(𝑡)≤𝑎(𝑡)𝑏(𝑡)×exp^𝑡𝛼₀^/Γ(1+𝛼)𝐻₂(𝑠Γ(1+𝛼))^1/𝛼+𝑏(𝑡)𝑑𝑠Γ(𝛼)×^𝑡₀(𝑡−𝜏)^𝛼−1𝐻₁(𝜏)(𝑡)××exp−𝑏_{𝑡𝛼𝜏/Γ(1+𝛼)𝛼()/Γ1+𝛼}𝐻₂×(𝑠Γ(1+𝛼))^1/𝛼𝑑𝑠𝑑𝜏^1/𝑝,𝑡≥0, (10)
where 𝐻₁(𝑡)=𝑔(𝑡)𝑝−𝑞𝑝𝐾^𝑞/𝑝+^𝑡₀ℎ(𝜉)𝑝−𝑟𝑝𝐾^𝑟/𝑝𝐻𝑑𝜉,₂(𝑡)(𝑡)𝑞=𝑔𝑝𝐾^{(𝑞−𝑝)/𝑝}+^𝑡₀ℎ(𝜉)𝑟𝑝𝐾^{(𝑟−𝑝)/𝑝}𝑑𝜉, (11) and 𝐾>0 is an arbitrary constant.

Proof. Fix 𝑇≥0, and let 𝑡∈[0,𝑇]. Denote 𝑣(𝑡)(𝑇)+𝑏(𝑇)=𝑎Γ(𝛼)×^𝑡₀(𝑡−𝑠)^𝛼−1𝑔(𝑠)𝑢^𝑞(𝑠)+^𝑠₀ℎ(𝜉)𝑢^𝑟(𝜉)𝑑𝜉𝑑𝑠. (12)
Then we have 𝑢^𝑝(𝑡)≤𝑣(𝑡),𝑡∈[0,𝑇]. (13)
Since 𝑢,𝑔, and ℎ are continuous, then there exists a constant 𝑀 such that |𝑔(𝑡)𝑢^𝑞(𝑡)+∫^𝑡₀ℎ(𝜉)𝑢^𝑟(𝜉)𝑑𝜉|≤𝑀 for 𝑡∈[0,𝜀], where 𝜀>0. So for 𝑡∈[0,𝜀], we have |∫^𝑡₀(𝑡−𝑠)^𝛼−1[𝑔(𝑠)𝑢^𝑞(𝑠)+∫^𝑠₀ℎ(𝜉)𝑢^𝑟(𝜉)𝑑𝜉]𝑑𝑠|≤𝑀∫^𝑡₀(𝑡−𝑠)^𝛼−1𝑑𝑠=(𝑀/𝛼)𝑡^𝛼. Then one can see 𝑣(0)=𝑎(𝑇). Furthermore, 𝐷^𝛼_𝑡𝑣(𝑡)=𝑏(𝑇)𝑔(𝑡)𝑢^𝑞(𝑡)+^𝑡₀ℎ(𝜉)𝑢^𝑟(𝜉)(𝑇𝑔(𝑡)𝑣𝑑𝜉≤𝑏)^𝑞/𝑝(𝑡)+^𝑡₀ℎ(𝜉)𝑣^𝑟/𝑝(𝜉)𝑑𝜉. (14)
By Lemma 3, we have 𝐷^𝛼_𝑡𝑣(𝑡)(𝑇𝑔(𝑡)𝑞≤𝑏)𝑝𝐾^{(𝑞−𝑝)/𝑝}𝑣(𝑡)+𝑝−𝑞𝑝𝐾^𝑞/𝑝+^𝑡₀ℎ(𝜉)𝑟𝑝𝐾^{(𝑟−𝑝)/𝑝}𝑣(𝜉)+𝑝−𝑟𝑝𝐾^𝑟/𝑝(𝑇𝑔(𝑡𝑑𝜉=𝑏))𝑝−𝑞𝑝𝐾^𝑞/𝑝+^𝑡₀ℎ(𝜉)𝑝−𝑟𝑝𝐾^𝑟/𝑝(𝑇𝑔(𝑡)𝑞𝑑𝜉+𝑏)𝑝𝐾^{(𝑞−𝑝)/𝑝}𝑣(𝑡)+^𝑡₀ℎ(𝜉)𝑟𝑝𝐾^{(𝑟−𝑝)/𝑝}𝑣(𝜉)(𝑇𝑔(𝑡)𝑑𝜉≤𝑏)𝑝−𝑞𝑝𝐾^𝑞/𝑝+^𝑡₀ℎ(𝜉)𝑝−𝑟𝑝𝐾^𝑟/𝑝(𝑇𝑔(𝑡)𝑞𝑑𝜉+𝑏)𝑝𝐾^{(𝑞−𝑝)/𝑝}+^𝑡₀ℎ(𝜉)𝑟𝑝𝐾^{(𝑟−𝑝)/𝑝}𝑣(𝑡)(𝑇)𝐻𝑑𝜉=𝑏₁(𝑡)+𝑏(𝑇)𝐻₂(𝑡)𝑣(𝑡),𝑡∈[0,𝑇], (15) where 𝐻₁(𝑡),𝐻₂(𝑡) are defined in (11). Applying Lemma 4 to (15), considering 𝑣(0)=𝑎(𝑇), we get that 𝑣(𝑡)≤𝑎(𝑇)𝑏(𝑇)×exp^𝑡𝛼₀^/Γ(1+𝛼)𝐻₂(𝑠Γ(1+𝛼))^1/𝛼+𝑏(𝑇)𝑑𝑠Γ(𝛼)×^𝑡₀(𝑡−𝜏)^𝛼−1𝐻₁(𝜏)×exp−𝑏(𝑇)^𝑡𝛼_𝜏^/Γ(1+𝛼)_{𝛼/Γ(1+𝛼)}𝐻₂×(𝑠Γ(1+𝛼))^1/𝛼[].𝑑𝑠𝑑𝜏,𝑡∈0,𝑇 (16)
Letting 𝑡=𝑇 in (16) and considering 𝑇≥0 is arbitrary, after substituting 𝑇 with 𝑡, we get that 𝑣(𝑡)≤𝑎(𝑡)𝑏(𝑡)×exp^𝑡𝛼₀^/Γ(1+𝛼)𝐻₂(𝑠Γ(1+𝛼))^1/𝛼+𝑏(𝑡)𝑑𝑠Γ(𝛼)^𝑡₀(𝑡−𝜏)^𝛼−1𝐻₁(𝜏)×exp−𝑏(𝑡)^𝑡𝛼_𝜏^/Γ(1+𝛼)_{𝛼/Γ(1+𝛼)}𝐻₂×(𝑠Γ(1+𝛼))^1/𝛼𝑑𝑠𝑑𝜏,𝑡≥0. (17)
Combining (13) and (17), we get (10).

Remark 6. In Lemma 5, if 𝑎,𝑏 are not necessarily nondecreasing, then one can let 𝑣(𝑡)=(1/Γ(𝛼))∫^𝑡₀(𝑡−𝑠)^𝛼−1[𝑔(𝑠)𝑢^𝑞(𝑠)+∫^𝑠₀ℎ(𝜉)𝑢^𝑟(𝜉)𝑑𝜉]𝑑𝑠 instead in the proof, and following in a similar process, obtain another explicit bound for 𝑢(𝑡): 𝑢(𝑡)≤𝑎(𝑡)+𝑏(𝑡)Γ(𝛼)×^𝑡₀(𝑡−𝜏)^𝛼−1𝐻₁(𝜏)×exp−^𝑡𝛼_𝜏^/Γ(1+𝛼)_{𝛼/Γ(1+𝛼)}𝐻₂×(𝑠Γ(1+𝛼))^1/𝛼𝑑𝑠𝑑𝜏^1/𝑝,𝑡≥0, (18) where 𝐻₁(𝑡)=𝑔(𝑡)𝑝−𝑞𝑝𝐾^𝑞/𝑝+𝑞𝑝𝐾^{(𝑞−𝑝)/𝑝}𝑎(𝑡)+^𝑡₀ℎ(𝜉)𝑝−𝑟𝑝𝐾^𝑟/𝑝+𝑟𝑝𝐾^{(𝑟−𝑝)/𝑝}𝑎(𝜉)𝐻𝑑𝜉,₂(𝑡)(𝑡𝑔(𝑡)𝑞=𝑏)𝑝𝐾^{(𝑞−𝑝)/𝑝}+^𝑡₀ℎ(𝜉)𝑟𝑝𝐾^{(𝑟−𝑝)/𝑝}𝑑𝜉. (19)

Remark 7. We note that if we take 𝑔(𝑡)≡1,ℎ(𝑡)≡0, and 𝑝=𝑞=1, then the inequality (9) in Lemma 5 reduces to the inequality in [39, Theorem 1]. So the present inequality is of more general form than that in [39]. Furthermore, the explicit bounds obtained for the function 𝑢(𝑡) above are essentially different from that in [39].

Theorem 8. Suppose that 𝛼>0, the functions 𝑢,𝑎,𝑏,𝑔,𝑝,𝑞, and 𝑟 are defined as in Lemma 5, and 𝑝≥1, 𝑚 is a nonnegative continuous function defined on 𝑡≥0. If the following inequality holds: 𝑢^𝑝(𝑡)≤𝑎(𝑡)+^𝑡₀𝑚(𝑠)𝑢^𝑝(𝑠)+𝑏(𝑡)𝑑𝑠Γ(𝛼)^𝑡₀(𝑡−𝑠)^𝛼−1×𝑔(𝑠)𝑢^𝑞(𝑠)+^𝑠₀ℎ(𝜉)𝑢^𝑟(𝜉)𝑑𝜉𝑑𝑠,𝑡≥0, (20)
then we have 𝑢(𝑡)1≤exp𝑝^𝑡₀𝑚(𝑠)×𝑎(𝑡)𝑏(𝑡)𝑑𝑠exp^𝑡𝛼₀^/Γ(1+𝛼)𝐻₂×(𝑠Γ(1+𝛼))^1/𝛼+𝑏(𝑡)𝑑𝑠Γ(𝛼)×^𝑡₀(𝑡−𝜏)^𝛼−1𝐻₁(𝜏)×exp−𝑏(𝑡)^𝑡𝛼_𝜏^/Γ(1+𝛼)_{𝛼/Γ(1+𝛼)}𝐻₂×(𝑠Γ(1+𝛼))^1/𝛼𝑑𝑠𝑑𝜏^1/𝑝,𝑡≥0, (21) where 𝐻₁(𝑡)=𝑔(𝑡)𝑝−𝑞𝑝𝐾^𝑞/𝑝+^𝑡₀ℎ(𝜉)𝑝−𝑟𝑝𝐾^𝑟/𝑝𝐻𝑑𝜉,₂(𝑡)=𝑔(𝑡)𝑞𝑝𝐾^{(𝑞−𝑝)/𝑝}+^𝑡₀ℎ(𝜉)𝑟𝑝𝐾^{(𝑟−𝑝)/𝑝}𝑔(𝑡)(𝑡)𝑞𝑑𝜉,=𝑔exp𝑝^𝑠₀𝑚(𝜏),ℎ(𝑡)(𝑡)𝑟𝑑𝜏=ℎexp𝑝^𝑡₀𝑚(𝜏)𝑑𝜏. (22)(23)(24)

Proof. Denote 𝑧^𝑝(𝑡) by 𝑎(𝑡) + (𝑏(𝑡)/Γ(𝛼))∫^𝑡₀(𝑡−𝑠)^𝛼−1[𝑔(𝑠)𝑢^𝑞(𝑠)+∫^𝑠₀ℎ(𝜉)𝑢^𝑟(𝜉)𝑑𝜉]𝑑𝑠. Then 𝑢^𝑝(𝑡)≤𝑧^𝑝(𝑡)+^𝑡₀𝑚(𝑠)𝑢^𝑝(𝑠)𝑑𝑠. (25)
Since 𝑎(𝑡),𝑏(𝑡) are both nondecreasing, then 𝑧(𝑡) is also nondecreasing, and subsequently we can deduce that 𝑢^𝑝(𝑡)≤𝑧^𝑝(𝑡)exp^𝑡₀𝑚(𝑠)𝑑𝑠. (26)
Furthermore,𝑧^𝑝(𝑡)(𝑡)+𝑏(𝑡)≤𝑎Γ(𝛼)×^𝑡₀(𝑡−𝑠)^𝛼−1×𝑔(𝑠)𝑧^𝑞(𝑠)𝑞×exp𝑝^𝑠₀𝑚(𝜏)+𝑑𝜏_𝑠0ℎ(𝜉)𝑧^𝑟(𝜉)𝑟exp𝑝^𝜉₀𝑚(𝜏)(𝑡)+𝑏(𝑡)𝑑𝜏𝑑𝜉𝑑𝑠=𝑎Γ(𝛼)^𝑡₀(𝑡−𝑠)^𝛼−1×𝑔(𝑠)𝑧^𝑞(𝑠)+^𝑠₀ℎ(𝜉)𝑧^𝑟(𝜉)𝑑𝜉𝑑𝑠, (27) where 𝑔(𝑡),ℎ(𝑡) are defined in (24). Then applying Lemma 5 to (23) yields 𝑧(𝑡)≤𝑎(𝑡)𝑏(𝑡)×exp^𝑡𝛼₀^/Γ(1+𝛼)𝐻₂(𝑠Γ(1+𝛼))^1/𝛼+𝑏(𝑡)𝑑𝑠Γ(𝛼)×^𝑡₀(𝑡−𝜏)^𝛼−1𝐻₁(𝜏)×exp−𝑏(𝑡)^𝑡𝛼_𝜏^/Γ(1+𝛼)_{𝛼/Γ(1+𝛼)}𝐻₂×(𝑠Γ(1+𝛼))^1/𝛼𝑑𝑠𝑑𝜏^1/𝑝,𝑡≥0, (28)
where 𝐻₁(𝑡),𝐻₂(𝑡) are defined in (22) and (23). Combining (26) and (28), we obtain the desired result.

Lemma 9. Suppose 𝛼>0, the functions 𝑢,𝑎,𝑏, and 𝑔 are defined as in Lemma 5, and 𝜔 is a nonnegative continuous function defined on 𝑡≥0 being nondecreasing, and 𝜔(𝑟)>0 for 𝑟>0. Define 𝐺(𝑣)=∫^𝑣₀(1/𝜔(𝑟))𝑑𝑟, and assume 𝐺(𝑣)<∞ for 𝑣<∞. If the following inequality holds: 𝑢(𝑡)(𝑡)+𝑏(𝑡)≤𝑎Γ(𝛼)^𝑡₀(𝑡−𝑠)^𝛼−1×𝑔(𝑠)𝜔(𝑢(𝑠))𝑑𝑠,𝑡≥0, (29)
then we have the following explicit estimate for 𝑢(𝑡): 𝑢(𝑡)≤𝐺⁻¹×𝐺(𝑎(𝑡+𝑏(𝑡)))Γ(𝛼)^𝑡₀(𝑡−𝜏)^𝛼−1𝑔(𝜏)𝑑𝜏,𝑡≥0. (30)

Proof. Fix 𝑇≥0, and let 𝑡∈[0,𝑇]. Denote 𝑣(𝑡)(𝑇)+𝑏(𝑇)=𝑎+𝛿Γ(𝛼)^𝑡₀(𝑡−𝑠)^𝛼−1𝑔(𝑠)𝜔(𝑢(𝑠))𝑑𝑠, (31) where 𝛿>0. Then we have 𝑢(𝑡)≤𝑣(𝑡),𝑡∈[0,𝑇]. (32)
Since 𝑢,𝑔, and 𝜔 are continuous, then there exists a constant 𝑀 such that |𝑔(𝑡)𝜔(𝑢(𝑡))|≤𝑀 for 𝑡∈[0,𝜀], where 𝜀>0. So for 𝑡∈[0,𝜀], we have |∫^𝑡₀(𝑡−𝑠)^𝛼−1𝑔(𝑠)𝜔(𝑢(𝑠))𝑑𝑠|≤𝑀∫^𝑡₀(𝑡−𝑠)^𝛼−1𝑑𝑠=(𝑀/𝛼)𝑡^𝛼. Then one can see 𝑣(0)=𝑎(𝑇)+𝛿, and 𝐷^𝛼_𝑡𝑣(𝑡)(𝑇)𝑔(𝑡)𝜔(𝑢(𝑡(𝑇)𝑔(𝑡)𝜔(𝑣(𝑡,=𝑏))≤𝑏)) (33)
which implies 𝐷^𝛼_𝑡𝑣(𝑡)𝜔(𝑣(𝑡(𝑇)𝑔(𝑡).))≤𝑏 (34)
That is, 𝐷^𝛼_𝑡𝐺(𝑣(𝑡))≤𝑏(𝑇)𝑔(𝑡). (35)
Substituting 𝑡 with 𝜏, fulfilling fractional integral of order 𝛼 for (35) with respect to 𝜏 from 0 to 𝑡, we deduce that 𝐺(𝑣(𝑡(𝑣(0≤𝑏(𝑇)))−𝐺))Γ(𝛼)^𝑡₀(𝑡−𝜏)^𝛼−1𝑔(𝜏)𝑑𝜏, (36)
which implies 𝑣(𝑡)≤𝐺⁻¹×𝐺(𝑎(𝑇))+𝑏(𝑇)+𝛿Γ(𝛼)^𝑡₀(𝑡−𝜏)^𝛼−1𝑔(𝜏)𝑑𝜏, (37)
and furthermore, 𝑢(𝑡)≤𝐺⁻¹𝐺(𝑎(𝑇))+𝑏(𝑇)+𝛿Γ(𝛼)×^𝑡₀(𝑡−𝜏)^𝛼−1𝑔(𝜏)𝑑𝜏,𝑡∈[0,𝑇]. (38)
Letting 𝑡=𝑇 in (38), we get that 𝑢(𝑇)≤𝐺⁻¹𝐺(𝑎(𝑇))+𝑏(𝑇)+𝛿Γ(𝛼)×^𝑇₀(𝑇−𝜏)^𝛼−1𝑔(𝜏)𝑑𝜏. (39)
Since 𝑇>0 is arbitrary, substituting 𝑇 with 𝑡 in (39) and after letting 𝛿→0, we can obtain the desired result.

Theorem 10. Suppose 𝛼>0, the functions 𝑢,𝑎,𝑏, and 𝑔 are defined as in Lemma 5, 𝑚 is a nonnegative continuous function defined on 𝑡≥0, and 𝜔 is defined as in Lemma 9, and furthermore, assume that 𝜔 is submultiplicative; that is, 𝜔(𝛼𝛽)≤𝜔(𝛼)𝜔(𝛽),𝛼,𝛽≥0. Define 𝐺(𝑣)=∫^𝑣₀(1/𝜔(𝑟))𝑑𝑟, and assume 𝐺(𝑣)<∞ for 𝑣<∞. If the following inequality holds: 𝑢(𝑡)(𝑡)+≤𝑎_𝑡0𝑚(𝑠)𝑢(𝑠)+𝑏(𝑡)𝑑𝑠Γ(𝛼)^𝑡₀(𝑡−𝑠)^𝛼−1𝑔(𝑠)𝜔(𝑢(𝑠))𝑑𝑠,𝑡≥0, (40)
then we have the following explicit estimate for 𝑢(𝑡): 𝑢(𝑡)≤𝐺⁻¹𝐺(𝑎(𝑡(𝑡)1))+𝑏Γ(𝛼)×^𝑡₀(𝑡−𝜏)^𝛼−1𝑔(𝜏)×exp^𝜏₀𝑚(𝜉)𝑑𝜉𝑑𝜏,𝑡≥0. (41)

Proof. Let 𝑧(𝑡)=𝑎(𝑡)+(𝑏(𝑡)/Γ(𝛼))∫^𝑡₀(𝑡−𝑠)^𝛼−1𝑔(𝑠)𝜔(𝑢(𝑠))𝑑𝑠.
Then we have 𝑢(𝑡)≤𝑧(𝑡)+^𝑡₀𝑚(𝑠)𝑢(𝑠)𝑑𝑠. (42)
Since 𝑧(𝑡) is nondecreasing, then furthermore we have 𝑢(𝑡)≤𝑧(𝑡)exp^𝑡₀𝑚(𝑠)𝑑𝑠. (43)
So 𝑧(𝑡)(𝑡)+𝑏(𝑡)≤𝑎Γ(𝛼)^𝑡₀(𝑡−𝑠)^𝛼−1𝑔(𝑠)𝜔(𝑧(𝑠))×exp^𝑠₀𝑚(𝜉)𝑑𝜉𝑑𝑠. (44)
Since 𝜔 is submultiplicative, then furthermore we get that 𝑧(𝑡)(𝑡)+𝑏(𝑡)≤𝑎Γ(𝛼)×^𝑡₀(𝑡−𝑠)^𝛼−1𝑔(𝑠)𝜔(𝑧(𝑠))×𝜔exp^𝑠₀𝑚(𝜉)𝑑𝜉𝑑𝑠. (45)
Noticing that the structure of (45) is similar to that of the inequality (29), a suitable application of Lemma 9 to (45) yields that 𝑧(𝑡)≤𝐺⁻¹𝐺(𝑎(𝑡(𝑡)1))+𝑏Γ(𝛼)×^𝑡₀(𝑡−𝜏)^𝛼−1𝑔(𝜏)𝜔exp^𝜏₀𝑚(𝜉)𝑑𝜉𝑑𝜏. (46)
Combining (43) and (46) we can obtain the desired result.

3. Applications

In this section, we apply the inequalities established above to research of boundedness, uniqueness, continuous dependence on the initial value for solutions to certain fractional differential and integral equations. Let us first consider the following IVP of fractional differential equation: 𝐷_𝑡^0.5𝑢³(𝑡)=𝐿𝑡,𝑢(𝑡),^𝑡₀𝑀((𝜉𝑢(0)𝜉,𝑢))𝑑𝜉,𝑡≥0,=𝐶, (47) where 𝑢∈𝐶([0,∞),𝑅),𝑀∈𝐶(𝑅×𝑅,𝑅), and 𝐿∈𝐶([0,∞)×𝑅²,𝑅).

Theorem 11. Suppose that 𝑢(𝑡) is a solution of the IVP (47). If |𝐿(𝑡,𝑢,𝑣)|≤𝑔(𝑡)|𝑢|³+|𝑣|, and |𝑀(𝑡,𝑢)|≤ℎ(𝑡)|𝑢|³, where 𝑔,ℎ are nonnegative continuous functions on [0,∞), then we have the following estimate for 𝑢(𝑡): |𝑢(𝑡≤)|₃|𝐶|exp^√₀^𝑡/Γ(1.5)𝐻₂(𝑠Γ(1.5))²𝑑𝑠,𝑡≥0, (48) where 𝐻₂(𝑡)=𝑔(𝑡)+∫^𝑡₀ℎ(𝜉)𝑑𝜉.

Proof. Similar to [28, Equation (5.1)], we can obtain the equivalent integral form of the IVP (47) as follows: 𝑢³(𝑡)1=𝐶+Γ(0.5)^𝑡₀(𝑡−𝑠)^−0.5×𝐿𝑠,𝑢(𝑠),^𝑠₀𝑀(𝜉,𝑢(𝜉))𝑑𝜉𝑑𝑠. (49)
So |𝑢(𝑡)|³≤|𝐶|+1Γ()×0.5_𝑠0(𝑡−𝑠)^−0.5×𝐿𝑠,𝑢(𝑠),^𝑠₀𝑀((𝜉≤|𝐶|+1𝜉,𝑢))𝑑𝜉𝑑𝑠Γ()×0.5_𝑠0(𝑡−𝑠)^−0.5×𝑔(𝑠)|𝑢(𝑠)|³+^𝑠₀𝑀((𝜉≤|𝐶|+1𝜉,𝑢))𝑑𝜉𝑑𝑠Γ()×0.5_𝑠0(𝑡−𝑠)^−0.5𝑔(𝑠)|𝑢(𝑠)|³+^𝑠₀𝑀((𝜉≤|𝐶|+1𝜉,𝑢))𝑑𝜉𝑑𝑠Γ()×0.5_𝑠0(𝑡−𝑠)^−0.5𝑔(𝑠)|𝑢(𝑠)|³+^𝑠₀ℎ(𝜉)𝑢(𝜉)³𝑑𝜉𝑑𝑠. (50)
Then a suitable application of Lemma 5 to (50) (with 𝛼=0.5,𝑝=𝑞=𝑟=3) yields the desired result.

Remark 12. At the end of the proof of Theorem 11, if we apply the result of Remark 6 instead of Lemma 5 to (50), then we obtain the following estimate: |𝑢(𝑡≤𝐶1)|||1+Γ(𝛼)×^𝑡₀(𝑡−𝜏)^𝛼−1𝐻₂(𝜏)×exp−^𝑡𝛼_𝜏^/Γ(1+𝛼)_{𝛼()/Γ1+𝛼}𝐻₂×(𝑠Γ(1+𝛼))^1/𝛼𝑑𝑠𝑑𝜏^1/3,𝑡≥0, (51) where 𝐻₂(𝑡) is defined as in Theorem 11.

Remark 13. In Theorem 11, if we change the conditions by |𝐿(𝑡,𝑢,𝑣)|≤𝑔(𝑡)|𝑢|+|𝑣|, and |𝑀(𝑡,𝑢)|≤ℎ(𝑡)|𝑢|, where 𝑔,ℎ are nonnegative continuous functions on [0,∞), then we can obtain the following estimate for 𝑢(𝑡): |𝑢(𝑡)|≤|𝐶|exp^√₀^𝑡/Γ(1.5)𝐻₂(𝑠Γ(1.5))²+1𝑑𝑠Γ(0.5)^𝑡₀(𝑡−𝜏)^−0.5𝐻₁(𝜏)×exp−^√_√^𝑡/Γ(1.5)_()𝜏/Γ1.5𝐻₂×(𝑠Γ(1.5))²𝑑𝑠𝑑𝜏^1/3,𝑡≥0, (52) where 𝐻₁(𝑡)=(2/3)𝐾^1/3[𝑔(𝑡)+∫^𝑡₀ℎ(𝜉)𝑑𝜉],𝐻₂(𝑡)=(1/3)𝐾^−2/3[𝑔(𝑡)+∫^𝑡₀ℎ(𝜉)𝑑𝜉], and 𝐾>0 is an arbitrary constant.

Theorem 14. If |𝐿(𝑡,𝑢₁,𝑣₁)−𝐿(𝑡,𝑢₂,𝑣₂)|≤𝑔(𝑡)|𝑢³₁−𝑢³₂|+|𝑣₁−𝑣₂|, |𝑀(𝑡,𝑢₁)−𝑀(𝑡,𝑢₂)|≤ℎ(𝑡)|𝑢³₁−𝑢³₂|, where 𝑔,ℎ are nonnegative continuous functions defined on [0,∞), then the IVP (47) has at most one solution.

Proof. Suppose that the IVP (47) has two solutions 𝑢₁(𝑡),𝑢₂(𝑡). Then similar to Theorem 11, we can obtain that 𝑢³₁(𝑡)1=𝐶+Γ(0.5)×^𝑡₀(𝑡−𝑠)^−0.5×𝐿𝑠,𝑢₁(𝑠),^𝑠₀𝑀𝜉,𝑢₂(𝜉𝑢)𝑑𝜉𝑑𝑠,₃₂(𝑡)1=𝐶+Γ()×0.5_𝑡0(𝑡−𝑠)^−0.5×𝐿𝑠,𝑢₂(𝑠),^𝑠₀𝑀𝜉,𝑢₂(𝜉)𝑑𝜉𝑑𝑠. (53)
Furthermore, 𝑢³₁(𝑡)−𝑢³₂(𝑡)=1Γ(0.5)^𝑡₀(𝑡−𝑠)^−0.5×𝐿𝑠,𝑢₁(𝑠),_𝑠0𝑀𝜉,𝑢₁(𝜉)𝑑𝜉−𝐿𝑠,𝑢₂(𝑠),_𝑠0𝑀𝜉,𝑢₂(𝜉)𝑑𝜉𝑑𝑠, (54)
which implies 𝑢³₁(𝑡)−𝑢³₂(𝑡≤1)Γ(0.5)^𝑡₀(𝑡−𝑠)^−0.5×𝐿𝑠,𝑢₁(𝑠),_𝑠0𝑀𝜉,𝑢₁(𝜉)𝑑𝜉−𝐿𝑠,𝑢₂(𝑠),_𝑠0𝑀𝜉,𝑢₂(𝜉)≤1𝑑𝜉𝑑𝑠Γ(0.5)^𝑡₀(𝑡−𝑠)^−0.5×𝑔(𝑠)𝑢³₁(𝑠)−𝑢³₂(𝑠+_𝑠0)𝑀𝜉,𝑢₁(𝜉)−𝑑𝜉_𝑠0𝑀𝜉,𝑢₂(𝜉)≤1𝑑𝜉𝑑𝑠Γ(0.5)^𝑡₀(𝑡−𝑠)^−0.5×𝑔(𝑠)𝑢³₁(𝑠)−𝑢³₂(𝑠+_𝑠0)ℎ(𝜉)𝑢³₁(𝜉)−𝑢³₂(𝜉)𝑑𝜉𝑑𝑠. (55)
Treating |𝑢³₁(𝑡)−𝑢³₂(𝑡)| as one whole function, a suitable application of Lemma 5 to (55) (with 𝛼=0.5) yields |𝑢³₁(𝑡)−𝑢³₂(𝑡)|≤0, which implies 𝑢₁(𝑡)≡𝑢₂(𝑡). So the proof is complete.

Now we research the continuous dependence on the initial value for the IVP (47).

Theorem 15. Suppose that 𝑢(𝑡) is the solution of the IVP (47), and 𝑢(𝑡) is the solution of following IVP of fractional differential equation: 𝐷_𝑡^0.5𝑢³(𝑡)=𝐿𝑡,𝑢(𝑡),^𝑡₀𝑀(𝑢(𝜉𝑢(0)=𝜉,))𝑑𝜉,𝑡≥0,𝐶, (56) where 𝑢∈𝐶([0,∞),𝑅),𝑀∈𝐶(𝑅×𝑅,𝑅), and 𝐿∈𝐶([0,∞)×𝑅²,𝑅). If |𝐿(𝑡,𝑢₁,𝑣₁)−𝐿(𝑡,𝑢₂,𝑣₂)|≤𝑔(𝑡)|𝑢³₁−𝑢³₂|+|𝑣₁−𝑣₂|, |𝑀(𝑡,𝑢₁)−𝑀(𝑡,𝑢₂)|≤ℎ(𝑡)|𝑢³₁−𝑢³₂|, where 𝑔,ℎ are nonnegative continuous functions on [0,∞) and |𝐶−𝐶|<𝜀, we have the following estimate: |𝑢(𝑡)|≤𝜀exp^√₀^𝑡/Γ(1.5)𝐻₂(𝑠Γ(1.5))²𝑑𝑠,𝑡≥0, (57)
where 𝐻₂(𝑡)=𝑔(𝑡)+∫^𝑡₀ℎ(𝜉)𝑑𝜉.

Proof. For the IVP (56), we have the following integral form: 𝑢³(𝑡)=1𝐶+Γ(0.5)×^𝑡₀(𝑡−𝑠)^−0.5×𝐿𝑠,𝑢(𝑠),^𝑠₀𝑀(𝜉,𝑢(𝜉))𝑑𝜉𝑑𝑠. (58)
So by (49) and (58), we deduce that 𝑢³(𝑡)−𝑢³(𝑡)1=𝐶−𝐶+Γ()×0.5_𝑡0(𝑡−𝑠)^−0.5×𝐿𝑠,𝑢_𝑠0(𝑠),𝑀((𝜉𝑢(𝑠),𝜉,𝑢))𝑑𝜉−𝐿𝑠,_𝑠0𝑀(𝜉,𝑢(𝜉))𝑑𝜉𝑑𝑠. (59)
Furthermore, 𝑢³(𝑡)−𝑢³(𝑡≤𝐶+1)𝐶−Γ()×0.5_𝑡0(𝑡−𝑠)^−0.5×𝐿𝑠,𝑢(𝑠),^𝑠₀𝑀((𝜉𝑢(𝑠),𝜉,𝑢))𝑑𝜉−𝐿𝑠,^𝑠₀𝑀(𝑢(𝜉1𝜉,))𝑑𝜉𝑑𝑠≤𝜀+Γ()×0.5_𝑡0(𝑡−𝑠)^−0.5×𝑔(𝑠)𝑢³(𝑠)−𝑢³(𝑠+_𝑠0)𝑀((𝜉−𝜉,𝑢))𝑑𝜉_𝑠0𝑀(𝑢(𝜉1𝜉,))𝑑𝜉𝑑𝑠≤𝜀+Γ()×0.5_𝑡0(𝑡−𝑠)^−0.5×𝑔(𝑠)𝑢³(𝑠)−𝑢³(𝑠+_𝑠0)ℎ(𝜉)𝑢³(𝜉)−𝑢³(𝜉)𝑑𝜉𝑑𝑠. (60)
A suitable application of Lemma 5 to (60) yields the desired result.

Remark 16. Theorem 15 indicates that the solution of the IVP (47) depends continuously on the initial value 𝑢(0)=𝐶.

Example 17. Consider the following fractional integral equation: 𝑢³(𝑡)=𝐶+^𝑡₀𝐿₁(𝑠,𝑢(𝑠))𝑑𝑠+𝐼^𝛼𝐿₂𝑡,𝑢(𝑡),^𝑡₀𝑀(𝜉,𝑢(𝜉))𝑑𝜉,0<𝛼<1. (61)

Theorem 18. Suppose that 𝑢(𝑡) is a solution of the fractional integral equation (61). If |𝐿₁(𝑡,𝑢)|≤𝑚(𝑡)|𝑢|³, |𝐿₂(𝑡,𝑢,𝑣)|≤𝑔(𝑡)|𝑢|+|𝑣|, and |𝑀(𝑡,𝑢)|≤ℎ(𝑡)|𝑢|, where 𝑚,𝑔, and ℎ are nonnegative continuous functions on [0,∞), then we have the following estimate for 𝑢(𝑡): |𝑢(𝑡1)|≤exp3^𝑡₀𝑚(𝑠)×𝐶|𝑑𝑠|exp^𝑡𝛼₀^/Γ(1+𝛼)𝐻₂(𝑠Γ(1+𝛼))^1/𝛼+1𝑑𝑠Γ(𝛼)^𝑡₀(𝑡−𝜏)^𝛼−1𝐻₁(𝜏)×exp−^𝑡𝛼_𝜏^/Γ(1+𝛼)_{𝛼()/Γ1+𝛼}𝐻₂×(𝑠Γ(1+𝛼))^1/𝛼𝑑𝑠𝑑𝜏^1/3,𝑡≥0, (62) where 𝐻₁(𝑡)=2𝐾^1/33𝑔(𝑡)+^𝑡₀ℎ(𝜉),𝐻𝑑𝜉₂(𝑡)=𝐾^−2/33𝑔(𝑡)+^𝑡₀ℎ(𝜉),𝑔(𝑡)(𝑡)1𝑑𝜉=𝑔exp3^𝑠₀𝑚(𝜏),ℎ(𝑡)(𝑡)1𝑑𝜏=ℎexp3^𝑡₀𝑚(𝜏)𝑑𝜏. (63)

Proof. From (61), we have 𝑢³(𝑡≤|𝐶|+)_𝑡0𝐿₁((𝑠+𝐼𝑠,𝑢))𝑑𝑠_𝛼𝐿₂𝑡,𝑢(𝑡),^𝑡₀𝑀((𝜉≤|𝐶|𝜉,𝑢))𝑑𝜉+^𝑡₀𝑚(𝑠)𝑢³(𝑠)+1𝑑𝑠Γ(𝛼)^𝑡₀(𝑡−𝑠)^𝛼−1×𝐿₂𝑠,𝑢(𝑠),^𝑠₀𝑀((𝜉≤|𝐶|+𝜉,𝑢))𝑑𝜉𝑑𝑠_𝑡0𝑚(𝑠)𝑢³(𝑠)+1𝑑𝑠Γ(𝛼)^𝑡₀(𝑡−𝑠)^𝛼−1×𝑔(𝑠)|𝑢(𝑠)|+^𝑠₀ℎ(𝜉)𝑢(𝜉)𝑑𝜉𝑑𝑠. (64)
Then a suitable application of Theorem 8 (with p = 3, q = r = 1) to (64) yields the desired estimate (62).

Theorem 19. For the fractional integral equation (62), if |𝐿₁(𝑡,𝑢₁)−𝐿₁(𝑡,𝑢₂)|≤𝑚(𝑡)|𝑢³₁−𝑢³₂||𝐿₂(𝑡,𝑢₁,𝑣₁)−𝐿₂(𝑡,𝑢₂,𝑣₂)|≤𝑔(𝑡)|𝑢³₁−𝑢³₂|+|𝑣₁−𝑣₂|, |𝑀(𝑡,𝑢₁)−𝑀(𝑡,𝑢₂)|≤ℎ(𝑡)|𝑢³₁−𝑢³₂|, where 𝑚,𝑔, and ℎ are nonnegative continuous functions on [0,∞), then the solution of (61) depends continuously on the initial value 𝐶.

Proof. Let 𝑢(𝑡) be the solution of the following fractional integral equation 𝑢³(𝑡)=𝐶+^𝑡₀𝐿₁(𝑠,𝑢(𝑠))𝑑𝑠+𝐼^𝛼𝐿₂𝑡,𝑢(𝑡),^𝑡₀𝑀(𝜉,𝑢(𝜉))𝑑𝜉,0<𝛼<1. (65)
A combination of (61) and (65) yields 𝑢³(𝑡)−𝑢³(𝑡)=𝐶−𝐶+^𝑡₀𝐿₁(𝑠,𝑢(𝑠))−𝐿₁(𝑠,𝑢(𝑠))𝑑𝑠+𝐼^𝛼𝐿₂𝑡,𝑢_𝑡0(𝑡),𝑀(𝜉,𝑢(𝜉))𝑑𝜉−𝐿₂𝑡,_𝑡0𝑢(𝑡),𝑀(𝜉,𝑢(𝜉))𝑑𝜉=𝐶−𝐶+^𝑡₀𝐿₁(𝑠,𝑢(𝑠))−𝐿₁(𝑢(𝑠+1𝑠,))𝑑𝑠Γ(𝛼)^𝑡₀(𝑡−𝑠)^𝛼−1×𝐿₂(𝑠),×𝑠,𝑢_𝑠0𝑀(𝜉,𝑢(𝜉))𝑑𝜉−𝐿₂𝑠,𝑢(𝑠),^𝑠₀𝑀(𝜉,𝑢(𝜉))𝑑𝜉𝑑𝑠. (66)
Furthermore, we have 𝑢³(𝑡)−𝑢³(𝑡≤𝐶+)𝐶−_𝑡0𝐿₁(𝑠,𝑢(𝑠))−𝐿₁(𝑢(𝑠+1𝑠,))𝑑𝑠Γ(𝛼)^𝑡₀(𝑡−𝑠)^𝛼−1×𝐿₂𝑠,𝑢(𝑠),^𝑠₀𝑀(𝜉,𝑢(𝜉))𝑑𝜉−𝐿₂𝑠,𝑢(𝑠),^𝑠₀𝑀(𝑢(𝜉≤𝐶+𝜉,))𝑑𝜉𝐶−_𝑡0𝑚(𝑠)𝑢³(𝑠)−𝑢³(𝑠+1)𝑑𝑠Γ(𝛼)^𝑡₀(𝑡−𝑠)^𝛼−1×𝑔(𝑠)𝑢³(𝑠)−𝑢³(𝑠+_𝑠0)𝑀((𝜉(𝑢(𝜉≤𝐶+𝜉,𝑢))−𝑀𝜉,))𝑑𝜉𝐶−_𝑡0𝑚(𝑠)𝑢³(𝑠)−𝑢³(𝑠+1)𝑑𝑠Γ(𝛼)^𝑡₀(𝑡−𝑠)^𝛼−1×𝑔(𝑠)𝑢³(𝑠)−𝑢³(𝑠+_𝑠0)ℎ(𝜉)𝑢³(𝜉)−𝑢³(𝜉)𝑑𝜉. (67)
Then treating |𝑢³(𝑡)−𝑢³(𝑡)| as one whole function, applying Theorem 8 to (67), we get that 𝑢³(𝑡)−𝑢³(𝑡≤𝐶)𝐶−exp^𝑡₀𝑚(𝑠)×𝑑𝑠exp^𝑡𝛼₀^/Γ(1+𝛼)𝐻₂×(𝑠Γ(1+𝛼))^1/𝛼𝑑𝑠,𝑡≥0, (68) where 𝐻₂(𝑡)=𝑔(𝑡)+^𝑡₀ℎ(𝜉)𝑔(𝑡)(𝑡)𝑑𝜉,=𝑔exp^𝑠₀𝑚(𝜏),ℎ(𝑡)(𝑡)𝑑𝜏=ℎexp^𝑡₀𝑚(𝜏)𝑑𝜏. (69)
So the continuous dependence on the initial value 𝐶 for the solution of (61) can be obtained from (68).

Remark 20. From the two examples presented above, one can see that the main results established in Section 2 are mainly used in the qualitative analysis as well as quantitative analysis of the solutions to some certain fractional differential or integral equations, such as the bound estimate, the number of the solutions, and the continuous dependence on the initial value for unknown solutions. On the other hand, by the variational iteration method and the homotopy perturbation method, approximate solutions for some fractional differential equations can be obtained (see the examples in [33–36], e.g.), while in few cases, the closed form of these approximate solutions can be obtained. So to this extent, we note that the starting point of establishing the main results in this paper is different from the variational iteration method and the homotopy perturbation method.

Acknowledgments

This work was partially supported by the Natural Science Foundation of Shandong Province (in China) (Grant no. ZR2013AQ009) and the Doctoral Initializing Foundation of Shandong University of Technology (in China) (Grant no. 4041-413030).