Hindawi Publishing CorporationJournal of Applied MathematicsVolume 2013 Article ID 419701 8 pageshttpdxdoiorg1011552013419701

Research ArticleThe Strong Fuzzy Henstock Integrals and Discontinuous FuzzyDifferential Equations

Yabin Shao12 and Huanhuan Zhang1

1 College of Mathematics and Computer Science Northwest University for Nationalities Lanzhou 730030 China2Department of Mathematics Sichuan University Chengdu 610065 China

Correspondence should be addressed to Yabin Shao yb-shao163com

Received 24 April 2013 Accepted 24 September 2013

Academic Editor Mehmet Sezer

Copyright copy 2013 Y Shao and H Zhang This is an open access article distributed under the Creative Commons AttributionLicense which permits unrestricted use distribution and reproduction in any medium provided the original work is properlycited

We generalized the existence theorems and the continuous dependence of a solution on parameters for initial problems of fuzzydiscontinuous differential equation by the strong fuzzy Henstock integral and its controlled convergence theorem

1 Introduction

The Cauchy problems for fuzzy differential equations havebeen studied by several authors [1ndash6] on the metric space(119864

119899 119863) of normal fuzzy convex set with the distance119863 givenby the maximum of the Hausdorff distance between thecorresponding level sets In [4] Nieto proved that the Cauchyproblem has a uniqueness result if 119891 is continuous andbounded In [1 3 7ndash9] the authors presented a uniquenessresult when 119891 satisfies a Lipschitz condition For a generalreference to fuzzy differential equations see a recent bookby Lakshmikantham and Mohapatra [10] and referencestherein In 2002 Xue and Fu [11] established the solutionsto fuzzy differential equations with right-hand side functionssatisfying Caratheodory conditions on a class of Lipschitzfuzzy setsHowever there are discontinuous systems inwhichthe right-hand side functions 119891 [119886 119887] times 119864

119899 rarr 119864119899 arenot integrable in the sense of Kaleva [1] on certain intervalsand their solutions are not absolute continuous functions Toillustrate we consider the following example

Example 1 Consider the following discontinuous system

1199091015840

(119905) = ℎ (119905) 119909 (0) = 119860

119892 (119905) =

2119905 sin 1

1199052minus2

119905cos 1

1199052 119905 = 0

0 119905 = 0

119860 (119904) =

119904 0 le 119904 le 1

2 minus 119904 1 lt 119904 le 2

0 others

ℎ (119905) = 120594|119892(119905)|

+ 119860

(1)

Then ℎ(119905) = 120594|119892(119905)|

+119860 is not integrable in the sense of KalevaHowever the above system has the following solution

119909 (119905) = 120594|119866(119905)|

+ 119860119905 (2)

where

119866 (119905) =

1199052 sin 1

1199052 119905 = 0

0 119905 = 0(3)

It is well known that the Henstock integral is designedto integrate highly oscillatory functions which the Lebesgueintegral fails to do It is known as nonabsolute integrationand is a powerful tool It is well known that the Henstockintegral includes the Riemann improper Riemann Lebesgueand Newton integrals [12 13] Though such an integral wasdefined by Denjoy in 1912 and also by Perron in 1914 itwas difficult to handle using their definitions But withthe Riemann-type definition introduced more recently byHenstock [12] in 1963 and also independently by Kurzweil

2 Journal of Applied Mathematics

[13] the definition is now simple and furthermore the proofinvolving the integral also turns out to be easy For moredetailed results about the Henstock integral we refer to[14] Recently Wu and Gong [15 16] have combined thefuzzy set theory and nonabsolute integration theory andthey discussed the fuzzyHenstock integrals of fuzzy-number-valued functions which extended Kaleva [1] integration Inorder to complete the theory of fuzzy calculus and tomeet thesolving need of transferring a fuzzy differential equation intoa fuzzy integral equation Gong and Shao [17 18] defined thestrong fuzzy Henstock integrals and discussed some of theirproperties and the controlled convergence theorem

In this paper according to the idea of [19] and usingthe concept of generalized differentiability [20] we willprove other controlled convergence theorems for the strongfuzzy Henstock integrals which will be of foundationalsignificance for studying the existence and uniqueness ofsolutions to the fuzzy discontinuous systems As we knowwe inevitably use the controlled convergence theorems forsolving the numerical solutions of differential equations Asthe main outcomes we will deal with the Cauchy problem ofdiscontinuous fuzzy systems as follows

1199091015840

(119905) = 119891 (119905 119909)

119909 (120591) = 120585 isin 119864119899

(4)

where 119891 119880 rarr 119864119899 is a strong fuzzy Henstock integrablefunction and

119880 = (119905 119909) |119905 minus 120591| le 119886 119909 isin 119864119899

119863 (119909 120585) le 119887 (5)

To make our analysis possible we will first recall somebasic results of fuzzy numbers and give some definitionsof absolutely continuous fuzzy-number-valued function Inaddition we present the concept of generalized differentia-bility In Section 3 we present the concept of strong fuzzyHenstock integrals and we prove a controlled convergencetheorem for the strong fuzzyHenstock integrals In Section 4we deal with the Cauchy problem of discontinuous fuzzysystems And in Section 5 we present some concludingremarks

2 Preliminaries

Let119875119896(119877119899) denote the family of all nonempty compact convex

subset of119877119899 and define the addition and scalarmultiplicationin 119875

119896(119877119899) as usual Let 119860 and 119861 be two nonempty bounded

subsets of 119877119899 The distance between119860 and 119861 is defined by theHausdorff metric [21] as follows

119889119867(119860 119861) = maxsup

119886isin119860

inf119887isin119861

119886 minus 119887 sup119887isin119861

inf119886isin119860

119887 minus 119886 (6)

Denote that 119864119899 = 119906 119877119899 rarr [0 1] | 119906 satisfies (1)ndash(4)below is a fuzzy number space where

(1) 119906 is normal that is there exists an 1199090isin 119877119899 such that

119906(1199090) = 1

(2) 119906 is fuzzy convex that is 119906(120582119909 + (1 minus 120582)119910) ge

min119906(119909) 119906(119910) for any 119909 119910 isin 119877119899 and 0 le 120582 le 1(3) 119906 is upper semicontinuous(4) [119906]0 = cl119909 isin 119877119899 | 119906(119909) gt 0 is compact

For 0 lt 120572 le 1 denote that [119906]120572 = 119909 isin 119877119899 | 119906(119909) ge 120572Then from the above (1)ndash(4) it follows that the 120572-level set[119906]

120572

isin 119875119896(119877119899) for all 0 le 120572 lt 1

According to Zadehrsquos extension principle we have addi-tion and scalar multiplication in fuzzy number space 119864119899 asfollows [21]

[119906 + V]120572 = [119906]120572 + [V]120572 [119896119906]120572

= 119896[119906]120572

(7)

where 119906 V isin 119864119899 and 0 le 120572 le 1Define119863 119864119899 times 119864119899 rarr [0infin)

119863 (119906 V) = sup 119889119867([119906]

120572

[V]120572) 120572 isin [0 1] (8)

where 119889 is the Hausdorff metric defined in 119875119896(119877119899) Then it is

easy to see that119863 is a metric in 119864119899 Using the results [22] weknow that

(1) (119864119899 119863) is a complete metric space(2) 119863(119906 + 119908 V + 119908) = 119863(119906 V) for all 119906 V 119908 isin 119864119899(3) 119863(120582119906 120582V) = |120582|119863(119906 V) for all 119906 V 119908 isin 119864119899 and 120582 isin 119877

Let 119909 119910 isin 119864119899 If there exist 119911 isin 119864119899 such that 119909 = 119910 + 119911then 119911 is called the 119867-difference of 119909 and 119910 and is denotedby 119909 minus

119867

119910 As mentioned above which always is called thecondition (119867) It is well known that the 119867-derivative forfuzzy-number-functions was initially introduced by Puri etal [5 23] and it is based on the condition (119867) of sets Wenote that this definition is fairly strong because the familyof fuzzy-number-valued functions 119867-differentiable is veryrestrictive For example the fuzzy-number-valued function119891 [119886 119887] rarr 119877F defined by 119891(119909) = 119862 sdot 119892(119909) where 119862 isa fuzzy number sdot is the scalar multiplication (in the fuzzycontext) and 119892 [119886 119887] rarr 119877

+ with 1198921015840(1199050) lt 0 is not 119867-

differentiable in 1199050(see [20 24]) To avoid the above difficulty

in this paper we consider a more general definition of aderivative for fuzzy-number-valued functions enlarging theclass of differentiable fuzzy-number-valued functions whichhas been introduced in [20]

Definition 2 (see [20]) Let 119891 (119886 119887) rarr 119864119899 and 119909

0isin

(119886 119887) One says that 119891 is differentiable at 1199090 if there exists

an element 1198911015840(1199050) isin 119864119899 such that

(1) for all ℎ gt 0 sufficiently small there exists 119891(1199090+

ℎ) minus119867119891(119909

0) 119891(119909

0) minus

119867119891(119909

0minus ℎ) and the limits (in

the metric119863)

limℎrarr0

119891 (1199090+ ℎ) minus

119867119891 (119909

0)

ℎ= lim

ℎrarr0

119891 (1199090) minus

119867119891 (119909

0minus ℎ)

ℎ

= 1198911015840

(1199090)

(9)

or

Journal of Applied Mathematics 3

(2) for all ℎ gt 0 sufficiently small there exists 119891(1199090) minus

119867119891(119909

0+ ℎ) 119891(119909

0minus ℎ) minus

119867119891(119909

0) and the limits

limℎrarr0

119891 (1199090) minus

119867119891 (119909

0+ ℎ)

minusℎ= lim

ℎrarr0

119891 (1199090minus ℎ) minus

119867119891 (119909

0)

minusℎ

= 1198911015840

(1199090)

(10)

or

(3) for all ℎ gt 0 sufficiently small there exists 119891(1199090+ℎ) minus

119867119891(119909

0) 119891(119909

0minus ℎ) minus

119867119891(119909

0) and the limits

limℎrarr0

119891 (1199090+ ℎ) minus

119867119891 (119909

0)

ℎ= lim

ℎrarr0

119891 (1199090minus ℎ) minus

119867119891 (119909

0)

minusℎ

= 1198911015840

(1199090)

(11)

or

(4) for all ℎ gt 0 sufficiently small there exists 119891(1199090) minus

119867119891(119909

0+ ℎ) 119891(119909

0) minus

119867119891(119909

0minus ℎ) and the limits

limℎrarr0

119891 (1199090) minus

119867119891 (119909

0+ ℎ)

minusℎ= lim

ℎrarr0

119891 (1199090) minus

119867119891 (119909

0minus ℎ)

ℎ

= 1198911015840

(1199090)

(12)

(ℎ and minusℎ at denominators mean (1ℎ)sdot and minus(1ℎ)sdotresp)

3 The Convergence Theorem of Strong FuzzyHenstock Integral

In this section we define the strong Henstock integrals offuzzy-number-valued functions in fuzzy number space 119864119899and we give some properties and controlled convergencetheorem of this integral by using new conditions

Definition 3 (see [18]) A fuzzy-number-valued function 119891

is said to be termed additive on [119886 119887] if for any division119879 119886 le 119909

1le 119909

2le sdot sdot sdot le 119909

119899le 119887 one has 119891([119909

119894 119909

119895]) (1 le

119894 lt 119895 le 119899) that exists and 119891([119909119894 119909

119895]) = sum

119895minus1

119896=119894119891([119909

119896 119909

119896+1]) or

119891([119909119895 119909

119894])(1 le 119894 lt 119895 le 119899) that exists and (minus1) sdot 119891([119909

119895 119909

119894]) =

(minus1) sdot sum119895minus1

119896=119894119891([119909

119896+1 119909

119896]) For convenience 119891([119904 119905]) denotes

119891(119905) minus119867119891(119904)

Definition 4 (see [17 18]) A fuzzy-number-valued function119891 is said to be strong Henstock integrable on [119886 119887] if thereexists a piecewise additive fuzzy-number-valued function 119865

on [119886 119887] such that for every 120576 gt 0 there is a function 120575(120585) gt 0and for any 120575-fine division 119875 = ([119906 V] 120585) of [119886 119887] one has

sum119894isin119870119899

119863(119891 (120585119894) (V

119894minus 119906

119894) 119865 ([119906

119894 V

119894]))

+ sum119895isin119868119899

119863(119891 (120585119895) (V

119895minus 119906

119895) (minus1) sdot 119865 ([119906

119895 V

119895minus1])) lt 120576

(13)

where 119870119899= 119894 isin 1 2 119899 such that 119865([119909

119894minus1 119909

119894]) is a fuzzy

number and 119868119899= 119895 isin 1 2 119899 such that 119865([119909

119895 119909

119895minus1]) is a

fuzzy number One writes 119891 isin SFH[119886 119887]

Definition 5 A fuzzy-number-valued function 119865 defined on119883 sub [119886 119887] is said to be 119860119862lowast

120575(119883) if for every 120576 gt 0 there exists

120578 gt 0 and 120575(120585) gt 0 such that for any 120575-fine partial division119875 = ([119906 V] 120585) with 120585 isin 119883

119894satisfyingsum119899

119894=1|V minus 119906| lt 120578 one has

sum119863(119865[119906 V]) lt 120576

Definition 6 A fuzzy-number-valued function 119865 is said to be119860119862119866lowast

120575on119883 sub [119886 119887] if119883 is the union of a sequence of closed

sets 119883119894 such that on each119883

119894 119865 is 119860119862lowast

120575(119883

119894)

Definition 7 The sequence of fuzzy-number-function 119865119899 is

119880119860119862119866lowast

120575on119883 sub [119886 119887] if119883 is the sequence of subsets119883

119894such

that 119865119899 is 119880119860119862lowast

120575for each 119894 independent of 119899

Definition 8 Let 119865119899 be a sequence of fuzzy-number-

function defined on [119886 119887] and let 119909 sub [119886 119887] be measurable

(i) The sequence of fuzzy-number-function 119865119899 is P-

Cauchy on 119864119899 if 119865119899 converges pointwise on 119883 and

if for each 120576 gt 0 there exist 120575(120585) gt 0 on 119883 and apositive integer 119873 such that 119863(119865

119898(119875) 119865

119899(119875)) lt 120576 for

all119898 119899 ge 119873 whenever 119875 is119883-subordinate to 120575(120585)(ii) The sequence of fuzzy-number-function 119865

119899 is gen-

eralized P-Cauchy on 119883 if 119883 can be written as acountable union of measurable sets on each of which119865

119899 isP-Cauchy

Theorem 9 Let the following conditions be satisfied

(i) 119891119899119883(119909) rarr 119891

119909ae on [119886 119887] as 119899 rarr infin where each

119891119899119883

is strong Fuzzy Henstock integrable on [119886 119887]

(ii) the primitives 119865119899119883

of 119891119899119883

are 119880119860119862lowast

120575with closed set119883

in [119886 119887]

Then 119891119883(119909) is strong fuzzy Henstock integrable on [119886 119887] with

the primitive 119865119883(119909)

Proof By (ii) for every 120576 gt 0 there exist a 120575(120585) gt 0 and 120578 gt 0such that for any 120575-fine partial division119875of119883 satisfyingsum |Vminus119906| lt 120578 we have sum119863(119865

119899119883(V 119906) 0) lt 120576 By Egoroff rsquos theorem

[18 Theorem 34] there is an open set 119866 with |119866| lt 120578 suchthat 119863(119891

119899(120585) 119891

119898(120585)) lt 120576 for 119899119898 ge 119873 and 120585 notin 119866 Consider

the following in which 119875 is a 120575-fine division of [119909 119910] and

4 Journal of Applied Mathematics

119875 = 1198751cup119875

2so that119875

1contains the intervalswith the associated

points 120585 notin 119866 and 1198752otherwise

119863(119865119899119883

(119909 119910) 119865119898119883

(119909 119910))

= (1198752)sum119863(119865

119899119883(119906 V) 119865

119898119883(119906 V))

le sum119863(119865119899119883

(119906 V) 119891119899119883

(120585) (V minus 119906))

+sum119863(119865119898119883

(119906 V) 119891119898119883

(120585) (V minus 119906))

+sum119863(119891119898119883

(120585) (V minus 119906) 119891119898119883

(120585) (V minus 119906))

+sum119863(119865119899119883

(V) 119865119899119883

(119906)) +sum119863(119865119898119883

(V) 119865119898119883

(119906))

lt 120576 (4 + 119887 minus 119886)

(14)

Hence for any 120575-fine partial division 119875 of [119886 119887] we have

10038161003816100381610038161003816sum119863(119865

119899119883(119906 V) 119865

119898119883(119906 V))

10038161003816100381610038161003816lt 120576 (15)

for 119898 119899 ge 119873 Therefore the fuzzy sequence 119865119899119883 is gen-

eralized P-Cauchy on [119886 119887] Then by (i) we have that 119891119883

is strong fuzzy Henstock integrable on [119886 119887] with primitive119865119883

Definition 10 (a) A sequence 119865119899 of fuzzy-number-valued

function is uniformly119860119862nabla on119883whenever to each 120576 gt 0 thereexist 120578 gt 0 and 120575(119909) gt 0 such that

(1) sup119899119863(sum

119869119896isin1198751

119865119899(119869119896) sum

119871ℎisin1198752

119865119899(119871

ℎ)) lt 120576 for each 119875

1

1198752isin prod(119883 120575)

(2) with |(cup1198751)Δ(cap119875

2)| lt 120578

(b) A sequence 119865119899 of fuzzy-number-valued function is

uniformly 119860119862119866nabla on [119886 119887] if [119886 119887] = cup119894119883119894 where 119883

119894are

measurable sets and 119865 is uniformly 119860119862nabla on each119883119894

Theorem 11 If 119865 is uniformly 119860119862119866nabla then 119865 is uniformly119860119862119866lowast

120575

Proof Let [119886 119887] = cup119894119883119894be such that 119865 is uniformly 119860119862119866nabla

on each 119883119894 So for each 120576 gt 0 there exist 120578 gt 0 and 120575(119909) gt 0

such that (1) holds in Definition 10 for each 1198751 119875

2isin prod(119883 120575)

satisfying condition (2) We take 119875 = ([119888119896 119889

119896] 119909

119896)119901

119894=1with

sum119896|119889

119896minus 119888

119896| lt 120578 and put 119875

1= ([119888

119896 119889

119896] 119909

119896) 119865

119899(119862

119896 119889

119896ge

0) and 1198752= ([119888

119896 119889

119896] 119909

119896) 119865

119899(119888119896 119889

119896lt 0) So we have

|(cup1198751)Δ(cap119875

2)| = sum

119901

119896=1|119889

119896minus 119888

119896| lt 120578 Then by condition (1)

in Definition 10 we have

sup119899

119901

sum119896=1

119863(119865119899(119888119896) 119865

119899(119889

119896))

= sup119863( sum

(119888119896 119889119896)isin1198751

119865119899(119888119896 119889

119896)

sum(119888119896119889119896)isin1198752

119865119899(119888119896 119889

119896)) lt 120576

(16)

Hence we have that 119865 is uniformly 119860119862119866lowast

120575

We get the following theorem byTheorems 9 and 11

Theorem 12 Let the following conditions be satisfied

(i) 119891119899119883

rarr 119891119883ae in [119886 119887]where each119891

119899119883is strong fuzzy

Henstock integrable on [119886 119887](ii) the primitives 119865

119899119883of 119891

119899119883are119880119860119862nabla(119883)with closed set

119883 in [119886 119887]

Then 119891119883

is strong fuzzy Henstock integrable on [119886 119887] withprimitive 119865

119883

Next we give the controlled convergence theorem forthe strong fuzzy Henstock integrals by the definition of the119880119860119862119866

120575for a fuzzy-number-valued function

Definition 13 Let 119865 [119886 119887] rarr 119864119899 and let 119883 sub [119886 119887]A fuzzy-number-valued function 119865 is 119860119862

120575on 119883 if for each

120576 gt 0 there exist 120578 gt 0 and 120575(119909) gt 0 on 119883 such thatsum119873

119894=1119863(119865(119888

119894) 119865(119889

119894)) lt 120576 forsum119873

119894=1(119889

119894minus119888

119894) lt 120578 A fuzzy-number-

valued function 119865 is 119860119862119866120575on [119886 119887] if [119886 119887] is the union of a

sequence of set 119883119894 such that the function 119865 is 119860119862

120575(119883

119894) for

each 119894

Definition 14 (see [18]) A fuzzy-number-valued function 119865defined on 119883 sub [119886 119887] is said to be 119860119862lowast(119883) if for every 120576 gt 0

there exists 120578 gt 0 such that for every finite sequence of non-overlapping intervals [119886

119894 119887119894] satisfying sum119899

119894=1|119887119894minus 119886

119894| lt 120578

where 119886119894 119887119894isin 119883 for all 119894 one has sum120596(119865 [119886

119894 119887119894]) lt 120576 where 120596

denotes the oscillation of 119865 over [119886119894 119887119894] that is 120596(119865 [119886

119894 119887119894]) =

sup119863(119865(119910) 119865(119909)) 119909 119910 isin [119886119894 119887119894] A fuzzy-number-valued

function 119865 is said to be 119860119862119866lowast on 119883 if 119883 is the union ofa sequence of closed sets 119883

119894 such that on each 119883

119894 119865 is

119860119862lowast(119883119894)

Theorem 15 A fuzzy-number-valued function 119865 is 119860119862119866120575if

and only if it is 119860119862119866lowast on [119886 119887]

Theorem 16 (controlled convergence theorem) Let the fol-lowing conditions be satisfied

(1) 119891119899(119909) rarr 119891(119909) almost everywhere in [119886 119887] as 119899 rarr infin

where each 119891119899is strong fuzzy Henstock integrable on

[119886 119887]

Journal of Applied Mathematics 5

(2) the primitives 119865119899(119909) = (SFH) int119909

119886

119891119899(119904)d119909 of 119891

119899are

119880119860119862119866120575uniformly in 119899

(3) the sequence 119865119899(119909) converges uniformly to a con-

tinuous function on [119886 119887] Then 119891(119909) is strong fuzzyHenstock integrable on [119886 119887] and one has

lim119899rarrinfin

(SFH) int119887

119886

119891119899(119909) d119909 = (SFH) int

119887

119886

119891 (119909) d119909 (17)

If conditions (1) and (2) are replaced by condition (4)(4) 119892(119909) le 119891(119909) le ℎ(119909) almost everywhere on [119886 119887]

where 119892(119909) and ℎ(119909) are strong fuzzy Henstock inte-grable

Proof By condition (2) there exists a sequence 119883119894 such that

119865119899isin 119880119860119862

120575in 119883

119894a bounded closed set with bounds 119886 and

119887 and put 119883 = 119883119894 We note that 119865

119899(119909) rarr 119865(119909) we have

119865 isin 119860119862120575on119883 and hence119865 isin 119860119862119866

120575on [119886 119887] ByTheorem 15

119865 isin 119860119862lowast on 119883 and hence 119865 isin 119860119862119866lowast on [119886 119887] and also 119865 isin

119860119862 on119883 and hence 119865 isin 119860119862119866 on [119886 119887]Now we prove that 1198651015840(119909) = 119891(119909) ae on [119886 119887] In fact

let 119866 [119886 119887] rarr 119864119899 equal 119865119899on 119883 and extend 119866

119899linearly

to the closed interval contiguous to119883 Likewise we define 119866from 119865 We see that119866

119899and119866 are119880119860119862 on [119886 119887] By condition

(3) we have 119866119899rarr 119866 on [119886 119887] Let [119888

119896 119889

119896] be the intervals

contiguous to 119883 Then we have 119863(1198661015840

119899) le 119872

119896 We define a

fuzzy-number-valued function as follows

1198661015840

(119909) =119866119899(119889

119896) minus

119867119866119899(119888119896)

119889119896minus 119888

119896

119909 isin (119888119896 119889

119896) (18)

Consequently 1198661015840

119899(119909) converges on (119888

119896 119889

119896) Hence 1198661015840

119899con-

verges on [119886 119887] ae Since 119866119899 isin 119860119862 on [119886 119887] then 1198661015840

119899(119909) =

119891119899

rarr 119891 on 119883 Therefore we have 1198661015840

119899(119909) = 119892(119909) =

119891(119909) = 1198651015840(119909) ae on 119883 Thus 1198651015840(119909) = 119891(119909) ae on [119886 119887]

by Theorem 15 Therefore there exists an 119860119862119866120575function on

[a 119887] such that 1198651015840(119909) = 119891(119909) ae on [119886 119887] Hence 119891 is strongfuzzy Henstock integrable on [119886 119887] and we have

lim119899rarrinfin

(SFH) int119887

119886

119891119899(119909) d119909 = (SFH) int

119887

119886

119891 (119909) d119909 (19)

4 The Generalized Solutions of DiscontinuousFuzzy Differential Equations

In this section a generalized fuzzy differential equation ofform (4) is defined by using strong fuzzy Henstock integralThemain results of this section are existence theorems for thegeneralized solution to the discontinuous fuzzy differentialequation

Definition 17 (see [11]) Let 120591 and 120585 be fixed and let a fuzzy-number-valued function 119891(119905 119909) be a Caratheodory functiondefined on a rectangle119880 |119905minus120591| le 119886 119863(119909 120585) le 119887 that is119891 iscontinuous in 119909 for almost all 119905 and measurable in 119905 for eachfixed 119909

Theorem 18 Let a fuzzy-number-valued function 119891 be afunction as given in Definition 17 then there exist two strongfuzzy Henstock integrable functions ℎ and 119892 defined on |119905minus120591| le119886 such that 119892(119905) le 119891(119905 119909) le ℎ(119905) for all (119905 119909) isin 119880

Proof Note that 119891 is a Caratheodory function Thus thereexist two measurable functions 119906(119905) and V(119905) defined on |119905 minus120591| le 119886 with values in 119863(119909 120585) le 119887 such that 119891(119905 119906(119905)) le

119891(119905 119909) le 119891(119905 V(119905)) for all (119905 119909) isin 119880 Next we will showthat 119891(119905 119906(119905)) and 119891(119905 V(119905)) are fuzzy Henstock integrable byusing controlled convergence Theorem 16 First there existsa sequence 119896

119899(119905) of step functions defined on |119905 minus 120591| le

119886 with values in 119863(119909 120585) le 119887 such that 119896119899(119905) rarr 119906(119905)

almost everywhere as 119899 rarr infin Let 119865119899(119905) = int

119905

120591

119891(119904 119896119899(119904))d119904

Then 119865119899(119905) is 119880119860119862119866

120575uniformly in 119899 and equicontinuous

By controlled convergence Theorem 16 119891(119905 119906(119905)) is strongfuzzy Henstock integrable Similarly 119891(119905 V(119905)) is strong fuzzyHenstock integrable

Definition 19 A fuzzy-number-valued function 119909(119905) 119868 rarr

119864119899 is said to be a solution of the discontinuous fuzzy differ-ential equation (4) if 119909(119905) satisfies the following conditions

(i) 119909(119905) is 119860119862119866120575on each compact subinterval of 119868

(ii) (119905 119909) isin 119880 for 119905 isin 119868(iii) 1199091015840(119905) for almost everywhere 119905 isin 119868

Now we will state the existence theorem for the general-ized solution of discontinuous fuzzy differential equation (4)

Theorem 20 Suppose that 119891 satisfies the condition ofTheorem 18 then there exists a generalized solution Φ of thediscontinuous fuzzy differential equation (4) on some interval|119905 minus 120591| le 119886 which satisfies Φ(120591) = 120585

Proof Given 119892(t) le 119891(119905 119909) le ℎ(119905) for all 119909 and almost all 119905with (119905 119909) isin 119880 we get 0 le 119891(119905 119909)minus

119867119892(119905) le ℎ(119905)minus

119867119892(119905) Let

119865 (119905 119909) = 119891(119905 119909 + int119905

120591

119892 (119904) d119904) minus119867119892 (119905) (20)

or

119865 (119905 119909) = 119891(119905 119909 + (minus1) sdot int119905

120591

119892 (119904) d119904) minus119867119892 (119905) (21)

Then 119865 is a Caratheodory function Furthermore 0 le

119865(119905 119909) le ℎ(119905) minus119867119892(119905) for all (119905 119909) isin 119880

0 where 119880

0sub 119880

such that

119863(119909 + int119905

120591

119892 (119904) d119904 120585) le 119887 forall (119905 119909) isin 1198800 (22)

By Caratheodory existence theorem (see Theorem 7 in [11])there is a fuzzy-number-valued function 120595 on some interval|119905 minus 120591| le 119886 such that 1205951015840(119905) = 119865(119905 120595(119905)) almost everywhere inthis interval and 120595(120591) = 120585 Let

120601 (119905)=120595 (119905)+int119905

120591

119892 (119904) d119904 or 120601 (119905)=120595 (119905)+(minus1) sdot int119905

120591

119892 (119904) d119904

(23)

6 Journal of Applied Mathematics

Then for almost all 119905 we have the following

Case 1 Consider

1206011015840

(119905) = 1205951015840

(119905) + 119892 (119905) = 119865 (119905 120595 (119905)) + 119892 (119905)

= 119891(119905 120595 (119905) + int119905

120591

119892 (119904) d119904) minus119867119892 (119905) + 119892 (119905)

= 119891 (119905 120601 (119905))

120601 (119905) = 120595 (120591) + int119905

120591

119892 (119904) d119904 = 120585

(24)

Case 2 Consider

1206011015840

(119905) = 1205951015840

(119905) + 119892 (119905) = 119865 (119905 120595 (119905)) + 119892 (119905)

= 119891(119905 120595 (119905) + (minus1) sdot int119905

120591

119892 (119904) d119904) minus119867119892 (119905) + 119892 (119905)

= 119891 (119905 120601 (119905))

120601 (119905) = 120595 (120591) + (minus1) sdot int119905

120591

119892 (119904) d119904 = 120585

(25)

The proof is complete

Example 21 Consider fuzzy differential equation 1199091015840 =

119891(119905 119909) = 119892(119905 119909) + ℎ(119905) where 119863(119892(119905 119909) 0) le 119863(1198921(119905) 0) for

all |119905| le 1 119863(119909 0) le 1 and 1198921(119905) is Kaleva integrable on |119905| le 1

and ℎ(119905) = 119860 sdot (119889119889119905)(1199052 sin 119905minus2) if 119905 = 0 and ℎ(0) = 0 Here 119860is defined in Example 1 Note that ℎ is strong fuzzy Henstockintegrable but not Kaleva integrable and

ℎ (119905) minus1198671198921(119905) le 119891 (119905 119909) le ℎ (119905) + 119892

1(119905)

for |119905| le 1 119863 (119909 0) le 1

(26)

Thus by Theorem 20 there exists a solution of 1199091015840 = 119891(119905 119909)

with 119909(0) = 0 For instance if 119892(119905 119909) = 1199052119909 then

120601 (119905) = 1198901199053

3

sdot int119905

0

119890minus1199043

3

ℎ (119904) d119904 (27)

is a solution by using integrating factor

We get the following existence theorem by Theorems 18and 20

Theorem 22 Let a fuzzy-number-valued function 119891 be aCaratheodory function defined on a rectangle 119880 |119905 minus

120591| le 119886119863(119909 120585) le 119887 Let 119891(119905 119906(119905)) be strong fuzzy Henstockintegrable on |119905 minus 120591| le 119886 for any step function 119906(119905) definedon |119905 minus 120591| le 119886 with values in 119863(119909 120585) le 119887 Denote that119865119906(119905) = int

119905

120591

119891(119904 119906(119904))d119904 If 119865119906 119906 is a step function is 119880119860119862119866

120575

uniformly in 119906 and equicontinuous on |119905 minus 120591| le 119886 then thereexists a solution 120601 of 1199091015840 = 119891(119905 119909) on some interval |119905 minus 120591| le 120573with 120601(120591) = 120585

Finally in this paper we will show the continuousdependence of a solution on parameters by using Theorems16 18 and 22

Let 119880119901be a connected set in 119880 Let 119888 gt 0 and let 120583

0be

fixed119868120583= 120583

1003816100381610038161003816120583 minus 12058301003816100381610038161003816 lt 119888

119880120583= (119905 119909 120583) (119905 119909) isin 119880

119901 120583 isin 119868

120583

(28)

Let 119891(119905 119909 120583) be a fuzzy-number-valued function definedon 119880

120583such that for each fixed 120583 the function 119891 is a

Caratheodory function defined on 119880119901for each fixed 119905 and

continuous at (119909 1205830) for every 119909 For 120583 = 120583

0 let

1199091015840

(119905) = 119891 (119905 119909 120583)

119909 (120591) = 120585 isin 119864119899

(29)

have a solution 1206010on [119886 119887] where 120591 isin [119886 119887] Let 119891(119905 119906(119905) 120583)

be strong fuzzy Henstock integrable on [119886 119887] for any stepfunction 119906(119905) with (119905 119906(119905)) isin 119880

119901for 119905 isin [119886 119887]

Definition 23 Denote

119865120583119906(119905) = int

119905

119886

119891 (119904 119906 (119904) 120583) d119904 (30)

The family 119865120583119906 is said to be equicontinuous in 119906 and near

1205830if for each 119905

0isin [119886 119887] there exists an interval |120583 minus 120583

0| lt 119903

0

such that the family 119865120583119906(119905) 119906 and 120583 with |120583 minus 120583

0| lt 119903

0 is

equicontinuous at 1199050

Theorem 24 Let 119891 be a fuzzy-number-valued function asgiven above If the primitive 119865

120583119906(119905) of 119891(119905 119909 120583) is 119860119862119866

120575

uniformly in119906 and120583 and equicontinuous in119906 and near1205830 then

there exists 120575 gt 0 such that for any fixed 120583 with |120583 minus 1205830| lt 120575

a solution 120601120583of discontinuous fuzzy differential equation (29)

exists over [119886 119887] and as 120583 rarr 1205830 120601

120583rarr 120601

0uniformly over

[119886 119887]

Proof Firstly we will consider the case 120591 isin (119886 119887) ByTheorem 22 and the equicontinuity of 119865

120583119906(119905) all solutions

120601120583of problem (29) with |120583 minus 120583

0| le 120575

1for some 120575

1gt 0

exist over some interval |119905 minus 120591| le 120572 with 120572 gt 0 Then119866 = 120601

120583 |120583 minus 120583

0| le 120575

1 is equicontinuous on |119905 minus 120591| le 120572

Because 120601120583(120591) = 120585 119866 is uniformly bounded Hence for all

sequence of119866with120583 rarr 1205830 there exists a subsequencewhich

converges uniformly That is to say 120601120583(119896)

rarr 120595 uniformly on|119905 minus 120591| le 120572 as 120583(119896) rarr 120583

0 Since we have

120601120583(119896)

(119905) = 120585 + int119905

120591

119891 (119904 120601120583(119896)

120583 (119896)) d119904 (31)

or

120601120583(119896)

(119905) = 120585 + (minus1) sdot int119905

120591

119891 (119904 120601120583(119896)

120583 (119896)) d119904 (32)

byTheorem 16 we get

120595 (119905) = 120585 + int119905

120591

119891 (119904 120595 (119904) 1205830) d119904 (33)

Journal of Applied Mathematics 7

or

120595 (119905) = 120585 + (minus1) sdot int119905

120591

119891 (119904 120595 (119904) 1205830) d119904 (34)

Thus 120595 is a solution of (29) for 120583 = 1205830 Hence 120595 = 120601

0on

|119905 minus 120591| le 120572 because the solution 1206010is unique Consequently

120601119906(119896)

rarr 1206010uniformly on |119905 minus 120591| le 120572 as 120583(119896) rarr 120583

0 By

Reductio ad absurdum 120601120583rarr 120601

0uniformly there as 120583 rarr

1205830Secondly we will extend the result over [119886 119887] We con-

sider the case [120591 119887] Assume that there exists 1199050isin [120591 119887) such

that the result is valid over [120591 1199050minus ℎ] but not [120591 119905

0+ ℎ]

Obviously 1199050ge 120591 + 120572 Let 120574 120573 be such that

119867 = (119905 119909) 1003816100381610038161003816119905 minus 1199050

1003816100381610038161003816 le 120572119863 (119909 1206010(119905 minus 120574) le 120573) (35)

is contained in 119880 Since 119865120583119906(119905) is equicontinuous in 119906 and

near 1205830 we may choose 120574 gt 0 small enough such that

119863(int119905

1199050

119891 (119904 119906 (119904) 120583) d119904 0) lt120573

3(36)

for all step functions 119906whenever |119905minus 1199050| le 120574 and |120583minus120583

0| lt 120575

2

Since 120601120583rarr 120601

0uniformly on [120591 119905

0minus ℎ] there exists a 120575 such

that

119863(120601120583(1199050minus 120574) 120601

0(1199050minus 120574)) lt

120573

3(37)

for |120583 minus 1205830| lt 120575 Hence we have

119863(int119905

1199050minus120574

119891 (119904 119906 (119904) 120583) d119904 120601119898119906(1199050minus 120574) minus

1198671206010(1199050minus 120574))

le 119863(int1199050

1199050minus120574

119891 (119904 119906 (119904) 120583) d119904 0)

+ 119863(int119905

1199050

119891 (119904 119906 (119904) 120583) d119904 0)

+ 119863 (120601119898119906(1199050minus 120574) 120601

0(1199050minus 120574)) lt 3 sdot

120573

3

(38)

Thus for each 119906 with |120583 minus 1205830| lt 120575 a solution of (29) with

119909(1199050minus 120574) = 120601

120583(1199050minus 120574) exists on |119905 minus 119905

0| le 120574 Hence 120601

120583can be

continued to 1199050+ 120574 So in the case [120591 minus 120572 120591 + 120572] 120601

120583rarr 120601

0

uniformly on [120591 1199050+ 120574] It leads to a contradiction similarly

for the cases 1199050= 119887 and [119886 120591] Therefore the theorem holds

over [119886 119887]

Example 25 Let

(119905) = 119860 sdot 119905

2 sin 119905minus2 119905 = 0

0 119905 = 0(39)

where fuzzy number 119860 is defined in Example 1 Let ℎ(119905) =1015840(119905) We define

ℎ120583(119905) =

ℎ (119905) 120583 isin [0 1) 119905 isin(minus2 2)

(minus120583 120583)

0 otherwise(40)

and ℎ120583(119905) = ℎ

minus120583(119905) for 120583 isin (minus1 0] Let 119891(119905 119909 120583) = 1199052119909 + ℎ

120583(119905)

defined on (minus2 2) times (minus1 1) Then we have

119863(119865120583119906(1199051) 119865

120583119906(1199052)) = 119863(int

1199052

1199051

119891 (119904 119906 (119904) 120583) d119904 0)

le int1199052

1199051

1199042d119904 + 119863(int

1199052

1199051

ℎ120583d119904 0)

119863(int1199052

1199051

ℎ120583d119904 0) le int

1199052

1199051

119863(ℎ (119904) d119904 0)

(41)

where 1199051 1199052isin (0 1] or 119905

1 1199052isin [minus1 0) and

119863(int119905

0

ℎ120583(119904) d119904 0)

=

119863(119860 sdot 1199052 sin 119905minus2 119860 sdot 1205832 sin 120583minus2) 120583 = 0

119863 (119860 sdot 1199052 sin 119905minus2 0) 120583 = 0

(42)

Note that ℎ is Kaleva integrable on every subinterval of [minus1 0)and (0 1] Therefore we have that 119865

120583119906(119905) is equicontinuous

on [minus1 1] in 119906 and near 1205830= 0 Furthermore 119865

120583119906(119905) is

119860119862120575(119883

119899) uniformly in 120583 and 119906 where 119883

119899= [1119899 1] 119899 =

1 2 On the other hand by using integrating factor for119905 isin [minus1 1] we have

120601120583(119905) = 119890

1199052

3

int119905

0

119890minus1199043

3

119860 sdot ℎ120583(119904) d119904 (43)

with 120601120583(0) = 0 Obviously 120601

0is uniqueThus byTheorem 24

120601120583rarr 120601

0uniformly on [minus1 1]

5 Conclusion

In this paper we give the definition of the119880119860119862119866120575for a fuzzy-

number-valued function and the nonabsolute fuzzy integraland its controlled convergence theorem In addition we dealwith the Cauchy problem and the continuous dependence ofa solution on parameters of discontinuous fuzzy differentialequations involving the strong fuzzy Henstock integral infuzzy number space The function governing the equationsis supposed to be discontinuous with respect to some vari-ables and satisfy nonabsolute fuzzy integrability Our resultimproves the result given in [1 11 19 20] (where uniformcontinuity was required) as well as those referred therein

Acknowledgments

The authors are very grateful to the anonymous refereesand Professor Mehmet Sezer for many valuable commentsand suggestions which helped to improve the presentationof the paper The authors would like to thank the NationalNatural Science Foundation of China (no 11161041) and theFundamental Research Fund for the Central Universities (no31920130010)

8 Journal of Applied Mathematics

References

[1] O Kaleva ldquoFuzzy differential equationsrdquo Fuzzy Sets and Sys-tems vol 24 no 3 pp 301ndash317 1987

[2] Z Gong and Y Shao ldquoGlobal existence and uniqueness ofsolutions for fuzzy differential equations under dissipative-typeconditionsrdquo Computers amp Mathematics with Applications vol56 no 10 pp 2716ndash2723 2008

[3] O Kaleva ldquoThe Cauchy problem for fuzzy differential equa-tionsrdquo Fuzzy Sets and Systems vol 35 no 3 pp 389ndash396 1990

[4] J J Nieto ldquoThe Cauchy problem for continuous fuzzy differen-tial equationsrdquo Fuzzy Sets and Systems vol 102 no 2 pp 259ndash262 1999

[5] M L Puri and D A Ralescu ldquoDifferentials of fuzzy functionsrdquoJournal of Mathematical Analysis and Applications vol 91 no 2pp 552ndash558 1983

[6] S Seikkala ldquoOn the fuzzy initial value problemrdquo Fuzzy Sets andSystems vol 24 no 3 pp 319ndash330 1987

[7] C Wu and S Song ldquoExistence theorem to the Cauchy problemof fuzzy differential equations under compactness-type condi-tionsrdquo Information Sciences vol 108 no 1ndash4 pp 123ndash134 1998

[8] C Wu S Song and Z Qi ldquoExistence and uniqueness for asolution on the closed subset to the Cauchy problem of fuzzydiffrential equationsrdquo Journal of Harbin Institute of Technologyvol 2 pp 1ndash7 1997

[9] CWu S Song and E S Lee ldquoApproximate solutions existenceand uniqueness of the Cauchy problem of fuzzy differentialequationsrdquo Journal of Mathematical Analysis and Applicationsvol 202 no 2 pp 629ndash644 1996

[10] V Lakshmikantham and R N Mohapatra Theory of FuzzyDifferential Equations and Inclusions vol 6 Taylor amp FrancisLondon UK 2003

[11] X Xue and Y Fu ldquoCaratheodory solutions of fuzzy differentialequationsrdquo Fuzzy Sets and Systems vol 125 no 2 pp 239ndash2432002

[12] R HenstockTheory of Integration Butterworths London UK1963

[13] J Kurzweil ldquoGeneralized ordinary differential equations andcontinuous dependence on a parameterrdquo Czechoslovak Mathe-matical Journal vol 7 no 82 pp 418ndash449 1957

[14] P Y LeeLanzhou Lectures onHenstock Integration vol 2WorldScientific Publishing Teaneck NJ USA 1989

[15] C Wu and Z Gong ldquoOn Henstock integrals of interval-valuedfunctions and fuzzy-valued functionsrdquo Fuzzy Sets and Systemsvol 115 no 3 pp 377ndash391 2000

[16] C Wu and Z Gong ldquoOn Henstock integral of fuzzy-number-valued functions Irdquo Fuzzy Sets and Systems vol 120 no 3 pp523ndash532 2001

[17] Z Gong ldquoOn the problem of characterizing derivatives for thefuzzy-valued functions II Almost everywhere differentiabilityand strong Henstock integralrdquo Fuzzy Sets and Systems vol 145no 3 pp 381ndash393 2004

[18] Z Gong and Y Shao ldquoThe controlled convergence theoremsfor the strong Henstock integrals of fuzzy-number-valuedfunctionsrdquo Fuzzy Sets and Systems vol 160 no 11 pp 1528ndash1546 2009

[19] T S Chew and F Flordeliza ldquoOn 1199091015840 = 119891(119905 119909) and Henstock-Kurzweil integralsrdquo Differential and Integral Equations vol 4no 4 pp 861ndash868 1991

[20] B Bede and S G Gal ldquoGeneralizations of the differentiabilityof fuzzy-number-valued functions with applications to fuzzy

differential equationsrdquo Fuzzy Sets and Systems vol 151 no 3pp 581ndash599 2005

[21] D Dubois and H Prade ldquoTowards fuzzy differential calculus IIntegration of fuzzy mappingsrdquo Fuzzy Sets and Systems vol 8no 1 pp 1ndash17 1982

[22] P Diamond and P KloedenMetric Spaces of Fuzzy Sets WorldScientific Publishing River Edge NJ USA 1994

[23] R Goetschel Jr and W Voxman ldquoElementary fuzzy calculusrdquoFuzzy Sets and Systems vol 18 no 1 pp 31ndash43 1986

[24] Y Chalco-Cano and H Roman-Flores ldquoOn new solutions offuzzy differential equationsrdquo Chaos Solitons and Fractals vol38 no 1 pp 112ndash119 2008

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Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

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Decision SciencesAdvances in

Discrete MathematicsJournal of

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Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

2 Journal of Applied Mathematics

[13] the definition is now simple and furthermore the proofinvolving the integral also turns out to be easy For moredetailed results about the Henstock integral we refer to[14] Recently Wu and Gong [15 16] have combined thefuzzy set theory and nonabsolute integration theory andthey discussed the fuzzyHenstock integrals of fuzzy-number-valued functions which extended Kaleva [1] integration Inorder to complete the theory of fuzzy calculus and tomeet thesolving need of transferring a fuzzy differential equation intoa fuzzy integral equation Gong and Shao [17 18] defined thestrong fuzzy Henstock integrals and discussed some of theirproperties and the controlled convergence theorem

In this paper according to the idea of [19] and usingthe concept of generalized differentiability [20] we willprove other controlled convergence theorems for the strongfuzzy Henstock integrals which will be of foundationalsignificance for studying the existence and uniqueness ofsolutions to the fuzzy discontinuous systems As we knowwe inevitably use the controlled convergence theorems forsolving the numerical solutions of differential equations Asthe main outcomes we will deal with the Cauchy problem ofdiscontinuous fuzzy systems as follows

1199091015840

(119905) = 119891 (119905 119909)

119909 (120591) = 120585 isin 119864119899

(4)

where 119891 119880 rarr 119864119899 is a strong fuzzy Henstock integrablefunction and

119880 = (119905 119909) |119905 minus 120591| le 119886 119909 isin 119864119899

119863 (119909 120585) le 119887 (5)

To make our analysis possible we will first recall somebasic results of fuzzy numbers and give some definitionsof absolutely continuous fuzzy-number-valued function Inaddition we present the concept of generalized differentia-bility In Section 3 we present the concept of strong fuzzyHenstock integrals and we prove a controlled convergencetheorem for the strong fuzzyHenstock integrals In Section 4we deal with the Cauchy problem of discontinuous fuzzysystems And in Section 5 we present some concludingremarks

2 Preliminaries

Let119875119896(119877119899) denote the family of all nonempty compact convex

subset of119877119899 and define the addition and scalarmultiplicationin 119875

119896(119877119899) as usual Let 119860 and 119861 be two nonempty bounded

subsets of 119877119899 The distance between119860 and 119861 is defined by theHausdorff metric [21] as follows

119889119867(119860 119861) = maxsup

119886isin119860

inf119887isin119861

119886 minus 119887 sup119887isin119861

inf119886isin119860

119887 minus 119886 (6)

Denote that 119864119899 = 119906 119877119899 rarr [0 1] | 119906 satisfies (1)ndash(4)below is a fuzzy number space where

(1) 119906 is normal that is there exists an 1199090isin 119877119899 such that

119906(1199090) = 1

(2) 119906 is fuzzy convex that is 119906(120582119909 + (1 minus 120582)119910) ge

min119906(119909) 119906(119910) for any 119909 119910 isin 119877119899 and 0 le 120582 le 1(3) 119906 is upper semicontinuous(4) [119906]0 = cl119909 isin 119877119899 | 119906(119909) gt 0 is compact

For 0 lt 120572 le 1 denote that [119906]120572 = 119909 isin 119877119899 | 119906(119909) ge 120572Then from the above (1)ndash(4) it follows that the 120572-level set[119906]

120572

isin 119875119896(119877119899) for all 0 le 120572 lt 1

According to Zadehrsquos extension principle we have addi-tion and scalar multiplication in fuzzy number space 119864119899 asfollows [21]

[119906 + V]120572 = [119906]120572 + [V]120572 [119896119906]120572

= 119896[119906]120572

(7)

where 119906 V isin 119864119899 and 0 le 120572 le 1Define119863 119864119899 times 119864119899 rarr [0infin)

119863 (119906 V) = sup 119889119867([119906]

120572

[V]120572) 120572 isin [0 1] (8)

where 119889 is the Hausdorff metric defined in 119875119896(119877119899) Then it is

easy to see that119863 is a metric in 119864119899 Using the results [22] weknow that

(1) (119864119899 119863) is a complete metric space(2) 119863(119906 + 119908 V + 119908) = 119863(119906 V) for all 119906 V 119908 isin 119864119899(3) 119863(120582119906 120582V) = |120582|119863(119906 V) for all 119906 V 119908 isin 119864119899 and 120582 isin 119877

Let 119909 119910 isin 119864119899 If there exist 119911 isin 119864119899 such that 119909 = 119910 + 119911then 119911 is called the 119867-difference of 119909 and 119910 and is denotedby 119909 minus

119867

119910 As mentioned above which always is called thecondition (119867) It is well known that the 119867-derivative forfuzzy-number-functions was initially introduced by Puri etal [5 23] and it is based on the condition (119867) of sets Wenote that this definition is fairly strong because the familyof fuzzy-number-valued functions 119867-differentiable is veryrestrictive For example the fuzzy-number-valued function119891 [119886 119887] rarr 119877F defined by 119891(119909) = 119862 sdot 119892(119909) where 119862 isa fuzzy number sdot is the scalar multiplication (in the fuzzycontext) and 119892 [119886 119887] rarr 119877

+ with 1198921015840(1199050) lt 0 is not 119867-

differentiable in 1199050(see [20 24]) To avoid the above difficulty

in this paper we consider a more general definition of aderivative for fuzzy-number-valued functions enlarging theclass of differentiable fuzzy-number-valued functions whichhas been introduced in [20]

Definition 2 (see [20]) Let 119891 (119886 119887) rarr 119864119899 and 119909

0isin

(119886 119887) One says that 119891 is differentiable at 1199090 if there exists

an element 1198911015840(1199050) isin 119864119899 such that

(1) for all ℎ gt 0 sufficiently small there exists 119891(1199090+

ℎ) minus119867119891(119909

0) 119891(119909

0) minus

119867119891(119909

0minus ℎ) and the limits (in

the metric119863)

limℎrarr0

119891 (1199090+ ℎ) minus

119867119891 (119909

0)

ℎ= lim

ℎrarr0

119891 (1199090) minus

119867119891 (119909

0minus ℎ)

ℎ

= 1198911015840

(1199090)

(9)

or

Journal of Applied Mathematics 3

(2) for all ℎ gt 0 sufficiently small there exists 119891(1199090) minus

119867119891(119909

0+ ℎ) 119891(119909

0minus ℎ) minus

119867119891(119909

0) and the limits

limℎrarr0

119891 (1199090) minus

119867119891 (119909

0+ ℎ)

minusℎ= lim

ℎrarr0

119891 (1199090minus ℎ) minus

119867119891 (119909

0)

minusℎ

= 1198911015840

(1199090)

(10)

or

(3) for all ℎ gt 0 sufficiently small there exists 119891(1199090+ℎ) minus

119867119891(119909

0) 119891(119909

0minus ℎ) minus

119867119891(119909

0) and the limits

limℎrarr0

119891 (1199090+ ℎ) minus

119867119891 (119909

0)

ℎ= lim

ℎrarr0

119891 (1199090minus ℎ) minus

119867119891 (119909

0)

minusℎ

= 1198911015840

(1199090)

(11)

or

(4) for all ℎ gt 0 sufficiently small there exists 119891(1199090) minus

119867119891(119909

0+ ℎ) 119891(119909

0) minus

119867119891(119909

0minus ℎ) and the limits

limℎrarr0

119891 (1199090) minus

119867119891 (119909

0+ ℎ)

minusℎ= lim

ℎrarr0

119891 (1199090) minus

119867119891 (119909

0minus ℎ)

ℎ

= 1198911015840

(1199090)

(12)

(ℎ and minusℎ at denominators mean (1ℎ)sdot and minus(1ℎ)sdotresp)

3 The Convergence Theorem of Strong FuzzyHenstock Integral

In this section we define the strong Henstock integrals offuzzy-number-valued functions in fuzzy number space 119864119899and we give some properties and controlled convergencetheorem of this integral by using new conditions

Definition 3 (see [18]) A fuzzy-number-valued function 119891

is said to be termed additive on [119886 119887] if for any division119879 119886 le 119909

1le 119909

2le sdot sdot sdot le 119909

119899le 119887 one has 119891([119909

119894 119909

119895]) (1 le

119894 lt 119895 le 119899) that exists and 119891([119909119894 119909

119895]) = sum

119895minus1

119896=119894119891([119909

119896 119909

119896+1]) or

119891([119909119895 119909

119894])(1 le 119894 lt 119895 le 119899) that exists and (minus1) sdot 119891([119909

119895 119909

119894]) =

(minus1) sdot sum119895minus1

119896=119894119891([119909

119896+1 119909

119896]) For convenience 119891([119904 119905]) denotes

119891(119905) minus119867119891(119904)

Definition 4 (see [17 18]) A fuzzy-number-valued function119891 is said to be strong Henstock integrable on [119886 119887] if thereexists a piecewise additive fuzzy-number-valued function 119865

on [119886 119887] such that for every 120576 gt 0 there is a function 120575(120585) gt 0and for any 120575-fine division 119875 = ([119906 V] 120585) of [119886 119887] one has

sum119894isin119870119899

119863(119891 (120585119894) (V

119894minus 119906

119894) 119865 ([119906

119894 V

119894]))

+ sum119895isin119868119899

119863(119891 (120585119895) (V

119895minus 119906

119895) (minus1) sdot 119865 ([119906

119895 V

119895minus1])) lt 120576

(13)

where 119870119899= 119894 isin 1 2 119899 such that 119865([119909

119894minus1 119909

119894]) is a fuzzy

number and 119868119899= 119895 isin 1 2 119899 such that 119865([119909

119895 119909

119895minus1]) is a

fuzzy number One writes 119891 isin SFH[119886 119887]

Definition 5 A fuzzy-number-valued function 119865 defined on119883 sub [119886 119887] is said to be 119860119862lowast

120575(119883) if for every 120576 gt 0 there exists

120578 gt 0 and 120575(120585) gt 0 such that for any 120575-fine partial division119875 = ([119906 V] 120585) with 120585 isin 119883

119894satisfyingsum119899

119894=1|V minus 119906| lt 120578 one has

sum119863(119865[119906 V]) lt 120576

Definition 6 A fuzzy-number-valued function 119865 is said to be119860119862119866lowast

120575on119883 sub [119886 119887] if119883 is the union of a sequence of closed

sets 119883119894 such that on each119883

119894 119865 is 119860119862lowast

120575(119883

119894)

Definition 7 The sequence of fuzzy-number-function 119865119899 is

119880119860119862119866lowast

120575on119883 sub [119886 119887] if119883 is the sequence of subsets119883

119894such

that 119865119899 is 119880119860119862lowast

120575for each 119894 independent of 119899

Definition 8 Let 119865119899 be a sequence of fuzzy-number-

function defined on [119886 119887] and let 119909 sub [119886 119887] be measurable

(i) The sequence of fuzzy-number-function 119865119899 is P-

Cauchy on 119864119899 if 119865119899 converges pointwise on 119883 and

if for each 120576 gt 0 there exist 120575(120585) gt 0 on 119883 and apositive integer 119873 such that 119863(119865

119898(119875) 119865

119899(119875)) lt 120576 for

all119898 119899 ge 119873 whenever 119875 is119883-subordinate to 120575(120585)(ii) The sequence of fuzzy-number-function 119865

119899 is gen-

eralized P-Cauchy on 119883 if 119883 can be written as acountable union of measurable sets on each of which119865

119899 isP-Cauchy

Theorem 9 Let the following conditions be satisfied

(i) 119891119899119883(119909) rarr 119891

119909ae on [119886 119887] as 119899 rarr infin where each

119891119899119883

is strong Fuzzy Henstock integrable on [119886 119887]

(ii) the primitives 119865119899119883

of 119891119899119883

are 119880119860119862lowast

120575with closed set119883

in [119886 119887]

Then 119891119883(119909) is strong fuzzy Henstock integrable on [119886 119887] with

the primitive 119865119883(119909)

Proof By (ii) for every 120576 gt 0 there exist a 120575(120585) gt 0 and 120578 gt 0such that for any 120575-fine partial division119875of119883 satisfyingsum |Vminus119906| lt 120578 we have sum119863(119865

119899119883(V 119906) 0) lt 120576 By Egoroff rsquos theorem

[18 Theorem 34] there is an open set 119866 with |119866| lt 120578 suchthat 119863(119891

119899(120585) 119891

119898(120585)) lt 120576 for 119899119898 ge 119873 and 120585 notin 119866 Consider

the following in which 119875 is a 120575-fine division of [119909 119910] and

4 Journal of Applied Mathematics

119875 = 1198751cup119875

2so that119875

1contains the intervalswith the associated

points 120585 notin 119866 and 1198752otherwise

119863(119865119899119883

(119909 119910) 119865119898119883

(119909 119910))

= (1198752)sum119863(119865

119899119883(119906 V) 119865

119898119883(119906 V))

le sum119863(119865119899119883

(119906 V) 119891119899119883

(120585) (V minus 119906))

+sum119863(119865119898119883

(119906 V) 119891119898119883

(120585) (V minus 119906))

+sum119863(119891119898119883

(120585) (V minus 119906) 119891119898119883

(120585) (V minus 119906))

+sum119863(119865119899119883

(V) 119865119899119883

(119906)) +sum119863(119865119898119883

(V) 119865119898119883

(119906))

lt 120576 (4 + 119887 minus 119886)

(14)

Hence for any 120575-fine partial division 119875 of [119886 119887] we have

10038161003816100381610038161003816sum119863(119865

119899119883(119906 V) 119865

119898119883(119906 V))

10038161003816100381610038161003816lt 120576 (15)

for 119898 119899 ge 119873 Therefore the fuzzy sequence 119865119899119883 is gen-

eralized P-Cauchy on [119886 119887] Then by (i) we have that 119891119883

is strong fuzzy Henstock integrable on [119886 119887] with primitive119865119883

Definition 10 (a) A sequence 119865119899 of fuzzy-number-valued

function is uniformly119860119862nabla on119883whenever to each 120576 gt 0 thereexist 120578 gt 0 and 120575(119909) gt 0 such that

(1) sup119899119863(sum

119869119896isin1198751

119865119899(119869119896) sum

119871ℎisin1198752

119865119899(119871

ℎ)) lt 120576 for each 119875

1

1198752isin prod(119883 120575)

(2) with |(cup1198751)Δ(cap119875

2)| lt 120578

(b) A sequence 119865119899 of fuzzy-number-valued function is

uniformly 119860119862119866nabla on [119886 119887] if [119886 119887] = cup119894119883119894 where 119883

119894are

measurable sets and 119865 is uniformly 119860119862nabla on each119883119894

Theorem 11 If 119865 is uniformly 119860119862119866nabla then 119865 is uniformly119860119862119866lowast

120575

Proof Let [119886 119887] = cup119894119883119894be such that 119865 is uniformly 119860119862119866nabla

on each 119883119894 So for each 120576 gt 0 there exist 120578 gt 0 and 120575(119909) gt 0

such that (1) holds in Definition 10 for each 1198751 119875

2isin prod(119883 120575)

satisfying condition (2) We take 119875 = ([119888119896 119889

119896] 119909

119896)119901

119894=1with

sum119896|119889

119896minus 119888

119896| lt 120578 and put 119875

1= ([119888

119896 119889

119896] 119909

119896) 119865

119899(119862

119896 119889

119896ge

0) and 1198752= ([119888

119896 119889

119896] 119909

119896) 119865

119899(119888119896 119889

119896lt 0) So we have

|(cup1198751)Δ(cap119875

2)| = sum

119901

119896=1|119889

119896minus 119888

119896| lt 120578 Then by condition (1)

in Definition 10 we have

sup119899

119901

sum119896=1

119863(119865119899(119888119896) 119865

119899(119889

119896))

= sup119863( sum

(119888119896 119889119896)isin1198751

119865119899(119888119896 119889

119896)

sum(119888119896119889119896)isin1198752

119865119899(119888119896 119889

119896)) lt 120576

(16)

Hence we have that 119865 is uniformly 119860119862119866lowast

120575

We get the following theorem byTheorems 9 and 11

Theorem 12 Let the following conditions be satisfied

(i) 119891119899119883

rarr 119891119883ae in [119886 119887]where each119891

119899119883is strong fuzzy

Henstock integrable on [119886 119887](ii) the primitives 119865

119899119883of 119891

119899119883are119880119860119862nabla(119883)with closed set

119883 in [119886 119887]

Then 119891119883

is strong fuzzy Henstock integrable on [119886 119887] withprimitive 119865

119883

Next we give the controlled convergence theorem forthe strong fuzzy Henstock integrals by the definition of the119880119860119862119866

120575for a fuzzy-number-valued function

Definition 13 Let 119865 [119886 119887] rarr 119864119899 and let 119883 sub [119886 119887]A fuzzy-number-valued function 119865 is 119860119862

120575on 119883 if for each

120576 gt 0 there exist 120578 gt 0 and 120575(119909) gt 0 on 119883 such thatsum119873

119894=1119863(119865(119888

119894) 119865(119889

119894)) lt 120576 forsum119873

119894=1(119889

119894minus119888

119894) lt 120578 A fuzzy-number-

valued function 119865 is 119860119862119866120575on [119886 119887] if [119886 119887] is the union of a

sequence of set 119883119894 such that the function 119865 is 119860119862

120575(119883

119894) for

each 119894

Definition 14 (see [18]) A fuzzy-number-valued function 119865defined on 119883 sub [119886 119887] is said to be 119860119862lowast(119883) if for every 120576 gt 0

there exists 120578 gt 0 such that for every finite sequence of non-overlapping intervals [119886

119894 119887119894] satisfying sum119899

119894=1|119887119894minus 119886

119894| lt 120578

where 119886119894 119887119894isin 119883 for all 119894 one has sum120596(119865 [119886

119894 119887119894]) lt 120576 where 120596

denotes the oscillation of 119865 over [119886119894 119887119894] that is 120596(119865 [119886

119894 119887119894]) =

sup119863(119865(119910) 119865(119909)) 119909 119910 isin [119886119894 119887119894] A fuzzy-number-valued

function 119865 is said to be 119860119862119866lowast on 119883 if 119883 is the union ofa sequence of closed sets 119883

119894 such that on each 119883

119894 119865 is

119860119862lowast(119883119894)

Theorem 15 A fuzzy-number-valued function 119865 is 119860119862119866120575if

and only if it is 119860119862119866lowast on [119886 119887]

Theorem 16 (controlled convergence theorem) Let the fol-lowing conditions be satisfied

(1) 119891119899(119909) rarr 119891(119909) almost everywhere in [119886 119887] as 119899 rarr infin

where each 119891119899is strong fuzzy Henstock integrable on

[119886 119887]

Journal of Applied Mathematics 5

(2) the primitives 119865119899(119909) = (SFH) int119909

119886

119891119899(119904)d119909 of 119891

119899are

119880119860119862119866120575uniformly in 119899

(3) the sequence 119865119899(119909) converges uniformly to a con-

tinuous function on [119886 119887] Then 119891(119909) is strong fuzzyHenstock integrable on [119886 119887] and one has

lim119899rarrinfin

(SFH) int119887

119886

119891119899(119909) d119909 = (SFH) int

119887

119886

119891 (119909) d119909 (17)

If conditions (1) and (2) are replaced by condition (4)(4) 119892(119909) le 119891(119909) le ℎ(119909) almost everywhere on [119886 119887]

where 119892(119909) and ℎ(119909) are strong fuzzy Henstock inte-grable

Proof By condition (2) there exists a sequence 119883119894 such that

119865119899isin 119880119860119862

120575in 119883

119894a bounded closed set with bounds 119886 and

119887 and put 119883 = 119883119894 We note that 119865

119899(119909) rarr 119865(119909) we have

119865 isin 119860119862120575on119883 and hence119865 isin 119860119862119866

120575on [119886 119887] ByTheorem 15

119865 isin 119860119862lowast on 119883 and hence 119865 isin 119860119862119866lowast on [119886 119887] and also 119865 isin

119860119862 on119883 and hence 119865 isin 119860119862119866 on [119886 119887]Now we prove that 1198651015840(119909) = 119891(119909) ae on [119886 119887] In fact

let 119866 [119886 119887] rarr 119864119899 equal 119865119899on 119883 and extend 119866

119899linearly

to the closed interval contiguous to119883 Likewise we define 119866from 119865 We see that119866

119899and119866 are119880119860119862 on [119886 119887] By condition

(3) we have 119866119899rarr 119866 on [119886 119887] Let [119888

119896 119889

119896] be the intervals

contiguous to 119883 Then we have 119863(1198661015840

119899) le 119872

119896 We define a

fuzzy-number-valued function as follows

1198661015840

(119909) =119866119899(119889

119896) minus

119867119866119899(119888119896)

119889119896minus 119888

119896

119909 isin (119888119896 119889

119896) (18)

Consequently 1198661015840

119899(119909) converges on (119888

119896 119889

119896) Hence 1198661015840

119899con-

verges on [119886 119887] ae Since 119866119899 isin 119860119862 on [119886 119887] then 1198661015840

119899(119909) =

119891119899

rarr 119891 on 119883 Therefore we have 1198661015840

119899(119909) = 119892(119909) =

119891(119909) = 1198651015840(119909) ae on 119883 Thus 1198651015840(119909) = 119891(119909) ae on [119886 119887]

by Theorem 15 Therefore there exists an 119860119862119866120575function on

[a 119887] such that 1198651015840(119909) = 119891(119909) ae on [119886 119887] Hence 119891 is strongfuzzy Henstock integrable on [119886 119887] and we have

lim119899rarrinfin

(SFH) int119887

119886

119891119899(119909) d119909 = (SFH) int

119887

119886

119891 (119909) d119909 (19)

4 The Generalized Solutions of DiscontinuousFuzzy Differential Equations

In this section a generalized fuzzy differential equation ofform (4) is defined by using strong fuzzy Henstock integralThemain results of this section are existence theorems for thegeneralized solution to the discontinuous fuzzy differentialequation

Definition 17 (see [11]) Let 120591 and 120585 be fixed and let a fuzzy-number-valued function 119891(119905 119909) be a Caratheodory functiondefined on a rectangle119880 |119905minus120591| le 119886 119863(119909 120585) le 119887 that is119891 iscontinuous in 119909 for almost all 119905 and measurable in 119905 for eachfixed 119909

Theorem 18 Let a fuzzy-number-valued function 119891 be afunction as given in Definition 17 then there exist two strongfuzzy Henstock integrable functions ℎ and 119892 defined on |119905minus120591| le119886 such that 119892(119905) le 119891(119905 119909) le ℎ(119905) for all (119905 119909) isin 119880

Proof Note that 119891 is a Caratheodory function Thus thereexist two measurable functions 119906(119905) and V(119905) defined on |119905 minus120591| le 119886 with values in 119863(119909 120585) le 119887 such that 119891(119905 119906(119905)) le

119891(119905 119909) le 119891(119905 V(119905)) for all (119905 119909) isin 119880 Next we will showthat 119891(119905 119906(119905)) and 119891(119905 V(119905)) are fuzzy Henstock integrable byusing controlled convergence Theorem 16 First there existsa sequence 119896

119899(119905) of step functions defined on |119905 minus 120591| le

119886 with values in 119863(119909 120585) le 119887 such that 119896119899(119905) rarr 119906(119905)

almost everywhere as 119899 rarr infin Let 119865119899(119905) = int

119905

120591

119891(119904 119896119899(119904))d119904

Then 119865119899(119905) is 119880119860119862119866

120575uniformly in 119899 and equicontinuous

By controlled convergence Theorem 16 119891(119905 119906(119905)) is strongfuzzy Henstock integrable Similarly 119891(119905 V(119905)) is strong fuzzyHenstock integrable

Definition 19 A fuzzy-number-valued function 119909(119905) 119868 rarr

119864119899 is said to be a solution of the discontinuous fuzzy differ-ential equation (4) if 119909(119905) satisfies the following conditions

(i) 119909(119905) is 119860119862119866120575on each compact subinterval of 119868

(ii) (119905 119909) isin 119880 for 119905 isin 119868(iii) 1199091015840(119905) for almost everywhere 119905 isin 119868

Now we will state the existence theorem for the general-ized solution of discontinuous fuzzy differential equation (4)

Theorem 20 Suppose that 119891 satisfies the condition ofTheorem 18 then there exists a generalized solution Φ of thediscontinuous fuzzy differential equation (4) on some interval|119905 minus 120591| le 119886 which satisfies Φ(120591) = 120585

Proof Given 119892(t) le 119891(119905 119909) le ℎ(119905) for all 119909 and almost all 119905with (119905 119909) isin 119880 we get 0 le 119891(119905 119909)minus

119867119892(119905) le ℎ(119905)minus

119867119892(119905) Let

119865 (119905 119909) = 119891(119905 119909 + int119905

120591

119892 (119904) d119904) minus119867119892 (119905) (20)

or

119865 (119905 119909) = 119891(119905 119909 + (minus1) sdot int119905

120591

119892 (119904) d119904) minus119867119892 (119905) (21)

Then 119865 is a Caratheodory function Furthermore 0 le

119865(119905 119909) le ℎ(119905) minus119867119892(119905) for all (119905 119909) isin 119880

0 where 119880

0sub 119880

such that

119863(119909 + int119905

120591

119892 (119904) d119904 120585) le 119887 forall (119905 119909) isin 1198800 (22)

By Caratheodory existence theorem (see Theorem 7 in [11])there is a fuzzy-number-valued function 120595 on some interval|119905 minus 120591| le 119886 such that 1205951015840(119905) = 119865(119905 120595(119905)) almost everywhere inthis interval and 120595(120591) = 120585 Let

120601 (119905)=120595 (119905)+int119905

120591

119892 (119904) d119904 or 120601 (119905)=120595 (119905)+(minus1) sdot int119905

120591

119892 (119904) d119904

(23)

6 Journal of Applied Mathematics

Then for almost all 119905 we have the following

Case 1 Consider

1206011015840

(119905) = 1205951015840

(119905) + 119892 (119905) = 119865 (119905 120595 (119905)) + 119892 (119905)

= 119891(119905 120595 (119905) + int119905

120591

119892 (119904) d119904) minus119867119892 (119905) + 119892 (119905)

= 119891 (119905 120601 (119905))

120601 (119905) = 120595 (120591) + int119905

120591

119892 (119904) d119904 = 120585

(24)

Case 2 Consider

1206011015840

(119905) = 1205951015840

(119905) + 119892 (119905) = 119865 (119905 120595 (119905)) + 119892 (119905)

= 119891(119905 120595 (119905) + (minus1) sdot int119905

120591

119892 (119904) d119904) minus119867119892 (119905) + 119892 (119905)

= 119891 (119905 120601 (119905))

120601 (119905) = 120595 (120591) + (minus1) sdot int119905

120591

119892 (119904) d119904 = 120585

(25)

The proof is complete

Example 21 Consider fuzzy differential equation 1199091015840 =

119891(119905 119909) = 119892(119905 119909) + ℎ(119905) where 119863(119892(119905 119909) 0) le 119863(1198921(119905) 0) for

all |119905| le 1 119863(119909 0) le 1 and 1198921(119905) is Kaleva integrable on |119905| le 1

and ℎ(119905) = 119860 sdot (119889119889119905)(1199052 sin 119905minus2) if 119905 = 0 and ℎ(0) = 0 Here 119860is defined in Example 1 Note that ℎ is strong fuzzy Henstockintegrable but not Kaleva integrable and

ℎ (119905) minus1198671198921(119905) le 119891 (119905 119909) le ℎ (119905) + 119892

1(119905)

for |119905| le 1 119863 (119909 0) le 1

(26)

Thus by Theorem 20 there exists a solution of 1199091015840 = 119891(119905 119909)

with 119909(0) = 0 For instance if 119892(119905 119909) = 1199052119909 then

120601 (119905) = 1198901199053

3

sdot int119905

0

119890minus1199043

3

ℎ (119904) d119904 (27)

is a solution by using integrating factor

We get the following existence theorem by Theorems 18and 20

Theorem 22 Let a fuzzy-number-valued function 119891 be aCaratheodory function defined on a rectangle 119880 |119905 minus

120591| le 119886119863(119909 120585) le 119887 Let 119891(119905 119906(119905)) be strong fuzzy Henstockintegrable on |119905 minus 120591| le 119886 for any step function 119906(119905) definedon |119905 minus 120591| le 119886 with values in 119863(119909 120585) le 119887 Denote that119865119906(119905) = int

119905

120591

119891(119904 119906(119904))d119904 If 119865119906 119906 is a step function is 119880119860119862119866

120575

uniformly in 119906 and equicontinuous on |119905 minus 120591| le 119886 then thereexists a solution 120601 of 1199091015840 = 119891(119905 119909) on some interval |119905 minus 120591| le 120573with 120601(120591) = 120585

Finally in this paper we will show the continuousdependence of a solution on parameters by using Theorems16 18 and 22

Let 119880119901be a connected set in 119880 Let 119888 gt 0 and let 120583

0be

fixed119868120583= 120583

1003816100381610038161003816120583 minus 12058301003816100381610038161003816 lt 119888

119880120583= (119905 119909 120583) (119905 119909) isin 119880

119901 120583 isin 119868

120583

(28)

Let 119891(119905 119909 120583) be a fuzzy-number-valued function definedon 119880

120583such that for each fixed 120583 the function 119891 is a

Caratheodory function defined on 119880119901for each fixed 119905 and

continuous at (119909 1205830) for every 119909 For 120583 = 120583

0 let

1199091015840

(119905) = 119891 (119905 119909 120583)

119909 (120591) = 120585 isin 119864119899

(29)

have a solution 1206010on [119886 119887] where 120591 isin [119886 119887] Let 119891(119905 119906(119905) 120583)

be strong fuzzy Henstock integrable on [119886 119887] for any stepfunction 119906(119905) with (119905 119906(119905)) isin 119880

119901for 119905 isin [119886 119887]

Definition 23 Denote

119865120583119906(119905) = int

119905

119886

119891 (119904 119906 (119904) 120583) d119904 (30)

The family 119865120583119906 is said to be equicontinuous in 119906 and near

1205830if for each 119905

0isin [119886 119887] there exists an interval |120583 minus 120583

0| lt 119903

0

such that the family 119865120583119906(119905) 119906 and 120583 with |120583 minus 120583

0| lt 119903

0 is

equicontinuous at 1199050

Theorem 24 Let 119891 be a fuzzy-number-valued function asgiven above If the primitive 119865

120583119906(119905) of 119891(119905 119909 120583) is 119860119862119866

120575

uniformly in119906 and120583 and equicontinuous in119906 and near1205830 then

there exists 120575 gt 0 such that for any fixed 120583 with |120583 minus 1205830| lt 120575

a solution 120601120583of discontinuous fuzzy differential equation (29)

exists over [119886 119887] and as 120583 rarr 1205830 120601

120583rarr 120601

0uniformly over

[119886 119887]

Proof Firstly we will consider the case 120591 isin (119886 119887) ByTheorem 22 and the equicontinuity of 119865

120583119906(119905) all solutions

120601120583of problem (29) with |120583 minus 120583

0| le 120575

1for some 120575

1gt 0

exist over some interval |119905 minus 120591| le 120572 with 120572 gt 0 Then119866 = 120601

120583 |120583 minus 120583

0| le 120575

1 is equicontinuous on |119905 minus 120591| le 120572

Because 120601120583(120591) = 120585 119866 is uniformly bounded Hence for all

sequence of119866with120583 rarr 1205830 there exists a subsequencewhich

converges uniformly That is to say 120601120583(119896)

rarr 120595 uniformly on|119905 minus 120591| le 120572 as 120583(119896) rarr 120583

0 Since we have

120601120583(119896)

(119905) = 120585 + int119905

120591

119891 (119904 120601120583(119896)

120583 (119896)) d119904 (31)

or

120601120583(119896)

(119905) = 120585 + (minus1) sdot int119905

120591

119891 (119904 120601120583(119896)

120583 (119896)) d119904 (32)

byTheorem 16 we get

120595 (119905) = 120585 + int119905

120591

119891 (119904 120595 (119904) 1205830) d119904 (33)

Journal of Applied Mathematics 7

or

120595 (119905) = 120585 + (minus1) sdot int119905

120591

119891 (119904 120595 (119904) 1205830) d119904 (34)

Thus 120595 is a solution of (29) for 120583 = 1205830 Hence 120595 = 120601

0on

|119905 minus 120591| le 120572 because the solution 1206010is unique Consequently

120601119906(119896)

rarr 1206010uniformly on |119905 minus 120591| le 120572 as 120583(119896) rarr 120583

0 By

Reductio ad absurdum 120601120583rarr 120601

0uniformly there as 120583 rarr

1205830Secondly we will extend the result over [119886 119887] We con-

sider the case [120591 119887] Assume that there exists 1199050isin [120591 119887) such

that the result is valid over [120591 1199050minus ℎ] but not [120591 119905

0+ ℎ]

Obviously 1199050ge 120591 + 120572 Let 120574 120573 be such that

119867 = (119905 119909) 1003816100381610038161003816119905 minus 1199050

1003816100381610038161003816 le 120572119863 (119909 1206010(119905 minus 120574) le 120573) (35)

is contained in 119880 Since 119865120583119906(119905) is equicontinuous in 119906 and

near 1205830 we may choose 120574 gt 0 small enough such that

119863(int119905

1199050

119891 (119904 119906 (119904) 120583) d119904 0) lt120573

3(36)

for all step functions 119906whenever |119905minus 1199050| le 120574 and |120583minus120583

0| lt 120575

2

Since 120601120583rarr 120601

0uniformly on [120591 119905

0minus ℎ] there exists a 120575 such

that

119863(120601120583(1199050minus 120574) 120601

0(1199050minus 120574)) lt

120573

3(37)

for |120583 minus 1205830| lt 120575 Hence we have

119863(int119905

1199050minus120574

119891 (119904 119906 (119904) 120583) d119904 120601119898119906(1199050minus 120574) minus

1198671206010(1199050minus 120574))

le 119863(int1199050

1199050minus120574

119891 (119904 119906 (119904) 120583) d119904 0)

+ 119863(int119905

1199050

119891 (119904 119906 (119904) 120583) d119904 0)

+ 119863 (120601119898119906(1199050minus 120574) 120601

0(1199050minus 120574)) lt 3 sdot

120573

3

(38)

Thus for each 119906 with |120583 minus 1205830| lt 120575 a solution of (29) with

119909(1199050minus 120574) = 120601

120583(1199050minus 120574) exists on |119905 minus 119905

0| le 120574 Hence 120601

120583can be

continued to 1199050+ 120574 So in the case [120591 minus 120572 120591 + 120572] 120601

120583rarr 120601

0

uniformly on [120591 1199050+ 120574] It leads to a contradiction similarly

for the cases 1199050= 119887 and [119886 120591] Therefore the theorem holds

over [119886 119887]

Example 25 Let

(119905) = 119860 sdot 119905

2 sin 119905minus2 119905 = 0

0 119905 = 0(39)

where fuzzy number 119860 is defined in Example 1 Let ℎ(119905) =1015840(119905) We define

ℎ120583(119905) =

ℎ (119905) 120583 isin [0 1) 119905 isin(minus2 2)

(minus120583 120583)

0 otherwise(40)

and ℎ120583(119905) = ℎ

minus120583(119905) for 120583 isin (minus1 0] Let 119891(119905 119909 120583) = 1199052119909 + ℎ

120583(119905)

defined on (minus2 2) times (minus1 1) Then we have

119863(119865120583119906(1199051) 119865

120583119906(1199052)) = 119863(int

1199052

1199051

119891 (119904 119906 (119904) 120583) d119904 0)

le int1199052

1199051

1199042d119904 + 119863(int

1199052

1199051

ℎ120583d119904 0)

119863(int1199052

1199051

ℎ120583d119904 0) le int

1199052

1199051

119863(ℎ (119904) d119904 0)

(41)

where 1199051 1199052isin (0 1] or 119905

1 1199052isin [minus1 0) and

119863(int119905

0

ℎ120583(119904) d119904 0)

=

119863(119860 sdot 1199052 sin 119905minus2 119860 sdot 1205832 sin 120583minus2) 120583 = 0

119863 (119860 sdot 1199052 sin 119905minus2 0) 120583 = 0

(42)

Note that ℎ is Kaleva integrable on every subinterval of [minus1 0)and (0 1] Therefore we have that 119865

120583119906(119905) is equicontinuous

on [minus1 1] in 119906 and near 1205830= 0 Furthermore 119865

120583119906(119905) is

119860119862120575(119883

119899) uniformly in 120583 and 119906 where 119883

119899= [1119899 1] 119899 =

1 2 On the other hand by using integrating factor for119905 isin [minus1 1] we have

120601120583(119905) = 119890

1199052

3

int119905

0

119890minus1199043

3

119860 sdot ℎ120583(119904) d119904 (43)

with 120601120583(0) = 0 Obviously 120601

0is uniqueThus byTheorem 24

120601120583rarr 120601

0uniformly on [minus1 1]

5 Conclusion

In this paper we give the definition of the119880119860119862119866120575for a fuzzy-

number-valued function and the nonabsolute fuzzy integraland its controlled convergence theorem In addition we dealwith the Cauchy problem and the continuous dependence ofa solution on parameters of discontinuous fuzzy differentialequations involving the strong fuzzy Henstock integral infuzzy number space The function governing the equationsis supposed to be discontinuous with respect to some vari-ables and satisfy nonabsolute fuzzy integrability Our resultimproves the result given in [1 11 19 20] (where uniformcontinuity was required) as well as those referred therein

Acknowledgments

The authors are very grateful to the anonymous refereesand Professor Mehmet Sezer for many valuable commentsand suggestions which helped to improve the presentationof the paper The authors would like to thank the NationalNatural Science Foundation of China (no 11161041) and theFundamental Research Fund for the Central Universities (no31920130010)

8 Journal of Applied Mathematics

References

[1] O Kaleva ldquoFuzzy differential equationsrdquo Fuzzy Sets and Sys-tems vol 24 no 3 pp 301ndash317 1987

[2] Z Gong and Y Shao ldquoGlobal existence and uniqueness ofsolutions for fuzzy differential equations under dissipative-typeconditionsrdquo Computers amp Mathematics with Applications vol56 no 10 pp 2716ndash2723 2008

[3] O Kaleva ldquoThe Cauchy problem for fuzzy differential equa-tionsrdquo Fuzzy Sets and Systems vol 35 no 3 pp 389ndash396 1990

[4] J J Nieto ldquoThe Cauchy problem for continuous fuzzy differen-tial equationsrdquo Fuzzy Sets and Systems vol 102 no 2 pp 259ndash262 1999

[5] M L Puri and D A Ralescu ldquoDifferentials of fuzzy functionsrdquoJournal of Mathematical Analysis and Applications vol 91 no 2pp 552ndash558 1983

[6] S Seikkala ldquoOn the fuzzy initial value problemrdquo Fuzzy Sets andSystems vol 24 no 3 pp 319ndash330 1987

[7] C Wu and S Song ldquoExistence theorem to the Cauchy problemof fuzzy differential equations under compactness-type condi-tionsrdquo Information Sciences vol 108 no 1ndash4 pp 123ndash134 1998

[8] C Wu S Song and Z Qi ldquoExistence and uniqueness for asolution on the closed subset to the Cauchy problem of fuzzydiffrential equationsrdquo Journal of Harbin Institute of Technologyvol 2 pp 1ndash7 1997

[9] CWu S Song and E S Lee ldquoApproximate solutions existenceand uniqueness of the Cauchy problem of fuzzy differentialequationsrdquo Journal of Mathematical Analysis and Applicationsvol 202 no 2 pp 629ndash644 1996

[10] V Lakshmikantham and R N Mohapatra Theory of FuzzyDifferential Equations and Inclusions vol 6 Taylor amp FrancisLondon UK 2003

[11] X Xue and Y Fu ldquoCaratheodory solutions of fuzzy differentialequationsrdquo Fuzzy Sets and Systems vol 125 no 2 pp 239ndash2432002

[12] R HenstockTheory of Integration Butterworths London UK1963

[13] J Kurzweil ldquoGeneralized ordinary differential equations andcontinuous dependence on a parameterrdquo Czechoslovak Mathe-matical Journal vol 7 no 82 pp 418ndash449 1957

[14] P Y LeeLanzhou Lectures onHenstock Integration vol 2WorldScientific Publishing Teaneck NJ USA 1989

[15] C Wu and Z Gong ldquoOn Henstock integrals of interval-valuedfunctions and fuzzy-valued functionsrdquo Fuzzy Sets and Systemsvol 115 no 3 pp 377ndash391 2000

[16] C Wu and Z Gong ldquoOn Henstock integral of fuzzy-number-valued functions Irdquo Fuzzy Sets and Systems vol 120 no 3 pp523ndash532 2001

[17] Z Gong ldquoOn the problem of characterizing derivatives for thefuzzy-valued functions II Almost everywhere differentiabilityand strong Henstock integralrdquo Fuzzy Sets and Systems vol 145no 3 pp 381ndash393 2004

[18] Z Gong and Y Shao ldquoThe controlled convergence theoremsfor the strong Henstock integrals of fuzzy-number-valuedfunctionsrdquo Fuzzy Sets and Systems vol 160 no 11 pp 1528ndash1546 2009

[19] T S Chew and F Flordeliza ldquoOn 1199091015840 = 119891(119905 119909) and Henstock-Kurzweil integralsrdquo Differential and Integral Equations vol 4no 4 pp 861ndash868 1991

[20] B Bede and S G Gal ldquoGeneralizations of the differentiabilityof fuzzy-number-valued functions with applications to fuzzy

differential equationsrdquo Fuzzy Sets and Systems vol 151 no 3pp 581ndash599 2005

[21] D Dubois and H Prade ldquoTowards fuzzy differential calculus IIntegration of fuzzy mappingsrdquo Fuzzy Sets and Systems vol 8no 1 pp 1ndash17 1982

[22] P Diamond and P KloedenMetric Spaces of Fuzzy Sets WorldScientific Publishing River Edge NJ USA 1994

[23] R Goetschel Jr and W Voxman ldquoElementary fuzzy calculusrdquoFuzzy Sets and Systems vol 18 no 1 pp 31ndash43 1986

[24] Y Chalco-Cano and H Roman-Flores ldquoOn new solutions offuzzy differential equationsrdquo Chaos Solitons and Fractals vol38 no 1 pp 112ndash119 2008

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

(2) for all ℎ gt 0 sufficiently small there exists 119891(1199090) minus

119867119891(119909

0+ ℎ) 119891(119909

0minus ℎ) minus

119867119891(119909

0) and the limits

limℎrarr0

119891 (1199090) minus

119867119891 (119909

0+ ℎ)

minusℎ= lim

ℎrarr0

119891 (1199090minus ℎ) minus

119867119891 (119909

0)

minusℎ

= 1198911015840

(1199090)

(10)

or

(3) for all ℎ gt 0 sufficiently small there exists 119891(1199090+ℎ) minus

119867119891(119909

0) 119891(119909

0minus ℎ) minus

119867119891(119909

0) and the limits

limℎrarr0

119891 (1199090+ ℎ) minus

119867119891 (119909

0)

ℎ= lim

ℎrarr0

119891 (1199090minus ℎ) minus

119867119891 (119909

0)

minusℎ

= 1198911015840

(1199090)

(11)

or

(4) for all ℎ gt 0 sufficiently small there exists 119891(1199090) minus

119867119891(119909

0+ ℎ) 119891(119909

0) minus

119867119891(119909

0minus ℎ) and the limits

limℎrarr0

119891 (1199090) minus

119867119891 (119909

0+ ℎ)

minusℎ= lim

ℎrarr0

119891 (1199090) minus

119867119891 (119909

0minus ℎ)

ℎ

= 1198911015840

(1199090)

(12)

(ℎ and minusℎ at denominators mean (1ℎ)sdot and minus(1ℎ)sdotresp)

3 The Convergence Theorem of Strong FuzzyHenstock Integral

In this section we define the strong Henstock integrals offuzzy-number-valued functions in fuzzy number space 119864119899and we give some properties and controlled convergencetheorem of this integral by using new conditions

Definition 3 (see [18]) A fuzzy-number-valued function 119891

is said to be termed additive on [119886 119887] if for any division119879 119886 le 119909

1le 119909

2le sdot sdot sdot le 119909

119899le 119887 one has 119891([119909

119894 119909

119895]) (1 le

119894 lt 119895 le 119899) that exists and 119891([119909119894 119909

119895]) = sum

119895minus1

119896=119894119891([119909

119896 119909

119896+1]) or

119891([119909119895 119909

119894])(1 le 119894 lt 119895 le 119899) that exists and (minus1) sdot 119891([119909

119895 119909

119894]) =

(minus1) sdot sum119895minus1

119896=119894119891([119909

119896+1 119909

119896]) For convenience 119891([119904 119905]) denotes

119891(119905) minus119867119891(119904)

Definition 4 (see [17 18]) A fuzzy-number-valued function119891 is said to be strong Henstock integrable on [119886 119887] if thereexists a piecewise additive fuzzy-number-valued function 119865

on [119886 119887] such that for every 120576 gt 0 there is a function 120575(120585) gt 0and for any 120575-fine division 119875 = ([119906 V] 120585) of [119886 119887] one has

sum119894isin119870119899

119863(119891 (120585119894) (V

119894minus 119906

119894) 119865 ([119906

119894 V

119894]))

+ sum119895isin119868119899

119863(119891 (120585119895) (V

119895minus 119906

119895) (minus1) sdot 119865 ([119906

119895 V

119895minus1])) lt 120576

(13)

where 119870119899= 119894 isin 1 2 119899 such that 119865([119909

119894minus1 119909

119894]) is a fuzzy

number and 119868119899= 119895 isin 1 2 119899 such that 119865([119909

119895 119909

119895minus1]) is a

fuzzy number One writes 119891 isin SFH[119886 119887]

Definition 5 A fuzzy-number-valued function 119865 defined on119883 sub [119886 119887] is said to be 119860119862lowast

120575(119883) if for every 120576 gt 0 there exists

120578 gt 0 and 120575(120585) gt 0 such that for any 120575-fine partial division119875 = ([119906 V] 120585) with 120585 isin 119883

119894satisfyingsum119899

119894=1|V minus 119906| lt 120578 one has

sum119863(119865[119906 V]) lt 120576

Definition 6 A fuzzy-number-valued function 119865 is said to be119860119862119866lowast

120575on119883 sub [119886 119887] if119883 is the union of a sequence of closed

sets 119883119894 such that on each119883

119894 119865 is 119860119862lowast

120575(119883

119894)

Definition 7 The sequence of fuzzy-number-function 119865119899 is

119880119860119862119866lowast

120575on119883 sub [119886 119887] if119883 is the sequence of subsets119883

119894such

that 119865119899 is 119880119860119862lowast

120575for each 119894 independent of 119899

Definition 8 Let 119865119899 be a sequence of fuzzy-number-

function defined on [119886 119887] and let 119909 sub [119886 119887] be measurable

(i) The sequence of fuzzy-number-function 119865119899 is P-

Cauchy on 119864119899 if 119865119899 converges pointwise on 119883 and

if for each 120576 gt 0 there exist 120575(120585) gt 0 on 119883 and apositive integer 119873 such that 119863(119865

119898(119875) 119865

119899(119875)) lt 120576 for

all119898 119899 ge 119873 whenever 119875 is119883-subordinate to 120575(120585)(ii) The sequence of fuzzy-number-function 119865

119899 is gen-

eralized P-Cauchy on 119883 if 119883 can be written as acountable union of measurable sets on each of which119865

119899 isP-Cauchy

Theorem 9 Let the following conditions be satisfied

(i) 119891119899119883(119909) rarr 119891

119909ae on [119886 119887] as 119899 rarr infin where each

119891119899119883

is strong Fuzzy Henstock integrable on [119886 119887]

(ii) the primitives 119865119899119883

of 119891119899119883

are 119880119860119862lowast

120575with closed set119883

in [119886 119887]

Then 119891119883(119909) is strong fuzzy Henstock integrable on [119886 119887] with

the primitive 119865119883(119909)

Proof By (ii) for every 120576 gt 0 there exist a 120575(120585) gt 0 and 120578 gt 0such that for any 120575-fine partial division119875of119883 satisfyingsum |Vminus119906| lt 120578 we have sum119863(119865

119899119883(V 119906) 0) lt 120576 By Egoroff rsquos theorem

[18 Theorem 34] there is an open set 119866 with |119866| lt 120578 suchthat 119863(119891

119899(120585) 119891

119898(120585)) lt 120576 for 119899119898 ge 119873 and 120585 notin 119866 Consider

the following in which 119875 is a 120575-fine division of [119909 119910] and

4 Journal of Applied Mathematics

119875 = 1198751cup119875

2so that119875

1contains the intervalswith the associated

points 120585 notin 119866 and 1198752otherwise

119863(119865119899119883

(119909 119910) 119865119898119883

(119909 119910))

= (1198752)sum119863(119865

119899119883(119906 V) 119865

119898119883(119906 V))

le sum119863(119865119899119883

(119906 V) 119891119899119883

(120585) (V minus 119906))

+sum119863(119865119898119883

(119906 V) 119891119898119883

(120585) (V minus 119906))

+sum119863(119891119898119883

(120585) (V minus 119906) 119891119898119883

(120585) (V minus 119906))

+sum119863(119865119899119883

(V) 119865119899119883

(119906)) +sum119863(119865119898119883

(V) 119865119898119883

(119906))

lt 120576 (4 + 119887 minus 119886)

(14)

Hence for any 120575-fine partial division 119875 of [119886 119887] we have

10038161003816100381610038161003816sum119863(119865

119899119883(119906 V) 119865

119898119883(119906 V))

10038161003816100381610038161003816lt 120576 (15)

for 119898 119899 ge 119873 Therefore the fuzzy sequence 119865119899119883 is gen-

eralized P-Cauchy on [119886 119887] Then by (i) we have that 119891119883

is strong fuzzy Henstock integrable on [119886 119887] with primitive119865119883

Definition 10 (a) A sequence 119865119899 of fuzzy-number-valued

function is uniformly119860119862nabla on119883whenever to each 120576 gt 0 thereexist 120578 gt 0 and 120575(119909) gt 0 such that

(1) sup119899119863(sum

119869119896isin1198751

119865119899(119869119896) sum

119871ℎisin1198752

119865119899(119871

ℎ)) lt 120576 for each 119875

1

1198752isin prod(119883 120575)

(2) with |(cup1198751)Δ(cap119875

2)| lt 120578

(b) A sequence 119865119899 of fuzzy-number-valued function is

uniformly 119860119862119866nabla on [119886 119887] if [119886 119887] = cup119894119883119894 where 119883

119894are

measurable sets and 119865 is uniformly 119860119862nabla on each119883119894

Theorem 11 If 119865 is uniformly 119860119862119866nabla then 119865 is uniformly119860119862119866lowast

120575

Proof Let [119886 119887] = cup119894119883119894be such that 119865 is uniformly 119860119862119866nabla

on each 119883119894 So for each 120576 gt 0 there exist 120578 gt 0 and 120575(119909) gt 0

such that (1) holds in Definition 10 for each 1198751 119875

2isin prod(119883 120575)

satisfying condition (2) We take 119875 = ([119888119896 119889

119896] 119909

119896)119901

119894=1with

sum119896|119889

119896minus 119888

119896| lt 120578 and put 119875

1= ([119888

119896 119889

119896] 119909

119896) 119865

119899(119862

119896 119889

119896ge

0) and 1198752= ([119888

119896 119889

119896] 119909

119896) 119865

119899(119888119896 119889

119896lt 0) So we have

|(cup1198751)Δ(cap119875

2)| = sum

119901

119896=1|119889

119896minus 119888

119896| lt 120578 Then by condition (1)

in Definition 10 we have

sup119899

119901

sum119896=1

119863(119865119899(119888119896) 119865

119899(119889

119896))

= sup119863( sum

(119888119896 119889119896)isin1198751

119865119899(119888119896 119889

119896)

sum(119888119896119889119896)isin1198752

119865119899(119888119896 119889

119896)) lt 120576

(16)

Hence we have that 119865 is uniformly 119860119862119866lowast

120575

We get the following theorem byTheorems 9 and 11

Theorem 12 Let the following conditions be satisfied

(i) 119891119899119883

rarr 119891119883ae in [119886 119887]where each119891

119899119883is strong fuzzy

Henstock integrable on [119886 119887](ii) the primitives 119865

119899119883of 119891

119899119883are119880119860119862nabla(119883)with closed set

119883 in [119886 119887]

Then 119891119883

is strong fuzzy Henstock integrable on [119886 119887] withprimitive 119865

119883

Next we give the controlled convergence theorem forthe strong fuzzy Henstock integrals by the definition of the119880119860119862119866

120575for a fuzzy-number-valued function

Definition 13 Let 119865 [119886 119887] rarr 119864119899 and let 119883 sub [119886 119887]A fuzzy-number-valued function 119865 is 119860119862

120575on 119883 if for each

120576 gt 0 there exist 120578 gt 0 and 120575(119909) gt 0 on 119883 such thatsum119873

119894=1119863(119865(119888

119894) 119865(119889

119894)) lt 120576 forsum119873

119894=1(119889

119894minus119888

119894) lt 120578 A fuzzy-number-

valued function 119865 is 119860119862119866120575on [119886 119887] if [119886 119887] is the union of a

sequence of set 119883119894 such that the function 119865 is 119860119862

120575(119883

119894) for

each 119894

Definition 14 (see [18]) A fuzzy-number-valued function 119865defined on 119883 sub [119886 119887] is said to be 119860119862lowast(119883) if for every 120576 gt 0

there exists 120578 gt 0 such that for every finite sequence of non-overlapping intervals [119886

119894 119887119894] satisfying sum119899

119894=1|119887119894minus 119886

119894| lt 120578

where 119886119894 119887119894isin 119883 for all 119894 one has sum120596(119865 [119886

119894 119887119894]) lt 120576 where 120596

denotes the oscillation of 119865 over [119886119894 119887119894] that is 120596(119865 [119886

119894 119887119894]) =

sup119863(119865(119910) 119865(119909)) 119909 119910 isin [119886119894 119887119894] A fuzzy-number-valued

function 119865 is said to be 119860119862119866lowast on 119883 if 119883 is the union ofa sequence of closed sets 119883

119894 such that on each 119883

119894 119865 is

119860119862lowast(119883119894)

Theorem 15 A fuzzy-number-valued function 119865 is 119860119862119866120575if

and only if it is 119860119862119866lowast on [119886 119887]

Theorem 16 (controlled convergence theorem) Let the fol-lowing conditions be satisfied

(1) 119891119899(119909) rarr 119891(119909) almost everywhere in [119886 119887] as 119899 rarr infin

where each 119891119899is strong fuzzy Henstock integrable on

[119886 119887]

Journal of Applied Mathematics 5

(2) the primitives 119865119899(119909) = (SFH) int119909

119886

119891119899(119904)d119909 of 119891

119899are

119880119860119862119866120575uniformly in 119899

(3) the sequence 119865119899(119909) converges uniformly to a con-

tinuous function on [119886 119887] Then 119891(119909) is strong fuzzyHenstock integrable on [119886 119887] and one has

lim119899rarrinfin

(SFH) int119887

119886

119891119899(119909) d119909 = (SFH) int

119887

119886

119891 (119909) d119909 (17)

If conditions (1) and (2) are replaced by condition (4)(4) 119892(119909) le 119891(119909) le ℎ(119909) almost everywhere on [119886 119887]

where 119892(119909) and ℎ(119909) are strong fuzzy Henstock inte-grable

Proof By condition (2) there exists a sequence 119883119894 such that

119865119899isin 119880119860119862

120575in 119883

119894a bounded closed set with bounds 119886 and

119887 and put 119883 = 119883119894 We note that 119865

119899(119909) rarr 119865(119909) we have

119865 isin 119860119862120575on119883 and hence119865 isin 119860119862119866

120575on [119886 119887] ByTheorem 15

119865 isin 119860119862lowast on 119883 and hence 119865 isin 119860119862119866lowast on [119886 119887] and also 119865 isin

119860119862 on119883 and hence 119865 isin 119860119862119866 on [119886 119887]Now we prove that 1198651015840(119909) = 119891(119909) ae on [119886 119887] In fact

let 119866 [119886 119887] rarr 119864119899 equal 119865119899on 119883 and extend 119866

119899linearly

to the closed interval contiguous to119883 Likewise we define 119866from 119865 We see that119866

119899and119866 are119880119860119862 on [119886 119887] By condition

(3) we have 119866119899rarr 119866 on [119886 119887] Let [119888

119896 119889

119896] be the intervals

contiguous to 119883 Then we have 119863(1198661015840

119899) le 119872

119896 We define a

fuzzy-number-valued function as follows

1198661015840

(119909) =119866119899(119889

119896) minus

119867119866119899(119888119896)

119889119896minus 119888

119896

119909 isin (119888119896 119889

119896) (18)

Consequently 1198661015840

119899(119909) converges on (119888

119896 119889

119896) Hence 1198661015840

119899con-

verges on [119886 119887] ae Since 119866119899 isin 119860119862 on [119886 119887] then 1198661015840

119899(119909) =

119891119899

rarr 119891 on 119883 Therefore we have 1198661015840

119899(119909) = 119892(119909) =

119891(119909) = 1198651015840(119909) ae on 119883 Thus 1198651015840(119909) = 119891(119909) ae on [119886 119887]

by Theorem 15 Therefore there exists an 119860119862119866120575function on

[a 119887] such that 1198651015840(119909) = 119891(119909) ae on [119886 119887] Hence 119891 is strongfuzzy Henstock integrable on [119886 119887] and we have

lim119899rarrinfin

(SFH) int119887

119886

119891119899(119909) d119909 = (SFH) int

119887

119886

119891 (119909) d119909 (19)

4 The Generalized Solutions of DiscontinuousFuzzy Differential Equations

In this section a generalized fuzzy differential equation ofform (4) is defined by using strong fuzzy Henstock integralThemain results of this section are existence theorems for thegeneralized solution to the discontinuous fuzzy differentialequation

Definition 17 (see [11]) Let 120591 and 120585 be fixed and let a fuzzy-number-valued function 119891(119905 119909) be a Caratheodory functiondefined on a rectangle119880 |119905minus120591| le 119886 119863(119909 120585) le 119887 that is119891 iscontinuous in 119909 for almost all 119905 and measurable in 119905 for eachfixed 119909

Theorem 18 Let a fuzzy-number-valued function 119891 be afunction as given in Definition 17 then there exist two strongfuzzy Henstock integrable functions ℎ and 119892 defined on |119905minus120591| le119886 such that 119892(119905) le 119891(119905 119909) le ℎ(119905) for all (119905 119909) isin 119880

Proof Note that 119891 is a Caratheodory function Thus thereexist two measurable functions 119906(119905) and V(119905) defined on |119905 minus120591| le 119886 with values in 119863(119909 120585) le 119887 such that 119891(119905 119906(119905)) le

119891(119905 119909) le 119891(119905 V(119905)) for all (119905 119909) isin 119880 Next we will showthat 119891(119905 119906(119905)) and 119891(119905 V(119905)) are fuzzy Henstock integrable byusing controlled convergence Theorem 16 First there existsa sequence 119896

119899(119905) of step functions defined on |119905 minus 120591| le

119886 with values in 119863(119909 120585) le 119887 such that 119896119899(119905) rarr 119906(119905)

almost everywhere as 119899 rarr infin Let 119865119899(119905) = int

119905

120591

119891(119904 119896119899(119904))d119904

Then 119865119899(119905) is 119880119860119862119866

120575uniformly in 119899 and equicontinuous

By controlled convergence Theorem 16 119891(119905 119906(119905)) is strongfuzzy Henstock integrable Similarly 119891(119905 V(119905)) is strong fuzzyHenstock integrable

Definition 19 A fuzzy-number-valued function 119909(119905) 119868 rarr

119864119899 is said to be a solution of the discontinuous fuzzy differ-ential equation (4) if 119909(119905) satisfies the following conditions

(i) 119909(119905) is 119860119862119866120575on each compact subinterval of 119868

(ii) (119905 119909) isin 119880 for 119905 isin 119868(iii) 1199091015840(119905) for almost everywhere 119905 isin 119868

Now we will state the existence theorem for the general-ized solution of discontinuous fuzzy differential equation (4)

Theorem 20 Suppose that 119891 satisfies the condition ofTheorem 18 then there exists a generalized solution Φ of thediscontinuous fuzzy differential equation (4) on some interval|119905 minus 120591| le 119886 which satisfies Φ(120591) = 120585

Proof Given 119892(t) le 119891(119905 119909) le ℎ(119905) for all 119909 and almost all 119905with (119905 119909) isin 119880 we get 0 le 119891(119905 119909)minus

119867119892(119905) le ℎ(119905)minus

119867119892(119905) Let

119865 (119905 119909) = 119891(119905 119909 + int119905

120591

119892 (119904) d119904) minus119867119892 (119905) (20)

or

119865 (119905 119909) = 119891(119905 119909 + (minus1) sdot int119905

120591

119892 (119904) d119904) minus119867119892 (119905) (21)

Then 119865 is a Caratheodory function Furthermore 0 le

119865(119905 119909) le ℎ(119905) minus119867119892(119905) for all (119905 119909) isin 119880

0 where 119880

0sub 119880

such that

119863(119909 + int119905

120591

119892 (119904) d119904 120585) le 119887 forall (119905 119909) isin 1198800 (22)

By Caratheodory existence theorem (see Theorem 7 in [11])there is a fuzzy-number-valued function 120595 on some interval|119905 minus 120591| le 119886 such that 1205951015840(119905) = 119865(119905 120595(119905)) almost everywhere inthis interval and 120595(120591) = 120585 Let

120601 (119905)=120595 (119905)+int119905

120591

119892 (119904) d119904 or 120601 (119905)=120595 (119905)+(minus1) sdot int119905

120591

119892 (119904) d119904

(23)

6 Journal of Applied Mathematics

Then for almost all 119905 we have the following

Case 1 Consider

1206011015840

(119905) = 1205951015840

(119905) + 119892 (119905) = 119865 (119905 120595 (119905)) + 119892 (119905)

= 119891(119905 120595 (119905) + int119905

120591

119892 (119904) d119904) minus119867119892 (119905) + 119892 (119905)

= 119891 (119905 120601 (119905))

120601 (119905) = 120595 (120591) + int119905

120591

119892 (119904) d119904 = 120585

(24)

Case 2 Consider

1206011015840

(119905) = 1205951015840

(119905) + 119892 (119905) = 119865 (119905 120595 (119905)) + 119892 (119905)

= 119891(119905 120595 (119905) + (minus1) sdot int119905

120591

119892 (119904) d119904) minus119867119892 (119905) + 119892 (119905)

= 119891 (119905 120601 (119905))

120601 (119905) = 120595 (120591) + (minus1) sdot int119905

120591

119892 (119904) d119904 = 120585

(25)

The proof is complete

Example 21 Consider fuzzy differential equation 1199091015840 =

119891(119905 119909) = 119892(119905 119909) + ℎ(119905) where 119863(119892(119905 119909) 0) le 119863(1198921(119905) 0) for

all |119905| le 1 119863(119909 0) le 1 and 1198921(119905) is Kaleva integrable on |119905| le 1

and ℎ(119905) = 119860 sdot (119889119889119905)(1199052 sin 119905minus2) if 119905 = 0 and ℎ(0) = 0 Here 119860is defined in Example 1 Note that ℎ is strong fuzzy Henstockintegrable but not Kaleva integrable and

ℎ (119905) minus1198671198921(119905) le 119891 (119905 119909) le ℎ (119905) + 119892

1(119905)

for |119905| le 1 119863 (119909 0) le 1

(26)

Thus by Theorem 20 there exists a solution of 1199091015840 = 119891(119905 119909)

with 119909(0) = 0 For instance if 119892(119905 119909) = 1199052119909 then

120601 (119905) = 1198901199053

3

sdot int119905

0

119890minus1199043

3

ℎ (119904) d119904 (27)

is a solution by using integrating factor

We get the following existence theorem by Theorems 18and 20

Theorem 22 Let a fuzzy-number-valued function 119891 be aCaratheodory function defined on a rectangle 119880 |119905 minus

120591| le 119886119863(119909 120585) le 119887 Let 119891(119905 119906(119905)) be strong fuzzy Henstockintegrable on |119905 minus 120591| le 119886 for any step function 119906(119905) definedon |119905 minus 120591| le 119886 with values in 119863(119909 120585) le 119887 Denote that119865119906(119905) = int

119905

120591

119891(119904 119906(119904))d119904 If 119865119906 119906 is a step function is 119880119860119862119866

120575

uniformly in 119906 and equicontinuous on |119905 minus 120591| le 119886 then thereexists a solution 120601 of 1199091015840 = 119891(119905 119909) on some interval |119905 minus 120591| le 120573with 120601(120591) = 120585

Finally in this paper we will show the continuousdependence of a solution on parameters by using Theorems16 18 and 22

Let 119880119901be a connected set in 119880 Let 119888 gt 0 and let 120583

0be

fixed119868120583= 120583

1003816100381610038161003816120583 minus 12058301003816100381610038161003816 lt 119888

119880120583= (119905 119909 120583) (119905 119909) isin 119880

119901 120583 isin 119868

120583

(28)

Let 119891(119905 119909 120583) be a fuzzy-number-valued function definedon 119880

120583such that for each fixed 120583 the function 119891 is a

Caratheodory function defined on 119880119901for each fixed 119905 and

continuous at (119909 1205830) for every 119909 For 120583 = 120583

0 let

1199091015840

(119905) = 119891 (119905 119909 120583)

119909 (120591) = 120585 isin 119864119899

(29)

have a solution 1206010on [119886 119887] where 120591 isin [119886 119887] Let 119891(119905 119906(119905) 120583)

be strong fuzzy Henstock integrable on [119886 119887] for any stepfunction 119906(119905) with (119905 119906(119905)) isin 119880

119901for 119905 isin [119886 119887]

Definition 23 Denote

119865120583119906(119905) = int

119905

119886

119891 (119904 119906 (119904) 120583) d119904 (30)

The family 119865120583119906 is said to be equicontinuous in 119906 and near

1205830if for each 119905

0isin [119886 119887] there exists an interval |120583 minus 120583

0| lt 119903

0

such that the family 119865120583119906(119905) 119906 and 120583 with |120583 minus 120583

0| lt 119903

0 is

equicontinuous at 1199050

Theorem 24 Let 119891 be a fuzzy-number-valued function asgiven above If the primitive 119865

120583119906(119905) of 119891(119905 119909 120583) is 119860119862119866

120575

uniformly in119906 and120583 and equicontinuous in119906 and near1205830 then

there exists 120575 gt 0 such that for any fixed 120583 with |120583 minus 1205830| lt 120575

a solution 120601120583of discontinuous fuzzy differential equation (29)

exists over [119886 119887] and as 120583 rarr 1205830 120601

120583rarr 120601

0uniformly over

[119886 119887]

Proof Firstly we will consider the case 120591 isin (119886 119887) ByTheorem 22 and the equicontinuity of 119865

120583119906(119905) all solutions

120601120583of problem (29) with |120583 minus 120583

0| le 120575

1for some 120575

1gt 0

exist over some interval |119905 minus 120591| le 120572 with 120572 gt 0 Then119866 = 120601

120583 |120583 minus 120583

0| le 120575

1 is equicontinuous on |119905 minus 120591| le 120572

Because 120601120583(120591) = 120585 119866 is uniformly bounded Hence for all

sequence of119866with120583 rarr 1205830 there exists a subsequencewhich

converges uniformly That is to say 120601120583(119896)

rarr 120595 uniformly on|119905 minus 120591| le 120572 as 120583(119896) rarr 120583

0 Since we have

120601120583(119896)

(119905) = 120585 + int119905

120591

119891 (119904 120601120583(119896)

120583 (119896)) d119904 (31)

or

120601120583(119896)

(119905) = 120585 + (minus1) sdot int119905

120591

119891 (119904 120601120583(119896)

120583 (119896)) d119904 (32)

byTheorem 16 we get

120595 (119905) = 120585 + int119905

120591

119891 (119904 120595 (119904) 1205830) d119904 (33)

Journal of Applied Mathematics 7

or

120595 (119905) = 120585 + (minus1) sdot int119905

120591

119891 (119904 120595 (119904) 1205830) d119904 (34)

Thus 120595 is a solution of (29) for 120583 = 1205830 Hence 120595 = 120601

0on

|119905 minus 120591| le 120572 because the solution 1206010is unique Consequently

120601119906(119896)

rarr 1206010uniformly on |119905 minus 120591| le 120572 as 120583(119896) rarr 120583

0 By

Reductio ad absurdum 120601120583rarr 120601

0uniformly there as 120583 rarr

1205830Secondly we will extend the result over [119886 119887] We con-

sider the case [120591 119887] Assume that there exists 1199050isin [120591 119887) such

that the result is valid over [120591 1199050minus ℎ] but not [120591 119905

0+ ℎ]

Obviously 1199050ge 120591 + 120572 Let 120574 120573 be such that

119867 = (119905 119909) 1003816100381610038161003816119905 minus 1199050

1003816100381610038161003816 le 120572119863 (119909 1206010(119905 minus 120574) le 120573) (35)

is contained in 119880 Since 119865120583119906(119905) is equicontinuous in 119906 and

near 1205830 we may choose 120574 gt 0 small enough such that

119863(int119905

1199050

119891 (119904 119906 (119904) 120583) d119904 0) lt120573

3(36)

for all step functions 119906whenever |119905minus 1199050| le 120574 and |120583minus120583

0| lt 120575

2

Since 120601120583rarr 120601

0uniformly on [120591 119905

0minus ℎ] there exists a 120575 such

that

119863(120601120583(1199050minus 120574) 120601

0(1199050minus 120574)) lt

120573

3(37)

for |120583 minus 1205830| lt 120575 Hence we have

119863(int119905

1199050minus120574

119891 (119904 119906 (119904) 120583) d119904 120601119898119906(1199050minus 120574) minus

1198671206010(1199050minus 120574))

le 119863(int1199050

1199050minus120574

119891 (119904 119906 (119904) 120583) d119904 0)

+ 119863(int119905

1199050

119891 (119904 119906 (119904) 120583) d119904 0)

+ 119863 (120601119898119906(1199050minus 120574) 120601

0(1199050minus 120574)) lt 3 sdot

120573

3

(38)

Thus for each 119906 with |120583 minus 1205830| lt 120575 a solution of (29) with

119909(1199050minus 120574) = 120601

120583(1199050minus 120574) exists on |119905 minus 119905

0| le 120574 Hence 120601

120583can be

continued to 1199050+ 120574 So in the case [120591 minus 120572 120591 + 120572] 120601

120583rarr 120601

0

uniformly on [120591 1199050+ 120574] It leads to a contradiction similarly

for the cases 1199050= 119887 and [119886 120591] Therefore the theorem holds

over [119886 119887]

Example 25 Let

(119905) = 119860 sdot 119905

2 sin 119905minus2 119905 = 0

0 119905 = 0(39)

where fuzzy number 119860 is defined in Example 1 Let ℎ(119905) =1015840(119905) We define

ℎ120583(119905) =

ℎ (119905) 120583 isin [0 1) 119905 isin(minus2 2)

(minus120583 120583)

0 otherwise(40)

and ℎ120583(119905) = ℎ

minus120583(119905) for 120583 isin (minus1 0] Let 119891(119905 119909 120583) = 1199052119909 + ℎ

120583(119905)

defined on (minus2 2) times (minus1 1) Then we have

119863(119865120583119906(1199051) 119865

120583119906(1199052)) = 119863(int

1199052

1199051

119891 (119904 119906 (119904) 120583) d119904 0)

le int1199052

1199051

1199042d119904 + 119863(int

1199052

1199051

ℎ120583d119904 0)

119863(int1199052

1199051

ℎ120583d119904 0) le int

1199052

1199051

119863(ℎ (119904) d119904 0)

(41)

where 1199051 1199052isin (0 1] or 119905

1 1199052isin [minus1 0) and

119863(int119905

0

ℎ120583(119904) d119904 0)

=

119863(119860 sdot 1199052 sin 119905minus2 119860 sdot 1205832 sin 120583minus2) 120583 = 0

119863 (119860 sdot 1199052 sin 119905minus2 0) 120583 = 0

(42)

Note that ℎ is Kaleva integrable on every subinterval of [minus1 0)and (0 1] Therefore we have that 119865

120583119906(119905) is equicontinuous

on [minus1 1] in 119906 and near 1205830= 0 Furthermore 119865

120583119906(119905) is

119860119862120575(119883

119899) uniformly in 120583 and 119906 where 119883

119899= [1119899 1] 119899 =

1 2 On the other hand by using integrating factor for119905 isin [minus1 1] we have

120601120583(119905) = 119890

1199052

3

int119905

0

119890minus1199043

3

119860 sdot ℎ120583(119904) d119904 (43)

with 120601120583(0) = 0 Obviously 120601

0is uniqueThus byTheorem 24

120601120583rarr 120601

0uniformly on [minus1 1]

5 Conclusion

In this paper we give the definition of the119880119860119862119866120575for a fuzzy-

number-valued function and the nonabsolute fuzzy integraland its controlled convergence theorem In addition we dealwith the Cauchy problem and the continuous dependence ofa solution on parameters of discontinuous fuzzy differentialequations involving the strong fuzzy Henstock integral infuzzy number space The function governing the equationsis supposed to be discontinuous with respect to some vari-ables and satisfy nonabsolute fuzzy integrability Our resultimproves the result given in [1 11 19 20] (where uniformcontinuity was required) as well as those referred therein

Acknowledgments

The authors are very grateful to the anonymous refereesand Professor Mehmet Sezer for many valuable commentsand suggestions which helped to improve the presentationof the paper The authors would like to thank the NationalNatural Science Foundation of China (no 11161041) and theFundamental Research Fund for the Central Universities (no31920130010)

8 Journal of Applied Mathematics

References

[1] O Kaleva ldquoFuzzy differential equationsrdquo Fuzzy Sets and Sys-tems vol 24 no 3 pp 301ndash317 1987

[2] Z Gong and Y Shao ldquoGlobal existence and uniqueness ofsolutions for fuzzy differential equations under dissipative-typeconditionsrdquo Computers amp Mathematics with Applications vol56 no 10 pp 2716ndash2723 2008

[3] O Kaleva ldquoThe Cauchy problem for fuzzy differential equa-tionsrdquo Fuzzy Sets and Systems vol 35 no 3 pp 389ndash396 1990

[4] J J Nieto ldquoThe Cauchy problem for continuous fuzzy differen-tial equationsrdquo Fuzzy Sets and Systems vol 102 no 2 pp 259ndash262 1999

[5] M L Puri and D A Ralescu ldquoDifferentials of fuzzy functionsrdquoJournal of Mathematical Analysis and Applications vol 91 no 2pp 552ndash558 1983

[6] S Seikkala ldquoOn the fuzzy initial value problemrdquo Fuzzy Sets andSystems vol 24 no 3 pp 319ndash330 1987

[7] C Wu and S Song ldquoExistence theorem to the Cauchy problemof fuzzy differential equations under compactness-type condi-tionsrdquo Information Sciences vol 108 no 1ndash4 pp 123ndash134 1998

[8] C Wu S Song and Z Qi ldquoExistence and uniqueness for asolution on the closed subset to the Cauchy problem of fuzzydiffrential equationsrdquo Journal of Harbin Institute of Technologyvol 2 pp 1ndash7 1997

[9] CWu S Song and E S Lee ldquoApproximate solutions existenceand uniqueness of the Cauchy problem of fuzzy differentialequationsrdquo Journal of Mathematical Analysis and Applicationsvol 202 no 2 pp 629ndash644 1996

[10] V Lakshmikantham and R N Mohapatra Theory of FuzzyDifferential Equations and Inclusions vol 6 Taylor amp FrancisLondon UK 2003

[11] X Xue and Y Fu ldquoCaratheodory solutions of fuzzy differentialequationsrdquo Fuzzy Sets and Systems vol 125 no 2 pp 239ndash2432002

[12] R HenstockTheory of Integration Butterworths London UK1963

[13] J Kurzweil ldquoGeneralized ordinary differential equations andcontinuous dependence on a parameterrdquo Czechoslovak Mathe-matical Journal vol 7 no 82 pp 418ndash449 1957

[14] P Y LeeLanzhou Lectures onHenstock Integration vol 2WorldScientific Publishing Teaneck NJ USA 1989

[15] C Wu and Z Gong ldquoOn Henstock integrals of interval-valuedfunctions and fuzzy-valued functionsrdquo Fuzzy Sets and Systemsvol 115 no 3 pp 377ndash391 2000

[16] C Wu and Z Gong ldquoOn Henstock integral of fuzzy-number-valued functions Irdquo Fuzzy Sets and Systems vol 120 no 3 pp523ndash532 2001

[17] Z Gong ldquoOn the problem of characterizing derivatives for thefuzzy-valued functions II Almost everywhere differentiabilityand strong Henstock integralrdquo Fuzzy Sets and Systems vol 145no 3 pp 381ndash393 2004

[18] Z Gong and Y Shao ldquoThe controlled convergence theoremsfor the strong Henstock integrals of fuzzy-number-valuedfunctionsrdquo Fuzzy Sets and Systems vol 160 no 11 pp 1528ndash1546 2009

[19] T S Chew and F Flordeliza ldquoOn 1199091015840 = 119891(119905 119909) and Henstock-Kurzweil integralsrdquo Differential and Integral Equations vol 4no 4 pp 861ndash868 1991

[20] B Bede and S G Gal ldquoGeneralizations of the differentiabilityof fuzzy-number-valued functions with applications to fuzzy

differential equationsrdquo Fuzzy Sets and Systems vol 151 no 3pp 581ndash599 2005

[21] D Dubois and H Prade ldquoTowards fuzzy differential calculus IIntegration of fuzzy mappingsrdquo Fuzzy Sets and Systems vol 8no 1 pp 1ndash17 1982

[22] P Diamond and P KloedenMetric Spaces of Fuzzy Sets WorldScientific Publishing River Edge NJ USA 1994

[23] R Goetschel Jr and W Voxman ldquoElementary fuzzy calculusrdquoFuzzy Sets and Systems vol 18 no 1 pp 31ndash43 1986

[24] Y Chalco-Cano and H Roman-Flores ldquoOn new solutions offuzzy differential equationsrdquo Chaos Solitons and Fractals vol38 no 1 pp 112ndash119 2008

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

119875 = 1198751cup119875

2so that119875

1contains the intervalswith the associated

points 120585 notin 119866 and 1198752otherwise

119863(119865119899119883

(119909 119910) 119865119898119883

(119909 119910))

= (1198752)sum119863(119865

119899119883(119906 V) 119865

119898119883(119906 V))

le sum119863(119865119899119883

(119906 V) 119891119899119883

(120585) (V minus 119906))

+sum119863(119865119898119883

(119906 V) 119891119898119883

(120585) (V minus 119906))

+sum119863(119891119898119883

(120585) (V minus 119906) 119891119898119883

(120585) (V minus 119906))

+sum119863(119865119899119883

(V) 119865119899119883

(119906)) +sum119863(119865119898119883

(V) 119865119898119883

(119906))

lt 120576 (4 + 119887 minus 119886)

(14)

Hence for any 120575-fine partial division 119875 of [119886 119887] we have

10038161003816100381610038161003816sum119863(119865

119899119883(119906 V) 119865

119898119883(119906 V))

10038161003816100381610038161003816lt 120576 (15)

for 119898 119899 ge 119873 Therefore the fuzzy sequence 119865119899119883 is gen-

eralized P-Cauchy on [119886 119887] Then by (i) we have that 119891119883

is strong fuzzy Henstock integrable on [119886 119887] with primitive119865119883

Definition 10 (a) A sequence 119865119899 of fuzzy-number-valued

function is uniformly119860119862nabla on119883whenever to each 120576 gt 0 thereexist 120578 gt 0 and 120575(119909) gt 0 such that

(1) sup119899119863(sum

119869119896isin1198751

119865119899(119869119896) sum

119871ℎisin1198752

119865119899(119871

ℎ)) lt 120576 for each 119875

1

1198752isin prod(119883 120575)

(2) with |(cup1198751)Δ(cap119875

2)| lt 120578

(b) A sequence 119865119899 of fuzzy-number-valued function is

uniformly 119860119862119866nabla on [119886 119887] if [119886 119887] = cup119894119883119894 where 119883

119894are

measurable sets and 119865 is uniformly 119860119862nabla on each119883119894

Theorem 11 If 119865 is uniformly 119860119862119866nabla then 119865 is uniformly119860119862119866lowast

120575

Proof Let [119886 119887] = cup119894119883119894be such that 119865 is uniformly 119860119862119866nabla

on each 119883119894 So for each 120576 gt 0 there exist 120578 gt 0 and 120575(119909) gt 0

such that (1) holds in Definition 10 for each 1198751 119875

2isin prod(119883 120575)

satisfying condition (2) We take 119875 = ([119888119896 119889

119896] 119909

119896)119901

119894=1with

sum119896|119889

119896minus 119888

119896| lt 120578 and put 119875

1= ([119888

119896 119889

119896] 119909

119896) 119865

119899(119862

119896 119889

119896ge

0) and 1198752= ([119888

119896 119889

119896] 119909

119896) 119865

119899(119888119896 119889

119896lt 0) So we have

|(cup1198751)Δ(cap119875

2)| = sum

119901

119896=1|119889

119896minus 119888

119896| lt 120578 Then by condition (1)

in Definition 10 we have

sup119899

119901

sum119896=1

119863(119865119899(119888119896) 119865

119899(119889

119896))

= sup119863( sum

(119888119896 119889119896)isin1198751

119865119899(119888119896 119889

119896)

sum(119888119896119889119896)isin1198752

119865119899(119888119896 119889

119896)) lt 120576

(16)

Hence we have that 119865 is uniformly 119860119862119866lowast

120575

We get the following theorem byTheorems 9 and 11

Theorem 12 Let the following conditions be satisfied

(i) 119891119899119883

rarr 119891119883ae in [119886 119887]where each119891

119899119883is strong fuzzy

Henstock integrable on [119886 119887](ii) the primitives 119865

119899119883of 119891

119899119883are119880119860119862nabla(119883)with closed set

119883 in [119886 119887]

Then 119891119883

is strong fuzzy Henstock integrable on [119886 119887] withprimitive 119865

119883

Next we give the controlled convergence theorem forthe strong fuzzy Henstock integrals by the definition of the119880119860119862119866

120575for a fuzzy-number-valued function

Definition 13 Let 119865 [119886 119887] rarr 119864119899 and let 119883 sub [119886 119887]A fuzzy-number-valued function 119865 is 119860119862

120575on 119883 if for each

120576 gt 0 there exist 120578 gt 0 and 120575(119909) gt 0 on 119883 such thatsum119873

119894=1119863(119865(119888

119894) 119865(119889

119894)) lt 120576 forsum119873

119894=1(119889

119894minus119888

119894) lt 120578 A fuzzy-number-

valued function 119865 is 119860119862119866120575on [119886 119887] if [119886 119887] is the union of a

sequence of set 119883119894 such that the function 119865 is 119860119862

120575(119883

119894) for

each 119894

Definition 14 (see [18]) A fuzzy-number-valued function 119865defined on 119883 sub [119886 119887] is said to be 119860119862lowast(119883) if for every 120576 gt 0

there exists 120578 gt 0 such that for every finite sequence of non-overlapping intervals [119886

119894 119887119894] satisfying sum119899

119894=1|119887119894minus 119886

119894| lt 120578

where 119886119894 119887119894isin 119883 for all 119894 one has sum120596(119865 [119886

119894 119887119894]) lt 120576 where 120596

denotes the oscillation of 119865 over [119886119894 119887119894] that is 120596(119865 [119886

119894 119887119894]) =

sup119863(119865(119910) 119865(119909)) 119909 119910 isin [119886119894 119887119894] A fuzzy-number-valued

function 119865 is said to be 119860119862119866lowast on 119883 if 119883 is the union ofa sequence of closed sets 119883

119894 such that on each 119883

119894 119865 is

119860119862lowast(119883119894)

Theorem 15 A fuzzy-number-valued function 119865 is 119860119862119866120575if

and only if it is 119860119862119866lowast on [119886 119887]

Theorem 16 (controlled convergence theorem) Let the fol-lowing conditions be satisfied

(1) 119891119899(119909) rarr 119891(119909) almost everywhere in [119886 119887] as 119899 rarr infin

where each 119891119899is strong fuzzy Henstock integrable on

[119886 119887]

Journal of Applied Mathematics 5

(2) the primitives 119865119899(119909) = (SFH) int119909

119886

119891119899(119904)d119909 of 119891

119899are

119880119860119862119866120575uniformly in 119899

(3) the sequence 119865119899(119909) converges uniformly to a con-

tinuous function on [119886 119887] Then 119891(119909) is strong fuzzyHenstock integrable on [119886 119887] and one has

lim119899rarrinfin

(SFH) int119887

119886

119891119899(119909) d119909 = (SFH) int

119887

119886

119891 (119909) d119909 (17)

If conditions (1) and (2) are replaced by condition (4)(4) 119892(119909) le 119891(119909) le ℎ(119909) almost everywhere on [119886 119887]

where 119892(119909) and ℎ(119909) are strong fuzzy Henstock inte-grable

Proof By condition (2) there exists a sequence 119883119894 such that

119865119899isin 119880119860119862

120575in 119883

119894a bounded closed set with bounds 119886 and

119887 and put 119883 = 119883119894 We note that 119865

119899(119909) rarr 119865(119909) we have

119865 isin 119860119862120575on119883 and hence119865 isin 119860119862119866

120575on [119886 119887] ByTheorem 15

119865 isin 119860119862lowast on 119883 and hence 119865 isin 119860119862119866lowast on [119886 119887] and also 119865 isin

119860119862 on119883 and hence 119865 isin 119860119862119866 on [119886 119887]Now we prove that 1198651015840(119909) = 119891(119909) ae on [119886 119887] In fact

let 119866 [119886 119887] rarr 119864119899 equal 119865119899on 119883 and extend 119866

119899linearly

to the closed interval contiguous to119883 Likewise we define 119866from 119865 We see that119866

119899and119866 are119880119860119862 on [119886 119887] By condition

(3) we have 119866119899rarr 119866 on [119886 119887] Let [119888

119896 119889

119896] be the intervals

contiguous to 119883 Then we have 119863(1198661015840

119899) le 119872

119896 We define a

fuzzy-number-valued function as follows

1198661015840

(119909) =119866119899(119889

119896) minus

119867119866119899(119888119896)

119889119896minus 119888

119896

119909 isin (119888119896 119889

119896) (18)

Consequently 1198661015840

119899(119909) converges on (119888

119896 119889

119896) Hence 1198661015840

119899con-

verges on [119886 119887] ae Since 119866119899 isin 119860119862 on [119886 119887] then 1198661015840

119899(119909) =

119891119899

rarr 119891 on 119883 Therefore we have 1198661015840

119899(119909) = 119892(119909) =

119891(119909) = 1198651015840(119909) ae on 119883 Thus 1198651015840(119909) = 119891(119909) ae on [119886 119887]

by Theorem 15 Therefore there exists an 119860119862119866120575function on

[a 119887] such that 1198651015840(119909) = 119891(119909) ae on [119886 119887] Hence 119891 is strongfuzzy Henstock integrable on [119886 119887] and we have

lim119899rarrinfin

(SFH) int119887

119886

119891119899(119909) d119909 = (SFH) int

119887

119886

119891 (119909) d119909 (19)

4 The Generalized Solutions of DiscontinuousFuzzy Differential Equations

In this section a generalized fuzzy differential equation ofform (4) is defined by using strong fuzzy Henstock integralThemain results of this section are existence theorems for thegeneralized solution to the discontinuous fuzzy differentialequation

Definition 17 (see [11]) Let 120591 and 120585 be fixed and let a fuzzy-number-valued function 119891(119905 119909) be a Caratheodory functiondefined on a rectangle119880 |119905minus120591| le 119886 119863(119909 120585) le 119887 that is119891 iscontinuous in 119909 for almost all 119905 and measurable in 119905 for eachfixed 119909

Theorem 18 Let a fuzzy-number-valued function 119891 be afunction as given in Definition 17 then there exist two strongfuzzy Henstock integrable functions ℎ and 119892 defined on |119905minus120591| le119886 such that 119892(119905) le 119891(119905 119909) le ℎ(119905) for all (119905 119909) isin 119880

Proof Note that 119891 is a Caratheodory function Thus thereexist two measurable functions 119906(119905) and V(119905) defined on |119905 minus120591| le 119886 with values in 119863(119909 120585) le 119887 such that 119891(119905 119906(119905)) le

119891(119905 119909) le 119891(119905 V(119905)) for all (119905 119909) isin 119880 Next we will showthat 119891(119905 119906(119905)) and 119891(119905 V(119905)) are fuzzy Henstock integrable byusing controlled convergence Theorem 16 First there existsa sequence 119896

119899(119905) of step functions defined on |119905 minus 120591| le

119886 with values in 119863(119909 120585) le 119887 such that 119896119899(119905) rarr 119906(119905)

almost everywhere as 119899 rarr infin Let 119865119899(119905) = int

119905

120591

119891(119904 119896119899(119904))d119904

Then 119865119899(119905) is 119880119860119862119866

120575uniformly in 119899 and equicontinuous

By controlled convergence Theorem 16 119891(119905 119906(119905)) is strongfuzzy Henstock integrable Similarly 119891(119905 V(119905)) is strong fuzzyHenstock integrable

Definition 19 A fuzzy-number-valued function 119909(119905) 119868 rarr

119864119899 is said to be a solution of the discontinuous fuzzy differ-ential equation (4) if 119909(119905) satisfies the following conditions

(i) 119909(119905) is 119860119862119866120575on each compact subinterval of 119868

(ii) (119905 119909) isin 119880 for 119905 isin 119868(iii) 1199091015840(119905) for almost everywhere 119905 isin 119868

Now we will state the existence theorem for the general-ized solution of discontinuous fuzzy differential equation (4)

Theorem 20 Suppose that 119891 satisfies the condition ofTheorem 18 then there exists a generalized solution Φ of thediscontinuous fuzzy differential equation (4) on some interval|119905 minus 120591| le 119886 which satisfies Φ(120591) = 120585

Proof Given 119892(t) le 119891(119905 119909) le ℎ(119905) for all 119909 and almost all 119905with (119905 119909) isin 119880 we get 0 le 119891(119905 119909)minus

119867119892(119905) le ℎ(119905)minus

119867119892(119905) Let

119865 (119905 119909) = 119891(119905 119909 + int119905

120591

119892 (119904) d119904) minus119867119892 (119905) (20)

or

119865 (119905 119909) = 119891(119905 119909 + (minus1) sdot int119905

120591

119892 (119904) d119904) minus119867119892 (119905) (21)

Then 119865 is a Caratheodory function Furthermore 0 le

119865(119905 119909) le ℎ(119905) minus119867119892(119905) for all (119905 119909) isin 119880

0 where 119880

0sub 119880

such that

119863(119909 + int119905

120591

119892 (119904) d119904 120585) le 119887 forall (119905 119909) isin 1198800 (22)

By Caratheodory existence theorem (see Theorem 7 in [11])there is a fuzzy-number-valued function 120595 on some interval|119905 minus 120591| le 119886 such that 1205951015840(119905) = 119865(119905 120595(119905)) almost everywhere inthis interval and 120595(120591) = 120585 Let

120601 (119905)=120595 (119905)+int119905

120591

119892 (119904) d119904 or 120601 (119905)=120595 (119905)+(minus1) sdot int119905

120591

119892 (119904) d119904

(23)

6 Journal of Applied Mathematics

Then for almost all 119905 we have the following

Case 1 Consider

1206011015840

(119905) = 1205951015840

(119905) + 119892 (119905) = 119865 (119905 120595 (119905)) + 119892 (119905)

= 119891(119905 120595 (119905) + int119905

120591

119892 (119904) d119904) minus119867119892 (119905) + 119892 (119905)

= 119891 (119905 120601 (119905))

120601 (119905) = 120595 (120591) + int119905

120591

119892 (119904) d119904 = 120585

(24)

Case 2 Consider

1206011015840

(119905) = 1205951015840

(119905) + 119892 (119905) = 119865 (119905 120595 (119905)) + 119892 (119905)

= 119891(119905 120595 (119905) + (minus1) sdot int119905

120591

119892 (119904) d119904) minus119867119892 (119905) + 119892 (119905)

= 119891 (119905 120601 (119905))

120601 (119905) = 120595 (120591) + (minus1) sdot int119905

120591

119892 (119904) d119904 = 120585

(25)

The proof is complete

Example 21 Consider fuzzy differential equation 1199091015840 =

119891(119905 119909) = 119892(119905 119909) + ℎ(119905) where 119863(119892(119905 119909) 0) le 119863(1198921(119905) 0) for

all |119905| le 1 119863(119909 0) le 1 and 1198921(119905) is Kaleva integrable on |119905| le 1

and ℎ(119905) = 119860 sdot (119889119889119905)(1199052 sin 119905minus2) if 119905 = 0 and ℎ(0) = 0 Here 119860is defined in Example 1 Note that ℎ is strong fuzzy Henstockintegrable but not Kaleva integrable and

ℎ (119905) minus1198671198921(119905) le 119891 (119905 119909) le ℎ (119905) + 119892

1(119905)

for |119905| le 1 119863 (119909 0) le 1

(26)

Thus by Theorem 20 there exists a solution of 1199091015840 = 119891(119905 119909)

with 119909(0) = 0 For instance if 119892(119905 119909) = 1199052119909 then

120601 (119905) = 1198901199053

3

sdot int119905

0

119890minus1199043

3

ℎ (119904) d119904 (27)

is a solution by using integrating factor

We get the following existence theorem by Theorems 18and 20

Theorem 22 Let a fuzzy-number-valued function 119891 be aCaratheodory function defined on a rectangle 119880 |119905 minus

120591| le 119886119863(119909 120585) le 119887 Let 119891(119905 119906(119905)) be strong fuzzy Henstockintegrable on |119905 minus 120591| le 119886 for any step function 119906(119905) definedon |119905 minus 120591| le 119886 with values in 119863(119909 120585) le 119887 Denote that119865119906(119905) = int

119905

120591

119891(119904 119906(119904))d119904 If 119865119906 119906 is a step function is 119880119860119862119866

120575

uniformly in 119906 and equicontinuous on |119905 minus 120591| le 119886 then thereexists a solution 120601 of 1199091015840 = 119891(119905 119909) on some interval |119905 minus 120591| le 120573with 120601(120591) = 120585

Finally in this paper we will show the continuousdependence of a solution on parameters by using Theorems16 18 and 22

Let 119880119901be a connected set in 119880 Let 119888 gt 0 and let 120583

0be

fixed119868120583= 120583

1003816100381610038161003816120583 minus 12058301003816100381610038161003816 lt 119888

119880120583= (119905 119909 120583) (119905 119909) isin 119880

119901 120583 isin 119868

120583

(28)

Let 119891(119905 119909 120583) be a fuzzy-number-valued function definedon 119880

120583such that for each fixed 120583 the function 119891 is a

Caratheodory function defined on 119880119901for each fixed 119905 and

continuous at (119909 1205830) for every 119909 For 120583 = 120583

0 let

1199091015840

(119905) = 119891 (119905 119909 120583)

119909 (120591) = 120585 isin 119864119899

(29)

have a solution 1206010on [119886 119887] where 120591 isin [119886 119887] Let 119891(119905 119906(119905) 120583)

be strong fuzzy Henstock integrable on [119886 119887] for any stepfunction 119906(119905) with (119905 119906(119905)) isin 119880

119901for 119905 isin [119886 119887]

Definition 23 Denote

119865120583119906(119905) = int

119905

119886

119891 (119904 119906 (119904) 120583) d119904 (30)

The family 119865120583119906 is said to be equicontinuous in 119906 and near

1205830if for each 119905

0isin [119886 119887] there exists an interval |120583 minus 120583

0| lt 119903

0

such that the family 119865120583119906(119905) 119906 and 120583 with |120583 minus 120583

0| lt 119903

0 is

equicontinuous at 1199050

Theorem 24 Let 119891 be a fuzzy-number-valued function asgiven above If the primitive 119865

120583119906(119905) of 119891(119905 119909 120583) is 119860119862119866

120575

uniformly in119906 and120583 and equicontinuous in119906 and near1205830 then

there exists 120575 gt 0 such that for any fixed 120583 with |120583 minus 1205830| lt 120575

a solution 120601120583of discontinuous fuzzy differential equation (29)

exists over [119886 119887] and as 120583 rarr 1205830 120601

120583rarr 120601

0uniformly over

[119886 119887]

Proof Firstly we will consider the case 120591 isin (119886 119887) ByTheorem 22 and the equicontinuity of 119865

120583119906(119905) all solutions

120601120583of problem (29) with |120583 minus 120583

0| le 120575

1for some 120575

1gt 0

exist over some interval |119905 minus 120591| le 120572 with 120572 gt 0 Then119866 = 120601

120583 |120583 minus 120583

0| le 120575

1 is equicontinuous on |119905 minus 120591| le 120572

Because 120601120583(120591) = 120585 119866 is uniformly bounded Hence for all

sequence of119866with120583 rarr 1205830 there exists a subsequencewhich

converges uniformly That is to say 120601120583(119896)

rarr 120595 uniformly on|119905 minus 120591| le 120572 as 120583(119896) rarr 120583

0 Since we have

120601120583(119896)

(119905) = 120585 + int119905

120591

119891 (119904 120601120583(119896)

120583 (119896)) d119904 (31)

or

120601120583(119896)

(119905) = 120585 + (minus1) sdot int119905

120591

119891 (119904 120601120583(119896)

120583 (119896)) d119904 (32)

byTheorem 16 we get

120595 (119905) = 120585 + int119905

120591

119891 (119904 120595 (119904) 1205830) d119904 (33)

Journal of Applied Mathematics 7

or

120595 (119905) = 120585 + (minus1) sdot int119905

120591

119891 (119904 120595 (119904) 1205830) d119904 (34)

Thus 120595 is a solution of (29) for 120583 = 1205830 Hence 120595 = 120601

0on

|119905 minus 120591| le 120572 because the solution 1206010is unique Consequently

120601119906(119896)

rarr 1206010uniformly on |119905 minus 120591| le 120572 as 120583(119896) rarr 120583

0 By

Reductio ad absurdum 120601120583rarr 120601

0uniformly there as 120583 rarr

1205830Secondly we will extend the result over [119886 119887] We con-

sider the case [120591 119887] Assume that there exists 1199050isin [120591 119887) such

that the result is valid over [120591 1199050minus ℎ] but not [120591 119905

0+ ℎ]

Obviously 1199050ge 120591 + 120572 Let 120574 120573 be such that

119867 = (119905 119909) 1003816100381610038161003816119905 minus 1199050

1003816100381610038161003816 le 120572119863 (119909 1206010(119905 minus 120574) le 120573) (35)

is contained in 119880 Since 119865120583119906(119905) is equicontinuous in 119906 and

near 1205830 we may choose 120574 gt 0 small enough such that

119863(int119905

1199050

119891 (119904 119906 (119904) 120583) d119904 0) lt120573

3(36)

for all step functions 119906whenever |119905minus 1199050| le 120574 and |120583minus120583

0| lt 120575

2

Since 120601120583rarr 120601

0uniformly on [120591 119905

0minus ℎ] there exists a 120575 such

that

119863(120601120583(1199050minus 120574) 120601

0(1199050minus 120574)) lt

120573

3(37)

for |120583 minus 1205830| lt 120575 Hence we have

119863(int119905

1199050minus120574

119891 (119904 119906 (119904) 120583) d119904 120601119898119906(1199050minus 120574) minus

1198671206010(1199050minus 120574))

le 119863(int1199050

1199050minus120574

119891 (119904 119906 (119904) 120583) d119904 0)

+ 119863(int119905

1199050

119891 (119904 119906 (119904) 120583) d119904 0)

+ 119863 (120601119898119906(1199050minus 120574) 120601

0(1199050minus 120574)) lt 3 sdot

120573

3

(38)

Thus for each 119906 with |120583 minus 1205830| lt 120575 a solution of (29) with

119909(1199050minus 120574) = 120601

120583(1199050minus 120574) exists on |119905 minus 119905

0| le 120574 Hence 120601

120583can be

continued to 1199050+ 120574 So in the case [120591 minus 120572 120591 + 120572] 120601

120583rarr 120601

0

uniformly on [120591 1199050+ 120574] It leads to a contradiction similarly

for the cases 1199050= 119887 and [119886 120591] Therefore the theorem holds

over [119886 119887]

Example 25 Let

(119905) = 119860 sdot 119905

2 sin 119905minus2 119905 = 0

0 119905 = 0(39)

where fuzzy number 119860 is defined in Example 1 Let ℎ(119905) =1015840(119905) We define

ℎ120583(119905) =

ℎ (119905) 120583 isin [0 1) 119905 isin(minus2 2)

(minus120583 120583)

0 otherwise(40)

and ℎ120583(119905) = ℎ

minus120583(119905) for 120583 isin (minus1 0] Let 119891(119905 119909 120583) = 1199052119909 + ℎ

120583(119905)

defined on (minus2 2) times (minus1 1) Then we have

119863(119865120583119906(1199051) 119865

120583119906(1199052)) = 119863(int

1199052

1199051

119891 (119904 119906 (119904) 120583) d119904 0)

le int1199052

1199051

1199042d119904 + 119863(int

1199052

1199051

ℎ120583d119904 0)

119863(int1199052

1199051

ℎ120583d119904 0) le int

1199052

1199051

119863(ℎ (119904) d119904 0)

(41)

where 1199051 1199052isin (0 1] or 119905

1 1199052isin [minus1 0) and

119863(int119905

0

ℎ120583(119904) d119904 0)

=

119863(119860 sdot 1199052 sin 119905minus2 119860 sdot 1205832 sin 120583minus2) 120583 = 0

119863 (119860 sdot 1199052 sin 119905minus2 0) 120583 = 0

(42)

Note that ℎ is Kaleva integrable on every subinterval of [minus1 0)and (0 1] Therefore we have that 119865

120583119906(119905) is equicontinuous

on [minus1 1] in 119906 and near 1205830= 0 Furthermore 119865

120583119906(119905) is

119860119862120575(119883

119899) uniformly in 120583 and 119906 where 119883

119899= [1119899 1] 119899 =

1 2 On the other hand by using integrating factor for119905 isin [minus1 1] we have

120601120583(119905) = 119890

1199052

3

int119905

0

119890minus1199043

3

119860 sdot ℎ120583(119904) d119904 (43)

with 120601120583(0) = 0 Obviously 120601

0is uniqueThus byTheorem 24

120601120583rarr 120601

0uniformly on [minus1 1]

5 Conclusion

In this paper we give the definition of the119880119860119862119866120575for a fuzzy-

number-valued function and the nonabsolute fuzzy integraland its controlled convergence theorem In addition we dealwith the Cauchy problem and the continuous dependence ofa solution on parameters of discontinuous fuzzy differentialequations involving the strong fuzzy Henstock integral infuzzy number space The function governing the equationsis supposed to be discontinuous with respect to some vari-ables and satisfy nonabsolute fuzzy integrability Our resultimproves the result given in [1 11 19 20] (where uniformcontinuity was required) as well as those referred therein

Acknowledgments

The authors are very grateful to the anonymous refereesand Professor Mehmet Sezer for many valuable commentsand suggestions which helped to improve the presentationof the paper The authors would like to thank the NationalNatural Science Foundation of China (no 11161041) and theFundamental Research Fund for the Central Universities (no31920130010)

8 Journal of Applied Mathematics

References

[1] O Kaleva ldquoFuzzy differential equationsrdquo Fuzzy Sets and Sys-tems vol 24 no 3 pp 301ndash317 1987

[2] Z Gong and Y Shao ldquoGlobal existence and uniqueness ofsolutions for fuzzy differential equations under dissipative-typeconditionsrdquo Computers amp Mathematics with Applications vol56 no 10 pp 2716ndash2723 2008

[3] O Kaleva ldquoThe Cauchy problem for fuzzy differential equa-tionsrdquo Fuzzy Sets and Systems vol 35 no 3 pp 389ndash396 1990

[4] J J Nieto ldquoThe Cauchy problem for continuous fuzzy differen-tial equationsrdquo Fuzzy Sets and Systems vol 102 no 2 pp 259ndash262 1999

[5] M L Puri and D A Ralescu ldquoDifferentials of fuzzy functionsrdquoJournal of Mathematical Analysis and Applications vol 91 no 2pp 552ndash558 1983

[6] S Seikkala ldquoOn the fuzzy initial value problemrdquo Fuzzy Sets andSystems vol 24 no 3 pp 319ndash330 1987

[7] C Wu and S Song ldquoExistence theorem to the Cauchy problemof fuzzy differential equations under compactness-type condi-tionsrdquo Information Sciences vol 108 no 1ndash4 pp 123ndash134 1998

[8] C Wu S Song and Z Qi ldquoExistence and uniqueness for asolution on the closed subset to the Cauchy problem of fuzzydiffrential equationsrdquo Journal of Harbin Institute of Technologyvol 2 pp 1ndash7 1997

[9] CWu S Song and E S Lee ldquoApproximate solutions existenceand uniqueness of the Cauchy problem of fuzzy differentialequationsrdquo Journal of Mathematical Analysis and Applicationsvol 202 no 2 pp 629ndash644 1996

[10] V Lakshmikantham and R N Mohapatra Theory of FuzzyDifferential Equations and Inclusions vol 6 Taylor amp FrancisLondon UK 2003

[11] X Xue and Y Fu ldquoCaratheodory solutions of fuzzy differentialequationsrdquo Fuzzy Sets and Systems vol 125 no 2 pp 239ndash2432002

[12] R HenstockTheory of Integration Butterworths London UK1963

[13] J Kurzweil ldquoGeneralized ordinary differential equations andcontinuous dependence on a parameterrdquo Czechoslovak Mathe-matical Journal vol 7 no 82 pp 418ndash449 1957

[14] P Y LeeLanzhou Lectures onHenstock Integration vol 2WorldScientific Publishing Teaneck NJ USA 1989

[15] C Wu and Z Gong ldquoOn Henstock integrals of interval-valuedfunctions and fuzzy-valued functionsrdquo Fuzzy Sets and Systemsvol 115 no 3 pp 377ndash391 2000

[16] C Wu and Z Gong ldquoOn Henstock integral of fuzzy-number-valued functions Irdquo Fuzzy Sets and Systems vol 120 no 3 pp523ndash532 2001

[17] Z Gong ldquoOn the problem of characterizing derivatives for thefuzzy-valued functions II Almost everywhere differentiabilityand strong Henstock integralrdquo Fuzzy Sets and Systems vol 145no 3 pp 381ndash393 2004

[18] Z Gong and Y Shao ldquoThe controlled convergence theoremsfor the strong Henstock integrals of fuzzy-number-valuedfunctionsrdquo Fuzzy Sets and Systems vol 160 no 11 pp 1528ndash1546 2009

[19] T S Chew and F Flordeliza ldquoOn 1199091015840 = 119891(119905 119909) and Henstock-Kurzweil integralsrdquo Differential and Integral Equations vol 4no 4 pp 861ndash868 1991

[20] B Bede and S G Gal ldquoGeneralizations of the differentiabilityof fuzzy-number-valued functions with applications to fuzzy

differential equationsrdquo Fuzzy Sets and Systems vol 151 no 3pp 581ndash599 2005

[21] D Dubois and H Prade ldquoTowards fuzzy differential calculus IIntegration of fuzzy mappingsrdquo Fuzzy Sets and Systems vol 8no 1 pp 1ndash17 1982

[22] P Diamond and P KloedenMetric Spaces of Fuzzy Sets WorldScientific Publishing River Edge NJ USA 1994

[23] R Goetschel Jr and W Voxman ldquoElementary fuzzy calculusrdquoFuzzy Sets and Systems vol 18 no 1 pp 31ndash43 1986

[24] Y Chalco-Cano and H Roman-Flores ldquoOn new solutions offuzzy differential equationsrdquo Chaos Solitons and Fractals vol38 no 1 pp 112ndash119 2008

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

(2) the primitives 119865119899(119909) = (SFH) int119909

119886

119891119899(119904)d119909 of 119891

119899are

119880119860119862119866120575uniformly in 119899

(3) the sequence 119865119899(119909) converges uniformly to a con-

tinuous function on [119886 119887] Then 119891(119909) is strong fuzzyHenstock integrable on [119886 119887] and one has

lim119899rarrinfin

(SFH) int119887

119886

119891119899(119909) d119909 = (SFH) int

119887

119886

119891 (119909) d119909 (17)

If conditions (1) and (2) are replaced by condition (4)(4) 119892(119909) le 119891(119909) le ℎ(119909) almost everywhere on [119886 119887]

where 119892(119909) and ℎ(119909) are strong fuzzy Henstock inte-grable

Proof By condition (2) there exists a sequence 119883119894 such that

119865119899isin 119880119860119862

120575in 119883

119894a bounded closed set with bounds 119886 and

119887 and put 119883 = 119883119894 We note that 119865

119899(119909) rarr 119865(119909) we have

119865 isin 119860119862120575on119883 and hence119865 isin 119860119862119866

120575on [119886 119887] ByTheorem 15

119865 isin 119860119862lowast on 119883 and hence 119865 isin 119860119862119866lowast on [119886 119887] and also 119865 isin

119860119862 on119883 and hence 119865 isin 119860119862119866 on [119886 119887]Now we prove that 1198651015840(119909) = 119891(119909) ae on [119886 119887] In fact

let 119866 [119886 119887] rarr 119864119899 equal 119865119899on 119883 and extend 119866

119899linearly

to the closed interval contiguous to119883 Likewise we define 119866from 119865 We see that119866

119899and119866 are119880119860119862 on [119886 119887] By condition

(3) we have 119866119899rarr 119866 on [119886 119887] Let [119888

119896 119889

119896] be the intervals

contiguous to 119883 Then we have 119863(1198661015840

119899) le 119872

119896 We define a

fuzzy-number-valued function as follows

1198661015840

(119909) =119866119899(119889

119896) minus

119867119866119899(119888119896)

119889119896minus 119888

119896

119909 isin (119888119896 119889

119896) (18)

Consequently 1198661015840

119899(119909) converges on (119888

119896 119889

119896) Hence 1198661015840

119899con-

verges on [119886 119887] ae Since 119866119899 isin 119860119862 on [119886 119887] then 1198661015840

119899(119909) =

119891119899

rarr 119891 on 119883 Therefore we have 1198661015840

119899(119909) = 119892(119909) =

119891(119909) = 1198651015840(119909) ae on 119883 Thus 1198651015840(119909) = 119891(119909) ae on [119886 119887]

by Theorem 15 Therefore there exists an 119860119862119866120575function on

[a 119887] such that 1198651015840(119909) = 119891(119909) ae on [119886 119887] Hence 119891 is strongfuzzy Henstock integrable on [119886 119887] and we have

lim119899rarrinfin

(SFH) int119887

119886

119891119899(119909) d119909 = (SFH) int

119887

119886

119891 (119909) d119909 (19)

4 The Generalized Solutions of DiscontinuousFuzzy Differential Equations

In this section a generalized fuzzy differential equation ofform (4) is defined by using strong fuzzy Henstock integralThemain results of this section are existence theorems for thegeneralized solution to the discontinuous fuzzy differentialequation

Definition 17 (see [11]) Let 120591 and 120585 be fixed and let a fuzzy-number-valued function 119891(119905 119909) be a Caratheodory functiondefined on a rectangle119880 |119905minus120591| le 119886 119863(119909 120585) le 119887 that is119891 iscontinuous in 119909 for almost all 119905 and measurable in 119905 for eachfixed 119909

Theorem 18 Let a fuzzy-number-valued function 119891 be afunction as given in Definition 17 then there exist two strongfuzzy Henstock integrable functions ℎ and 119892 defined on |119905minus120591| le119886 such that 119892(119905) le 119891(119905 119909) le ℎ(119905) for all (119905 119909) isin 119880

Proof Note that 119891 is a Caratheodory function Thus thereexist two measurable functions 119906(119905) and V(119905) defined on |119905 minus120591| le 119886 with values in 119863(119909 120585) le 119887 such that 119891(119905 119906(119905)) le

119891(119905 119909) le 119891(119905 V(119905)) for all (119905 119909) isin 119880 Next we will showthat 119891(119905 119906(119905)) and 119891(119905 V(119905)) are fuzzy Henstock integrable byusing controlled convergence Theorem 16 First there existsa sequence 119896

119899(119905) of step functions defined on |119905 minus 120591| le

119886 with values in 119863(119909 120585) le 119887 such that 119896119899(119905) rarr 119906(119905)

almost everywhere as 119899 rarr infin Let 119865119899(119905) = int

119905

120591

119891(119904 119896119899(119904))d119904

Then 119865119899(119905) is 119880119860119862119866

120575uniformly in 119899 and equicontinuous

By controlled convergence Theorem 16 119891(119905 119906(119905)) is strongfuzzy Henstock integrable Similarly 119891(119905 V(119905)) is strong fuzzyHenstock integrable

Definition 19 A fuzzy-number-valued function 119909(119905) 119868 rarr

119864119899 is said to be a solution of the discontinuous fuzzy differ-ential equation (4) if 119909(119905) satisfies the following conditions

(i) 119909(119905) is 119860119862119866120575on each compact subinterval of 119868

(ii) (119905 119909) isin 119880 for 119905 isin 119868(iii) 1199091015840(119905) for almost everywhere 119905 isin 119868

Now we will state the existence theorem for the general-ized solution of discontinuous fuzzy differential equation (4)

Theorem 20 Suppose that 119891 satisfies the condition ofTheorem 18 then there exists a generalized solution Φ of thediscontinuous fuzzy differential equation (4) on some interval|119905 minus 120591| le 119886 which satisfies Φ(120591) = 120585

Proof Given 119892(t) le 119891(119905 119909) le ℎ(119905) for all 119909 and almost all 119905with (119905 119909) isin 119880 we get 0 le 119891(119905 119909)minus

119867119892(119905) le ℎ(119905)minus

119867119892(119905) Let

119865 (119905 119909) = 119891(119905 119909 + int119905

120591

119892 (119904) d119904) minus119867119892 (119905) (20)

or

119865 (119905 119909) = 119891(119905 119909 + (minus1) sdot int119905

120591

119892 (119904) d119904) minus119867119892 (119905) (21)

Then 119865 is a Caratheodory function Furthermore 0 le

119865(119905 119909) le ℎ(119905) minus119867119892(119905) for all (119905 119909) isin 119880

0 where 119880

0sub 119880

such that

119863(119909 + int119905

120591

119892 (119904) d119904 120585) le 119887 forall (119905 119909) isin 1198800 (22)

By Caratheodory existence theorem (see Theorem 7 in [11])there is a fuzzy-number-valued function 120595 on some interval|119905 minus 120591| le 119886 such that 1205951015840(119905) = 119865(119905 120595(119905)) almost everywhere inthis interval and 120595(120591) = 120585 Let

120601 (119905)=120595 (119905)+int119905

120591

119892 (119904) d119904 or 120601 (119905)=120595 (119905)+(minus1) sdot int119905

120591

119892 (119904) d119904

(23)

6 Journal of Applied Mathematics

Then for almost all 119905 we have the following

Case 1 Consider

1206011015840

(119905) = 1205951015840

(119905) + 119892 (119905) = 119865 (119905 120595 (119905)) + 119892 (119905)

= 119891(119905 120595 (119905) + int119905

120591

119892 (119904) d119904) minus119867119892 (119905) + 119892 (119905)

= 119891 (119905 120601 (119905))

120601 (119905) = 120595 (120591) + int119905

120591

119892 (119904) d119904 = 120585

(24)

Case 2 Consider

1206011015840

(119905) = 1205951015840

(119905) + 119892 (119905) = 119865 (119905 120595 (119905)) + 119892 (119905)

= 119891(119905 120595 (119905) + (minus1) sdot int119905

120591

119892 (119904) d119904) minus119867119892 (119905) + 119892 (119905)

= 119891 (119905 120601 (119905))

120601 (119905) = 120595 (120591) + (minus1) sdot int119905

120591

119892 (119904) d119904 = 120585

(25)

The proof is complete

Example 21 Consider fuzzy differential equation 1199091015840 =

119891(119905 119909) = 119892(119905 119909) + ℎ(119905) where 119863(119892(119905 119909) 0) le 119863(1198921(119905) 0) for

all |119905| le 1 119863(119909 0) le 1 and 1198921(119905) is Kaleva integrable on |119905| le 1

and ℎ(119905) = 119860 sdot (119889119889119905)(1199052 sin 119905minus2) if 119905 = 0 and ℎ(0) = 0 Here 119860is defined in Example 1 Note that ℎ is strong fuzzy Henstockintegrable but not Kaleva integrable and

ℎ (119905) minus1198671198921(119905) le 119891 (119905 119909) le ℎ (119905) + 119892

1(119905)

for |119905| le 1 119863 (119909 0) le 1

(26)

Thus by Theorem 20 there exists a solution of 1199091015840 = 119891(119905 119909)

with 119909(0) = 0 For instance if 119892(119905 119909) = 1199052119909 then

120601 (119905) = 1198901199053

3

sdot int119905

0

119890minus1199043

3

ℎ (119904) d119904 (27)

is a solution by using integrating factor

We get the following existence theorem by Theorems 18and 20

Theorem 22 Let a fuzzy-number-valued function 119891 be aCaratheodory function defined on a rectangle 119880 |119905 minus

120591| le 119886119863(119909 120585) le 119887 Let 119891(119905 119906(119905)) be strong fuzzy Henstockintegrable on |119905 minus 120591| le 119886 for any step function 119906(119905) definedon |119905 minus 120591| le 119886 with values in 119863(119909 120585) le 119887 Denote that119865119906(119905) = int

119905

120591

119891(119904 119906(119904))d119904 If 119865119906 119906 is a step function is 119880119860119862119866

120575

uniformly in 119906 and equicontinuous on |119905 minus 120591| le 119886 then thereexists a solution 120601 of 1199091015840 = 119891(119905 119909) on some interval |119905 minus 120591| le 120573with 120601(120591) = 120585

Finally in this paper we will show the continuousdependence of a solution on parameters by using Theorems16 18 and 22

Let 119880119901be a connected set in 119880 Let 119888 gt 0 and let 120583

0be

fixed119868120583= 120583

1003816100381610038161003816120583 minus 12058301003816100381610038161003816 lt 119888

119880120583= (119905 119909 120583) (119905 119909) isin 119880

119901 120583 isin 119868

120583

(28)

Let 119891(119905 119909 120583) be a fuzzy-number-valued function definedon 119880

120583such that for each fixed 120583 the function 119891 is a

Caratheodory function defined on 119880119901for each fixed 119905 and

continuous at (119909 1205830) for every 119909 For 120583 = 120583

0 let

1199091015840

(119905) = 119891 (119905 119909 120583)

119909 (120591) = 120585 isin 119864119899

(29)

have a solution 1206010on [119886 119887] where 120591 isin [119886 119887] Let 119891(119905 119906(119905) 120583)

be strong fuzzy Henstock integrable on [119886 119887] for any stepfunction 119906(119905) with (119905 119906(119905)) isin 119880

119901for 119905 isin [119886 119887]

Definition 23 Denote

119865120583119906(119905) = int

119905

119886

119891 (119904 119906 (119904) 120583) d119904 (30)

The family 119865120583119906 is said to be equicontinuous in 119906 and near

1205830if for each 119905

0isin [119886 119887] there exists an interval |120583 minus 120583

0| lt 119903

0

such that the family 119865120583119906(119905) 119906 and 120583 with |120583 minus 120583

0| lt 119903

0 is

equicontinuous at 1199050

Theorem 24 Let 119891 be a fuzzy-number-valued function asgiven above If the primitive 119865

120583119906(119905) of 119891(119905 119909 120583) is 119860119862119866

120575

uniformly in119906 and120583 and equicontinuous in119906 and near1205830 then

there exists 120575 gt 0 such that for any fixed 120583 with |120583 minus 1205830| lt 120575

a solution 120601120583of discontinuous fuzzy differential equation (29)

exists over [119886 119887] and as 120583 rarr 1205830 120601

120583rarr 120601

0uniformly over

[119886 119887]

Proof Firstly we will consider the case 120591 isin (119886 119887) ByTheorem 22 and the equicontinuity of 119865

120583119906(119905) all solutions

120601120583of problem (29) with |120583 minus 120583

0| le 120575

1for some 120575

1gt 0

exist over some interval |119905 minus 120591| le 120572 with 120572 gt 0 Then119866 = 120601

120583 |120583 minus 120583

0| le 120575

1 is equicontinuous on |119905 minus 120591| le 120572

Because 120601120583(120591) = 120585 119866 is uniformly bounded Hence for all

sequence of119866with120583 rarr 1205830 there exists a subsequencewhich

converges uniformly That is to say 120601120583(119896)

rarr 120595 uniformly on|119905 minus 120591| le 120572 as 120583(119896) rarr 120583

0 Since we have

120601120583(119896)

(119905) = 120585 + int119905

120591

119891 (119904 120601120583(119896)

120583 (119896)) d119904 (31)

or

120601120583(119896)

(119905) = 120585 + (minus1) sdot int119905

120591

119891 (119904 120601120583(119896)

120583 (119896)) d119904 (32)

byTheorem 16 we get

120595 (119905) = 120585 + int119905

120591

119891 (119904 120595 (119904) 1205830) d119904 (33)

Journal of Applied Mathematics 7

or

120595 (119905) = 120585 + (minus1) sdot int119905

120591

119891 (119904 120595 (119904) 1205830) d119904 (34)

Thus 120595 is a solution of (29) for 120583 = 1205830 Hence 120595 = 120601

0on

|119905 minus 120591| le 120572 because the solution 1206010is unique Consequently

120601119906(119896)

rarr 1206010uniformly on |119905 minus 120591| le 120572 as 120583(119896) rarr 120583

0 By

Reductio ad absurdum 120601120583rarr 120601

0uniformly there as 120583 rarr

1205830Secondly we will extend the result over [119886 119887] We con-

sider the case [120591 119887] Assume that there exists 1199050isin [120591 119887) such

that the result is valid over [120591 1199050minus ℎ] but not [120591 119905

0+ ℎ]

Obviously 1199050ge 120591 + 120572 Let 120574 120573 be such that

119867 = (119905 119909) 1003816100381610038161003816119905 minus 1199050

1003816100381610038161003816 le 120572119863 (119909 1206010(119905 minus 120574) le 120573) (35)

is contained in 119880 Since 119865120583119906(119905) is equicontinuous in 119906 and

near 1205830 we may choose 120574 gt 0 small enough such that

119863(int119905

1199050

119891 (119904 119906 (119904) 120583) d119904 0) lt120573

3(36)

for all step functions 119906whenever |119905minus 1199050| le 120574 and |120583minus120583

0| lt 120575

2

Since 120601120583rarr 120601

0uniformly on [120591 119905

0minus ℎ] there exists a 120575 such

that

119863(120601120583(1199050minus 120574) 120601

0(1199050minus 120574)) lt

120573

3(37)

for |120583 minus 1205830| lt 120575 Hence we have

119863(int119905

1199050minus120574

119891 (119904 119906 (119904) 120583) d119904 120601119898119906(1199050minus 120574) minus

1198671206010(1199050minus 120574))

le 119863(int1199050

1199050minus120574

119891 (119904 119906 (119904) 120583) d119904 0)

+ 119863(int119905

1199050

119891 (119904 119906 (119904) 120583) d119904 0)

+ 119863 (120601119898119906(1199050minus 120574) 120601

0(1199050minus 120574)) lt 3 sdot

120573

3

(38)

Thus for each 119906 with |120583 minus 1205830| lt 120575 a solution of (29) with

119909(1199050minus 120574) = 120601

120583(1199050minus 120574) exists on |119905 minus 119905

0| le 120574 Hence 120601

120583can be

continued to 1199050+ 120574 So in the case [120591 minus 120572 120591 + 120572] 120601

120583rarr 120601

0

uniformly on [120591 1199050+ 120574] It leads to a contradiction similarly

for the cases 1199050= 119887 and [119886 120591] Therefore the theorem holds

over [119886 119887]

Example 25 Let

(119905) = 119860 sdot 119905

2 sin 119905minus2 119905 = 0

0 119905 = 0(39)

where fuzzy number 119860 is defined in Example 1 Let ℎ(119905) =1015840(119905) We define

ℎ120583(119905) =

ℎ (119905) 120583 isin [0 1) 119905 isin(minus2 2)

(minus120583 120583)

0 otherwise(40)

and ℎ120583(119905) = ℎ

minus120583(119905) for 120583 isin (minus1 0] Let 119891(119905 119909 120583) = 1199052119909 + ℎ

120583(119905)

defined on (minus2 2) times (minus1 1) Then we have

119863(119865120583119906(1199051) 119865

120583119906(1199052)) = 119863(int

1199052

1199051

119891 (119904 119906 (119904) 120583) d119904 0)

le int1199052

1199051

1199042d119904 + 119863(int

1199052

1199051

ℎ120583d119904 0)

119863(int1199052

1199051

ℎ120583d119904 0) le int

1199052

1199051

119863(ℎ (119904) d119904 0)

(41)

where 1199051 1199052isin (0 1] or 119905

1 1199052isin [minus1 0) and

119863(int119905

0

ℎ120583(119904) d119904 0)

=

119863(119860 sdot 1199052 sin 119905minus2 119860 sdot 1205832 sin 120583minus2) 120583 = 0

119863 (119860 sdot 1199052 sin 119905minus2 0) 120583 = 0

(42)

Note that ℎ is Kaleva integrable on every subinterval of [minus1 0)and (0 1] Therefore we have that 119865

120583119906(119905) is equicontinuous

on [minus1 1] in 119906 and near 1205830= 0 Furthermore 119865

120583119906(119905) is

119860119862120575(119883

119899) uniformly in 120583 and 119906 where 119883

119899= [1119899 1] 119899 =

1 2 On the other hand by using integrating factor for119905 isin [minus1 1] we have

120601120583(119905) = 119890

1199052

3

int119905

0

119890minus1199043

3

119860 sdot ℎ120583(119904) d119904 (43)

with 120601120583(0) = 0 Obviously 120601

0is uniqueThus byTheorem 24

120601120583rarr 120601

0uniformly on [minus1 1]

5 Conclusion

In this paper we give the definition of the119880119860119862119866120575for a fuzzy-

number-valued function and the nonabsolute fuzzy integraland its controlled convergence theorem In addition we dealwith the Cauchy problem and the continuous dependence ofa solution on parameters of discontinuous fuzzy differentialequations involving the strong fuzzy Henstock integral infuzzy number space The function governing the equationsis supposed to be discontinuous with respect to some vari-ables and satisfy nonabsolute fuzzy integrability Our resultimproves the result given in [1 11 19 20] (where uniformcontinuity was required) as well as those referred therein

Acknowledgments

The authors are very grateful to the anonymous refereesand Professor Mehmet Sezer for many valuable commentsand suggestions which helped to improve the presentationof the paper The authors would like to thank the NationalNatural Science Foundation of China (no 11161041) and theFundamental Research Fund for the Central Universities (no31920130010)

8 Journal of Applied Mathematics

References

[1] O Kaleva ldquoFuzzy differential equationsrdquo Fuzzy Sets and Sys-tems vol 24 no 3 pp 301ndash317 1987

[2] Z Gong and Y Shao ldquoGlobal existence and uniqueness ofsolutions for fuzzy differential equations under dissipative-typeconditionsrdquo Computers amp Mathematics with Applications vol56 no 10 pp 2716ndash2723 2008

[3] O Kaleva ldquoThe Cauchy problem for fuzzy differential equa-tionsrdquo Fuzzy Sets and Systems vol 35 no 3 pp 389ndash396 1990

[4] J J Nieto ldquoThe Cauchy problem for continuous fuzzy differen-tial equationsrdquo Fuzzy Sets and Systems vol 102 no 2 pp 259ndash262 1999

[5] M L Puri and D A Ralescu ldquoDifferentials of fuzzy functionsrdquoJournal of Mathematical Analysis and Applications vol 91 no 2pp 552ndash558 1983

[6] S Seikkala ldquoOn the fuzzy initial value problemrdquo Fuzzy Sets andSystems vol 24 no 3 pp 319ndash330 1987

[7] C Wu and S Song ldquoExistence theorem to the Cauchy problemof fuzzy differential equations under compactness-type condi-tionsrdquo Information Sciences vol 108 no 1ndash4 pp 123ndash134 1998

[8] C Wu S Song and Z Qi ldquoExistence and uniqueness for asolution on the closed subset to the Cauchy problem of fuzzydiffrential equationsrdquo Journal of Harbin Institute of Technologyvol 2 pp 1ndash7 1997

[9] CWu S Song and E S Lee ldquoApproximate solutions existenceand uniqueness of the Cauchy problem of fuzzy differentialequationsrdquo Journal of Mathematical Analysis and Applicationsvol 202 no 2 pp 629ndash644 1996

[10] V Lakshmikantham and R N Mohapatra Theory of FuzzyDifferential Equations and Inclusions vol 6 Taylor amp FrancisLondon UK 2003

[11] X Xue and Y Fu ldquoCaratheodory solutions of fuzzy differentialequationsrdquo Fuzzy Sets and Systems vol 125 no 2 pp 239ndash2432002

[12] R HenstockTheory of Integration Butterworths London UK1963

[13] J Kurzweil ldquoGeneralized ordinary differential equations andcontinuous dependence on a parameterrdquo Czechoslovak Mathe-matical Journal vol 7 no 82 pp 418ndash449 1957

[14] P Y LeeLanzhou Lectures onHenstock Integration vol 2WorldScientific Publishing Teaneck NJ USA 1989

[15] C Wu and Z Gong ldquoOn Henstock integrals of interval-valuedfunctions and fuzzy-valued functionsrdquo Fuzzy Sets and Systemsvol 115 no 3 pp 377ndash391 2000

[16] C Wu and Z Gong ldquoOn Henstock integral of fuzzy-number-valued functions Irdquo Fuzzy Sets and Systems vol 120 no 3 pp523ndash532 2001

[17] Z Gong ldquoOn the problem of characterizing derivatives for thefuzzy-valued functions II Almost everywhere differentiabilityand strong Henstock integralrdquo Fuzzy Sets and Systems vol 145no 3 pp 381ndash393 2004

[18] Z Gong and Y Shao ldquoThe controlled convergence theoremsfor the strong Henstock integrals of fuzzy-number-valuedfunctionsrdquo Fuzzy Sets and Systems vol 160 no 11 pp 1528ndash1546 2009

[19] T S Chew and F Flordeliza ldquoOn 1199091015840 = 119891(119905 119909) and Henstock-Kurzweil integralsrdquo Differential and Integral Equations vol 4no 4 pp 861ndash868 1991

[20] B Bede and S G Gal ldquoGeneralizations of the differentiabilityof fuzzy-number-valued functions with applications to fuzzy

differential equationsrdquo Fuzzy Sets and Systems vol 151 no 3pp 581ndash599 2005

[21] D Dubois and H Prade ldquoTowards fuzzy differential calculus IIntegration of fuzzy mappingsrdquo Fuzzy Sets and Systems vol 8no 1 pp 1ndash17 1982

[22] P Diamond and P KloedenMetric Spaces of Fuzzy Sets WorldScientific Publishing River Edge NJ USA 1994

[23] R Goetschel Jr and W Voxman ldquoElementary fuzzy calculusrdquoFuzzy Sets and Systems vol 18 no 1 pp 31ndash43 1986

[24] Y Chalco-Cano and H Roman-Flores ldquoOn new solutions offuzzy differential equationsrdquo Chaos Solitons and Fractals vol38 no 1 pp 112ndash119 2008

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

6 Journal of Applied Mathematics

Then for almost all 119905 we have the following

Case 1 Consider

1206011015840

(119905) = 1205951015840

(119905) + 119892 (119905) = 119865 (119905 120595 (119905)) + 119892 (119905)

= 119891(119905 120595 (119905) + int119905

120591

119892 (119904) d119904) minus119867119892 (119905) + 119892 (119905)

= 119891 (119905 120601 (119905))

120601 (119905) = 120595 (120591) + int119905

120591

119892 (119904) d119904 = 120585

(24)

Case 2 Consider

1206011015840

(119905) = 1205951015840

(119905) + 119892 (119905) = 119865 (119905 120595 (119905)) + 119892 (119905)

= 119891(119905 120595 (119905) + (minus1) sdot int119905

120591

119892 (119904) d119904) minus119867119892 (119905) + 119892 (119905)

= 119891 (119905 120601 (119905))

120601 (119905) = 120595 (120591) + (minus1) sdot int119905

120591

119892 (119904) d119904 = 120585

(25)

The proof is complete

Example 21 Consider fuzzy differential equation 1199091015840 =

119891(119905 119909) = 119892(119905 119909) + ℎ(119905) where 119863(119892(119905 119909) 0) le 119863(1198921(119905) 0) for

all |119905| le 1 119863(119909 0) le 1 and 1198921(119905) is Kaleva integrable on |119905| le 1

and ℎ(119905) = 119860 sdot (119889119889119905)(1199052 sin 119905minus2) if 119905 = 0 and ℎ(0) = 0 Here 119860is defined in Example 1 Note that ℎ is strong fuzzy Henstockintegrable but not Kaleva integrable and

ℎ (119905) minus1198671198921(119905) le 119891 (119905 119909) le ℎ (119905) + 119892

1(119905)

for |119905| le 1 119863 (119909 0) le 1

(26)

Thus by Theorem 20 there exists a solution of 1199091015840 = 119891(119905 119909)

with 119909(0) = 0 For instance if 119892(119905 119909) = 1199052119909 then

120601 (119905) = 1198901199053

3

sdot int119905

0

119890minus1199043

3

ℎ (119904) d119904 (27)

is a solution by using integrating factor

We get the following existence theorem by Theorems 18and 20

Theorem 22 Let a fuzzy-number-valued function 119891 be aCaratheodory function defined on a rectangle 119880 |119905 minus

120591| le 119886119863(119909 120585) le 119887 Let 119891(119905 119906(119905)) be strong fuzzy Henstockintegrable on |119905 minus 120591| le 119886 for any step function 119906(119905) definedon |119905 minus 120591| le 119886 with values in 119863(119909 120585) le 119887 Denote that119865119906(119905) = int

119905

120591

119891(119904 119906(119904))d119904 If 119865119906 119906 is a step function is 119880119860119862119866

120575

uniformly in 119906 and equicontinuous on |119905 minus 120591| le 119886 then thereexists a solution 120601 of 1199091015840 = 119891(119905 119909) on some interval |119905 minus 120591| le 120573with 120601(120591) = 120585

Finally in this paper we will show the continuousdependence of a solution on parameters by using Theorems16 18 and 22

Let 119880119901be a connected set in 119880 Let 119888 gt 0 and let 120583

0be

fixed119868120583= 120583

1003816100381610038161003816120583 minus 12058301003816100381610038161003816 lt 119888

119880120583= (119905 119909 120583) (119905 119909) isin 119880

119901 120583 isin 119868

120583

(28)

Let 119891(119905 119909 120583) be a fuzzy-number-valued function definedon 119880

120583such that for each fixed 120583 the function 119891 is a

Caratheodory function defined on 119880119901for each fixed 119905 and

continuous at (119909 1205830) for every 119909 For 120583 = 120583

0 let

1199091015840

(119905) = 119891 (119905 119909 120583)

119909 (120591) = 120585 isin 119864119899

(29)

have a solution 1206010on [119886 119887] where 120591 isin [119886 119887] Let 119891(119905 119906(119905) 120583)

be strong fuzzy Henstock integrable on [119886 119887] for any stepfunction 119906(119905) with (119905 119906(119905)) isin 119880

119901for 119905 isin [119886 119887]

Definition 23 Denote

119865120583119906(119905) = int

119905

119886

119891 (119904 119906 (119904) 120583) d119904 (30)

The family 119865120583119906 is said to be equicontinuous in 119906 and near

1205830if for each 119905

0isin [119886 119887] there exists an interval |120583 minus 120583

0| lt 119903

0

such that the family 119865120583119906(119905) 119906 and 120583 with |120583 minus 120583

0| lt 119903

0 is

equicontinuous at 1199050

Theorem 24 Let 119891 be a fuzzy-number-valued function asgiven above If the primitive 119865

120583119906(119905) of 119891(119905 119909 120583) is 119860119862119866

120575

uniformly in119906 and120583 and equicontinuous in119906 and near1205830 then

there exists 120575 gt 0 such that for any fixed 120583 with |120583 minus 1205830| lt 120575

a solution 120601120583of discontinuous fuzzy differential equation (29)

exists over [119886 119887] and as 120583 rarr 1205830 120601

120583rarr 120601

0uniformly over

[119886 119887]

Proof Firstly we will consider the case 120591 isin (119886 119887) ByTheorem 22 and the equicontinuity of 119865

120583119906(119905) all solutions

120601120583of problem (29) with |120583 minus 120583

0| le 120575

1for some 120575

1gt 0

exist over some interval |119905 minus 120591| le 120572 with 120572 gt 0 Then119866 = 120601

120583 |120583 minus 120583

0| le 120575

1 is equicontinuous on |119905 minus 120591| le 120572

Because 120601120583(120591) = 120585 119866 is uniformly bounded Hence for all

sequence of119866with120583 rarr 1205830 there exists a subsequencewhich

converges uniformly That is to say 120601120583(119896)

rarr 120595 uniformly on|119905 minus 120591| le 120572 as 120583(119896) rarr 120583

0 Since we have

120601120583(119896)

(119905) = 120585 + int119905

120591

119891 (119904 120601120583(119896)

120583 (119896)) d119904 (31)

or

120601120583(119896)

(119905) = 120585 + (minus1) sdot int119905

120591

119891 (119904 120601120583(119896)

120583 (119896)) d119904 (32)

byTheorem 16 we get

120595 (119905) = 120585 + int119905

120591

119891 (119904 120595 (119904) 1205830) d119904 (33)

Journal of Applied Mathematics 7

or

120595 (119905) = 120585 + (minus1) sdot int119905

120591

119891 (119904 120595 (119904) 1205830) d119904 (34)

Thus 120595 is a solution of (29) for 120583 = 1205830 Hence 120595 = 120601

0on

|119905 minus 120591| le 120572 because the solution 1206010is unique Consequently

120601119906(119896)

rarr 1206010uniformly on |119905 minus 120591| le 120572 as 120583(119896) rarr 120583

0 By

Reductio ad absurdum 120601120583rarr 120601

0uniformly there as 120583 rarr

1205830Secondly we will extend the result over [119886 119887] We con-

sider the case [120591 119887] Assume that there exists 1199050isin [120591 119887) such

that the result is valid over [120591 1199050minus ℎ] but not [120591 119905

0+ ℎ]

Obviously 1199050ge 120591 + 120572 Let 120574 120573 be such that

119867 = (119905 119909) 1003816100381610038161003816119905 minus 1199050

1003816100381610038161003816 le 120572119863 (119909 1206010(119905 minus 120574) le 120573) (35)

is contained in 119880 Since 119865120583119906(119905) is equicontinuous in 119906 and

near 1205830 we may choose 120574 gt 0 small enough such that

119863(int119905

1199050

119891 (119904 119906 (119904) 120583) d119904 0) lt120573

3(36)

for all step functions 119906whenever |119905minus 1199050| le 120574 and |120583minus120583

0| lt 120575

2

Since 120601120583rarr 120601

0uniformly on [120591 119905

0minus ℎ] there exists a 120575 such

that

119863(120601120583(1199050minus 120574) 120601

0(1199050minus 120574)) lt

120573

3(37)

for |120583 minus 1205830| lt 120575 Hence we have

119863(int119905

1199050minus120574

119891 (119904 119906 (119904) 120583) d119904 120601119898119906(1199050minus 120574) minus

1198671206010(1199050minus 120574))

le 119863(int1199050

1199050minus120574

119891 (119904 119906 (119904) 120583) d119904 0)

+ 119863(int119905

1199050

119891 (119904 119906 (119904) 120583) d119904 0)

+ 119863 (120601119898119906(1199050minus 120574) 120601

0(1199050minus 120574)) lt 3 sdot

120573

3

(38)

Thus for each 119906 with |120583 minus 1205830| lt 120575 a solution of (29) with

119909(1199050minus 120574) = 120601

120583(1199050minus 120574) exists on |119905 minus 119905

0| le 120574 Hence 120601

120583can be

continued to 1199050+ 120574 So in the case [120591 minus 120572 120591 + 120572] 120601

120583rarr 120601

0

uniformly on [120591 1199050+ 120574] It leads to a contradiction similarly

for the cases 1199050= 119887 and [119886 120591] Therefore the theorem holds

over [119886 119887]

Example 25 Let

(119905) = 119860 sdot 119905

2 sin 119905minus2 119905 = 0

0 119905 = 0(39)

where fuzzy number 119860 is defined in Example 1 Let ℎ(119905) =1015840(119905) We define

ℎ120583(119905) =

ℎ (119905) 120583 isin [0 1) 119905 isin(minus2 2)

(minus120583 120583)

0 otherwise(40)

and ℎ120583(119905) = ℎ

minus120583(119905) for 120583 isin (minus1 0] Let 119891(119905 119909 120583) = 1199052119909 + ℎ

120583(119905)

defined on (minus2 2) times (minus1 1) Then we have

119863(119865120583119906(1199051) 119865

120583119906(1199052)) = 119863(int

1199052

1199051

119891 (119904 119906 (119904) 120583) d119904 0)

le int1199052

1199051

1199042d119904 + 119863(int

1199052

1199051

ℎ120583d119904 0)

119863(int1199052

1199051

ℎ120583d119904 0) le int

1199052

1199051

119863(ℎ (119904) d119904 0)

(41)

where 1199051 1199052isin (0 1] or 119905

1 1199052isin [minus1 0) and

119863(int119905

0

ℎ120583(119904) d119904 0)

=

119863(119860 sdot 1199052 sin 119905minus2 119860 sdot 1205832 sin 120583minus2) 120583 = 0

119863 (119860 sdot 1199052 sin 119905minus2 0) 120583 = 0

(42)

Note that ℎ is Kaleva integrable on every subinterval of [minus1 0)and (0 1] Therefore we have that 119865

120583119906(119905) is equicontinuous

on [minus1 1] in 119906 and near 1205830= 0 Furthermore 119865

120583119906(119905) is

119860119862120575(119883

119899) uniformly in 120583 and 119906 where 119883

119899= [1119899 1] 119899 =

1 2 On the other hand by using integrating factor for119905 isin [minus1 1] we have

120601120583(119905) = 119890

1199052

3

int119905

0

119890minus1199043

3

119860 sdot ℎ120583(119904) d119904 (43)

with 120601120583(0) = 0 Obviously 120601

0is uniqueThus byTheorem 24

120601120583rarr 120601

0uniformly on [minus1 1]

5 Conclusion

In this paper we give the definition of the119880119860119862119866120575for a fuzzy-

number-valued function and the nonabsolute fuzzy integraland its controlled convergence theorem In addition we dealwith the Cauchy problem and the continuous dependence ofa solution on parameters of discontinuous fuzzy differentialequations involving the strong fuzzy Henstock integral infuzzy number space The function governing the equationsis supposed to be discontinuous with respect to some vari-ables and satisfy nonabsolute fuzzy integrability Our resultimproves the result given in [1 11 19 20] (where uniformcontinuity was required) as well as those referred therein

Acknowledgments

The authors are very grateful to the anonymous refereesand Professor Mehmet Sezer for many valuable commentsand suggestions which helped to improve the presentationof the paper The authors would like to thank the NationalNatural Science Foundation of China (no 11161041) and theFundamental Research Fund for the Central Universities (no31920130010)

8 Journal of Applied Mathematics

References

[1] O Kaleva ldquoFuzzy differential equationsrdquo Fuzzy Sets and Sys-tems vol 24 no 3 pp 301ndash317 1987

[2] Z Gong and Y Shao ldquoGlobal existence and uniqueness ofsolutions for fuzzy differential equations under dissipative-typeconditionsrdquo Computers amp Mathematics with Applications vol56 no 10 pp 2716ndash2723 2008

[3] O Kaleva ldquoThe Cauchy problem for fuzzy differential equa-tionsrdquo Fuzzy Sets and Systems vol 35 no 3 pp 389ndash396 1990

[4] J J Nieto ldquoThe Cauchy problem for continuous fuzzy differen-tial equationsrdquo Fuzzy Sets and Systems vol 102 no 2 pp 259ndash262 1999

[5] M L Puri and D A Ralescu ldquoDifferentials of fuzzy functionsrdquoJournal of Mathematical Analysis and Applications vol 91 no 2pp 552ndash558 1983

[6] S Seikkala ldquoOn the fuzzy initial value problemrdquo Fuzzy Sets andSystems vol 24 no 3 pp 319ndash330 1987

[7] C Wu and S Song ldquoExistence theorem to the Cauchy problemof fuzzy differential equations under compactness-type condi-tionsrdquo Information Sciences vol 108 no 1ndash4 pp 123ndash134 1998

[8] C Wu S Song and Z Qi ldquoExistence and uniqueness for asolution on the closed subset to the Cauchy problem of fuzzydiffrential equationsrdquo Journal of Harbin Institute of Technologyvol 2 pp 1ndash7 1997

[9] CWu S Song and E S Lee ldquoApproximate solutions existenceand uniqueness of the Cauchy problem of fuzzy differentialequationsrdquo Journal of Mathematical Analysis and Applicationsvol 202 no 2 pp 629ndash644 1996

[10] V Lakshmikantham and R N Mohapatra Theory of FuzzyDifferential Equations and Inclusions vol 6 Taylor amp FrancisLondon UK 2003

[11] X Xue and Y Fu ldquoCaratheodory solutions of fuzzy differentialequationsrdquo Fuzzy Sets and Systems vol 125 no 2 pp 239ndash2432002

[12] R HenstockTheory of Integration Butterworths London UK1963

[13] J Kurzweil ldquoGeneralized ordinary differential equations andcontinuous dependence on a parameterrdquo Czechoslovak Mathe-matical Journal vol 7 no 82 pp 418ndash449 1957

[14] P Y LeeLanzhou Lectures onHenstock Integration vol 2WorldScientific Publishing Teaneck NJ USA 1989

[15] C Wu and Z Gong ldquoOn Henstock integrals of interval-valuedfunctions and fuzzy-valued functionsrdquo Fuzzy Sets and Systemsvol 115 no 3 pp 377ndash391 2000

[16] C Wu and Z Gong ldquoOn Henstock integral of fuzzy-number-valued functions Irdquo Fuzzy Sets and Systems vol 120 no 3 pp523ndash532 2001

[17] Z Gong ldquoOn the problem of characterizing derivatives for thefuzzy-valued functions II Almost everywhere differentiabilityand strong Henstock integralrdquo Fuzzy Sets and Systems vol 145no 3 pp 381ndash393 2004

[18] Z Gong and Y Shao ldquoThe controlled convergence theoremsfor the strong Henstock integrals of fuzzy-number-valuedfunctionsrdquo Fuzzy Sets and Systems vol 160 no 11 pp 1528ndash1546 2009

[19] T S Chew and F Flordeliza ldquoOn 1199091015840 = 119891(119905 119909) and Henstock-Kurzweil integralsrdquo Differential and Integral Equations vol 4no 4 pp 861ndash868 1991

[20] B Bede and S G Gal ldquoGeneralizations of the differentiabilityof fuzzy-number-valued functions with applications to fuzzy

differential equationsrdquo Fuzzy Sets and Systems vol 151 no 3pp 581ndash599 2005

[21] D Dubois and H Prade ldquoTowards fuzzy differential calculus IIntegration of fuzzy mappingsrdquo Fuzzy Sets and Systems vol 8no 1 pp 1ndash17 1982

[22] P Diamond and P KloedenMetric Spaces of Fuzzy Sets WorldScientific Publishing River Edge NJ USA 1994

[23] R Goetschel Jr and W Voxman ldquoElementary fuzzy calculusrdquoFuzzy Sets and Systems vol 18 no 1 pp 31ndash43 1986

[24] Y Chalco-Cano and H Roman-Flores ldquoOn new solutions offuzzy differential equationsrdquo Chaos Solitons and Fractals vol38 no 1 pp 112ndash119 2008

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 7

or

120595 (119905) = 120585 + (minus1) sdot int119905

120591

119891 (119904 120595 (119904) 1205830) d119904 (34)

Thus 120595 is a solution of (29) for 120583 = 1205830 Hence 120595 = 120601

0on

|119905 minus 120591| le 120572 because the solution 1206010is unique Consequently

120601119906(119896)

rarr 1206010uniformly on |119905 minus 120591| le 120572 as 120583(119896) rarr 120583

0 By

Reductio ad absurdum 120601120583rarr 120601

0uniformly there as 120583 rarr

1205830Secondly we will extend the result over [119886 119887] We con-

sider the case [120591 119887] Assume that there exists 1199050isin [120591 119887) such

that the result is valid over [120591 1199050minus ℎ] but not [120591 119905

0+ ℎ]

Obviously 1199050ge 120591 + 120572 Let 120574 120573 be such that

119867 = (119905 119909) 1003816100381610038161003816119905 minus 1199050

1003816100381610038161003816 le 120572119863 (119909 1206010(119905 minus 120574) le 120573) (35)

is contained in 119880 Since 119865120583119906(119905) is equicontinuous in 119906 and

near 1205830 we may choose 120574 gt 0 small enough such that

119863(int119905

1199050

119891 (119904 119906 (119904) 120583) d119904 0) lt120573

3(36)

for all step functions 119906whenever |119905minus 1199050| le 120574 and |120583minus120583

0| lt 120575

2

Since 120601120583rarr 120601

0uniformly on [120591 119905

0minus ℎ] there exists a 120575 such

that

119863(120601120583(1199050minus 120574) 120601

0(1199050minus 120574)) lt

120573

3(37)

for |120583 minus 1205830| lt 120575 Hence we have

119863(int119905

1199050minus120574

119891 (119904 119906 (119904) 120583) d119904 120601119898119906(1199050minus 120574) minus

1198671206010(1199050minus 120574))

le 119863(int1199050

1199050minus120574

119891 (119904 119906 (119904) 120583) d119904 0)

+ 119863(int119905

1199050

119891 (119904 119906 (119904) 120583) d119904 0)

+ 119863 (120601119898119906(1199050minus 120574) 120601

0(1199050minus 120574)) lt 3 sdot

120573

3

(38)

Thus for each 119906 with |120583 minus 1205830| lt 120575 a solution of (29) with

119909(1199050minus 120574) = 120601

120583(1199050minus 120574) exists on |119905 minus 119905

0| le 120574 Hence 120601

120583can be

continued to 1199050+ 120574 So in the case [120591 minus 120572 120591 + 120572] 120601

120583rarr 120601

0

uniformly on [120591 1199050+ 120574] It leads to a contradiction similarly

for the cases 1199050= 119887 and [119886 120591] Therefore the theorem holds

over [119886 119887]

Example 25 Let

(119905) = 119860 sdot 119905

2 sin 119905minus2 119905 = 0

0 119905 = 0(39)

where fuzzy number 119860 is defined in Example 1 Let ℎ(119905) =1015840(119905) We define

ℎ120583(119905) =

ℎ (119905) 120583 isin [0 1) 119905 isin(minus2 2)

(minus120583 120583)

0 otherwise(40)

and ℎ120583(119905) = ℎ

minus120583(119905) for 120583 isin (minus1 0] Let 119891(119905 119909 120583) = 1199052119909 + ℎ

120583(119905)

defined on (minus2 2) times (minus1 1) Then we have

119863(119865120583119906(1199051) 119865

120583119906(1199052)) = 119863(int

1199052

1199051

119891 (119904 119906 (119904) 120583) d119904 0)

le int1199052

1199051

1199042d119904 + 119863(int

1199052

1199051

ℎ120583d119904 0)

119863(int1199052

1199051

ℎ120583d119904 0) le int

1199052

1199051

119863(ℎ (119904) d119904 0)

(41)

where 1199051 1199052isin (0 1] or 119905

1 1199052isin [minus1 0) and

119863(int119905

0

ℎ120583(119904) d119904 0)

=

119863(119860 sdot 1199052 sin 119905minus2 119860 sdot 1205832 sin 120583minus2) 120583 = 0

119863 (119860 sdot 1199052 sin 119905minus2 0) 120583 = 0

(42)

Note that ℎ is Kaleva integrable on every subinterval of [minus1 0)and (0 1] Therefore we have that 119865

120583119906(119905) is equicontinuous

on [minus1 1] in 119906 and near 1205830= 0 Furthermore 119865

120583119906(119905) is

119860119862120575(119883

119899) uniformly in 120583 and 119906 where 119883

119899= [1119899 1] 119899 =

1 2 On the other hand by using integrating factor for119905 isin [minus1 1] we have

120601120583(119905) = 119890

1199052

3

int119905

0

119890minus1199043

3

119860 sdot ℎ120583(119904) d119904 (43)

with 120601120583(0) = 0 Obviously 120601

0is uniqueThus byTheorem 24

120601120583rarr 120601

0uniformly on [minus1 1]

5 Conclusion

In this paper we give the definition of the119880119860119862119866120575for a fuzzy-

number-valued function and the nonabsolute fuzzy integraland its controlled convergence theorem In addition we dealwith the Cauchy problem and the continuous dependence ofa solution on parameters of discontinuous fuzzy differentialequations involving the strong fuzzy Henstock integral infuzzy number space The function governing the equationsis supposed to be discontinuous with respect to some vari-ables and satisfy nonabsolute fuzzy integrability Our resultimproves the result given in [1 11 19 20] (where uniformcontinuity was required) as well as those referred therein

Acknowledgments

The authors are very grateful to the anonymous refereesand Professor Mehmet Sezer for many valuable commentsand suggestions which helped to improve the presentationof the paper The authors would like to thank the NationalNatural Science Foundation of China (no 11161041) and theFundamental Research Fund for the Central Universities (no31920130010)

8 Journal of Applied Mathematics

References

[1] O Kaleva ldquoFuzzy differential equationsrdquo Fuzzy Sets and Sys-tems vol 24 no 3 pp 301ndash317 1987

[2] Z Gong and Y Shao ldquoGlobal existence and uniqueness ofsolutions for fuzzy differential equations under dissipative-typeconditionsrdquo Computers amp Mathematics with Applications vol56 no 10 pp 2716ndash2723 2008

[3] O Kaleva ldquoThe Cauchy problem for fuzzy differential equa-tionsrdquo Fuzzy Sets and Systems vol 35 no 3 pp 389ndash396 1990

[4] J J Nieto ldquoThe Cauchy problem for continuous fuzzy differen-tial equationsrdquo Fuzzy Sets and Systems vol 102 no 2 pp 259ndash262 1999

[5] M L Puri and D A Ralescu ldquoDifferentials of fuzzy functionsrdquoJournal of Mathematical Analysis and Applications vol 91 no 2pp 552ndash558 1983

[6] S Seikkala ldquoOn the fuzzy initial value problemrdquo Fuzzy Sets andSystems vol 24 no 3 pp 319ndash330 1987

[7] C Wu and S Song ldquoExistence theorem to the Cauchy problemof fuzzy differential equations under compactness-type condi-tionsrdquo Information Sciences vol 108 no 1ndash4 pp 123ndash134 1998

[8] C Wu S Song and Z Qi ldquoExistence and uniqueness for asolution on the closed subset to the Cauchy problem of fuzzydiffrential equationsrdquo Journal of Harbin Institute of Technologyvol 2 pp 1ndash7 1997

[9] CWu S Song and E S Lee ldquoApproximate solutions existenceand uniqueness of the Cauchy problem of fuzzy differentialequationsrdquo Journal of Mathematical Analysis and Applicationsvol 202 no 2 pp 629ndash644 1996

[10] V Lakshmikantham and R N Mohapatra Theory of FuzzyDifferential Equations and Inclusions vol 6 Taylor amp FrancisLondon UK 2003

[11] X Xue and Y Fu ldquoCaratheodory solutions of fuzzy differentialequationsrdquo Fuzzy Sets and Systems vol 125 no 2 pp 239ndash2432002

[12] R HenstockTheory of Integration Butterworths London UK1963

[13] J Kurzweil ldquoGeneralized ordinary differential equations andcontinuous dependence on a parameterrdquo Czechoslovak Mathe-matical Journal vol 7 no 82 pp 418ndash449 1957

[14] P Y LeeLanzhou Lectures onHenstock Integration vol 2WorldScientific Publishing Teaneck NJ USA 1989

[15] C Wu and Z Gong ldquoOn Henstock integrals of interval-valuedfunctions and fuzzy-valued functionsrdquo Fuzzy Sets and Systemsvol 115 no 3 pp 377ndash391 2000

[16] C Wu and Z Gong ldquoOn Henstock integral of fuzzy-number-valued functions Irdquo Fuzzy Sets and Systems vol 120 no 3 pp523ndash532 2001

[17] Z Gong ldquoOn the problem of characterizing derivatives for thefuzzy-valued functions II Almost everywhere differentiabilityand strong Henstock integralrdquo Fuzzy Sets and Systems vol 145no 3 pp 381ndash393 2004

[18] Z Gong and Y Shao ldquoThe controlled convergence theoremsfor the strong Henstock integrals of fuzzy-number-valuedfunctionsrdquo Fuzzy Sets and Systems vol 160 no 11 pp 1528ndash1546 2009

[19] T S Chew and F Flordeliza ldquoOn 1199091015840 = 119891(119905 119909) and Henstock-Kurzweil integralsrdquo Differential and Integral Equations vol 4no 4 pp 861ndash868 1991

[20] B Bede and S G Gal ldquoGeneralizations of the differentiabilityof fuzzy-number-valued functions with applications to fuzzy

differential equationsrdquo Fuzzy Sets and Systems vol 151 no 3pp 581ndash599 2005

[21] D Dubois and H Prade ldquoTowards fuzzy differential calculus IIntegration of fuzzy mappingsrdquo Fuzzy Sets and Systems vol 8no 1 pp 1ndash17 1982

[22] P Diamond and P KloedenMetric Spaces of Fuzzy Sets WorldScientific Publishing River Edge NJ USA 1994

[23] R Goetschel Jr and W Voxman ldquoElementary fuzzy calculusrdquoFuzzy Sets and Systems vol 18 no 1 pp 31ndash43 1986

[24] Y Chalco-Cano and H Roman-Flores ldquoOn new solutions offuzzy differential equationsrdquo Chaos Solitons and Fractals vol38 no 1 pp 112ndash119 2008

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

8 Journal of Applied Mathematics

References

[1] O Kaleva ldquoFuzzy differential equationsrdquo Fuzzy Sets and Sys-tems vol 24 no 3 pp 301ndash317 1987

[2] Z Gong and Y Shao ldquoGlobal existence and uniqueness ofsolutions for fuzzy differential equations under dissipative-typeconditionsrdquo Computers amp Mathematics with Applications vol56 no 10 pp 2716ndash2723 2008

[3] O Kaleva ldquoThe Cauchy problem for fuzzy differential equa-tionsrdquo Fuzzy Sets and Systems vol 35 no 3 pp 389ndash396 1990

[4] J J Nieto ldquoThe Cauchy problem for continuous fuzzy differen-tial equationsrdquo Fuzzy Sets and Systems vol 102 no 2 pp 259ndash262 1999

[5] M L Puri and D A Ralescu ldquoDifferentials of fuzzy functionsrdquoJournal of Mathematical Analysis and Applications vol 91 no 2pp 552ndash558 1983

[6] S Seikkala ldquoOn the fuzzy initial value problemrdquo Fuzzy Sets andSystems vol 24 no 3 pp 319ndash330 1987

[7] C Wu and S Song ldquoExistence theorem to the Cauchy problemof fuzzy differential equations under compactness-type condi-tionsrdquo Information Sciences vol 108 no 1ndash4 pp 123ndash134 1998

[8] C Wu S Song and Z Qi ldquoExistence and uniqueness for asolution on the closed subset to the Cauchy problem of fuzzydiffrential equationsrdquo Journal of Harbin Institute of Technologyvol 2 pp 1ndash7 1997

[9] CWu S Song and E S Lee ldquoApproximate solutions existenceand uniqueness of the Cauchy problem of fuzzy differentialequationsrdquo Journal of Mathematical Analysis and Applicationsvol 202 no 2 pp 629ndash644 1996

[10] V Lakshmikantham and R N Mohapatra Theory of FuzzyDifferential Equations and Inclusions vol 6 Taylor amp FrancisLondon UK 2003

[11] X Xue and Y Fu ldquoCaratheodory solutions of fuzzy differentialequationsrdquo Fuzzy Sets and Systems vol 125 no 2 pp 239ndash2432002

[12] R HenstockTheory of Integration Butterworths London UK1963

[13] J Kurzweil ldquoGeneralized ordinary differential equations andcontinuous dependence on a parameterrdquo Czechoslovak Mathe-matical Journal vol 7 no 82 pp 418ndash449 1957

[14] P Y LeeLanzhou Lectures onHenstock Integration vol 2WorldScientific Publishing Teaneck NJ USA 1989

[15] C Wu and Z Gong ldquoOn Henstock integrals of interval-valuedfunctions and fuzzy-valued functionsrdquo Fuzzy Sets and Systemsvol 115 no 3 pp 377ndash391 2000

[16] C Wu and Z Gong ldquoOn Henstock integral of fuzzy-number-valued functions Irdquo Fuzzy Sets and Systems vol 120 no 3 pp523ndash532 2001

[17] Z Gong ldquoOn the problem of characterizing derivatives for thefuzzy-valued functions II Almost everywhere differentiabilityand strong Henstock integralrdquo Fuzzy Sets and Systems vol 145no 3 pp 381ndash393 2004

[18] Z Gong and Y Shao ldquoThe controlled convergence theoremsfor the strong Henstock integrals of fuzzy-number-valuedfunctionsrdquo Fuzzy Sets and Systems vol 160 no 11 pp 1528ndash1546 2009

[19] T S Chew and F Flordeliza ldquoOn 1199091015840 = 119891(119905 119909) and Henstock-Kurzweil integralsrdquo Differential and Integral Equations vol 4no 4 pp 861ndash868 1991

[20] B Bede and S G Gal ldquoGeneralizations of the differentiabilityof fuzzy-number-valued functions with applications to fuzzy

differential equationsrdquo Fuzzy Sets and Systems vol 151 no 3pp 581ndash599 2005

[21] D Dubois and H Prade ldquoTowards fuzzy differential calculus IIntegration of fuzzy mappingsrdquo Fuzzy Sets and Systems vol 8no 1 pp 1ndash17 1982

[22] P Diamond and P KloedenMetric Spaces of Fuzzy Sets WorldScientific Publishing River Edge NJ USA 1994

[23] R Goetschel Jr and W Voxman ldquoElementary fuzzy calculusrdquoFuzzy Sets and Systems vol 18 no 1 pp 31ndash43 1986

[24] Y Chalco-Cano and H Roman-Flores ldquoOn new solutions offuzzy differential equationsrdquo Chaos Solitons and Fractals vol38 no 1 pp 112ndash119 2008

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

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