# Random Fuzzy Differential Equations with Impulses

Authors: Ho Vu
Journal: Complexity
 (2017)

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## Abstract

We consider the random fuzzy differential equations (RFDEs) with impulses. Using Picard method of successive approximations, we shall prove the existence and uniqueness of solutions to RFDEs with impulses under suitable conditions. Some of the properties of solution of RFDEs with impulses are studied. Finally, an example is presented to illustrate the results.


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## Body

## 1. Introduction

Impulsive differential equations (IDEs) are a new branch of differential equations. IDEs can find numerous applications in different branches of optimal control, electronics, economics, physics, chemistry, and biological sciences. We refer to [1–4] and the references therein. As we know, the real systems are often faced with two kinds of uncertainties (fuzziness and randomness). Therefore, this topic has extensively been studied by mathematicians in recent years. Investigations of dynamic systems with fuzziness have been developed in connection with fuzzy differential equations (FDEs). Evidence of FDEs for such areas as control theory, differential inclusions, and fuzzy differential equations can be found in the papers of [5–8], the books and monographs [9], and references therein. In [10], Lakshmikantham and McRae combined the theories of impulsive differential equations and fuzzy differential equations. There are a few papers on the latter topic; see [10–12].Moreover, the class of random fuzzy differential equations (RFDEs) could be applicable in the investigation of numerous engineering and economics problems where the phenomena are simultaneously subjected to two kinds of uncertainties, that is, fuzziness and randomness, simultaneously (see, e.g., Malinowski [13–16], Feng [17, 18], and Fei [19, 20]). Feng [17] introduced the concepts by the mean-square derivative and mean-square integral of second-order fuzzy stochastic processes. Using the results, the author [18] investigated the properties of solutions of the fuzzy stochastic differential systems, including the existence and uniqueness of solution, the dependence of the solution of the initial condition, and the continuity and the boundedness of solution of systems when there are perturbations of the coefficients and the initial conditions. In [19, 20], Fei proved the existence and uniqueness of solution of fuzzy random differential equation (FRDE). The author also discussed the dependence of solution to FRDE on initial values. Finally, the nonconfluence property of the solution for FRDE is studied.In [13], Malinowski considered the following random fuzzy differential equations:(1)DHxt,ω=t0,t0+p,P.1ft,xt,ω,xt0,ω=P.1x0ω∈Ed,where f:Ω×t0,t0+p×Ed→Ed and the symbol DH denotes the fuzzy Hukuhara derivative. The author proved the existence and uniqueness of the solution for RFDEs under Lipschitz condition. Malinowski [14, 15] studied two kinds of solutions to the RFDEs with two kinds of fuzzy derivatives. For both cases the author established the existence and uniqueness of local solutions to RFDEs. In addition, the author also presented some examples being simple illustrations of the theory of RFDEs.Inspired and motivated by Fei [19], Feng [18], Malinowski [14], and other authors as in [3, 10, 21], in this paper, we consider the RFDEs with impulses under Hukuhara derivative. The paper is organized as follows: in Section 2, we summarize some preliminary facts and properties of the fuzzy set space, fuzzy differentiation, and integration. We also recall the notions of fuzzy random variable and fuzzy stochastic process. In Section 3, we discuss the RFDEs with impulses. Under suitable conditions, we prove the existence and uniqueness of solutions to RFDEs with impulses. In Section 4, we give some examples to illustrate these results.


## 2. Preliminaries

In this section, we give some definitions and properties and introduce the necessary notation which will be used throughout the paper. We denoteEd=u:Rd→0,1∣usatisfiesi–ivstatedbelow, where(i)
u is normal; that is, there exists an x0∈Rd such that ux0=1;(ii)
u is fuzzy convex; that is, for 0≤λ≤1, uλx1+1-λx2≥min⁡ux1,ux2, for any x1,x2∈Rd;(iii)
u is upper semicontinuous;(iv)
clx∈Rd:uz>0 is compact set.Then Ed is called the space of fuzzy numbers.For0<α≤1, we denote uα=x∈Rd∣ux≥α and u0=clx∈R∣ux>0. For d=1 and from conditions (i)–(iv), we infer that the α-level cut of u, denoted by uα, is a bounded closed interval for any α∈0,1 and u∈Ed, and uα=uαl,uαr, where uαl and uαr are the lower and upper branches of u.Foru,v∈Ed, the Hausdorff distance between u and v is defined by(2)d∞x,y=supα∈0,1⁡max⁡dHuα,dHvαand Ed,d∞ is a complete metric space.If we defineD:Ed×Ed→R+ by the expression(3)Du,v=supt∈a,b⁡d∞ut,vt,then it is well-known that D is metric in Ed and Ed,D is also a complete metric space.Some properties are well-known for the metric HausdorffD defined on Ed as follows:(4)Du+w,v+w=Du,v,Dλu,λv=λDu,v,Du,v≤Du,w+Dv,w,for every u,v,w∈Ed and λ∈R+.Definition 1 (see [22]).
Letu,v∈Ed. If there exists w∈Ed such that u=v+w, then w is called the Hukuhara difference of u,v and it is denoted by u⊖v.Definition 2 (see [22]).
Letf:a,b→Ed and t∈a,b. We say that f is differentiable at t if there exists an element DHft∈Ed such that the limits(5)limh→0+ft+h⊖fth=limh→0+ft⊖ft-hhexist and are equal to DHft.Definition 3 (see [22]).
Letf:a,b→Ed. The integral of f on a,b, denoted by ∫abftdt, is defined levelwise by the equation(6)∫abftdtα=∫abftαdt=∫abf~tdt∣f~:a,b⟶Ris  a  measurable  selection  forf·α,for all α∈0,1.Definition 4 (see [22]).
A fuzzy mappingf:a,b→Ed is integrable if f is integrable bounded and strongly measurable.The following are some properties of integrability of fuzzy mapping (see [22]):(a)
Iff:a,b→Ed is continuous then it is integrable.(b)
Iff:a,b→Ed is integrable and c∈a,b then ∫abfsds=∫acfsds+∫cbfsds.(c)
Letf,g:a,b→Ed be integrable and λ>0. Then
(i)
∫abfs+gsds=∫abfsds+∫abgsds,(ii)
∫abλfsds=λ∫abfsds,(iii)
Df,g is integrable and D∫abfsds,∫abgsds≤∫abDfs,gsds.LetΩ,F,P be a complete probability space. A function x:Ω→Ed is called a fuzzy random variable, if the set-valued mapping x·α:Ω→KcRd is a measurable multifunction for all α∈0,1; that is,(7)ω∈Ω∣xωα∩B≠∅∈Ffor every closed set B⊂Rd.Definition 5 (see [13]).
A mappingx:a,b×Ω→Ed is said to be a fuzzy stochastic process if x·,ω is a fuzzy-set-valued function with any fixed ω∈Ω and xt,· is a fuzzy random variable for any fixed t∈a,b.Definition 6 (see [13]).
A fuzzy stochastic processx:a,b×Ω→Ed is called continuous if there exists Ω0⊂Ω with PΩ0=1 and such that, for every ω∈Ω0, the trajectory x·,ω is a continuous function on a,b with respect to the metric D.For convenience, from now on, we shall writexω=P.1yω to replace Pω∣xω=yω=1 for short, where x, y are random elements, and similarly for inequalities. Also we shall write xt,ω=a,b,P.1yt,ω to replace Pω∣xt,ω=yt,ω,∀t∈a,b=1 for short, where x, y are some stochastic processes, and similarly for inequalities.


## 3. Existence and Uniqueness for RFDEs with Impulses

In this section, we consider the following random fuzzy differential equation with impulses:(8)DHxt,ω=P.1ft,xt,ω,ω,t∈J≔t0,t0+p,t≠tk,xtk+,ω=P.1Ikxtk,ω,ω,k=1,m¯,t=tk,xt0+,ω=P.1x0ω∈Ed,where f:J×Ed×Ω→Ed, Ik:Ed×Ω→Ed is continuous with P.1, and tk, k=1,m¯, are points of impulses such that t0≤⋯<tk<tk+1≤t0+p and x0:Ω→Ed is fuzzy random variable.Lemma 7.
Letx:J×Ω→Ed be a fuzzy stochastic process. Then x is the solution of problem (8) if and only if x is a continuous fuzzy stochastic process and satisfy the following random impulsive fuzzy integral equation:(9)xt,ω=x0ω+∫t0tfs,xs,ω,ωds+∑i=1kIixti,ω,ω.Proof.
We divide the proof into two steps.
Step  1. If xt satisfies problem (8), then it will be expressed as (9). Indeed, for every t∈t0,t1 we have(10)DHxt,ω=P.1ft,xt,ω,ω.By Lemma  3.1 in [13], we obtain(11)xt1,ω=t0,t1,P.1x0ω+∫t0t1fs,xs,ω,ωds.
Ift∈t1,t2 and by Lemma  3.1 in [13], we have(12)xt,ω=P.1xt1+,ω+∫t1tfs,xs,ω,ωds=P.1I1x1t,ω,ω+∫t1tfs,xs,ω,ωds=P.1I1x1t,ω,ω+x0ω+∫t0t1fs,xs,ω,ωds+∫t1tfs,xs,ω,ωds.If we assume that(13)xt,ω=tk-1,tk,P.1x0ω+∫t0tfs,xs,ω,ωds+∑i=1k-1Iixti,ω,ω,
then we have(14)xt,ω=tk,tk+1,P.1xtk+,ω+∫tktfs,xs,ω,ωds=tk,tk+1,P.1I1xkt,ω,ω+∫tktfs,xs,ω,ωds⋮=tk,tk+1,P.1∑i=1kIixti,ω,ω+x0ω+∫t0tfs,xs,ω,ωds.
It follows by mathematical induction that (13) holds for any k≥1.
Step  2. Conversely, if a fuzzy stochastic process x satisfies the random fuzzy integral equation (9), then it is equivalent to problem (8). Indeed, if t∈t0,t1 we easily see that xt0,ω=P.1x0ω and the Hukuhara difference x0ω+∫t0tfs,us,ω,ωds exists, with P.1. By Lemma  3.2 in [13] we have(15)DHxt,ω=t0,t1,P.1ft,xt,ω,ω.Let h>0 small enough such that t-h∈t1,t2 for every t∈t1,t2; we have(16)xt,ω⊖xt-h,ω=P.1∫t1tfs,xs,ω,ωds⊖∫t1t-hfs,xs,ω,ωds=P.1∫t-htfs,xs,ω,ωds.
Similarly, leth>0 small enough such that t+h∈t1,t2 for every t∈t1,t2; we obtain(17)xt+h,ω⊖xt,ω=P.1∫t1tfs,xs,ω,ωds⊖∫t1t+hfs,xs,ω,ωds=P.1∫tt+hfs,xs,ω,ωds.Multiplying both sides of (16) and (17) by 1/h and passing to the limit with h→0+, we obtain(18)limh→0+xt,ω⊖xt-h,ωh=limh→0+1h∫t-htfs,xs,ω,ωds=DHxt,ω,limh→0+xt+h,ω⊖xt,ωh=limh→0+1h∫tt+hfs,xs,ω,ωds=DHxt,ω.
This allows us to claim thatx is differentiable on t1,t2 and consequently (19)DHxt,ω=t1,t2,P.1ft,xt,ω,ω.By mathematical induction, if t∈tk,tk+1, k=1,m¯, we get(20)DHxt,ω=tk,tk+1,P.1ft,xt,ω,ω.
Also, we can easily show that(21)Δxtk,ω=P.1Ikxtk,ω,ω,k=1,m¯.The proof is complete.Lemma 8.
LetΩ,F,P be a probability space. Let A:Ω→R+, Bi:Ω→R+, i=0,1,2,…, and stochastic processes X,Y:J×Ω→R be such that(i)
X·,ω is nonnegative and continuous with P.1 and ti are the points of discontinuity of the first of X·,ω with P.1,(ii)
Y·,ω is locally Lebesgue integrable with P.1.
If(22)Xt,ω≤J,P.1Aω+∫t0tXs,ωYs,ωds+∑t0≤ti<tBiωXti,ω,then we have(23)Xt,ω≤J,P.1Aω∏t0≤ti<t1+Biωexp⁡∫t0tYs,ωds.Now, we show the main results of this paper.Theorem 9.
Let the mappingf:J×Ed×Ω→Ed be continuous with P.1 and Ik:Ed×Ω→Ed. Assume the following conditions hold:(A1)
There exists a nonnegative constantL1 such that Dft,φ,ω,ft,ϕ,ω≤L1Dφ,ϕ, for every t∈J and φ,ϕ∈Ed with P.1.(A2)
There exists a nonnegative constantL2,k such that DIkφ,ω,Ikϕ,ω≤L2,kDφ,ϕ, for k=0,1,2,…,m, for every t∈J and φ,ϕ∈Ed with P.1.(A3)
There exists a nonnegative constantM1 such that Dft,x0ω,ω,0^≤M1, for every t∈J and x0∈Ed with P.1.
Then the random fuzzy differential equation with impulses (9) has a unique solution, provided that(24)L1pnn!+∑i=1kL2,i<1,for  anyn∈N.Proof.
Define a sequence of the functionsxn:J×Ω→Ed, n=0,1,2,… as follows: for every ω∈Ω let us put(25)x0t,ω=x0ω,xnt,ω=x0ω+∫t0tfs,xn-1s,ω,ωds+∑i=1kIixn-1ti,ω,ω.For every t∈J and ω∈Ω, we have(26)d∞x1t,ω,x0t,ω=d∞∫t0tfs,x0s,ω,ωds,0^+d∞∑i=1kIix0ti,ω,ω,0^≤∫t0td∞fs,x0ω,ω,0^ds+∑i=1kd∞Iix0ω,ω,0^≤M1t-t0+∑i=1kd∞Iix0ω,ω,0^≤pM1+∑i=1kd∞Iix0ω,ω,0^≔M0;
it follows thatd∞x1t,ω,x0t,ω≤J,P.1M0. Furthermore, by assumptions (A1)-(A2) and (25), we can find that(27)d∞xnt,ω,xn-1t,ω≤J,P.1∫t0td∞fs,xn-1s,ω,ω,fs,xn-2s,ω,ωds+∑i=1kd∞Iixn-1ti,ω,ω,Iixn-2ti,ω,ω≤J,P.1L1∫t0td∞xn-1s,ω,xn-2s,ωds+∑i=1kL2,id∞xn-1ti,ω,xn-2ti,ω≤J,P.1L1∫t0tsups∈J⁡d∞xn-1s,ω,xn-2s,ωds+∑i=1kL2,id∞xn-1ti,ω,xn-2ti,ω≤J,P.1L1t-t0n-1n-1!+∑i=1kL2,iDxn-1ω,xn-2ω,which implies that(28)Dxnt,ω,xn-1t,ω≤J,P.1L1t-t0n-1n-1!+∑i=1kL2,iDxn-1ω,xn-2ω.
Now, we need to prove that for allt∈J with P.1 the following inequality holds: for any n=1,2,…,(29)Dxnt,ω,xn-1t,ω≤J,P.1L1t-t0n-1n-1!+∑i=1kL2,iDxn-1ω,xn-2ω.
Indeed, inequality (29) holds for n=1. Further, if inequality (29) is true for any n=m≥1, then using (25) and assumptions (A1)-(A2), we have(30)Dxm+1t,ω,xmt,ω≤J,P.1∫t0tsups∈J⁡d∞fs,xms,ω,ω,fs,xm-1s,ω,ωds+∑i=1kd∞Iixmti,ω,ω,Iixm-1ti,ω,ω≤J,P.1L1∫t0tsups∈J⁡d∞xms,ω,xm-1s,ωds+∑i=1kL2,id∞xmti,ω,xm-1ti,ω≤J,P.1L1∫t0tsups∈J⁡d∞xms,ω,xm-1s,ωds+∑i=1kL2,id∞xmti,ω,xm-1ti,ω≤J,P.1L1t-t0mm!+∑i=1kL2,iDxmω,xm-1ω.Thus, inequality (29) is true for every t∈J with P.1.
Next, we see thatx0t,ω does not depend on t and for the right-side continuity of x1·,ω, one obtains(31)Dx1t+h,ω,x1t,ω≤t0,t0+p,P.1∫tt+hDfs,x0s,ω,ω,0^ds+∑i=1kDIix0ti+h,ω,ω,Iix0ti,ω,ω.From the assumption (A3) and DIix0ti+h,ω,ω,Iix0ti,ω,ω→0 as h→0+ with P.1, we imply that d∞x1t+h,ω,x1t,ω→0 as h→0+ with P.1.
For everyn≥2, we deduce that(32)Dxnt+h,ω,xnt,ω≤t0,t0+p,P.1∫tt+hDfs,x0s,ω,ω,0^+∑q=1n-1Dfs,xqs,ω,ω,fs,xq-1s,ω,ωds+∑i=1kDIix0ti+h,ω,ω,Iix0ti,ω,ω.Using inequality (29) and assumption (A3), we get(33)Dxnt+h,ω,xnt,ω⟶0ash⟶0+withP.1.
Similar for the left-side continuity, we haved∞xnt-h,ω,xnt,ω→0  ash→0+. Hence the functions xn·,ω, n≥2, are continuous with P.1.
Forn∈N and t∈J the function xnt,· defined by (25) is fuzzy random variable. Indeed, xn·α is measurable multifunction for every α∈0,1; it remains to show the same for the mapping ω↦∫t0tfs,xn-1s,ω,ωds+∑i=1kIixn-1ti,ω,ωα which is a measurable multifunction with every α∈0,1, n∈N, and t∈J. Let α∈0,1 be fixed. By virtue of the definition of fuzzy integral and theorem of Nguyen [23] we obtain(34)∫t0tfs,xn-1s,ω,ωds+∑i=1kIixn-1ti,ω,ωα=∫t0tfs,xn-1s,ωα,ωds+∑i=1kIixn-1ti,ωα,ω.As the integrand is a multifunction continuous in s and measurable in ω, with any t∈J, the mapping(35)ω⟼∫t0tfs,xn-1s,ωα,ωds+∑i=1kIixn-1ti,ωα,ω
is a measurable multifunction forn∈N. Therefore, for every t∈J, the sequence xnt,· is a sequence of fuzzy random variable. Consequently, xnt,ω is a sequence of fuzzy stochastic process.
In the sequel, for anyn∈N, we shall prove that the sequence xnt,ω is a Cauchy sequence uniformly on the variable t with P.1 and then xn·,ω is uniformly convergent with P.1.
For anyn∈N and by inequality (29), we obtain(36)Dxn+1t,ω,xnt,ω≤J,P.1MDxnω,xn-1ω≤J,P.1MnDx1ω,x0ω.Notice now that, for every m>n>0, we have(37)Dxmt,ω,xnt,ω≤J,P.1∑l=nm-1Dxl+1t,ω,xlt,ω≤J,P.1Mn+Mn+1+⋯+Mm-1Dx1ω,x0ω≤J,P.1Mn1-MDx1ω,x0ω.
Form>n>0 large enough, it follows from the above inequalities with M<1 that(38)Dxmt,ω,xnt,ω→P.10.Since Ed,D is a complete metric space and (38) holds, then Dxnt,ω,xt,ω→P.10, which means that there exists Ω0⊂Ω such that PΩ=1 and for every ω∈Ω0 the sequence xn·,ω is uniformly convergent.
In the following, we shall show thatxt,ω is solution of the random impulsive fuzzy integral equation (8). Let n∈N. Observe that(39)D∫t0tfs,xn-1s,ω,ωds,∫t0tfs,xs,ω,ωds≤J,P.1∫t0tDfs,xn-1s,ω,ω,fs,xs,ω,ωds≤J,P.1L1∫t0tDxn-1s,ω,xs,ωds.Since the sequence xnt,ω converges uniformly to xt,ω on the variable t∈J with P.1 as n→+∞, Thus for any ε>0 there is n0>0 large enough such that, for all n>n0, we derive (40)Dxn-1t,ω,xt,ω≤J,P.1min⁡n-1!L1pn-1ε,∑i=1kL2,i-1ε.
Therefore,(41)D∫t0tfs,xn-1s,ω,ωds,∫t0tfs,xs,ω,ωds≤J,P.1ε,D∑i=1kIixn-1ti,ω,ω,∑i=1kIixti,ω,ω≤J,P.1∑i=1kL2,iDxn-1ti,ω,xti,ω≤J,P.1ε.On the other hand, we have (42)Dxt,ω,x0ω+∫t0tfs,xs,ω,ωds+∑i=1kIixti,ω,ω≤J,P.1Dxt,ω,xnt,ω+d∞xnt,ω,x0ω+∫t0tfs,xn-1s,ω,ωds+∑i=1kIixn-1ti,ω,ω+D∫t0tfs,xn-1s,ω,ωds,∫t0tfs,xs,ω,ωds+D∑i=1kIixn-1ti,ω,ω,∑i=1kIixti,ω,ω.
Thus, in view of the convergence of the two previous equations and (41), one obtains that(43)Dxt,ω,x0ω+∫t0tfs,xs,ω,ωds+∑i=1kIixti,ω,ω=J,P.10.It means the fuzzy stochastic process xt,ω is solution of problem (8).
To prove the uniqueness, let us assume thatx,y:J×Ω→Ed are the two continuous fuzzy stochastic processes which are solutions of problem (8). Note that(44)Dxt,ω,yt,ω=J,P.1D∫t0tfs,xs,ω,ωds,∫t0tfs,ys,ω,ωds+D∑i=1kIixti,ω,ω,∑i=1kIiyti,ω,ω≤J,P.1L1pnn!+∑i=1kL2,iDxω,yω.
By Lemma8, we get(45)Dxt,ω,yt,ω≤J,P.10.The uniqueness is proved. The proof is complete.


## 4. Some of the Properties of Solution of RFDEs with Impulses

Theorem 10.
Suppose that the mappingsf:J×Ed×Ω→Ed and Ik:Ed×Ω→Ed satisfy all the conditions of Theorem 9. Then we have(46)Dxt,ω,0^≤J,P.1Dx0ω,0^+t-t0M1∏t0≤ti<t1+L2,iexp⁡L1t-t0,where L1, L2,i are constants nonnegative for any i=0,1,2,3,….Proof.
Letxt,ω be solution of problem (8). For every t∈t0,t1 and ω∈Ω, we have(47)Dxt,ω,0^≤Dx0ω,0^+D∫t0tfs,xs,ω,ωds,0^≤Dx0ω,0^+∫t0tDfs,xs,ω,ω,0^ds≤Dx0ω,0^+∫t0tDfs,0^,ω,0^+Dfs,xs,ω,ω,fs,0^,ωds≤Dx0ω,0^+t-t0Dfs,0^,ω,0^+L1∫t0tDxs,ω,0^ds≤Dx0ω,0^+t-t0M1+∫t0tL1Dxs,ω,0^ds.For every t∈tk,tk+1, k=1,2,3,…, and ω∈Ω, we have(48)Dxt,ω,0^≤d∞x0ω,0^+D∫t0tfs,xs,ω,ωds,0^+D∑i=1kIixti,ω,ω,0^≤Dx0ω,0^+∫t0tDfs,0^,ω,0^+Dfs,xs,ω,ω,fs,0^,ωds+∑i=1kDIixti,ω,ω,0^≤Dx0ω,0^+t-t0M1+∫t0tL1Dxs,ω,0^ds+∑i=1kL2,kDxti,ω,0^.
If we letξt,ω=Dxt,ω,0^, t∈tk,tk+1, and k=0,1,2,3,…, then we have(49)ξt,ω≤P.1ξ0ω+t-t0M1+∫t0tL1ξs,ωds+∑i=1kL2,iξti,ω.By virtue of Lemma 8, one obtains (50)ξt,ω≤P.1ξ0ω+t-t0M1∏t0≤ti<t1+L2,iexp⁡L1t-t0.
The proof is complete.Theorem 11.
Suppose that the mappingsf:J×Ed×Ω→Ed and Ik:Ed×Ω→Ed satisfy all the conditions of Theorem 9. Then we have(51)Dxt,ω,yt,ω≤J,P.1Dx0ω,y0ω∏t0≤ti<t1+L2,iexp⁡L1t-t0,where L1, L2,i are constants nonnegative for any i=0,1,2,3,….Proof.
Letxt,ω and yt,ω be solutions of problem (8). For every t∈t0,t1 and ω∈Ω, we have(52)Dxt,ω,yt,ω≤d∞x0ω,y0ω+D∫t0tfs,xs,ω,ωds,∫t0tfs,ys,ω,ωds≤Dx0ω,y0ω+∫t0tDfs,xs,ω,ω,fs,ys,ω,ωds=Dx0ω,y0ω+L1∫t0tDxs,ω,ys,ωds.For every t∈tk,tk+1, k=1,2,3,…, and ω∈Ω, we have(53)Dxt,ω,yt,ω≤d∞x0ω,y0ω+D∫t0tfs,xs,ω,ωds,∫t0tfs,ys,ω,ωds+D∑i=1kIixti,ω,ω,∑i=1kIiyti,ω,ω≤Dx0ω,y0ω+∫t0tDfs,xs,ω,ω,fs,ys,ω,ωds+∑i=1kd∞Iixti,ω,ω,Iiyti,ω,ω=Dx0ω,y0ω+∫t0tL1Dxs,ω,ys,ωds+∑i=1kL2,iDxti,ω,yti,ω.
If we letξt,ω=Dx0ω,y0ω, t∈tk,tk+1, and k=0,1,2,3,…, then we have(54)ξt,ω≤P.1ξ0ω+∫t0tL1ξs,ωds+∑i=1kL2,iξti,ω.By virtue of Lemma 8, one obtains (55)ξt,ω≤P.1ξ0ω∏t0≤ti<t1+L2,iexp⁡L1t-t0.
The proof is complete.


## 5. Illustrative Examples

In this section, we shall consider two examples. First, we give an example to illustrate the existence and uniqueness results obtained in Section3. Second, we will find explicit representation of solutions RFDEs with impulses.Example 1.
LetΩ=0,1, F-Borel σ-algebra of subsets of Ω, and P-Lebesgue measure on Ω,F. Let us consider the problem as follows:(56)DHxt,ω=0,1,P.1exp⁡-t5+exp⁡t1+xt,ω,t≠tk,xtk+,ω=P.1xtk,ω2+xtk,ω,t=tk,k=0,1,2,…,m,x0,ω=P.1-1,0,1,2ω∈E1,where x:0,1×Ω→E1 is a fuzzy stochastic process.
Set(57)ft,xt,ω,ω=exp⁡-t5+exp⁡t1+xt,ω,for  everyt∈0,1,t≠tk,k=0,1,2,…,m,Ikxkt,ω,ω=xtk,ω2+xtk,ω,for  everyt∈0,1,t=tk,k=0,1,2,…,m.
For everyt∈0,1, t≠tk, k=0,1,2,…,m, we have(58)d∞ft,xt,ω,ω,ft,yt,ω,ω=d∞exp⁡-t5+exp⁡t1+xt,ω,exp⁡-t5+exp⁡t1+yt,ω=0,1,P.1exp⁡-t5+exp⁡td∞11+xt,ω,11+yt,ω=0,1,P.1exp⁡-t5+exp⁡tsupα∈0,1⁡max⁡11+xlαt,ω-11+ylαt,ω,11+xrαt,ω-11+yrαt,ω=0,1,P.1exp⁡-t5+exp⁡tsupα∈0,1⁡max⁡xlαt,ω-ylαt,ω1+xlαt,ω1+ylαt,ω,xrαt,ω-yrαt,ω1+xrαt,ω1+yrαt,ω≤0,1,P.116supα∈0,1⁡max⁡xlαt,ω-ylαt,ω,xrαt,ω-yrαt,ω≤0,1,P.116d∞x,y,
whereL1=supt∈0,1⁡exp⁡-t/5+exp⁡t=1/6.
Using a similar calculation as above, for everyt∈0,1, t=tk, k=0,1,2,…,m, we obtain(59)d∞Ikxkt,ω,ω,Ikykt,ω,ω≤0,1,P.112d∞x,y,whereL2,k=12.
By a direct calculation, one obtains that(60)d∞exp⁡-t5+exp⁡t1+xt,ω,0^=0,1,P.1exp⁡-t5+exp⁡td∞11+xt,ω,0^=0,1,P.1exp⁡-t5+exp⁡tsupα∈0,1⁡max⁡11+xlαt,ω,11+xrαt,ω≤0,1,P.116and for any n=1,2,…, (61)L1pnn!+∑i=1kL2,i=16nn!+12<1.We can see that conditions (A1)–(A4) are satisfied. Hence, by Theorem9, problem (56) has a solution defined on 0,1.Example 2.
LetΩ=0,1, F-Borel σ-algebra of subsets of Ω, and P-Lebesgue measure on Ω,F. Consider the RFDEs with impulses as follows:(62)DHxt,ω=0,T,P.1λωxt,ω,t≠tk,k=0,1,2,…,m,xtk+,ω=P.1xktk,ω+Ikxkt,ω,ω,t=tk,k=0,1,2,…,m,x0,ω=P.1x0ω,where λ:Ω→R+ is a random variable and x:0,1×Ω→E1 is a fuzzy stochastic process. In this example, we suppose that t∈0,2 and λω=1 with P.1 and for every α∈0,1(63)xt,ωα=xlαt,ω,xrαt,ωxktk,ωα=ω,2ω,3ωα=1+αω,3-αω,t=k,k=1,2,
and initial conditionsx0ωα=-ω,0,ωα=α-1ω,1-αω, where xlα,xrα:0,∞×Ω→R are the crisp stochastic process.
Problem (62) can translate this into the following system of random differential equation with impulses:(64)xlα′t,ω=xlαt,ω,t∈0,2,t≠tk,xrα′t,ω=xrαt,ω,t∈0,2,t≠tk,xlαtk+,ω=1+αω,t=k,k=1,2,xrαtk+,ω=3-αω,t=k,k=1,2,xlα0,ω=α-1ω,xrα0,ω=1-αω.Solving system (64) on 0,2, we obtain (65)xlαt,ω=α-1ωexp⁡t,fort∈0,1,1+αωexp⁡t-1,fort∈0,2,xrαt,ω=1-αωexp⁡t,fort∈0,1,3-αωexp⁡t-1,fort∈0,2.
It is easy to see that the diameter of solutionxt,ω of (62) is an increasing function with P.1 for every t∈0,2. Hence we infer that the solution x:0,2×Ω→E1 to (62) is as follows:(66)xt,ω=α-1,1-αωexp⁡t,fort∈0,1,1+α,3-αωexp⁡t-1,fort∈0,2
or(67)xt,ω=-1,0,1ωexp⁡t,fort∈0,1,1,2,3ωexp⁡t-1,fort∈0,2.
Note that the existence of a unique solution is guaranteed. Therefore, this procedure can be continued to be the solution on eachm,m+1, for every m∈N, m≥3.


## 6. Conclusion

Under suitable conditions, we investigated the existence and uniqueness of solutions to random fuzzy differential equation with impulses by using the method of successive approximations. Moreover, we studied some of the properties of solution of RFDEs with impulses. Finally, some examples are given to illustrate the main theorems. In the future, we shall study the class of random fuzzy differential equations in the quotient space of fuzzy numbers, introduced by Qiu et al. in [24].



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