Research ArticleUlam-Hyers Stability and Ulam-Hyers-Rassias Stability forFuzzy Integrodifferential Equation

Nguyen Ngoc Phung1 Bao Quoc Ta2 and Ho Vu 34

1Faculty of Mathematical Economics Banking University of Ho Chi Minh City Vietnam2Center for Advanced Chemistry Institute of Research and Development Duy Tan University Da Nang 550000 Vietnam3Division of Computational Mathematics and Engineering Institute for Computational Science Ton Duc ang UniversityHo Chi Minh City Vietnam

4Faculty of Mathematics and Statistics Ton Ducang University Ho Chi Minh City Vietnam

Correspondence should be addressed to Ho Vu hovutdtueduvn

Received 6 November 2018 Accepted 2 March 2019 Published 26 March 2019

Academic Editor Eric Campos-Canton

Copyright copy 2019 Nguyen Ngoc Phung et al 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

In this paper we establish the Ulam-Hyers stability and Ulam-Hyers-Rassias stability for fuzzy integrodifferential equations byusing the fixed point method and the successive approximation method

1 Introduction

Fuzzy differential and integrodifferential equation (FD-FIDE) are a natural way to model dynamical systems subjectto uncertainties In the past few years the study of fuzzyintegrodifferential equations is an area of mathematics thathas recently received a lot of attention (see eg [1ndash11])Alikhani et al [3] studied the existence and uniqueness ofglobal solutions for fuzzy initial value problems via inte-grodifferential operators of Volterra type Alikhani et al [2]introduced a new concept of upper and lower solutions forFIDE Using this concept authors proved the existence anduniqueness of global solutions of an initial value problemfor first-order nonlinear FIDE under generalized differen-tiability Additionally authors [4] studied the two kinds ofsolutions to FIDE with delay by the usage of two differentconcepts of fuzzy derivative

In recent years accompanied by the development of theUlam-Hyers stability of differential equation (see eg [12ndash18]) Ulam-Hyers stability of fuzzy differential equation hasattracted the attention of reseachers the reader is refferredto [19ndash22] Ren [22] studied the Ulam-Hyers stability ofthe Hermite fuzzy differential equation associated with theinhomogeneous Hermite fuzzy differential equation undersome suitable conditions However the fixed point method

has been successfully used to study the Ulam stability offuzzy differential equation by [19ndash21] In [19 20] Shenconsidered the Ulam stability of the first order linear (partial)fuzzy differential equation under generalized differentiabilityUsing the fixed point technique authors [21] studied theUlam stability of fuzzy differential equations and we see thatUlam stability of this problem requires various prerequisitesunder different types of differentiability

To the best of our knowledge up to now the numberof papers dealing with Ulam stability for FD-FIDE is ratherlimited as opposed to the amount of publications concerningFD-FIDE From this reason we choose the study of Ulamstability of FIDE by using the fixed point technique and themethod of successive approximation

The rest of this paper is organized as follows In Sec-tion 2 some notations and preparation results are givenIn Section 3 Ulam-Hyers stability and Ulam-Hyers-Rassiasstability criteria for FIDE defined on a bounded and closedinterval are obtained in terms of the method of successiveapproximation and the fixed point theorem in [23]

2 Preliminaries

In this section we introduce some basic definitions theo-rems and lemmas which are required throughout this paper

HindawiComplexityVolume 2019 Article ID 8275979 10 pageshttpsdoiorg10115520198275979

2 Complexity

Definition 1 (see [23]) A function 119889 X times X 997888rarr [0 +infin) iscalled a generalized metric on 119883 if and only if 119889 satisfies

(1) 119889(119909 119910) = 0 if and only if 119909 = 119910(2) 119889(119909 119910) = 119889(119910 119909) for all 119909 119910 119911 isin X(3) 119889(119909 119911) le 119889(119909 119910) + 119889(119910 119911) for all 119909 119910 119911 isin X

Theorem 2 (see [23]) Let 119889 X times X 997888rarr [0 +infin) be ageneralized metric on X and (X 119889) is a generalized completemetric space Assume that 119879 X 997888rarr X is a strictly contractiveoperator with the Lipschitz constant 119871 lt 1 If there exists anonnegative integer 119899 such that 119889(119879119899+1119909 119879119899119909) lt infin for some119909 isin X then the following are true

(i) the sequence 119879119899119909 converges to a fixed point 119909lowast of 119879(ii) 119909lowast is the unique fixed point of 119879 in

Xlowast = 119910 isin X | 119889 (119879119899119909 119910) lt infin (1)

(iii) if 119910 isin Xlowast then we have

119889 (119910 119909lowast) le 11 minus 119871119889 (119879119910 119910) (2)

Lemma 3 Let 120601 119869 997888rarr [0 +infin) be a continuous functionWe define the set

X fl 119909 119869 997888rarr RF | 119909 is continuous function on 119869 (3)

equipped with the metric

119889 (119909 119910) = inf 120578 isin [0 +infin)cup +infin | 119863 (119909 (119905) 119910 (119905) le 120578120593 (119905) forall119905 isin 119869 (4)

en (X 119889) is a complete generalized metric space

Proof The proof of this lemma can be found in Shen andWang [21]

Denote by RF the class of fuzzy sets 119906 R 997888rarr [0 1]with the following properties (i) 119906 is normal ie there exists1199090 isin R such that 119906(1199090) = 1 (ii) 119906 is fuzzy convex that is119906(120582119909 + (1 minus 120582)119909) ge min119906(119909) 119906(119910) for any 119909 119910 isin R 119906 isin RF

and 120582 isin [0 1] (iii) 119906 is upper semi-continuous (iv) 119888119897119909 isinR 119906(119909) gt 0 is compact where 119888119897 denotes the closure of aset

Usually the set RF is called the space of fuzzy numbersand it is easy to see that R sub RF For 120572 isin (0 1] we denote[119906]120572 = 119909 isin R 119906(119909) ge 120572 and [119906]0 = 119909 isin R 119906(119909) gt 0Then it follows from the conditions (i)-(iv) that the 120572-levelset [119906]120572 is a non-empty compact interval for all 120572 isin [0 1] andeach 119906 isin RF For any 119906 V isin RF and 120582 isin R the addition119906 + V and scalar multiplication 120582119906 can be defined levelwiseby [119906 + V]120572 = [119906]120572 + [V]120572 and [120582119906] = 120582[119906]120572 for all 120572 isin [0 1]

The supremum metric between 119906 and V is defined by

119863 RF timesRF 997888rarr R+ cup 0 119863 (119906 V) = sup

120572isin[01]

119889119867 ([119906]120572 [V]120572)= sup120572isin[01]

max 1003816100381610038161003816119906120572 minus V1205721003816100381610038161003816 1003816100381610038161003816119906120572 minus V1205721003816100381610038161003816 (5)

It is easy to see that (RF 119863) is a complete metric space Itis well known that the supremum metric has the followingproperties

(D1) 119863(119906 + 119908 V + 119908) = 119863(119906 V) for any 119906 V 119908 isin RF(D2) 119863(120582119906 120582V) = 120582119863(119906 V) for any 120582 isin R+ 119906 V isin RF(D3) 119863(119906 + V 119908 + 119890) le 119863(119906119908) +119863(V 119890) for any 119906 V 119908 119890 isin

RF

Definition 4 (see [24]) Let 119906 V isin RF If there exists 119908 isin RF

such that 119906 = V+119908 then119908 is called the H-difference of 119906 andV and it is denoted by 119906 ⊖ V

Throughout this paper the symbol ldquo⊖rdquo always stands forthe H-difference In general 119906 ⊖ V = 119906 + (minus1)VDefinition 5 (see [24]) Let 119891 (119886 119887) 997888rarr RF and 1199050 isin (119886 119887)We say 119891 is generalized differential at 1199050 if there exists anelement 119863119892119867119891(1199050) isin RF such that

(1) for all ℎ gt 0 sufficiently small there exists 119891(1199050 + ℎ) ⊖119891(1199050) 119891(1199050) ⊖ 119891(1199050 minus ℎ) and then limits (in metric 119863)limℎ997888rarr0

119891 (1199050 + ℎ) ⊖ 119891 (1199050)ℎ = limℎ997888rarr0

119891 (1199050) ⊖ 119891 (1199050 minus ℎ)ℎ= 119863119892119867119891 (1199050)

(6)

(2) for all ℎ gt 0 sufficiently small there exists 119891(1199050) ⊖119891(1199050 + ℎ) 119891(1199050 minus ℎ) ⊖ 119891(1199050) and then limits (in metric119863)limℎ997888rarr0

119891 (1199050) ⊖ 119891 (1199050 + ℎ)minusℎ = limℎ997888rarr0

119891 (1199050 minus ℎ) ⊖ 119891 (1199050)minusℎ= 119863119892119867119891 (1199050)

(7)

Note that Bede and Gal [24] considered four cases in thedefinition of derivative In this paper we consider only thetwo first cases of Definition 5 in [24] In the other cases thederivative reduces to a crisp element that is 119863119892119867119891(1199050) isin R

Theorem 6 (see [24]) Let 119891 (119886 119887) 997888rarr RF and denote[119891(119905)]120572 = [1198911205721 (119905) 1198911205722 (119905)] for each 120572 isin [0 1] 119905 isin (119886 119887)(i) If 119891 is (1)-differentiable at all 119905 isin (119886 119887) then 1198911205721 (119905) and1198911205722 (119905) are differentiable functions and we have

[119863119892119867119891 (119905)]120572 = [(1198911205721 (119905))1015840 (1198911205722 (119905))1015840] (8)

(ii) If 119891 is (2)-differentiable at all 119905 isin (119886 119887) then 1198911205721 (119905) and1198911205722 (119905) are differentiable functions and we have[119863119892119867119891 (119905)]120572 = [(1198911205722 (119905))1015840 (1198911205721 (119905))1015840] (9)

InTheorem 6 we see that if 119891 is (1)-differentiable then itis not (2)-differentiable and vice versa

Theorem 7 (see [24]) Let 119891 (119886 119887) 997888rarr RF be differentiableon (119886 119887) and assume that derivative 119863119892119867119891 is integrable over(119886 119887) For each 119905 isin (119886 119887) we have

Complexity 3

(i) If 119891 is (1)-differentiable then

119891 (119905) = 119891 (119886) + int119905119886119863119892119867119891 (119904) 119889119904 (10)

(ii) If 119891 is (2)-differentiable then

119891 (119905) = 119891 (119886) ⊖ (minus1)int119905119886119863119892119867119891 (119904) 119889119904 (11)

Let 119869 fl [119886 119887] (with 119879 gt 0) be a compact interval of RWe denote by

119862 (119869RF) = 119906 | 119906 119869997888rarr RF is a countinuous functions on 119869 (12)

On the space 119862(119869RF) we consider the supremummetric asfollows

119863 (119906 V) = sup119905isin119869

119863[119906 (119905) V (119905)] (13)

In next section we consider the following fuzzy integrodif-ferential equation

119863119892119867119906 (119905) = 119891 (119905 119906 (119905)) + int119905

119886119892 (119905 119904 119906 (119904)) 119889119904 119905 isin 119869

119906 (119886) = 1199060 isin RF(14)

where the symbol119863119892119867 is generalized Hukuhara derivative themapping 119891 119869 times RF times RF 997888rarr RF is continuous on 119869 and119892 119869 times 119869 timesRF 997888rarr RF is a continuous function on 119869 times 119869Definition 8 (see [3]) We say that a mapping 119906 119869 997888rarr RF

is solution to the problem (14) if 119906 is generalized Hukuharadifferentiable on 119869 and119863119892119867119906(119905) = 119891(119905 119906(119905)) + int119905119886 119892(119905 119904 119906(119904))119889119904for any 119905 isin 119869 119891 isin 119862(119869RF) 119892 isin 119862(119869 times 119869RF)Lemma9 (see [3]) Let 119906 119869 997888rarr RF be a continuous functionon 119869 Problem (14) is equivalent to one of the following the fuzzyintegro integral equations

(S1) if 119906 is (1)-differentiable on 119869 then119906 (119905) = 1199060 + int

119905

119886119891 (119904 119906 (119904)) 119889119904

+ int119905119886int119904119886119892 (119904 119903 119906 (119903)) 119889119903119889119904

(15)

(S2) if 119906 is (2)-differentiable on 119869 then119906 (119905) = 1199060 ⊖ (minus1) int

119905

119886119891 (119904 119906 (119904)) 119889119904 ⊖ (minus1)

sdot int119905119886int119904119886119892 (119904 119903 119906 (119903)) 119889119903119889119904

(16)

3 Main Results

Definition 10 We say that problem (14) is Ulam-Hyers stableif there exists a real number 119870119891 gt 0 such that for 120576 gt 0 andfor each V isin 1198621(119869RF) to the problem

119863[119863119892119867V (119905) 119891 (119905 V (119905)) + int119905

119886119892 (119905 119904 V (119904)) 119889119904] le 120576 (17)

there exists a solution to problem (14) with

119863[V (119905) 119906 (119905)] le 119870119891120576 (18)

for all 119905 isin 119869 We call 119870119891 a Ulam-Hyers stability constant of(14)

Definition 11 We say that problem (14) is Ulam-Hyers-Rassias stable if there exists a real number 119862119891 gt 0 such thatfor 120576 gt 0 and for each V isin 1198621(119869RF) to the problem

119863[119863119892119867V (119905) 119891 (119905 V (119905)) + int119905

0119892 (119905 119904 V (119904)) 119889119904] le 120593 (119905) (19)

there exists a solution to problem (14) with

119863[V (119905) 119906 (119905)] le 119862119891120593 (119905) (20)

for all 119905 isin 119869Firstly we prove that problem (14) is Hyers-Ulam stable

via the method of successive approximation We consider thefollowing inequality

119863[119863119892119867V (119905) 119891 (119905 V (119905)) + int119905

119886119892 (119905 119904 V (119904)) 119889119904] le 120576

for V isin RF(21)

Definition 12 We say that

(a) A function V isin 1198621(119869RF) is a (S1)-solution of theinequality (21) if and only if there exists a function1205751 isin 119862(119869RF) such that

(i) 119863[1205751(119905) 0] le 120576 for any 119905 isin 119869(ii) 119863119892119867V(119905) = 119891(119905 V(119905)) + int119905119886 119892(119905 119904 V(119904))119889119904 + 1205751(119905) for

any 119905 isin 119869(b) A function V isin 1198621(119869RF) is a (S2)-solution of the

inequality (21) if and only if there exists a function1205752 isin 119862(119869RF) such that

(i) 119863[1205752(119905) 0] le 120576 for any 119905 isin 119869(ii) 119863119892119867V(119905) = 119891(119905 V(119905)) + int119905119886 119892(119905 119904 V(119904))119889119904 + 1205752(119905) for

any 119905 isin 119869Theorem 13 Assume that 119891 119869 times RF 997888rarr RF and 119892 119869 times119869 times RF 997888rarr RF are a continuous function that satisfies thefollowing conditions (i) there exists a constant 119871119891119892 gt 0 suchthat

max 119863 [119891 (119905 119906) 119891 (119905 V)] 119863 [119892 (119905 119904 119906) 119892 (119905 119904 V)]le 119871119891119892119863 [119906 V] (22)

4 Complexity

for any (119905 119906) (119905 V) isin 119869 times RF (119905 119904 119906) (119905 119904 V) isin 119869 times 119869 times RF(ii) for each 120576 gt 0 if a continuously (2)-differentiable functionV 119869 997888rarr RF satisfies

119863[119863119892119867V (119905) 119891 (119905 V (119905)) + int119905

119886119892 (119905 119904 V (119904)) 119889119904] le 120576

forall119905 isin 119869(23)

then there exists a (S2)-solution 119906 119869 997888rarr RF of (14) with1199060 = V0 such that

119863[V (119905) 119906 (119905)] le 120576120582119871119891119892 119890(119887minus119886)(1+119871119891119892) (24)

where 120582 is a maximum balancing constant

Proof For each 120576 gt 0 let a continuously (2)-differentiablefunction V 119869 997888rarr RF satisfy inequality (21) for any 119905 isin 119869V isin RF By the part (b) of Definition 12 we have

119863[1205752 (119905) 0] le 120576 (25)

and

119863119892119867V (119905) = 119891 (119905 V (119905)) + int119905

119886119892 (119905 119904 V (119904)) 119889119904 + 1205752 (119905) (26)

for any 119905 isin 119869 and 1205752 isin 119862(119869RF)If a function V 119869 997888rarr RF is continuous and (2)-

differentiable on 119869 then by Lemma 9 it satisfies equivalentlythe following fuzzy integrointegral equation

V (119905) = V0 ⊖ (minus1)int119905

119886119891 (119904 V (119904)) 119889119904 ⊖ (minus1)

sdot int119905119886int119904119886119892 (119904 119903 V (119903)) 119889119903119889119904 + int119905

1198861205752 (119904) 119889119904

forall119905 isin 119869(27)

Let us define

1199060 (119905) = V (119905) forall119905 isin 119869 (28)

and sequence functions 119906119899 119869 997888rarr RF 119899 = 1 2 ofsuccessive approximations as follows

119906119899 (119905) = V0 ⊖ (minus1)int119905

119886119891 (119904 119906119899minus1 (119904)) 119889119904 ⊖ (minus1)

sdot int119905119886int119904119886119892 (119904 119903 119906119899minus1 (119903)) 119889119903119889119904 forall119905 isin 119869

(29)

By virtue of the properties of 119863 and assumption (i) we have

119863[1199061 (119905) 1199060 (119905)] = 119863 [V0 ⊖ (minus1)

sdot int119905119886119891 (119904 1199060 (119904)) 119889119904 ⊖ (minus1)

sdot int119905119886int119904119886119892 (119904 119903 1199060 (119903)) 119889119903119889119904 V (119905)] = 119863[V0 ⊖ (minus1)

sdot int119905119886119891 (119904 V0 (119904)) 119889119904 ⊖ (minus1)

sdot int119905119886int119904119886119892 (119904 119903 V0 (119903)) 119889119903119889119904 V (119905)]

= 119863[int1199051198861205752 (119904) 119889119904 0] le int

119905

119886119863[1205752 (119904) 0] 119889119904 le 120576 (119905

minus 119886) forall119905 isin 119869(30)

Therefore

119863[1199061 (119905) 1199060 (119905)] le 120576 (119905 minus 119886) forall119905 isin 119869 (31)

Observe that for 119899 = 1 2 and for 119905 isin 119869 one has119863 [119906119899+1 (119905) 119906119899 (119905)] = 119863[V0 ⊖ (minus1)

sdot int119905119886119891 (119904 119906119899 (119904)) 119889119904 ⊖ (minus1)

sdot int119905119886int119904119886119892 (119904 119903 119906119899 (119903)) 119889119903119889119904 V0 ⊖ (minus1)

sdot int119905119886119891 (119904 119906119899minus1 (119904)) 119889119904 ⊖ (minus1)

sdot int119905119886int119904119886119892 (119904 119903 119906119899minus1 (119903)) 119889119903119889119904]

le 119871119891119892 int119905

119886119863[119906119899 (119904) 119906119899minus1 (119904)] 119889119904

+ 119871119891119892 int119905

119886int119904119886119863 [119906119899 (119903) 119906119899minus1 (119903)] 119889119903119889119904

(32)

In particular

119863[1199062 (119905) 1199061 (119905)]le 119871119891119892 int

119905

119886119863[1199061 (119904) 1199060 (119904)] 119889119904

+ 119871119891119892 int119905

119886int119904119886119863[1199061 (119903) 1199060 (119903)] 119889119903119889119904

= 120576119871119891119892 int119905

119886(119904 minus 119886) 119889119904 + 120576119871119891119892 int

119905

119886int119904119886(119903 minus 119886) 119889119903119889119904

= 120576119871119891119892 ((119905 minus 119886)2

2 + (119905 minus 119886)33 )

= 120576119871119891119892

(119871119891119892 (119905 minus 119886))22 (1 + 119905 minus 119886

3 )

(33)

and so119863 [1199063 (119905) 1199062 (119905)]

le 119871119891119892 int119905

119886119863[1199062 (119904) 1199061 (119904)] 119889119904

+ 119871119891119892 int119905

119886int119904119886119863 [1199062 (119903) 1199061 (119903)] 119889119903119889119904

Complexity 5

= 1205761198712119891119892 int119905

119886((119904 minus 119886)22 + (119904 minus 119886)3

3 ) 119889119904

+ 1205761198712119891119892 int119905

119886int119904119886((119903 minus 119886)22 + (119903 minus 119886)3

3 ) 119889119903119889119904

= 1205761198712119891119892 ((119905 minus 119886)3

3 + 2(119905 minus 119886)44 + (119905 minus 119886)55 )

= 120576119871119891119892

(119871119891119892 (119905 minus 119886))33 (1 + 2119905 minus 1198864 + (119905 minus 119886)2

5 )(34)

and for 119899 gt 4 we have119863[119906119899 (119905) 119906119899minus1 (119905)] le 120576119871119899minus1119891119892 ((119905 minus 119886)

119899

119899 + 1205821 (119905 minus 119886)119899+1

(119899 + 1)+ + 120582119899 (119905 minus 119886)

2119899

(2119899) + (119905 minus 119886)2119899+1(2119899 + 1) )

(35)

where 1205821 120582119899 are balancing constantsWe choose120582 = max1 1205821 120582119899 (called120582 is amaximum

balancing constant) and then estimation (35) can be rewrittenas follows

119863 [119906119899 (119905) 119906119899minus1 (119905)] le 120576120582119871119891119892

(119871119891119892 (119905 minus 119886))119899119899 (1 + 119905 minus 119886

119899 + 1+ (119905 minus 119886)2(119899 + 1) (119899 + 2) + +

(119905 minus 119886)119899(119899 + 1) (119899 + 2) 2119899

+ (119905 minus 119886)119899+1(119899 + 1) (119899 + 2) 2119899 (2119899 + 1)) le 120576120582

119871119891119892sdot (119871119891119892 (119905 minus 119886))

119899

119899 (1 + 119905 minus 1198861 + (119905 minus 119886)2

2 +

+ (119905 minus 119886)119899119899 + (119905 minus 119886)119899+1

(119899 + 1) ) le 120576120582119871119891119892

(119871119891119892 (119905 minus 119886))119899119899

sdot 119890119905minus119886

(36)

Further if we assume that

119863[119906119899 (119905) 119906119899minus1 (119905)] le 120576120582119871119891119892

(119871119891119892 (119905 minus 119886))119899119899 119890119905minus119886

forall119905 isin 119869(37)

then we obtain

119863[119906119899+1 (119905) 119906119899 (119905)] le 120576120582119871119891119892

(119871119891119892 (119905 minus 119886))119899+1(119899 + 1) 119890119905minus119886

forall119905 isin 119869(38)

By the principle ofmathematical induction that (37) holds forevery 119899 gt 1 and now using estimation (37) for any 119905 isin 119869 weget

infinsum119899=1

119863 [119906119899 (119905) 119906119899minus1 (119905)] le 120576120582119871119891119892infinsum119899=1

(119871119891119892 (119905 minus 119886))119899119899 119890119905minus119886 (39)

The series suminfin119895=1(119911119896119896) is convergent for every 119911 isin R Hencefor every 120576 gt 0 we infer that the series suminfin119899=1119863[119906119899(119905) 119906119899minus1(119905)]is uniformly convergent on 119869 with respect to metric 119863 and

infinsum119899=1

119863 [119906119899 (119905) 119906119899minus1 (119905)] le 120576120582119871119891119892infinsum119899=1

(119871119891119892 (119905 minus 119886))119899119899 119890119905minus119886

le 120576120582119871119891119892infinsum119899=1

(119871119891119892 (119887 minus 119886))119899119899 119890119887minus119886 = 120576120582

119871119891119892 119890119871119891119892(119887minus119886)119890119887minus119886

= 120576120582119871119891119892 119890(119887minus119886)(1+119871119891119892)

(40)

For 119905 isin 119869 we have119863 [119906119899 (119905) 119906 (119905)] le 120576120582

119871119891119892infinsum119899=1

(119871119891119892 (119905 minus 119886))119899119899 119890119905minus119886 (41)

and the following estimation

119863[int119905119886119891 (119904 119906119899 (119904)) 119889119904

+ int119905119886int119904119886119892 (119904 119903 119906119899 (119903)) 119889119903119889119904 int

119905

119886119891 (119904 119906 (119904)) 119889119904

+ int119905119886int119904119886119892 (119904 119903 119906 (119903)) 119889119903119889119904]

le int119905119886119863[119891 (119904 119906119899 (119904)) 119891 (119904 119906 (119904))] 119889119904

+ int119905119886int119904119886119863 [119892 (119904 119903 119906119899 (119903)) 119892 (119904 119903 119906 (119903))] 119889119903119889119904

le 119871119891119892 int119905

119886119863 [119906119899 (119904) 119906 (119904)] 119889119904

+ 119871119891119892 int119905

119886int119904119886119863[119906119899 (119903) 119906 (119903)] 119889119903119889119904 forall119905 isin 119869

(42)

Combining estimation (41) and inequality (42) we obtainthat

119863[int119905119886119891 (119904 119906119899 (119904)) 119889119904 ⊖ (minus1)

sdot int119905119886int119904119886119892 (119904 119903 119906119899 (119903)) 119889119903119889119904 int

119905

119886119891 (119904 119906 (119904)) 119889119904 ⊖ (minus1)

sdot int119905119886int119904119886119892 (119904 119903 119906 (119903)) 119889119903119889119904]

(43)

6 Complexity

converges to 0 uniformly as 119899 997888rarr +infin Therefore 119906(119905) is a(S2)-solution of (14) with initial condition 1199060

Finally we shall prove that problem (14) has a unique(S2)-solution Assume that is another (S2)-solution of (14)with initial condition 1199060 Then we have for any 119905 isin 119869

119863 [119906 (119905) (119905)] le 119871119891119892 int119905

119886119863 [119906 (119904) (119904)] 119889119904

+ 119871119891119892 int119905

119886int119905119886119863[119906 (119903) (119903)] 119889119903119889119904

(44)

If we let 120585(119905) = 119863[119906(119905) (119905)] for any 119905 isin 119869 then120585 (119905) le 119871119891119892 int

119905

119886120585 (119904) 119889119904 + 119871119891119892 int

119905

119886int119905119886120585 (119903) 119889119903119889119904 (45)

Applying Lemma 23 in Hoa et al [4] we obtain 120585(119905) = 0 forany 119905 isin 119869 This completes the proof

Example 14 Consider the following fuzzy integro differentialequation

119863119892119867119906 (119905) = (minus2 0 2) + int119905

0

119906 (119904)1 + 119906 (119904)119889119904 (46)

119906 (0) = (minus2 0 2) (47)

where 119906 isin 119862([0 1]RF) 119891 [0 1] times RF 997888rarr RF and 119892 [0 1] times [0 1] timesRF 997888rarr RFLet

119891 (119905 119906) = (minus2 0 2) 119892 (119905 119904 119906) = 119906

1 + 119906 forall119905 119904 isin [0 1] 119906 isin RF (48)

It is easy to see that 119891 119892 satisfy Lipschitz condition withLipschitz constant 119871119891119892 = 1 Indeed for any 119905 119904 isin [0 1] and119906 isin RF

119863 [119892 (119905 119904 119906) 119892 (119905 119904 V)] = 119863 [ 1199061 + 119906

V1 + V

]= sup120572isin[01]

max10038161003816100381610038161003816100381610038161003816119906

1 + 119906 minusV

1 + V

10038161003816100381610038161003816100381610038161003816 10038161003816100381610038161003816100381610038161003816119906

1 + 119906 minusV

1 + V

10038161003816100381610038161003816100381610038161003816

= sup120572isin[01]

max 1003816100381610038161003816119906 minus V100381610038161003816100381610038161003816100381610038161 + 1199061003816100381610038161003816 10038161003816100381610038161 + V1003816100381610038161003816 |119906 minus V|

|1 + 119906| |1 + V|le sup120572isin[01]

max 1003816100381610038161003816119906 minus V1003816100381610038161003816 |119906 minus V| = 119863 [119906 V]

(49)

Hence byTheorem 15 the problem (46)-(47)has unique (S1)-solution or (S2)-solution on [0 1]

Moreover if a continuously (2)-differentiable function V [0 1] 997888rarr RF satisfies the following inequation

119863[119863119892119867V (119905) (minus2 0 2) + int119905

0

V (119904)1 + V (119904)119889119904] le 120576 (50)

then as shown in Theorem 15 there exists a (S2)-solution 119906 [0 1] 997888rarr RF of (46) such that

119863 [V (119905) 119906 (119905)] le 1205761205821198902 forall119905 isin [0 1] (51)

where 120582 is the maximum balancing constant

Secondly we shall prove the Ulam-Hyers-Rassias stabilityof the FIDE (14) defined on a bounded and closed interval

Theorem 15 Assume that 119891 119869 997888rarr RF 997888rarr RF and119892 119869 times 119869 times RF 997888rarr RF are continuous function satisfyingthe following conditions (i) there exists a constant 119871119891119892 gt0 such that max119863[119891(119905 119906) 119891(119905 V)]119863[119892(119905 119904 119906) 119892(119905 119904 V)] le119871119891119892119863[119906 V] for each (119905 119904 119906) (119905 119904 V) isin 119869 times 119869 times RF (ii) thereexists a constant 119862 gt 0 such that 0 lt 119871119891119892(119862 + 1198622) lt 1 Let120593 119869 997888rarr (119886 +infin) be a continuous function and increasing on119869 with

int119905119886120593 (119904) 119889119904 le 119862120593 (119905) for each 119905 isin 119869 (52)

If a continuously (1)-differentiable function 119906 119869 997888rarr RF

satisfies the following inequality

119863[119863119892119867119906 (119905) 119891 (119905 119906 (119905)) + int119905

119886119892 (119905 119904 119906 (119904)) 119889119904] le 120593 (119905) (53)

for any 119905 isin 119869 then there exists a unique (S1)-solution 119869 997888rarrRF of (14) such that

(119905) = 0 + int119905

119886119891 (119904 (119904)) 119889119904

+ int119905119886int119904119886119892 (119904 119903 (119903)) 119889119903119889s

(54)

and

119863[ (119905) 119906 (119905)] le 1198621 minus 119871119891119892 (119862 + 1198622)120593 (119905) (55)

for any 119905 isin 119869Proof Let us define a setX of all continuous fuzzy functions119908 119869 997888rarr RF by

X = 119908 119869 997888rarr RF | 119908 is continuous on 119869 (56)

equipped with the metric

119889 (V 119908) = inf 119862 isin [0 +infin) cup +infin | 119863 [V (119905) 119908 (119905)]le 119862120593 (119905) forall119905 isin 119869 (57)

It is easy to see that (X 119889) is also a complete generalizedmetric space (see Lemma 3)

The operator P X 997888rarr X is defined as follows

(PV) (119905) = V0 + int119905

119886119891 (119904 V (119904)) 119889119904

+ int119905119886int119904119886119892 (119904 119903 V (119903)) 119889119903119889119904 forall119905 isin 119869

(58)

Based on Lemma 32 and 33 in [3] we infer that PV is (1)-differentiable and so PV isin X

The operator P is strict contractive on X Indeed for anyV 119908 isin X and letting 119862V119908 isin [0 +infin) cup +infin be an arbitrary

Complexity 7

constant with 119889(V 119908) le 119862V119908 that is by the definition of themetric 119889 we have

119863 [V (119905) 119908 (119905)] le 119862V119908120593 (119905) forall119905 isin 119869 (59)

From the definition of the metric 119863 and assumptions (i)-(ii)of Theorem 15 and inequality (59) we infer that

119863[(PV) (119905) (P119908) (119905)] = 119863[V0 + int119905

119886119891 (119904 V (119904)) 119889119904

+ int119905119886int119904119886119892 (119904 119903 V (119903)) 119889119903119889119904 V0 + int

119905

119886119891 (119904 119908 (119904)) 119889119904

+ int119905119886int119904119886119892 (119904 119903 119908 (119903)) 119889119903119889119904]

le int119905119886119863[119891 (119904 V (119904)) 119891 (119904 119908 (119904))] 119889119904

+ int119905119886int119904119886119863[119892 (119904 119903 V (119903)) 119892 (119904 119903 119908 (119903))] 119889119903119889119904

le 119871119891119892 int119905

119886119863 [V (119904) 119908 (119904)] 119889119904

+ 119871119891119892 int119905

119886int119904119886119863[V (119903) 119908 (119903)] 119889119903119889119904

le 119871119891119892119862V119908 int119905

119886120593 (119904) 119889119904

+ 119871119891119892119862V119908 int119905

119886int119904119886120593 (119903) 119889119903119889119904 le 119871119891119892 (119862 + 1198622)

sdot 119862V119908120593 (119905)

(60)

for any 119905 isin 119869For each V 119908 isin X and by the definition of metric 119889 we

get

119889 ((PV) (119905) (P119908) (119905)) le 119871119891119892 (119862 + 1198622)119862V119908120593 (119905) (61)

for any 119905 isin 119869 Hence by (59) we can conclude that

119889 ((PV) (119905) (P119908) (119905)) le 119871119891119892 (119862 + 1198622) 119889 (V 119908) (62)

for any 119905 isin 119869It follows form the definitions of X and the operator P

that for arbitrary 119908 isin X there exists a constant 0 lt 119862 lt +infinsuch that

119863[(P119908) (119905) 119908 (119905)] = 119863 [1199080 + int119905

119886119891 (119904 119908 (119904)) 119889119904

+ int119905119886int119904119886119892 (119904 119903 119908 (119903)) 119889119903119889119904 119908 (119905)] le 119862120593 (119905)

(63)

for any 119905 isin 119869 since 119891(119905 119908(119905)) 119892(119905 119904 119908(119904)) and 119908(119905) arebounded on 119869 and min119905isin119869120593(119905) gt 0 Thus by definition of 119889 itis implied that

119889 (P119908119908) le 119862 lt +infin (64)

Therefore according toTheorem 2 there exists a continuousfunction on 119869 such that 119869119899119908 997888rarr as 119899 997888rarr +infin in the space(X 119889) and 119869119899 = that is satisfies (58) for each 119905 isin 119869

Observe that

X = 119908 isin X | 119889 (119908 119908) lt +infin (65)

Indeed for any 119908 isin X there exists a constant 0 lt 119862 lt +infinsuch that

119863 [119908 (119905) 119908 (119905)] le 119862120593 (119905) (66)

since119908 and119908 are bounded on 119869 andmin119905isin119869120593(119905) gt 0 It followsfrom the preceding inequality that

119889 (119908119908) lt +infin (67)

for all 119908 isin X Hence we obtained that X = 119908 isin X |119889(119908119908) lt +infinFrom Theorem 2 we infer that 119908 is a unique fixed point

of 119869 inX It is obvious that 119908 is a unique fuzzy function in Xwhich satisfies equation 119869119908 = 119908

On the other hand we have

119863[119906 (119905) 1199060 + int119905

119886119891 (119904 119906 (119904)) 119889119904

+ int119905119886int119904119886119892 (119904 119903 119906 (119903)) 119889119903119889119904] = 119863[119906 (119905)

⊖ 1199060 int119905

119886119891 (119904 119906 (119904)) 119889119904 + int119905

119886int119904119886119892 (119904 119903 119906 (119903)) 119889119903119889119904]

= 119863[int119905119886119863119892119867119906 (119904) 119889119904 int

119905

119886119891 (119904 119906 (119904)) 119889119904

+ int119905119886int119904119886119892 (119904 119903 119906 (119903)) 119889119903119889119904] le int119905

119886119863

sdot [119863119892119867119906 (119904) 119891 (119904 119906 (119904)) + int119904

119886119892 (119904 119903 119906 (119903)) 119889119903] 119889119904

le int119905119886120593 (119904) 119889119904 le 119862120593 (119905)

(68)

for any 119905 isin 119869 This means that

119889 (119906P119906) le 119862 (69)

Finally byTheorem 2 and inequation (69) we deduce that

119889 (119906 ) le 11 minus 119871119891119892 (119862 + 1198622)119889 (119906P119906)

le 1198621 minus 119871119891119892 (119862 + 1198622)120593 (119905)

(70)

for any 119905 isin 119869 which means that inequality (55) is true for all119905 isin 119869Corollary 16 Assume that 119891 and 119892 satisfy the assumption (i)of eorem 15 and 0 lt 119871119891119892(119887 minus 119886) + 119871119891119892((119887 minus 119886)22) lt 1

8 Complexity

For a given 120576 gt 0 if a continuously (1)-differentiable function119906 119869 997888rarr RF satisfies the following inequality

119863[119863119892119867119906 (119905) 119891 (119905 119906 (119905)) + int119905

119886119892 (119905 119904 119906 (119904)) 119889119904] le 120576 (71)

for any 119905 isin 119869 then there exists a unique (S1)-solution 119869 997888rarrRF of (14) such that

(119905) = 0 + int119905

119886119891 (119904 (119904)) 119889119904

+ int119905119886int119904119886119892 (119904 119903 (119903)) 119889119903119889119904

(72)

and

119863[ (119905) 119906 (119905)]le (119887 minus 119886) 1205761 minus (119871119891119892 (119887 minus 119886) + 119871119891119892 ((119887 minus 119886)2 2))

(73)

for any 119905 isin 119869Theorem 17 Suppose the functions 119891 119892 and 120593 satisfy allconditions as ineorem 15 If a continuously (2)-differentiablefunction 119906 119869 997888rarr RF satisfies inequality (53) in eorem 15for any 119905 isin 119869 then there exists a unique (S2)-solution 119869 997888rarrRF of (14) such that

(119905) = 0 ⊖ (minus1) int119905

119886119891 (119904 (119904)) 119889119904 ⊖ (minus1)

sdot int119905119886int119904119886119892 (119904 119903 (119903)) 119889119903119889119904

(74)

and the estimation (55) as in eorem 15 on 119869Proof Similar to the proof of Theorem 15 Consider theoperator H X 997888rarr X defined by

(HV) (119905) = V0 ⊖ (minus1)int119905

119886119891 (119904 V (119904)) 119889119904 ⊖ (minus1)

sdot int119905119886int119904119886119892 (119904 119903 V (119903)) 119889119903119889119904

(75)

for all V isin X Based on Lemmas 32 and 33 in [3] it is easy tosee that HV is (2)-differentiable and so HV isin X

We check the operator H is strict contractive on X LetV 119908 isin X and let 119862V119908 isin [0 +infin) cup +infin be an arbitraryconstant with 119889(V 119908) le 119862V119908 we have

119863 [V (119905) 119908 (119905)] le 119862V119908120593 (119905) forall119905 isin 119869 (76)

From the assumptions (i)-(ii) in Theorem 15 and (76) we get

119863 [(HV) (119905) (H119908) (119905)] = 119863[V0 ⊖ (minus1)

sdot int119905119886119891 (119904 V (119904)) 119889119904 ⊖ (minus1)int119905

119886int119904119886119892 (119904 119903 V (119903)) 119889119903119889119904 V0

⊖ (minus1)int119905119886119891 (119904 119908 (119904)) 119889119904 ⊖ (minus1)

sdot int119905119886int119904119886119892 (119904 119903 119908 (119903)) 119889119903119889119904]

le int119905119886119863[119891 (119904 V (119904)) 119891 (119904 119908 (119904))] 119889119904

+ int119905119886int119904119886119863 [119892 (119904 119903 V (119903)) 119892 (119904 119903 119908 (119903))] 119889119903119889119904

le 119871119891119892 int119905

119886119863[V (119904) 119908 (119904)] 119889119904

+ 119871119891119892 int119905

119886int119904119886119863[V (119903) 119908 (119903)] 119889119903119889119904

le 119871119891119892119862V119908 int119905

119886120593 (119904) 119889119904

+ 119871119891119892119862V119908 int119905

119886int119904119886120593 (119903) 119889119903119889119904 le 119871119891119892 (119862 + 1198622)

sdot 119862V119908120593 (119905)(77)

for any 119905 isin 119869 Hence by (76) we conclude that119889 ((HV) (119905) (H119908) (119905)) le 119862119871119891119892119889 (V 119908) forall119905 isin 119869 (78)

By the definitions of X and P we have for arbitrary 119908 isin Xthere exists a constant 0 lt 119862 lt +infin such that

119863[(H119908) (119905) 119908 (119905)] = 119863 [1199080 ⊖ (minus1)

sdot int119905119886119891 (119904 119908 (119904)) 119889119904 ⊖ (minus1)

sdot int119905119886int119904119886119892 (119904 119903 119908 (119903)) 119889119903119889119904 119908 (119905)] le 119862120593 (119905) forall119905 isin 119869

(79)

119891119892 and 119908 are bounded on 119869 and min119905isin119869120593(119905) gt 0 Thus bythe definition of 119889 we have 119889(119908 119908) lt +infin for all 119908 isin XHence we infer that X = 119908 isin X | 119889(119908119908) lt +infin FromTheorem 2 we deduce that 119908 is a unique fixed point of 119869 in119883 It is obvious that 119908 is a unique fuzzy function in X whichsatisfies the equality 119869119908 = 119908

On the other hand Hukuhara difference 1199060 ⊖ 119906(119905) existsfor all 119905 isin 119869 and from (52) and the definition of H that

119863[119906 (119905) 1199060 ⊖ (minus1) int119905

119886119891 (119904 119906 (119904)) 119889119904 ⊖ (minus1)

sdot int119905119886int119904119886119892 (119904 119903 119906 (119903)) 119889119903119889119904] = 119863[1199060 ⊖ 119906 (119905)

minus (int119905119886119891 (119904 119906 (119904)) 119889119904 + int119905

119886int119904119886119892 (119904 119903 119906 (119903)) 119889119903119889119904)]

= 119863[minusint119905119886119863119892119867119906 (119904) 119889119904

Complexity 9

minus (int119905119886119891 (119904 119906 (119904)) 119889119904 + int119905

119886int119904119886119892 (119904 119903 119906 (119903)) 119889119903119889119904)]

le int119905119886119863[119863119892119867119906 (119904) 119891 (119904 119906 (119904))

+ int119904119886119892 (119904 119903 119906 (119903)) 119889119903] 119889119904 le int119905

119886120593 (119904) 119889119904 le 119862120593 (119905)

(80)

for any 119905 isin 119869 which implies that

119889 (119906H119906) le 119862 (81)

ByTheorem 2 and inequation (81) we deduce that

119889 (119906 ) le 11 minus 119871119891119892 (119862 + 1198622)119889 (119906H119906)

le 1198621 minus 119871119891119892 (119862 + 1198622)120593 (119905)

(82)

for any 119905 isin 119869 which means that inequality (55) is true for all119905 isin 119869Corollary 18 Assume that 119891 and 119892 satisfy the assumption (i)of eorem 15 and 0 lt 119871119891119892(119887 minus 119886) + 119871119891119892((119887 minus 119886)22) lt 1For a given 120576 gt 0 if a continuously (2)-differentiable function119906 119869 997888rarr RF satisfies the following inequation

119863[119863g119867119906 (119905) 119891 (119905 119906 (119905)) + int

119905

119886119892 (119905 119904 119906 (119904)) 119889119904] le 120576 (83)

for any 119905 isin 119869 then there exists a unique (S2)-solution 119869 997888rarrRF of (14) such that

(119905) = 0 ⊖ (minus1) int119905

119886119891 (119904 (119904)) 119889119904 ⊖ (minus1)

sdot int119905119886int119904119886119892 (119904 119903 (119903)) 119889119903119889119904

(84)

and

119863[ (119905) 119906 (119905)]le (119887 minus 119886) 1205761 minus (119871119891119892 (119887 minus 119886) + 119871119891119892 ((119887 minus 119886)2 2))

(85)

for any 119905 isin 119869Example 19 We consider the fuzzy intergodifferential equa-tion as follows

119863119892119867119906 (119905) = 12 int119905

01199051199042119906 (119904) 119889119904 forall119905 isin [0 1] (86)

and the following inequality

119863[119863119892119867 (119905) 12 int119905

01199051199042 (119904) 119889119904] le 11989005119905 forall119905 119904 isin [0 1] (87)

where 119906 is a continuously (1)-differentiable (or (2)-differenti-able) function

It is easy to check that the functions 119891 119892 satisfy Lipschitzcondition with 119871119891119892 = 12 Choosing 120593(119905) = 1205761198902119905 with 120576 gt 0and 119862 = 05 we have

int1199050120593 (119904) 119889119904 = int119905

01205761198902119904119889119904 = 120576 (051198902119905 minus 1) le 051205761198902119905

= 1198621205761198902119905 forall119905 isin [0 1] (88)

Now all assumptions in Theorem 15 (or Theorem 17) aresatisfied problem (86) has a unique solution and (86) isUlam-Hyers-Rassias stable with

119863[ (119905) 119906 (119905)] le 1198621 minus 119871119891119892 (119862 + 1198622)120593 (119905) =

451205761198902119905 (89)

for any 119905 isin [0 1]In particular if we choose 120593(119905) = 120576 then we have

119863( (119905) 119906 (119905))le (119887 minus 119886) 1205761 minus (119871119891119892 (119887 minus 119886) + 119871119891119892 ((119887 minus 119886)2 2)) = 4120576

(90)

for all 119905 isin [0 1]4 Conlusion

In this study the Ulam-Hyers stability and Ulam-Hyers-Rassias stability for fuzzy intergodifferential equation via thefixed point technique and successive approximation methodare studied Moreover some illustrative examples are givenIn future work we will study Ulam stability for fuzzyintergodifferential equation in the quotient space of fuzzynumbers introduced by [25ndash27]

Data Availability

No data were used to support this study

Conflicts of Interest

The authors declare that they have no conflicts of interest

References

[1] O AbuArqub ldquoAdaptation of reproducing kernel algorithm forsolving fuzzy Fredholm-Volterra integrodifferential equationsrdquoNeural Computing and Applications vol 28 no 7 pp 1591ndash16102017

[2] R Alikhani and F Bahrami ldquoGlobal solutions of fuzzy integro-differential equations under generalized differentiability by themethod of upper and lower solutionsrdquo Information Sciences vol295 pp 600ndash608 2015

[3] R Alikhani F Bahrami and A Jabbari ldquoExistence of globalsolutions to nonlinear fuzzy Volterra integro-differential equa-tionsrdquo Nonlinear Analysis eory Methods amp Applications vol75 no 4 pp 1810ndash1821 2012

10 Complexity

[4] N V Hoa and N D Phu ldquoFuzzy functional integro-differentialequations under generalized h-differentiabilityrdquo Journal of Intel-ligent Fuzzy Systems vol 26 no 1 pp 2073ndash2085 2014

[5] R M Shabestari R Ezzati and T Allahviranloo ldquoSolvingfuzzy volterra integrodifferential equations of fractional orderby bernoulli wavelet methodrdquo Advances in Fuzzy Systems vol2018 Article ID 5603560 11 pages 2018

[6] H Vu ldquoRandom fuzzy differential equations with impulsesrdquoComplexity vol 2017 Article ID 4056016 11 pages 2017

[7] O Abu Arqub S Momani S Al-Mezel and M Kutbi ldquoExis-tence uniqueness and characterization theorems for nonlinearfuzzy integrodifferential equations of volterra typerdquoMathemat-ical Problems in Engineering vol 2015 Article ID 835891 13pages 2015

[8] E Eljaoui S Melliani and L S Chadli ldquoAumann fuzzy im-proper integral and its application to solve fuzzy integro-differential equations by laplace transform methodrdquo Advancesin Fuzzy Systems vol 2018 Article ID 9730502 10 pages 2018

[9] N V Hoa and N D Phu ldquoOn maximal and minimal solutionsfor set-valued differential equations with feedback controlrdquoAbstract and Applied Analysis vol 2012 Article ID 816218 11pages 2012

[10] H Vu ldquoExistence results for fuzzy Volterra integral equationrdquoJournal of Intelligent amp Fuzzy Systems Applications in Engineer-ing and Technology vol 33 no 1 pp 207ndash213 2017

[11] H Vu L S Dong andNN Phung ldquoApplication of contractive-like mapping principles to impulsive fuzzy functional dif-ferential equationrdquo Journal of Intelligent amp Fuzzy SystemsApplications in Engineering and Technology vol 33 no 2 pp753ndash759 2017

[12] J Vanterler da C Sousa and E Capelas de Oliveira ldquoUlam-Hyers stability of a nonlinear fractional Volterra integro-differential equationrdquo Applied Mathematics Letters vol 81 pp50ndash56 2018

[13] D Popa and I Rasa ldquoOn the Hyers-Ulam stability of the lineardifferential equationrdquo Journal of Mathematical Analysis andApplications vol 381 no 2 pp 530ndash537 2011

[14] G Wang M Zhou and L Sun ldquoHyers-Ulam stability oflinear differential equations of first orderrdquo Applied MathematicsLetters vol 21 no 10 pp 1024ndash1028 2008

[15] J R Wang L L Lv and Y Zhou ldquoNew concepts and results instability of fractional differential equationsrdquoCommunications inNonlinear Science and Numerical Simulation vol 17 no 6 pp2530ndash2538 2012

[16] E C de Oliveira and J V Sousa ldquoUlam-Hyers-Rassias stabilityfor a class of fractional integro-differential equationsrdquo Results inMathematics vol 111 no 72 pp 1ndash16 2018

[17] J V D C Sousa and E C D Oliveira ldquoFractional order pseu-doparabolic partial differential equation ulamndashhyers stabilityrdquoBulletin of the Brazilian Mathematical Society pp 1ndash16 2018

[18] J Vanterler da C Sousa K D Kucche and E C de OliveiraldquoStability of 120595-Hilfer impulsive fractional differential equa-tionsrdquo Applied Mathematics Letters vol 88 pp 73ndash80 2019

[19] Y Shen ldquoHyers-Ulam-Rassias stability of first order linearpartial fuzzy differential equations under generalized differ-entiabilityrdquo Advances in Difference Equations vol 2015 no 1article no 351 pp 1ndash18 2015

[20] Y Shen ldquoOn the Ulam stability of first order linear fuzzydifferential equations under generalized differentiabilityrdquo FuzzySets and Systems vol 280 no C pp 27ndash57 2015

[21] Y Shen and F Wang ldquoA fixed point approach to the Ulamstability of fuzzy differential equations under generalized differ-entiabilityrdquo Journal of Intelligent amp Fuzzy Systems Applicationsin Engineering and Technology vol 30 no 6 pp 3253ndash32602016

[22] W Ren Z Yang X Sun and M Qi ldquoHyers-Ulam stability ofHermite fuzzy differential equations and fuzzy Mellin trans-formrdquo Journal of Intelligent amp Fuzzy Systems Applications inEngineering and Technology vol 35 no 3 pp 3721ndash3731 2018

[23] J B Diaz and B Margolis ldquoA fixed point theorem of thealternative for contractions on a generalized complete metricspacerdquo Bulletin (New Series) of the American MathematicalSociety vol 74 no 2 pp 305ndash309 1968

[24] B Bede and S G Gal ldquoGeneralizations of the differentiabilityof fuzzy-number-valued functions with applications to fuzzydifferential equationsrdquo Fuzzy Sets and Systems vol 151 no 3pp 581ndash599 2005

[25] D Qiu W Zhang and C Lu ldquoOn fuzzy differential equationsin the quotient space of fuzzy numbersrdquo Fuzzy Sets and Systemsvol 295 pp 72ndash98 2016

[26] D Qiu andW Zhang ldquoSymmetric fuzzy numbers and additiveequivalence of fuzzy numbersrdquo So13 Computing vol 17 no 8 pp1471ndash1477 2013

[27] D Qiu C Lu W Zhang and Y Lan ldquoAlgebraic properties andtopological properties of the quotient space of fuzzy numbersbased on Mares equivalence relationrdquo Fuzzy Sets and Systemsvol 245 pp 63ndash82 2014

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Complexity 3

(i) If 119891 is (1)-differentiable then

119891 (119905) = 119891 (119886) + int119905119886119863119892119867119891 (119904) 119889119904 (10)

(ii) If 119891 is (2)-differentiable then

119891 (119905) = 119891 (119886) ⊖ (minus1)int119905119886119863119892119867119891 (119904) 119889119904 (11)

Let 119869 fl [119886 119887] (with 119879 gt 0) be a compact interval of RWe denote by

119862 (119869RF) = 119906 | 119906 119869997888rarr RF is a countinuous functions on 119869 (12)

On the space 119862(119869RF) we consider the supremummetric asfollows

119863 (119906 V) = sup119905isin119869

119863[119906 (119905) V (119905)] (13)

In next section we consider the following fuzzy integrodif-ferential equation

119863119892119867119906 (119905) = 119891 (119905 119906 (119905)) + int119905

119886119892 (119905 119904 119906 (119904)) 119889119904 119905 isin 119869

119906 (119886) = 1199060 isin RF(14)

where the symbol119863119892119867 is generalized Hukuhara derivative themapping 119891 119869 times RF times RF 997888rarr RF is continuous on 119869 and119892 119869 times 119869 timesRF 997888rarr RF is a continuous function on 119869 times 119869Definition 8 (see [3]) We say that a mapping 119906 119869 997888rarr RF

is solution to the problem (14) if 119906 is generalized Hukuharadifferentiable on 119869 and119863119892119867119906(119905) = 119891(119905 119906(119905)) + int119905119886 119892(119905 119904 119906(119904))119889119904for any 119905 isin 119869 119891 isin 119862(119869RF) 119892 isin 119862(119869 times 119869RF)Lemma9 (see [3]) Let 119906 119869 997888rarr RF be a continuous functionon 119869 Problem (14) is equivalent to one of the following the fuzzyintegro integral equations

(S1) if 119906 is (1)-differentiable on 119869 then119906 (119905) = 1199060 + int

119905

119886119891 (119904 119906 (119904)) 119889119904

+ int119905119886int119904119886119892 (119904 119903 119906 (119903)) 119889119903119889119904

(15)

(S2) if 119906 is (2)-differentiable on 119869 then119906 (119905) = 1199060 ⊖ (minus1) int

119905

119886119891 (119904 119906 (119904)) 119889119904 ⊖ (minus1)

sdot int119905119886int119904119886119892 (119904 119903 119906 (119903)) 119889119903119889119904

(16)

3 Main Results

Definition 10 We say that problem (14) is Ulam-Hyers stableif there exists a real number 119870119891 gt 0 such that for 120576 gt 0 andfor each V isin 1198621(119869RF) to the problem

119863[119863119892119867V (119905) 119891 (119905 V (119905)) + int119905

119886119892 (119905 119904 V (119904)) 119889119904] le 120576 (17)

there exists a solution to problem (14) with

119863[V (119905) 119906 (119905)] le 119870119891120576 (18)

for all 119905 isin 119869 We call 119870119891 a Ulam-Hyers stability constant of(14)

Definition 11 We say that problem (14) is Ulam-Hyers-Rassias stable if there exists a real number 119862119891 gt 0 such thatfor 120576 gt 0 and for each V isin 1198621(119869RF) to the problem

119863[119863119892119867V (119905) 119891 (119905 V (119905)) + int119905

0119892 (119905 119904 V (119904)) 119889119904] le 120593 (119905) (19)

there exists a solution to problem (14) with

119863[V (119905) 119906 (119905)] le 119862119891120593 (119905) (20)

for all 119905 isin 119869Firstly we prove that problem (14) is Hyers-Ulam stable

via the method of successive approximation We consider thefollowing inequality

119863[119863119892119867V (119905) 119891 (119905 V (119905)) + int119905

119886119892 (119905 119904 V (119904)) 119889119904] le 120576

for V isin RF(21)

Definition 12 We say that

(a) A function V isin 1198621(119869RF) is a (S1)-solution of theinequality (21) if and only if there exists a function1205751 isin 119862(119869RF) such that

(i) 119863[1205751(119905) 0] le 120576 for any 119905 isin 119869(ii) 119863119892119867V(119905) = 119891(119905 V(119905)) + int119905119886 119892(119905 119904 V(119904))119889119904 + 1205751(119905) for

any 119905 isin 119869(b) A function V isin 1198621(119869RF) is a (S2)-solution of the

inequality (21) if and only if there exists a function1205752 isin 119862(119869RF) such that

(i) 119863[1205752(119905) 0] le 120576 for any 119905 isin 119869(ii) 119863119892119867V(119905) = 119891(119905 V(119905)) + int119905119886 119892(119905 119904 V(119904))119889119904 + 1205752(119905) for

any 119905 isin 119869Theorem 13 Assume that 119891 119869 times RF 997888rarr RF and 119892 119869 times119869 times RF 997888rarr RF are a continuous function that satisfies thefollowing conditions (i) there exists a constant 119871119891119892 gt 0 suchthat

max 119863 [119891 (119905 119906) 119891 (119905 V)] 119863 [119892 (119905 119904 119906) 119892 (119905 119904 V)]le 119871119891119892119863 [119906 V] (22)

4 Complexity

for any (119905 119906) (119905 V) isin 119869 times RF (119905 119904 119906) (119905 119904 V) isin 119869 times 119869 times RF(ii) for each 120576 gt 0 if a continuously (2)-differentiable functionV 119869 997888rarr RF satisfies

119863[119863119892119867V (119905) 119891 (119905 V (119905)) + int119905

119886119892 (119905 119904 V (119904)) 119889119904] le 120576

forall119905 isin 119869(23)

then there exists a (S2)-solution 119906 119869 997888rarr RF of (14) with1199060 = V0 such that

119863[V (119905) 119906 (119905)] le 120576120582119871119891119892 119890(119887minus119886)(1+119871119891119892) (24)

where 120582 is a maximum balancing constant

Proof For each 120576 gt 0 let a continuously (2)-differentiablefunction V 119869 997888rarr RF satisfy inequality (21) for any 119905 isin 119869V isin RF By the part (b) of Definition 12 we have

119863[1205752 (119905) 0] le 120576 (25)

and

119863119892119867V (119905) = 119891 (119905 V (119905)) + int119905

119886119892 (119905 119904 V (119904)) 119889119904 + 1205752 (119905) (26)

for any 119905 isin 119869 and 1205752 isin 119862(119869RF)If a function V 119869 997888rarr RF is continuous and (2)-

differentiable on 119869 then by Lemma 9 it satisfies equivalentlythe following fuzzy integrointegral equation

V (119905) = V0 ⊖ (minus1)int119905

119886119891 (119904 V (119904)) 119889119904 ⊖ (minus1)

sdot int119905119886int119904119886119892 (119904 119903 V (119903)) 119889119903119889119904 + int119905

1198861205752 (119904) 119889119904

forall119905 isin 119869(27)

Let us define

1199060 (119905) = V (119905) forall119905 isin 119869 (28)

and sequence functions 119906119899 119869 997888rarr RF 119899 = 1 2 ofsuccessive approximations as follows

119906119899 (119905) = V0 ⊖ (minus1)int119905

119886119891 (119904 119906119899minus1 (119904)) 119889119904 ⊖ (minus1)

sdot int119905119886int119904119886119892 (119904 119903 119906119899minus1 (119903)) 119889119903119889119904 forall119905 isin 119869

(29)

By virtue of the properties of 119863 and assumption (i) we have

119863[1199061 (119905) 1199060 (119905)] = 119863 [V0 ⊖ (minus1)

sdot int119905119886119891 (119904 1199060 (119904)) 119889119904 ⊖ (minus1)

sdot int119905119886int119904119886119892 (119904 119903 1199060 (119903)) 119889119903119889119904 V (119905)] = 119863[V0 ⊖ (minus1)

sdot int119905119886119891 (119904 V0 (119904)) 119889119904 ⊖ (minus1)

sdot int119905119886int119904119886119892 (119904 119903 V0 (119903)) 119889119903119889119904 V (119905)]

= 119863[int1199051198861205752 (119904) 119889119904 0] le int

119905

119886119863[1205752 (119904) 0] 119889119904 le 120576 (119905

minus 119886) forall119905 isin 119869(30)

Therefore

119863[1199061 (119905) 1199060 (119905)] le 120576 (119905 minus 119886) forall119905 isin 119869 (31)

Observe that for 119899 = 1 2 and for 119905 isin 119869 one has119863 [119906119899+1 (119905) 119906119899 (119905)] = 119863[V0 ⊖ (minus1)

sdot int119905119886119891 (119904 119906119899 (119904)) 119889119904 ⊖ (minus1)

sdot int119905119886int119904119886119892 (119904 119903 119906119899 (119903)) 119889119903119889119904 V0 ⊖ (minus1)

sdot int119905119886119891 (119904 119906119899minus1 (119904)) 119889119904 ⊖ (minus1)

sdot int119905119886int119904119886119892 (119904 119903 119906119899minus1 (119903)) 119889119903119889119904]

le 119871119891119892 int119905

119886119863[119906119899 (119904) 119906119899minus1 (119904)] 119889119904

+ 119871119891119892 int119905

119886int119904119886119863 [119906119899 (119903) 119906119899minus1 (119903)] 119889119903119889119904

(32)

In particular

119863[1199062 (119905) 1199061 (119905)]le 119871119891119892 int

119905

119886119863[1199061 (119904) 1199060 (119904)] 119889119904

+ 119871119891119892 int119905

119886int119904119886119863[1199061 (119903) 1199060 (119903)] 119889119903119889119904

= 120576119871119891119892 int119905

119886(119904 minus 119886) 119889119904 + 120576119871119891119892 int

119905

119886int119904119886(119903 minus 119886) 119889119903119889119904

= 120576119871119891119892 ((119905 minus 119886)2

2 + (119905 minus 119886)33 )

= 120576119871119891119892

(119871119891119892 (119905 minus 119886))22 (1 + 119905 minus 119886

3 )

(33)

and so119863 [1199063 (119905) 1199062 (119905)]

le 119871119891119892 int119905

119886119863[1199062 (119904) 1199061 (119904)] 119889119904

+ 119871119891119892 int119905

119886int119904119886119863 [1199062 (119903) 1199061 (119903)] 119889119903119889119904

Complexity 5

= 1205761198712119891119892 int119905

119886((119904 minus 119886)22 + (119904 minus 119886)3

3 ) 119889119904

+ 1205761198712119891119892 int119905

119886int119904119886((119903 minus 119886)22 + (119903 minus 119886)3

3 ) 119889119903119889119904

= 1205761198712119891119892 ((119905 minus 119886)3

3 + 2(119905 minus 119886)44 + (119905 minus 119886)55 )

= 120576119871119891119892

(119871119891119892 (119905 minus 119886))33 (1 + 2119905 minus 1198864 + (119905 minus 119886)2

5 )(34)

and for 119899 gt 4 we have119863[119906119899 (119905) 119906119899minus1 (119905)] le 120576119871119899minus1119891119892 ((119905 minus 119886)

119899

119899 + 1205821 (119905 minus 119886)119899+1

(119899 + 1)+ + 120582119899 (119905 minus 119886)

2119899

(2119899) + (119905 minus 119886)2119899+1(2119899 + 1) )

(35)

where 1205821 120582119899 are balancing constantsWe choose120582 = max1 1205821 120582119899 (called120582 is amaximum

balancing constant) and then estimation (35) can be rewrittenas follows

119863 [119906119899 (119905) 119906119899minus1 (119905)] le 120576120582119871119891119892

(119871119891119892 (119905 minus 119886))119899119899 (1 + 119905 minus 119886

119899 + 1+ (119905 minus 119886)2(119899 + 1) (119899 + 2) + +

(119905 minus 119886)119899(119899 + 1) (119899 + 2) 2119899

+ (119905 minus 119886)119899+1(119899 + 1) (119899 + 2) 2119899 (2119899 + 1)) le 120576120582

119871119891119892sdot (119871119891119892 (119905 minus 119886))

119899

119899 (1 + 119905 minus 1198861 + (119905 minus 119886)2

2 +

+ (119905 minus 119886)119899119899 + (119905 minus 119886)119899+1

(119899 + 1) ) le 120576120582119871119891119892

(119871119891119892 (119905 minus 119886))119899119899

sdot 119890119905minus119886

(36)

Further if we assume that

119863[119906119899 (119905) 119906119899minus1 (119905)] le 120576120582119871119891119892

(119871119891119892 (119905 minus 119886))119899119899 119890119905minus119886

forall119905 isin 119869(37)

then we obtain

119863[119906119899+1 (119905) 119906119899 (119905)] le 120576120582119871119891119892

(119871119891119892 (119905 minus 119886))119899+1(119899 + 1) 119890119905minus119886

forall119905 isin 119869(38)

By the principle ofmathematical induction that (37) holds forevery 119899 gt 1 and now using estimation (37) for any 119905 isin 119869 weget

infinsum119899=1

119863 [119906119899 (119905) 119906119899minus1 (119905)] le 120576120582119871119891119892infinsum119899=1

(119871119891119892 (119905 minus 119886))119899119899 119890119905minus119886 (39)

The series suminfin119895=1(119911119896119896) is convergent for every 119911 isin R Hencefor every 120576 gt 0 we infer that the series suminfin119899=1119863[119906119899(119905) 119906119899minus1(119905)]is uniformly convergent on 119869 with respect to metric 119863 and

infinsum119899=1

119863 [119906119899 (119905) 119906119899minus1 (119905)] le 120576120582119871119891119892infinsum119899=1

(119871119891119892 (119905 minus 119886))119899119899 119890119905minus119886

le 120576120582119871119891119892infinsum119899=1

(119871119891119892 (119887 minus 119886))119899119899 119890119887minus119886 = 120576120582

119871119891119892 119890119871119891119892(119887minus119886)119890119887minus119886

= 120576120582119871119891119892 119890(119887minus119886)(1+119871119891119892)

(40)

For 119905 isin 119869 we have119863 [119906119899 (119905) 119906 (119905)] le 120576120582

119871119891119892infinsum119899=1

(119871119891119892 (119905 minus 119886))119899119899 119890119905minus119886 (41)

and the following estimation

119863[int119905119886119891 (119904 119906119899 (119904)) 119889119904

+ int119905119886int119904119886119892 (119904 119903 119906119899 (119903)) 119889119903119889119904 int

119905

119886119891 (119904 119906 (119904)) 119889119904

+ int119905119886int119904119886119892 (119904 119903 119906 (119903)) 119889119903119889119904]

le int119905119886119863[119891 (119904 119906119899 (119904)) 119891 (119904 119906 (119904))] 119889119904

+ int119905119886int119904119886119863 [119892 (119904 119903 119906119899 (119903)) 119892 (119904 119903 119906 (119903))] 119889119903119889119904

le 119871119891119892 int119905

119886119863 [119906119899 (119904) 119906 (119904)] 119889119904

+ 119871119891119892 int119905

119886int119904119886119863[119906119899 (119903) 119906 (119903)] 119889119903119889119904 forall119905 isin 119869

(42)

Combining estimation (41) and inequality (42) we obtainthat

119863[int119905119886119891 (119904 119906119899 (119904)) 119889119904 ⊖ (minus1)

sdot int119905119886int119904119886119892 (119904 119903 119906119899 (119903)) 119889119903119889119904 int

119905

119886119891 (119904 119906 (119904)) 119889119904 ⊖ (minus1)

sdot int119905119886int119904119886119892 (119904 119903 119906 (119903)) 119889119903119889119904]

(43)

6 Complexity

converges to 0 uniformly as 119899 997888rarr +infin Therefore 119906(119905) is a(S2)-solution of (14) with initial condition 1199060

Finally we shall prove that problem (14) has a unique(S2)-solution Assume that is another (S2)-solution of (14)with initial condition 1199060 Then we have for any 119905 isin 119869

119863 [119906 (119905) (119905)] le 119871119891119892 int119905

119886119863 [119906 (119904) (119904)] 119889119904

+ 119871119891119892 int119905

119886int119905119886119863[119906 (119903) (119903)] 119889119903119889119904

(44)

If we let 120585(119905) = 119863[119906(119905) (119905)] for any 119905 isin 119869 then120585 (119905) le 119871119891119892 int

119905

119886120585 (119904) 119889119904 + 119871119891119892 int

119905

119886int119905119886120585 (119903) 119889119903119889119904 (45)

Applying Lemma 23 in Hoa et al [4] we obtain 120585(119905) = 0 forany 119905 isin 119869 This completes the proof

Example 14 Consider the following fuzzy integro differentialequation

119863119892119867119906 (119905) = (minus2 0 2) + int119905

0

119906 (119904)1 + 119906 (119904)119889119904 (46)

119906 (0) = (minus2 0 2) (47)

where 119906 isin 119862([0 1]RF) 119891 [0 1] times RF 997888rarr RF and 119892 [0 1] times [0 1] timesRF 997888rarr RFLet

119891 (119905 119906) = (minus2 0 2) 119892 (119905 119904 119906) = 119906

1 + 119906 forall119905 119904 isin [0 1] 119906 isin RF (48)

It is easy to see that 119891 119892 satisfy Lipschitz condition withLipschitz constant 119871119891119892 = 1 Indeed for any 119905 119904 isin [0 1] and119906 isin RF

119863 [119892 (119905 119904 119906) 119892 (119905 119904 V)] = 119863 [ 1199061 + 119906

V1 + V

]= sup120572isin[01]

max10038161003816100381610038161003816100381610038161003816119906

1 + 119906 minusV

1 + V

10038161003816100381610038161003816100381610038161003816 10038161003816100381610038161003816100381610038161003816119906

1 + 119906 minusV

1 + V

10038161003816100381610038161003816100381610038161003816

= sup120572isin[01]

max 1003816100381610038161003816119906 minus V100381610038161003816100381610038161003816100381610038161 + 1199061003816100381610038161003816 10038161003816100381610038161 + V1003816100381610038161003816 |119906 minus V|

|1 + 119906| |1 + V|le sup120572isin[01]

max 1003816100381610038161003816119906 minus V1003816100381610038161003816 |119906 minus V| = 119863 [119906 V]

(49)

Hence byTheorem 15 the problem (46)-(47)has unique (S1)-solution or (S2)-solution on [0 1]

Moreover if a continuously (2)-differentiable function V [0 1] 997888rarr RF satisfies the following inequation

119863[119863119892119867V (119905) (minus2 0 2) + int119905

0

V (119904)1 + V (119904)119889119904] le 120576 (50)

then as shown in Theorem 15 there exists a (S2)-solution 119906 [0 1] 997888rarr RF of (46) such that

119863 [V (119905) 119906 (119905)] le 1205761205821198902 forall119905 isin [0 1] (51)

where 120582 is the maximum balancing constant

Secondly we shall prove the Ulam-Hyers-Rassias stabilityof the FIDE (14) defined on a bounded and closed interval

Theorem 15 Assume that 119891 119869 997888rarr RF 997888rarr RF and119892 119869 times 119869 times RF 997888rarr RF are continuous function satisfyingthe following conditions (i) there exists a constant 119871119891119892 gt0 such that max119863[119891(119905 119906) 119891(119905 V)]119863[119892(119905 119904 119906) 119892(119905 119904 V)] le119871119891119892119863[119906 V] for each (119905 119904 119906) (119905 119904 V) isin 119869 times 119869 times RF (ii) thereexists a constant 119862 gt 0 such that 0 lt 119871119891119892(119862 + 1198622) lt 1 Let120593 119869 997888rarr (119886 +infin) be a continuous function and increasing on119869 with

int119905119886120593 (119904) 119889119904 le 119862120593 (119905) for each 119905 isin 119869 (52)

If a continuously (1)-differentiable function 119906 119869 997888rarr RF

satisfies the following inequality

119863[119863119892119867119906 (119905) 119891 (119905 119906 (119905)) + int119905

119886119892 (119905 119904 119906 (119904)) 119889119904] le 120593 (119905) (53)

for any 119905 isin 119869 then there exists a unique (S1)-solution 119869 997888rarrRF of (14) such that

(119905) = 0 + int119905

119886119891 (119904 (119904)) 119889119904

+ int119905119886int119904119886119892 (119904 119903 (119903)) 119889119903119889s

(54)

and

119863[ (119905) 119906 (119905)] le 1198621 minus 119871119891119892 (119862 + 1198622)120593 (119905) (55)

for any 119905 isin 119869Proof Let us define a setX of all continuous fuzzy functions119908 119869 997888rarr RF by

X = 119908 119869 997888rarr RF | 119908 is continuous on 119869 (56)

equipped with the metric

119889 (V 119908) = inf 119862 isin [0 +infin) cup +infin | 119863 [V (119905) 119908 (119905)]le 119862120593 (119905) forall119905 isin 119869 (57)

It is easy to see that (X 119889) is also a complete generalizedmetric space (see Lemma 3)

The operator P X 997888rarr X is defined as follows

(PV) (119905) = V0 + int119905

119886119891 (119904 V (119904)) 119889119904

+ int119905119886int119904119886119892 (119904 119903 V (119903)) 119889119903119889119904 forall119905 isin 119869

(58)

Based on Lemma 32 and 33 in [3] we infer that PV is (1)-differentiable and so PV isin X

The operator P is strict contractive on X Indeed for anyV 119908 isin X and letting 119862V119908 isin [0 +infin) cup +infin be an arbitrary

Complexity 7

constant with 119889(V 119908) le 119862V119908 that is by the definition of themetric 119889 we have

119863 [V (119905) 119908 (119905)] le 119862V119908120593 (119905) forall119905 isin 119869 (59)

From the definition of the metric 119863 and assumptions (i)-(ii)of Theorem 15 and inequality (59) we infer that

119863[(PV) (119905) (P119908) (119905)] = 119863[V0 + int119905

119886119891 (119904 V (119904)) 119889119904

+ int119905119886int119904119886119892 (119904 119903 V (119903)) 119889119903119889119904 V0 + int

119905

119886119891 (119904 119908 (119904)) 119889119904

+ int119905119886int119904119886119892 (119904 119903 119908 (119903)) 119889119903119889119904]

le int119905119886119863[119891 (119904 V (119904)) 119891 (119904 119908 (119904))] 119889119904

+ int119905119886int119904119886119863[119892 (119904 119903 V (119903)) 119892 (119904 119903 119908 (119903))] 119889119903119889119904

le 119871119891119892 int119905

119886119863 [V (119904) 119908 (119904)] 119889119904

+ 119871119891119892 int119905

119886int119904119886119863[V (119903) 119908 (119903)] 119889119903119889119904

le 119871119891119892119862V119908 int119905

119886120593 (119904) 119889119904

+ 119871119891119892119862V119908 int119905

119886int119904119886120593 (119903) 119889119903119889119904 le 119871119891119892 (119862 + 1198622)

sdot 119862V119908120593 (119905)

(60)

for any 119905 isin 119869For each V 119908 isin X and by the definition of metric 119889 we

get

119889 ((PV) (119905) (P119908) (119905)) le 119871119891119892 (119862 + 1198622)119862V119908120593 (119905) (61)

for any 119905 isin 119869 Hence by (59) we can conclude that

119889 ((PV) (119905) (P119908) (119905)) le 119871119891119892 (119862 + 1198622) 119889 (V 119908) (62)

for any 119905 isin 119869It follows form the definitions of X and the operator P

that for arbitrary 119908 isin X there exists a constant 0 lt 119862 lt +infinsuch that

119863[(P119908) (119905) 119908 (119905)] = 119863 [1199080 + int119905

119886119891 (119904 119908 (119904)) 119889119904

+ int119905119886int119904119886119892 (119904 119903 119908 (119903)) 119889119903119889119904 119908 (119905)] le 119862120593 (119905)

(63)

for any 119905 isin 119869 since 119891(119905 119908(119905)) 119892(119905 119904 119908(119904)) and 119908(119905) arebounded on 119869 and min119905isin119869120593(119905) gt 0 Thus by definition of 119889 itis implied that

119889 (P119908119908) le 119862 lt +infin (64)

Therefore according toTheorem 2 there exists a continuousfunction on 119869 such that 119869119899119908 997888rarr as 119899 997888rarr +infin in the space(X 119889) and 119869119899 = that is satisfies (58) for each 119905 isin 119869

Observe that

X = 119908 isin X | 119889 (119908 119908) lt +infin (65)

Indeed for any 119908 isin X there exists a constant 0 lt 119862 lt +infinsuch that

119863 [119908 (119905) 119908 (119905)] le 119862120593 (119905) (66)

since119908 and119908 are bounded on 119869 andmin119905isin119869120593(119905) gt 0 It followsfrom the preceding inequality that

119889 (119908119908) lt +infin (67)

for all 119908 isin X Hence we obtained that X = 119908 isin X |119889(119908119908) lt +infinFrom Theorem 2 we infer that 119908 is a unique fixed point

of 119869 inX It is obvious that 119908 is a unique fuzzy function in Xwhich satisfies equation 119869119908 = 119908

On the other hand we have

119863[119906 (119905) 1199060 + int119905

119886119891 (119904 119906 (119904)) 119889119904

+ int119905119886int119904119886119892 (119904 119903 119906 (119903)) 119889119903119889119904] = 119863[119906 (119905)

⊖ 1199060 int119905

119886119891 (119904 119906 (119904)) 119889119904 + int119905

119886int119904119886119892 (119904 119903 119906 (119903)) 119889119903119889119904]

= 119863[int119905119886119863119892119867119906 (119904) 119889119904 int

119905

119886119891 (119904 119906 (119904)) 119889119904

+ int119905119886int119904119886119892 (119904 119903 119906 (119903)) 119889119903119889119904] le int119905

119886119863

sdot [119863119892119867119906 (119904) 119891 (119904 119906 (119904)) + int119904

119886119892 (119904 119903 119906 (119903)) 119889119903] 119889119904

le int119905119886120593 (119904) 119889119904 le 119862120593 (119905)

(68)

for any 119905 isin 119869 This means that

119889 (119906P119906) le 119862 (69)

Finally byTheorem 2 and inequation (69) we deduce that

119889 (119906 ) le 11 minus 119871119891119892 (119862 + 1198622)119889 (119906P119906)

le 1198621 minus 119871119891119892 (119862 + 1198622)120593 (119905)

(70)

for any 119905 isin 119869 which means that inequality (55) is true for all119905 isin 119869Corollary 16 Assume that 119891 and 119892 satisfy the assumption (i)of eorem 15 and 0 lt 119871119891119892(119887 minus 119886) + 119871119891119892((119887 minus 119886)22) lt 1

8 Complexity

For a given 120576 gt 0 if a continuously (1)-differentiable function119906 119869 997888rarr RF satisfies the following inequality

119863[119863119892119867119906 (119905) 119891 (119905 119906 (119905)) + int119905

119886119892 (119905 119904 119906 (119904)) 119889119904] le 120576 (71)

for any 119905 isin 119869 then there exists a unique (S1)-solution 119869 997888rarrRF of (14) such that

(119905) = 0 + int119905

119886119891 (119904 (119904)) 119889119904

+ int119905119886int119904119886119892 (119904 119903 (119903)) 119889119903119889119904

(72)

and

119863[ (119905) 119906 (119905)]le (119887 minus 119886) 1205761 minus (119871119891119892 (119887 minus 119886) + 119871119891119892 ((119887 minus 119886)2 2))

(73)

for any 119905 isin 119869Theorem 17 Suppose the functions 119891 119892 and 120593 satisfy allconditions as ineorem 15 If a continuously (2)-differentiablefunction 119906 119869 997888rarr RF satisfies inequality (53) in eorem 15for any 119905 isin 119869 then there exists a unique (S2)-solution 119869 997888rarrRF of (14) such that

(119905) = 0 ⊖ (minus1) int119905

119886119891 (119904 (119904)) 119889119904 ⊖ (minus1)

sdot int119905119886int119904119886119892 (119904 119903 (119903)) 119889119903119889119904

(74)

and the estimation (55) as in eorem 15 on 119869Proof Similar to the proof of Theorem 15 Consider theoperator H X 997888rarr X defined by

(HV) (119905) = V0 ⊖ (minus1)int119905

119886119891 (119904 V (119904)) 119889119904 ⊖ (minus1)

sdot int119905119886int119904119886119892 (119904 119903 V (119903)) 119889119903119889119904

(75)

for all V isin X Based on Lemmas 32 and 33 in [3] it is easy tosee that HV is (2)-differentiable and so HV isin X

We check the operator H is strict contractive on X LetV 119908 isin X and let 119862V119908 isin [0 +infin) cup +infin be an arbitraryconstant with 119889(V 119908) le 119862V119908 we have

119863 [V (119905) 119908 (119905)] le 119862V119908120593 (119905) forall119905 isin 119869 (76)

From the assumptions (i)-(ii) in Theorem 15 and (76) we get

119863 [(HV) (119905) (H119908) (119905)] = 119863[V0 ⊖ (minus1)

sdot int119905119886119891 (119904 V (119904)) 119889119904 ⊖ (minus1)int119905

119886int119904119886119892 (119904 119903 V (119903)) 119889119903119889119904 V0

⊖ (minus1)int119905119886119891 (119904 119908 (119904)) 119889119904 ⊖ (minus1)

sdot int119905119886int119904119886119892 (119904 119903 119908 (119903)) 119889119903119889119904]

le int119905119886119863[119891 (119904 V (119904)) 119891 (119904 119908 (119904))] 119889119904

+ int119905119886int119904119886119863 [119892 (119904 119903 V (119903)) 119892 (119904 119903 119908 (119903))] 119889119903119889119904

le 119871119891119892 int119905

119886119863[V (119904) 119908 (119904)] 119889119904

+ 119871119891119892 int119905

119886int119904119886119863[V (119903) 119908 (119903)] 119889119903119889119904

le 119871119891119892119862V119908 int119905

119886120593 (119904) 119889119904

+ 119871119891119892119862V119908 int119905

119886int119904119886120593 (119903) 119889119903119889119904 le 119871119891119892 (119862 + 1198622)

sdot 119862V119908120593 (119905)(77)

for any 119905 isin 119869 Hence by (76) we conclude that119889 ((HV) (119905) (H119908) (119905)) le 119862119871119891119892119889 (V 119908) forall119905 isin 119869 (78)

By the definitions of X and P we have for arbitrary 119908 isin Xthere exists a constant 0 lt 119862 lt +infin such that

119863[(H119908) (119905) 119908 (119905)] = 119863 [1199080 ⊖ (minus1)

sdot int119905119886119891 (119904 119908 (119904)) 119889119904 ⊖ (minus1)

sdot int119905119886int119904119886119892 (119904 119903 119908 (119903)) 119889119903119889119904 119908 (119905)] le 119862120593 (119905) forall119905 isin 119869

(79)

119891119892 and 119908 are bounded on 119869 and min119905isin119869120593(119905) gt 0 Thus bythe definition of 119889 we have 119889(119908 119908) lt +infin for all 119908 isin XHence we infer that X = 119908 isin X | 119889(119908119908) lt +infin FromTheorem 2 we deduce that 119908 is a unique fixed point of 119869 in119883 It is obvious that 119908 is a unique fuzzy function in X whichsatisfies the equality 119869119908 = 119908

On the other hand Hukuhara difference 1199060 ⊖ 119906(119905) existsfor all 119905 isin 119869 and from (52) and the definition of H that

119863[119906 (119905) 1199060 ⊖ (minus1) int119905

119886119891 (119904 119906 (119904)) 119889119904 ⊖ (minus1)

sdot int119905119886int119904119886119892 (119904 119903 119906 (119903)) 119889119903119889119904] = 119863[1199060 ⊖ 119906 (119905)

minus (int119905119886119891 (119904 119906 (119904)) 119889119904 + int119905

119886int119904119886119892 (119904 119903 119906 (119903)) 119889119903119889119904)]

= 119863[minusint119905119886119863119892119867119906 (119904) 119889119904

Complexity 9

minus (int119905119886119891 (119904 119906 (119904)) 119889119904 + int119905

119886int119904119886119892 (119904 119903 119906 (119903)) 119889119903119889119904)]

le int119905119886119863[119863119892119867119906 (119904) 119891 (119904 119906 (119904))

+ int119904119886119892 (119904 119903 119906 (119903)) 119889119903] 119889119904 le int119905

119886120593 (119904) 119889119904 le 119862120593 (119905)

(80)

for any 119905 isin 119869 which implies that

119889 (119906H119906) le 119862 (81)

ByTheorem 2 and inequation (81) we deduce that

119889 (119906 ) le 11 minus 119871119891119892 (119862 + 1198622)119889 (119906H119906)

le 1198621 minus 119871119891119892 (119862 + 1198622)120593 (119905)

(82)

for any 119905 isin 119869 which means that inequality (55) is true for all119905 isin 119869Corollary 18 Assume that 119891 and 119892 satisfy the assumption (i)of eorem 15 and 0 lt 119871119891119892(119887 minus 119886) + 119871119891119892((119887 minus 119886)22) lt 1For a given 120576 gt 0 if a continuously (2)-differentiable function119906 119869 997888rarr RF satisfies the following inequation

119863[119863g119867119906 (119905) 119891 (119905 119906 (119905)) + int

119905

119886119892 (119905 119904 119906 (119904)) 119889119904] le 120576 (83)

for any 119905 isin 119869 then there exists a unique (S2)-solution 119869 997888rarrRF of (14) such that

(119905) = 0 ⊖ (minus1) int119905

119886119891 (119904 (119904)) 119889119904 ⊖ (minus1)

sdot int119905119886int119904119886119892 (119904 119903 (119903)) 119889119903119889119904

(84)

and

119863[ (119905) 119906 (119905)]le (119887 minus 119886) 1205761 minus (119871119891119892 (119887 minus 119886) + 119871119891119892 ((119887 minus 119886)2 2))

(85)

for any 119905 isin 119869Example 19 We consider the fuzzy intergodifferential equa-tion as follows

119863119892119867119906 (119905) = 12 int119905

01199051199042119906 (119904) 119889119904 forall119905 isin [0 1] (86)

and the following inequality

119863[119863119892119867 (119905) 12 int119905

01199051199042 (119904) 119889119904] le 11989005119905 forall119905 119904 isin [0 1] (87)

where 119906 is a continuously (1)-differentiable (or (2)-differenti-able) function

It is easy to check that the functions 119891 119892 satisfy Lipschitzcondition with 119871119891119892 = 12 Choosing 120593(119905) = 1205761198902119905 with 120576 gt 0and 119862 = 05 we have

int1199050120593 (119904) 119889119904 = int119905

01205761198902119904119889119904 = 120576 (051198902119905 minus 1) le 051205761198902119905

= 1198621205761198902119905 forall119905 isin [0 1] (88)

Now all assumptions in Theorem 15 (or Theorem 17) aresatisfied problem (86) has a unique solution and (86) isUlam-Hyers-Rassias stable with

119863[ (119905) 119906 (119905)] le 1198621 minus 119871119891119892 (119862 + 1198622)120593 (119905) =

451205761198902119905 (89)

for any 119905 isin [0 1]In particular if we choose 120593(119905) = 120576 then we have

119863( (119905) 119906 (119905))le (119887 minus 119886) 1205761 minus (119871119891119892 (119887 minus 119886) + 119871119891119892 ((119887 minus 119886)2 2)) = 4120576

(90)

for all 119905 isin [0 1]4 Conlusion

In this study the Ulam-Hyers stability and Ulam-Hyers-Rassias stability for fuzzy intergodifferential equation via thefixed point technique and successive approximation methodare studied Moreover some illustrative examples are givenIn future work we will study Ulam stability for fuzzyintergodifferential equation in the quotient space of fuzzynumbers introduced by [25ndash27]

Data Availability

No data were used to support this study

Conflicts of Interest

The authors declare that they have no conflicts of interest

References

[1] O AbuArqub ldquoAdaptation of reproducing kernel algorithm forsolving fuzzy Fredholm-Volterra integrodifferential equationsrdquoNeural Computing and Applications vol 28 no 7 pp 1591ndash16102017

[2] R Alikhani and F Bahrami ldquoGlobal solutions of fuzzy integro-differential equations under generalized differentiability by themethod of upper and lower solutionsrdquo Information Sciences vol295 pp 600ndash608 2015

[3] R Alikhani F Bahrami and A Jabbari ldquoExistence of globalsolutions to nonlinear fuzzy Volterra integro-differential equa-tionsrdquo Nonlinear Analysis eory Methods amp Applications vol75 no 4 pp 1810ndash1821 2012

10 Complexity

[4] N V Hoa and N D Phu ldquoFuzzy functional integro-differentialequations under generalized h-differentiabilityrdquo Journal of Intel-ligent Fuzzy Systems vol 26 no 1 pp 2073ndash2085 2014

[5] R M Shabestari R Ezzati and T Allahviranloo ldquoSolvingfuzzy volterra integrodifferential equations of fractional orderby bernoulli wavelet methodrdquo Advances in Fuzzy Systems vol2018 Article ID 5603560 11 pages 2018

[6] H Vu ldquoRandom fuzzy differential equations with impulsesrdquoComplexity vol 2017 Article ID 4056016 11 pages 2017

[7] O Abu Arqub S Momani S Al-Mezel and M Kutbi ldquoExis-tence uniqueness and characterization theorems for nonlinearfuzzy integrodifferential equations of volterra typerdquoMathemat-ical Problems in Engineering vol 2015 Article ID 835891 13pages 2015

[8] E Eljaoui S Melliani and L S Chadli ldquoAumann fuzzy im-proper integral and its application to solve fuzzy integro-differential equations by laplace transform methodrdquo Advancesin Fuzzy Systems vol 2018 Article ID 9730502 10 pages 2018

[9] N V Hoa and N D Phu ldquoOn maximal and minimal solutionsfor set-valued differential equations with feedback controlrdquoAbstract and Applied Analysis vol 2012 Article ID 816218 11pages 2012

[10] H Vu ldquoExistence results for fuzzy Volterra integral equationrdquoJournal of Intelligent amp Fuzzy Systems Applications in Engineer-ing and Technology vol 33 no 1 pp 207ndash213 2017

[11] H Vu L S Dong andNN Phung ldquoApplication of contractive-like mapping principles to impulsive fuzzy functional dif-ferential equationrdquo Journal of Intelligent amp Fuzzy SystemsApplications in Engineering and Technology vol 33 no 2 pp753ndash759 2017

[12] J Vanterler da C Sousa and E Capelas de Oliveira ldquoUlam-Hyers stability of a nonlinear fractional Volterra integro-differential equationrdquo Applied Mathematics Letters vol 81 pp50ndash56 2018

[13] D Popa and I Rasa ldquoOn the Hyers-Ulam stability of the lineardifferential equationrdquo Journal of Mathematical Analysis andApplications vol 381 no 2 pp 530ndash537 2011

[14] G Wang M Zhou and L Sun ldquoHyers-Ulam stability oflinear differential equations of first orderrdquo Applied MathematicsLetters vol 21 no 10 pp 1024ndash1028 2008

[15] J R Wang L L Lv and Y Zhou ldquoNew concepts and results instability of fractional differential equationsrdquoCommunications inNonlinear Science and Numerical Simulation vol 17 no 6 pp2530ndash2538 2012

[16] E C de Oliveira and J V Sousa ldquoUlam-Hyers-Rassias stabilityfor a class of fractional integro-differential equationsrdquo Results inMathematics vol 111 no 72 pp 1ndash16 2018

[17] J V D C Sousa and E C D Oliveira ldquoFractional order pseu-doparabolic partial differential equation ulamndashhyers stabilityrdquoBulletin of the Brazilian Mathematical Society pp 1ndash16 2018

[18] J Vanterler da C Sousa K D Kucche and E C de OliveiraldquoStability of 120595-Hilfer impulsive fractional differential equa-tionsrdquo Applied Mathematics Letters vol 88 pp 73ndash80 2019

[19] Y Shen ldquoHyers-Ulam-Rassias stability of first order linearpartial fuzzy differential equations under generalized differ-entiabilityrdquo Advances in Difference Equations vol 2015 no 1article no 351 pp 1ndash18 2015

[20] Y Shen ldquoOn the Ulam stability of first order linear fuzzydifferential equations under generalized differentiabilityrdquo FuzzySets and Systems vol 280 no C pp 27ndash57 2015

[21] Y Shen and F Wang ldquoA fixed point approach to the Ulamstability of fuzzy differential equations under generalized differ-entiabilityrdquo Journal of Intelligent amp Fuzzy Systems Applicationsin Engineering and Technology vol 30 no 6 pp 3253ndash32602016

[22] W Ren Z Yang X Sun and M Qi ldquoHyers-Ulam stability ofHermite fuzzy differential equations and fuzzy Mellin trans-formrdquo Journal of Intelligent amp Fuzzy Systems Applications inEngineering and Technology vol 35 no 3 pp 3721ndash3731 2018

[23] J B Diaz and B Margolis ldquoA fixed point theorem of thealternative for contractions on a generalized complete metricspacerdquo Bulletin (New Series) of the American MathematicalSociety vol 74 no 2 pp 305ndash309 1968

[24] B Bede and S G Gal ldquoGeneralizations of the differentiabilityof fuzzy-number-valued functions with applications to fuzzydifferential equationsrdquo Fuzzy Sets and Systems vol 151 no 3pp 581ndash599 2005

[25] D Qiu W Zhang and C Lu ldquoOn fuzzy differential equationsin the quotient space of fuzzy numbersrdquo Fuzzy Sets and Systemsvol 295 pp 72ndash98 2016

[26] D Qiu andW Zhang ldquoSymmetric fuzzy numbers and additiveequivalence of fuzzy numbersrdquo So13 Computing vol 17 no 8 pp1471ndash1477 2013

[27] D Qiu C Lu W Zhang and Y Lan ldquoAlgebraic properties andtopological properties of the quotient space of fuzzy numbersbased on Mares equivalence relationrdquo Fuzzy Sets and Systemsvol 245 pp 63ndash82 2014

Hindawiwwwhindawicom Volume 2018

MathematicsJournal of

Hindawiwwwhindawicom Volume 2018

Mathematical Problems in Engineering

Applied MathematicsJournal of

Hindawiwwwhindawicom Volume 2018

Probability and StatisticsHindawiwwwhindawicom Volume 2018

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Mathematical PhysicsAdvances in

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OptimizationJournal of

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Hindawiwwwhindawicom Volume 2018

Engineering Mathematics

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4 Complexity

for any (119905 119906) (119905 V) isin 119869 times RF (119905 119904 119906) (119905 119904 V) isin 119869 times 119869 times RF(ii) for each 120576 gt 0 if a continuously (2)-differentiable functionV 119869 997888rarr RF satisfies

119863[119863119892119867V (119905) 119891 (119905 V (119905)) + int119905

119886119892 (119905 119904 V (119904)) 119889119904] le 120576

forall119905 isin 119869(23)

then there exists a (S2)-solution 119906 119869 997888rarr RF of (14) with1199060 = V0 such that

119863[V (119905) 119906 (119905)] le 120576120582119871119891119892 119890(119887minus119886)(1+119871119891119892) (24)

where 120582 is a maximum balancing constant

Proof For each 120576 gt 0 let a continuously (2)-differentiablefunction V 119869 997888rarr RF satisfy inequality (21) for any 119905 isin 119869V isin RF By the part (b) of Definition 12 we have

119863[1205752 (119905) 0] le 120576 (25)

and

119863119892119867V (119905) = 119891 (119905 V (119905)) + int119905

119886119892 (119905 119904 V (119904)) 119889119904 + 1205752 (119905) (26)

for any 119905 isin 119869 and 1205752 isin 119862(119869RF)If a function V 119869 997888rarr RF is continuous and (2)-

differentiable on 119869 then by Lemma 9 it satisfies equivalentlythe following fuzzy integrointegral equation

V (119905) = V0 ⊖ (minus1)int119905

119886119891 (119904 V (119904)) 119889119904 ⊖ (minus1)

sdot int119905119886int119904119886119892 (119904 119903 V (119903)) 119889119903119889119904 + int119905

1198861205752 (119904) 119889119904

forall119905 isin 119869(27)

Let us define

1199060 (119905) = V (119905) forall119905 isin 119869 (28)

and sequence functions 119906119899 119869 997888rarr RF 119899 = 1 2 ofsuccessive approximations as follows

119906119899 (119905) = V0 ⊖ (minus1)int119905

119886119891 (119904 119906119899minus1 (119904)) 119889119904 ⊖ (minus1)

sdot int119905119886int119904119886119892 (119904 119903 119906119899minus1 (119903)) 119889119903119889119904 forall119905 isin 119869

(29)

By virtue of the properties of 119863 and assumption (i) we have

119863[1199061 (119905) 1199060 (119905)] = 119863 [V0 ⊖ (minus1)

sdot int119905119886119891 (119904 1199060 (119904)) 119889119904 ⊖ (minus1)

sdot int119905119886int119904119886119892 (119904 119903 1199060 (119903)) 119889119903119889119904 V (119905)] = 119863[V0 ⊖ (minus1)

sdot int119905119886119891 (119904 V0 (119904)) 119889119904 ⊖ (minus1)

sdot int119905119886int119904119886119892 (119904 119903 V0 (119903)) 119889119903119889119904 V (119905)]

= 119863[int1199051198861205752 (119904) 119889119904 0] le int

119905

119886119863[1205752 (119904) 0] 119889119904 le 120576 (119905

minus 119886) forall119905 isin 119869(30)

Therefore

119863[1199061 (119905) 1199060 (119905)] le 120576 (119905 minus 119886) forall119905 isin 119869 (31)

Observe that for 119899 = 1 2 and for 119905 isin 119869 one has119863 [119906119899+1 (119905) 119906119899 (119905)] = 119863[V0 ⊖ (minus1)

sdot int119905119886119891 (119904 119906119899 (119904)) 119889119904 ⊖ (minus1)

sdot int119905119886int119904119886119892 (119904 119903 119906119899 (119903)) 119889119903119889119904 V0 ⊖ (minus1)

sdot int119905119886119891 (119904 119906119899minus1 (119904)) 119889119904 ⊖ (minus1)

sdot int119905119886int119904119886119892 (119904 119903 119906119899minus1 (119903)) 119889119903119889119904]

le 119871119891119892 int119905

119886119863[119906119899 (119904) 119906119899minus1 (119904)] 119889119904

+ 119871119891119892 int119905

119886int119904119886119863 [119906119899 (119903) 119906119899minus1 (119903)] 119889119903119889119904

(32)

In particular

119863[1199062 (119905) 1199061 (119905)]le 119871119891119892 int

119905

119886119863[1199061 (119904) 1199060 (119904)] 119889119904

+ 119871119891119892 int119905

119886int119904119886119863[1199061 (119903) 1199060 (119903)] 119889119903119889119904

= 120576119871119891119892 int119905

119886(119904 minus 119886) 119889119904 + 120576119871119891119892 int

119905

119886int119904119886(119903 minus 119886) 119889119903119889119904

= 120576119871119891119892 ((119905 minus 119886)2

2 + (119905 minus 119886)33 )

= 120576119871119891119892

(119871119891119892 (119905 minus 119886))22 (1 + 119905 minus 119886

3 )

(33)

and so119863 [1199063 (119905) 1199062 (119905)]

le 119871119891119892 int119905

119886119863[1199062 (119904) 1199061 (119904)] 119889119904

+ 119871119891119892 int119905

119886int119904119886119863 [1199062 (119903) 1199061 (119903)] 119889119903119889119904

Complexity 5

= 1205761198712119891119892 int119905

119886((119904 minus 119886)22 + (119904 minus 119886)3

3 ) 119889119904

+ 1205761198712119891119892 int119905

119886int119904119886((119903 minus 119886)22 + (119903 minus 119886)3

3 ) 119889119903119889119904

= 1205761198712119891119892 ((119905 minus 119886)3

3 + 2(119905 minus 119886)44 + (119905 minus 119886)55 )

= 120576119871119891119892

(119871119891119892 (119905 minus 119886))33 (1 + 2119905 minus 1198864 + (119905 minus 119886)2

5 )(34)

and for 119899 gt 4 we have119863[119906119899 (119905) 119906119899minus1 (119905)] le 120576119871119899minus1119891119892 ((119905 minus 119886)

119899

119899 + 1205821 (119905 minus 119886)119899+1

(119899 + 1)+ + 120582119899 (119905 minus 119886)

2119899

(2119899) + (119905 minus 119886)2119899+1(2119899 + 1) )

(35)

where 1205821 120582119899 are balancing constantsWe choose120582 = max1 1205821 120582119899 (called120582 is amaximum

balancing constant) and then estimation (35) can be rewrittenas follows

119863 [119906119899 (119905) 119906119899minus1 (119905)] le 120576120582119871119891119892

(119871119891119892 (119905 minus 119886))119899119899 (1 + 119905 minus 119886

119899 + 1+ (119905 minus 119886)2(119899 + 1) (119899 + 2) + +

(119905 minus 119886)119899(119899 + 1) (119899 + 2) 2119899

+ (119905 minus 119886)119899+1(119899 + 1) (119899 + 2) 2119899 (2119899 + 1)) le 120576120582

119871119891119892sdot (119871119891119892 (119905 minus 119886))

119899

119899 (1 + 119905 minus 1198861 + (119905 minus 119886)2

2 +

+ (119905 minus 119886)119899119899 + (119905 minus 119886)119899+1

(119899 + 1) ) le 120576120582119871119891119892

(119871119891119892 (119905 minus 119886))119899119899

sdot 119890119905minus119886

(36)

Further if we assume that

119863[119906119899 (119905) 119906119899minus1 (119905)] le 120576120582119871119891119892

(119871119891119892 (119905 minus 119886))119899119899 119890119905minus119886

forall119905 isin 119869(37)

then we obtain

119863[119906119899+1 (119905) 119906119899 (119905)] le 120576120582119871119891119892

(119871119891119892 (119905 minus 119886))119899+1(119899 + 1) 119890119905minus119886

forall119905 isin 119869(38)

By the principle ofmathematical induction that (37) holds forevery 119899 gt 1 and now using estimation (37) for any 119905 isin 119869 weget

infinsum119899=1

119863 [119906119899 (119905) 119906119899minus1 (119905)] le 120576120582119871119891119892infinsum119899=1

(119871119891119892 (119905 minus 119886))119899119899 119890119905minus119886 (39)

The series suminfin119895=1(119911119896119896) is convergent for every 119911 isin R Hencefor every 120576 gt 0 we infer that the series suminfin119899=1119863[119906119899(119905) 119906119899minus1(119905)]is uniformly convergent on 119869 with respect to metric 119863 and

infinsum119899=1

119863 [119906119899 (119905) 119906119899minus1 (119905)] le 120576120582119871119891119892infinsum119899=1

(119871119891119892 (119905 minus 119886))119899119899 119890119905minus119886

le 120576120582119871119891119892infinsum119899=1

(119871119891119892 (119887 minus 119886))119899119899 119890119887minus119886 = 120576120582

119871119891119892 119890119871119891119892(119887minus119886)119890119887minus119886

= 120576120582119871119891119892 119890(119887minus119886)(1+119871119891119892)

(40)

For 119905 isin 119869 we have119863 [119906119899 (119905) 119906 (119905)] le 120576120582

119871119891119892infinsum119899=1

(119871119891119892 (119905 minus 119886))119899119899 119890119905minus119886 (41)

and the following estimation

119863[int119905119886119891 (119904 119906119899 (119904)) 119889119904

+ int119905119886int119904119886119892 (119904 119903 119906119899 (119903)) 119889119903119889119904 int

119905

119886119891 (119904 119906 (119904)) 119889119904

+ int119905119886int119904119886119892 (119904 119903 119906 (119903)) 119889119903119889119904]

le int119905119886119863[119891 (119904 119906119899 (119904)) 119891 (119904 119906 (119904))] 119889119904

+ int119905119886int119904119886119863 [119892 (119904 119903 119906119899 (119903)) 119892 (119904 119903 119906 (119903))] 119889119903119889119904

le 119871119891119892 int119905

119886119863 [119906119899 (119904) 119906 (119904)] 119889119904

+ 119871119891119892 int119905

119886int119904119886119863[119906119899 (119903) 119906 (119903)] 119889119903119889119904 forall119905 isin 119869

(42)

Combining estimation (41) and inequality (42) we obtainthat

119863[int119905119886119891 (119904 119906119899 (119904)) 119889119904 ⊖ (minus1)

sdot int119905119886int119904119886119892 (119904 119903 119906119899 (119903)) 119889119903119889119904 int

119905

119886119891 (119904 119906 (119904)) 119889119904 ⊖ (minus1)

sdot int119905119886int119904119886119892 (119904 119903 119906 (119903)) 119889119903119889119904]

(43)

6 Complexity

converges to 0 uniformly as 119899 997888rarr +infin Therefore 119906(119905) is a(S2)-solution of (14) with initial condition 1199060

Finally we shall prove that problem (14) has a unique(S2)-solution Assume that is another (S2)-solution of (14)with initial condition 1199060 Then we have for any 119905 isin 119869

119863 [119906 (119905) (119905)] le 119871119891119892 int119905

119886119863 [119906 (119904) (119904)] 119889119904

+ 119871119891119892 int119905

119886int119905119886119863[119906 (119903) (119903)] 119889119903119889119904

(44)

If we let 120585(119905) = 119863[119906(119905) (119905)] for any 119905 isin 119869 then120585 (119905) le 119871119891119892 int

119905

119886120585 (119904) 119889119904 + 119871119891119892 int

119905

119886int119905119886120585 (119903) 119889119903119889119904 (45)

Applying Lemma 23 in Hoa et al [4] we obtain 120585(119905) = 0 forany 119905 isin 119869 This completes the proof

Example 14 Consider the following fuzzy integro differentialequation

119863119892119867119906 (119905) = (minus2 0 2) + int119905

0

119906 (119904)1 + 119906 (119904)119889119904 (46)

119906 (0) = (minus2 0 2) (47)

where 119906 isin 119862([0 1]RF) 119891 [0 1] times RF 997888rarr RF and 119892 [0 1] times [0 1] timesRF 997888rarr RFLet

119891 (119905 119906) = (minus2 0 2) 119892 (119905 119904 119906) = 119906

1 + 119906 forall119905 119904 isin [0 1] 119906 isin RF (48)

It is easy to see that 119891 119892 satisfy Lipschitz condition withLipschitz constant 119871119891119892 = 1 Indeed for any 119905 119904 isin [0 1] and119906 isin RF

119863 [119892 (119905 119904 119906) 119892 (119905 119904 V)] = 119863 [ 1199061 + 119906

V1 + V

]= sup120572isin[01]

max10038161003816100381610038161003816100381610038161003816119906

1 + 119906 minusV

1 + V

10038161003816100381610038161003816100381610038161003816 10038161003816100381610038161003816100381610038161003816119906

1 + 119906 minusV

1 + V

10038161003816100381610038161003816100381610038161003816

= sup120572isin[01]

max 1003816100381610038161003816119906 minus V100381610038161003816100381610038161003816100381610038161 + 1199061003816100381610038161003816 10038161003816100381610038161 + V1003816100381610038161003816 |119906 minus V|

|1 + 119906| |1 + V|le sup120572isin[01]

max 1003816100381610038161003816119906 minus V1003816100381610038161003816 |119906 minus V| = 119863 [119906 V]

(49)

Hence byTheorem 15 the problem (46)-(47)has unique (S1)-solution or (S2)-solution on [0 1]

Moreover if a continuously (2)-differentiable function V [0 1] 997888rarr RF satisfies the following inequation

119863[119863119892119867V (119905) (minus2 0 2) + int119905

0

V (119904)1 + V (119904)119889119904] le 120576 (50)

then as shown in Theorem 15 there exists a (S2)-solution 119906 [0 1] 997888rarr RF of (46) such that

119863 [V (119905) 119906 (119905)] le 1205761205821198902 forall119905 isin [0 1] (51)

where 120582 is the maximum balancing constant

Secondly we shall prove the Ulam-Hyers-Rassias stabilityof the FIDE (14) defined on a bounded and closed interval

Theorem 15 Assume that 119891 119869 997888rarr RF 997888rarr RF and119892 119869 times 119869 times RF 997888rarr RF are continuous function satisfyingthe following conditions (i) there exists a constant 119871119891119892 gt0 such that max119863[119891(119905 119906) 119891(119905 V)]119863[119892(119905 119904 119906) 119892(119905 119904 V)] le119871119891119892119863[119906 V] for each (119905 119904 119906) (119905 119904 V) isin 119869 times 119869 times RF (ii) thereexists a constant 119862 gt 0 such that 0 lt 119871119891119892(119862 + 1198622) lt 1 Let120593 119869 997888rarr (119886 +infin) be a continuous function and increasing on119869 with

int119905119886120593 (119904) 119889119904 le 119862120593 (119905) for each 119905 isin 119869 (52)

If a continuously (1)-differentiable function 119906 119869 997888rarr RF

satisfies the following inequality

119863[119863119892119867119906 (119905) 119891 (119905 119906 (119905)) + int119905

119886119892 (119905 119904 119906 (119904)) 119889119904] le 120593 (119905) (53)

for any 119905 isin 119869 then there exists a unique (S1)-solution 119869 997888rarrRF of (14) such that

(119905) = 0 + int119905

119886119891 (119904 (119904)) 119889119904

+ int119905119886int119904119886119892 (119904 119903 (119903)) 119889119903119889s

(54)

and

119863[ (119905) 119906 (119905)] le 1198621 minus 119871119891119892 (119862 + 1198622)120593 (119905) (55)

for any 119905 isin 119869Proof Let us define a setX of all continuous fuzzy functions119908 119869 997888rarr RF by

X = 119908 119869 997888rarr RF | 119908 is continuous on 119869 (56)

equipped with the metric

119889 (V 119908) = inf 119862 isin [0 +infin) cup +infin | 119863 [V (119905) 119908 (119905)]le 119862120593 (119905) forall119905 isin 119869 (57)

It is easy to see that (X 119889) is also a complete generalizedmetric space (see Lemma 3)

The operator P X 997888rarr X is defined as follows

(PV) (119905) = V0 + int119905

119886119891 (119904 V (119904)) 119889119904

+ int119905119886int119904119886119892 (119904 119903 V (119903)) 119889119903119889119904 forall119905 isin 119869

(58)

Based on Lemma 32 and 33 in [3] we infer that PV is (1)-differentiable and so PV isin X

The operator P is strict contractive on X Indeed for anyV 119908 isin X and letting 119862V119908 isin [0 +infin) cup +infin be an arbitrary

Complexity 7

constant with 119889(V 119908) le 119862V119908 that is by the definition of themetric 119889 we have

119863 [V (119905) 119908 (119905)] le 119862V119908120593 (119905) forall119905 isin 119869 (59)

From the definition of the metric 119863 and assumptions (i)-(ii)of Theorem 15 and inequality (59) we infer that

119863[(PV) (119905) (P119908) (119905)] = 119863[V0 + int119905

119886119891 (119904 V (119904)) 119889119904

+ int119905119886int119904119886119892 (119904 119903 V (119903)) 119889119903119889119904 V0 + int

119905

119886119891 (119904 119908 (119904)) 119889119904

+ int119905119886int119904119886119892 (119904 119903 119908 (119903)) 119889119903119889119904]

le int119905119886119863[119891 (119904 V (119904)) 119891 (119904 119908 (119904))] 119889119904

+ int119905119886int119904119886119863[119892 (119904 119903 V (119903)) 119892 (119904 119903 119908 (119903))] 119889119903119889119904

le 119871119891119892 int119905

119886119863 [V (119904) 119908 (119904)] 119889119904

+ 119871119891119892 int119905

119886int119904119886119863[V (119903) 119908 (119903)] 119889119903119889119904

le 119871119891119892119862V119908 int119905

119886120593 (119904) 119889119904

+ 119871119891119892119862V119908 int119905

119886int119904119886120593 (119903) 119889119903119889119904 le 119871119891119892 (119862 + 1198622)

sdot 119862V119908120593 (119905)

(60)

for any 119905 isin 119869For each V 119908 isin X and by the definition of metric 119889 we

get

119889 ((PV) (119905) (P119908) (119905)) le 119871119891119892 (119862 + 1198622)119862V119908120593 (119905) (61)

for any 119905 isin 119869 Hence by (59) we can conclude that

119889 ((PV) (119905) (P119908) (119905)) le 119871119891119892 (119862 + 1198622) 119889 (V 119908) (62)

for any 119905 isin 119869It follows form the definitions of X and the operator P

that for arbitrary 119908 isin X there exists a constant 0 lt 119862 lt +infinsuch that

119863[(P119908) (119905) 119908 (119905)] = 119863 [1199080 + int119905

119886119891 (119904 119908 (119904)) 119889119904

+ int119905119886int119904119886119892 (119904 119903 119908 (119903)) 119889119903119889119904 119908 (119905)] le 119862120593 (119905)

(63)

for any 119905 isin 119869 since 119891(119905 119908(119905)) 119892(119905 119904 119908(119904)) and 119908(119905) arebounded on 119869 and min119905isin119869120593(119905) gt 0 Thus by definition of 119889 itis implied that

119889 (P119908119908) le 119862 lt +infin (64)

Therefore according toTheorem 2 there exists a continuousfunction on 119869 such that 119869119899119908 997888rarr as 119899 997888rarr +infin in the space(X 119889) and 119869119899 = that is satisfies (58) for each 119905 isin 119869

Observe that

X = 119908 isin X | 119889 (119908 119908) lt +infin (65)

Indeed for any 119908 isin X there exists a constant 0 lt 119862 lt +infinsuch that

119863 [119908 (119905) 119908 (119905)] le 119862120593 (119905) (66)

since119908 and119908 are bounded on 119869 andmin119905isin119869120593(119905) gt 0 It followsfrom the preceding inequality that

119889 (119908119908) lt +infin (67)

for all 119908 isin X Hence we obtained that X = 119908 isin X |119889(119908119908) lt +infinFrom Theorem 2 we infer that 119908 is a unique fixed point

of 119869 inX It is obvious that 119908 is a unique fuzzy function in Xwhich satisfies equation 119869119908 = 119908

On the other hand we have

119863[119906 (119905) 1199060 + int119905

119886119891 (119904 119906 (119904)) 119889119904

+ int119905119886int119904119886119892 (119904 119903 119906 (119903)) 119889119903119889119904] = 119863[119906 (119905)

⊖ 1199060 int119905

119886119891 (119904 119906 (119904)) 119889119904 + int119905

119886int119904119886119892 (119904 119903 119906 (119903)) 119889119903119889119904]

= 119863[int119905119886119863119892119867119906 (119904) 119889119904 int

119905

119886119891 (119904 119906 (119904)) 119889119904

+ int119905119886int119904119886119892 (119904 119903 119906 (119903)) 119889119903119889119904] le int119905

119886119863

sdot [119863119892119867119906 (119904) 119891 (119904 119906 (119904)) + int119904

119886119892 (119904 119903 119906 (119903)) 119889119903] 119889119904

le int119905119886120593 (119904) 119889119904 le 119862120593 (119905)

(68)

for any 119905 isin 119869 This means that

119889 (119906P119906) le 119862 (69)

Finally byTheorem 2 and inequation (69) we deduce that

119889 (119906 ) le 11 minus 119871119891119892 (119862 + 1198622)119889 (119906P119906)

le 1198621 minus 119871119891119892 (119862 + 1198622)120593 (119905)

(70)

for any 119905 isin 119869 which means that inequality (55) is true for all119905 isin 119869Corollary 16 Assume that 119891 and 119892 satisfy the assumption (i)of eorem 15 and 0 lt 119871119891119892(119887 minus 119886) + 119871119891119892((119887 minus 119886)22) lt 1

8 Complexity

For a given 120576 gt 0 if a continuously (1)-differentiable function119906 119869 997888rarr RF satisfies the following inequality

119863[119863119892119867119906 (119905) 119891 (119905 119906 (119905)) + int119905

119886119892 (119905 119904 119906 (119904)) 119889119904] le 120576 (71)

for any 119905 isin 119869 then there exists a unique (S1)-solution 119869 997888rarrRF of (14) such that

(119905) = 0 + int119905

119886119891 (119904 (119904)) 119889119904

+ int119905119886int119904119886119892 (119904 119903 (119903)) 119889119903119889119904

(72)

and

119863[ (119905) 119906 (119905)]le (119887 minus 119886) 1205761 minus (119871119891119892 (119887 minus 119886) + 119871119891119892 ((119887 minus 119886)2 2))

(73)

for any 119905 isin 119869Theorem 17 Suppose the functions 119891 119892 and 120593 satisfy allconditions as ineorem 15 If a continuously (2)-differentiablefunction 119906 119869 997888rarr RF satisfies inequality (53) in eorem 15for any 119905 isin 119869 then there exists a unique (S2)-solution 119869 997888rarrRF of (14) such that

(119905) = 0 ⊖ (minus1) int119905

119886119891 (119904 (119904)) 119889119904 ⊖ (minus1)

sdot int119905119886int119904119886119892 (119904 119903 (119903)) 119889119903119889119904

(74)

and the estimation (55) as in eorem 15 on 119869Proof Similar to the proof of Theorem 15 Consider theoperator H X 997888rarr X defined by

(HV) (119905) = V0 ⊖ (minus1)int119905

119886119891 (119904 V (119904)) 119889119904 ⊖ (minus1)

sdot int119905119886int119904119886119892 (119904 119903 V (119903)) 119889119903119889119904

(75)

for all V isin X Based on Lemmas 32 and 33 in [3] it is easy tosee that HV is (2)-differentiable and so HV isin X

We check the operator H is strict contractive on X LetV 119908 isin X and let 119862V119908 isin [0 +infin) cup +infin be an arbitraryconstant with 119889(V 119908) le 119862V119908 we have

119863 [V (119905) 119908 (119905)] le 119862V119908120593 (119905) forall119905 isin 119869 (76)

From the assumptions (i)-(ii) in Theorem 15 and (76) we get

119863 [(HV) (119905) (H119908) (119905)] = 119863[V0 ⊖ (minus1)

sdot int119905119886119891 (119904 V (119904)) 119889119904 ⊖ (minus1)int119905

119886int119904119886119892 (119904 119903 V (119903)) 119889119903119889119904 V0

⊖ (minus1)int119905119886119891 (119904 119908 (119904)) 119889119904 ⊖ (minus1)

sdot int119905119886int119904119886119892 (119904 119903 119908 (119903)) 119889119903119889119904]

le int119905119886119863[119891 (119904 V (119904)) 119891 (119904 119908 (119904))] 119889119904

+ int119905119886int119904119886119863 [119892 (119904 119903 V (119903)) 119892 (119904 119903 119908 (119903))] 119889119903119889119904

le 119871119891119892 int119905

119886119863[V (119904) 119908 (119904)] 119889119904

+ 119871119891119892 int119905

119886int119904119886119863[V (119903) 119908 (119903)] 119889119903119889119904

le 119871119891119892119862V119908 int119905

119886120593 (119904) 119889119904

+ 119871119891119892119862V119908 int119905

119886int119904119886120593 (119903) 119889119903119889119904 le 119871119891119892 (119862 + 1198622)

sdot 119862V119908120593 (119905)(77)

for any 119905 isin 119869 Hence by (76) we conclude that119889 ((HV) (119905) (H119908) (119905)) le 119862119871119891119892119889 (V 119908) forall119905 isin 119869 (78)

By the definitions of X and P we have for arbitrary 119908 isin Xthere exists a constant 0 lt 119862 lt +infin such that

119863[(H119908) (119905) 119908 (119905)] = 119863 [1199080 ⊖ (minus1)

sdot int119905119886119891 (119904 119908 (119904)) 119889119904 ⊖ (minus1)

sdot int119905119886int119904119886119892 (119904 119903 119908 (119903)) 119889119903119889119904 119908 (119905)] le 119862120593 (119905) forall119905 isin 119869

(79)

119891119892 and 119908 are bounded on 119869 and min119905isin119869120593(119905) gt 0 Thus bythe definition of 119889 we have 119889(119908 119908) lt +infin for all 119908 isin XHence we infer that X = 119908 isin X | 119889(119908119908) lt +infin FromTheorem 2 we deduce that 119908 is a unique fixed point of 119869 in119883 It is obvious that 119908 is a unique fuzzy function in X whichsatisfies the equality 119869119908 = 119908

On the other hand Hukuhara difference 1199060 ⊖ 119906(119905) existsfor all 119905 isin 119869 and from (52) and the definition of H that

119863[119906 (119905) 1199060 ⊖ (minus1) int119905

119886119891 (119904 119906 (119904)) 119889119904 ⊖ (minus1)

sdot int119905119886int119904119886119892 (119904 119903 119906 (119903)) 119889119903119889119904] = 119863[1199060 ⊖ 119906 (119905)

minus (int119905119886119891 (119904 119906 (119904)) 119889119904 + int119905

119886int119904119886119892 (119904 119903 119906 (119903)) 119889119903119889119904)]

= 119863[minusint119905119886119863119892119867119906 (119904) 119889119904

Complexity 9

minus (int119905119886119891 (119904 119906 (119904)) 119889119904 + int119905

119886int119904119886119892 (119904 119903 119906 (119903)) 119889119903119889119904)]

le int119905119886119863[119863119892119867119906 (119904) 119891 (119904 119906 (119904))

+ int119904119886119892 (119904 119903 119906 (119903)) 119889119903] 119889119904 le int119905

119886120593 (119904) 119889119904 le 119862120593 (119905)

(80)

for any 119905 isin 119869 which implies that

119889 (119906H119906) le 119862 (81)

ByTheorem 2 and inequation (81) we deduce that

119889 (119906 ) le 11 minus 119871119891119892 (119862 + 1198622)119889 (119906H119906)

le 1198621 minus 119871119891119892 (119862 + 1198622)120593 (119905)

(82)

for any 119905 isin 119869 which means that inequality (55) is true for all119905 isin 119869Corollary 18 Assume that 119891 and 119892 satisfy the assumption (i)of eorem 15 and 0 lt 119871119891119892(119887 minus 119886) + 119871119891119892((119887 minus 119886)22) lt 1For a given 120576 gt 0 if a continuously (2)-differentiable function119906 119869 997888rarr RF satisfies the following inequation

119863[119863g119867119906 (119905) 119891 (119905 119906 (119905)) + int

119905

119886119892 (119905 119904 119906 (119904)) 119889119904] le 120576 (83)

for any 119905 isin 119869 then there exists a unique (S2)-solution 119869 997888rarrRF of (14) such that

(119905) = 0 ⊖ (minus1) int119905

119886119891 (119904 (119904)) 119889119904 ⊖ (minus1)

sdot int119905119886int119904119886119892 (119904 119903 (119903)) 119889119903119889119904

(84)

and

119863[ (119905) 119906 (119905)]le (119887 minus 119886) 1205761 minus (119871119891119892 (119887 minus 119886) + 119871119891119892 ((119887 minus 119886)2 2))

(85)

for any 119905 isin 119869Example 19 We consider the fuzzy intergodifferential equa-tion as follows

119863119892119867119906 (119905) = 12 int119905

01199051199042119906 (119904) 119889119904 forall119905 isin [0 1] (86)

and the following inequality

119863[119863119892119867 (119905) 12 int119905

01199051199042 (119904) 119889119904] le 11989005119905 forall119905 119904 isin [0 1] (87)

where 119906 is a continuously (1)-differentiable (or (2)-differenti-able) function

It is easy to check that the functions 119891 119892 satisfy Lipschitzcondition with 119871119891119892 = 12 Choosing 120593(119905) = 1205761198902119905 with 120576 gt 0and 119862 = 05 we have

int1199050120593 (119904) 119889119904 = int119905

01205761198902119904119889119904 = 120576 (051198902119905 minus 1) le 051205761198902119905

= 1198621205761198902119905 forall119905 isin [0 1] (88)

Now all assumptions in Theorem 15 (or Theorem 17) aresatisfied problem (86) has a unique solution and (86) isUlam-Hyers-Rassias stable with

119863[ (119905) 119906 (119905)] le 1198621 minus 119871119891119892 (119862 + 1198622)120593 (119905) =

451205761198902119905 (89)

for any 119905 isin [0 1]In particular if we choose 120593(119905) = 120576 then we have

119863( (119905) 119906 (119905))le (119887 minus 119886) 1205761 minus (119871119891119892 (119887 minus 119886) + 119871119891119892 ((119887 minus 119886)2 2)) = 4120576

(90)

for all 119905 isin [0 1]4 Conlusion

In this study the Ulam-Hyers stability and Ulam-Hyers-Rassias stability for fuzzy intergodifferential equation via thefixed point technique and successive approximation methodare studied Moreover some illustrative examples are givenIn future work we will study Ulam stability for fuzzyintergodifferential equation in the quotient space of fuzzynumbers introduced by [25ndash27]

Data Availability

No data were used to support this study

Conflicts of Interest

The authors declare that they have no conflicts of interest

References

[1] O AbuArqub ldquoAdaptation of reproducing kernel algorithm forsolving fuzzy Fredholm-Volterra integrodifferential equationsrdquoNeural Computing and Applications vol 28 no 7 pp 1591ndash16102017

[2] R Alikhani and F Bahrami ldquoGlobal solutions of fuzzy integro-differential equations under generalized differentiability by themethod of upper and lower solutionsrdquo Information Sciences vol295 pp 600ndash608 2015

[3] R Alikhani F Bahrami and A Jabbari ldquoExistence of globalsolutions to nonlinear fuzzy Volterra integro-differential equa-tionsrdquo Nonlinear Analysis eory Methods amp Applications vol75 no 4 pp 1810ndash1821 2012

10 Complexity

[4] N V Hoa and N D Phu ldquoFuzzy functional integro-differentialequations under generalized h-differentiabilityrdquo Journal of Intel-ligent Fuzzy Systems vol 26 no 1 pp 2073ndash2085 2014

[5] R M Shabestari R Ezzati and T Allahviranloo ldquoSolvingfuzzy volterra integrodifferential equations of fractional orderby bernoulli wavelet methodrdquo Advances in Fuzzy Systems vol2018 Article ID 5603560 11 pages 2018

[6] H Vu ldquoRandom fuzzy differential equations with impulsesrdquoComplexity vol 2017 Article ID 4056016 11 pages 2017

[7] O Abu Arqub S Momani S Al-Mezel and M Kutbi ldquoExis-tence uniqueness and characterization theorems for nonlinearfuzzy integrodifferential equations of volterra typerdquoMathemat-ical Problems in Engineering vol 2015 Article ID 835891 13pages 2015

[8] E Eljaoui S Melliani and L S Chadli ldquoAumann fuzzy im-proper integral and its application to solve fuzzy integro-differential equations by laplace transform methodrdquo Advancesin Fuzzy Systems vol 2018 Article ID 9730502 10 pages 2018

[9] N V Hoa and N D Phu ldquoOn maximal and minimal solutionsfor set-valued differential equations with feedback controlrdquoAbstract and Applied Analysis vol 2012 Article ID 816218 11pages 2012

[10] H Vu ldquoExistence results for fuzzy Volterra integral equationrdquoJournal of Intelligent amp Fuzzy Systems Applications in Engineer-ing and Technology vol 33 no 1 pp 207ndash213 2017

[11] H Vu L S Dong andNN Phung ldquoApplication of contractive-like mapping principles to impulsive fuzzy functional dif-ferential equationrdquo Journal of Intelligent amp Fuzzy SystemsApplications in Engineering and Technology vol 33 no 2 pp753ndash759 2017

[12] J Vanterler da C Sousa and E Capelas de Oliveira ldquoUlam-Hyers stability of a nonlinear fractional Volterra integro-differential equationrdquo Applied Mathematics Letters vol 81 pp50ndash56 2018

[13] D Popa and I Rasa ldquoOn the Hyers-Ulam stability of the lineardifferential equationrdquo Journal of Mathematical Analysis andApplications vol 381 no 2 pp 530ndash537 2011

[14] G Wang M Zhou and L Sun ldquoHyers-Ulam stability oflinear differential equations of first orderrdquo Applied MathematicsLetters vol 21 no 10 pp 1024ndash1028 2008

[15] J R Wang L L Lv and Y Zhou ldquoNew concepts and results instability of fractional differential equationsrdquoCommunications inNonlinear Science and Numerical Simulation vol 17 no 6 pp2530ndash2538 2012

[16] E C de Oliveira and J V Sousa ldquoUlam-Hyers-Rassias stabilityfor a class of fractional integro-differential equationsrdquo Results inMathematics vol 111 no 72 pp 1ndash16 2018

[17] J V D C Sousa and E C D Oliveira ldquoFractional order pseu-doparabolic partial differential equation ulamndashhyers stabilityrdquoBulletin of the Brazilian Mathematical Society pp 1ndash16 2018

[18] J Vanterler da C Sousa K D Kucche and E C de OliveiraldquoStability of 120595-Hilfer impulsive fractional differential equa-tionsrdquo Applied Mathematics Letters vol 88 pp 73ndash80 2019

[19] Y Shen ldquoHyers-Ulam-Rassias stability of first order linearpartial fuzzy differential equations under generalized differ-entiabilityrdquo Advances in Difference Equations vol 2015 no 1article no 351 pp 1ndash18 2015

[20] Y Shen ldquoOn the Ulam stability of first order linear fuzzydifferential equations under generalized differentiabilityrdquo FuzzySets and Systems vol 280 no C pp 27ndash57 2015

[21] Y Shen and F Wang ldquoA fixed point approach to the Ulamstability of fuzzy differential equations under generalized differ-entiabilityrdquo Journal of Intelligent amp Fuzzy Systems Applicationsin Engineering and Technology vol 30 no 6 pp 3253ndash32602016

[22] W Ren Z Yang X Sun and M Qi ldquoHyers-Ulam stability ofHermite fuzzy differential equations and fuzzy Mellin trans-formrdquo Journal of Intelligent amp Fuzzy Systems Applications inEngineering and Technology vol 35 no 3 pp 3721ndash3731 2018

[23] J B Diaz and B Margolis ldquoA fixed point theorem of thealternative for contractions on a generalized complete metricspacerdquo Bulletin (New Series) of the American MathematicalSociety vol 74 no 2 pp 305ndash309 1968

[24] B Bede and S G Gal ldquoGeneralizations of the differentiabilityof fuzzy-number-valued functions with applications to fuzzydifferential equationsrdquo Fuzzy Sets and Systems vol 151 no 3pp 581ndash599 2005

[25] D Qiu W Zhang and C Lu ldquoOn fuzzy differential equationsin the quotient space of fuzzy numbersrdquo Fuzzy Sets and Systemsvol 295 pp 72ndash98 2016

[26] D Qiu andW Zhang ldquoSymmetric fuzzy numbers and additiveequivalence of fuzzy numbersrdquo So13 Computing vol 17 no 8 pp1471ndash1477 2013

[27] D Qiu C Lu W Zhang and Y Lan ldquoAlgebraic properties andtopological properties of the quotient space of fuzzy numbersbased on Mares equivalence relationrdquo Fuzzy Sets and Systemsvol 245 pp 63ndash82 2014

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Complexity 5

= 1205761198712119891119892 int119905

119886((119904 minus 119886)22 + (119904 minus 119886)3

3 ) 119889119904

+ 1205761198712119891119892 int119905

119886int119904119886((119903 minus 119886)22 + (119903 minus 119886)3

3 ) 119889119903119889119904

= 1205761198712119891119892 ((119905 minus 119886)3

3 + 2(119905 minus 119886)44 + (119905 minus 119886)55 )

= 120576119871119891119892

(119871119891119892 (119905 minus 119886))33 (1 + 2119905 minus 1198864 + (119905 minus 119886)2

5 )(34)

and for 119899 gt 4 we have119863[119906119899 (119905) 119906119899minus1 (119905)] le 120576119871119899minus1119891119892 ((119905 minus 119886)

119899

119899 + 1205821 (119905 minus 119886)119899+1

(119899 + 1)+ + 120582119899 (119905 minus 119886)

2119899

(2119899) + (119905 minus 119886)2119899+1(2119899 + 1) )

(35)

where 1205821 120582119899 are balancing constantsWe choose120582 = max1 1205821 120582119899 (called120582 is amaximum

balancing constant) and then estimation (35) can be rewrittenas follows

119863 [119906119899 (119905) 119906119899minus1 (119905)] le 120576120582119871119891119892

(119871119891119892 (119905 minus 119886))119899119899 (1 + 119905 minus 119886

119899 + 1+ (119905 minus 119886)2(119899 + 1) (119899 + 2) + +

(119905 minus 119886)119899(119899 + 1) (119899 + 2) 2119899

+ (119905 minus 119886)119899+1(119899 + 1) (119899 + 2) 2119899 (2119899 + 1)) le 120576120582

119871119891119892sdot (119871119891119892 (119905 minus 119886))

119899

119899 (1 + 119905 minus 1198861 + (119905 minus 119886)2

2 +

+ (119905 minus 119886)119899119899 + (119905 minus 119886)119899+1

(119899 + 1) ) le 120576120582119871119891119892

(119871119891119892 (119905 minus 119886))119899119899

sdot 119890119905minus119886

(36)

Further if we assume that

119863[119906119899 (119905) 119906119899minus1 (119905)] le 120576120582119871119891119892

(119871119891119892 (119905 minus 119886))119899119899 119890119905minus119886

forall119905 isin 119869(37)

then we obtain

119863[119906119899+1 (119905) 119906119899 (119905)] le 120576120582119871119891119892

(119871119891119892 (119905 minus 119886))119899+1(119899 + 1) 119890119905minus119886

forall119905 isin 119869(38)

By the principle ofmathematical induction that (37) holds forevery 119899 gt 1 and now using estimation (37) for any 119905 isin 119869 weget

infinsum119899=1

119863 [119906119899 (119905) 119906119899minus1 (119905)] le 120576120582119871119891119892infinsum119899=1

(119871119891119892 (119905 minus 119886))119899119899 119890119905minus119886 (39)

The series suminfin119895=1(119911119896119896) is convergent for every 119911 isin R Hencefor every 120576 gt 0 we infer that the series suminfin119899=1119863[119906119899(119905) 119906119899minus1(119905)]is uniformly convergent on 119869 with respect to metric 119863 and

infinsum119899=1

119863 [119906119899 (119905) 119906119899minus1 (119905)] le 120576120582119871119891119892infinsum119899=1

(119871119891119892 (119905 minus 119886))119899119899 119890119905minus119886

le 120576120582119871119891119892infinsum119899=1

(119871119891119892 (119887 minus 119886))119899119899 119890119887minus119886 = 120576120582

119871119891119892 119890119871119891119892(119887minus119886)119890119887minus119886

= 120576120582119871119891119892 119890(119887minus119886)(1+119871119891119892)

(40)

For 119905 isin 119869 we have119863 [119906119899 (119905) 119906 (119905)] le 120576120582

119871119891119892infinsum119899=1

(119871119891119892 (119905 minus 119886))119899119899 119890119905minus119886 (41)

and the following estimation

119863[int119905119886119891 (119904 119906119899 (119904)) 119889119904

+ int119905119886int119904119886119892 (119904 119903 119906119899 (119903)) 119889119903119889119904 int

119905

119886119891 (119904 119906 (119904)) 119889119904

+ int119905119886int119904119886119892 (119904 119903 119906 (119903)) 119889119903119889119904]

le int119905119886119863[119891 (119904 119906119899 (119904)) 119891 (119904 119906 (119904))] 119889119904

+ int119905119886int119904119886119863 [119892 (119904 119903 119906119899 (119903)) 119892 (119904 119903 119906 (119903))] 119889119903119889119904

le 119871119891119892 int119905

119886119863 [119906119899 (119904) 119906 (119904)] 119889119904

+ 119871119891119892 int119905

119886int119904119886119863[119906119899 (119903) 119906 (119903)] 119889119903119889119904 forall119905 isin 119869

(42)

Combining estimation (41) and inequality (42) we obtainthat

119863[int119905119886119891 (119904 119906119899 (119904)) 119889119904 ⊖ (minus1)

sdot int119905119886int119904119886119892 (119904 119903 119906119899 (119903)) 119889119903119889119904 int

119905

119886119891 (119904 119906 (119904)) 119889119904 ⊖ (minus1)

sdot int119905119886int119904119886119892 (119904 119903 119906 (119903)) 119889119903119889119904]

(43)

6 Complexity

converges to 0 uniformly as 119899 997888rarr +infin Therefore 119906(119905) is a(S2)-solution of (14) with initial condition 1199060

Finally we shall prove that problem (14) has a unique(S2)-solution Assume that is another (S2)-solution of (14)with initial condition 1199060 Then we have for any 119905 isin 119869

119863 [119906 (119905) (119905)] le 119871119891119892 int119905

119886119863 [119906 (119904) (119904)] 119889119904

+ 119871119891119892 int119905

119886int119905119886119863[119906 (119903) (119903)] 119889119903119889119904

(44)

If we let 120585(119905) = 119863[119906(119905) (119905)] for any 119905 isin 119869 then120585 (119905) le 119871119891119892 int

119905

119886120585 (119904) 119889119904 + 119871119891119892 int

119905

119886int119905119886120585 (119903) 119889119903119889119904 (45)

Applying Lemma 23 in Hoa et al [4] we obtain 120585(119905) = 0 forany 119905 isin 119869 This completes the proof

Example 14 Consider the following fuzzy integro differentialequation

119863119892119867119906 (119905) = (minus2 0 2) + int119905

0

119906 (119904)1 + 119906 (119904)119889119904 (46)

119906 (0) = (minus2 0 2) (47)

where 119906 isin 119862([0 1]RF) 119891 [0 1] times RF 997888rarr RF and 119892 [0 1] times [0 1] timesRF 997888rarr RFLet

119891 (119905 119906) = (minus2 0 2) 119892 (119905 119904 119906) = 119906

1 + 119906 forall119905 119904 isin [0 1] 119906 isin RF (48)

It is easy to see that 119891 119892 satisfy Lipschitz condition withLipschitz constant 119871119891119892 = 1 Indeed for any 119905 119904 isin [0 1] and119906 isin RF

119863 [119892 (119905 119904 119906) 119892 (119905 119904 V)] = 119863 [ 1199061 + 119906

V1 + V

]= sup120572isin[01]

max10038161003816100381610038161003816100381610038161003816119906

1 + 119906 minusV

1 + V

10038161003816100381610038161003816100381610038161003816 10038161003816100381610038161003816100381610038161003816119906

1 + 119906 minusV

1 + V

10038161003816100381610038161003816100381610038161003816

= sup120572isin[01]

max 1003816100381610038161003816119906 minus V100381610038161003816100381610038161003816100381610038161 + 1199061003816100381610038161003816 10038161003816100381610038161 + V1003816100381610038161003816 |119906 minus V|

|1 + 119906| |1 + V|le sup120572isin[01]

max 1003816100381610038161003816119906 minus V1003816100381610038161003816 |119906 minus V| = 119863 [119906 V]

(49)

Hence byTheorem 15 the problem (46)-(47)has unique (S1)-solution or (S2)-solution on [0 1]

Moreover if a continuously (2)-differentiable function V [0 1] 997888rarr RF satisfies the following inequation

119863[119863119892119867V (119905) (minus2 0 2) + int119905

0

V (119904)1 + V (119904)119889119904] le 120576 (50)

then as shown in Theorem 15 there exists a (S2)-solution 119906 [0 1] 997888rarr RF of (46) such that

119863 [V (119905) 119906 (119905)] le 1205761205821198902 forall119905 isin [0 1] (51)

where 120582 is the maximum balancing constant

Secondly we shall prove the Ulam-Hyers-Rassias stabilityof the FIDE (14) defined on a bounded and closed interval

Theorem 15 Assume that 119891 119869 997888rarr RF 997888rarr RF and119892 119869 times 119869 times RF 997888rarr RF are continuous function satisfyingthe following conditions (i) there exists a constant 119871119891119892 gt0 such that max119863[119891(119905 119906) 119891(119905 V)]119863[119892(119905 119904 119906) 119892(119905 119904 V)] le119871119891119892119863[119906 V] for each (119905 119904 119906) (119905 119904 V) isin 119869 times 119869 times RF (ii) thereexists a constant 119862 gt 0 such that 0 lt 119871119891119892(119862 + 1198622) lt 1 Let120593 119869 997888rarr (119886 +infin) be a continuous function and increasing on119869 with

int119905119886120593 (119904) 119889119904 le 119862120593 (119905) for each 119905 isin 119869 (52)

If a continuously (1)-differentiable function 119906 119869 997888rarr RF

satisfies the following inequality

119863[119863119892119867119906 (119905) 119891 (119905 119906 (119905)) + int119905

119886119892 (119905 119904 119906 (119904)) 119889119904] le 120593 (119905) (53)

for any 119905 isin 119869 then there exists a unique (S1)-solution 119869 997888rarrRF of (14) such that

(119905) = 0 + int119905

119886119891 (119904 (119904)) 119889119904

+ int119905119886int119904119886119892 (119904 119903 (119903)) 119889119903119889s

(54)

and

119863[ (119905) 119906 (119905)] le 1198621 minus 119871119891119892 (119862 + 1198622)120593 (119905) (55)

for any 119905 isin 119869Proof Let us define a setX of all continuous fuzzy functions119908 119869 997888rarr RF by

X = 119908 119869 997888rarr RF | 119908 is continuous on 119869 (56)

equipped with the metric

119889 (V 119908) = inf 119862 isin [0 +infin) cup +infin | 119863 [V (119905) 119908 (119905)]le 119862120593 (119905) forall119905 isin 119869 (57)

It is easy to see that (X 119889) is also a complete generalizedmetric space (see Lemma 3)

The operator P X 997888rarr X is defined as follows

(PV) (119905) = V0 + int119905

119886119891 (119904 V (119904)) 119889119904

+ int119905119886int119904119886119892 (119904 119903 V (119903)) 119889119903119889119904 forall119905 isin 119869

(58)

Based on Lemma 32 and 33 in [3] we infer that PV is (1)-differentiable and so PV isin X

The operator P is strict contractive on X Indeed for anyV 119908 isin X and letting 119862V119908 isin [0 +infin) cup +infin be an arbitrary

Complexity 7

constant with 119889(V 119908) le 119862V119908 that is by the definition of themetric 119889 we have

119863 [V (119905) 119908 (119905)] le 119862V119908120593 (119905) forall119905 isin 119869 (59)

From the definition of the metric 119863 and assumptions (i)-(ii)of Theorem 15 and inequality (59) we infer that

119863[(PV) (119905) (P119908) (119905)] = 119863[V0 + int119905

119886119891 (119904 V (119904)) 119889119904

+ int119905119886int119904119886119892 (119904 119903 V (119903)) 119889119903119889119904 V0 + int

119905

119886119891 (119904 119908 (119904)) 119889119904

+ int119905119886int119904119886119892 (119904 119903 119908 (119903)) 119889119903119889119904]

le int119905119886119863[119891 (119904 V (119904)) 119891 (119904 119908 (119904))] 119889119904

+ int119905119886int119904119886119863[119892 (119904 119903 V (119903)) 119892 (119904 119903 119908 (119903))] 119889119903119889119904

le 119871119891119892 int119905

119886119863 [V (119904) 119908 (119904)] 119889119904

+ 119871119891119892 int119905

119886int119904119886119863[V (119903) 119908 (119903)] 119889119903119889119904

le 119871119891119892119862V119908 int119905

119886120593 (119904) 119889119904

+ 119871119891119892119862V119908 int119905

119886int119904119886120593 (119903) 119889119903119889119904 le 119871119891119892 (119862 + 1198622)

sdot 119862V119908120593 (119905)

(60)

for any 119905 isin 119869For each V 119908 isin X and by the definition of metric 119889 we

get

119889 ((PV) (119905) (P119908) (119905)) le 119871119891119892 (119862 + 1198622)119862V119908120593 (119905) (61)

for any 119905 isin 119869 Hence by (59) we can conclude that

119889 ((PV) (119905) (P119908) (119905)) le 119871119891119892 (119862 + 1198622) 119889 (V 119908) (62)

for any 119905 isin 119869It follows form the definitions of X and the operator P

that for arbitrary 119908 isin X there exists a constant 0 lt 119862 lt +infinsuch that

119863[(P119908) (119905) 119908 (119905)] = 119863 [1199080 + int119905

119886119891 (119904 119908 (119904)) 119889119904

+ int119905119886int119904119886119892 (119904 119903 119908 (119903)) 119889119903119889119904 119908 (119905)] le 119862120593 (119905)

(63)

for any 119905 isin 119869 since 119891(119905 119908(119905)) 119892(119905 119904 119908(119904)) and 119908(119905) arebounded on 119869 and min119905isin119869120593(119905) gt 0 Thus by definition of 119889 itis implied that

119889 (P119908119908) le 119862 lt +infin (64)

Therefore according toTheorem 2 there exists a continuousfunction on 119869 such that 119869119899119908 997888rarr as 119899 997888rarr +infin in the space(X 119889) and 119869119899 = that is satisfies (58) for each 119905 isin 119869

Observe that

X = 119908 isin X | 119889 (119908 119908) lt +infin (65)

Indeed for any 119908 isin X there exists a constant 0 lt 119862 lt +infinsuch that

119863 [119908 (119905) 119908 (119905)] le 119862120593 (119905) (66)

since119908 and119908 are bounded on 119869 andmin119905isin119869120593(119905) gt 0 It followsfrom the preceding inequality that

119889 (119908119908) lt +infin (67)

for all 119908 isin X Hence we obtained that X = 119908 isin X |119889(119908119908) lt +infinFrom Theorem 2 we infer that 119908 is a unique fixed point

of 119869 inX It is obvious that 119908 is a unique fuzzy function in Xwhich satisfies equation 119869119908 = 119908

On the other hand we have

119863[119906 (119905) 1199060 + int119905

119886119891 (119904 119906 (119904)) 119889119904

+ int119905119886int119904119886119892 (119904 119903 119906 (119903)) 119889119903119889119904] = 119863[119906 (119905)

⊖ 1199060 int119905

119886119891 (119904 119906 (119904)) 119889119904 + int119905

119886int119904119886119892 (119904 119903 119906 (119903)) 119889119903119889119904]

= 119863[int119905119886119863119892119867119906 (119904) 119889119904 int

119905

119886119891 (119904 119906 (119904)) 119889119904

+ int119905119886int119904119886119892 (119904 119903 119906 (119903)) 119889119903119889119904] le int119905

119886119863

sdot [119863119892119867119906 (119904) 119891 (119904 119906 (119904)) + int119904

119886119892 (119904 119903 119906 (119903)) 119889119903] 119889119904

le int119905119886120593 (119904) 119889119904 le 119862120593 (119905)

(68)

for any 119905 isin 119869 This means that

119889 (119906P119906) le 119862 (69)

Finally byTheorem 2 and inequation (69) we deduce that

119889 (119906 ) le 11 minus 119871119891119892 (119862 + 1198622)119889 (119906P119906)

le 1198621 minus 119871119891119892 (119862 + 1198622)120593 (119905)

(70)

for any 119905 isin 119869 which means that inequality (55) is true for all119905 isin 119869Corollary 16 Assume that 119891 and 119892 satisfy the assumption (i)of eorem 15 and 0 lt 119871119891119892(119887 minus 119886) + 119871119891119892((119887 minus 119886)22) lt 1

8 Complexity

For a given 120576 gt 0 if a continuously (1)-differentiable function119906 119869 997888rarr RF satisfies the following inequality

119863[119863119892119867119906 (119905) 119891 (119905 119906 (119905)) + int119905

119886119892 (119905 119904 119906 (119904)) 119889119904] le 120576 (71)

for any 119905 isin 119869 then there exists a unique (S1)-solution 119869 997888rarrRF of (14) such that

(119905) = 0 + int119905

119886119891 (119904 (119904)) 119889119904

+ int119905119886int119904119886119892 (119904 119903 (119903)) 119889119903119889119904

(72)

and

119863[ (119905) 119906 (119905)]le (119887 minus 119886) 1205761 minus (119871119891119892 (119887 minus 119886) + 119871119891119892 ((119887 minus 119886)2 2))

(73)

for any 119905 isin 119869Theorem 17 Suppose the functions 119891 119892 and 120593 satisfy allconditions as ineorem 15 If a continuously (2)-differentiablefunction 119906 119869 997888rarr RF satisfies inequality (53) in eorem 15for any 119905 isin 119869 then there exists a unique (S2)-solution 119869 997888rarrRF of (14) such that

(119905) = 0 ⊖ (minus1) int119905

119886119891 (119904 (119904)) 119889119904 ⊖ (minus1)

sdot int119905119886int119904119886119892 (119904 119903 (119903)) 119889119903119889119904

(74)

and the estimation (55) as in eorem 15 on 119869Proof Similar to the proof of Theorem 15 Consider theoperator H X 997888rarr X defined by

(HV) (119905) = V0 ⊖ (minus1)int119905

119886119891 (119904 V (119904)) 119889119904 ⊖ (minus1)

sdot int119905119886int119904119886119892 (119904 119903 V (119903)) 119889119903119889119904

(75)

for all V isin X Based on Lemmas 32 and 33 in [3] it is easy tosee that HV is (2)-differentiable and so HV isin X

We check the operator H is strict contractive on X LetV 119908 isin X and let 119862V119908 isin [0 +infin) cup +infin be an arbitraryconstant with 119889(V 119908) le 119862V119908 we have

119863 [V (119905) 119908 (119905)] le 119862V119908120593 (119905) forall119905 isin 119869 (76)

From the assumptions (i)-(ii) in Theorem 15 and (76) we get

119863 [(HV) (119905) (H119908) (119905)] = 119863[V0 ⊖ (minus1)

sdot int119905119886119891 (119904 V (119904)) 119889119904 ⊖ (minus1)int119905

119886int119904119886119892 (119904 119903 V (119903)) 119889119903119889119904 V0

⊖ (minus1)int119905119886119891 (119904 119908 (119904)) 119889119904 ⊖ (minus1)

sdot int119905119886int119904119886119892 (119904 119903 119908 (119903)) 119889119903119889119904]

le int119905119886119863[119891 (119904 V (119904)) 119891 (119904 119908 (119904))] 119889119904

+ int119905119886int119904119886119863 [119892 (119904 119903 V (119903)) 119892 (119904 119903 119908 (119903))] 119889119903119889119904

le 119871119891119892 int119905

119886119863[V (119904) 119908 (119904)] 119889119904

+ 119871119891119892 int119905

119886int119904119886119863[V (119903) 119908 (119903)] 119889119903119889119904

le 119871119891119892119862V119908 int119905

119886120593 (119904) 119889119904

+ 119871119891119892119862V119908 int119905

119886int119904119886120593 (119903) 119889119903119889119904 le 119871119891119892 (119862 + 1198622)

sdot 119862V119908120593 (119905)(77)

for any 119905 isin 119869 Hence by (76) we conclude that119889 ((HV) (119905) (H119908) (119905)) le 119862119871119891119892119889 (V 119908) forall119905 isin 119869 (78)

By the definitions of X and P we have for arbitrary 119908 isin Xthere exists a constant 0 lt 119862 lt +infin such that

119863[(H119908) (119905) 119908 (119905)] = 119863 [1199080 ⊖ (minus1)

sdot int119905119886119891 (119904 119908 (119904)) 119889119904 ⊖ (minus1)

sdot int119905119886int119904119886119892 (119904 119903 119908 (119903)) 119889119903119889119904 119908 (119905)] le 119862120593 (119905) forall119905 isin 119869

(79)

119891119892 and 119908 are bounded on 119869 and min119905isin119869120593(119905) gt 0 Thus bythe definition of 119889 we have 119889(119908 119908) lt +infin for all 119908 isin XHence we infer that X = 119908 isin X | 119889(119908119908) lt +infin FromTheorem 2 we deduce that 119908 is a unique fixed point of 119869 in119883 It is obvious that 119908 is a unique fuzzy function in X whichsatisfies the equality 119869119908 = 119908

On the other hand Hukuhara difference 1199060 ⊖ 119906(119905) existsfor all 119905 isin 119869 and from (52) and the definition of H that

119863[119906 (119905) 1199060 ⊖ (minus1) int119905

119886119891 (119904 119906 (119904)) 119889119904 ⊖ (minus1)

sdot int119905119886int119904119886119892 (119904 119903 119906 (119903)) 119889119903119889119904] = 119863[1199060 ⊖ 119906 (119905)

minus (int119905119886119891 (119904 119906 (119904)) 119889119904 + int119905

119886int119904119886119892 (119904 119903 119906 (119903)) 119889119903119889119904)]

= 119863[minusint119905119886119863119892119867119906 (119904) 119889119904

Complexity 9

minus (int119905119886119891 (119904 119906 (119904)) 119889119904 + int119905

119886int119904119886119892 (119904 119903 119906 (119903)) 119889119903119889119904)]

le int119905119886119863[119863119892119867119906 (119904) 119891 (119904 119906 (119904))

+ int119904119886119892 (119904 119903 119906 (119903)) 119889119903] 119889119904 le int119905

119886120593 (119904) 119889119904 le 119862120593 (119905)

(80)

for any 119905 isin 119869 which implies that

119889 (119906H119906) le 119862 (81)

ByTheorem 2 and inequation (81) we deduce that

119889 (119906 ) le 11 minus 119871119891119892 (119862 + 1198622)119889 (119906H119906)

le 1198621 minus 119871119891119892 (119862 + 1198622)120593 (119905)

(82)

for any 119905 isin 119869 which means that inequality (55) is true for all119905 isin 119869Corollary 18 Assume that 119891 and 119892 satisfy the assumption (i)of eorem 15 and 0 lt 119871119891119892(119887 minus 119886) + 119871119891119892((119887 minus 119886)22) lt 1For a given 120576 gt 0 if a continuously (2)-differentiable function119906 119869 997888rarr RF satisfies the following inequation

119863[119863g119867119906 (119905) 119891 (119905 119906 (119905)) + int

119905

119886119892 (119905 119904 119906 (119904)) 119889119904] le 120576 (83)

for any 119905 isin 119869 then there exists a unique (S2)-solution 119869 997888rarrRF of (14) such that

(119905) = 0 ⊖ (minus1) int119905

119886119891 (119904 (119904)) 119889119904 ⊖ (minus1)

sdot int119905119886int119904119886119892 (119904 119903 (119903)) 119889119903119889119904

(84)

and

119863[ (119905) 119906 (119905)]le (119887 minus 119886) 1205761 minus (119871119891119892 (119887 minus 119886) + 119871119891119892 ((119887 minus 119886)2 2))

(85)

for any 119905 isin 119869Example 19 We consider the fuzzy intergodifferential equa-tion as follows

119863119892119867119906 (119905) = 12 int119905

01199051199042119906 (119904) 119889119904 forall119905 isin [0 1] (86)

and the following inequality

119863[119863119892119867 (119905) 12 int119905

01199051199042 (119904) 119889119904] le 11989005119905 forall119905 119904 isin [0 1] (87)

where 119906 is a continuously (1)-differentiable (or (2)-differenti-able) function

It is easy to check that the functions 119891 119892 satisfy Lipschitzcondition with 119871119891119892 = 12 Choosing 120593(119905) = 1205761198902119905 with 120576 gt 0and 119862 = 05 we have

int1199050120593 (119904) 119889119904 = int119905

01205761198902119904119889119904 = 120576 (051198902119905 minus 1) le 051205761198902119905

= 1198621205761198902119905 forall119905 isin [0 1] (88)

Now all assumptions in Theorem 15 (or Theorem 17) aresatisfied problem (86) has a unique solution and (86) isUlam-Hyers-Rassias stable with

119863[ (119905) 119906 (119905)] le 1198621 minus 119871119891119892 (119862 + 1198622)120593 (119905) =

451205761198902119905 (89)

for any 119905 isin [0 1]In particular if we choose 120593(119905) = 120576 then we have

119863( (119905) 119906 (119905))le (119887 minus 119886) 1205761 minus (119871119891119892 (119887 minus 119886) + 119871119891119892 ((119887 minus 119886)2 2)) = 4120576

(90)

for all 119905 isin [0 1]4 Conlusion

In this study the Ulam-Hyers stability and Ulam-Hyers-Rassias stability for fuzzy intergodifferential equation via thefixed point technique and successive approximation methodare studied Moreover some illustrative examples are givenIn future work we will study Ulam stability for fuzzyintergodifferential equation in the quotient space of fuzzynumbers introduced by [25ndash27]

Data Availability

No data were used to support this study

Conflicts of Interest

The authors declare that they have no conflicts of interest

References

[1] O AbuArqub ldquoAdaptation of reproducing kernel algorithm forsolving fuzzy Fredholm-Volterra integrodifferential equationsrdquoNeural Computing and Applications vol 28 no 7 pp 1591ndash16102017

[2] R Alikhani and F Bahrami ldquoGlobal solutions of fuzzy integro-differential equations under generalized differentiability by themethod of upper and lower solutionsrdquo Information Sciences vol295 pp 600ndash608 2015

[3] R Alikhani F Bahrami and A Jabbari ldquoExistence of globalsolutions to nonlinear fuzzy Volterra integro-differential equa-tionsrdquo Nonlinear Analysis eory Methods amp Applications vol75 no 4 pp 1810ndash1821 2012

10 Complexity

[4] N V Hoa and N D Phu ldquoFuzzy functional integro-differentialequations under generalized h-differentiabilityrdquo Journal of Intel-ligent Fuzzy Systems vol 26 no 1 pp 2073ndash2085 2014

[5] R M Shabestari R Ezzati and T Allahviranloo ldquoSolvingfuzzy volterra integrodifferential equations of fractional orderby bernoulli wavelet methodrdquo Advances in Fuzzy Systems vol2018 Article ID 5603560 11 pages 2018

[6] H Vu ldquoRandom fuzzy differential equations with impulsesrdquoComplexity vol 2017 Article ID 4056016 11 pages 2017

[7] O Abu Arqub S Momani S Al-Mezel and M Kutbi ldquoExis-tence uniqueness and characterization theorems for nonlinearfuzzy integrodifferential equations of volterra typerdquoMathemat-ical Problems in Engineering vol 2015 Article ID 835891 13pages 2015

[8] E Eljaoui S Melliani and L S Chadli ldquoAumann fuzzy im-proper integral and its application to solve fuzzy integro-differential equations by laplace transform methodrdquo Advancesin Fuzzy Systems vol 2018 Article ID 9730502 10 pages 2018

[9] N V Hoa and N D Phu ldquoOn maximal and minimal solutionsfor set-valued differential equations with feedback controlrdquoAbstract and Applied Analysis vol 2012 Article ID 816218 11pages 2012

[10] H Vu ldquoExistence results for fuzzy Volterra integral equationrdquoJournal of Intelligent amp Fuzzy Systems Applications in Engineer-ing and Technology vol 33 no 1 pp 207ndash213 2017

[11] H Vu L S Dong andNN Phung ldquoApplication of contractive-like mapping principles to impulsive fuzzy functional dif-ferential equationrdquo Journal of Intelligent amp Fuzzy SystemsApplications in Engineering and Technology vol 33 no 2 pp753ndash759 2017

[12] J Vanterler da C Sousa and E Capelas de Oliveira ldquoUlam-Hyers stability of a nonlinear fractional Volterra integro-differential equationrdquo Applied Mathematics Letters vol 81 pp50ndash56 2018

[13] D Popa and I Rasa ldquoOn the Hyers-Ulam stability of the lineardifferential equationrdquo Journal of Mathematical Analysis andApplications vol 381 no 2 pp 530ndash537 2011

[14] G Wang M Zhou and L Sun ldquoHyers-Ulam stability oflinear differential equations of first orderrdquo Applied MathematicsLetters vol 21 no 10 pp 1024ndash1028 2008

[15] J R Wang L L Lv and Y Zhou ldquoNew concepts and results instability of fractional differential equationsrdquoCommunications inNonlinear Science and Numerical Simulation vol 17 no 6 pp2530ndash2538 2012

[16] E C de Oliveira and J V Sousa ldquoUlam-Hyers-Rassias stabilityfor a class of fractional integro-differential equationsrdquo Results inMathematics vol 111 no 72 pp 1ndash16 2018

[17] J V D C Sousa and E C D Oliveira ldquoFractional order pseu-doparabolic partial differential equation ulamndashhyers stabilityrdquoBulletin of the Brazilian Mathematical Society pp 1ndash16 2018

[18] J Vanterler da C Sousa K D Kucche and E C de OliveiraldquoStability of 120595-Hilfer impulsive fractional differential equa-tionsrdquo Applied Mathematics Letters vol 88 pp 73ndash80 2019

[19] Y Shen ldquoHyers-Ulam-Rassias stability of first order linearpartial fuzzy differential equations under generalized differ-entiabilityrdquo Advances in Difference Equations vol 2015 no 1article no 351 pp 1ndash18 2015

[20] Y Shen ldquoOn the Ulam stability of first order linear fuzzydifferential equations under generalized differentiabilityrdquo FuzzySets and Systems vol 280 no C pp 27ndash57 2015

[21] Y Shen and F Wang ldquoA fixed point approach to the Ulamstability of fuzzy differential equations under generalized differ-entiabilityrdquo Journal of Intelligent amp Fuzzy Systems Applicationsin Engineering and Technology vol 30 no 6 pp 3253ndash32602016

[22] W Ren Z Yang X Sun and M Qi ldquoHyers-Ulam stability ofHermite fuzzy differential equations and fuzzy Mellin trans-formrdquo Journal of Intelligent amp Fuzzy Systems Applications inEngineering and Technology vol 35 no 3 pp 3721ndash3731 2018

[23] J B Diaz and B Margolis ldquoA fixed point theorem of thealternative for contractions on a generalized complete metricspacerdquo Bulletin (New Series) of the American MathematicalSociety vol 74 no 2 pp 305ndash309 1968

[24] B Bede and S G Gal ldquoGeneralizations of the differentiabilityof fuzzy-number-valued functions with applications to fuzzydifferential equationsrdquo Fuzzy Sets and Systems vol 151 no 3pp 581ndash599 2005

[25] D Qiu W Zhang and C Lu ldquoOn fuzzy differential equationsin the quotient space of fuzzy numbersrdquo Fuzzy Sets and Systemsvol 295 pp 72ndash98 2016

[26] D Qiu andW Zhang ldquoSymmetric fuzzy numbers and additiveequivalence of fuzzy numbersrdquo So13 Computing vol 17 no 8 pp1471ndash1477 2013

[27] D Qiu C Lu W Zhang and Y Lan ldquoAlgebraic properties andtopological properties of the quotient space of fuzzy numbersbased on Mares equivalence relationrdquo Fuzzy Sets and Systemsvol 245 pp 63ndash82 2014

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6 Complexity

converges to 0 uniformly as 119899 997888rarr +infin Therefore 119906(119905) is a(S2)-solution of (14) with initial condition 1199060

Finally we shall prove that problem (14) has a unique(S2)-solution Assume that is another (S2)-solution of (14)with initial condition 1199060 Then we have for any 119905 isin 119869

119863 [119906 (119905) (119905)] le 119871119891119892 int119905

119886119863 [119906 (119904) (119904)] 119889119904

+ 119871119891119892 int119905

119886int119905119886119863[119906 (119903) (119903)] 119889119903119889119904

(44)

If we let 120585(119905) = 119863[119906(119905) (119905)] for any 119905 isin 119869 then120585 (119905) le 119871119891119892 int

119905

119886120585 (119904) 119889119904 + 119871119891119892 int

119905

119886int119905119886120585 (119903) 119889119903119889119904 (45)

Applying Lemma 23 in Hoa et al [4] we obtain 120585(119905) = 0 forany 119905 isin 119869 This completes the proof

Example 14 Consider the following fuzzy integro differentialequation

119863119892119867119906 (119905) = (minus2 0 2) + int119905

0

119906 (119904)1 + 119906 (119904)119889119904 (46)

119906 (0) = (minus2 0 2) (47)

where 119906 isin 119862([0 1]RF) 119891 [0 1] times RF 997888rarr RF and 119892 [0 1] times [0 1] timesRF 997888rarr RFLet

119891 (119905 119906) = (minus2 0 2) 119892 (119905 119904 119906) = 119906

1 + 119906 forall119905 119904 isin [0 1] 119906 isin RF (48)

It is easy to see that 119891 119892 satisfy Lipschitz condition withLipschitz constant 119871119891119892 = 1 Indeed for any 119905 119904 isin [0 1] and119906 isin RF

119863 [119892 (119905 119904 119906) 119892 (119905 119904 V)] = 119863 [ 1199061 + 119906

V1 + V

]= sup120572isin[01]

max10038161003816100381610038161003816100381610038161003816119906

1 + 119906 minusV

1 + V

10038161003816100381610038161003816100381610038161003816 10038161003816100381610038161003816100381610038161003816119906

1 + 119906 minusV

1 + V

10038161003816100381610038161003816100381610038161003816

= sup120572isin[01]

max 1003816100381610038161003816119906 minus V100381610038161003816100381610038161003816100381610038161 + 1199061003816100381610038161003816 10038161003816100381610038161 + V1003816100381610038161003816 |119906 minus V|

|1 + 119906| |1 + V|le sup120572isin[01]

max 1003816100381610038161003816119906 minus V1003816100381610038161003816 |119906 minus V| = 119863 [119906 V]

(49)

Hence byTheorem 15 the problem (46)-(47)has unique (S1)-solution or (S2)-solution on [0 1]

Moreover if a continuously (2)-differentiable function V [0 1] 997888rarr RF satisfies the following inequation

119863[119863119892119867V (119905) (minus2 0 2) + int119905

0

V (119904)1 + V (119904)119889119904] le 120576 (50)

then as shown in Theorem 15 there exists a (S2)-solution 119906 [0 1] 997888rarr RF of (46) such that

119863 [V (119905) 119906 (119905)] le 1205761205821198902 forall119905 isin [0 1] (51)

where 120582 is the maximum balancing constant

Secondly we shall prove the Ulam-Hyers-Rassias stabilityof the FIDE (14) defined on a bounded and closed interval

Theorem 15 Assume that 119891 119869 997888rarr RF 997888rarr RF and119892 119869 times 119869 times RF 997888rarr RF are continuous function satisfyingthe following conditions (i) there exists a constant 119871119891119892 gt0 such that max119863[119891(119905 119906) 119891(119905 V)]119863[119892(119905 119904 119906) 119892(119905 119904 V)] le119871119891119892119863[119906 V] for each (119905 119904 119906) (119905 119904 V) isin 119869 times 119869 times RF (ii) thereexists a constant 119862 gt 0 such that 0 lt 119871119891119892(119862 + 1198622) lt 1 Let120593 119869 997888rarr (119886 +infin) be a continuous function and increasing on119869 with

int119905119886120593 (119904) 119889119904 le 119862120593 (119905) for each 119905 isin 119869 (52)

If a continuously (1)-differentiable function 119906 119869 997888rarr RF

satisfies the following inequality

119863[119863119892119867119906 (119905) 119891 (119905 119906 (119905)) + int119905

119886119892 (119905 119904 119906 (119904)) 119889119904] le 120593 (119905) (53)

for any 119905 isin 119869 then there exists a unique (S1)-solution 119869 997888rarrRF of (14) such that

(119905) = 0 + int119905

119886119891 (119904 (119904)) 119889119904

+ int119905119886int119904119886119892 (119904 119903 (119903)) 119889119903119889s

(54)

and

119863[ (119905) 119906 (119905)] le 1198621 minus 119871119891119892 (119862 + 1198622)120593 (119905) (55)

for any 119905 isin 119869Proof Let us define a setX of all continuous fuzzy functions119908 119869 997888rarr RF by

X = 119908 119869 997888rarr RF | 119908 is continuous on 119869 (56)

equipped with the metric

119889 (V 119908) = inf 119862 isin [0 +infin) cup +infin | 119863 [V (119905) 119908 (119905)]le 119862120593 (119905) forall119905 isin 119869 (57)

It is easy to see that (X 119889) is also a complete generalizedmetric space (see Lemma 3)

The operator P X 997888rarr X is defined as follows

(PV) (119905) = V0 + int119905

119886119891 (119904 V (119904)) 119889119904

+ int119905119886int119904119886119892 (119904 119903 V (119903)) 119889119903119889119904 forall119905 isin 119869

(58)

Based on Lemma 32 and 33 in [3] we infer that PV is (1)-differentiable and so PV isin X

The operator P is strict contractive on X Indeed for anyV 119908 isin X and letting 119862V119908 isin [0 +infin) cup +infin be an arbitrary

Complexity 7

constant with 119889(V 119908) le 119862V119908 that is by the definition of themetric 119889 we have

119863 [V (119905) 119908 (119905)] le 119862V119908120593 (119905) forall119905 isin 119869 (59)

From the definition of the metric 119863 and assumptions (i)-(ii)of Theorem 15 and inequality (59) we infer that

119863[(PV) (119905) (P119908) (119905)] = 119863[V0 + int119905

119886119891 (119904 V (119904)) 119889119904

+ int119905119886int119904119886119892 (119904 119903 V (119903)) 119889119903119889119904 V0 + int

119905

119886119891 (119904 119908 (119904)) 119889119904

+ int119905119886int119904119886119892 (119904 119903 119908 (119903)) 119889119903119889119904]

le int119905119886119863[119891 (119904 V (119904)) 119891 (119904 119908 (119904))] 119889119904

+ int119905119886int119904119886119863[119892 (119904 119903 V (119903)) 119892 (119904 119903 119908 (119903))] 119889119903119889119904

le 119871119891119892 int119905

119886119863 [V (119904) 119908 (119904)] 119889119904

+ 119871119891119892 int119905

119886int119904119886119863[V (119903) 119908 (119903)] 119889119903119889119904

le 119871119891119892119862V119908 int119905

119886120593 (119904) 119889119904

+ 119871119891119892119862V119908 int119905

119886int119904119886120593 (119903) 119889119903119889119904 le 119871119891119892 (119862 + 1198622)

sdot 119862V119908120593 (119905)

(60)

for any 119905 isin 119869For each V 119908 isin X and by the definition of metric 119889 we

get

119889 ((PV) (119905) (P119908) (119905)) le 119871119891119892 (119862 + 1198622)119862V119908120593 (119905) (61)

for any 119905 isin 119869 Hence by (59) we can conclude that

119889 ((PV) (119905) (P119908) (119905)) le 119871119891119892 (119862 + 1198622) 119889 (V 119908) (62)

for any 119905 isin 119869It follows form the definitions of X and the operator P

that for arbitrary 119908 isin X there exists a constant 0 lt 119862 lt +infinsuch that

119863[(P119908) (119905) 119908 (119905)] = 119863 [1199080 + int119905

119886119891 (119904 119908 (119904)) 119889119904

+ int119905119886int119904119886119892 (119904 119903 119908 (119903)) 119889119903119889119904 119908 (119905)] le 119862120593 (119905)

(63)

for any 119905 isin 119869 since 119891(119905 119908(119905)) 119892(119905 119904 119908(119904)) and 119908(119905) arebounded on 119869 and min119905isin119869120593(119905) gt 0 Thus by definition of 119889 itis implied that

119889 (P119908119908) le 119862 lt +infin (64)

Therefore according toTheorem 2 there exists a continuousfunction on 119869 such that 119869119899119908 997888rarr as 119899 997888rarr +infin in the space(X 119889) and 119869119899 = that is satisfies (58) for each 119905 isin 119869

Observe that

X = 119908 isin X | 119889 (119908 119908) lt +infin (65)

Indeed for any 119908 isin X there exists a constant 0 lt 119862 lt +infinsuch that

119863 [119908 (119905) 119908 (119905)] le 119862120593 (119905) (66)

since119908 and119908 are bounded on 119869 andmin119905isin119869120593(119905) gt 0 It followsfrom the preceding inequality that

119889 (119908119908) lt +infin (67)

for all 119908 isin X Hence we obtained that X = 119908 isin X |119889(119908119908) lt +infinFrom Theorem 2 we infer that 119908 is a unique fixed point

of 119869 inX It is obvious that 119908 is a unique fuzzy function in Xwhich satisfies equation 119869119908 = 119908

On the other hand we have

119863[119906 (119905) 1199060 + int119905

119886119891 (119904 119906 (119904)) 119889119904

+ int119905119886int119904119886119892 (119904 119903 119906 (119903)) 119889119903119889119904] = 119863[119906 (119905)

⊖ 1199060 int119905

119886119891 (119904 119906 (119904)) 119889119904 + int119905

119886int119904119886119892 (119904 119903 119906 (119903)) 119889119903119889119904]

= 119863[int119905119886119863119892119867119906 (119904) 119889119904 int

119905

119886119891 (119904 119906 (119904)) 119889119904

+ int119905119886int119904119886119892 (119904 119903 119906 (119903)) 119889119903119889119904] le int119905

119886119863

sdot [119863119892119867119906 (119904) 119891 (119904 119906 (119904)) + int119904

119886119892 (119904 119903 119906 (119903)) 119889119903] 119889119904

le int119905119886120593 (119904) 119889119904 le 119862120593 (119905)

(68)

for any 119905 isin 119869 This means that

119889 (119906P119906) le 119862 (69)

Finally byTheorem 2 and inequation (69) we deduce that

119889 (119906 ) le 11 minus 119871119891119892 (119862 + 1198622)119889 (119906P119906)

le 1198621 minus 119871119891119892 (119862 + 1198622)120593 (119905)

(70)

for any 119905 isin 119869 which means that inequality (55) is true for all119905 isin 119869Corollary 16 Assume that 119891 and 119892 satisfy the assumption (i)of eorem 15 and 0 lt 119871119891119892(119887 minus 119886) + 119871119891119892((119887 minus 119886)22) lt 1

8 Complexity

For a given 120576 gt 0 if a continuously (1)-differentiable function119906 119869 997888rarr RF satisfies the following inequality

119863[119863119892119867119906 (119905) 119891 (119905 119906 (119905)) + int119905

119886119892 (119905 119904 119906 (119904)) 119889119904] le 120576 (71)

for any 119905 isin 119869 then there exists a unique (S1)-solution 119869 997888rarrRF of (14) such that

(119905) = 0 + int119905

119886119891 (119904 (119904)) 119889119904

+ int119905119886int119904119886119892 (119904 119903 (119903)) 119889119903119889119904

(72)

and

119863[ (119905) 119906 (119905)]le (119887 minus 119886) 1205761 minus (119871119891119892 (119887 minus 119886) + 119871119891119892 ((119887 minus 119886)2 2))

(73)

for any 119905 isin 119869Theorem 17 Suppose the functions 119891 119892 and 120593 satisfy allconditions as ineorem 15 If a continuously (2)-differentiablefunction 119906 119869 997888rarr RF satisfies inequality (53) in eorem 15for any 119905 isin 119869 then there exists a unique (S2)-solution 119869 997888rarrRF of (14) such that

(119905) = 0 ⊖ (minus1) int119905

119886119891 (119904 (119904)) 119889119904 ⊖ (minus1)

sdot int119905119886int119904119886119892 (119904 119903 (119903)) 119889119903119889119904

(74)

and the estimation (55) as in eorem 15 on 119869Proof Similar to the proof of Theorem 15 Consider theoperator H X 997888rarr X defined by

(HV) (119905) = V0 ⊖ (minus1)int119905

119886119891 (119904 V (119904)) 119889119904 ⊖ (minus1)

sdot int119905119886int119904119886119892 (119904 119903 V (119903)) 119889119903119889119904

(75)

for all V isin X Based on Lemmas 32 and 33 in [3] it is easy tosee that HV is (2)-differentiable and so HV isin X

We check the operator H is strict contractive on X LetV 119908 isin X and let 119862V119908 isin [0 +infin) cup +infin be an arbitraryconstant with 119889(V 119908) le 119862V119908 we have

119863 [V (119905) 119908 (119905)] le 119862V119908120593 (119905) forall119905 isin 119869 (76)

From the assumptions (i)-(ii) in Theorem 15 and (76) we get

119863 [(HV) (119905) (H119908) (119905)] = 119863[V0 ⊖ (minus1)

sdot int119905119886119891 (119904 V (119904)) 119889119904 ⊖ (minus1)int119905

119886int119904119886119892 (119904 119903 V (119903)) 119889119903119889119904 V0

⊖ (minus1)int119905119886119891 (119904 119908 (119904)) 119889119904 ⊖ (minus1)

sdot int119905119886int119904119886119892 (119904 119903 119908 (119903)) 119889119903119889119904]

le int119905119886119863[119891 (119904 V (119904)) 119891 (119904 119908 (119904))] 119889119904

+ int119905119886int119904119886119863 [119892 (119904 119903 V (119903)) 119892 (119904 119903 119908 (119903))] 119889119903119889119904

le 119871119891119892 int119905

119886119863[V (119904) 119908 (119904)] 119889119904

+ 119871119891119892 int119905

119886int119904119886119863[V (119903) 119908 (119903)] 119889119903119889119904

le 119871119891119892119862V119908 int119905

119886120593 (119904) 119889119904

+ 119871119891119892119862V119908 int119905

119886int119904119886120593 (119903) 119889119903119889119904 le 119871119891119892 (119862 + 1198622)

sdot 119862V119908120593 (119905)(77)

for any 119905 isin 119869 Hence by (76) we conclude that119889 ((HV) (119905) (H119908) (119905)) le 119862119871119891119892119889 (V 119908) forall119905 isin 119869 (78)

By the definitions of X and P we have for arbitrary 119908 isin Xthere exists a constant 0 lt 119862 lt +infin such that

119863[(H119908) (119905) 119908 (119905)] = 119863 [1199080 ⊖ (minus1)

sdot int119905119886119891 (119904 119908 (119904)) 119889119904 ⊖ (minus1)

sdot int119905119886int119904119886119892 (119904 119903 119908 (119903)) 119889119903119889119904 119908 (119905)] le 119862120593 (119905) forall119905 isin 119869

(79)

119891119892 and 119908 are bounded on 119869 and min119905isin119869120593(119905) gt 0 Thus bythe definition of 119889 we have 119889(119908 119908) lt +infin for all 119908 isin XHence we infer that X = 119908 isin X | 119889(119908119908) lt +infin FromTheorem 2 we deduce that 119908 is a unique fixed point of 119869 in119883 It is obvious that 119908 is a unique fuzzy function in X whichsatisfies the equality 119869119908 = 119908

On the other hand Hukuhara difference 1199060 ⊖ 119906(119905) existsfor all 119905 isin 119869 and from (52) and the definition of H that

119863[119906 (119905) 1199060 ⊖ (minus1) int119905

119886119891 (119904 119906 (119904)) 119889119904 ⊖ (minus1)

sdot int119905119886int119904119886119892 (119904 119903 119906 (119903)) 119889119903119889119904] = 119863[1199060 ⊖ 119906 (119905)

minus (int119905119886119891 (119904 119906 (119904)) 119889119904 + int119905

119886int119904119886119892 (119904 119903 119906 (119903)) 119889119903119889119904)]

= 119863[minusint119905119886119863119892119867119906 (119904) 119889119904

Complexity 9

minus (int119905119886119891 (119904 119906 (119904)) 119889119904 + int119905

119886int119904119886119892 (119904 119903 119906 (119903)) 119889119903119889119904)]

le int119905119886119863[119863119892119867119906 (119904) 119891 (119904 119906 (119904))

+ int119904119886119892 (119904 119903 119906 (119903)) 119889119903] 119889119904 le int119905

119886120593 (119904) 119889119904 le 119862120593 (119905)

(80)

for any 119905 isin 119869 which implies that

119889 (119906H119906) le 119862 (81)

ByTheorem 2 and inequation (81) we deduce that

119889 (119906 ) le 11 minus 119871119891119892 (119862 + 1198622)119889 (119906H119906)

le 1198621 minus 119871119891119892 (119862 + 1198622)120593 (119905)

(82)

for any 119905 isin 119869 which means that inequality (55) is true for all119905 isin 119869Corollary 18 Assume that 119891 and 119892 satisfy the assumption (i)of eorem 15 and 0 lt 119871119891119892(119887 minus 119886) + 119871119891119892((119887 minus 119886)22) lt 1For a given 120576 gt 0 if a continuously (2)-differentiable function119906 119869 997888rarr RF satisfies the following inequation

119863[119863g119867119906 (119905) 119891 (119905 119906 (119905)) + int

119905

119886119892 (119905 119904 119906 (119904)) 119889119904] le 120576 (83)

for any 119905 isin 119869 then there exists a unique (S2)-solution 119869 997888rarrRF of (14) such that

(119905) = 0 ⊖ (minus1) int119905

119886119891 (119904 (119904)) 119889119904 ⊖ (minus1)

sdot int119905119886int119904119886119892 (119904 119903 (119903)) 119889119903119889119904

(84)

and

119863[ (119905) 119906 (119905)]le (119887 minus 119886) 1205761 minus (119871119891119892 (119887 minus 119886) + 119871119891119892 ((119887 minus 119886)2 2))

(85)

for any 119905 isin 119869Example 19 We consider the fuzzy intergodifferential equa-tion as follows

119863119892119867119906 (119905) = 12 int119905

01199051199042119906 (119904) 119889119904 forall119905 isin [0 1] (86)

and the following inequality

119863[119863119892119867 (119905) 12 int119905

01199051199042 (119904) 119889119904] le 11989005119905 forall119905 119904 isin [0 1] (87)

where 119906 is a continuously (1)-differentiable (or (2)-differenti-able) function

It is easy to check that the functions 119891 119892 satisfy Lipschitzcondition with 119871119891119892 = 12 Choosing 120593(119905) = 1205761198902119905 with 120576 gt 0and 119862 = 05 we have

int1199050120593 (119904) 119889119904 = int119905

01205761198902119904119889119904 = 120576 (051198902119905 minus 1) le 051205761198902119905

= 1198621205761198902119905 forall119905 isin [0 1] (88)

Now all assumptions in Theorem 15 (or Theorem 17) aresatisfied problem (86) has a unique solution and (86) isUlam-Hyers-Rassias stable with

119863[ (119905) 119906 (119905)] le 1198621 minus 119871119891119892 (119862 + 1198622)120593 (119905) =

451205761198902119905 (89)

for any 119905 isin [0 1]In particular if we choose 120593(119905) = 120576 then we have

119863( (119905) 119906 (119905))le (119887 minus 119886) 1205761 minus (119871119891119892 (119887 minus 119886) + 119871119891119892 ((119887 minus 119886)2 2)) = 4120576

(90)

for all 119905 isin [0 1]4 Conlusion

In this study the Ulam-Hyers stability and Ulam-Hyers-Rassias stability for fuzzy intergodifferential equation via thefixed point technique and successive approximation methodare studied Moreover some illustrative examples are givenIn future work we will study Ulam stability for fuzzyintergodifferential equation in the quotient space of fuzzynumbers introduced by [25ndash27]

Data Availability

No data were used to support this study

Conflicts of Interest

The authors declare that they have no conflicts of interest

References

[1] O AbuArqub ldquoAdaptation of reproducing kernel algorithm forsolving fuzzy Fredholm-Volterra integrodifferential equationsrdquoNeural Computing and Applications vol 28 no 7 pp 1591ndash16102017

[2] R Alikhani and F Bahrami ldquoGlobal solutions of fuzzy integro-differential equations under generalized differentiability by themethod of upper and lower solutionsrdquo Information Sciences vol295 pp 600ndash608 2015

[3] R Alikhani F Bahrami and A Jabbari ldquoExistence of globalsolutions to nonlinear fuzzy Volterra integro-differential equa-tionsrdquo Nonlinear Analysis eory Methods amp Applications vol75 no 4 pp 1810ndash1821 2012

10 Complexity

[4] N V Hoa and N D Phu ldquoFuzzy functional integro-differentialequations under generalized h-differentiabilityrdquo Journal of Intel-ligent Fuzzy Systems vol 26 no 1 pp 2073ndash2085 2014

[5] R M Shabestari R Ezzati and T Allahviranloo ldquoSolvingfuzzy volterra integrodifferential equations of fractional orderby bernoulli wavelet methodrdquo Advances in Fuzzy Systems vol2018 Article ID 5603560 11 pages 2018

[6] H Vu ldquoRandom fuzzy differential equations with impulsesrdquoComplexity vol 2017 Article ID 4056016 11 pages 2017

[7] O Abu Arqub S Momani S Al-Mezel and M Kutbi ldquoExis-tence uniqueness and characterization theorems for nonlinearfuzzy integrodifferential equations of volterra typerdquoMathemat-ical Problems in Engineering vol 2015 Article ID 835891 13pages 2015

[8] E Eljaoui S Melliani and L S Chadli ldquoAumann fuzzy im-proper integral and its application to solve fuzzy integro-differential equations by laplace transform methodrdquo Advancesin Fuzzy Systems vol 2018 Article ID 9730502 10 pages 2018

[9] N V Hoa and N D Phu ldquoOn maximal and minimal solutionsfor set-valued differential equations with feedback controlrdquoAbstract and Applied Analysis vol 2012 Article ID 816218 11pages 2012

[10] H Vu ldquoExistence results for fuzzy Volterra integral equationrdquoJournal of Intelligent amp Fuzzy Systems Applications in Engineer-ing and Technology vol 33 no 1 pp 207ndash213 2017

[11] H Vu L S Dong andNN Phung ldquoApplication of contractive-like mapping principles to impulsive fuzzy functional dif-ferential equationrdquo Journal of Intelligent amp Fuzzy SystemsApplications in Engineering and Technology vol 33 no 2 pp753ndash759 2017

[12] J Vanterler da C Sousa and E Capelas de Oliveira ldquoUlam-Hyers stability of a nonlinear fractional Volterra integro-differential equationrdquo Applied Mathematics Letters vol 81 pp50ndash56 2018

[13] D Popa and I Rasa ldquoOn the Hyers-Ulam stability of the lineardifferential equationrdquo Journal of Mathematical Analysis andApplications vol 381 no 2 pp 530ndash537 2011

[14] G Wang M Zhou and L Sun ldquoHyers-Ulam stability oflinear differential equations of first orderrdquo Applied MathematicsLetters vol 21 no 10 pp 1024ndash1028 2008

[15] J R Wang L L Lv and Y Zhou ldquoNew concepts and results instability of fractional differential equationsrdquoCommunications inNonlinear Science and Numerical Simulation vol 17 no 6 pp2530ndash2538 2012

[16] E C de Oliveira and J V Sousa ldquoUlam-Hyers-Rassias stabilityfor a class of fractional integro-differential equationsrdquo Results inMathematics vol 111 no 72 pp 1ndash16 2018

[17] J V D C Sousa and E C D Oliveira ldquoFractional order pseu-doparabolic partial differential equation ulamndashhyers stabilityrdquoBulletin of the Brazilian Mathematical Society pp 1ndash16 2018

[18] J Vanterler da C Sousa K D Kucche and E C de OliveiraldquoStability of 120595-Hilfer impulsive fractional differential equa-tionsrdquo Applied Mathematics Letters vol 88 pp 73ndash80 2019

[19] Y Shen ldquoHyers-Ulam-Rassias stability of first order linearpartial fuzzy differential equations under generalized differ-entiabilityrdquo Advances in Difference Equations vol 2015 no 1article no 351 pp 1ndash18 2015

[20] Y Shen ldquoOn the Ulam stability of first order linear fuzzydifferential equations under generalized differentiabilityrdquo FuzzySets and Systems vol 280 no C pp 27ndash57 2015

[21] Y Shen and F Wang ldquoA fixed point approach to the Ulamstability of fuzzy differential equations under generalized differ-entiabilityrdquo Journal of Intelligent amp Fuzzy Systems Applicationsin Engineering and Technology vol 30 no 6 pp 3253ndash32602016

[22] W Ren Z Yang X Sun and M Qi ldquoHyers-Ulam stability ofHermite fuzzy differential equations and fuzzy Mellin trans-formrdquo Journal of Intelligent amp Fuzzy Systems Applications inEngineering and Technology vol 35 no 3 pp 3721ndash3731 2018

[23] J B Diaz and B Margolis ldquoA fixed point theorem of thealternative for contractions on a generalized complete metricspacerdquo Bulletin (New Series) of the American MathematicalSociety vol 74 no 2 pp 305ndash309 1968

[24] B Bede and S G Gal ldquoGeneralizations of the differentiabilityof fuzzy-number-valued functions with applications to fuzzydifferential equationsrdquo Fuzzy Sets and Systems vol 151 no 3pp 581ndash599 2005

[25] D Qiu W Zhang and C Lu ldquoOn fuzzy differential equationsin the quotient space of fuzzy numbersrdquo Fuzzy Sets and Systemsvol 295 pp 72ndash98 2016

[26] D Qiu andW Zhang ldquoSymmetric fuzzy numbers and additiveequivalence of fuzzy numbersrdquo So13 Computing vol 17 no 8 pp1471ndash1477 2013

[27] D Qiu C Lu W Zhang and Y Lan ldquoAlgebraic properties andtopological properties of the quotient space of fuzzy numbersbased on Mares equivalence relationrdquo Fuzzy Sets and Systemsvol 245 pp 63ndash82 2014

Hindawiwwwhindawicom Volume 2018

MathematicsJournal of

Hindawiwwwhindawicom Volume 2018

Mathematical Problems in Engineering

Applied MathematicsJournal of

Hindawiwwwhindawicom Volume 2018

Probability and StatisticsHindawiwwwhindawicom Volume 2018

Journal of

Hindawiwwwhindawicom Volume 2018

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawiwwwhindawicom Volume 2018

OptimizationJournal of

Hindawiwwwhindawicom Volume 2018

Hindawiwwwhindawicom Volume 2018

Engineering Mathematics

International Journal of

Hindawiwwwhindawicom Volume 2018

Operations ResearchAdvances in

Journal of

Hindawiwwwhindawicom Volume 2018

Function SpacesAbstract and Applied AnalysisHindawiwwwhindawicom Volume 2018

International Journal of Mathematics and Mathematical Sciences

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Volume 2018

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Numerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisAdvances inAdvances in Discrete Dynamics in

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Stochastic AnalysisInternational Journal of

Submit your manuscripts atwwwhindawicom

Complexity 7

constant with 119889(V 119908) le 119862V119908 that is by the definition of themetric 119889 we have

119863 [V (119905) 119908 (119905)] le 119862V119908120593 (119905) forall119905 isin 119869 (59)

From the definition of the metric 119863 and assumptions (i)-(ii)of Theorem 15 and inequality (59) we infer that

119863[(PV) (119905) (P119908) (119905)] = 119863[V0 + int119905

119886119891 (119904 V (119904)) 119889119904

+ int119905119886int119904119886119892 (119904 119903 V (119903)) 119889119903119889119904 V0 + int

119905

119886119891 (119904 119908 (119904)) 119889119904

+ int119905119886int119904119886119892 (119904 119903 119908 (119903)) 119889119903119889119904]

le int119905119886119863[119891 (119904 V (119904)) 119891 (119904 119908 (119904))] 119889119904

+ int119905119886int119904119886119863[119892 (119904 119903 V (119903)) 119892 (119904 119903 119908 (119903))] 119889119903119889119904

le 119871119891119892 int119905

119886119863 [V (119904) 119908 (119904)] 119889119904

+ 119871119891119892 int119905

119886int119904119886119863[V (119903) 119908 (119903)] 119889119903119889119904

le 119871119891119892119862V119908 int119905

119886120593 (119904) 119889119904

+ 119871119891119892119862V119908 int119905

119886int119904119886120593 (119903) 119889119903119889119904 le 119871119891119892 (119862 + 1198622)

sdot 119862V119908120593 (119905)

(60)

for any 119905 isin 119869For each V 119908 isin X and by the definition of metric 119889 we

get

119889 ((PV) (119905) (P119908) (119905)) le 119871119891119892 (119862 + 1198622)119862V119908120593 (119905) (61)

for any 119905 isin 119869 Hence by (59) we can conclude that

119889 ((PV) (119905) (P119908) (119905)) le 119871119891119892 (119862 + 1198622) 119889 (V 119908) (62)

for any 119905 isin 119869It follows form the definitions of X and the operator P

that for arbitrary 119908 isin X there exists a constant 0 lt 119862 lt +infinsuch that

119863[(P119908) (119905) 119908 (119905)] = 119863 [1199080 + int119905

119886119891 (119904 119908 (119904)) 119889119904

+ int119905119886int119904119886119892 (119904 119903 119908 (119903)) 119889119903119889119904 119908 (119905)] le 119862120593 (119905)

(63)

for any 119905 isin 119869 since 119891(119905 119908(119905)) 119892(119905 119904 119908(119904)) and 119908(119905) arebounded on 119869 and min119905isin119869120593(119905) gt 0 Thus by definition of 119889 itis implied that

119889 (P119908119908) le 119862 lt +infin (64)

Therefore according toTheorem 2 there exists a continuousfunction on 119869 such that 119869119899119908 997888rarr as 119899 997888rarr +infin in the space(X 119889) and 119869119899 = that is satisfies (58) for each 119905 isin 119869

Observe that

X = 119908 isin X | 119889 (119908 119908) lt +infin (65)

Indeed for any 119908 isin X there exists a constant 0 lt 119862 lt +infinsuch that

119863 [119908 (119905) 119908 (119905)] le 119862120593 (119905) (66)

since119908 and119908 are bounded on 119869 andmin119905isin119869120593(119905) gt 0 It followsfrom the preceding inequality that

119889 (119908119908) lt +infin (67)

for all 119908 isin X Hence we obtained that X = 119908 isin X |119889(119908119908) lt +infinFrom Theorem 2 we infer that 119908 is a unique fixed point

of 119869 inX It is obvious that 119908 is a unique fuzzy function in Xwhich satisfies equation 119869119908 = 119908

On the other hand we have

119863[119906 (119905) 1199060 + int119905

119886119891 (119904 119906 (119904)) 119889119904

+ int119905119886int119904119886119892 (119904 119903 119906 (119903)) 119889119903119889119904] = 119863[119906 (119905)

⊖ 1199060 int119905

119886119891 (119904 119906 (119904)) 119889119904 + int119905

119886int119904119886119892 (119904 119903 119906 (119903)) 119889119903119889119904]

= 119863[int119905119886119863119892119867119906 (119904) 119889119904 int

119905

119886119891 (119904 119906 (119904)) 119889119904

+ int119905119886int119904119886119892 (119904 119903 119906 (119903)) 119889119903119889119904] le int119905

119886119863

sdot [119863119892119867119906 (119904) 119891 (119904 119906 (119904)) + int119904

119886119892 (119904 119903 119906 (119903)) 119889119903] 119889119904

le int119905119886120593 (119904) 119889119904 le 119862120593 (119905)

(68)

for any 119905 isin 119869 This means that

119889 (119906P119906) le 119862 (69)

Finally byTheorem 2 and inequation (69) we deduce that

119889 (119906 ) le 11 minus 119871119891119892 (119862 + 1198622)119889 (119906P119906)

le 1198621 minus 119871119891119892 (119862 + 1198622)120593 (119905)

(70)

for any 119905 isin 119869 which means that inequality (55) is true for all119905 isin 119869Corollary 16 Assume that 119891 and 119892 satisfy the assumption (i)of eorem 15 and 0 lt 119871119891119892(119887 minus 119886) + 119871119891119892((119887 minus 119886)22) lt 1

8 Complexity

For a given 120576 gt 0 if a continuously (1)-differentiable function119906 119869 997888rarr RF satisfies the following inequality

119863[119863119892119867119906 (119905) 119891 (119905 119906 (119905)) + int119905

119886119892 (119905 119904 119906 (119904)) 119889119904] le 120576 (71)

for any 119905 isin 119869 then there exists a unique (S1)-solution 119869 997888rarrRF of (14) such that

(119905) = 0 + int119905

119886119891 (119904 (119904)) 119889119904

+ int119905119886int119904119886119892 (119904 119903 (119903)) 119889119903119889119904

(72)

and

119863[ (119905) 119906 (119905)]le (119887 minus 119886) 1205761 minus (119871119891119892 (119887 minus 119886) + 119871119891119892 ((119887 minus 119886)2 2))

(73)

for any 119905 isin 119869Theorem 17 Suppose the functions 119891 119892 and 120593 satisfy allconditions as ineorem 15 If a continuously (2)-differentiablefunction 119906 119869 997888rarr RF satisfies inequality (53) in eorem 15for any 119905 isin 119869 then there exists a unique (S2)-solution 119869 997888rarrRF of (14) such that

(119905) = 0 ⊖ (minus1) int119905

119886119891 (119904 (119904)) 119889119904 ⊖ (minus1)

sdot int119905119886int119904119886119892 (119904 119903 (119903)) 119889119903119889119904

(74)

and the estimation (55) as in eorem 15 on 119869Proof Similar to the proof of Theorem 15 Consider theoperator H X 997888rarr X defined by

(HV) (119905) = V0 ⊖ (minus1)int119905

119886119891 (119904 V (119904)) 119889119904 ⊖ (minus1)

sdot int119905119886int119904119886119892 (119904 119903 V (119903)) 119889119903119889119904

(75)

for all V isin X Based on Lemmas 32 and 33 in [3] it is easy tosee that HV is (2)-differentiable and so HV isin X

We check the operator H is strict contractive on X LetV 119908 isin X and let 119862V119908 isin [0 +infin) cup +infin be an arbitraryconstant with 119889(V 119908) le 119862V119908 we have

119863 [V (119905) 119908 (119905)] le 119862V119908120593 (119905) forall119905 isin 119869 (76)

From the assumptions (i)-(ii) in Theorem 15 and (76) we get

119863 [(HV) (119905) (H119908) (119905)] = 119863[V0 ⊖ (minus1)

sdot int119905119886119891 (119904 V (119904)) 119889119904 ⊖ (minus1)int119905

119886int119904119886119892 (119904 119903 V (119903)) 119889119903119889119904 V0

⊖ (minus1)int119905119886119891 (119904 119908 (119904)) 119889119904 ⊖ (minus1)

sdot int119905119886int119904119886119892 (119904 119903 119908 (119903)) 119889119903119889119904]

le int119905119886119863[119891 (119904 V (119904)) 119891 (119904 119908 (119904))] 119889119904

+ int119905119886int119904119886119863 [119892 (119904 119903 V (119903)) 119892 (119904 119903 119908 (119903))] 119889119903119889119904

le 119871119891119892 int119905

119886119863[V (119904) 119908 (119904)] 119889119904

+ 119871119891119892 int119905

119886int119904119886119863[V (119903) 119908 (119903)] 119889119903119889119904

le 119871119891119892119862V119908 int119905

119886120593 (119904) 119889119904

+ 119871119891119892119862V119908 int119905

119886int119904119886120593 (119903) 119889119903119889119904 le 119871119891119892 (119862 + 1198622)

sdot 119862V119908120593 (119905)(77)

for any 119905 isin 119869 Hence by (76) we conclude that119889 ((HV) (119905) (H119908) (119905)) le 119862119871119891119892119889 (V 119908) forall119905 isin 119869 (78)

By the definitions of X and P we have for arbitrary 119908 isin Xthere exists a constant 0 lt 119862 lt +infin such that

119863[(H119908) (119905) 119908 (119905)] = 119863 [1199080 ⊖ (minus1)

sdot int119905119886119891 (119904 119908 (119904)) 119889119904 ⊖ (minus1)

sdot int119905119886int119904119886119892 (119904 119903 119908 (119903)) 119889119903119889119904 119908 (119905)] le 119862120593 (119905) forall119905 isin 119869

(79)

119891119892 and 119908 are bounded on 119869 and min119905isin119869120593(119905) gt 0 Thus bythe definition of 119889 we have 119889(119908 119908) lt +infin for all 119908 isin XHence we infer that X = 119908 isin X | 119889(119908119908) lt +infin FromTheorem 2 we deduce that 119908 is a unique fixed point of 119869 in119883 It is obvious that 119908 is a unique fuzzy function in X whichsatisfies the equality 119869119908 = 119908

On the other hand Hukuhara difference 1199060 ⊖ 119906(119905) existsfor all 119905 isin 119869 and from (52) and the definition of H that

119863[119906 (119905) 1199060 ⊖ (minus1) int119905

119886119891 (119904 119906 (119904)) 119889119904 ⊖ (minus1)

sdot int119905119886int119904119886119892 (119904 119903 119906 (119903)) 119889119903119889119904] = 119863[1199060 ⊖ 119906 (119905)

minus (int119905119886119891 (119904 119906 (119904)) 119889119904 + int119905

119886int119904119886119892 (119904 119903 119906 (119903)) 119889119903119889119904)]

= 119863[minusint119905119886119863119892119867119906 (119904) 119889119904

Complexity 9

minus (int119905119886119891 (119904 119906 (119904)) 119889119904 + int119905

119886int119904119886119892 (119904 119903 119906 (119903)) 119889119903119889119904)]

le int119905119886119863[119863119892119867119906 (119904) 119891 (119904 119906 (119904))

+ int119904119886119892 (119904 119903 119906 (119903)) 119889119903] 119889119904 le int119905

119886120593 (119904) 119889119904 le 119862120593 (119905)

(80)

for any 119905 isin 119869 which implies that

119889 (119906H119906) le 119862 (81)

ByTheorem 2 and inequation (81) we deduce that

119889 (119906 ) le 11 minus 119871119891119892 (119862 + 1198622)119889 (119906H119906)

le 1198621 minus 119871119891119892 (119862 + 1198622)120593 (119905)

(82)

for any 119905 isin 119869 which means that inequality (55) is true for all119905 isin 119869Corollary 18 Assume that 119891 and 119892 satisfy the assumption (i)of eorem 15 and 0 lt 119871119891119892(119887 minus 119886) + 119871119891119892((119887 minus 119886)22) lt 1For a given 120576 gt 0 if a continuously (2)-differentiable function119906 119869 997888rarr RF satisfies the following inequation

119863[119863g119867119906 (119905) 119891 (119905 119906 (119905)) + int

119905

119886119892 (119905 119904 119906 (119904)) 119889119904] le 120576 (83)

for any 119905 isin 119869 then there exists a unique (S2)-solution 119869 997888rarrRF of (14) such that

(119905) = 0 ⊖ (minus1) int119905

119886119891 (119904 (119904)) 119889119904 ⊖ (minus1)

sdot int119905119886int119904119886119892 (119904 119903 (119903)) 119889119903119889119904

(84)

and

119863[ (119905) 119906 (119905)]le (119887 minus 119886) 1205761 minus (119871119891119892 (119887 minus 119886) + 119871119891119892 ((119887 minus 119886)2 2))

(85)

for any 119905 isin 119869Example 19 We consider the fuzzy intergodifferential equa-tion as follows

119863119892119867119906 (119905) = 12 int119905

01199051199042119906 (119904) 119889119904 forall119905 isin [0 1] (86)

and the following inequality

119863[119863119892119867 (119905) 12 int119905

01199051199042 (119904) 119889119904] le 11989005119905 forall119905 119904 isin [0 1] (87)

where 119906 is a continuously (1)-differentiable (or (2)-differenti-able) function

It is easy to check that the functions 119891 119892 satisfy Lipschitzcondition with 119871119891119892 = 12 Choosing 120593(119905) = 1205761198902119905 with 120576 gt 0and 119862 = 05 we have

int1199050120593 (119904) 119889119904 = int119905

01205761198902119904119889119904 = 120576 (051198902119905 minus 1) le 051205761198902119905

= 1198621205761198902119905 forall119905 isin [0 1] (88)

Now all assumptions in Theorem 15 (or Theorem 17) aresatisfied problem (86) has a unique solution and (86) isUlam-Hyers-Rassias stable with

119863[ (119905) 119906 (119905)] le 1198621 minus 119871119891119892 (119862 + 1198622)120593 (119905) =

451205761198902119905 (89)

for any 119905 isin [0 1]In particular if we choose 120593(119905) = 120576 then we have

119863( (119905) 119906 (119905))le (119887 minus 119886) 1205761 minus (119871119891119892 (119887 minus 119886) + 119871119891119892 ((119887 minus 119886)2 2)) = 4120576

(90)

for all 119905 isin [0 1]4 Conlusion

In this study the Ulam-Hyers stability and Ulam-Hyers-Rassias stability for fuzzy intergodifferential equation via thefixed point technique and successive approximation methodare studied Moreover some illustrative examples are givenIn future work we will study Ulam stability for fuzzyintergodifferential equation in the quotient space of fuzzynumbers introduced by [25ndash27]

Data Availability

No data were used to support this study

Conflicts of Interest

The authors declare that they have no conflicts of interest

References

[1] O AbuArqub ldquoAdaptation of reproducing kernel algorithm forsolving fuzzy Fredholm-Volterra integrodifferential equationsrdquoNeural Computing and Applications vol 28 no 7 pp 1591ndash16102017

[2] R Alikhani and F Bahrami ldquoGlobal solutions of fuzzy integro-differential equations under generalized differentiability by themethod of upper and lower solutionsrdquo Information Sciences vol295 pp 600ndash608 2015

[3] R Alikhani F Bahrami and A Jabbari ldquoExistence of globalsolutions to nonlinear fuzzy Volterra integro-differential equa-tionsrdquo Nonlinear Analysis eory Methods amp Applications vol75 no 4 pp 1810ndash1821 2012

10 Complexity

[4] N V Hoa and N D Phu ldquoFuzzy functional integro-differentialequations under generalized h-differentiabilityrdquo Journal of Intel-ligent Fuzzy Systems vol 26 no 1 pp 2073ndash2085 2014

[5] R M Shabestari R Ezzati and T Allahviranloo ldquoSolvingfuzzy volterra integrodifferential equations of fractional orderby bernoulli wavelet methodrdquo Advances in Fuzzy Systems vol2018 Article ID 5603560 11 pages 2018

[6] H Vu ldquoRandom fuzzy differential equations with impulsesrdquoComplexity vol 2017 Article ID 4056016 11 pages 2017

[7] O Abu Arqub S Momani S Al-Mezel and M Kutbi ldquoExis-tence uniqueness and characterization theorems for nonlinearfuzzy integrodifferential equations of volterra typerdquoMathemat-ical Problems in Engineering vol 2015 Article ID 835891 13pages 2015

[8] E Eljaoui S Melliani and L S Chadli ldquoAumann fuzzy im-proper integral and its application to solve fuzzy integro-differential equations by laplace transform methodrdquo Advancesin Fuzzy Systems vol 2018 Article ID 9730502 10 pages 2018

[9] N V Hoa and N D Phu ldquoOn maximal and minimal solutionsfor set-valued differential equations with feedback controlrdquoAbstract and Applied Analysis vol 2012 Article ID 816218 11pages 2012

[10] H Vu ldquoExistence results for fuzzy Volterra integral equationrdquoJournal of Intelligent amp Fuzzy Systems Applications in Engineer-ing and Technology vol 33 no 1 pp 207ndash213 2017

[11] H Vu L S Dong andNN Phung ldquoApplication of contractive-like mapping principles to impulsive fuzzy functional dif-ferential equationrdquo Journal of Intelligent amp Fuzzy SystemsApplications in Engineering and Technology vol 33 no 2 pp753ndash759 2017

[12] J Vanterler da C Sousa and E Capelas de Oliveira ldquoUlam-Hyers stability of a nonlinear fractional Volterra integro-differential equationrdquo Applied Mathematics Letters vol 81 pp50ndash56 2018

[13] D Popa and I Rasa ldquoOn the Hyers-Ulam stability of the lineardifferential equationrdquo Journal of Mathematical Analysis andApplications vol 381 no 2 pp 530ndash537 2011

[14] G Wang M Zhou and L Sun ldquoHyers-Ulam stability oflinear differential equations of first orderrdquo Applied MathematicsLetters vol 21 no 10 pp 1024ndash1028 2008

[15] J R Wang L L Lv and Y Zhou ldquoNew concepts and results instability of fractional differential equationsrdquoCommunications inNonlinear Science and Numerical Simulation vol 17 no 6 pp2530ndash2538 2012

[16] E C de Oliveira and J V Sousa ldquoUlam-Hyers-Rassias stabilityfor a class of fractional integro-differential equationsrdquo Results inMathematics vol 111 no 72 pp 1ndash16 2018

[17] J V D C Sousa and E C D Oliveira ldquoFractional order pseu-doparabolic partial differential equation ulamndashhyers stabilityrdquoBulletin of the Brazilian Mathematical Society pp 1ndash16 2018

[18] J Vanterler da C Sousa K D Kucche and E C de OliveiraldquoStability of 120595-Hilfer impulsive fractional differential equa-tionsrdquo Applied Mathematics Letters vol 88 pp 73ndash80 2019

[19] Y Shen ldquoHyers-Ulam-Rassias stability of first order linearpartial fuzzy differential equations under generalized differ-entiabilityrdquo Advances in Difference Equations vol 2015 no 1article no 351 pp 1ndash18 2015

[20] Y Shen ldquoOn the Ulam stability of first order linear fuzzydifferential equations under generalized differentiabilityrdquo FuzzySets and Systems vol 280 no C pp 27ndash57 2015

[21] Y Shen and F Wang ldquoA fixed point approach to the Ulamstability of fuzzy differential equations under generalized differ-entiabilityrdquo Journal of Intelligent amp Fuzzy Systems Applicationsin Engineering and Technology vol 30 no 6 pp 3253ndash32602016

[22] W Ren Z Yang X Sun and M Qi ldquoHyers-Ulam stability ofHermite fuzzy differential equations and fuzzy Mellin trans-formrdquo Journal of Intelligent amp Fuzzy Systems Applications inEngineering and Technology vol 35 no 3 pp 3721ndash3731 2018

[23] J B Diaz and B Margolis ldquoA fixed point theorem of thealternative for contractions on a generalized complete metricspacerdquo Bulletin (New Series) of the American MathematicalSociety vol 74 no 2 pp 305ndash309 1968

[24] B Bede and S G Gal ldquoGeneralizations of the differentiabilityof fuzzy-number-valued functions with applications to fuzzydifferential equationsrdquo Fuzzy Sets and Systems vol 151 no 3pp 581ndash599 2005

[25] D Qiu W Zhang and C Lu ldquoOn fuzzy differential equationsin the quotient space of fuzzy numbersrdquo Fuzzy Sets and Systemsvol 295 pp 72ndash98 2016

[26] D Qiu andW Zhang ldquoSymmetric fuzzy numbers and additiveequivalence of fuzzy numbersrdquo So13 Computing vol 17 no 8 pp1471ndash1477 2013

[27] D Qiu C Lu W Zhang and Y Lan ldquoAlgebraic properties andtopological properties of the quotient space of fuzzy numbersbased on Mares equivalence relationrdquo Fuzzy Sets and Systemsvol 245 pp 63ndash82 2014

Hindawiwwwhindawicom Volume 2018

MathematicsJournal of

Hindawiwwwhindawicom Volume 2018

Mathematical Problems in Engineering

Applied MathematicsJournal of

Hindawiwwwhindawicom Volume 2018

Probability and StatisticsHindawiwwwhindawicom Volume 2018

Journal of

Hindawiwwwhindawicom Volume 2018

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawiwwwhindawicom Volume 2018

OptimizationJournal of

Hindawiwwwhindawicom Volume 2018

Hindawiwwwhindawicom Volume 2018

Engineering Mathematics

International Journal of

Hindawiwwwhindawicom Volume 2018

Operations ResearchAdvances in

Journal of

Hindawiwwwhindawicom Volume 2018

Function SpacesAbstract and Applied AnalysisHindawiwwwhindawicom Volume 2018

International Journal of Mathematics and Mathematical Sciences

Hindawiwwwhindawicom Volume 2018

Hindawi Publishing Corporation httpwwwhindawicom Volume 2013Hindawiwwwhindawicom

The Scientific World Journal

Volume 2018

Hindawiwwwhindawicom Volume 2018Volume 2018

Numerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisAdvances inAdvances in Discrete Dynamics in

Nature and SocietyHindawiwwwhindawicom Volume 2018

Hindawiwwwhindawicom

Dierential EquationsInternational Journal of

Volume 2018

Hindawiwwwhindawicom Volume 2018

Decision SciencesAdvances in

Hindawiwwwhindawicom Volume 2018

AnalysisInternational Journal of

Hindawiwwwhindawicom Volume 2018

Stochastic AnalysisInternational Journal of

Submit your manuscripts atwwwhindawicom

8 Complexity

For a given 120576 gt 0 if a continuously (1)-differentiable function119906 119869 997888rarr RF satisfies the following inequality

119863[119863119892119867119906 (119905) 119891 (119905 119906 (119905)) + int119905

119886119892 (119905 119904 119906 (119904)) 119889119904] le 120576 (71)

for any 119905 isin 119869 then there exists a unique (S1)-solution 119869 997888rarrRF of (14) such that

(119905) = 0 + int119905

119886119891 (119904 (119904)) 119889119904

+ int119905119886int119904119886119892 (119904 119903 (119903)) 119889119903119889119904

(72)

and

119863[ (119905) 119906 (119905)]le (119887 minus 119886) 1205761 minus (119871119891119892 (119887 minus 119886) + 119871119891119892 ((119887 minus 119886)2 2))

(73)

for any 119905 isin 119869Theorem 17 Suppose the functions 119891 119892 and 120593 satisfy allconditions as ineorem 15 If a continuously (2)-differentiablefunction 119906 119869 997888rarr RF satisfies inequality (53) in eorem 15for any 119905 isin 119869 then there exists a unique (S2)-solution 119869 997888rarrRF of (14) such that

(119905) = 0 ⊖ (minus1) int119905

119886119891 (119904 (119904)) 119889119904 ⊖ (minus1)

sdot int119905119886int119904119886119892 (119904 119903 (119903)) 119889119903119889119904

(74)

and the estimation (55) as in eorem 15 on 119869Proof Similar to the proof of Theorem 15 Consider theoperator H X 997888rarr X defined by

(HV) (119905) = V0 ⊖ (minus1)int119905

119886119891 (119904 V (119904)) 119889119904 ⊖ (minus1)

sdot int119905119886int119904119886119892 (119904 119903 V (119903)) 119889119903119889119904

(75)

for all V isin X Based on Lemmas 32 and 33 in [3] it is easy tosee that HV is (2)-differentiable and so HV isin X

We check the operator H is strict contractive on X LetV 119908 isin X and let 119862V119908 isin [0 +infin) cup +infin be an arbitraryconstant with 119889(V 119908) le 119862V119908 we have

119863 [V (119905) 119908 (119905)] le 119862V119908120593 (119905) forall119905 isin 119869 (76)

From the assumptions (i)-(ii) in Theorem 15 and (76) we get

119863 [(HV) (119905) (H119908) (119905)] = 119863[V0 ⊖ (minus1)

sdot int119905119886119891 (119904 V (119904)) 119889119904 ⊖ (minus1)int119905

119886int119904119886119892 (119904 119903 V (119903)) 119889119903119889119904 V0

⊖ (minus1)int119905119886119891 (119904 119908 (119904)) 119889119904 ⊖ (minus1)

sdot int119905119886int119904119886119892 (119904 119903 119908 (119903)) 119889119903119889119904]

le int119905119886119863[119891 (119904 V (119904)) 119891 (119904 119908 (119904))] 119889119904

+ int119905119886int119904119886119863 [119892 (119904 119903 V (119903)) 119892 (119904 119903 119908 (119903))] 119889119903119889119904

le 119871119891119892 int119905

119886119863[V (119904) 119908 (119904)] 119889119904

+ 119871119891119892 int119905

119886int119904119886119863[V (119903) 119908 (119903)] 119889119903119889119904

le 119871119891119892119862V119908 int119905

119886120593 (119904) 119889119904

+ 119871119891119892119862V119908 int119905

119886int119904119886120593 (119903) 119889119903119889119904 le 119871119891119892 (119862 + 1198622)

sdot 119862V119908120593 (119905)(77)

for any 119905 isin 119869 Hence by (76) we conclude that119889 ((HV) (119905) (H119908) (119905)) le 119862119871119891119892119889 (V 119908) forall119905 isin 119869 (78)

By the definitions of X and P we have for arbitrary 119908 isin Xthere exists a constant 0 lt 119862 lt +infin such that

119863[(H119908) (119905) 119908 (119905)] = 119863 [1199080 ⊖ (minus1)

sdot int119905119886119891 (119904 119908 (119904)) 119889119904 ⊖ (minus1)

sdot int119905119886int119904119886119892 (119904 119903 119908 (119903)) 119889119903119889119904 119908 (119905)] le 119862120593 (119905) forall119905 isin 119869

(79)

119891119892 and 119908 are bounded on 119869 and min119905isin119869120593(119905) gt 0 Thus bythe definition of 119889 we have 119889(119908 119908) lt +infin for all 119908 isin XHence we infer that X = 119908 isin X | 119889(119908119908) lt +infin FromTheorem 2 we deduce that 119908 is a unique fixed point of 119869 in119883 It is obvious that 119908 is a unique fuzzy function in X whichsatisfies the equality 119869119908 = 119908

On the other hand Hukuhara difference 1199060 ⊖ 119906(119905) existsfor all 119905 isin 119869 and from (52) and the definition of H that

119863[119906 (119905) 1199060 ⊖ (minus1) int119905

119886119891 (119904 119906 (119904)) 119889119904 ⊖ (minus1)

sdot int119905119886int119904119886119892 (119904 119903 119906 (119903)) 119889119903119889119904] = 119863[1199060 ⊖ 119906 (119905)

minus (int119905119886119891 (119904 119906 (119904)) 119889119904 + int119905

119886int119904119886119892 (119904 119903 119906 (119903)) 119889119903119889119904)]

= 119863[minusint119905119886119863119892119867119906 (119904) 119889119904

Complexity 9

minus (int119905119886119891 (119904 119906 (119904)) 119889119904 + int119905

119886int119904119886119892 (119904 119903 119906 (119903)) 119889119903119889119904)]

le int119905119886119863[119863119892119867119906 (119904) 119891 (119904 119906 (119904))

+ int119904119886119892 (119904 119903 119906 (119903)) 119889119903] 119889119904 le int119905

119886120593 (119904) 119889119904 le 119862120593 (119905)

(80)

for any 119905 isin 119869 which implies that

119889 (119906H119906) le 119862 (81)

ByTheorem 2 and inequation (81) we deduce that

119889 (119906 ) le 11 minus 119871119891119892 (119862 + 1198622)119889 (119906H119906)

le 1198621 minus 119871119891119892 (119862 + 1198622)120593 (119905)

(82)

for any 119905 isin 119869 which means that inequality (55) is true for all119905 isin 119869Corollary 18 Assume that 119891 and 119892 satisfy the assumption (i)of eorem 15 and 0 lt 119871119891119892(119887 minus 119886) + 119871119891119892((119887 minus 119886)22) lt 1For a given 120576 gt 0 if a continuously (2)-differentiable function119906 119869 997888rarr RF satisfies the following inequation

119863[119863g119867119906 (119905) 119891 (119905 119906 (119905)) + int

119905

119886119892 (119905 119904 119906 (119904)) 119889119904] le 120576 (83)

for any 119905 isin 119869 then there exists a unique (S2)-solution 119869 997888rarrRF of (14) such that

(119905) = 0 ⊖ (minus1) int119905

119886119891 (119904 (119904)) 119889119904 ⊖ (minus1)

sdot int119905119886int119904119886119892 (119904 119903 (119903)) 119889119903119889119904

(84)

and

119863[ (119905) 119906 (119905)]le (119887 minus 119886) 1205761 minus (119871119891119892 (119887 minus 119886) + 119871119891119892 ((119887 minus 119886)2 2))

(85)

for any 119905 isin 119869Example 19 We consider the fuzzy intergodifferential equa-tion as follows

119863119892119867119906 (119905) = 12 int119905

01199051199042119906 (119904) 119889119904 forall119905 isin [0 1] (86)

and the following inequality

119863[119863119892119867 (119905) 12 int119905

01199051199042 (119904) 119889119904] le 11989005119905 forall119905 119904 isin [0 1] (87)

where 119906 is a continuously (1)-differentiable (or (2)-differenti-able) function

It is easy to check that the functions 119891 119892 satisfy Lipschitzcondition with 119871119891119892 = 12 Choosing 120593(119905) = 1205761198902119905 with 120576 gt 0and 119862 = 05 we have

int1199050120593 (119904) 119889119904 = int119905

01205761198902119904119889119904 = 120576 (051198902119905 minus 1) le 051205761198902119905

= 1198621205761198902119905 forall119905 isin [0 1] (88)

Now all assumptions in Theorem 15 (or Theorem 17) aresatisfied problem (86) has a unique solution and (86) isUlam-Hyers-Rassias stable with

119863[ (119905) 119906 (119905)] le 1198621 minus 119871119891119892 (119862 + 1198622)120593 (119905) =

451205761198902119905 (89)

for any 119905 isin [0 1]In particular if we choose 120593(119905) = 120576 then we have

119863( (119905) 119906 (119905))le (119887 minus 119886) 1205761 minus (119871119891119892 (119887 minus 119886) + 119871119891119892 ((119887 minus 119886)2 2)) = 4120576

(90)

for all 119905 isin [0 1]4 Conlusion

In this study the Ulam-Hyers stability and Ulam-Hyers-Rassias stability for fuzzy intergodifferential equation via thefixed point technique and successive approximation methodare studied Moreover some illustrative examples are givenIn future work we will study Ulam stability for fuzzyintergodifferential equation in the quotient space of fuzzynumbers introduced by [25ndash27]

Data Availability

No data were used to support this study

Conflicts of Interest

The authors declare that they have no conflicts of interest

References

[1] O AbuArqub ldquoAdaptation of reproducing kernel algorithm forsolving fuzzy Fredholm-Volterra integrodifferential equationsrdquoNeural Computing and Applications vol 28 no 7 pp 1591ndash16102017

[2] R Alikhani and F Bahrami ldquoGlobal solutions of fuzzy integro-differential equations under generalized differentiability by themethod of upper and lower solutionsrdquo Information Sciences vol295 pp 600ndash608 2015

[3] R Alikhani F Bahrami and A Jabbari ldquoExistence of globalsolutions to nonlinear fuzzy Volterra integro-differential equa-tionsrdquo Nonlinear Analysis eory Methods amp Applications vol75 no 4 pp 1810ndash1821 2012

10 Complexity

[4] N V Hoa and N D Phu ldquoFuzzy functional integro-differentialequations under generalized h-differentiabilityrdquo Journal of Intel-ligent Fuzzy Systems vol 26 no 1 pp 2073ndash2085 2014

[5] R M Shabestari R Ezzati and T Allahviranloo ldquoSolvingfuzzy volterra integrodifferential equations of fractional orderby bernoulli wavelet methodrdquo Advances in Fuzzy Systems vol2018 Article ID 5603560 11 pages 2018

[6] H Vu ldquoRandom fuzzy differential equations with impulsesrdquoComplexity vol 2017 Article ID 4056016 11 pages 2017

[7] O Abu Arqub S Momani S Al-Mezel and M Kutbi ldquoExis-tence uniqueness and characterization theorems for nonlinearfuzzy integrodifferential equations of volterra typerdquoMathemat-ical Problems in Engineering vol 2015 Article ID 835891 13pages 2015

[8] E Eljaoui S Melliani and L S Chadli ldquoAumann fuzzy im-proper integral and its application to solve fuzzy integro-differential equations by laplace transform methodrdquo Advancesin Fuzzy Systems vol 2018 Article ID 9730502 10 pages 2018

[9] N V Hoa and N D Phu ldquoOn maximal and minimal solutionsfor set-valued differential equations with feedback controlrdquoAbstract and Applied Analysis vol 2012 Article ID 816218 11pages 2012

[10] H Vu ldquoExistence results for fuzzy Volterra integral equationrdquoJournal of Intelligent amp Fuzzy Systems Applications in Engineer-ing and Technology vol 33 no 1 pp 207ndash213 2017

[11] H Vu L S Dong andNN Phung ldquoApplication of contractive-like mapping principles to impulsive fuzzy functional dif-ferential equationrdquo Journal of Intelligent amp Fuzzy SystemsApplications in Engineering and Technology vol 33 no 2 pp753ndash759 2017

[12] J Vanterler da C Sousa and E Capelas de Oliveira ldquoUlam-Hyers stability of a nonlinear fractional Volterra integro-differential equationrdquo Applied Mathematics Letters vol 81 pp50ndash56 2018

[13] D Popa and I Rasa ldquoOn the Hyers-Ulam stability of the lineardifferential equationrdquo Journal of Mathematical Analysis andApplications vol 381 no 2 pp 530ndash537 2011

[14] G Wang M Zhou and L Sun ldquoHyers-Ulam stability oflinear differential equations of first orderrdquo Applied MathematicsLetters vol 21 no 10 pp 1024ndash1028 2008

[15] J R Wang L L Lv and Y Zhou ldquoNew concepts and results instability of fractional differential equationsrdquoCommunications inNonlinear Science and Numerical Simulation vol 17 no 6 pp2530ndash2538 2012

[16] E C de Oliveira and J V Sousa ldquoUlam-Hyers-Rassias stabilityfor a class of fractional integro-differential equationsrdquo Results inMathematics vol 111 no 72 pp 1ndash16 2018

[17] J V D C Sousa and E C D Oliveira ldquoFractional order pseu-doparabolic partial differential equation ulamndashhyers stabilityrdquoBulletin of the Brazilian Mathematical Society pp 1ndash16 2018

[18] J Vanterler da C Sousa K D Kucche and E C de OliveiraldquoStability of 120595-Hilfer impulsive fractional differential equa-tionsrdquo Applied Mathematics Letters vol 88 pp 73ndash80 2019

[19] Y Shen ldquoHyers-Ulam-Rassias stability of first order linearpartial fuzzy differential equations under generalized differ-entiabilityrdquo Advances in Difference Equations vol 2015 no 1article no 351 pp 1ndash18 2015

[20] Y Shen ldquoOn the Ulam stability of first order linear fuzzydifferential equations under generalized differentiabilityrdquo FuzzySets and Systems vol 280 no C pp 27ndash57 2015

[21] Y Shen and F Wang ldquoA fixed point approach to the Ulamstability of fuzzy differential equations under generalized differ-entiabilityrdquo Journal of Intelligent amp Fuzzy Systems Applicationsin Engineering and Technology vol 30 no 6 pp 3253ndash32602016

[22] W Ren Z Yang X Sun and M Qi ldquoHyers-Ulam stability ofHermite fuzzy differential equations and fuzzy Mellin trans-formrdquo Journal of Intelligent amp Fuzzy Systems Applications inEngineering and Technology vol 35 no 3 pp 3721ndash3731 2018

[23] J B Diaz and B Margolis ldquoA fixed point theorem of thealternative for contractions on a generalized complete metricspacerdquo Bulletin (New Series) of the American MathematicalSociety vol 74 no 2 pp 305ndash309 1968

[24] B Bede and S G Gal ldquoGeneralizations of the differentiabilityof fuzzy-number-valued functions with applications to fuzzydifferential equationsrdquo Fuzzy Sets and Systems vol 151 no 3pp 581ndash599 2005

[25] D Qiu W Zhang and C Lu ldquoOn fuzzy differential equationsin the quotient space of fuzzy numbersrdquo Fuzzy Sets and Systemsvol 295 pp 72ndash98 2016

[26] D Qiu andW Zhang ldquoSymmetric fuzzy numbers and additiveequivalence of fuzzy numbersrdquo So13 Computing vol 17 no 8 pp1471ndash1477 2013

[27] D Qiu C Lu W Zhang and Y Lan ldquoAlgebraic properties andtopological properties of the quotient space of fuzzy numbersbased on Mares equivalence relationrdquo Fuzzy Sets and Systemsvol 245 pp 63ndash82 2014

Hindawiwwwhindawicom Volume 2018

MathematicsJournal of

Hindawiwwwhindawicom Volume 2018

Mathematical Problems in Engineering

Applied MathematicsJournal of

Hindawiwwwhindawicom Volume 2018

Probability and StatisticsHindawiwwwhindawicom Volume 2018

Journal of

Hindawiwwwhindawicom Volume 2018

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawiwwwhindawicom Volume 2018

OptimizationJournal of

Hindawiwwwhindawicom Volume 2018

Hindawiwwwhindawicom Volume 2018

Engineering Mathematics

International Journal of

Hindawiwwwhindawicom Volume 2018

Operations ResearchAdvances in

Journal of

Hindawiwwwhindawicom Volume 2018

Function SpacesAbstract and Applied AnalysisHindawiwwwhindawicom Volume 2018

International Journal of Mathematics and Mathematical Sciences

Hindawiwwwhindawicom Volume 2018

Hindawi Publishing Corporation httpwwwhindawicom Volume 2013Hindawiwwwhindawicom

The Scientific World Journal

Volume 2018

Hindawiwwwhindawicom Volume 2018Volume 2018

Numerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisAdvances inAdvances in Discrete Dynamics in

Nature and SocietyHindawiwwwhindawicom Volume 2018

Hindawiwwwhindawicom

Dierential EquationsInternational Journal of

Volume 2018

Hindawiwwwhindawicom Volume 2018

Decision SciencesAdvances in

Hindawiwwwhindawicom Volume 2018

AnalysisInternational Journal of

Hindawiwwwhindawicom Volume 2018

Stochastic AnalysisInternational Journal of

Submit your manuscripts atwwwhindawicom

Complexity 9

minus (int119905119886119891 (119904 119906 (119904)) 119889119904 + int119905

119886int119904119886119892 (119904 119903 119906 (119903)) 119889119903119889119904)]

le int119905119886119863[119863119892119867119906 (119904) 119891 (119904 119906 (119904))

+ int119904119886119892 (119904 119903 119906 (119903)) 119889119903] 119889119904 le int119905

119886120593 (119904) 119889119904 le 119862120593 (119905)

(80)

for any 119905 isin 119869 which implies that

119889 (119906H119906) le 119862 (81)

ByTheorem 2 and inequation (81) we deduce that

119889 (119906 ) le 11 minus 119871119891119892 (119862 + 1198622)119889 (119906H119906)

le 1198621 minus 119871119891119892 (119862 + 1198622)120593 (119905)

(82)

for any 119905 isin 119869 which means that inequality (55) is true for all119905 isin 119869Corollary 18 Assume that 119891 and 119892 satisfy the assumption (i)of eorem 15 and 0 lt 119871119891119892(119887 minus 119886) + 119871119891119892((119887 minus 119886)22) lt 1For a given 120576 gt 0 if a continuously (2)-differentiable function119906 119869 997888rarr RF satisfies the following inequation

119863[119863g119867119906 (119905) 119891 (119905 119906 (119905)) + int

119905

119886119892 (119905 119904 119906 (119904)) 119889119904] le 120576 (83)

for any 119905 isin 119869 then there exists a unique (S2)-solution 119869 997888rarrRF of (14) such that

(119905) = 0 ⊖ (minus1) int119905

119886119891 (119904 (119904)) 119889119904 ⊖ (minus1)

sdot int119905119886int119904119886119892 (119904 119903 (119903)) 119889119903119889119904

(84)

and

119863[ (119905) 119906 (119905)]le (119887 minus 119886) 1205761 minus (119871119891119892 (119887 minus 119886) + 119871119891119892 ((119887 minus 119886)2 2))

(85)

for any 119905 isin 119869Example 19 We consider the fuzzy intergodifferential equa-tion as follows

119863119892119867119906 (119905) = 12 int119905

01199051199042119906 (119904) 119889119904 forall119905 isin [0 1] (86)

and the following inequality

119863[119863119892119867 (119905) 12 int119905

01199051199042 (119904) 119889119904] le 11989005119905 forall119905 119904 isin [0 1] (87)

where 119906 is a continuously (1)-differentiable (or (2)-differenti-able) function

It is easy to check that the functions 119891 119892 satisfy Lipschitzcondition with 119871119891119892 = 12 Choosing 120593(119905) = 1205761198902119905 with 120576 gt 0and 119862 = 05 we have

int1199050120593 (119904) 119889119904 = int119905

01205761198902119904119889119904 = 120576 (051198902119905 minus 1) le 051205761198902119905

= 1198621205761198902119905 forall119905 isin [0 1] (88)

Now all assumptions in Theorem 15 (or Theorem 17) aresatisfied problem (86) has a unique solution and (86) isUlam-Hyers-Rassias stable with

119863[ (119905) 119906 (119905)] le 1198621 minus 119871119891119892 (119862 + 1198622)120593 (119905) =

451205761198902119905 (89)

for any 119905 isin [0 1]In particular if we choose 120593(119905) = 120576 then we have

119863( (119905) 119906 (119905))le (119887 minus 119886) 1205761 minus (119871119891119892 (119887 minus 119886) + 119871119891119892 ((119887 minus 119886)2 2)) = 4120576

(90)

for all 119905 isin [0 1]4 Conlusion

In this study the Ulam-Hyers stability and Ulam-Hyers-Rassias stability for fuzzy intergodifferential equation via thefixed point technique and successive approximation methodare studied Moreover some illustrative examples are givenIn future work we will study Ulam stability for fuzzyintergodifferential equation in the quotient space of fuzzynumbers introduced by [25ndash27]

Data Availability

No data were used to support this study

Conflicts of Interest

The authors declare that they have no conflicts of interest

References

[1] O AbuArqub ldquoAdaptation of reproducing kernel algorithm forsolving fuzzy Fredholm-Volterra integrodifferential equationsrdquoNeural Computing and Applications vol 28 no 7 pp 1591ndash16102017

[2] R Alikhani and F Bahrami ldquoGlobal solutions of fuzzy integro-differential equations under generalized differentiability by themethod of upper and lower solutionsrdquo Information Sciences vol295 pp 600ndash608 2015

[3] R Alikhani F Bahrami and A Jabbari ldquoExistence of globalsolutions to nonlinear fuzzy Volterra integro-differential equa-tionsrdquo Nonlinear Analysis eory Methods amp Applications vol75 no 4 pp 1810ndash1821 2012

10 Complexity

[4] N V Hoa and N D Phu ldquoFuzzy functional integro-differentialequations under generalized h-differentiabilityrdquo Journal of Intel-ligent Fuzzy Systems vol 26 no 1 pp 2073ndash2085 2014

[5] R M Shabestari R Ezzati and T Allahviranloo ldquoSolvingfuzzy volterra integrodifferential equations of fractional orderby bernoulli wavelet methodrdquo Advances in Fuzzy Systems vol2018 Article ID 5603560 11 pages 2018

[6] H Vu ldquoRandom fuzzy differential equations with impulsesrdquoComplexity vol 2017 Article ID 4056016 11 pages 2017

[7] O Abu Arqub S Momani S Al-Mezel and M Kutbi ldquoExis-tence uniqueness and characterization theorems for nonlinearfuzzy integrodifferential equations of volterra typerdquoMathemat-ical Problems in Engineering vol 2015 Article ID 835891 13pages 2015

[8] E Eljaoui S Melliani and L S Chadli ldquoAumann fuzzy im-proper integral and its application to solve fuzzy integro-differential equations by laplace transform methodrdquo Advancesin Fuzzy Systems vol 2018 Article ID 9730502 10 pages 2018

[9] N V Hoa and N D Phu ldquoOn maximal and minimal solutionsfor set-valued differential equations with feedback controlrdquoAbstract and Applied Analysis vol 2012 Article ID 816218 11pages 2012

[10] H Vu ldquoExistence results for fuzzy Volterra integral equationrdquoJournal of Intelligent amp Fuzzy Systems Applications in Engineer-ing and Technology vol 33 no 1 pp 207ndash213 2017

[11] H Vu L S Dong andNN Phung ldquoApplication of contractive-like mapping principles to impulsive fuzzy functional dif-ferential equationrdquo Journal of Intelligent amp Fuzzy SystemsApplications in Engineering and Technology vol 33 no 2 pp753ndash759 2017

[12] J Vanterler da C Sousa and E Capelas de Oliveira ldquoUlam-Hyers stability of a nonlinear fractional Volterra integro-differential equationrdquo Applied Mathematics Letters vol 81 pp50ndash56 2018

[13] D Popa and I Rasa ldquoOn the Hyers-Ulam stability of the lineardifferential equationrdquo Journal of Mathematical Analysis andApplications vol 381 no 2 pp 530ndash537 2011

[14] G Wang M Zhou and L Sun ldquoHyers-Ulam stability oflinear differential equations of first orderrdquo Applied MathematicsLetters vol 21 no 10 pp 1024ndash1028 2008

[15] J R Wang L L Lv and Y Zhou ldquoNew concepts and results instability of fractional differential equationsrdquoCommunications inNonlinear Science and Numerical Simulation vol 17 no 6 pp2530ndash2538 2012

[16] E C de Oliveira and J V Sousa ldquoUlam-Hyers-Rassias stabilityfor a class of fractional integro-differential equationsrdquo Results inMathematics vol 111 no 72 pp 1ndash16 2018

[17] J V D C Sousa and E C D Oliveira ldquoFractional order pseu-doparabolic partial differential equation ulamndashhyers stabilityrdquoBulletin of the Brazilian Mathematical Society pp 1ndash16 2018

[18] J Vanterler da C Sousa K D Kucche and E C de OliveiraldquoStability of 120595-Hilfer impulsive fractional differential equa-tionsrdquo Applied Mathematics Letters vol 88 pp 73ndash80 2019

[19] Y Shen ldquoHyers-Ulam-Rassias stability of first order linearpartial fuzzy differential equations under generalized differ-entiabilityrdquo Advances in Difference Equations vol 2015 no 1article no 351 pp 1ndash18 2015

[20] Y Shen ldquoOn the Ulam stability of first order linear fuzzydifferential equations under generalized differentiabilityrdquo FuzzySets and Systems vol 280 no C pp 27ndash57 2015

[21] Y Shen and F Wang ldquoA fixed point approach to the Ulamstability of fuzzy differential equations under generalized differ-entiabilityrdquo Journal of Intelligent amp Fuzzy Systems Applicationsin Engineering and Technology vol 30 no 6 pp 3253ndash32602016

[22] W Ren Z Yang X Sun and M Qi ldquoHyers-Ulam stability ofHermite fuzzy differential equations and fuzzy Mellin trans-formrdquo Journal of Intelligent amp Fuzzy Systems Applications inEngineering and Technology vol 35 no 3 pp 3721ndash3731 2018

[23] J B Diaz and B Margolis ldquoA fixed point theorem of thealternative for contractions on a generalized complete metricspacerdquo Bulletin (New Series) of the American MathematicalSociety vol 74 no 2 pp 305ndash309 1968

[24] B Bede and S G Gal ldquoGeneralizations of the differentiabilityof fuzzy-number-valued functions with applications to fuzzydifferential equationsrdquo Fuzzy Sets and Systems vol 151 no 3pp 581ndash599 2005

[25] D Qiu W Zhang and C Lu ldquoOn fuzzy differential equationsin the quotient space of fuzzy numbersrdquo Fuzzy Sets and Systemsvol 295 pp 72ndash98 2016

[26] D Qiu andW Zhang ldquoSymmetric fuzzy numbers and additiveequivalence of fuzzy numbersrdquo So13 Computing vol 17 no 8 pp1471ndash1477 2013

[27] D Qiu C Lu W Zhang and Y Lan ldquoAlgebraic properties andtopological properties of the quotient space of fuzzy numbersbased on Mares equivalence relationrdquo Fuzzy Sets and Systemsvol 245 pp 63ndash82 2014

Hindawiwwwhindawicom Volume 2018

MathematicsJournal of

Hindawiwwwhindawicom Volume 2018

Mathematical Problems in Engineering

Applied MathematicsJournal of

Hindawiwwwhindawicom Volume 2018

Probability and StatisticsHindawiwwwhindawicom Volume 2018

Journal of

Hindawiwwwhindawicom Volume 2018

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawiwwwhindawicom Volume 2018

OptimizationJournal of

Hindawiwwwhindawicom Volume 2018

Hindawiwwwhindawicom Volume 2018

Engineering Mathematics

International Journal of

Hindawiwwwhindawicom Volume 2018

Operations ResearchAdvances in

Journal of

Hindawiwwwhindawicom Volume 2018

Function SpacesAbstract and Applied AnalysisHindawiwwwhindawicom Volume 2018

International Journal of Mathematics and Mathematical Sciences

Hindawiwwwhindawicom Volume 2018

Hindawi Publishing Corporation httpwwwhindawicom Volume 2013Hindawiwwwhindawicom

The Scientific World Journal

Volume 2018

Hindawiwwwhindawicom Volume 2018Volume 2018

Numerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisAdvances inAdvances in Discrete Dynamics in

Nature and SocietyHindawiwwwhindawicom Volume 2018

Hindawiwwwhindawicom

Dierential EquationsInternational Journal of

Volume 2018

Hindawiwwwhindawicom Volume 2018

Decision SciencesAdvances in

Hindawiwwwhindawicom Volume 2018

AnalysisInternational Journal of

Hindawiwwwhindawicom Volume 2018

Stochastic AnalysisInternational Journal of

Submit your manuscripts atwwwhindawicom

10 Complexity

[4] N V Hoa and N D Phu ldquoFuzzy functional integro-differentialequations under generalized h-differentiabilityrdquo Journal of Intel-ligent Fuzzy Systems vol 26 no 1 pp 2073ndash2085 2014

[5] R M Shabestari R Ezzati and T Allahviranloo ldquoSolvingfuzzy volterra integrodifferential equations of fractional orderby bernoulli wavelet methodrdquo Advances in Fuzzy Systems vol2018 Article ID 5603560 11 pages 2018

[6] H Vu ldquoRandom fuzzy differential equations with impulsesrdquoComplexity vol 2017 Article ID 4056016 11 pages 2017

[7] O Abu Arqub S Momani S Al-Mezel and M Kutbi ldquoExis-tence uniqueness and characterization theorems for nonlinearfuzzy integrodifferential equations of volterra typerdquoMathemat-ical Problems in Engineering vol 2015 Article ID 835891 13pages 2015

[8] E Eljaoui S Melliani and L S Chadli ldquoAumann fuzzy im-proper integral and its application to solve fuzzy integro-differential equations by laplace transform methodrdquo Advancesin Fuzzy Systems vol 2018 Article ID 9730502 10 pages 2018

[9] N V Hoa and N D Phu ldquoOn maximal and minimal solutionsfor set-valued differential equations with feedback controlrdquoAbstract and Applied Analysis vol 2012 Article ID 816218 11pages 2012

[10] H Vu ldquoExistence results for fuzzy Volterra integral equationrdquoJournal of Intelligent amp Fuzzy Systems Applications in Engineer-ing and Technology vol 33 no 1 pp 207ndash213 2017

[11] H Vu L S Dong andNN Phung ldquoApplication of contractive-like mapping principles to impulsive fuzzy functional dif-ferential equationrdquo Journal of Intelligent amp Fuzzy SystemsApplications in Engineering and Technology vol 33 no 2 pp753ndash759 2017

[12] J Vanterler da C Sousa and E Capelas de Oliveira ldquoUlam-Hyers stability of a nonlinear fractional Volterra integro-differential equationrdquo Applied Mathematics Letters vol 81 pp50ndash56 2018

[13] D Popa and I Rasa ldquoOn the Hyers-Ulam stability of the lineardifferential equationrdquo Journal of Mathematical Analysis andApplications vol 381 no 2 pp 530ndash537 2011

[14] G Wang M Zhou and L Sun ldquoHyers-Ulam stability oflinear differential equations of first orderrdquo Applied MathematicsLetters vol 21 no 10 pp 1024ndash1028 2008

[15] J R Wang L L Lv and Y Zhou ldquoNew concepts and results instability of fractional differential equationsrdquoCommunications inNonlinear Science and Numerical Simulation vol 17 no 6 pp2530ndash2538 2012

[16] E C de Oliveira and J V Sousa ldquoUlam-Hyers-Rassias stabilityfor a class of fractional integro-differential equationsrdquo Results inMathematics vol 111 no 72 pp 1ndash16 2018

[17] J V D C Sousa and E C D Oliveira ldquoFractional order pseu-doparabolic partial differential equation ulamndashhyers stabilityrdquoBulletin of the Brazilian Mathematical Society pp 1ndash16 2018

[18] J Vanterler da C Sousa K D Kucche and E C de OliveiraldquoStability of 120595-Hilfer impulsive fractional differential equa-tionsrdquo Applied Mathematics Letters vol 88 pp 73ndash80 2019

[19] Y Shen ldquoHyers-Ulam-Rassias stability of first order linearpartial fuzzy differential equations under generalized differ-entiabilityrdquo Advances in Difference Equations vol 2015 no 1article no 351 pp 1ndash18 2015

[20] Y Shen ldquoOn the Ulam stability of first order linear fuzzydifferential equations under generalized differentiabilityrdquo FuzzySets and Systems vol 280 no C pp 27ndash57 2015

[21] Y Shen and F Wang ldquoA fixed point approach to the Ulamstability of fuzzy differential equations under generalized differ-entiabilityrdquo Journal of Intelligent amp Fuzzy Systems Applicationsin Engineering and Technology vol 30 no 6 pp 3253ndash32602016

[22] W Ren Z Yang X Sun and M Qi ldquoHyers-Ulam stability ofHermite fuzzy differential equations and fuzzy Mellin trans-formrdquo Journal of Intelligent amp Fuzzy Systems Applications inEngineering and Technology vol 35 no 3 pp 3721ndash3731 2018

[23] J B Diaz and B Margolis ldquoA fixed point theorem of thealternative for contractions on a generalized complete metricspacerdquo Bulletin (New Series) of the American MathematicalSociety vol 74 no 2 pp 305ndash309 1968

[24] B Bede and S G Gal ldquoGeneralizations of the differentiabilityof fuzzy-number-valued functions with applications to fuzzydifferential equationsrdquo Fuzzy Sets and Systems vol 151 no 3pp 581ndash599 2005

[25] D Qiu W Zhang and C Lu ldquoOn fuzzy differential equationsin the quotient space of fuzzy numbersrdquo Fuzzy Sets and Systemsvol 295 pp 72ndash98 2016

[26] D Qiu andW Zhang ldquoSymmetric fuzzy numbers and additiveequivalence of fuzzy numbersrdquo So13 Computing vol 17 no 8 pp1471ndash1477 2013

[27] D Qiu C Lu W Zhang and Y Lan ldquoAlgebraic properties andtopological properties of the quotient space of fuzzy numbersbased on Mares equivalence relationrdquo Fuzzy Sets and Systemsvol 245 pp 63ndash82 2014

Hindawiwwwhindawicom Volume 2018

MathematicsJournal of

Hindawiwwwhindawicom Volume 2018

Mathematical Problems in Engineering

Applied MathematicsJournal of

Hindawiwwwhindawicom Volume 2018

Probability and StatisticsHindawiwwwhindawicom Volume 2018

Journal of

Hindawiwwwhindawicom Volume 2018

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawiwwwhindawicom Volume 2018

OptimizationJournal of

Hindawiwwwhindawicom Volume 2018

Hindawiwwwhindawicom Volume 2018

Engineering Mathematics

International Journal of

Hindawiwwwhindawicom Volume 2018

Operations ResearchAdvances in

Journal of

Hindawiwwwhindawicom Volume 2018

Function SpacesAbstract and Applied AnalysisHindawiwwwhindawicom Volume 2018

International Journal of Mathematics and Mathematical Sciences

Hindawiwwwhindawicom Volume 2018

Hindawi Publishing Corporation httpwwwhindawicom Volume 2013Hindawiwwwhindawicom

The Scientific World Journal

Volume 2018

Hindawiwwwhindawicom Volume 2018Volume 2018

Numerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisAdvances inAdvances in Discrete Dynamics in

Nature and SocietyHindawiwwwhindawicom Volume 2018

Hindawiwwwhindawicom

Dierential EquationsInternational Journal of

Volume 2018

Hindawiwwwhindawicom Volume 2018

Decision SciencesAdvances in

Hindawiwwwhindawicom Volume 2018

AnalysisInternational Journal of

Hindawiwwwhindawicom Volume 2018

Stochastic AnalysisInternational Journal of

Submit your manuscripts atwwwhindawicom

Hindawiwwwhindawicom Volume 2018

MathematicsJournal of

Hindawiwwwhindawicom Volume 2018

Mathematical Problems in Engineering

Applied MathematicsJournal of

Hindawiwwwhindawicom Volume 2018

Probability and StatisticsHindawiwwwhindawicom Volume 2018

Journal of

Hindawiwwwhindawicom Volume 2018

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawiwwwhindawicom Volume 2018

OptimizationJournal of

Hindawiwwwhindawicom Volume 2018

Hindawiwwwhindawicom Volume 2018

Engineering Mathematics

International Journal of

Hindawiwwwhindawicom Volume 2018

Operations ResearchAdvances in

Journal of

Hindawiwwwhindawicom Volume 2018

Function SpacesAbstract and Applied AnalysisHindawiwwwhindawicom Volume 2018

International Journal of Mathematics and Mathematical Sciences

Hindawiwwwhindawicom Volume 2018

Hindawi Publishing Corporation httpwwwhindawicom Volume 2013Hindawiwwwhindawicom

The Scientific World Journal

Volume 2018

Hindawiwwwhindawicom Volume 2018Volume 2018

Numerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisAdvances inAdvances in Discrete Dynamics in

Nature and SocietyHindawiwwwhindawicom Volume 2018

Hindawiwwwhindawicom

Dierential EquationsInternational Journal of

Volume 2018

Hindawiwwwhindawicom Volume 2018

Decision SciencesAdvances in

Hindawiwwwhindawicom Volume 2018

AnalysisInternational Journal of

Hindawiwwwhindawicom Volume 2018

Stochastic AnalysisInternational Journal of

Submit your manuscripts atwwwhindawicom