Open Access

Explicit bounds derived by some new inequalities and applications in fractional integral equations

Journal of Inequalities and Applications20142014:4

https://doi.org/10.1186/1029-242X-2014-4

Received: 8 June 2013

Accepted: 1 December 2013

Published: 2 January 2014

Abstract

In this paper, we present some new Gronwall-type inequalities. Explicit bounds for the unknown functions concerned are derived based on these inequalities and the properties of the modified Riemann-Liouville fractional derivative. The inequalities established are of new forms compared with the existing results so far in the literature. For illustrating the validity of the inequalities established, we apply them to research the boundedness, quantitative property, and continuous dependence on the initial value for the solution to a certain fractional integral equation.

MSC:26D10.

Keywords

Gronwall-type inequalityexplicit boundfractional differential equationqualitative analysisquantitative analysis

1 Introduction

Recently, with the development of the theory of differential equations, many authors have researched various inequalities and investigated the boundedness, global existence, uniqueness, stability, and continuous dependence on the initial value and parameters of solutions to differential equations, integral equations as well as difference equations. The Gronwall-Bellman inequality [1, 2] is widely used in the qualitative and quantitative analysis of differential equations, as it can provide explicit bound for an unknown function lying in the inequality. In the last few decades, many authors have researched various generalizations of the Gronwall-Bellman inequality; for example, we refer the reader to [328] and the references therein. These Gronwall-type inequalities established can be used as a handy tool in the research of the theory of differential and integral equations as well as difference equations. However, we notice that the existing results in the literature are inadequate for researching the qualitative and quantitative properties of solutions to some fractional integral equations, for example, the following fractional integral equation:
u ( t ) = u ( 0 ) + I α ( f ( t , u ( t ) ) ) + 1 Γ ( α ) 0 T ( T s ) α 1 f ( s , u ( s ) ) d s ,

where 0 < α < 1 , T 0 is a constant, I α denotes the Riemann-Liouville fractional integral of order α.

So it is necessary to establish some new Gronwall-type inequalities in order to fulfill the desired analysis result.

The modified Riemann-Liouville fractional derivative, presented by Jumarie in [29, 30], is defined by the following expression.

Definition 1 The modified Riemann-Liouville derivative of order α is defined by the following expression:
D t α f ( t ) = { 1 Γ ( 1 α ) d d t 0 t ( t ξ ) α ( f ( ξ ) f ( 0 ) ) d ξ , 0 < α < 1 , ( f ( n ) ( t ) ) ( α n ) , n α < n + 1 , n 1 .
Definition 2 The Riemann-Liouville fractional integral of order α on the interval [ 0 , t ] is defined by
I α f ( t ) = 1 Γ ( 1 + α ) 0 t f ( s ) ( d s ) α = 1 Γ ( α ) 0 t ( t s ) α 1 f ( s ) d s .
Some important properties for the modified Riemann-Liouville derivative and fractional integral are listed as follows (see [31, 32] and the interval concerned below is always defined by [ 0 , t ] ):
  1. (a)

    D t α t r = Γ ( 1 + r ) Γ ( 1 + r α ) t r α .

     
  2. (b)

    D t α ( f ( t ) g ( t ) ) = g ( t ) D t α f ( t ) + f ( t ) D t α g ( t ) .

     
  3. (c)

    D t α f [ g ( t ) ] = f g [ g ( t ) ] D t α g ( t ) = D g α f [ g ( t ) ] ( g ( t ) ) α .

     
  4. (d)

    I α ( D t α f ( t ) ) = f ( t ) f ( 0 ) .

     
  5. (e)

    I α ( g ( t ) D t α f ( t ) ) = f ( t ) g ( t ) f ( 0 ) g ( 0 ) I α ( f ( t ) D t α g ( t ) ) .

     
  6. (f)

    D t α C = 0 , where C is a constant.

     

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 [32, 33], the authors sought exact solutions for some types of fractional differential equations based on the modified Riemann-Liouville fractional derivative, and in [34], the modified Riemann-Liouville fractional derivative was used in fractional calculus of variations, where a fractional basic problem of the calculus of variations with free boundary conditions as well as problems with isoperimetric and holonomic constraints were considered. In [35], 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 [3638], the fractional variational iteration method based on the modified Riemann-Liouville fractional derivative was concerned. In [39], a fractional variational homotopy perturbation iteration method was proposed.

Based on the analysis above, in Section 2, we present some new Gronwall-type inequalities, based on which and some basic properties of the modified Riemann-Liouville fractional derivative, we derive explicit bounds for the unknown functions concerned in these inequalities. In Section 3, we apply the results established in Section 2 to research boundedness, quantitative property, and continuous dependence on the initial data for the solution to a certain fractional integral equation.

2 Main results

Lemma 1 Suppose 0 < α < 1 , f is a continuous function, then D α ( I t α f ( t ) ) = f ( t ) .

Proof Since f is continuous, then there exists a constant M such that | f ( t ) | M for t [ 0 , ε ] , where ε > 0 . So, for t [ 0 , ε ] , we have | I t α f ( t ) | = | 1 Γ ( α ) 0 t ( t s ) α 1 f ( s ) d s | M Γ ( α ) 0 t ( t s ) α 1 d s = M α Γ ( α ) t α . Then one can see I t α f ( 0 ) = 0 . Therefore,
D α ( I t α f ( t ) ) = 1 Γ ( 1 α ) d d t { 0 t ( t ξ ) α ( I t α f ( ξ ) I t α f ( 0 ) ) d ξ } = 1 Γ ( α ) Γ ( 1 α ) d d t { 0 t ( t ξ ) α 0 ξ ( ξ s ) α 1 f ( s ) d s d ξ } = 1 Γ ( α ) Γ ( 1 α ) d d t { 0 t 0 ξ ( t ξ ) α ( ξ s ) α 1 f ( s ) d s d ξ } = 1 Γ ( α ) Γ ( 1 α ) d d t { 0 t f ( s ) s t ( t ξ ) α ( ξ s ) α 1 d ξ d s } .
Letting ξ = s + ( t s ) x , we obtain that
D α ( I t α f ( t ) ) = 1 Γ ( α ) Γ ( 1 α ) d d t { 0 t f ( s ) 0 1 ( 1 x ) α x α 1 d x d s } = B ( α , 1 α ) Γ ( α ) Γ ( 1 α ) d d t { 0 t f ( s ) d s } = f ( t ) ,

where B ( , ) denotes the beta function. The proof is complete. □

Theorem 2 Suppose 0 < α < 1 , the functions u, g are nonnegative continuous functions defined on t 0 , T 0 is a constant. If the following inequality is satisfied
u ( t ) C + 1 Γ ( α ) 0 t ( t s ) α 1 g ( s ) u ( s ) d s + 1 Γ ( α ) 0 T ( T s ) α 1 g ( s ) u ( s ) d s , t [ 0 , T ] ,
(1)
then we have the following explicit estimate for u ( t ) :
u ( t ) C exp [ 0 t α Γ ( 1 + α ) g ( ( s Γ ( 1 + α ) ) 1 α ) d s ] 2 exp [ 0 T α Γ ( 1 + α ) g ( ( s Γ ( 1 + α ) ) 1 α ) d s ] , t [ 0 , T ] ,
(2)

provided that exp [ 0 T α Γ ( 1 + α ) g ( ( s Γ ( 1 + α ) ) 1 α ) d s ] < 2 .

Proof Denote the right-hand side of (1) by v ( t ) . Then we have
u ( t ) v ( t ) , t [ 0 , T ] ,
(3)
and, by use of Lemma 1 and the property ( f ) , we obtain
D t α v ( t ) = g ( t ) u ( t ) g ( t ) v ( t ) .
Furthermore, by the properties (a), (b), (c), we have:
D t α { v ( t ) exp [ 0 t α Γ ( 1 + α ) g ( ( s Γ ( 1 + α ) ) 1 α ) d s ] } = exp [ 0 t α Γ ( 1 + α ) g ( ( s Γ ( 1 + α ) ) 1 α ) d s ] D t α v ( t ) + v ( t ) D t α { exp [ 0 t α Γ ( 1 + α ) g ( ( s Γ ( 1 + α ) ) 1 α ) d s ] } = exp [ 0 t α Γ ( 1 + α ) g ( ( s Γ ( 1 + α ) ) 1 α ) d s ] D t α v ( t ) g ( t ) v ( t ) exp [ 0 t α Γ ( 1 + α ) g ( ( s Γ ( 1 + α ) ) 1 α ) d s ] D t α ( t α Γ ( 1 + α ) ) = exp [ 0 t α Γ ( 1 + α ) g ( ( s Γ ( 1 + α ) ) 1 α ) d s ] [ D t α v ( t ) g ( t ) v ( t ) ] 0 .
(4)
Substituting t with τ, fulfilling a fractional integral of order α for (3) with respect to τ from 0 to t, we deduce that
v ( t ) exp [ 0 t α Γ ( 1 + α ) g ( ( s Γ ( 1 + α ) ) 1 α ) d s ] v ( 0 ) ,
which implies
v ( t ) exp [ 0 t α Γ ( 1 + α ) g ( ( s Γ ( 1 + α ) ) 1 α ) d s ] v ( 0 ) , t [ 0 , T ] .
(5)
On the other hand, we have
2 v ( 0 ) C = v ( T ) exp [ 0 T α Γ ( 1 + α ) g ( ( s Γ ( 1 + α ) ) 1 α ) d s ] v ( 0 ) ,
which is followed by
v ( 0 ) C 2 exp [ 0 T α Γ ( 1 + α ) g ( ( s Γ ( 1 + α ) ) 1 α ) d s ] .
(6)

Combining (3), (5), (6), we can get the desired result. □

Now we study the inequality of the following form:
u p ( t ) C + 0 t h ( s ) u p ( s ) d s + 1 Γ ( α ) 0 t ( t s ) α 1 g ( s ) u q ( s ) d s + 0 T h ( s ) u p ( s ) d s + 1 Γ ( α ) 0 T ( T s ) α 1 g ( s ) u q ( s ) d s , t [ 0 , T ] ,
(7)

where 0 < α < 1 , the functions u, g, h are nonnegative continuous functions defined on t 0 , and T 0 is a constant, p, q are constants with p q > 0 .

The following lemma is useful in deriving explicit bound for the function u ( t ) in (7).

Lemma 3 [24]

Assume that a 0 , p q 0 , and p 0 , then for any K > 0 ,
a q p q p K q p p a + p q p K q p .
Theorem 4 The inequality admits the following explicit estimate for u ( t ) :
u ( t ) { { p q p K q p 1 Γ ( α ) 0 t ( t s ) α 1 g ( s ) exp [ q p 0 s h ( ξ ) d ξ ] d s + C + [ p q p K q p 1 Γ ( α ) 0 T ( T s ) α 1 g ( s ) exp [ q p 0 s h ( ξ ) d ξ ] d s ] exp [ 0 T h ( s ) d s ] 2 exp [ 0 T h ( s ) d s ] + exp [ 0 T h ( s ) d s ] × ( { 1 Γ ( α ) exp [ 0 T α Γ ( 1 + α ) g ˜ ( ( s Γ ( 1 + α ) ) 1 α ) d s ] 0 T ( T τ ) α 1 a ( τ ) g ˜ ( τ ) × exp { 0 τ α Γ ( 1 + α ) g ˜ ( ( s Γ ( 1 + α ) ) 1 α ) d s } d τ } ) / ( 2 exp [ 0 T h ( s ) d s ] ) + 1 Γ ( α ) exp [ 0 t α Γ ( 1 + α ) g ˜ ( ( s Γ ( 1 + α ) ) 1 α ) d s ] 0 t ( t τ ) α 1 a ( τ ) g ˜ ( τ ) × exp { 0 τ α Γ ( 1 + α ) g ˜ ( ( s Γ ( 1 + α ) ) 1 α ) d s } d τ } × exp [ 0 t h ( s ) d s ] } 1 p , t [ 0 , T ] ,
(8)
provided that exp [ 0 T h ( s ) d s ] < 2 , where K > 0 , and
g ˜ ( t ) = q p K q p p g ( t ) exp [ q p 0 t h ( ξ ) d ξ ] .
Proof Denote the right-hand side of (7) by v ( t ) . Then we have
u ( t ) v 1 p ( t ) , t [ 0 , T ] ,
(9)
and considering v ( 0 ) = C + 0 T h ( s ) u p ( s ) d s + 1 Γ ( α ) 0 T ( T s ) α 1 g ( s ) u q ( s ) d s , it follows that
v ( t ) v ( 0 ) + 0 t h ( s ) v ( s ) d s + 1 Γ ( α ) 0 t ( t s ) α 1 g ( s ) v q p ( s ) d s , t [ 0 , T ] .
(10)
Let z ( t ) = v ( 0 ) + 1 Γ ( α ) 0 t ( t s ) α 1 g ( s ) v q p ( s ) d s . Then
v ( t ) z ( t ) + 0 t h ( s ) v ( s ) d s , t [ 0 , T ] ,
which implies that
v ( t ) z ( t ) exp [ 0 t h ( s ) d s ] , t [ 0 , T ] .
(11)
So
z ( t ) v ( 0 ) + 1 Γ ( α ) 0 t ( t s ) α 1 g ( s ) exp [ q p 0 s h ( ξ ) d ξ ] z q p ( s ) d s , t [ 0 , T ] .
Using Lemma 3, we get that
z ( t ) v ( 0 ) + 1 Γ ( α ) 0 t ( t s ) α 1 g ( s ) exp [ q p 0 s h ( ξ ) d ξ ] [ q p K q p p z ( s ) + p q p K q p ] d s = v ( 0 ) + p q p K q p 1 Γ ( α ) 0 t ( t s ) α 1 g ( s ) exp [ q p 0 s h ( ξ ) d ξ ] d s + q p K q p p 1 Γ ( α ) 0 t ( t s ) α 1 g ( s ) exp [ q p 0 s h ( ξ ) d ξ ] z ( s ) d s = a ( t ) + 1 Γ ( α ) 0 t ( t s ) α 1 g ˜ ( s ) z ( s ) d s , t [ 0 , T ] ,
where g ˜ ( t ) is defined as above, and
a ( t ) = v ( 0 ) + p q p K q p 1 Γ ( α ) 0 t ( t s ) α 1 g ( s ) exp [ q p 0 s h ( ξ ) d ξ ] d s .
Let w ( t ) = 1 Γ ( α ) 0 t ( t s ) α 1 g ˜ ( s ) z ( s ) d s . Then
z ( t ) a ( t ) + w ( t ) , t [ 0 , T ] ,
(12)
and
D t α w ( t ) = g ˜ ( t ) z ( t ) a ( t ) g ˜ ( t ) + g ˜ ( t ) w ( t ) .
By the properties (a), (b), and (c), we get that
D t α { w ( t ) exp [ 0 t α Γ ( 1 + α ) g ˜ ( ( s Γ ( 1 + α ) ) 1 α ) d s ] } = exp [ 0 t α Γ ( 1 + α ) g ˜ ( ( s Γ ( 1 + α ) ) 1 α ) d s ] D t α w ( t ) + w ( t ) D t α { exp [ 0 t α Γ ( 1 + α ) g ˜ ( ( s Γ ( 1 + α ) ) 1 α ) d s ] } = exp [ 0 t α Γ ( 1 + α ) g ˜ ( ( s Γ ( 1 + α ) ) 1 α ) d s ] D t α w ( t ) g ˜ ( t ) w ( t ) exp [ 0 t α Γ ( 1 + α ) g ˜ ( ( s Γ ( 1 + α ) ) 1 α ) d s ] D t α ( t α Γ ( 1 + α ) ) = exp [ 0 t α Γ ( 1 + α ) g ˜ ( ( s Γ ( 1 + α ) ) 1 α ) d s ] [ D t α w ( t ) g ˜ ( t ) w ( t ) ] a ( t ) g ˜ ( t ) exp [ 0 t α Γ ( 1 + α ) g ˜ ( ( s Γ ( 1 + α ) ) 1 α ) d s ] , t [ 0 , T ] .
(13)
Substituting t with τ, fulfilling a fractional integral of order α for (13) with respect to τ from 0 to t, and using w ( 0 ) = 0 , we deduce that
w ( t ) exp { 0 t α Γ ( 1 + α ) g ˜ ( ( s Γ ( 1 + α ) ) 1 α ) d s } 1 Γ ( α ) 0 t ( t τ ) α 1 a ( τ ) g ˜ ( τ ) exp [ 0 τ α Γ ( 1 + α ) g ˜ ( ( s Γ ( 1 + α ) ) 1 α ) d s ] d τ ,
which implies
w ( t ) 1 Γ ( α ) exp [ 0 t α Γ ( 1 + α ) g ˜ ( ( s Γ ( 1 + α ) ) 1 α ) d s ] × 0 t ( t τ ) α 1 a ( τ ) g ˜ ( τ ) exp { 0 τ α Γ ( 1 + α ) g ˜ ( ( s Γ ( 1 + α ) ) 1 α ) d s } d τ .
(14)
Combining (11), (12), and (14), we get that
2 v ( 0 ) C = v ( T ) z ( T ) exp [ 0 T h ( s ) d s ] [ a ( T ) + w ( T ) ] exp [ 0 T h ( s ) d s ] { v ( 0 ) + p q p K q p 1 Γ ( α ) 0 T ( T s ) α 1 g ( s ) exp [ q p 0 s h ( ξ ) d ξ ] d s + 1 Γ ( α ) exp [ 0 T α Γ ( 1 + α ) g ˜ ( ( s Γ ( 1 + α ) ) 1 α ) d s ] × 0 T ( T τ ) α 1 a ( τ ) g ˜ ( τ ) exp { 0 τ α Γ ( 1 + α ) g ˜ ( ( s Γ ( 1 + α ) ) 1 α ) d s } d τ } × exp [ 0 T h ( s ) d s ] ,
which implies
v ( 0 ) C + [ p q p K q p 1 Γ ( α ) 0 T ( T s ) α 1 g ( s ) exp [ q p 0 s h ( ξ ) d ξ ] d s ] exp [ 0 T h ( s ) d s ] 2 exp [ 0 T h ( s ) d s ] + exp [ 0 T h ( s ) d s ] × ( { 1 Γ ( α ) exp [ 0 T α Γ ( 1 + α ) g ˜ ( ( s Γ ( 1 + α ) ) 1 α ) d s ] 0 T ( T τ ) α 1 a ( τ ) g ˜ ( τ ) × exp { 0 τ α Γ ( 1 + α ) g ˜ ( ( s Γ ( 1 + α ) ) 1 α ) d s } d τ } ) / ( 2 exp [ 0 T h ( s ) d s ] ) ,
(15)

under the condition exp [ 0 T h ( s ) d s ] < 2 .

The desired result can be obtained by the combination of (11), (12), (14), and (15). □

Theorem 5 Suppose 0 < α < 1 , the function u is a nonnegative continuous function defined on t 0 , p, T are constants with p 1 , T 0 , L C ( R + 2 , R + ) satisfying 0 L ( t , u ) L ( t , v ) M ( u v ) for u v , t 0 , where M > 0 is a constant. If the following inequality is satisfied
u p ( t ) C + 1 Γ ( α ) 0 t ( t s ) α 1 L ( s , u ( s ) ) d s + 1 Γ ( α ) 0 T ( T s ) α 1 L ( s , u ( s ) ) d s , t [ 0 , T ] ,
(16)
then we have the following explicit estimate for u ( t ) :
u ( t ) exp [ M t α p Γ ( 1 + α ) K 1 p p ] 2 exp [ M T α p Γ ( 1 + α ) K 1 p p ] [ C + 2 T α α Γ ( α ) L ( s , p 1 p K 1 p ) ] , t [ 0 , T ] ,
(17)

provided that exp [ M T α p Γ ( 1 + α ) K 1 p p ] < 2 .

Proof Denote the right-hand side of (16) by v ( t ) . Then we have
u ( t ) v 1 p ( t ) , t [ 0 , T ] ,
(18)
and
v ( t ) C + 1 Γ ( α ) 0 t ( t s ) α 1 L ( s , v 1 p ( s ) ) d s + 1 Γ ( α ) 0 T ( T s ) α 1 L ( s , v 1 p ( s ) ) d s C + 1 Γ ( α ) 0 t ( t s ) α 1 L ( s , 1 p K 1 p p v ( s ) + p 1 p K 1 p ) d s + 1 Γ ( α ) 0 T ( T s ) α 1 L ( s , 1 p K 1 p p v ( s ) + p 1 p K 1 p ) d s = C + 1 Γ ( α ) 0 t ( t s ) α 1 [ L ( s , 1 p K 1 p p v ( s ) + p 1 p K 1 p ) L ( s , p 1 p K 1 p ) + L ( s , p 1 p K 1 p ) ] d s + 1 Γ ( α ) 0 T ( T s ) α 1 [ L ( s , 1 p K 1 p p v ( s ) + p 1 p K 1 p ) L ( s , p 1 p K 1 p ) + L ( s , p 1 p K 1 p ) ] d s C + 1 Γ ( α ) 0 t ( t s ) α 1 L ( s , p 1 p K 1 p ) d s + M p Γ ( α ) K 1 p p 0 t ( t s ) α 1 v ( s ) d s + 1 Γ ( α ) 0 T ( T s ) α 1 L ( s , p 1 p K 1 p ) d s + M p Γ ( α ) K 1 p p 0 T ( T s ) α 1 v ( s ) d s = C + t α α Γ ( α ) L ( s , p 1 p K 1 p ) + M p Γ ( α ) K 1 p p 0 t ( t s ) α 1 v ( s ) d s + T α α Γ ( α ) L ( s , p 1 p K 1 p ) + M p Γ ( α ) K 1 p p 0 T ( T s ) α 1 v ( s ) d s C + 2 T α α Γ ( α ) L ( s , p 1 p K 1 p ) + M p Γ ( α ) K 1 p p 0 t ( t s ) α 1 v ( s ) d s + M p Γ ( α ) K 1 p p 0 T ( T s ) α 1 v ( s ) d s , t [ 0 , T ] .
(19)

Then a suitable application of Theorem 2 to (19) yields the desired result. □

3 Applications

In this section, we present one example for the results established above, in which the boundedness, quantitative property, and continuous dependence on the initial value for the solutions to one certain fractional integral equation are researched.

Example Consider the following fractional integral equation:
u ( t ) = u ( 0 ) + I α ( f ( t , u ( t ) ) ) + 1 Γ ( α ) 0 T ( T s ) α 1 f ( s , u ( s ) ) d s , t [ 0 , T ] ,
(20)

where 0 < α < 1 , f C ( R × R , R ) , T 0 is a constant, I α denotes the Riemann-Liouville fractional integral of order α on the interval [ 0 , t ] as defined in Definition 2.

Theorem 6 For Eq. (20), if | f ( t , u ) | M | u | , where g C ( R , R + ) , then under the condition exp [ M T α Γ ( 1 + α ) ] < 2 , we have the following estimate:
| u ( t ) | | u ( 0 ) | exp [ M t α Γ ( 1 + α ) ] 2 exp [ M T α Γ ( 1 + α ) ] , t [ 0 , T ] .
(21)
Proof By Eq. (20) we in fact have
u ( t ) = u ( 0 ) + 1 Γ ( α ) 0 t ( t s ) α 1 f ( s , u ( s ) ) d s + 1 Γ ( α ) 0 T ( T s ) α 1 f ( s , u ( s ) ) d s , t [ 0 , T ] .
So,
| u ( t ) | | u ( 0 ) | + 1 Γ ( α ) 0 t ( t s ) α 1 | f ( s , u ( s ) ) | d s + 1 Γ ( α ) 0 T ( T s ) α 1 | f ( s , u ( s ) ) | d s | u ( 0 ) | + M Γ ( α ) 0 t ( t s ) α 1 | u ( s ) | d s + M Γ ( α ) 0 T ( T s ) α 1 | u ( s ) | d s , t [ 0 , T ] .
(22)

Then a suitable application of Theorem 2 to (22) yields the desired result. □

Remark 1 The result of Theorem 6 shows that the trivial solution to Eq. (20) is uniformly stable on the initial value.

Theorem 7 If the function f satisfies the Lipschitz condition with | f ( t , u ) f ( t , v ) | A | u v | , where A is the Lipschitz constant, then under the condition of the same initial value, Eq. (20) has at most one solution.

Proof Suppose that Eq. (20) has two solutions u 1 ( t ) , u 2 ( t ) with the same initial value u ( 0 ) . Then we have
u 1 ( t ) = u ( 0 ) + 1 Γ ( α ) 0 t ( t s ) α 1 f ( s , u 1 ( s ) ) d s + 1 Γ ( α ) 0 T ( T s ) α 1 f ( s , u 1 ( s ) ) d s ,
(23)
u 2 ( t ) = u ( 0 ) + 1 Γ ( α ) 0 t ( t s ) α 1 f ( s , u 2 ( s ) ) d s + 1 Γ ( α ) 0 T ( T s ) α 1 f ( s , u 2 ( s ) ) d s .
(24)
Furthermore,
u 1 ( t ) u 2 ( t ) = 1 Γ ( α ) 0 t ( t s ) α 1 [ f ( s , u 1 ( s ) ) f ( s , u 2 ( s ) ) ] d s + 1 Γ ( α ) 0 T ( T s ) α 1 [ f ( s , u 1 ( s ) ) f ( s , u 2 ( s ) ) ] d s ,
(25)
which implies
| u 1 ( t ) u 2 ( t ) | 1 Γ ( α ) 0 t ( t s ) α 1 | f ( s , u 1 ( s ) ) f ( s , u 2 ( s ) ) | d s + 1 Γ ( α ) 0 T ( T s ) α 1 | f ( s , u 1 ( s ) ) f ( s , u 2 ( s ) ) | d s A Γ ( α ) 0 t ( t s ) α 1 | u 1 ( s ) u 2 ( s ) | d s + A Γ ( α ) 0 T ( T s ) α 1 | u 1 ( s ) u 2 ( s ) | d s .
(26)

After a suitable application of Theorem 2 to (26) (with | u 1 ( t ) u 2 ( t ) | being treated as one independent function), we obtain that | u 1 ( t ) u 2 ( t ) | 0 , which implies u 1 ( t ) u 2 ( t ) . So the proof is complete. □

Theorem 8 Let u ( t ) be the solution of Eq. (20), and let u ˜ ( t ) be the solution of the following fractional integral equation:
u ˜ ( t ) = u ˜ ( 0 ) + I α ( f ( t , u ˜ ( t ) ) ) + 1 Γ ( α ) 0 T ( T s ) α 1 f ( s , u ˜ ( s ) ) d s , t [ 0 , T ] .
(27)
If f satisfies the Lipschitz condition with A being the Lipschitz constant, then we have the following estimate:
| u ( t ) u ˜ ( t ) | | u ( 0 ) u ˜ ( 0 ) | exp [ M t α Γ ( 1 + α ) ] 2 exp [ M T α Γ ( 1 + α ) ] , t [ 0 , T ] .
(28)
Proof By Eq. (27) we have
u ˜ ( t ) = u ˜ ( 0 ) + 1 Γ ( α ) 0 t ( t s ) α 1 f ( s , u ˜ ( s ) ) d s + 1 Γ ( α ) 0 T ( T s ) α 1 f ( s , u ˜ ( s ) ) d s .
(29)
So, we have
u ( t ) u ˜ ( t ) = u ( 0 ) u ˜ ( 0 ) + 1 Γ ( α ) 0 t ( t s ) α 1 [ f ( s , u ( s ) ) f ( s , u ˜ ( s ) ) ] d s + 1 Γ ( α ) 0 T ( T s ) α 1 [ f ( s , u ( s ) ) f ( s , u ˜ ( s ) ) ] d s .
(30)
Furthermore,
| u ( t ) u ˜ ( t ) | | u ( 0 ) u ˜ ( 0 ) | + 1 Γ ( α ) 0 t ( t s ) α 1 | f ( s , u ( s ) ) f ( s , u ˜ ( s ) ) | d s + 1 Γ ( α ) 0 T ( T s ) α 1 | f ( s , u ( s ) ) f ( s , u ˜ ( s ) ) | d s | u ( 0 ) u ˜ ( 0 ) | + A Γ ( α ) 0 t ( t s ) α 1 | u ( s ) u ˜ ( s ) | d s + A Γ ( α ) 0 T ( T s ) α 1 | u ( s ) u ˜ ( s ) | d s .
(31)

Applying Theorem 2 to (31), after some basic computation, we can get the desired result. □

Remark 2 The result of Theorem 8 shows that the solution to Eq. (20) depends continuously on the initial value.

4 Conclusions

In this paper, we have derived new explicit bounds for the unknown functions concerned in some new Gronwall-type inequalities. In the proof for the main results, we have used the properties of the modified Riemann-Liouville fractional derivative. As for applications, we have presented one example, in which the boundedness, uniqueness, and continuous dependence on the initial value for the solution to a certain fractional integral equation are investigated. Finally, we note that these inequalities can be generalized to more general forms, as well as be generalized to 2D cases.

Declarations

Acknowledgements

The authors would thank the referees very much for their valuable suggestions on improving this paper. This work was partially supported by the Natural Science Foundation of Shandong Province (in China) (grant No. ZR2013AQ009), and Doctoral Initializing Foundation of Shandong University of Technology (in China) (grant No. 4041-413030).

Authors’ Affiliations

(1)
School of Science, Shandong University of Technology

References

  1. Gronwall TH: Note on the derivatives with respect to a parameter of solutions of a system of differential equations. Ann. Math. 1919, 20: 292-296. 10.2307/1967124MathSciNetView ArticleMATHGoogle Scholar
  2. Bellman R: The stability of solutions of linear differential equations. Duke Math. J. 1943, 10: 643-647. 10.1215/S0012-7094-43-01059-2MathSciNetView ArticleMATHGoogle Scholar
  3. Ma QH: Estimates on some power nonlinear Volterra-Fredholm type discrete inequalities and their applications. J. Comput. Appl. Math. 2010, 233: 2170-2180. 10.1016/j.cam.2009.10.002MathSciNetView ArticleMATHGoogle Scholar
  4. Pachpatte BG: Inequalities for Differential and Integral Equations. Academic Press, New York; 1998.MATHGoogle Scholar
  5. Sun YG: On retarded integral inequalities and their applications. J. Math. Anal. Appl. 2005, 301: 265-275. 10.1016/j.jmaa.2004.07.020MathSciNetView ArticleMATHGoogle Scholar
  6. Agarwal RP, Deng SF, Zhang WN: Generalization of a retarded Gronwall-like inequality and its applications. Appl. Math. Comput. 2005, 165: 599-612. 10.1016/j.amc.2004.04.067MathSciNetView ArticleMATHGoogle Scholar
  7. Li LZ, Meng FW, Ju PJ: Some new integral inequalities and their applications in studying the stability of nonlinear integro-differential equations with time delay. J. Math. Anal. Appl. 2010, 377: 853-862.MathSciNetView ArticleMATHGoogle Scholar
  8. Gallo A, Piccirillo AM: About some new generalizations of Bellman-Bihari results for integro-functional inequalities with discontinuous functions and applications. Nonlinear Anal. 2009, 71: e2276-e2287. 10.1016/j.na.2009.05.019MathSciNetView ArticleMATHGoogle Scholar
  9. Ma QH, Pečarić J: The bounds on the solutions of certain two-dimensional delay dynamic systems on time scales. Comput. Math. Appl. 2011, 61: 2158-2163. 10.1016/j.camwa.2010.09.001MathSciNetView ArticleMATHGoogle Scholar
  10. Lipovan O: Integral inequalities for retarded Volterra equations. J. Math. Anal. Appl. 2006, 322: 349-358. 10.1016/j.jmaa.2005.08.097MathSciNetView ArticleMATHGoogle Scholar
  11. Feng QH, Zheng B: Generalized Gronwall-Bellman-type delay dynamic inequalities on time scales and their applications. Appl. Math. Comput. 2012, 218: 7880-7892. 10.1016/j.amc.2012.02.006MathSciNetView ArticleMATHGoogle Scholar
  12. Kim YH: Gronwall, Bellman and Pachpatte type integral inequalities with applications. Nonlinear Anal. 2009, 71: e2641-e2656. 10.1016/j.na.2009.06.009View ArticleMathSciNetMATHGoogle Scholar
  13. Pachpatte BG: Explicit bounds on certain integral inequalities. J. Math. Anal. Appl. 2002, 267: 48-61. 10.1006/jmaa.2001.7743MathSciNetView ArticleMATHGoogle Scholar
  14. Agarwal RP, Bohner M, Peterson A: Inequalities on time scales: a survey. Math. Inequal. Appl. 2001,4(4):535-557.MathSciNetMATHGoogle Scholar
  15. Wang WS: Some retarded nonlinear integral inequalities and their applications in retarded differential equations. J. Inequal. Appl. 2012,2012(75):1-8.MathSciNetGoogle Scholar
  16. Li WN: Some delay integral inequalities on time scales. Comput. Math. Appl. 2010, 59: 1929-1936. 10.1016/j.camwa.2009.11.006MathSciNetView ArticleMATHGoogle Scholar
  17. Saker SH: Some nonlinear dynamic inequalities on time scales. Math. Inequal. Appl. 2011, 14: 633-645.MathSciNetMATHGoogle Scholar
  18. Feng QH, Meng FW, Zhang YM: Generalized Gronwall-Bellman-type discrete inequalities and their applications. J. Inequal. Appl. 2011,2011(47):1-21.MathSciNetMATHGoogle Scholar
  19. Feng QH, Meng FW, Zheng B: Gronwall-Bellman type nonlinear delay integral inequalities on time scales. J. Math. Anal. Appl. 2011, 382: 772-784. 10.1016/j.jmaa.2011.04.077MathSciNetView ArticleMATHGoogle Scholar
  20. Wang WS: A class of retarded nonlinear integral inequalities and its application in nonlinear differential-integral equation. J. Inequal. Appl. 2012,2012(154):1-10.MathSciNetGoogle Scholar
  21. Saker SH: Some nonlinear dynamic inequalities on time scales and applications. J. Math. Inequal. 2010, 4: 561-579.MathSciNetView ArticleMATHGoogle Scholar
  22. Zheng B, Feng QH, Meng FW, Zhang YM: Some new Gronwall-Bellman type nonlinear dynamic inequalities containing integration on infinite intervals on time scales. J. Inequal. Appl. 2012,2012(201):1-20.MathSciNetMATHGoogle Scholar
  23. Li WN, Han MA, Meng FW: Some new delay integral inequalities and their applications. J. Comput. Appl. Math. 2005, 180: 191-200. 10.1016/j.cam.2004.10.011MathSciNetView ArticleMATHGoogle Scholar
  24. Jiang FC, Meng FW: Explicit bounds on some new nonlinear integral inequality with delay. J. Comput. Appl. Math. 2007, 205: 479-486. 10.1016/j.cam.2006.05.038MathSciNetView ArticleMATHGoogle Scholar
  25. Feng QH, Meng FW, Zhang YM, Zheng B, Zhou JC: Some nonlinear delay integral inequalities on time scales arising in the theory of dynamics equations. J. Inequal. Appl. 2011,2011(29):1-14.MathSciNetMATHGoogle Scholar
  26. Ferreira RAC, Torres DFM: Generalized retarded integral inequalities. Appl. Math. Lett. 2009, 22: 876-881. 10.1016/j.aml.2008.08.022MathSciNetView ArticleMATHGoogle Scholar
  27. Cheung WS, Ren JL: Discrete non-linear inequalities and applications to boundary value problems. J. Math. Anal. Appl. 2006, 319: 708-724. 10.1016/j.jmaa.2005.06.064MathSciNetView ArticleMATHGoogle Scholar
  28. Ye HP, Gao JM, Ding YS: A generalized Gronwall inequality and ins application to a fractional differential equation. J. Math. Anal. Appl. 2007, 328: 1075-1081. 10.1016/j.jmaa.2006.05.061MathSciNetView ArticleMATHGoogle Scholar
  29. Jumarie G: Modified Riemann-Liouville derivative and fractional Taylor series of nondifferentiable functions further results. Comput. Math. Appl. 2006, 51: 1367-1376. 10.1016/j.camwa.2006.02.001MathSciNetView ArticleMATHGoogle Scholar
  30. Jumarie G: Table of some basic fractional calculus formulae derived from a modified Riemann-Liouville derivative for non-differentiable functions. Appl. Math. Lett. 2009, 22: 378-385. 10.1016/j.aml.2008.06.003MathSciNetView ArticleMATHGoogle Scholar
  31. Wu GC, Lee EWM: Fractional variational iteration method and its application. Phys. Lett. A 2010, 374: 2506-2509. 10.1016/j.physleta.2010.04.034MathSciNetView ArticleMATHGoogle Scholar
  32. Zheng B: ( G / G ) -expansion method for solving fractional partial differential equations in the theory of mathematical physics. Commun. Theor. Phys. 2012, 58: 623-630. 10.1088/0253-6102/58/5/02View ArticleMathSciNetMATHGoogle Scholar
  33. Feng QH: Exact solutions for fractional differential-difference equations by an extended Riccati Sub-ODE method. Commun. Theor. Phys. 2013, 59: 521-527. 10.1088/0253-6102/59/5/01View ArticleMathSciNetGoogle Scholar
  34. Almeida R, Torres DFM: Fractional variational calculus for nondifferentiable functions. Comput. Math. Appl. 2011, 61: 3097-3104. 10.1016/j.camwa.2011.03.098MathSciNetView ArticleMATHGoogle Scholar
  35. Khan Y, Wu Q, Faraz N, Yildirim A, Madani M: A new fractional analytical approach via a modified Riemann-Liouville derivative. Appl. Math. Lett. 2012, 25: 1340-1346. 10.1016/j.aml.2011.11.041MathSciNetView ArticleMATHGoogle Scholar
  36. Faraz N, Khan Y, Jafari H, Yildirim A, Madani M: Fractional variational iteration method via modified Riemann-Liouville derivative. J. King Saud Univ., Sci. 2011, 23: 413-417. 10.1016/j.jksus.2010.07.025View ArticleGoogle Scholar
  37. Khana Y, Faraz N, Yildirim A, Wu Q: Fractional variational iteration method for fractional initial-boundary value problems arising in the application of nonlinear science. Comput. Math. Appl. 2011, 62: 2273-2278. 10.1016/j.camwa.2011.07.014MathSciNetView ArticleMATHGoogle Scholar
  38. Merdan M: Analytical approximate solutions of fractional convection-diffusion equation with modified Riemann-Liouville derivative by means of fractional variational iteration method. Iran. J. Sci. Technol., Trans. A, Sci. 2013,37(1):83-92.MathSciNetMATHGoogle Scholar
  39. Guo S, Mei L, Li Y: Fractional variational homotopy perturbation iteration method and its application to a fractional diffusion equation. Appl. Math. Comput. 2013, 219: 5909-5917. 10.1016/j.amc.2012.12.003MathSciNetView ArticleMATHGoogle Scholar

Copyright

© Zheng; licensee Springer. 2014

This article is published under license to BioMed Central Ltd. This is an open access article distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/2.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.