Lin Journal of Inequalities and Applications 2013, 2013:549http://www.journalofinequalitiesandapplications.com/content/2013/1/549

R E S E A R C H Open Access

Generalized Gronwall inequalities and theirapplications to fractional differentialequationsShi-you Lin*

*Correspondence:linsy1111@aliyun.comSchool of Mathematics andStatistics, Hainan Normal University,Haikou, Hainan 571158, People’sRepublic of China

AbstractIn this paper, we provide several generalizations of the Gronwall inequality andpresent their applications to prove the uniqueness of solutions for fractionaldifferential equations with various derivatives.MSC: 26A33; 34A08

Keywords: fractional derivative; differential equation; fractional order integral;generalized Gronwall inequality

1 IntroductionThe Gronwall inequality has an important role in numerous differential and integral equa-tions. The classical form of this inequality is described as follows, cf. [].

Theorem . For any t ∈ [t, T),

u(t) ≤ a(t) +∫ t

t

b(s)u(s) ds,

where b ≥ , then

u(t) ≤ a(t) +∫ t

t

a(s)b(s) exp

[∫ t

sb(u) du

]ds, t ∈ [t, T).

In particular, if a(t) is not decreasing, then

u(t) ≤ a(t) exp

[∫ t

t

b(s) ds]

, t ∈ [t, T).

In recent years, an increasing number of Gronwall inequality generalizations have beendiscovered to address difficulties encountered in differential equations, cf. [–]. Amongthese generalizations, we focus on the works of Ye, Gao and Qian, Gong, Li, the general-ized Gronwall inequality with Riemann-Liouville fractional derivative and the Hadamardderivative which are presented as follows.

©2013 Lin; licensee Springer. This is an Open Access article distributed under the terms of the Creative Commons Attribu-tion License (http://creativecommons.org/licenses/by/2.0), which permits unrestricted use, distribution, and reproduction in anymedium, provided the original work is properly cited.

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Theorem . ([]) For any t ∈ [, T),

u(t) ≤ a(t) + b(t)∫ t

(t – s)β–u(s) ds,

where all the functions are not negative and continuous. The constant β > . b is a boundedand monotonic increasing function on [, T), then

u(t) ≤ a(t) +∫ t

[ ∞∑n=

(b(t)�(β))n

�(nβ)(t – s)nβ–a(s)

]ds, t ∈ [, T).

Theorem . ([]) For any t ∈ [, T),

u(t) ≤ a(t) + b(t)∫ t

(ln

ts

)β– u(s)s

ds,

where all the functions are not negative and continuous. The constant β > . b is a boundedand monotonic increasing function on [, T), then

u(t) ≤ a(t) +∫ t

[ ∞∑n=

(b(t)�(β))n

�(nβ)

(ln

ts

)nβ–

a(s)

]dss

, t ∈ [, T).

The aforementioned inequalities are obtained using the estimation method of the com-position operators. This method is usually applied in studying qualitative theory of frac-tional differential equations. However, this method is not suitable for more complex situ-ations. Therefore, we shall use a simpler technique to prove the main results obtained inthis work.

Theorem . For any t ∈ [, T),

u(t) ≤ a(t) +n∑

i=

bi(t)∫ t

(t – s)βi–u(s) ds,

where all the functions are not negative and continuous. The constants βi > . bi (i =, , . . . , n) are the bounded and monotonic increasing functions on [, T), then

u(t) ≤ a(t) +∞∑

k=

( n∑′ ,′ ,...,k′=

∏ki=[bi′ (t)�(βi′ )]�(

∑ki= βi′ )

∫ t

(t – s)

∑ki= βi′ –a(s) ds

).

Theorem . For any t ∈ [, T),

u(t) ≤ a(t) +n∑

i=

bi(t)∫ t

(ln

ts

)βi– u(s)s

ds,

where all the functions are not negative and continuous. The constants βi > . bi (i =, , . . . , n) are the bounded and monotonic increasing functions on [, T), then

u(t) ≤ a(t) +∞∑

k=

( n∑′ ,′ ,...,k′=

∏ki=[bi′ (t)�(βi′ )]�(

∑ki= βi′ )

∫ t

[(ln

ts

)∑ki= βi′ –

a(s)]

dss

).

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2 The proof of the main resultsIn this section, we use the following critical lemmas to prove our main results.

Lemma . For any t ∈ [, T),

H(t) ≥n∑

i=

bi(t)∫ t

(t – s)βi–H(s) ds,

where all the functions are continuous. The constants βi > . bi (i = , , . . . , n) are thebounded, not negative, and monotonic increasing functions on [, T), then H(t) ≥ , t ∈[, T).

Proof Obviously, H() ≥ . If the proposition is false, that is,

{t|t ∈ [, T), H(t) <

} �= �,

where � is an empty set, then a number t exists on [, T) which satisfies H|[,t] ≥ ,H(t) = . H is a strictly monotonic decreasing function on (t, t + ξ) ⊂ [, T). Here,ξ > . Therefore, for any t ∈ (t, t + ξ), H(t) < and

H(t) ≥n∑

i=

bi(t)∫ t

(t – s)βi–H(s) ds

≥n∑

i=

bi(t)∫ t

t

(t – s)βi–H(s) ds

≥n∑

i=

bi(t)H(t)∫ t

t

(t – s)βi– ds

= H(t) ·n∑

i=

bi(t)(t – t)βi

βi,

which implies that

n∑i=

bi(t)(t – t)βi

βi≥ .

Let t → t, then we have a contradiction, that is, ≥ . This process completes the proofof Lemma .. �

The next lemma is given following the same method as for the previous lemma. Theproving process is relatively similar, thus we do not include it in this paper.

Lemma . For any t ∈ [, T),

H(t) ≥n∑

i=

bi(t)∫ t

(ln

ts

)βi– H(s)s

ds,

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where all the functions are continuous. The constants βi > . bi (i = , , . . . , n) are thebounded, not negative, and monotonic increasing functions on [, T), then H(t) ≥ , t ∈[, T).

Our next task is proving our main results. To prove Theorem ., we initially supposethat

g(t) = a(t) +∞∑

k=

( n∑′ ,′ ,...,k′=

∏ki=[bi′ (t)�(βi′ )]�(

∑ki= βi′ )

∫ t

(t – s)

∑ki= βi′ –a(s) ds

).

Then, the following equality is given:∫ t

∫ s

(t – s)βj–(s – u)

∑ki= βi′ –a(u) du ds

=∫ t

∫ t

u(t – s)βj–(s – u)

∑ki= βi′ –a(u) ds du

=�(βj)�(

∑ki= βi′ )

�(βj +∑k

i= βi′ )

∫ t

(t – s)βj+

∑ki= βi′ –a(s) ds.

This equality, combined with the fact that bi (i = , , . . . , n) are the monotonic increasingfunctions on [, T), yields

n∑i=

bi(t)∫ t

(t – s)βi–g(s) ds

≤∞∑

k=

n∑j=

n∑′ ,′ ,...,k′=

bj(t)∫ t

∫ s

∏ki=[bi′ (s)�(βi′ )]�(

∑ki= βi′ )

(t – s)βj–(s – u)∑k

i= βi′ – · a(u) du ds

+n∑

i=

bi(t)∫ t

(t – s)βi–a(s) ds

≤∞∑

k=

n∑j=

n∑′ ,′ ,...,k′=

bj(t)∫ t

∫ s

∏ki=[bi′ (t)�(βi′ )]�(

∑ki= βi′ )

(t – s)βj–(s – u)∑k

i= βi′ – · a(u) du ds

+n∑

i=

bi(t)∫ t

(t – s)βi–a(s) ds

≤∞∑

k=

( n∑′ ,′ ,...,k′=

∏ki=[bi′ (t)�(βi′ )]�(

∑ki= βi′ )

∫ t

(t – s)

∑ki= βi′ –a(s) ds

)

= g(t) – a(t),

which indicates that

u(t) –n∑

i=

bi(t)∫ t

(t – s)βi–u(s) ds ≤ a(t) ≤ g(t) –

n∑i=

bi(t)∫ t

(t – s)βi–g(s) ds.

Let H(t) = g(t) – u(t), then we obtain

H(t) ≥n∑

i=

bi(t)∫ t

(t – s)βi–H(s) ds.

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According to Lemma ., H(t) ≥ . That is, u(t) ≤ g(t) and t ∈ [, T). This process com-pletes the proof of Theorem ..

We can prove Theorem . by applying Lemma . in the same manner as in the previoustheorem. We conclude the main results of this work.

3 ApplicationsIn this section, we apply the main results of this work to demonstrate the uniqueness ofthe solution for fractional differential equations. First, the following initial value problemswith the Riemann-Liouville fractional derivative are considered:⎧⎨

⎩∑n

i= DβiR u(t) = f (t, u(t)),∑n

i= I–βiR u(t)|t= = δ,

(.)

where < β < β < · · · < βn < , t ∈ [, T), Dβ

R and Iβ

R denote the Riemann-Liouville frac-tional derivative and fractional integral operators, respectively.

Definition . ([–]) The βth Riemann-Liouville-type fractional order integral of afunction u : [, +∞) →R is defined by

Iβ

R u(t) =

�(β)

∫ t

(t – s)β–u(s) ds,

where β > and � is the gamma function.

Definition . ([, , –]) For any < β < , the βth Riemann-Liouville-type frac-tional order derivative of a function u : [, +∞) →R is defined by

Dβ

Ru(t) =

�( – β)ddt

∫ t

(t – s)–βu(s) ds.

Lemma . ([, , ]) For any < β < ,

Iβ

R Dβ

Ru(t) = u(t) + c · tβ–,

where c is a constant in R.

Lemma . ([]) For any α,β > ,

IαR Iβ

R u(t) = Iα+β

R u(t).

We can then state the next theorem.

Theorem . For any t ∈ [, T), suppose that |f (t, y) – f (t, z)| ≤ γ (t)|y – z|, and γ (t) ≥ isa bounded and monotonic increasing function. If initial value problem (.) has a solution,then the solution is unique.

Proof Since < β < β < · · · < βn < , then according to Lemma ., we can suppose that

IβiR Dβi

R u(t) = u(t) + citβ–,

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where ci, i = , , . . . , n, are some constants. This equality, combined with Lemma ., en-ables us to change problem (.) into the following:

IβnR f

(t, u(t)

)=

n∑i=

IβnR Dβi

R u(t) =n∑

i=

Iβn–βiR

(Iβi

R DβiR u(t)

)

=n∑

i=

Iβn–βiR

(u(t) + citβi–)

=n∑

i=

Iβn–βiR u(t) +

tβn–

�(βn)·

n∑i=

(ci�(βi)

).

Therefore,

= I–βnR Iβn

R f |t=

=n∑

i=

I–βnR Iβn–βi

R u|t= +I–βn

R (tβn–)|t=

�(βn)·

n∑i=

(ci�(βi)

)

=n∑

i=

I–βiR u|t= +

n∑i=

(ci�(βi)

)

= δ +n∑

i=

(ci�(βi)

).

That is,

n∑i=

(ci�(βi)

)= –δ,

which implies that

u(t) = IβnR f

(t, u(t)

)–

n–∑i=

Iβn–βiR u(t) + δ · tβn–

�(βn).

If u(t) and u(t) are two solutions to problem (.), then they also satisfy the above equal-ity. Thus we have

∣∣u(t) – u(t)∣∣ =

∣∣∣∣∣IβnR

[f(t, u(t)

)– f

(t, u(t)

)]–

n–∑i=

Iβn–βiR

[u(t) – u(t)

]∣∣∣∣∣≤

�(βn)

∫ t

(t – s)βn–∣∣f (s, u(s)

)– f

(s, u(s)

)∣∣ds

+n–∑i=

�(βn – βi)

∫ t

(t – s)βn–βi–∣∣u(s) – u(s)

∣∣ds

≤ γ (t)�(βn)

∫ t

(t – s)βn–∣∣u(s) – u(s)

∣∣ds

+n–∑i=

�(βn – βi)

∫ t

(t – s)βn–βi–∣∣u(s) – u(s)

∣∣ds.

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According to Theorem ., we can conclude that

∣∣u(t) – u(t)∣∣ ≤ ,

which indicates that

u(t) = u(t), t ∈ [, T).

This process completes the proof of Theorem .. �

Next, we study the uniqueness of the solution for the following initial value problemswith the Hadamard-type fractional derivative:

⎧⎨⎩

∑ni= Dαi

H v(t) = g(t, v(t)),∑ni= I–αi

H v(t)|t= = η,(.)

where < α < α < · · · < αn < , t ∈ [, T). For any α ∈ (, ), DαH and Iα

H are defined asfollows:⎧⎨

⎩DαHv(t) =

�(–α) (t ddt )

∫ t (ln t

s )–α v(s)s ds,

IαHv(t) =

�(α)∫ t

(ln ts )α– v(s)

s ds.(.)

From [], we obtain

IαnH g

(t, v(t)

)=

n∑i=

IαnH Dαi

H v(t)

=n∑

i=

Iαn–αiH

(Iαi

H DαiH v(t)

)

=n∑

i=

Iαn–αiH

[v(t) + di · (log t)αi–]

=n∑

i=

Iαn–αiH v(t) +

(log t)αn–

�(αn)·

n∑i=

(di�(αi)

), (.)

where di, i = , , . . . , n, are some constants. Therefore,

=

�()

∫ t

(ln

ts

)– g(s, v(s))s

ds∣∣∣∣t=

= I–αnH Iαn

H g|t=

=n∑

i=

I–αnH Iαn–αi

H v|t= +I–αn

H ((log t)αn–)|t=

�(αn)·

n∑i=

(di�(αi)

)

=n∑

i=

I–αiH v|t= +

n∑i=

(di�(αi)

)

= η +n∑

i=

(di�(αi)

).

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When the aforementioned equality is plugged into problem (.), we obtain

v(t) = IαnH g

(t, v(t)

)–

n–∑i=

Iαn–αiH v(t) +

(log t)αn–

�(αn)· η. (.)

This equality, combined with Theorem ., can derive the next theorem. The procedureis relatively similar to the proof of Theorem ., thus we do not include it in this paper.

Theorem . For any t ∈ [, T), suppose that |g(t, y) – g(t, z)| ≤ ζ (t)|y – z|, and ζ (t) ≥ isa bounded and monotonic increasing function. If initial value problem (.) has a solution,then the solution is unique.

4 Concluding remarksIn this work, we have obtained generalizations of the Gronwall inequality using severalmathematical techniques. In addition, we have listed the initial value problems, namely(.) and (.), and proved the uniqueness of solutions to these problems by applying thegeneralized Gronwall inequalities.

Competing interestsThe author declares that they have no competing interests.

AcknowledgementsThis work was supported by the National Natural Science Foundation of China (Grant Nos. 11226167 and 11361020), theNatural Science Foundation of Hainan Province (No. 111005) and the Ph.D. Scientific Research Starting Foundation ofHainan Normal University (No. HSBS1016).

Received: 23 July 2013 Accepted: 18 October 2013 Published: 22 Nov 2013

References1. Corduneanu, C: Principles of Differential and Integral Equations. Allyn & Bacon, Boston (1971)2. Abdeldaim, A, Yakout, M: On some new integral inequalities of Gronwall-Bellman-Pachpatte type. Appl. Math.

Comput. 217, 7887-7899 (2011)3. Feng, Q, Zheng, B: Generalized Gronwall-Bellman-type delay dynamic inequalities on time scales and their

applications. Appl. Math. Comput. 218, 7880-7892 (2012)4. Li, L, Meng, F, He, L: Some generalized integral inequalities and their applications. J. Math. Anal. Appl. 372, 339-349

(2010)5. Qian, D, Gong, Z, Li, C: A generalized Gronwall inequality and its application to fractional differential equations with

Hadamard derivatives. http://nsc10.cankaya.edu.tr/proceedings/PAPERS/Symp2-Fractional%20Calculus%20Applications/Paper26.pdf

6. Ye, H, Gao, J, Ding, Y: A generalized Gronwall inequality and its application to a fractional differential equation.J. Math. Anal. Appl. 328, 1075-1081 (2007)

7. Ye, H, Gao, J: Henry-Gronwall type retarded integral inequalities and their applications to fractional differentialequations with delay. Appl. Math. Comput. 218, 4125-4160 (2011)

8. Ahmad, B, Ntouyas, S, Alsaedi, A: New existence results for nonlinear fractional differential equations with three-pointintegral boundary conditions. Adv. Differ. Equ. 2011, Article ID 107384 (2011)

9. Benchohra, M, Hamani, S, Ntouyas, S: Boundary value problems for differential equations with fractional order. Surv.Math. Appl. 3, 1-12 (2008)

10. Feng, M, Liu, X, Feng, H: The existence of positive solution to a nonlinear fractional differential equation with integralboundary conditions. Adv. Differ. Equ. 2011, Article ID 546038 (2011)

11. Furati, K, Tatar, N: An existence result for a nonlocal fractional differential problem. J. Fract. Calc. 26, 43-51 (2004)12. Lv, Z, Liang, J, Xiao, T: Solutions to fractional differential equations with nonlocal initial condition in Banach spaces.

Adv. Differ. Equ. 2010, Article ID 340349 (2010)13. Momani, S, Jameel, A, Azawi, S: Local and global uniqueness theorems on fractional integro-differential equations via

Bihari’s and Gronwall’s inequalities. Soochow J. Math. 33(4), 619-627 (2007)14. Lv, Z: Positive solutions ofm-point boundary value problems for fractional differential equations. Adv. Differ. Equ.

2011, Article ID 571804 (2011)15. Dixit, S, Singh, O, Kumar, S: An analytic algorithm for solving system of fractional differential equations. J. Mod.

Methods Numer. Math. 1(1), 12-26 (2010)16. Kilbas, A, Srivastava, H, Trujillo, J: Theory and Applications of Fractional Differential Equations. Elsevier, Amsterdam

(2006)

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10.1186/1029-242X-2013-549Cite this article as: Lin: Generalized Gronwall inequalities and their applications to fractional differential equations.Journal of Inequalities and Applications 2013, 2013:549