# Ulam-Hyers Stability and Ulam-Hyers-Rassias Stability for Fuzzy Integrodifferential Equation

Authors: Nguyen Ngoc Phung; Bao Quoc Ta; Ho Vu
Journal: Complexity
 (2019)

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## Abstract

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


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## Body

## 1. Introduction

Fuzzy differential and integrodifferential equation (FD-FIDE) are a natural way to model dynamical systems subject to uncertainties. In the past few years, the study of fuzzy integrodifferential equations is an area of mathematics that has recently received a lot of attention (see, e.g., [1–11]). Alikhani et al. [3] studied the existence and uniqueness of global solutions for fuzzy initial value problems via integrodifferential operators of Volterra type. Alikhani et al. [2] introduced a new concept of upper and lower solutions for FIDE. Using this concept, authors proved the existence and uniqueness of global solutions of an initial value problem for first-order nonlinear FIDE under generalized differentiability. Additionally, authors [4] studied the two kinds of solutions to FIDE with delay by the usage of two different concepts of fuzzy derivative.In recent years, accompanied by the development of the Ulam-Hyers stability of differential equation (see, e.g., [12–18]). Ulam-Hyers stability of fuzzy differential equation has attracted the attention of reseachers; the reader is refferred to [19–22]. Ren [22] studied the Ulam-Hyers stability of the Hermite fuzzy differential equation associated with the inhomogeneous Hermite fuzzy differential equation under some suitable conditions. However, the fixed point method has been successfully used to study the Ulam stability of fuzzy differential equation by [19–21]. In [19, 20], Shen considered the Ulam stability of the first order linear (partial) fuzzy differential equation under generalized differentiability. Using the fixed point technique, authors [21] studied the Ulam stability of fuzzy differential equations and we see that Ulam stability of this problem requires various prerequisites under different types of differentiability.To the best of our knowledge, up to now, the number of papers dealing with Ulam stability for FD-FIDE is rather limited as opposed to the amount of publications concerning FD-FIDE. From this reason, we choose the study of Ulam stability of FIDE by using the fixed point technique and the method of successive approximation.The rest of this paper is organized as follows. In Section2, some notations and preparation results are given. In Section 3, Ulam-Hyers stability and Ulam-Hyers-Rassias stability criteria for FIDE defined on a bounded and closed interval are obtained in terms of the method of successive approximation and the fixed point theorem in [23].


## 2. Preliminaries

In this section, we introduce some basic definitions, theorems, and lemmas, which are required throughout this paper.Definition 1 (see [23]).
A functiond:X×X→[0,+∞) is called a generalized metric on X if and only if d satisfies (1)
d(x,y)=0 if and only if x=y;(2)
d(x,y)=d(y,x) for all x,y,z∈X;(3)
d(x,z)≤d(x,y)+d(y,z) for all x,y,z∈X.Theorem 2 (see [23]).
Letd:X×X→[0,+∞) be a generalized metric on X and (X,d) is a generalized complete metric space. Assume that T:X→X is a strictly contractive operator with the Lipschitz constant L<1. If there exists a nonnegative integer n such that d(Tn+1x,Tnx)<∞ for some x∈X, then the following are true: (i)
the sequence{Tnx} converges to a fixed point x∗ of T;(ii)
x∗ is the unique fixed point of T in (1)X∗=y∈X∣dTnx,y<∞;(iii)
ify∈X∗, then we have (2)dy,x∗≤11-LdTy,y.Lemma 3.
Letϕ:J→[0,+∞) be a continuous function. We define the set (3)X≔x:J→RF∣xiscontinuousfunctiononJequipped with the metric (4)dx,y=inf⁡η∈0,+∞∪+∞∣Dxt,yt≤ηφt,∀t∈J.Then, (X,d) is a complete generalized metric space.Proof.
The proof of this lemma can be found in Shen and Wang [21].Denote byRF the class of fuzzy sets u:R→[0,1] with the following properties: (i) u is normal, i.e., there exists x0∈R such that u(x0)=1; (ii) u is fuzzy convex; that is, u(λx+(1-λ)x)≥min{u(x),u(y)} for any x,y∈R, u∈RF and λ∈[0,1]; (iii) u is upper semi-continuous; (iv) cl{x∈R:u(x)>0} is compact, where cl denotes the closure of a set.Usually, the setRF is called the space of fuzzy numbers and it is easy to see that R⊂RF. For α∈(0,1], we denote [u]α={x∈R:u(x)≥α} and [u]0={x∈R:u(x)>0}. Then it follows from the conditions (i)-(iv) that the α-level set [u]α is a non-empty compact interval for all α∈[0,1] and each u∈RF. For any u,v∈RF and λ∈R, the addition u+v and scalar multiplication λu can be defined, levelwise, by [u+v]α=[u]α+[v]α and [λu]=λ[u]α for all α∈[0,1].The supremum metric betweenu and v is defined by (5)D:RF×RF→R+∪0,Du,v=supα∈0,1dHuα,vα=supα∈0,1maxu_α-v_α,u¯α-v¯α.It is easy to see that (RF,D) is a complete metric space. It is well known that the supremum metric has the following properties: (D1)
D(u+w,v+w)=D(u,v) for any u,v,w∈RF;(D2)
D(λu,λv)=λD(u,v) for any λ∈R+, u,v∈RF;(D3)
D(u+v,w+e)≤D(u,w)+D(v,e) for any u,v,w,e∈RF.Definition 4 (see [24]).
Letu,v∈RF. If there exists w∈RF such that u=v+w, then w is called the H-difference of u and v, and it is denoted by u⊖v.Throughout this paper, the symbol “⊖” always stands for the H-difference. In general, u⊖v=u+(-1)v.Definition 5 (see [24]).
Letf:(a,b)→RF and t0∈(a,b). We say f is generalized differential at t0, if there exists an element DHgf(t0)∈RF, such that (1)
for allh>0 sufficiently small, there exists f(t0+h)⊖f(t0), f(t0)⊖f(t0-h) and then limits (in metric D) (6)limh→0ft0+h⊖ft0h=limh→0ft0⊖ft0-hh=DHgft0.(2)
for allh>0 sufficiently small, there exists f(t0)⊖f(t0+h), f(t0-h)⊖f(t0) and then limits (in metric D) (7)limh→0ft0⊖ft0+h-h=limh→0ft0-h⊖ft0-h=DHgft0.Note that Bede and Gal [24] considered four cases in the definition of derivative. In this paper, we consider only the two first cases of Definition 5 in [24]. In the other cases, the derivative reduces to a crisp element; that is, DHgf(t0)∈R.Theorem 6 (see [24]).
Letf:(a,b)→RF and denote [f(t)]α=[f1α(t),f2α(t)] for each α∈[0,1], t∈(a,b). (i)
Iff is (1)-differentiable at all t∈(a,b), then f1α(t) and f2α(t) are differentiable functions and we have (8)DHgftα=f1αt′,f2αt′(ii)
Iff is (2)-differentiable at all t∈(a,b), then f1α(t) and f2α(t) are differentiable functions and we have (9)DHgftα=f2αt′,f1αt′In Theorem6, we see that if f is (1)-differentiable, then it is not (2)-differentiable and vice versa.Theorem 7 (see [24]).
Letf:(a,b)→RF be differentiable on (a,b) and assume that derivative DHgf is integrable over (a,b). For each t∈(a,b), we have (i)
Iff is (1)-differentiable, then (10)ft=fa+∫atDHgfsds;(ii)
Iff is (2)-differentiable, then (11)ft=fa⊖-1∫atDHgfsds.LetJ≔[a,b] (with T>0) be a compact interval of R. We denote by (12)CJ,RF=u∣u:J→RFisacountinuousfunctionsonJ.On the space C(J,RF), we consider the supremum metric as follows: (13)Du,v=supt∈JDut,vt.In next section, we consider the following fuzzy integrodifferential equation:(14)DHgut=ft,ut+∫atgt,s,usds,t∈J,ua=u0∈RF,where the symbol DHg is generalized Hukuhara derivative, the mapping f:J×RF×RF→RF is continuous on J, and g:J×J×RF→RF is a continuous function on J×J.Definition 8 (see [3]).
We say that a mappingu:J→RF is solution to the problem (14) if u is generalized Hukuhara differentiable on J and DHgu(t)=f(t,u(t))+∫at‍g(t,s,u(s))ds for any t∈J, f∈C(J,RF), g∈C(J×J,RF).Lemma 9 (see [3]).
Letu:J→RF be a continuous function on J. Problem (14) is equivalent to one of the following the fuzzy integro integral equations: (S1)
ifu is (1)-differentiable on J, then (15)ut=u0+∫atfs,usds+∫at∫asgs,r,urdrds(S2)
ifu is (2)-differentiable on J, then (16)ut=u0⊖-1∫atfs,usds⊖-1∫at∫asgs,r,urdrds


## 3. Main Results

Definition 10.
We say that problem (14) is Ulam-Hyers stable if there exists a real number Kf>0 such that for ε>0 and for each v∈C1(J,RF) to the problem (17)DDHgvt,ft,vt+∫atgt,s,vsds≤εthere exists a solution to problem (14) with (18)Dvt,ut≤Kfεfor all t∈J. We call Kf 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 Cf>0 such that for ε>0 and for each v∈C1(J,RF) to the problem (19)DDHgvt,ft,vt+∫0tgt,s,vsds≤φtthere exists a solution to problem (14) with (20)Dvt,ut≤Cfφtfor all t∈J.Firstly, we prove that problem (14) is Hyers-Ulam stable via the method of successive approximation. We consider the following inequality:(21)DDHgvt,ft,vt+∫atgt,s,vsds≤εforv∈RF.Definition 12.
We say that(a)
A functionv∈C1(J,RF) is a (S1)-solution of the inequality (21) if and only if there exists a function δ1∈C(J,RF) such that
(i)
D[δ1(t),0^]≤ε for any t∈J;(ii)
DHgv(t)=f(t,v(t))+∫at‍g(t,s,v(s))ds+δ1(t) for any t∈J.(b)
A functionv∈C1(J,RF) is a (S2)-solution of the inequality (21) if and only if there exists a function δ2∈C(J,RF) such that
(i)
D[δ2(t),0^]≤ε for any t∈J;(ii)
DHgv(t)=f(t,v(t))+∫at‍g(t,s,v(s))ds+δ2(t) for any t∈J.Theorem 13.
Assume thatf:J×RF→RF and g:J×J×RF→RF are a continuous function that satisfies the following conditions: (i) there exists a constant Lfg>0 such that (22)maxDft,u,ft,v,Dgt,s,u,gt,s,v≤LfgDu,vfor any (t,u),(t,v)∈J×RF, (t,s,u),(t,s,v)∈J×J×RF; (ii) for each ε>0, if a continuously (2)-differentiable function v:J→RF satisfies(23)DDHgvt,ft,vt+∫atgt,s,vsds≤ε,∀t∈J,then there exists a (S2)-solution u:J→RF of (14) with u0=v0 such that(24)Dvt,ut≤ελLfgeb-a1+Lfg,where λ is a maximum balancing constant.Proof.
For eachε>0, let a continuously (2)-differentiable function v:J→RF satisfy inequality (21) for any t∈J, v∈RF. By the part (b) of Definition 12, we have (25)Dδ2t,0^≤εand(26)DHgvt=ft,vt+∫atgt,s,vsds+δ2tfor any t∈J and δ2∈C(J,RF).
If a functionv:J→RF is continuous and (2)-differentiable on J, then by Lemma 9 it satisfies equivalently the following fuzzy integrointegral equation: (27)vt=v0⊖-1∫atfs,vsds⊖-1∫at∫asgs,r,vrdrds+∫atδ2sds,∀t∈J.Let us define (28)u0t=vt,∀t∈Jand sequence functions un:J→RF, n=1,2,… of successive approximations as follows(29)unt=v0⊖-1∫atfs,un-1sds⊖-1∫at∫asgs,r,un-1rdrds,∀t∈J.By virtue of the properties of D and assumption (i), we have (30)Du1t,u0t=Dv0⊖-1∫atfs,u0sds⊖-1∫at∫asgs,r,u0rdrds,vt=Dv0⊖-1∫atfs,v0sds⊖-1∫at∫asgs,r,v0rdrds,vt=D∫atδ2sds,0^≤∫atDδ2s,0^ds≤εt-a,∀t∈J.Therefore, (31)Du1t,u0t≤εt-a,∀t∈J.Observe that for n=1,2,… and for t∈J, one has (32)Dun+1t,unt=Dv0⊖-1∫atfs,unsds⊖-1∫at∫asgs,r,unrdrds,v0⊖-1∫atfs,un-1sds⊖-1∫at∫asgs,r,un-1rdrds≤Lfg∫atDuns,un-1sds+Lfg∫at∫asDunr,un-1rdrds.In particular, (33)Du2t,u1t≤Lfg∫atDu1s,u0sds+Lfg∫at∫asDu1r,u0rdrds=εLfg∫ats-ads+εLfg∫at∫asr-adrds=εLfgt-a22!+t-a33!=εLfgLfgt-a22!1+t-a3and so (34)Du3t,u2t≤Lfg∫atDu2s,u1sds+Lfg∫at∫asDu2r,u1rdrds=εLfg2∫ats-a22!+s-a33!ds+εLfg2∫at∫asr-a22!+r-a33!drds=εLfg2t-a33!+2t-a44!+t-a55!=εLfgLfgt-a33!1+2t-a4+t-a25and for n>4 we have(35)Dunt,un-1t≤εLfgn-1t-ann!+λ1t-an+1n+1!+…+λnt-a2n2n!+t-a2n+12n+1!,where λ1,…,λn are balancing constants.
We chooseλ=max{1,λ1,…,λn} (called λ is a maximum balancing constant) and then estimation (35) can be rewritten as follows: (36)Dunt,un-1t≤ελLfgLfgt-ann!1+t-an+1+t-a2n+1n+2+…+t-ann+1n+2…2n+t-an+1n+1n+2…2n2n+1≤ελLfgLfgt-ann!1+t-a1!+t-a22!+…+t-ann!+t-an+1n+1!≤ελLfgLfgt-ann!et-a.Further, if we assume that(37)Dunt,un-1t≤ελLfgLfgt-ann!et-a,∀t∈J,then we obtain (38)Dun+1t,unt≤ελLfgLfgt-an+1n+1!et-a,∀t∈J.By the principle of mathematical induction that (37) holds for every n>1 and now using estimation (37) for any t∈J, we get (39)∑n=1∞Dunt,un-1t≤ελLfg∑n=1∞Lfgt-ann!et-a.The series ∑j=1∞‍zk/k! is convergent for every z∈R. Hence, for every ε>0 we infer that the series ∑n=1∞‍D[un(t),un-1(t)] is uniformly convergent on J with respect to metric D and (40)∑n=1∞Dunt,un-1t≤ελLfg∑n=1∞Lfgt-ann!et-a≤ελLfg∑n=1∞Lfgb-ann!eb-a=ελLfgeLfgb-aeb-a=ελLfgeb-a1+Lfg.For t∈J, we have(41)Dunt,ut≤ελLfg∑n=1∞Lfgt-ann!et-a.and the following estimation(42)D∫atfs,unsds+∫at∫asgs,r,unrdrds,∫atfs,usds+∫at∫asgs,r,urdrds≤∫atDfs,uns,fs,usds+∫at∫asDgs,r,unr,gs,r,urdrds≤Lfg∫atDuns,usds+Lfg∫at∫asDunr,urdrds,∀t∈J.Combining estimation (41) and inequality (42), we obtain that (43)D∫atfs,unsds⊖-1∫at∫asgs,r,unrdrds,∫atfs,usds⊖-1∫at∫asgs,r,urdrdsconverges to 0 uniformly as n→+∞. Therefore, u(t) is a (S2)-solution of (14) with initial condition u0.
Finally, we shall prove that problem (14) has a unique (S2)-solution. Assume that u~ is another (S2)-solution of (14) with initial condition u0. Then, we have for any t∈J(44)Dut,u~t≤Lfg∫atDus,u~sds+Lfg∫at∫atDur,u~rdrdsIf we let ξ(t)=D[u(t),u~(t)] for any t∈J, then (45)ξt≤Lfg∫atξsds+Lfg∫at∫atξrdrds.Applying Lemma 2.3 in Hoa et al. [4], we obtain ξ(t)=0 for any t∈J. This completes the proof.Example 14.
Consider the following fuzzy integro differential equation:(46)DHgut=-2,0,2+∫0tus1+usds(47)u0=-2,0,2where u∈C([0,1],RF), f:[0,1]×RF→RF and g:[0,1]×[0,1]×RF→RF.
Let(48)ft,u=-2,0,2,gt,s,u=u1+u,∀t,s∈0,1,u∈RF.It is easy to see that f,g satisfy Lipschitz condition with Lipschitz constant Lfg=1. Indeed, for any t,s∈[0,1] and u∈RF(49)Dgt,s,u,gt,s,v=Du1+u,v1+v=supα∈0,1maxu_1+u_-v_1+v_,u¯1+u¯-v¯1+v¯=supα∈0,1maxu_-v_1+u_1+v_,u¯-v¯1+u¯1+v¯≤supα∈0,1maxu_-v_,u¯-v¯=Du,v.Hence, by Theorem 15, the problem (46)-(47) has unique (S1)-solution or (S2)-solution on [0,1].
Moreover, if a continuously (2)-differentiable functionv:[0,1]→RF satisfies the following inequation: (50)DDHgvt,-2,0,2+∫0tvs1+vsds≤ε;then as shown in Theorem 15, there exists a (S2)-solution u:[0,1]→RF of (46) such that (51)Dvt,ut≤ελe2,∀t∈0,1,where λ is the maximum balancing constant.Secondly, we shall prove the Ulam-Hyers-Rassias stability of the FIDE (14) defined on a bounded and closed interval.Theorem 15.
Assume thatf:J→RF→RF and g:J×J×RF→RF are continuous function satisfying the following conditions: (i) there exists a constant Lfg>0 such that max{D[f(t,u),f(t,v)],D[g(t,s,u),g(t,s,v)]}≤LfgD[u,v] for each (t,s,u),(t,s,v)∈J×J×RF; (ii) there exists a constant C>0 such that 0<Lfg(C+C2)<1. Let φ:J→(a,+∞) be a continuous function and increasing on J with(52)∫atφsds≤Cφt,foreacht∈J.If a continuously (1)-differentiable function u:J→RF satisfies the following inequality(53)DDHgut,ft,ut+∫atgt,s,usds≤φtfor any t∈J, then there exists a unique (S1)-solution u~:J→RF of (14) such that(54)u~t=u~0+∫atfs,u~sds+∫at∫asgs,r,u~rdrdsand(55)Du~t,ut≤C1-LfgC+C2φtfor any t∈J.Proof.
Let us define a setX of all continuous fuzzy functions w:J→RF by (56)X=w:J→RF∣wiscontinuousonJequipped with the metric (57)dv,w=infC∈0,+∞∪+∞∣Dvt,wt≤Cφt,∀t∈J.It is easy to see that (X,d) is also a complete generalized metric space (see Lemma 3).
The operatorP:X→X is defined as follows:(58)Pvt=v0+∫atfs,vsds+∫at∫asgs,r,vrdrds,∀t∈J.Based on Lemma 3.2 and 3.3 in [3], we infer that Pv is (1)-differentiable and so Pv∈X.
The operatorP is strict contractive on X. Indeed, for any v,w∈X and letting Cv,w∈[0,+∞)∪{+∞} be an arbitrary constant with d(v,w)≤Cv,w, that is, by the definition of the metric d, we have(59)Dvt,wt≤Cv,wφt,∀t∈J.From the definition of the metric D and assumptions (i)-(ii) of Theorem 15 and inequality (59), we infer that (60)DPvt,Pwt=Dv0+∫atfs,vsds+∫at∫asgs,r,vrdrds,v0+∫atfs,wsds+∫at∫asgs,r,wrdrds≤∫atDfs,vs,fs,wsds+∫at∫asDgs,r,vr,gs,r,wrdrds≤Lfg∫atDvs,wsds+Lfg∫at∫asDvr,wrdrds≤LfgCv,w∫atφsds+LfgCv,w∫at∫asφrdrds≤LfgC+C2Cv,wφtfor any t∈J.
For eachv,w∈X and by the definition of metric d, we get (61)dPvt,Pwt≤LfgC+C2Cv,wφtfor any t∈J. Hence, by (59), we can conclude that (62)dPvt,Pwt≤LfgC+C2dv,wfor any t∈J.
It follows form the definitions ofX and the operator P that, for arbitrary w~∈X, there exists a constant 0<C<+∞ such that (63)DPw~t,w~t=Dw~0+∫atfs,w~sds+∫at∫asgs,r,w~rdrds,w~t≤Cφtfor any t∈J, since f(t,w~(t)), g(t,s,w~(s)), and w~(t) are bounded on J, and mint∈Jφ(t)>0. Thus, by definition of d it is implied that (64)dPw~,w~≤C<+∞.Therefore, according to Theorem 2, there exists a continuous function u~ on J such that Jnw~→u~ as n→+∞ in the space (X,d) and Jnu~=u~; that is, u~ satisfies (58) for each t∈J.
Observe that(65)X=w∈X∣dw~,w<+∞.Indeed, for any w∈X, there exists a constant 0<C<+∞ such that (66)Dw~t,wt≤Cφt,since w~ and w are bounded on J and mint∈Jφ(t)>0. It follows from the preceding inequality that (67)dw~,w<+∞for all w∈X. Hence, we obtained that X={w∈X∣d(w~,w)<+∞}.
From Theorem2, we infer that w~ is a unique fixed point of J in X. It is obvious that w~ is a unique fuzzy function in X which satisfies equation Jw~=w~.
On the other hand, we have(68)Dut,u0+∫atfs,usds+∫at∫asgs,r,urdrds=Dut⊖u0,∫atfs,usds+∫at∫asgs,r,urdrds=D∫atDHgusds,∫atfs,usds+∫at∫asgs,r,urdrds≤∫atDDHgus,fs,us+∫asgs,r,urdrds≤∫atφsds≤Cφtfor any t∈J. This means that(69)du,Pu≤C.Finally, by Theorem 2 and inequation (69), we deduce that (70)du,u~≤11-LfgC+C2du,Pu≤C1-LfgC+C2φtfor any t∈J, which means that inequality (55) is true for all t∈J.Corollary 16.
Assume thatf and g satisfy the assumption (i) of Theorem 15 and 0<Lfg(b-a)+Lfg(b-a)2/2<1. For a given ε>0, if a continuously (1)-differentiable function u:J→RF satisfies the following inequality (71)DDHgut,ft,ut+∫atgt,s,usds≤εfor any t∈J, then there exists a unique (S1)-solution u~:J→RF of (14) such that (72)u~t=u~0+∫atfs,u~sds+∫at∫asgs,r,u~rdrdsand (73)Du~t,ut≤b-aε1-Lfgb-a+Lfgb-a2/2for any t∈J.Theorem 17.
Suppose the functionsf,g and φ satisfy all conditions as in Theorem 15. If a continuously (2)-differentiable function u:J→RF satisfies inequality (53) in Theorem 15 for any t∈J, then there exists a unique (S2)-solution u~:J→RF of (14) such that(74)u~t=u~0⊖-1∫atfs,u~sds⊖-1∫at∫asgs,r,u~rdrdsand the estimation (55) as in Theorem 15 on J.Proof.
Similar to the proof of Theorem15. Consider the operator H:X→X defined by(75)Hvt=v0⊖-1∫atfs,vsds⊖-1∫at∫asgs,r,vrdrdsfor all v∈X. Based on Lemmas 3.2 and 3.3 in [3], it is easy to see that Hv is (2)-differentiable and so Hv∈X.
We check the operatorH is strict contractive on X. Let v,w∈X and let Cv,w∈[0,+∞)∪{+∞} be an arbitrary constant with d(v,w)≤Cv,w; we have(76)Dvt,wt≤Cv,wφt,∀t∈J.From the assumptions (i)-(ii) in Theorem 15 and (76), we get (77)DHvt,Hwt=Dv0⊖-1∫atfs,vsds⊖-1∫at∫asgs,r,vrdrds,v0⊖-1∫atfs,wsds⊖-1∫at∫asgs,r,wrdrds≤∫atDfs,vs,fs,wsds+∫at∫asDgs,r,vr,gs,r,wrdrds≤Lfg∫atDvs,wsds+Lfg∫at∫asDvr,wrdrds≤LfgCv,w∫atφsds+LfgCv,w∫at∫asφrdrds≤LfgC+C2Cv,wφtfor any t∈J. Hence, by (76), we conclude that (78)dHvt,Hwt≤CLfgdv,w∀t∈J.By the definitions of X and P, we have for arbitrary w~∈X, there exists a constant 0<C<+∞ such that (79)DHw~t,w~t=Dw~0⊖-1∫atfs,w~sds⊖-1∫at∫asgs,r,w~rdrds,w~t≤Cφt,∀t∈J.f,g and w~ are bounded on J, and mint∈Jφ(t)>0. Thus, by the definition of d, we have d(w~,w)<+∞ for all w∈X. Hence, we infer that X={w∈X∣d(w~,w)<+∞}. From Theorem 2, we deduce that w~ is a unique fixed point of J in X. It is obvious that w~ is a unique fuzzy function in X which satisfies the equality Jw~=w~.
On the other hand, Hukuhara differenceu0⊖u(t) exists for all t∈J and from (52) and the definition of H that (80)Dut,u0⊖-1∫atfs,usds⊖-1∫at∫asgs,r,urdrds=Du0⊖ut,-∫atfs,usds+∫at∫asgs,r,urdrds=D-∫atDHgusds,-∫atfs,usds+∫at∫asgs,r,urdrds≤∫atDDHgus,fs,us+∫asgs,r,urdrds≤∫atφsds≤Cφtfor any t∈J, which implies that(81)du,Hu≤C.By Theorem 2 and inequation (81), we deduce that (82)du,u~≤11-LfgC+C2du,Hu≤C1-LfgC+C2φtfor any t∈J, which means that inequality (55) is true for all t∈J.Corollary 18.
Assume thatf and g satisfy the assumption (i) of Theorem 15 and 0<Lfg(b-a)+Lfg(b-a)2/2<1. For a given ε>0, if a continuously (2)-differentiable function u:J→RF satisfies the following inequation (83)DDHgut,ft,ut+∫atgt,s,usds≤εfor any t∈J, then there exists a unique (S2)-solution u~:J→RF of (14) such that (84)u~t=u~0⊖-1∫atfs,u~sds⊖-1∫at∫asgs,r,u~rdrdsand (85)Du~t,ut≤b-aε1-Lfgb-a+Lfgb-a2/2for any t∈J.Example 19.
We consider the fuzzy intergodifferential equation as follows:(86)DHgut=12∫0tts2usds,∀t∈0,1,and the following inequality (87)DDHgu~t,12∫0tts2u~sds≤e0.5t,∀t,s∈0,1,where u is a continuously (1)-differentiable (or (2)-differentiable) function.
It is easy to check that the functionsf,g satisfy Lipschitz condition with Lfg=1/2. Choosing φ(t)=εe2t with ε>0 and C=0.5, we have (88)∫0tφsds=∫0tεe2sds=ε0.5e2t-1≤0.5εe2t=Cεe2t,∀t∈0,1.Now all assumptions in Theorem 15 (or Theorem 17) are satisfied, problem (86) has a unique solution and (86) is Ulam-Hyers-Rassias stable with (89)Du~t,ut≤C1-LfgC+C2φt=45εe2tfor any t∈[0,1].
In particular, if we chooseφ(t)=ε, then we have (90)Du~t,ut≤b-aε1-Lfgb-a+Lfgb-a2/2=4ε,for all t∈[0,1].


## 4. Conlusion

In this study, the Ulam-Hyers stability and Ulam-Hyers-Rassias stability for fuzzy intergodifferential equation via the fixed point technique and successive approximation method are studied. Moreover, some illustrative examples are given. In future work, we will study Ulam stability for fuzzy intergodifferential equation in the quotient space of fuzzy numbers, introduced by [25–27].



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