Research ArticleNew Integral Inequalities with Weakly Singular Kernel forDiscontinuous Functions and Their Applications to ImpulsiveFractional Differential Systems

Jing Shao12

1 Department of Mathematics Jining University Qufu Shandong 273155 China2 College of Mathematical Sciences Qufu Normal University Qufu Shandong 273165 China

Correspondence should be addressed to Jing Shao shaojing99500163com

Received 27 February 2014 Accepted 5 May 2014 Published 12 May 2014

Academic Editor Kai Diethelm

Copyright copy 2014 Jing Shao This is an open access article distributed under the Creative Commons Attribution License whichpermits unrestricted use distribution and reproduction in any medium provided the original work is properly cited

Some new integral inequalities with weakly singular kernel for discontinuous functions are established using the method ofsuccessive iteration and properties of Mittag-Leffler function which can be used in the qualitative analysis of the solutions tocertain impulsive fractional differential systems

1 Introduction

Differential equations of fractional order have recentlyproved to be valuable tools in the modeling of manyphenomena in various fields of engineering physics andeconomics There has been a significant development in thestudy of fractional differential equations in recent years seethemonographs ofKilbas et al [1] Lakshmikanthamet al [2]and Podlubny [3] and the survey by Diethelm [4] Integralinequalities with weakly singular kernels play an importantrole in the qualitative analysis of the solutions to fractionaldifferential equationsWith the development of fractional dif-ferential equations integral inequalities with weakly singularkernels have drawnmore andmore researchersrsquo attention andlead to inspiring results see for example [5ndash8]

Impulsive differential equations that is differential equa-tions involving impulse effect appear as a natural descriptionof observed evolution phenomena of several real worldproblems Many processes studied in applied sciences arerepresented by impulsive differential equations However thesituation is quite different in many physical phenomena thathave a sudden change in their states such as mechanicalsystems with impact biological systems such as heart beatsblood flow population dynamics theoretical physics phar-macokinetics mathematical economy chemical technology

electric technology metallurgy ecology industrial roboticsand biotechnology processes (see the monographs [9ndash12] fordetails)

The theory of impulsive differential equations is animportant branch of differential equations In spite of itsimportance the development of the theory has been quiteslow due to special features possessed by impulsive differen-tial equations in general such as pulse phenomena conflu-ence and loss of autonomy Among these results integrosuminequalities for discontinuous functions play increasinglyimportant roles in the study of quantitative properties of solu-tions of impulsive differential systems In 2005 Borysenko etal [13] considered some integrosum inequalities and devotedthem to investigate the properties of motion representedby essential nonlinear system of differential equations withimpulsive effect In 2007 and 2009 Gallo and Piccirillo [1415] presented some new nonlinear integral inequalities likeGronwall-Bellman-Bihari type with delay for discontinuousfunctions and applied them to investigate the properties ofsolutions of impulsive differential systems

The theory of impulsive fractional differential equationsis a new topic of research which involve both the fractionalorder integral (or differentiation) and the impulsive effectmost of the results related to this topic are the existence ofsolutions (see [16ndash19] and the references therein) To our best

Hindawi Publishing CorporationJournal of Applied MathematicsVolume 2014 Article ID 252946 5 pageshttpdxdoiorg1011552014252946

2 Journal of Applied Mathematics

knowledge there is no result on other qualitative properties(such as boundedness and stability) and impulsive fractionaldifferential equations involving the Caputo fractional deriva-tive have not been studied very perfectly so we set up a newkind of integral inequalities with weakly singular kernel fordiscontinuous functions and use the new inequalities to studythe qualitative properties of the solutions to certain impulsivefractional differential systems

On the basis of previous studies in this paper we considerthe following integral inequalities withweakly singular kernelfor discontinuous functions

119906 (119905) le 119886 + 119887int

119905

1199050

(119905 minus 119904)120573minus1119906 (119904) 119889119904 + sum

1199050lt119905119894lt119905

120574119894119906 (119905minus

119894) (1)

where 119886 119887 and 120574119894are constants 119886 ge 0 119887 ge 0 120574

119894ge 0 and 119906(119905)

is a nonnegative piecewise-continuous function with the 1stkind of discontinuous points 119905

0lt 1199051lt sdot sdot sdot lim

119894rarrinfin119905119894= infin

119906(119905minus

119894) = lim

119905rarr 119905119894minus0119906(119905) In general due to the existence of weak

singular integral kernel the methods of these inequalitiesfor discontinuous functions are quite different to that ofclassical Gronwall-Bellman-Bihari inequalities We use theproperties of the Mittag-Leffler function 119864

120573(sdot) defined by

119864120573(119911) = sum

infin

119896=0(119911119896Γ(119896120573 + 1)) and the successive iterative

technique to establish the new type of integral inequalitiesfor discontinuous functionsThese inequalities are applied toinvestigate the qualitative analysis of the solutions to certainimpulsive fractional differential systems

2 Preliminary Knowledge

In this section we give some definitions symbols and knowninequalities whichwill be used in the remainder of this paper

Definition 1 Given an interval [119886 119887] of R the fractional(arbitrary) order integral of the function ℎ isin 1198711([119886 119887]R) oforder 120572 isin R

+is defined by

119868120572

119886ℎ (119905) = int

119905

119886

(119905 minus 119904)120572minus1

Γ (120572)

ℎ (119904) 119889119904 (2)

where Γ(sdot) is the gamma function When 119886 = 0 we write119868120572ℎ(119905) = [ℎ lowast 120593

120572](119905) where 120593

120572(119905) = 119905

120572minus1Γ(120572) for 119905 gt 0

120593120572(119905) = 0 for 119905 le 0 and 120593

120572rarr 120575(119905) as 120572 rarr 0 where 120575

is the delta function

Definition 2 For a given function ℎ on the interval [119886 119887] the120572-order Caputo fractional order derivative of ℎ is defined by

(119888119863120572

119886+ℎ) (119905) =

1

Γ (119898 minus 120572)

int

119905

119886

(119905 minus 119904)119898minus120572minus1

ℎ(119898)(119904) 119889119904 (3)

where119898 = [120572] + 1

For calculation simplification the symbols are defined asfollows

119878 (119905 119905119894) = 119864120573(119887Γ (120573) (119905 minus 119905

119894)120573

) for 119905 gt 119905119894

119878119894119895= 119878 (119905

119894 119905119895) = 119864120573(119887Γ (120573) (119905

119894minus 119905119895)

120573

) for 119905119894gt 119905119895

(4)

where 119864120573(sdot) is the Mittag-Leffler function

Lemma 3 (see [6]) Suppose that 120573 gt 0 119886(119905) is a nonnegativeand nondecreasing function which is locally integrable on 0 le119905 lt 119879 (for some 119879 le +infin) and 119892(119905) is a nonnegativenondecreasing continuous function defined on 0 le 119905 lt 119879119892(119905) le 119872 (constant) and suppose that 119906(119905) is a nonnegativeand locally integrable function on 0 le 119905 lt 119879 which satisfies

119906 (119905) le 119886 (119905) + 119892 (119905) int

119905

0

(119905 minus 119904)120573minus1119906 (119904) 119889119904 (5)

and then

119906 (119905) le 119886 (119905) 119864120573(119892 (119905) Γ (120573) 119905

120573) (6)

Using Lemma 3 we can easily get the following corollary

Corollary 4 Let 119886 119887 be constants 119886 ge 0 and 119887 ge 0 Andsuppose 119906(119905) is a nonnegative and locally integrable functionon 1199050le 119905 lt 119879 with

119906 (119905) le 119886 + 119887int

119905

1199050

(119905 minus 119904)120573minus1119906 (119904) 119889119904 (7)

and then

119906 (119905) le 119886119878 (119905 1199050) (8)

Lemma 5 Let 120572 ge 0 and 120573 ge 0 Then

int

1199051

1199050

(119905 minus 119904)120572minus1(119904 minus 1199050)120573minus1

119889119904 le 119861 (120572 120573) (119905 minus 1199050)120572+120573minus1

119905 ge 1199051gt 1199050

(9)

where 119861(sdot sdot) is the Beta function

Proof Let 119904 = 1199050+ 120585(119905 minus 119905

0) we obtain for 119905 ge 119905

1

int

1199051

1199050

(119905 minus 119904)120572minus1(119904 minus 1199050)120573minus1

119889119904

= int

(1199051minus1199050)(119905minus1199050)

0

(1 minus 120585)120572minus1(119905 minus 1199050)120572minus1

(119905 minus 1199050)120573minus1

120585120573minus1119889120585

le (119905 minus 1199050)120572+120573minus1

int

1

0

(1 minus 120585)120572minus1120585120573minus1119889120585

le 119861 (120572 120573) (119905 minus 1199050)120572+120573minus1

(10)

which is the desired result

Remark 6 If we replace the integration interval [1199050 1199051] by

[119905119894 119905119894+1] for 119894 = 0 1 2 we can get the following equality

int

119905119894+1

119905119894

(119905 minus 119904)120572minus1(119904 minus 119905119894)120573minus1

119889119904 le 119861 (120572 120573) (119905 minus 119905119894)120572+120573minus1

119905 ge 119905119894+1gt 119905119894 119894 = 0 1 2

(11)

Corollary 7 Let 120573 ge 0 Then

119887int

1199051

1199050

(119905 minus 119904)120573minus1119878 (119904 1199050) 119889119904 le 119878 (119905 119905

0) minus 1 119905 ge 119905

1gt 1199050

(12)

Journal of Applied Mathematics 3

Proof By the definition of 119878(119904 1199050) we have

119887int

1199051

1199050

(119905 minus 119904)120573minus1119878 (119904 1199050) 119889119904

= 119887int

1199051

1199050

(119905 minus 119904)120573minus1

infin

sum

119896=0

119887119896Γ119896(120573) (119904 minus 119905

0)119896120573

Γ (119896120573 + 1)

119889119904

(13)

Since the series of function is the Mittag-Leffler function119864120573(119887Γ(120573)(119904 minus 119905

0)120573) which is convergent uniformly on 119904 isin

[1199050 1199051] we permute the sum and the integral to obtain

119887int

1199051

1199050

(119905 minus 119904)120573minus1119878 (119904 1199050) 119889119904

=

infin

sum

119896=0

119887119896+1Γ119896(120573)

Γ (119896120573 + 1)

int

1199051

1199050

(119905 minus 119904)120573minus1(119904 minus 1199050)119896120573

119889119904

(14)

Using Lemma 5 we get

119887int

1199051

1199050

(119905 minus 119904)120573minus1119878 (119904 1199050) 119889119904

le

infin

sum

119896=0

119887119896+1Γ119896(120573)

Γ (119896120573 + 1)

(119905 minus 1199050)(119896+1)120573

119861 (120573 119896120573 + 1)

=

infin

sum

119896=0

119887119896+1Γ119896+1(120573) (119905 minus 119905

0)(119896+1)120573

Γ ((119896 + 1) 120573 + 1)

=

infin

sum

119896=1

119887119896Γ119896(120573) (119905 minus 119905

0)119896120573

Γ (119896120573 + 1)

le

infin

sum

119896=0

119887119896Γ119896(120573) (119905 minus 119905

0)119896120573

Γ (119896120573 + 1)

minus 1 = 119878 (119905 1199050) minus 1

(15)

Remark 8 If we replace the integration interval [1199050 1199051] by

[119905119894 119905119894+1] we can get the similar conclusion

119887int

119905119894+1

119905119894

(119905 minus 119904)120573minus1119878 (119904 1199050) 119889119904 le 119878 (119905 119905

119894) minus 1

119905 ge 119905119894+1gt 119905119894 119894 = 0 1 2

(16)

3 Main Results

Theorem 9 Let 119886 119887 be constants 119886 ge 0 119887 ge 0 and 119906(119905) is anonnegative piecewise-continuous function with the 1st kind ofdiscontinuous points 119905

0lt 1199051lt sdot sdot sdot and lim

119894rarrinfin119905119894= infin If

119906 (119905) le 119886 + 119887int

119905

1199050

(119905 minus 119904)120573minus1119906 (119904) 119889119904 + sum

1199050lt119905119894lt119905

120574119894119906 (119905minus

119894) 119905 isin 119868

(17)

where 120573 gt 0 then the following assertions hold

119906 (119905) le 119886 (1 + 1205741) sdot sdot sdot (1 + 120574

119894) 119878119894+10119878119894+11

sdot sdot sdot 119878119894+1119894minus1

119878 (119905 119905119894)

119905 isin (119905119894 119905119894+1]

119906 (119905) le 119886119878 (119905 1199050) 119905 isin [119905

0 1199051]

(18)

Proof If 119905 isin [1199050 1199051] the inequality (17) is reduced to the

following form

119906 (119905) le 119886 + 119887int

119905

1199050

(119905 minus 119904)120573minus1119906 (119904) 119889119904 (19)

Using Lemma 3 we get

119906 (119905) le 119886119878 (119905 1199050)

119906 (119905minus

1) le 119886119878 (119905

1 1199050) = 119886119878

10

(20)

If 119905 isin (1199051 1199052] then

119906 (119905) le 119886 + 1205741119906 (119905minus

1) + 119887int

119905

1199050

(119905 minus 119904)120573minus1119906 (119904) 119889119904

le 119886 + 1205741119906 (119905minus

1) + 119887int

1199051

1199050

(119905 minus 119904)120573minus1119906 (119904) 119889119904

+ 119887int

119905

1199051

(119905 minus 119904)120573minus1119906 (119904) 119889119904

le 119886 + 119886120574111987810+ 119886119887int

1199051

1199050

(119905 minus 119904)120573minus1119878 (119904 1199050) 119889119904

+ 119887int

119905

1199051

(119905 minus 119904)120573minus1119906 (119904) 119889119904

(21)

Using Corollary 7 we get

119906 (119905) le 119886 + 119886120574111987810+ 119886119878 (119905 119905

0) minus 119886 + 119887int

119905

1199051

(119905 minus 119904)120573minus1119906 (119904) 119889119904

le 119886 (120574111987810+ 11987820) + 119887int

119905

1199051

(119905 minus 119904)120573minus1119906 (119904) 119889119904

(22)

Applying Lemma 3 we obtain that

119906 (119905) le 119886 (120574111987810+ 11987820) 119878 (119905 119905

1) le 119886 (1 + 120574

1) 11987820119878 (119905 1199051)

119906 (119905minus

2) le 119886 (1 + 120574

1) 1198782011987821

(23)

Similarly for 119905 isin (1199052 1199053] using Remark 8 we have

119906 (119905) le 119886 + 1205741119906 (119905minus

1) + 1205742119906 (119905minus

2) + 119887int

119905

1199050

(119905 minus 119904)120573minus1119906 (119904) 119889119904

le 119886 + 119886120574111987810+ 1198861205742(1 + 120574

1) 1198782011987821

+ 119887int

1199051

1199050

(119905 minus 119904)120573minus1119906 (119904) 119889119904

+ 119887int

1199052

1199051

(119905 minus 119904)120573minus1119906 (119904) 119889119904 + 119887int

119905

1199052

(119905 minus 119904)120573minus1119906 (119904) 119889119904

le 119886 (1 + 120574111987810+ 1205742(1 + 120574

1) 1198782011987821+ (11987830minus 1)

+ (1 + 1205741) 11987820(11987831minus 1)) + 119887int

119905

1199052

(119905 minus 119904)120573minus1119906 (119904) 119889119904

4 Journal of Applied Mathematics

le 119886 (1 + 120574111987830+ 1205742(1 + 120574

1) 1198783011987831+ (11987830minus 1)

+ (1 + 1205741) 11987830(11987831minus 1)) + 119887int

119905

1199052

(119905 minus 119904)120573minus1119906 (119904) 119889119904

= 119886 (1 + 1205741) (1 + 120574

2) 1198783011987831+ 119887int

119905

1199052

(119905 minus 119904)120573minus1119906 (119904) 119889119904

(24)

Again Lemma 3 implies that

119906 (119905) le 119886 (1 + 1205741) (1 + 120574

2) 1198783011987831119878 (119905 1199052) 119905 isin (119905

2 1199053]

(25)

Using the inductivemethod suppose that for 119905 isin (119905119894minus1 119905119894] 119894 =

1 2 3

119906 (119905) le 119886 (1 + 1205741) (1 + 120574

2) sdot sdot sdot (1 + 120574

119894minus1) 119878119894011987811989411198781198942

sdot sdot sdot 119878119894119894minus2119878 (119905 119905119894minus1)

(26)

and then for each 119905 isin (119905119894 119905119894+1] and from (26) we have

119906 (119905)

le 119886 + 1205741119906 (119905minus

1) + 1205742119906 (119905minus

2) + sdot sdot sdot + 120574

119894119906 (119905minus

119894minus1)

+ 119887int

119905

1199050

(119905 minus 119904)120573minus1119906 (119904) 119889119904

= 119886 + 119886120574111987810+ 1198861205742(1 + 120574

1) 1198782011987821+ sdot sdot sdot

+ 119886120574119894(1 + 120574

1) sdot sdot sdot (1 + 120574

119894minus1) 11987811989401198781198941sdot sdot sdot 119878119894119894minus1

+

119894minus1

sum

119895=0

119887int

119905119895+1

119905119895

(119905 minus 119904)120573minus1119906 (119904) 119889119904

+ int

119905

119905119894

(119905 minus 119904)120573minus1119906 (119904) 119889119904

le 119886 + 119886120574111987810+ 1198861205742(1 + 120574

1) 1198782011987821+ sdot sdot sdot

+ 119886120574119894(1 + 120574

1) sdot sdot sdot (1 + 120574

119894minus1) 11987811989401198781198941sdot sdot sdot 119878119894119894minus1

+ 119886 (119878119894+10

minus 1) + 119886 (1 + 1205741) 11987820(119878119894+11

minus 1) + sdot sdot sdot

+ 119886 (1 + 1205741) sdot sdot sdot (1 + 120574

119894minus1) 11987811989401198781198941sdot sdot sdot 119878119894119894minus2

(119878119894+1119894minus1

minus 1)

+ int

119905

119905119894

(119905 minus 119904)120573minus1119906 (119904) 119889119904

le 119886 (1 + 1205741119878119894+10

+ 1205742(1 + 120574

1) 119878119894+10119878119894+11

+ sdot sdot sdot

+ 120574119894(1 + 120574

1) sdot sdot sdot (1 + 120574

119894minus1) 119878119894+10119878119894+11

sdot sdot sdot 119878119894+1119894minus1

+ 119878119894+10

minus 1 + (1 + 1205741) 119878119894+10

(119878119894+11

minus 1) + sdot sdot sdot

+ (1 + 1205741) sdot sdot sdot (1 + 120574

119894minus1) 11987811989401198781198941sdot sdot sdot 119878119894119894minus2

(119878119894+1119894minus1

minus 1))

+ int

119905

119905119894

(119905 minus 119904)120573minus1119906 (119904) 119889119904

le 119886 (1 + 1205741) (1 + 120574

2) sdot sdot sdot (1 + 120574

119894minus1) 11987811989401198781198941sdot sdot sdot 119878119894119894minus2

+ int

119905

119905119894

(119905 minus 119904)120573minus1119906 (119904) 119889119904

(27)

Then applying Lemma 3 we obtain the inequality (17) Thiscompletes the proof of Theorem 9

Remark 10 The results of Theorem 9 are valid when thefunction has only finite number of discontinuities points1199050 1199051 119905

119894

4 Application to Impulsive FractionalDifferential Systems

Let us consider the following system of Caputo fractionaldifferential equations

119862119863120573

0+119909 (119905) = 119891 (119905 119909 (119905)) 119905 isin 119869 [0 119879] 119905

1 119905

119896

Δ119909 (119905119894) = 119909 (119905

+

119894) minus 119909 (119905

minus

119894) = 119868119894(119909 (119905minus

119894)) 119894 = 1 2 119896

119909 (0) = 1205940

(28)

where 1198621198631205730+ is the Caputo fractional derivatives 0 = 119905

0lt

1199051lt sdot sdot sdot lt 119905

119896lt 119905119896+1

= 119879 119891 [0 119879] times R119899 rarr R119899 isLebesgue measurable with respect to 119905 on [0 119879] and 119891(119905 119909)is continuous with respect to 119909 on R119899 119868

119894 R119899 rarr R119899 are

continuous for 119894 = 1 2 119896 119909(119905+119894) = lim

120576rarr0+119909(119905119894+ 120576) and

119909(119905minus

119894) = lim

120576rarr0minus119909(119905119894minus 120576) represent the right and left limits of

119909(119905) at 119905 = 119905119894 and 120594

0isin R119899 is a constant-valued vector

Using Lemma 62 in [3] 119909(119905) is a solution of Cauchyproblem for system (28) if only if 119909(119905) satisfies

119909 (119905) = 1205940+ sum

1199050lt119905119894lt119905

119868119894(119909 (119905minus

119894))

+

1

Γ (120573)

int

119905

0

(119905 minus 119904)120573minus1119891 (119904 119909 (119904)) 119889119904

(29)

Theorem 11 Suppose that 119891(119905 119909) le 119887119909 and 119868119894(119909(119905minus

119894)) le

120574119894119909(119905minus

119894) where 119887 120574

119894are constants 119887 ge 0 and 120574

119894ge 0 If 119909(119905) is

any solution of the initial value problem (28) then the followingestimations hold

119909 (119905)

le

10038171003817100381710038171205940

1003817100381710038171003817119878lowast(119905 1199050) 119905 isin [119905

0 1199051]

10038171003817100381710038171205940

1003817100381710038171003817(1 + 120574

1) sdot sdot sdot (1 + 120574

119894) 119878lowast

119894+10119878lowast

119894+11

sdot sdot sdot 119878lowast

119894+1119894minus1119878lowast(119905 119905119894) 119905 isin [119905

119894minus1 119905119894]

(30)

where 119878lowast(119905 119905119894) = 119864120573(119887(119905minus119905

119895)120573) 119878lowast(119905

119894 119905119895) = 119864120573(119887(119905119894minus119905119895)120573) and

sdot is a suitable complete norm in R

Journal of Applied Mathematics 5

Proof From (29) it is easy to obtain that

119909 (119905) le10038171003817100381710038171205940

1003817100381710038171003817+ sum

1199050lt119905119894lt119905

120574119894

1003817100381710038171003817119909 (119905minus

119894)1003817100381710038171003817

+ int

119905

0

119887

Γ (120573)

(119905 minus 119904)120573minus1119909 (119904) 119889119904 119905 ge 0

(31)

UsingTheorem 9 we get the desired conclusion

Remark 12 In [13 15] during considering the qualitativeproperties of the impulsive differential equations the func-tions 119891(119905 119909) and 119868

119894(119909) are defined in the domain

119863 = (119905 119909) 119905 isin [1199050 119879] 119879 le infin 119909 le ℎ (32)

for some ℎ gt 0 However we do not add such a restrictionsince our conclusion inTheorem 11 does not involve 119909

Corollary 13 Let the right-hand side of the initial valueproblem (28) satisfy the following conditions

(i) 119891(119905 119909) le 119887119909 119887 is a constant and 119887 ge 0(ii) 119868

119894(119909(119905minus

119894)) le 120574

119894119909(119905minus

119894) 120574119894are constants and 120574

119894ge 0

(iii) there exists a constant 120585 such that 119878lowast(119905 119905119894) le 120585 lt infin

119905 gt 119905119894 119894 = 0 1 2 119896 Then all solutions of the

problem (28) are bounded and the trivial solution ofthe problem (28) is stable in the sense of Lyapunovstability

Corollary 14 Suppose that

(i) 119891(119905 119909) minus 119891(119905 119909) le 119887119909 minus 119910119897 0 lt 119897 lt 1 for all 119909 119910 isin1198751198621

(ii) 119868119894(119909) minus 119868

119894(119910) le 120574

119894119909 minus 119910 where 119887 120574

119894are constants

119887 ge 0 120574119894ge 0 1 le 119894 le 119896

(iii) there exists a constant 120585 such that 119878lowast(119905 119905119894) le 120585 lt infin 119905 gt

119905119894 119894 = 0 1 2 119896 Then all solutions of the problem

(28) are bounded

Conflict of Interests

The author declares that there is no conflict of interests

Acknowledgments

The author sincerely thanks the referees for their constructivesuggestions and corrections This research was partially sup-ported by the NSF of China (Grants 11171178 and 11271225)the NSF of Shandong Province (Grant ZR2013AL005) theScience and Technology Project of High Schools of ShandongProvince (Grant J12LI52) and the Key Projects of Jining Uni-versity Servicing the Local Economics (Grant 2013FWDF14)

References

[1] A A Kilbas H M Srivastava and J J Trujillo Theoryand Applications of Fractional Differential Equations vol 204Elsevier Science Amsterdam The Netherlands 2006

[2] V Lakshmikantham S Leela and D J Vasundhara Theory ofFractional Dynamic Systems Cambridge Scientific Publishers2009

[3] I Podlubny Fractional Differential Equations vol 198 Aca-demic Press San Diego Calif USA 1999

[4] K Diethelm The Analysis of Fractional Differential Equationsvol 2004 Springer Berlin Germany 2010

[5] Z Denton and A S Vatsala ldquoFractional integral inequalitiesand applicationsrdquo Computers amp Mathematics with Applicationsvol 59 no 3 pp 1087ndash1094 2010

[6] H Ye J Gao and Y Ding ldquoA generalized Gronwall inequalityand its application to a fractional differential equationrdquo Journalof Mathematical Analysis and Applications vol 328 no 2 pp1075ndash1081 2007

[7] Q-H Ma and J Pecaric ldquoSome new explicit bounds for weaklysingular integral inequalities with applications to fractionaldifferential and integral equationsrdquo Journal of MathematicalAnalysis and Applications vol 341 no 2 pp 894ndash905 2008

[8] J Shao and F Meng ldquoGronwall-Bellman type inequalities andtheir applications to fractional differential equationsrdquo Abstractand Applied Analysis vol 2013 Article ID 217641 7 pages 2013

[9] A Halanay and D Wexler Qualitative Theory of ImpulsiveSystems Academia Romana Bucharest Romania 1968

[10] A M Samoilenko and N PerestyukDifferential Equations withImpulse Effect Visha Shkola Kyiv Ukraine 1987

[11] V Lakshmikantham D D Baınov and P S Simeonov Theoryof Impulsive Differential Equations vol 6 of Series in ModernApplied Mathematics World Scientific Publishing Teaneck NJUSA 1989

[12] D Xu Y Hueng and L Ling ldquoExistence of positive solutionsof an impulsive delay fishing modelrdquo Bulletin of MathematicalAnalysis and Applications vol 3 no 2 pp 89ndash94 2011

[13] S D Borysenko M Ciarletta and G Iovane ldquoIntegro-suminequalities and motion stability of systems with impulse per-turbationsrdquoNonlinearAnalysisTheoryMethodsampApplicationsvol 62 no 3 pp 417ndash428 2005

[14] AGallo andAM Piccirillo ldquoAbout new analogies ofGronwall-Bellman-Bihari type inequalities for discontinuous functionsand estimated solutions for impulsive differential systemsrdquoNonlinear Analysis Theory Methods amp Applications vol 67 no5 pp 1550ndash1559 2007

[15] A Gallo and A M Piccirillo ldquoOn some generalizationsBellman-Bihari result for integro-functional inequalities fordiscontinuous functions and their applicationsrdquo BoundaryValue Problems vol 2009 Article ID 808124 14 pages 2009

[16] K Balachandran S Kiruthika and J J Trujillo ldquoOn fractionalimpulsive equations of Sobolev type with nonlocal conditionin Banach spacesrdquoComputers ampMathematics with Applicationsvol 62 no 3 pp 1157ndash1165 2011

[17] H Jiang ldquoExistence results for fractional order functionaldifferential equations with impulserdquo Computers ampMathematicswith Applications vol 64 no 10 pp 3477ndash3483 2012

[18] X Li F Chen and X Li ldquoGeneralized anti-periodic boundaryvalue problems of impulsive fractional differential equationsrdquoCommunications in Nonlinear Science and Numerical Simula-tion vol 18 no 1 pp 28ndash41 2013

[19] LMahto S Abbas andA Favini ldquoAnalysis of Caputo impulsivefractional order differential equations with applicationsrdquo Inter-national Journal of Differential Equations vol 2013 Article ID704547 11 pages 2013

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Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

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Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

2 Journal of Applied Mathematics

knowledge there is no result on other qualitative properties(such as boundedness and stability) and impulsive fractionaldifferential equations involving the Caputo fractional deriva-tive have not been studied very perfectly so we set up a newkind of integral inequalities with weakly singular kernel fordiscontinuous functions and use the new inequalities to studythe qualitative properties of the solutions to certain impulsivefractional differential systems

On the basis of previous studies in this paper we considerthe following integral inequalities withweakly singular kernelfor discontinuous functions

119906 (119905) le 119886 + 119887int

119905

1199050

(119905 minus 119904)120573minus1119906 (119904) 119889119904 + sum

1199050lt119905119894lt119905

120574119894119906 (119905minus

119894) (1)

where 119886 119887 and 120574119894are constants 119886 ge 0 119887 ge 0 120574

119894ge 0 and 119906(119905)

is a nonnegative piecewise-continuous function with the 1stkind of discontinuous points 119905

0lt 1199051lt sdot sdot sdot lim

119894rarrinfin119905119894= infin

119906(119905minus

119894) = lim

119905rarr 119905119894minus0119906(119905) In general due to the existence of weak

singular integral kernel the methods of these inequalitiesfor discontinuous functions are quite different to that ofclassical Gronwall-Bellman-Bihari inequalities We use theproperties of the Mittag-Leffler function 119864

120573(sdot) defined by

119864120573(119911) = sum

infin

119896=0(119911119896Γ(119896120573 + 1)) and the successive iterative

technique to establish the new type of integral inequalitiesfor discontinuous functionsThese inequalities are applied toinvestigate the qualitative analysis of the solutions to certainimpulsive fractional differential systems

2 Preliminary Knowledge

In this section we give some definitions symbols and knowninequalities whichwill be used in the remainder of this paper

Definition 1 Given an interval [119886 119887] of R the fractional(arbitrary) order integral of the function ℎ isin 1198711([119886 119887]R) oforder 120572 isin R

+is defined by

119868120572

119886ℎ (119905) = int

119905

119886

(119905 minus 119904)120572minus1

Γ (120572)

ℎ (119904) 119889119904 (2)

where Γ(sdot) is the gamma function When 119886 = 0 we write119868120572ℎ(119905) = [ℎ lowast 120593

120572](119905) where 120593

120572(119905) = 119905

120572minus1Γ(120572) for 119905 gt 0

120593120572(119905) = 0 for 119905 le 0 and 120593

120572rarr 120575(119905) as 120572 rarr 0 where 120575

is the delta function

Definition 2 For a given function ℎ on the interval [119886 119887] the120572-order Caputo fractional order derivative of ℎ is defined by

(119888119863120572

119886+ℎ) (119905) =

1

Γ (119898 minus 120572)

int

119905

119886

(119905 minus 119904)119898minus120572minus1

ℎ(119898)(119904) 119889119904 (3)

where119898 = [120572] + 1

For calculation simplification the symbols are defined asfollows

119878 (119905 119905119894) = 119864120573(119887Γ (120573) (119905 minus 119905

119894)120573

) for 119905 gt 119905119894

119878119894119895= 119878 (119905

119894 119905119895) = 119864120573(119887Γ (120573) (119905

119894minus 119905119895)

120573

) for 119905119894gt 119905119895

(4)

where 119864120573(sdot) is the Mittag-Leffler function

Lemma 3 (see [6]) Suppose that 120573 gt 0 119886(119905) is a nonnegativeand nondecreasing function which is locally integrable on 0 le119905 lt 119879 (for some 119879 le +infin) and 119892(119905) is a nonnegativenondecreasing continuous function defined on 0 le 119905 lt 119879119892(119905) le 119872 (constant) and suppose that 119906(119905) is a nonnegativeand locally integrable function on 0 le 119905 lt 119879 which satisfies

119906 (119905) le 119886 (119905) + 119892 (119905) int

119905

0

(119905 minus 119904)120573minus1119906 (119904) 119889119904 (5)

and then

119906 (119905) le 119886 (119905) 119864120573(119892 (119905) Γ (120573) 119905

120573) (6)

Using Lemma 3 we can easily get the following corollary

Corollary 4 Let 119886 119887 be constants 119886 ge 0 and 119887 ge 0 Andsuppose 119906(119905) is a nonnegative and locally integrable functionon 1199050le 119905 lt 119879 with

119906 (119905) le 119886 + 119887int

119905

1199050

(119905 minus 119904)120573minus1119906 (119904) 119889119904 (7)

and then

119906 (119905) le 119886119878 (119905 1199050) (8)

Lemma 5 Let 120572 ge 0 and 120573 ge 0 Then

int

1199051

1199050

(119905 minus 119904)120572minus1(119904 minus 1199050)120573minus1

119889119904 le 119861 (120572 120573) (119905 minus 1199050)120572+120573minus1

119905 ge 1199051gt 1199050

(9)

where 119861(sdot sdot) is the Beta function

Proof Let 119904 = 1199050+ 120585(119905 minus 119905

0) we obtain for 119905 ge 119905

1

int

1199051

1199050

(119905 minus 119904)120572minus1(119904 minus 1199050)120573minus1

119889119904

= int

(1199051minus1199050)(119905minus1199050)

0

(1 minus 120585)120572minus1(119905 minus 1199050)120572minus1

(119905 minus 1199050)120573minus1

120585120573minus1119889120585

le (119905 minus 1199050)120572+120573minus1

int

1

0

(1 minus 120585)120572minus1120585120573minus1119889120585

le 119861 (120572 120573) (119905 minus 1199050)120572+120573minus1

(10)

which is the desired result

Remark 6 If we replace the integration interval [1199050 1199051] by

[119905119894 119905119894+1] for 119894 = 0 1 2 we can get the following equality

int

119905119894+1

119905119894

(119905 minus 119904)120572minus1(119904 minus 119905119894)120573minus1

119889119904 le 119861 (120572 120573) (119905 minus 119905119894)120572+120573minus1

119905 ge 119905119894+1gt 119905119894 119894 = 0 1 2

(11)

Corollary 7 Let 120573 ge 0 Then

119887int

1199051

1199050

(119905 minus 119904)120573minus1119878 (119904 1199050) 119889119904 le 119878 (119905 119905

0) minus 1 119905 ge 119905

1gt 1199050

(12)

Journal of Applied Mathematics 3

Proof By the definition of 119878(119904 1199050) we have

119887int

1199051

1199050

(119905 minus 119904)120573minus1119878 (119904 1199050) 119889119904

= 119887int

1199051

1199050

(119905 minus 119904)120573minus1

infin

sum

119896=0

119887119896Γ119896(120573) (119904 minus 119905

0)119896120573

Γ (119896120573 + 1)

119889119904

(13)

Since the series of function is the Mittag-Leffler function119864120573(119887Γ(120573)(119904 minus 119905

0)120573) which is convergent uniformly on 119904 isin

[1199050 1199051] we permute the sum and the integral to obtain

119887int

1199051

1199050

(119905 minus 119904)120573minus1119878 (119904 1199050) 119889119904

=

infin

sum

119896=0

119887119896+1Γ119896(120573)

Γ (119896120573 + 1)

int

1199051

1199050

(119905 minus 119904)120573minus1(119904 minus 1199050)119896120573

119889119904

(14)

Using Lemma 5 we get

119887int

1199051

1199050

(119905 minus 119904)120573minus1119878 (119904 1199050) 119889119904

le

infin

sum

119896=0

119887119896+1Γ119896(120573)

Γ (119896120573 + 1)

(119905 minus 1199050)(119896+1)120573

119861 (120573 119896120573 + 1)

=

infin

sum

119896=0

119887119896+1Γ119896+1(120573) (119905 minus 119905

0)(119896+1)120573

Γ ((119896 + 1) 120573 + 1)

=

infin

sum

119896=1

119887119896Γ119896(120573) (119905 minus 119905

0)119896120573

Γ (119896120573 + 1)

le

infin

sum

119896=0

119887119896Γ119896(120573) (119905 minus 119905

0)119896120573

Γ (119896120573 + 1)

minus 1 = 119878 (119905 1199050) minus 1

(15)

Remark 8 If we replace the integration interval [1199050 1199051] by

[119905119894 119905119894+1] we can get the similar conclusion

119887int

119905119894+1

119905119894

(119905 minus 119904)120573minus1119878 (119904 1199050) 119889119904 le 119878 (119905 119905

119894) minus 1

119905 ge 119905119894+1gt 119905119894 119894 = 0 1 2

(16)

3 Main Results

Theorem 9 Let 119886 119887 be constants 119886 ge 0 119887 ge 0 and 119906(119905) is anonnegative piecewise-continuous function with the 1st kind ofdiscontinuous points 119905

0lt 1199051lt sdot sdot sdot and lim

119894rarrinfin119905119894= infin If

119906 (119905) le 119886 + 119887int

119905

1199050

(119905 minus 119904)120573minus1119906 (119904) 119889119904 + sum

1199050lt119905119894lt119905

120574119894119906 (119905minus

119894) 119905 isin 119868

(17)

where 120573 gt 0 then the following assertions hold

119906 (119905) le 119886 (1 + 1205741) sdot sdot sdot (1 + 120574

119894) 119878119894+10119878119894+11

sdot sdot sdot 119878119894+1119894minus1

119878 (119905 119905119894)

119905 isin (119905119894 119905119894+1]

119906 (119905) le 119886119878 (119905 1199050) 119905 isin [119905

0 1199051]

(18)

Proof If 119905 isin [1199050 1199051] the inequality (17) is reduced to the

following form

119906 (119905) le 119886 + 119887int

119905

1199050

(119905 minus 119904)120573minus1119906 (119904) 119889119904 (19)

Using Lemma 3 we get

119906 (119905) le 119886119878 (119905 1199050)

119906 (119905minus

1) le 119886119878 (119905

1 1199050) = 119886119878

10

(20)

If 119905 isin (1199051 1199052] then

119906 (119905) le 119886 + 1205741119906 (119905minus

1) + 119887int

119905

1199050

(119905 minus 119904)120573minus1119906 (119904) 119889119904

le 119886 + 1205741119906 (119905minus

1) + 119887int

1199051

1199050

(119905 minus 119904)120573minus1119906 (119904) 119889119904

+ 119887int

119905

1199051

(119905 minus 119904)120573minus1119906 (119904) 119889119904

le 119886 + 119886120574111987810+ 119886119887int

1199051

1199050

(119905 minus 119904)120573minus1119878 (119904 1199050) 119889119904

+ 119887int

119905

1199051

(119905 minus 119904)120573minus1119906 (119904) 119889119904

(21)

Using Corollary 7 we get

119906 (119905) le 119886 + 119886120574111987810+ 119886119878 (119905 119905

0) minus 119886 + 119887int

119905

1199051

(119905 minus 119904)120573minus1119906 (119904) 119889119904

le 119886 (120574111987810+ 11987820) + 119887int

119905

1199051

(119905 minus 119904)120573minus1119906 (119904) 119889119904

(22)

Applying Lemma 3 we obtain that

119906 (119905) le 119886 (120574111987810+ 11987820) 119878 (119905 119905

1) le 119886 (1 + 120574

1) 11987820119878 (119905 1199051)

119906 (119905minus

2) le 119886 (1 + 120574

1) 1198782011987821

(23)

Similarly for 119905 isin (1199052 1199053] using Remark 8 we have

119906 (119905) le 119886 + 1205741119906 (119905minus

1) + 1205742119906 (119905minus

2) + 119887int

119905

1199050

(119905 minus 119904)120573minus1119906 (119904) 119889119904

le 119886 + 119886120574111987810+ 1198861205742(1 + 120574

1) 1198782011987821

+ 119887int

1199051

1199050

(119905 minus 119904)120573minus1119906 (119904) 119889119904

+ 119887int

1199052

1199051

(119905 minus 119904)120573minus1119906 (119904) 119889119904 + 119887int

119905

1199052

(119905 minus 119904)120573minus1119906 (119904) 119889119904

le 119886 (1 + 120574111987810+ 1205742(1 + 120574

1) 1198782011987821+ (11987830minus 1)

+ (1 + 1205741) 11987820(11987831minus 1)) + 119887int

119905

1199052

(119905 minus 119904)120573minus1119906 (119904) 119889119904

4 Journal of Applied Mathematics

le 119886 (1 + 120574111987830+ 1205742(1 + 120574

1) 1198783011987831+ (11987830minus 1)

+ (1 + 1205741) 11987830(11987831minus 1)) + 119887int

119905

1199052

(119905 minus 119904)120573minus1119906 (119904) 119889119904

= 119886 (1 + 1205741) (1 + 120574

2) 1198783011987831+ 119887int

119905

1199052

(119905 minus 119904)120573minus1119906 (119904) 119889119904

(24)

Again Lemma 3 implies that

119906 (119905) le 119886 (1 + 1205741) (1 + 120574

2) 1198783011987831119878 (119905 1199052) 119905 isin (119905

2 1199053]

(25)

Using the inductivemethod suppose that for 119905 isin (119905119894minus1 119905119894] 119894 =

1 2 3

119906 (119905) le 119886 (1 + 1205741) (1 + 120574

2) sdot sdot sdot (1 + 120574

119894minus1) 119878119894011987811989411198781198942

sdot sdot sdot 119878119894119894minus2119878 (119905 119905119894minus1)

(26)

and then for each 119905 isin (119905119894 119905119894+1] and from (26) we have

119906 (119905)

le 119886 + 1205741119906 (119905minus

1) + 1205742119906 (119905minus

2) + sdot sdot sdot + 120574

119894119906 (119905minus

119894minus1)

+ 119887int

119905

1199050

(119905 minus 119904)120573minus1119906 (119904) 119889119904

= 119886 + 119886120574111987810+ 1198861205742(1 + 120574

1) 1198782011987821+ sdot sdot sdot

+ 119886120574119894(1 + 120574

1) sdot sdot sdot (1 + 120574

119894minus1) 11987811989401198781198941sdot sdot sdot 119878119894119894minus1

+

119894minus1

sum

119895=0

119887int

119905119895+1

119905119895

(119905 minus 119904)120573minus1119906 (119904) 119889119904

+ int

119905

119905119894

(119905 minus 119904)120573minus1119906 (119904) 119889119904

le 119886 + 119886120574111987810+ 1198861205742(1 + 120574

1) 1198782011987821+ sdot sdot sdot

+ 119886120574119894(1 + 120574

1) sdot sdot sdot (1 + 120574

119894minus1) 11987811989401198781198941sdot sdot sdot 119878119894119894minus1

+ 119886 (119878119894+10

minus 1) + 119886 (1 + 1205741) 11987820(119878119894+11

minus 1) + sdot sdot sdot

+ 119886 (1 + 1205741) sdot sdot sdot (1 + 120574

119894minus1) 11987811989401198781198941sdot sdot sdot 119878119894119894minus2

(119878119894+1119894minus1

minus 1)

+ int

119905

119905119894

(119905 minus 119904)120573minus1119906 (119904) 119889119904

le 119886 (1 + 1205741119878119894+10

+ 1205742(1 + 120574

1) 119878119894+10119878119894+11

+ sdot sdot sdot

+ 120574119894(1 + 120574

1) sdot sdot sdot (1 + 120574

119894minus1) 119878119894+10119878119894+11

sdot sdot sdot 119878119894+1119894minus1

+ 119878119894+10

minus 1 + (1 + 1205741) 119878119894+10

(119878119894+11

minus 1) + sdot sdot sdot

+ (1 + 1205741) sdot sdot sdot (1 + 120574

119894minus1) 11987811989401198781198941sdot sdot sdot 119878119894119894minus2

(119878119894+1119894minus1

minus 1))

+ int

119905

119905119894

(119905 minus 119904)120573minus1119906 (119904) 119889119904

le 119886 (1 + 1205741) (1 + 120574

2) sdot sdot sdot (1 + 120574

119894minus1) 11987811989401198781198941sdot sdot sdot 119878119894119894minus2

+ int

119905

119905119894

(119905 minus 119904)120573minus1119906 (119904) 119889119904

(27)

Then applying Lemma 3 we obtain the inequality (17) Thiscompletes the proof of Theorem 9

Remark 10 The results of Theorem 9 are valid when thefunction has only finite number of discontinuities points1199050 1199051 119905

119894

4 Application to Impulsive FractionalDifferential Systems

Let us consider the following system of Caputo fractionaldifferential equations

119862119863120573

0+119909 (119905) = 119891 (119905 119909 (119905)) 119905 isin 119869 [0 119879] 119905

1 119905

119896

Δ119909 (119905119894) = 119909 (119905

+

119894) minus 119909 (119905

minus

119894) = 119868119894(119909 (119905minus

119894)) 119894 = 1 2 119896

119909 (0) = 1205940

(28)

where 1198621198631205730+ is the Caputo fractional derivatives 0 = 119905

0lt

1199051lt sdot sdot sdot lt 119905

119896lt 119905119896+1

= 119879 119891 [0 119879] times R119899 rarr R119899 isLebesgue measurable with respect to 119905 on [0 119879] and 119891(119905 119909)is continuous with respect to 119909 on R119899 119868

119894 R119899 rarr R119899 are

continuous for 119894 = 1 2 119896 119909(119905+119894) = lim

120576rarr0+119909(119905119894+ 120576) and

119909(119905minus

119894) = lim

120576rarr0minus119909(119905119894minus 120576) represent the right and left limits of

119909(119905) at 119905 = 119905119894 and 120594

0isin R119899 is a constant-valued vector

Using Lemma 62 in [3] 119909(119905) is a solution of Cauchyproblem for system (28) if only if 119909(119905) satisfies

119909 (119905) = 1205940+ sum

1199050lt119905119894lt119905

119868119894(119909 (119905minus

119894))

+

1

Γ (120573)

int

119905

0

(119905 minus 119904)120573minus1119891 (119904 119909 (119904)) 119889119904

(29)

Theorem 11 Suppose that 119891(119905 119909) le 119887119909 and 119868119894(119909(119905minus

119894)) le

120574119894119909(119905minus

119894) where 119887 120574

119894are constants 119887 ge 0 and 120574

119894ge 0 If 119909(119905) is

any solution of the initial value problem (28) then the followingestimations hold

119909 (119905)

le

10038171003817100381710038171205940

1003817100381710038171003817119878lowast(119905 1199050) 119905 isin [119905

0 1199051]

10038171003817100381710038171205940

1003817100381710038171003817(1 + 120574

1) sdot sdot sdot (1 + 120574

119894) 119878lowast

119894+10119878lowast

119894+11

sdot sdot sdot 119878lowast

119894+1119894minus1119878lowast(119905 119905119894) 119905 isin [119905

119894minus1 119905119894]

(30)

where 119878lowast(119905 119905119894) = 119864120573(119887(119905minus119905

119895)120573) 119878lowast(119905

119894 119905119895) = 119864120573(119887(119905119894minus119905119895)120573) and

sdot is a suitable complete norm in R

Journal of Applied Mathematics 5

Proof From (29) it is easy to obtain that

119909 (119905) le10038171003817100381710038171205940

1003817100381710038171003817+ sum

1199050lt119905119894lt119905

120574119894

1003817100381710038171003817119909 (119905minus

119894)1003817100381710038171003817

+ int

119905

0

119887

Γ (120573)

(119905 minus 119904)120573minus1119909 (119904) 119889119904 119905 ge 0

(31)

UsingTheorem 9 we get the desired conclusion

Remark 12 In [13 15] during considering the qualitativeproperties of the impulsive differential equations the func-tions 119891(119905 119909) and 119868

119894(119909) are defined in the domain

119863 = (119905 119909) 119905 isin [1199050 119879] 119879 le infin 119909 le ℎ (32)

for some ℎ gt 0 However we do not add such a restrictionsince our conclusion inTheorem 11 does not involve 119909

Corollary 13 Let the right-hand side of the initial valueproblem (28) satisfy the following conditions

(i) 119891(119905 119909) le 119887119909 119887 is a constant and 119887 ge 0(ii) 119868

119894(119909(119905minus

119894)) le 120574

119894119909(119905minus

119894) 120574119894are constants and 120574

119894ge 0

(iii) there exists a constant 120585 such that 119878lowast(119905 119905119894) le 120585 lt infin

119905 gt 119905119894 119894 = 0 1 2 119896 Then all solutions of the

problem (28) are bounded and the trivial solution ofthe problem (28) is stable in the sense of Lyapunovstability

Corollary 14 Suppose that

(i) 119891(119905 119909) minus 119891(119905 119909) le 119887119909 minus 119910119897 0 lt 119897 lt 1 for all 119909 119910 isin1198751198621

(ii) 119868119894(119909) minus 119868

119894(119910) le 120574

119894119909 minus 119910 where 119887 120574

119894are constants

119887 ge 0 120574119894ge 0 1 le 119894 le 119896

(iii) there exists a constant 120585 such that 119878lowast(119905 119905119894) le 120585 lt infin 119905 gt

119905119894 119894 = 0 1 2 119896 Then all solutions of the problem

(28) are bounded

Conflict of Interests

The author declares that there is no conflict of interests

Acknowledgments

The author sincerely thanks the referees for their constructivesuggestions and corrections This research was partially sup-ported by the NSF of China (Grants 11171178 and 11271225)the NSF of Shandong Province (Grant ZR2013AL005) theScience and Technology Project of High Schools of ShandongProvince (Grant J12LI52) and the Key Projects of Jining Uni-versity Servicing the Local Economics (Grant 2013FWDF14)

References

[1] A A Kilbas H M Srivastava and J J Trujillo Theoryand Applications of Fractional Differential Equations vol 204Elsevier Science Amsterdam The Netherlands 2006

[2] V Lakshmikantham S Leela and D J Vasundhara Theory ofFractional Dynamic Systems Cambridge Scientific Publishers2009

[3] I Podlubny Fractional Differential Equations vol 198 Aca-demic Press San Diego Calif USA 1999

[4] K Diethelm The Analysis of Fractional Differential Equationsvol 2004 Springer Berlin Germany 2010

[5] Z Denton and A S Vatsala ldquoFractional integral inequalitiesand applicationsrdquo Computers amp Mathematics with Applicationsvol 59 no 3 pp 1087ndash1094 2010

[6] H Ye J Gao and Y Ding ldquoA generalized Gronwall inequalityand its application to a fractional differential equationrdquo Journalof Mathematical Analysis and Applications vol 328 no 2 pp1075ndash1081 2007

[7] Q-H Ma and J Pecaric ldquoSome new explicit bounds for weaklysingular integral inequalities with applications to fractionaldifferential and integral equationsrdquo Journal of MathematicalAnalysis and Applications vol 341 no 2 pp 894ndash905 2008

[8] J Shao and F Meng ldquoGronwall-Bellman type inequalities andtheir applications to fractional differential equationsrdquo Abstractand Applied Analysis vol 2013 Article ID 217641 7 pages 2013

[9] A Halanay and D Wexler Qualitative Theory of ImpulsiveSystems Academia Romana Bucharest Romania 1968

[10] A M Samoilenko and N PerestyukDifferential Equations withImpulse Effect Visha Shkola Kyiv Ukraine 1987

[11] V Lakshmikantham D D Baınov and P S Simeonov Theoryof Impulsive Differential Equations vol 6 of Series in ModernApplied Mathematics World Scientific Publishing Teaneck NJUSA 1989

[12] D Xu Y Hueng and L Ling ldquoExistence of positive solutionsof an impulsive delay fishing modelrdquo Bulletin of MathematicalAnalysis and Applications vol 3 no 2 pp 89ndash94 2011

[13] S D Borysenko M Ciarletta and G Iovane ldquoIntegro-suminequalities and motion stability of systems with impulse per-turbationsrdquoNonlinearAnalysisTheoryMethodsampApplicationsvol 62 no 3 pp 417ndash428 2005

[14] AGallo andAM Piccirillo ldquoAbout new analogies ofGronwall-Bellman-Bihari type inequalities for discontinuous functionsand estimated solutions for impulsive differential systemsrdquoNonlinear Analysis Theory Methods amp Applications vol 67 no5 pp 1550ndash1559 2007

[15] A Gallo and A M Piccirillo ldquoOn some generalizationsBellman-Bihari result for integro-functional inequalities fordiscontinuous functions and their applicationsrdquo BoundaryValue Problems vol 2009 Article ID 808124 14 pages 2009

[16] K Balachandran S Kiruthika and J J Trujillo ldquoOn fractionalimpulsive equations of Sobolev type with nonlocal conditionin Banach spacesrdquoComputers ampMathematics with Applicationsvol 62 no 3 pp 1157ndash1165 2011

[17] H Jiang ldquoExistence results for fractional order functionaldifferential equations with impulserdquo Computers ampMathematicswith Applications vol 64 no 10 pp 3477ndash3483 2012

[18] X Li F Chen and X Li ldquoGeneralized anti-periodic boundaryvalue problems of impulsive fractional differential equationsrdquoCommunications in Nonlinear Science and Numerical Simula-tion vol 18 no 1 pp 28ndash41 2013

[19] LMahto S Abbas andA Favini ldquoAnalysis of Caputo impulsivefractional order differential equations with applicationsrdquo Inter-national Journal of Differential Equations vol 2013 Article ID704547 11 pages 2013

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

Journal of Applied Mathematics 3

Proof By the definition of 119878(119904 1199050) we have

119887int

1199051

1199050

(119905 minus 119904)120573minus1119878 (119904 1199050) 119889119904

= 119887int

1199051

1199050

(119905 minus 119904)120573minus1

infin

sum

119896=0

119887119896Γ119896(120573) (119904 minus 119905

0)119896120573

Γ (119896120573 + 1)

119889119904

(13)

Since the series of function is the Mittag-Leffler function119864120573(119887Γ(120573)(119904 minus 119905

0)120573) which is convergent uniformly on 119904 isin

[1199050 1199051] we permute the sum and the integral to obtain

119887int

1199051

1199050

(119905 minus 119904)120573minus1119878 (119904 1199050) 119889119904

=

infin

sum

119896=0

119887119896+1Γ119896(120573)

Γ (119896120573 + 1)

int

1199051

1199050

(119905 minus 119904)120573minus1(119904 minus 1199050)119896120573

119889119904

(14)

Using Lemma 5 we get

119887int

1199051

1199050

(119905 minus 119904)120573minus1119878 (119904 1199050) 119889119904

le

infin

sum

119896=0

119887119896+1Γ119896(120573)

Γ (119896120573 + 1)

(119905 minus 1199050)(119896+1)120573

119861 (120573 119896120573 + 1)

=

infin

sum

119896=0

119887119896+1Γ119896+1(120573) (119905 minus 119905

0)(119896+1)120573

Γ ((119896 + 1) 120573 + 1)

=

infin

sum

119896=1

119887119896Γ119896(120573) (119905 minus 119905

0)119896120573

Γ (119896120573 + 1)

le

infin

sum

119896=0

119887119896Γ119896(120573) (119905 minus 119905

0)119896120573

Γ (119896120573 + 1)

minus 1 = 119878 (119905 1199050) minus 1

(15)

Remark 8 If we replace the integration interval [1199050 1199051] by

[119905119894 119905119894+1] we can get the similar conclusion

119887int

119905119894+1

119905119894

(119905 minus 119904)120573minus1119878 (119904 1199050) 119889119904 le 119878 (119905 119905

119894) minus 1

119905 ge 119905119894+1gt 119905119894 119894 = 0 1 2

(16)

3 Main Results

Theorem 9 Let 119886 119887 be constants 119886 ge 0 119887 ge 0 and 119906(119905) is anonnegative piecewise-continuous function with the 1st kind ofdiscontinuous points 119905

0lt 1199051lt sdot sdot sdot and lim

119894rarrinfin119905119894= infin If

119906 (119905) le 119886 + 119887int

119905

1199050

(119905 minus 119904)120573minus1119906 (119904) 119889119904 + sum

1199050lt119905119894lt119905

120574119894119906 (119905minus

119894) 119905 isin 119868

(17)

where 120573 gt 0 then the following assertions hold

119906 (119905) le 119886 (1 + 1205741) sdot sdot sdot (1 + 120574

119894) 119878119894+10119878119894+11

sdot sdot sdot 119878119894+1119894minus1

119878 (119905 119905119894)

119905 isin (119905119894 119905119894+1]

119906 (119905) le 119886119878 (119905 1199050) 119905 isin [119905

0 1199051]

(18)

Proof If 119905 isin [1199050 1199051] the inequality (17) is reduced to the

following form

119906 (119905) le 119886 + 119887int

119905

1199050

(119905 minus 119904)120573minus1119906 (119904) 119889119904 (19)

Using Lemma 3 we get

119906 (119905) le 119886119878 (119905 1199050)

119906 (119905minus

1) le 119886119878 (119905

1 1199050) = 119886119878

10

(20)

If 119905 isin (1199051 1199052] then

119906 (119905) le 119886 + 1205741119906 (119905minus

1) + 119887int

119905

1199050

(119905 minus 119904)120573minus1119906 (119904) 119889119904

le 119886 + 1205741119906 (119905minus

1) + 119887int

1199051

1199050

(119905 minus 119904)120573minus1119906 (119904) 119889119904

+ 119887int

119905

1199051

(119905 minus 119904)120573minus1119906 (119904) 119889119904

le 119886 + 119886120574111987810+ 119886119887int

1199051

1199050

(119905 minus 119904)120573minus1119878 (119904 1199050) 119889119904

+ 119887int

119905

1199051

(119905 minus 119904)120573minus1119906 (119904) 119889119904

(21)

Using Corollary 7 we get

119906 (119905) le 119886 + 119886120574111987810+ 119886119878 (119905 119905

0) minus 119886 + 119887int

119905

1199051

(119905 minus 119904)120573minus1119906 (119904) 119889119904

le 119886 (120574111987810+ 11987820) + 119887int

119905

1199051

(119905 minus 119904)120573minus1119906 (119904) 119889119904

(22)

Applying Lemma 3 we obtain that

119906 (119905) le 119886 (120574111987810+ 11987820) 119878 (119905 119905

1) le 119886 (1 + 120574

1) 11987820119878 (119905 1199051)

119906 (119905minus

2) le 119886 (1 + 120574

1) 1198782011987821

(23)

Similarly for 119905 isin (1199052 1199053] using Remark 8 we have

119906 (119905) le 119886 + 1205741119906 (119905minus

1) + 1205742119906 (119905minus

2) + 119887int

119905

1199050

(119905 minus 119904)120573minus1119906 (119904) 119889119904

le 119886 + 119886120574111987810+ 1198861205742(1 + 120574

1) 1198782011987821

+ 119887int

1199051

1199050

(119905 minus 119904)120573minus1119906 (119904) 119889119904

+ 119887int

1199052

1199051

(119905 minus 119904)120573minus1119906 (119904) 119889119904 + 119887int

119905

1199052

(119905 minus 119904)120573minus1119906 (119904) 119889119904

le 119886 (1 + 120574111987810+ 1205742(1 + 120574

1) 1198782011987821+ (11987830minus 1)

+ (1 + 1205741) 11987820(11987831minus 1)) + 119887int

119905

1199052

(119905 minus 119904)120573minus1119906 (119904) 119889119904

4 Journal of Applied Mathematics

le 119886 (1 + 120574111987830+ 1205742(1 + 120574

1) 1198783011987831+ (11987830minus 1)

+ (1 + 1205741) 11987830(11987831minus 1)) + 119887int

119905

1199052

(119905 minus 119904)120573minus1119906 (119904) 119889119904

= 119886 (1 + 1205741) (1 + 120574

2) 1198783011987831+ 119887int

119905

1199052

(119905 minus 119904)120573minus1119906 (119904) 119889119904

(24)

Again Lemma 3 implies that

119906 (119905) le 119886 (1 + 1205741) (1 + 120574

2) 1198783011987831119878 (119905 1199052) 119905 isin (119905

2 1199053]

(25)

Using the inductivemethod suppose that for 119905 isin (119905119894minus1 119905119894] 119894 =

1 2 3

119906 (119905) le 119886 (1 + 1205741) (1 + 120574

2) sdot sdot sdot (1 + 120574

119894minus1) 119878119894011987811989411198781198942

sdot sdot sdot 119878119894119894minus2119878 (119905 119905119894minus1)

(26)

and then for each 119905 isin (119905119894 119905119894+1] and from (26) we have

119906 (119905)

le 119886 + 1205741119906 (119905minus

1) + 1205742119906 (119905minus

2) + sdot sdot sdot + 120574

119894119906 (119905minus

119894minus1)

+ 119887int

119905

1199050

(119905 minus 119904)120573minus1119906 (119904) 119889119904

= 119886 + 119886120574111987810+ 1198861205742(1 + 120574

1) 1198782011987821+ sdot sdot sdot

+ 119886120574119894(1 + 120574

1) sdot sdot sdot (1 + 120574

119894minus1) 11987811989401198781198941sdot sdot sdot 119878119894119894minus1

+

119894minus1

sum

119895=0

119887int

119905119895+1

119905119895

(119905 minus 119904)120573minus1119906 (119904) 119889119904

+ int

119905

119905119894

(119905 minus 119904)120573minus1119906 (119904) 119889119904

le 119886 + 119886120574111987810+ 1198861205742(1 + 120574

1) 1198782011987821+ sdot sdot sdot

+ 119886120574119894(1 + 120574

1) sdot sdot sdot (1 + 120574

119894minus1) 11987811989401198781198941sdot sdot sdot 119878119894119894minus1

+ 119886 (119878119894+10

minus 1) + 119886 (1 + 1205741) 11987820(119878119894+11

minus 1) + sdot sdot sdot

+ 119886 (1 + 1205741) sdot sdot sdot (1 + 120574

119894minus1) 11987811989401198781198941sdot sdot sdot 119878119894119894minus2

(119878119894+1119894minus1

minus 1)

+ int

119905

119905119894

(119905 minus 119904)120573minus1119906 (119904) 119889119904

le 119886 (1 + 1205741119878119894+10

+ 1205742(1 + 120574

1) 119878119894+10119878119894+11

+ sdot sdot sdot

+ 120574119894(1 + 120574

1) sdot sdot sdot (1 + 120574

119894minus1) 119878119894+10119878119894+11

sdot sdot sdot 119878119894+1119894minus1

+ 119878119894+10

minus 1 + (1 + 1205741) 119878119894+10

(119878119894+11

minus 1) + sdot sdot sdot

+ (1 + 1205741) sdot sdot sdot (1 + 120574

119894minus1) 11987811989401198781198941sdot sdot sdot 119878119894119894minus2

(119878119894+1119894minus1

minus 1))

+ int

119905

119905119894

(119905 minus 119904)120573minus1119906 (119904) 119889119904

le 119886 (1 + 1205741) (1 + 120574

2) sdot sdot sdot (1 + 120574

119894minus1) 11987811989401198781198941sdot sdot sdot 119878119894119894minus2

+ int

119905

119905119894

(119905 minus 119904)120573minus1119906 (119904) 119889119904

(27)

Then applying Lemma 3 we obtain the inequality (17) Thiscompletes the proof of Theorem 9

Remark 10 The results of Theorem 9 are valid when thefunction has only finite number of discontinuities points1199050 1199051 119905

119894

4 Application to Impulsive FractionalDifferential Systems

Let us consider the following system of Caputo fractionaldifferential equations

119862119863120573

0+119909 (119905) = 119891 (119905 119909 (119905)) 119905 isin 119869 [0 119879] 119905

1 119905

119896

Δ119909 (119905119894) = 119909 (119905

+

119894) minus 119909 (119905

minus

119894) = 119868119894(119909 (119905minus

119894)) 119894 = 1 2 119896

119909 (0) = 1205940

(28)

where 1198621198631205730+ is the Caputo fractional derivatives 0 = 119905

0lt

1199051lt sdot sdot sdot lt 119905

119896lt 119905119896+1

= 119879 119891 [0 119879] times R119899 rarr R119899 isLebesgue measurable with respect to 119905 on [0 119879] and 119891(119905 119909)is continuous with respect to 119909 on R119899 119868

119894 R119899 rarr R119899 are

continuous for 119894 = 1 2 119896 119909(119905+119894) = lim

120576rarr0+119909(119905119894+ 120576) and

119909(119905minus

119894) = lim

120576rarr0minus119909(119905119894minus 120576) represent the right and left limits of

119909(119905) at 119905 = 119905119894 and 120594

0isin R119899 is a constant-valued vector

Using Lemma 62 in [3] 119909(119905) is a solution of Cauchyproblem for system (28) if only if 119909(119905) satisfies

119909 (119905) = 1205940+ sum

1199050lt119905119894lt119905

119868119894(119909 (119905minus

119894))

+

1

Γ (120573)

int

119905

0

(119905 minus 119904)120573minus1119891 (119904 119909 (119904)) 119889119904

(29)

Theorem 11 Suppose that 119891(119905 119909) le 119887119909 and 119868119894(119909(119905minus

119894)) le

120574119894119909(119905minus

119894) where 119887 120574

119894are constants 119887 ge 0 and 120574

119894ge 0 If 119909(119905) is

any solution of the initial value problem (28) then the followingestimations hold

119909 (119905)

le

10038171003817100381710038171205940

1003817100381710038171003817119878lowast(119905 1199050) 119905 isin [119905

0 1199051]

10038171003817100381710038171205940

1003817100381710038171003817(1 + 120574

1) sdot sdot sdot (1 + 120574

119894) 119878lowast

119894+10119878lowast

119894+11

sdot sdot sdot 119878lowast

119894+1119894minus1119878lowast(119905 119905119894) 119905 isin [119905

119894minus1 119905119894]

(30)

where 119878lowast(119905 119905119894) = 119864120573(119887(119905minus119905

119895)120573) 119878lowast(119905

119894 119905119895) = 119864120573(119887(119905119894minus119905119895)120573) and

sdot is a suitable complete norm in R

Journal of Applied Mathematics 5

Proof From (29) it is easy to obtain that

119909 (119905) le10038171003817100381710038171205940

1003817100381710038171003817+ sum

1199050lt119905119894lt119905

120574119894

1003817100381710038171003817119909 (119905minus

119894)1003817100381710038171003817

+ int

119905

0

119887

Γ (120573)

(119905 minus 119904)120573minus1119909 (119904) 119889119904 119905 ge 0

(31)

UsingTheorem 9 we get the desired conclusion

Remark 12 In [13 15] during considering the qualitativeproperties of the impulsive differential equations the func-tions 119891(119905 119909) and 119868

119894(119909) are defined in the domain

119863 = (119905 119909) 119905 isin [1199050 119879] 119879 le infin 119909 le ℎ (32)

for some ℎ gt 0 However we do not add such a restrictionsince our conclusion inTheorem 11 does not involve 119909

Corollary 13 Let the right-hand side of the initial valueproblem (28) satisfy the following conditions

(i) 119891(119905 119909) le 119887119909 119887 is a constant and 119887 ge 0(ii) 119868

119894(119909(119905minus

119894)) le 120574

119894119909(119905minus

119894) 120574119894are constants and 120574

119894ge 0

(iii) there exists a constant 120585 such that 119878lowast(119905 119905119894) le 120585 lt infin

119905 gt 119905119894 119894 = 0 1 2 119896 Then all solutions of the

problem (28) are bounded and the trivial solution ofthe problem (28) is stable in the sense of Lyapunovstability

Corollary 14 Suppose that

(i) 119891(119905 119909) minus 119891(119905 119909) le 119887119909 minus 119910119897 0 lt 119897 lt 1 for all 119909 119910 isin1198751198621

(ii) 119868119894(119909) minus 119868

119894(119910) le 120574

119894119909 minus 119910 where 119887 120574

119894are constants

119887 ge 0 120574119894ge 0 1 le 119894 le 119896

(iii) there exists a constant 120585 such that 119878lowast(119905 119905119894) le 120585 lt infin 119905 gt

119905119894 119894 = 0 1 2 119896 Then all solutions of the problem

(28) are bounded

Conflict of Interests

The author declares that there is no conflict of interests

Acknowledgments

The author sincerely thanks the referees for their constructivesuggestions and corrections This research was partially sup-ported by the NSF of China (Grants 11171178 and 11271225)the NSF of Shandong Province (Grant ZR2013AL005) theScience and Technology Project of High Schools of ShandongProvince (Grant J12LI52) and the Key Projects of Jining Uni-versity Servicing the Local Economics (Grant 2013FWDF14)

References

[1] A A Kilbas H M Srivastava and J J Trujillo Theoryand Applications of Fractional Differential Equations vol 204Elsevier Science Amsterdam The Netherlands 2006

[2] V Lakshmikantham S Leela and D J Vasundhara Theory ofFractional Dynamic Systems Cambridge Scientific Publishers2009

[3] I Podlubny Fractional Differential Equations vol 198 Aca-demic Press San Diego Calif USA 1999

[4] K Diethelm The Analysis of Fractional Differential Equationsvol 2004 Springer Berlin Germany 2010

[5] Z Denton and A S Vatsala ldquoFractional integral inequalitiesand applicationsrdquo Computers amp Mathematics with Applicationsvol 59 no 3 pp 1087ndash1094 2010

[6] H Ye J Gao and Y Ding ldquoA generalized Gronwall inequalityand its application to a fractional differential equationrdquo Journalof Mathematical Analysis and Applications vol 328 no 2 pp1075ndash1081 2007

[7] Q-H Ma and J Pecaric ldquoSome new explicit bounds for weaklysingular integral inequalities with applications to fractionaldifferential and integral equationsrdquo Journal of MathematicalAnalysis and Applications vol 341 no 2 pp 894ndash905 2008

[8] J Shao and F Meng ldquoGronwall-Bellman type inequalities andtheir applications to fractional differential equationsrdquo Abstractand Applied Analysis vol 2013 Article ID 217641 7 pages 2013

[9] A Halanay and D Wexler Qualitative Theory of ImpulsiveSystems Academia Romana Bucharest Romania 1968

[10] A M Samoilenko and N PerestyukDifferential Equations withImpulse Effect Visha Shkola Kyiv Ukraine 1987

[11] V Lakshmikantham D D Baınov and P S Simeonov Theoryof Impulsive Differential Equations vol 6 of Series in ModernApplied Mathematics World Scientific Publishing Teaneck NJUSA 1989

[12] D Xu Y Hueng and L Ling ldquoExistence of positive solutionsof an impulsive delay fishing modelrdquo Bulletin of MathematicalAnalysis and Applications vol 3 no 2 pp 89ndash94 2011

[13] S D Borysenko M Ciarletta and G Iovane ldquoIntegro-suminequalities and motion stability of systems with impulse per-turbationsrdquoNonlinearAnalysisTheoryMethodsampApplicationsvol 62 no 3 pp 417ndash428 2005

[14] AGallo andAM Piccirillo ldquoAbout new analogies ofGronwall-Bellman-Bihari type inequalities for discontinuous functionsand estimated solutions for impulsive differential systemsrdquoNonlinear Analysis Theory Methods amp Applications vol 67 no5 pp 1550ndash1559 2007

[15] A Gallo and A M Piccirillo ldquoOn some generalizationsBellman-Bihari result for integro-functional inequalities fordiscontinuous functions and their applicationsrdquo BoundaryValue Problems vol 2009 Article ID 808124 14 pages 2009

[16] K Balachandran S Kiruthika and J J Trujillo ldquoOn fractionalimpulsive equations of Sobolev type with nonlocal conditionin Banach spacesrdquoComputers ampMathematics with Applicationsvol 62 no 3 pp 1157ndash1165 2011

[17] H Jiang ldquoExistence results for fractional order functionaldifferential equations with impulserdquo Computers ampMathematicswith Applications vol 64 no 10 pp 3477ndash3483 2012

[18] X Li F Chen and X Li ldquoGeneralized anti-periodic boundaryvalue problems of impulsive fractional differential equationsrdquoCommunications in Nonlinear Science and Numerical Simula-tion vol 18 no 1 pp 28ndash41 2013

[19] LMahto S Abbas andA Favini ldquoAnalysis of Caputo impulsivefractional order differential equations with applicationsrdquo Inter-national Journal of Differential Equations vol 2013 Article ID704547 11 pages 2013

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

4 Journal of Applied Mathematics

le 119886 (1 + 120574111987830+ 1205742(1 + 120574

1) 1198783011987831+ (11987830minus 1)

+ (1 + 1205741) 11987830(11987831minus 1)) + 119887int

119905

1199052

(119905 minus 119904)120573minus1119906 (119904) 119889119904

= 119886 (1 + 1205741) (1 + 120574

2) 1198783011987831+ 119887int

119905

1199052

(119905 minus 119904)120573minus1119906 (119904) 119889119904

(24)

Again Lemma 3 implies that

119906 (119905) le 119886 (1 + 1205741) (1 + 120574

2) 1198783011987831119878 (119905 1199052) 119905 isin (119905

2 1199053]

(25)

Using the inductivemethod suppose that for 119905 isin (119905119894minus1 119905119894] 119894 =

1 2 3

119906 (119905) le 119886 (1 + 1205741) (1 + 120574

2) sdot sdot sdot (1 + 120574

119894minus1) 119878119894011987811989411198781198942

sdot sdot sdot 119878119894119894minus2119878 (119905 119905119894minus1)

(26)

and then for each 119905 isin (119905119894 119905119894+1] and from (26) we have

119906 (119905)

le 119886 + 1205741119906 (119905minus

1) + 1205742119906 (119905minus

2) + sdot sdot sdot + 120574

119894119906 (119905minus

119894minus1)

+ 119887int

119905

1199050

(119905 minus 119904)120573minus1119906 (119904) 119889119904

= 119886 + 119886120574111987810+ 1198861205742(1 + 120574

1) 1198782011987821+ sdot sdot sdot

+ 119886120574119894(1 + 120574

1) sdot sdot sdot (1 + 120574

119894minus1) 11987811989401198781198941sdot sdot sdot 119878119894119894minus1

+

119894minus1

sum

119895=0

119887int

119905119895+1

119905119895

(119905 minus 119904)120573minus1119906 (119904) 119889119904

+ int

119905

119905119894

(119905 minus 119904)120573minus1119906 (119904) 119889119904

le 119886 + 119886120574111987810+ 1198861205742(1 + 120574

1) 1198782011987821+ sdot sdot sdot

+ 119886120574119894(1 + 120574

1) sdot sdot sdot (1 + 120574

119894minus1) 11987811989401198781198941sdot sdot sdot 119878119894119894minus1

+ 119886 (119878119894+10

minus 1) + 119886 (1 + 1205741) 11987820(119878119894+11

minus 1) + sdot sdot sdot

+ 119886 (1 + 1205741) sdot sdot sdot (1 + 120574

119894minus1) 11987811989401198781198941sdot sdot sdot 119878119894119894minus2

(119878119894+1119894minus1

minus 1)

+ int

119905

119905119894

(119905 minus 119904)120573minus1119906 (119904) 119889119904

le 119886 (1 + 1205741119878119894+10

+ 1205742(1 + 120574

1) 119878119894+10119878119894+11

+ sdot sdot sdot

+ 120574119894(1 + 120574

1) sdot sdot sdot (1 + 120574

119894minus1) 119878119894+10119878119894+11

sdot sdot sdot 119878119894+1119894minus1

+ 119878119894+10

minus 1 + (1 + 1205741) 119878119894+10

(119878119894+11

minus 1) + sdot sdot sdot

+ (1 + 1205741) sdot sdot sdot (1 + 120574

119894minus1) 11987811989401198781198941sdot sdot sdot 119878119894119894minus2

(119878119894+1119894minus1

minus 1))

+ int

119905

119905119894

(119905 minus 119904)120573minus1119906 (119904) 119889119904

le 119886 (1 + 1205741) (1 + 120574

2) sdot sdot sdot (1 + 120574

119894minus1) 11987811989401198781198941sdot sdot sdot 119878119894119894minus2

+ int

119905

119905119894

(119905 minus 119904)120573minus1119906 (119904) 119889119904

(27)

Then applying Lemma 3 we obtain the inequality (17) Thiscompletes the proof of Theorem 9

Remark 10 The results of Theorem 9 are valid when thefunction has only finite number of discontinuities points1199050 1199051 119905

119894

4 Application to Impulsive FractionalDifferential Systems

Let us consider the following system of Caputo fractionaldifferential equations

119862119863120573

0+119909 (119905) = 119891 (119905 119909 (119905)) 119905 isin 119869 [0 119879] 119905

1 119905

119896

Δ119909 (119905119894) = 119909 (119905

+

119894) minus 119909 (119905

minus

119894) = 119868119894(119909 (119905minus

119894)) 119894 = 1 2 119896

119909 (0) = 1205940

(28)

where 1198621198631205730+ is the Caputo fractional derivatives 0 = 119905

0lt

1199051lt sdot sdot sdot lt 119905

119896lt 119905119896+1

= 119879 119891 [0 119879] times R119899 rarr R119899 isLebesgue measurable with respect to 119905 on [0 119879] and 119891(119905 119909)is continuous with respect to 119909 on R119899 119868

119894 R119899 rarr R119899 are

continuous for 119894 = 1 2 119896 119909(119905+119894) = lim

120576rarr0+119909(119905119894+ 120576) and

119909(119905minus

119894) = lim

120576rarr0minus119909(119905119894minus 120576) represent the right and left limits of

119909(119905) at 119905 = 119905119894 and 120594

0isin R119899 is a constant-valued vector

Using Lemma 62 in [3] 119909(119905) is a solution of Cauchyproblem for system (28) if only if 119909(119905) satisfies

119909 (119905) = 1205940+ sum

1199050lt119905119894lt119905

119868119894(119909 (119905minus

119894))

+

1

Γ (120573)

int

119905

0

(119905 minus 119904)120573minus1119891 (119904 119909 (119904)) 119889119904

(29)

Theorem 11 Suppose that 119891(119905 119909) le 119887119909 and 119868119894(119909(119905minus

119894)) le

120574119894119909(119905minus

119894) where 119887 120574

119894are constants 119887 ge 0 and 120574

119894ge 0 If 119909(119905) is

any solution of the initial value problem (28) then the followingestimations hold

119909 (119905)

le

10038171003817100381710038171205940

1003817100381710038171003817119878lowast(119905 1199050) 119905 isin [119905

0 1199051]

10038171003817100381710038171205940

1003817100381710038171003817(1 + 120574

1) sdot sdot sdot (1 + 120574

119894) 119878lowast

119894+10119878lowast

119894+11

sdot sdot sdot 119878lowast

119894+1119894minus1119878lowast(119905 119905119894) 119905 isin [119905

119894minus1 119905119894]

(30)

where 119878lowast(119905 119905119894) = 119864120573(119887(119905minus119905

119895)120573) 119878lowast(119905

119894 119905119895) = 119864120573(119887(119905119894minus119905119895)120573) and

sdot is a suitable complete norm in R

Journal of Applied Mathematics 5

Proof From (29) it is easy to obtain that

119909 (119905) le10038171003817100381710038171205940

1003817100381710038171003817+ sum

1199050lt119905119894lt119905

120574119894

1003817100381710038171003817119909 (119905minus

119894)1003817100381710038171003817

+ int

119905

0

119887

Γ (120573)

(119905 minus 119904)120573minus1119909 (119904) 119889119904 119905 ge 0

(31)

UsingTheorem 9 we get the desired conclusion

Remark 12 In [13 15] during considering the qualitativeproperties of the impulsive differential equations the func-tions 119891(119905 119909) and 119868

119894(119909) are defined in the domain

119863 = (119905 119909) 119905 isin [1199050 119879] 119879 le infin 119909 le ℎ (32)

for some ℎ gt 0 However we do not add such a restrictionsince our conclusion inTheorem 11 does not involve 119909

Corollary 13 Let the right-hand side of the initial valueproblem (28) satisfy the following conditions

(i) 119891(119905 119909) le 119887119909 119887 is a constant and 119887 ge 0(ii) 119868

119894(119909(119905minus

119894)) le 120574

119894119909(119905minus

119894) 120574119894are constants and 120574

119894ge 0

(iii) there exists a constant 120585 such that 119878lowast(119905 119905119894) le 120585 lt infin

119905 gt 119905119894 119894 = 0 1 2 119896 Then all solutions of the

problem (28) are bounded and the trivial solution ofthe problem (28) is stable in the sense of Lyapunovstability

Corollary 14 Suppose that

(i) 119891(119905 119909) minus 119891(119905 119909) le 119887119909 minus 119910119897 0 lt 119897 lt 1 for all 119909 119910 isin1198751198621

(ii) 119868119894(119909) minus 119868

119894(119910) le 120574

119894119909 minus 119910 where 119887 120574

119894are constants

119887 ge 0 120574119894ge 0 1 le 119894 le 119896

(iii) there exists a constant 120585 such that 119878lowast(119905 119905119894) le 120585 lt infin 119905 gt

119905119894 119894 = 0 1 2 119896 Then all solutions of the problem

(28) are bounded

Conflict of Interests

The author declares that there is no conflict of interests

Acknowledgments

The author sincerely thanks the referees for their constructivesuggestions and corrections This research was partially sup-ported by the NSF of China (Grants 11171178 and 11271225)the NSF of Shandong Province (Grant ZR2013AL005) theScience and Technology Project of High Schools of ShandongProvince (Grant J12LI52) and the Key Projects of Jining Uni-versity Servicing the Local Economics (Grant 2013FWDF14)

References

[1] A A Kilbas H M Srivastava and J J Trujillo Theoryand Applications of Fractional Differential Equations vol 204Elsevier Science Amsterdam The Netherlands 2006

[2] V Lakshmikantham S Leela and D J Vasundhara Theory ofFractional Dynamic Systems Cambridge Scientific Publishers2009

[3] I Podlubny Fractional Differential Equations vol 198 Aca-demic Press San Diego Calif USA 1999

[4] K Diethelm The Analysis of Fractional Differential Equationsvol 2004 Springer Berlin Germany 2010

[5] Z Denton and A S Vatsala ldquoFractional integral inequalitiesand applicationsrdquo Computers amp Mathematics with Applicationsvol 59 no 3 pp 1087ndash1094 2010

[6] H Ye J Gao and Y Ding ldquoA generalized Gronwall inequalityand its application to a fractional differential equationrdquo Journalof Mathematical Analysis and Applications vol 328 no 2 pp1075ndash1081 2007

[7] Q-H Ma and J Pecaric ldquoSome new explicit bounds for weaklysingular integral inequalities with applications to fractionaldifferential and integral equationsrdquo Journal of MathematicalAnalysis and Applications vol 341 no 2 pp 894ndash905 2008

[8] J Shao and F Meng ldquoGronwall-Bellman type inequalities andtheir applications to fractional differential equationsrdquo Abstractand Applied Analysis vol 2013 Article ID 217641 7 pages 2013

[9] A Halanay and D Wexler Qualitative Theory of ImpulsiveSystems Academia Romana Bucharest Romania 1968

[10] A M Samoilenko and N PerestyukDifferential Equations withImpulse Effect Visha Shkola Kyiv Ukraine 1987

[11] V Lakshmikantham D D Baınov and P S Simeonov Theoryof Impulsive Differential Equations vol 6 of Series in ModernApplied Mathematics World Scientific Publishing Teaneck NJUSA 1989

[12] D Xu Y Hueng and L Ling ldquoExistence of positive solutionsof an impulsive delay fishing modelrdquo Bulletin of MathematicalAnalysis and Applications vol 3 no 2 pp 89ndash94 2011

[13] S D Borysenko M Ciarletta and G Iovane ldquoIntegro-suminequalities and motion stability of systems with impulse per-turbationsrdquoNonlinearAnalysisTheoryMethodsampApplicationsvol 62 no 3 pp 417ndash428 2005

[14] AGallo andAM Piccirillo ldquoAbout new analogies ofGronwall-Bellman-Bihari type inequalities for discontinuous functionsand estimated solutions for impulsive differential systemsrdquoNonlinear Analysis Theory Methods amp Applications vol 67 no5 pp 1550ndash1559 2007

[15] A Gallo and A M Piccirillo ldquoOn some generalizationsBellman-Bihari result for integro-functional inequalities fordiscontinuous functions and their applicationsrdquo BoundaryValue Problems vol 2009 Article ID 808124 14 pages 2009

[16] K Balachandran S Kiruthika and J J Trujillo ldquoOn fractionalimpulsive equations of Sobolev type with nonlocal conditionin Banach spacesrdquoComputers ampMathematics with Applicationsvol 62 no 3 pp 1157ndash1165 2011

[17] H Jiang ldquoExistence results for fractional order functionaldifferential equations with impulserdquo Computers ampMathematicswith Applications vol 64 no 10 pp 3477ndash3483 2012

[18] X Li F Chen and X Li ldquoGeneralized anti-periodic boundaryvalue problems of impulsive fractional differential equationsrdquoCommunications in Nonlinear Science and Numerical Simula-tion vol 18 no 1 pp 28ndash41 2013

[19] LMahto S Abbas andA Favini ldquoAnalysis of Caputo impulsivefractional order differential equations with applicationsrdquo Inter-national Journal of Differential Equations vol 2013 Article ID704547 11 pages 2013

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

Journal of Applied Mathematics 5

Proof From (29) it is easy to obtain that

119909 (119905) le10038171003817100381710038171205940

1003817100381710038171003817+ sum

1199050lt119905119894lt119905

120574119894

1003817100381710038171003817119909 (119905minus

119894)1003817100381710038171003817

+ int

119905

0

119887

Γ (120573)

(119905 minus 119904)120573minus1119909 (119904) 119889119904 119905 ge 0

(31)

UsingTheorem 9 we get the desired conclusion

Remark 12 In [13 15] during considering the qualitativeproperties of the impulsive differential equations the func-tions 119891(119905 119909) and 119868

119894(119909) are defined in the domain

119863 = (119905 119909) 119905 isin [1199050 119879] 119879 le infin 119909 le ℎ (32)

for some ℎ gt 0 However we do not add such a restrictionsince our conclusion inTheorem 11 does not involve 119909

Corollary 13 Let the right-hand side of the initial valueproblem (28) satisfy the following conditions

(i) 119891(119905 119909) le 119887119909 119887 is a constant and 119887 ge 0(ii) 119868

119894(119909(119905minus

119894)) le 120574

119894119909(119905minus

119894) 120574119894are constants and 120574

119894ge 0

(iii) there exists a constant 120585 such that 119878lowast(119905 119905119894) le 120585 lt infin

119905 gt 119905119894 119894 = 0 1 2 119896 Then all solutions of the

problem (28) are bounded and the trivial solution ofthe problem (28) is stable in the sense of Lyapunovstability

Corollary 14 Suppose that

(i) 119891(119905 119909) minus 119891(119905 119909) le 119887119909 minus 119910119897 0 lt 119897 lt 1 for all 119909 119910 isin1198751198621

(ii) 119868119894(119909) minus 119868

119894(119910) le 120574

119894119909 minus 119910 where 119887 120574

119894are constants

119887 ge 0 120574119894ge 0 1 le 119894 le 119896

(iii) there exists a constant 120585 such that 119878lowast(119905 119905119894) le 120585 lt infin 119905 gt

119905119894 119894 = 0 1 2 119896 Then all solutions of the problem

(28) are bounded

Conflict of Interests

The author declares that there is no conflict of interests

Acknowledgments

The author sincerely thanks the referees for their constructivesuggestions and corrections This research was partially sup-ported by the NSF of China (Grants 11171178 and 11271225)the NSF of Shandong Province (Grant ZR2013AL005) theScience and Technology Project of High Schools of ShandongProvince (Grant J12LI52) and the Key Projects of Jining Uni-versity Servicing the Local Economics (Grant 2013FWDF14)

References

[1] A A Kilbas H M Srivastava and J J Trujillo Theoryand Applications of Fractional Differential Equations vol 204Elsevier Science Amsterdam The Netherlands 2006

[2] V Lakshmikantham S Leela and D J Vasundhara Theory ofFractional Dynamic Systems Cambridge Scientific Publishers2009

[3] I Podlubny Fractional Differential Equations vol 198 Aca-demic Press San Diego Calif USA 1999

[4] K Diethelm The Analysis of Fractional Differential Equationsvol 2004 Springer Berlin Germany 2010

[5] Z Denton and A S Vatsala ldquoFractional integral inequalitiesand applicationsrdquo Computers amp Mathematics with Applicationsvol 59 no 3 pp 1087ndash1094 2010

[6] H Ye J Gao and Y Ding ldquoA generalized Gronwall inequalityand its application to a fractional differential equationrdquo Journalof Mathematical Analysis and Applications vol 328 no 2 pp1075ndash1081 2007

[7] Q-H Ma and J Pecaric ldquoSome new explicit bounds for weaklysingular integral inequalities with applications to fractionaldifferential and integral equationsrdquo Journal of MathematicalAnalysis and Applications vol 341 no 2 pp 894ndash905 2008

[8] J Shao and F Meng ldquoGronwall-Bellman type inequalities andtheir applications to fractional differential equationsrdquo Abstractand Applied Analysis vol 2013 Article ID 217641 7 pages 2013

[9] A Halanay and D Wexler Qualitative Theory of ImpulsiveSystems Academia Romana Bucharest Romania 1968

[10] A M Samoilenko and N PerestyukDifferential Equations withImpulse Effect Visha Shkola Kyiv Ukraine 1987

[11] V Lakshmikantham D D Baınov and P S Simeonov Theoryof Impulsive Differential Equations vol 6 of Series in ModernApplied Mathematics World Scientific Publishing Teaneck NJUSA 1989

[12] D Xu Y Hueng and L Ling ldquoExistence of positive solutionsof an impulsive delay fishing modelrdquo Bulletin of MathematicalAnalysis and Applications vol 3 no 2 pp 89ndash94 2011

[13] S D Borysenko M Ciarletta and G Iovane ldquoIntegro-suminequalities and motion stability of systems with impulse per-turbationsrdquoNonlinearAnalysisTheoryMethodsampApplicationsvol 62 no 3 pp 417ndash428 2005

[14] AGallo andAM Piccirillo ldquoAbout new analogies ofGronwall-Bellman-Bihari type inequalities for discontinuous functionsand estimated solutions for impulsive differential systemsrdquoNonlinear Analysis Theory Methods amp Applications vol 67 no5 pp 1550ndash1559 2007

[15] A Gallo and A M Piccirillo ldquoOn some generalizationsBellman-Bihari result for integro-functional inequalities fordiscontinuous functions and their applicationsrdquo BoundaryValue Problems vol 2009 Article ID 808124 14 pages 2009

[16] K Balachandran S Kiruthika and J J Trujillo ldquoOn fractionalimpulsive equations of Sobolev type with nonlocal conditionin Banach spacesrdquoComputers ampMathematics with Applicationsvol 62 no 3 pp 1157ndash1165 2011

[17] H Jiang ldquoExistence results for fractional order functionaldifferential equations with impulserdquo Computers ampMathematicswith Applications vol 64 no 10 pp 3477ndash3483 2012

[18] X Li F Chen and X Li ldquoGeneralized anti-periodic boundaryvalue problems of impulsive fractional differential equationsrdquoCommunications in Nonlinear Science and Numerical Simula-tion vol 18 no 1 pp 28ndash41 2013

[19] LMahto S Abbas andA Favini ldquoAnalysis of Caputo impulsivefractional order differential equations with applicationsrdquo Inter-national Journal of Differential Equations vol 2013 Article ID704547 11 pages 2013

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

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Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of