Fuzzy Optim Decis Making (2012) 11:99–111DOI 10.1007/s10700-011-9112-7

Fuzzy delay differential equations

Vasile Lupulescu · Umber Abbas

Published online: 19 November 2011© Springer Science+Business Media, LLC 2011

Abstract In this paper, we prove a local existence and uniqueness result for fuzzydelay differential equations driven by Liu process. We also establish continuous depen-dence of solution with respect to initial data.

Keywords Fuzzy delay differential equations · Fuzzy Liu process ·Existence and uniqueness

1 Introduction

Differential equations with delay (or memory), known also as functional differentialinclusions, express the fact that the velocity of the system depends not only on thestate of the system at a given instant but depends upon the history of the trajectoryuntil this instant. The class of differential equations with delay encompasses a largevariety of differential equations. Differential equations with delay play an importantrole in an increasing number of system models in biology, engineering, physics andother sciences. There exists an extensive literature dealing with functional differen-tial equations and their applications. We refer to the monographs (Hale 1997), andreferences therein. On the other hand, when a dynamical system is modeled by deter-ministic ordinary differential equations we cannot usually be sure that the model isperfect because, in general, knowledge of dynamical system is often incomplete or

V. Lupulescu (B)Constantin Brancusi University, Republicii 1, 210152, Targu-Jiu, Romaniae-mail: lupulescu_v@yahoo.com

U. AbbasAbdus Salam School of Mathematical Sciences (ASSMS), Government College University,Lahore, Pakistane-mail: umberabbas@yahoo.com

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100 V. Lupulescu, U. Abbas

vague. If the underlying structure of the model depends upon subjective choices, oneway to incorporate these into the model, is to utilize the aspect of fuzziness, whichleads to the consideration of fuzzy differential equations.

The concept of fuzzy set was initiated by Zadeh (1965) via membership function.In order to define a self-dual measure for fuzzy event, Liu and Liu (2002) presentedthe concept of credibility measure. In addition, a sufficient and necessary condition forcredibility measure was given by Li and Liu (2006). Credibility theory was foundedby Liu (2004) and refined by Liu (2007) as a branch of mathematics for studyingthe behavior of fuzzy phenomena. The multidimensional Liu process are defined andstudied in You (2007). Also, Liu (2007) founded an uncertainty theory that is a branchof mathematics based on normality, monotonicity, self-duality, and countable subad-ditivity axioms. In the paper (Chen and Ralescu 2009), a formula for computing thetruth value of independent uncertain propositions is proved. A hybrid variable wasintroduced by Liu (2006) as a tool to describe the quantities with fuzziness and ran-domness. Fuzzy random variable and random fuzzy variable are instances of hybridvariable. In order to measure hybrid events, a concept of chance measure was intro-duced by Li and Liu (2009) (see also Liu 2008). The reflection principle of Liu processis proved in Dai (2007). Based on the random fuzzy theory, a renewal process withrandom fuzzy interarrival times is proposed in the papers (Li et al. 2009) and (Zhaoand Liu 2003).

Fuzzy differential equation was proposed by Li and Liu (2006) (see also, Chen2008; You 2008) as a type of differential equation driven by Liu process just likethat stochastic differential equations driven by Brownian motion (Mohammed 1984).The existence and uniqueness theorem for homogeneous fuzzy differential equationwas proved in Liu (2007) (see also, Chen 2008). In the paper (Chen and Liu 2010) theauthors prove existence and uniqueness of solution for uncertain differential equations(see also, Liu 2007). Some concepts of stability for fuzzy differential equations aregiven in the paper (Zhu 2010).

The aim of this paper is to prove the existence and uniqueness theorem for fuzzydelay differential equation driven by Liu process. We also establish continuous depen-dence of solution with respect to initial data.

2 Preliminaries

Let Ω be a nonempty set, and P the family of all subsets of Ω . Each element A in Pis called an event.

A mapping Cr : P → [0, 1] is called credibility measure if it satisfies the followingaxioms Liu and Liu (2002):

(A1) (Normality) Cr (Ω) = 1(A2) (Monotonicity) Cr (A) ≤ Cr (B) if A ⊂ B(A3) (Self-Duality) Cr (A) + Cr (Ac) = 1 for any event A. Here Ac =

{ω ∈ Ω;ω /∈ A}(A4) (Maximality) Cr(

⋂

i∈IAi ) = sup

i∈ICr (Ai ) for any events Ai , i ∈ I , with

supi∈I

Cr (Ai ) < 0, 5.

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Fuzzy delay differential equations 101

The triplet (Ω,P,Cr) is called a credibility space.A fuzzy variable is defined as a (measurable) function ξ : (Ω,P,Cr) → R. We

denote by F(Ω) the space of fuzzy variables. If ξ is a fuzzy variable then its mem-bership function is derived from the credibility measure by

μ(x) = max {1, 2Cr (ξ = x)} , x ∈ R

Let T ⊂ R be an index set and (Ω,P,Cr) be a credibility space. A fuzzy processLiu (2008) is a function X : T ×Ω → R. Note that for each t ∈ T fixed we have afuzzy variable ω �→ X (t, ω) : (Ω,P,Cr) → R. On other hand, fixing ω ∈ Ω wecan consider the function t �→ X (t, ω) which is called a path of X . A fuzzy processX is said to be continuous if the function t �→ X (t, ω) is continuous for all ω ∈ Ω .In the following, we use the notation X (t) instead of X (t, ω).

Let ξ be a fuzzy variable. Then the expected value of ξ is defined by Liu and Liu(2002)

E [ξ ] =+∞∫

0

Cr (ξ ≥ r) dr −0∫

−∞Cr (ξ ≤ r) dr,

provided that at least one of the two integrals is finite. The variance of ξ is defined by

V [ξ ] = E[(ξ − E [ξ ])2

]

A fuzzy process X is said to have independent increments if

X (t1)− X (t0), X (t2)− X (t1), . . . , X (tk)− X (tk−1)

are independent fuzzy variables for any times t0 < t1 < · · · < tk ; that is,

Cr

(k⋂

i=1

X (ti ) ∈ Bi

)

= min1≤i≤k

Cr (X (ti ) ∈ Bi )

for any sets B1, B2, . . . , Bk of real numbers.A fuzzy process X is said to have stationary increments if, for any given t > 0, the

X (s + t)− X (s) are identically distributed fuzzy variables for all s > 0.A fuzzy process C is said to be a Liu process if Liu (2008)

(L1) C(0) = 0,(L2) C (t) has stationary and independent increments,(L3) every increment C (s + t)−C (s) is a normally distributed fuzzy variable with

expected value et and variance σ 2t2 whose membership functions is

μ (x) = 2

(

1 + exp

(π |x − et |√

6σ t

))−1

,−∞ < x < +∞.

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102 V. Lupulescu, U. Abbas

The parameters e and σ are called the drift and diffusion coefficients, respectively.A Liu process is said to be standard if e = 0 and σ = 1. Let C be a standard Liuprocess, and dt an infinitesimal time interval. Then dC(t) = C(t + dt) − C(t) is afuzzy process such that, for each t , the dC(t) is a normally distributed fuzzy variablewith E[dC(t)] = 0.

Let X be a fuzzy process and let dC(t) be a standard Liu process. For any partitionof closed interval [a, b] with a = t1 < t2 < ···< tk+1 = b, the mesh size is written as

� = max1≤i≤k

|ti+1 − ti |.

Then the Liu integral (Liu 2008) of X with respect to C(t) is defined by

b∫

a

X (t)dC(t) = lim�→0

k∑

i=1

X (ti )[C(ti+1)− C(ti )],

provided that the limit exists almost surely and it is a fuzzy variable.Let X (t) be a fuzzy process and let C(t) be a standard Liu process. If Liu integral

∫ ba X (t)dC(t) exists and it is a fuzzy variable, then X (t) is called Liu integrable. We

know that any continuous fuzzy process is Liu integrable (You 2007).Also, we remark that a Liu process C is Lipschitz-continuous (Dai 2007), that is,

for every given ω ∈ Ω , there exists K (ω) > 0 such that

|C(t, ω)− C(s, ω)| ≤ K (ω)|t − s|, for all t, s ≥ 0. (1)

Using the same idea that in Chen and Liu (2010), we can obtain the following resultfor Liu integrable process.

Lemma 1 Suppose that C(t) is a standard Liu process and X (t)is a fuzzy process on[a, b] with respect to t . If K (ω) > 0 is the Lipschitz constant for path t �→ C(t, ω)with ω ∈ Ω fixed, then we have

∣∣∣∣∣∣

b∫

a

X (t)dC(t)

∣∣∣∣∣∣≤ K (ω)

b∫

a

|X (t)| dt. (2)

Proof Let a = t1 < t2 <···< tk+1 = b be a partition of [a, b] and � = max1≤i≤k

|ti+1 − ti |. Then, by (1), we have

∣∣∣∣∣∣

b∫

a

X (t)dC(t)

∣∣∣∣∣∣=

∣∣∣∣∣

lim�→0

k∑

i=1

X (ti )[C(ti+1)− C(ti )]∣∣∣∣∣

≤ K (ω) lim�→0

k∑

i=1

|X (ti )| · |ti+1 − ti |

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Fuzzy delay differential equations 103

≤ K (ω)

b∫

a

|X (t)| dt.

�

3 Fuzzy delay differential equation

For a positive number q, we denote by Cq the space C ([−q, 0] , R). Then Cq is aBanach space with the respect to the supremum norm: ‖ϕ‖ = sup

t∈[−q,0]|ϕ (t)|. Let

X : [−q,∞) × Ω → R be a fuzzy process. For each t ≥ 0 we can define a fuzzysegment process Xt : [−q, 0] ×Ω → R given by

Xt (s, ω) = X (t + s, ω) ∀ s ∈ [−q, 0] and ω ∈ Ω.

Xt is called a fuzzy process with delay (or memory) of the fuzzy process X at momentt ≥ 0. In the following, we use the notation Xt (s) instead of Xt (s, ω).

Lemma 2 If X is a continuous fuzzy process, then t �→ Xt : [0,∞) → Cq is alsocontinuous.

Proof Let us fixed τ ∈ [0,∞) and ε > 0. Since X is continuous, then it is uniformlycontinuous on the compact interval I = [max{−q, τ−q−δ1}, τ+δ1] for some δ1 > 0.Hence, there exists δ2 > 0 such that, for every t1, t2 ∈ I with |t1 − t2| < δ2, we havethat |X (t1) − X (t2)| < ε. Then, for every s ∈ [−q, 0], we have that τ + s ∈ I andt + s ∈ I if t ≥ 0 and |t −τ | < δ1. Let δ := min{δ1, δ2}. Since |(t + s)− (τ + s)| < δ,it follows that

‖Xt − Xτ‖ = sup−q≤s≤0

|Xt (s)− Xτ (s)| = sup−q≤s≤0

|X (t + s)− X (τ + s)| ≤ ε,

and so, t �→ Xt is continuous. �Corollary 3 If F : [0,∞) × Cq → R is a jointly continuous function and X is acontinuous fuzzy process, then t �→ F(t, Xt ) is also continuous.

Remark 4 If F : [0,∞) × Cq → R is a jointly continuous function and X :[−q,∞) → R is a continuous fuzzy process, then t �→ F(t, Xt ) is integrable oneach compact interval [τ, T ]. Also, if F : [0,∞) × Cq → R is a jointly continuousfunction and X : [−q,∞) → R is a continuous fuzzy process, then the functiont �→ F(t, Xt ) : [0,∞) → R is bounded on each compact interval [0, T ]. Also, thefunction t �→ F(t, 0) : [0,∞) → R is bounded on each compact interval [0, T ].

We say that the function F : [0,∞) × Cq → R is locally Lipschitz if for alla, b ∈ [0,∞) and ρ > 0, there exists L > 0 such that

|F(t, ϕ)− F(t, ψ)| ≤ L ‖ϕ − ψ‖ , a ≤ t ≤ b, ϕ, ψ ∈ Bρ,

where Bρ := {ϕ ∈ Cq; ||ϕ|| ≤ ρ}.

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104 V. Lupulescu, U. Abbas

Lemma 5 Assume that F : [0,∞) × Cq → R is continuous and locally Lipschitz.Then, for each compact interval J ⊂ [0,∞) and ρ > 0, there exists M > 0 such that

|F(t, ϕ)| ≤ M, t ∈ J, ϕ ∈ Bρ.

Proof Indeed, for t ∈ J , we have

|F(t, ϕ)| ≤ |F(t, ϕ)− F(t, 0)| + |F(t, 0)|≤ L ‖ϕ‖ + |F(t, 0)| ≤ ρL + η,

where η := supt∈J |F(t, 0)|.Suppose that C is a standard Liu process and F,G : [0,∞) × Cq → R are some

given function.Consider the following fuzzy delay differential equation

{d X (t) = F (t, Xt ) dt + G (t, Xt ) dC (t), t ≥ τ

X (t) = ϕ(t − τ), τ − q ≤ t ≤ τ(3)

By solution of fuzzy delay differential equation (3) on some interval [τ, b), we wemean a continuous fuzzy process X : [τ−q, b)×Ω → R, such that X (t) = ϕ(t −τ),for τ − q ≤ t ≤ τ and d X (t) = F (t, Xt ) dt + G (t, Xt ) dC (t), for all t ≥ τ . Weremark that if F,G : [0,∞)×Cq → R are continuous, then X : [τ −q, b)×Ω → Ris a solution for (3) if and only if

X (t) ={ϕ(t − τ), τ − q ≤ t ≤ τ

ϕ(0)+ ∫ tτ

F (s, Xs) ds + ∫ tτ

G (s, Xs) dC (s), τ ≤ t < b

To solve a fuzzy differential equations with delay, we use the method of steps (Hale1997). �Example 6 Consider the following fuzzy differential equation:

{d X (t) = μX (t − τ)dt + σdC (t), t ≥ 0X (t) = 1,−τ ≤ t ≤ 0,

(4)

where μ and σ are constants, and C (t) is a standard Liu process.If 0 ≤ t ≤ τ , then −τ ≤ t −τ ≤ 0, and so X (t −τ) = 1. Therefore, for 0 ≤ t ≤ τ ,

we have to solve the following differential equation

{d X (t) = μdt + σdC (t)X (0) = 1.

We obtain

X (t) = X (0)+t∫

0

μdt + σC (t),

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Fuzzy delay differential equations 105

and so

X (t) = 1 + μt + σC(t) for 0 ≤ t ≤ τ

If τ ≤ t ≤ 2τ , then 0 ≤ t − τ ≤ τ , and so X (t − τ) = 1 + μ(t − τ) + σC(t − τ).Therefore, for τ ≤ t ≤ 2τ , we have to solve the following differential equation

{d X (t) = μ[1 + μ(t − τ)+ σC(t − τ)]dt + σdC (t)X (τ ) = 1 + μτ + σC (τ ).

Then

X (t) = X (τ )+t∫

τ

μ[1 + μ(s − τ)+ σC (s − τ)]ds + σC (t)

and so

X (t) = 1 + μτ + μ(t − τ)+ μ2 (t − τ)2

2+ σC (t)+ μσ

t∫

τ

C (s − τ) ds

for τ ≤ t ≤ 2τ . Clearly, we can continue this method, finding the expression for X (t)on each interval [nτ, (n + 1)τ ] with n ≥ 0.

Example 7 (Ehrlich ascities tumor model).In Schuster and Schuster (1995), the following form of delay logistic equation was

proposed to describe the Ehrlich ascities tumor:

X ′(t) = αX (t − q)

(

1 − X (t − q)

K

)

where α is the net reproduction of the rate tumor and K is the caring capacity. Here,q reflects the duration of the cell cycle. In generally, the deterministic growth modelsdo not necessarily give a satisfactory deterministic prediction of mean trends. Fluctu-ations of the growth dynamics for a cell tumor population can be influenced by manyindependent characteristics of the state variables, such as: medical treatment, mentalstatus, diet, physical activity, age, etc.. The specific value of these characteristics notalways can be evaluated or measured in classical sense, which are uncertain and wecan only conjecture intuitively. Therefore, we consider that a more realistic growthmodel for the Ehrlich ascities tumor should be the following fuzzy model:

d X (t) = αX (t − q)

(

1 − X (t − q)

K

)

dt + σdC (t), (5)

where σ is a constant controlling the amplitude of noise, and C(t) is the standard Liuprocess (white noise). We can associate with the above fuzzy differential equationsthe initial condition

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106 V. Lupulescu, U. Abbas

X (t) = ϕ (t),−q ≤ t ≤ 0. (6)

The initial value problem (5)–(6) can be solved using the method of steps (Hale 1997).Given the importance of such issues in the modeling of growth dynamics for a celltumor population, a detailed study of it will be done in a next paper.

Theorem 8 Suppose that F,G : [0,∞) × Cq → R are continuous and satisfy thefollowing locally Lipschitz continuous: for all a, b ∈ [0,∞) and ρ > 0 there existsL > 0 such that

|F (t, ϕ)− F (t, ψ)| + |G (t, ϕ)− G (t, ψ)| ≤ L ‖ϕ − ψ‖ , (7)

for a ≤ t ≤ b and ϕ,ψ ∈ Bρ . Then, for each (τ, ϕ) ∈ [0,∞) × Cq, there existsT > τ such that the fuzzy delay differential equation (3) has a unique solution on[τ − q, T ].

Proof Let ρ > 0 be any positive number. Since F and G satisfy the locally Lipschitzcondition (7), then there exists L > 0 such that

|F (t, ϕ)− F (t, ψ)| + |G (t, ϕ)− G (t, ψ)| ≤ L ‖ϕ − ψ‖ , (8)

for τ ≤ t ≤ τ + h, ϕ, ψ ∈ Bρ and for some h > 0. By Lemma 5, there exists M > 0such that

max{|F (t, ϕ) |, |G (t, ϕ) |} ≤ M f or (t, ϕ) ∈ [τ, τ + h] × B2ρ.

It is known that K (ω) > 0 is bounded on Ω (see Dai 2009, Theorem 2). We chooseT := τ + min{h, ρ

K M }, where K > 0 is a constant such that 1 + K (ω) ≤ K forall ω ∈ Ω . To solve (3) we construct a solution process via approximation by Picarditerations. For this, we defined a sequence of functions Xm : [τ − q, T ] → E startingwith the initial continuous function

X0(t) ={ϕ(t − τ), for τ − q ≤ t ≤ τ

ϕ(0), for τ ≤ t ≤ T

and we define

Xm+1(t) ={ϕ(t − τ), τ − q ≤ t ≤ τ

ϕ(0)+ ∫ tτ

F(s, Xm

s

)ds + ∫ t

τG

(s, Xm

s

)dC (s), τ ≤ t ≤ T

(9)

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Fuzzy delay differential equations 107

if m = 0, 1, . . .. Clearly, |X0(t)| ≤ 2ρ on [τ, T ] . Suppose that |Xm(t)| ≤ 2ρ on[τ, T ]. Then by Lemma 1, we have

∣∣∣Xm+1(t)

∣∣∣ ≤

∣∣∣X0(t)

∣∣∣ +

∣∣∣∣∣∣

t∫

τ

F(s, Xm

s

)ds

∣∣∣∣∣∣+

∣∣∣∣∣∣

t∫

τ

G(s, Xm

s

)dC(s)

∣∣∣∣∣∣

≤ ρ +t∫

τ

|F (s, Xm

s

) |ds + K (ω)

t∫

τ

|G (s, Xm

s

) |ds

≤ ρ + M[1 + K (ω)](t − τ)

≤ ρ + M[1 + K (ω)](T − τ) < ρ + M K (T − τ) < 2ρ,

and so, |Xm+1(t)| ≤ 2ρ on [τ, T ]. Therefore, Xm(t) ∈ B2ρ for all t ∈ [τ, T ] andm ≥ 0. By (8) and (9), we find that

∣∣∣Xm+1(t)− Xm(t)

∣∣∣ ≤

∣∣∣∣∣∣

t∫

τ

[F(s, Xm

s

)ds − F

(s, Xm−1

s

)]ds

∣∣∣∣∣∣

+∣∣∣∣∣∣

t∫

τ

[G (s, Xm

s

)dC(s)− G

(s, Xm−1

s

)]dC(s)

∣∣∣∣∣∣

≤t∫

τ

∣∣∣F

(s, Xm

s

) − F(

s, Xm−1s

)∣∣∣ ds

+K (ω)

t∫

τ

∣∣∣G

(s, Xm

s

) − G(

s, Xm−1s

)∣∣∣ ds

≤ L[1 + K (ω)]t∫

τ

∥∥∥Xm

s − Xm−1s

∥∥∥ ds

≤ L K

t∫

τ

supθ∈[−q,0]

|Xms (θ)− Xm−1

s (θ)|ds

= L K

t∫

τ

supθ∈[−q,0]

|Xm(θ + s)− Xm−1(θ + s)|ds

= L K

t∫

τ

supζ∈[s−q,s]

|Xm(ζ )− Xm−1(ζ )|ds, t ∈ [τ, T ].

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108 V. Lupulescu, U. Abbas

In particular,

∣∣∣X1(t)− X0(t)

∣∣∣ ≤

∣∣∣∣∣∣

t∫

τ

F(

s, X0s

)ds

∣∣∣∣∣∣+

∣∣∣∣∣∣

t∫

τ

G(

s, X0s

)dC(s)

∣∣∣∣∣∣

≤t∫

τ

|F(

s, X0s

)|ds + K (ω)

t∫

τ

|G(

s, X0s

)|ds ≤ M K (t − τ),

and so,

∣∣∣X2(t)− X1(t)

∣∣∣ ≤ L[1 + K (ω)]

t∫

τ

supζ∈[s−q,s]

|X1(ζ )− X0(ζ )|ds

= L M K 2

t∫

τ

(s − τ)ds ≤ M

L

L2 K 2

2(t − τ)2, t ∈ [τ, T ] .

Further, if we assume that

|Xm(t)− Xm−1(t)| ≤ M

L

Lm K m

m! (t − τ)m, t ∈ [τ, T ] (10)

then we have

|Xm+1(t)− Xm(t)| ≤ L(1 + K (ω))

t∫

τ

supζ∈[s−q,s]

|Xm(ζ )− Xm−1(ζ )|ds

= L[1 + K (ω)]t∫

τ

M

L

Lm K m

m! (t − τ)mds

= M

L

Lm+1 K m+1

(m + 1)! (t − τ)m+1,

for t ∈ [τ, T ]. If follows by mathematical induction that (10) holds for any m ≥ 1.

Consequently, the series∞∑

m=1|Xm(t) − Xm−1(t)| is uniformly convergent on [τ, T ],

and so is the sequence {Xm}m≥0. It follows that there exists X : [τ, T ] → R such that|Xm(t)− X (t)| → 0 as m → ∞. Since

∣∣F

(s, Xm

s

) − F (s, Xs)∣∣ + ∣

∣G(s, Xm

s

) − G (s, Xs)∣∣

≤ L∥∥Xm

s − Xs∥∥ ≤ L sup

τ≤t≤T|Xm(t)− X (t)|,

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Fuzzy delay differential equations 109

we deduce that

∣∣F

(s, Xm

s

) − F (s, Xs)∣∣ → 0 and

∣∣G

(s, Xm

s

) − G (s, Xs)∣∣ → 0

uniformly on [τ, T ] as m → ∞. Therefore, since

∣∣∣∣

t∫

τ

F(s, Xm

s

)ds −

t∫

τ

F (s, Xs) ds

∣∣∣∣ ≤

t∫

τ

∣∣F

(s, Xm

s

) − F (s, Xs)∣∣ ds

it follows that limm→∞

∫ tτ

F(s, Xm

s

)ds = ∫ t

τF (s, Xs) ds, t ∈ [τ, T ]. Also, by

Lemma 1, we obtain

∣∣∣∣∣∣

t∫

τ

G(s, Xm

s

)dC(s)−

t∫

τ

G (s, Xs) dC(s)

∣∣∣∣∣∣

≤ K (ω)

t∫

τ

∣∣G

(s, Xm

s

) − G (s, Xs)∣∣ ds,

and so, limm→∞

∫ tτ

G(s, Xm

s

)dC(s) = ∫ t

τG (s, Xs) dC(s), t ∈ [τ, T ]. Extending X to

[τ − q, τ ] in the usual way by X (t) = ϕ(t − τ) for t ∈ [τ − q, τ ], then by (9) weobtain that

X (t) =⎧⎨

⎩

ϕ(t − τ), τ − q ≤ t ≤ τ

ϕ(0)+t∫

τ

F (s, Xs) ds +t∫

τ

G (s, Xs) dC(s), τ ≤ t ≤ T .

Next, we prove that X is continuous. Obviously, X is sample continuous on[τ − q, τ ]. For t > s > τ , we have

|X (t)− X (s)| =∣∣∣∣∣∣

t∫

τ

F(ξ, Xξ

)dξ +

t∫

τ

G(ξ, Xξ

)dC(ξ)

∣∣∣∣∣∣

≤t∫

τ

∣∣F

(ξ, Xξ

)∣∣ dξ + K (ω)

t∫

τ

∣∣G

(ξ, Xξ

)∣∣ dξ

≤ M[1 + K (ω)](t − s) ≤ M K (t − s),

and so, |X (t) − X (s)| → 0 as s → t . Hence X is continuous. Therefore, X (t) is asolution for (3). To prove the uniqueness, assume that X and Y are two solutions of(3). Then for every t ∈ [τ, T ] we have

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110 V. Lupulescu, U. Abbas

|X (t)− Y (t)|

=∣∣∣∣∣∣

t∫

τ

[F (s, Xs) ds − F (s,Ys)]ds +t∫

τ

[G (s, Xs) ds − G (s,Ys)]dC(s)

∣∣∣∣∣∣

≤t∫

τ

|F (s, Xs)− F (s,Ys) |ds + K (ω)

t∫

τ

|G (s, Xs)− G (s,Ys)| ds

≤ L[1 + K (ω)]t∫

τ

L ‖Xs − Ys‖ ds ≤ L K

t∫

τ

supζ∈[s−q,s]

|X (ζ )− Y (ζ )| ds

If we let ξ(s) := supζ∈[s−q,s]

|X (ζ )− Y (ζ )| , s ∈ [τ, T ], by Gronwall’s lemma we obtain

that X (t) = Y (t) on [τ, T ]. This proves the uniqueness of the solution of (3). �Theorem 9 Assume that the functions F,G : [0,∞) × Cq → R are continuousand satisfy the locally Lipschitz (7). If (τ, ϕ), (τ, ψ) ∈ [0,∞) × Cq and X (ϕ) :[τ − q, T1) → R and X (ψ) : [τ − q, T2) → R are unique solutions of ( 3) withX (t) = ϕ(t − τ) and X (t) = ψ(t − τ) on [τ − q, τ ], respectively. Then

|X (ϕ)(t)− X (ψ)(t)| ≤ ‖ϕ − ψ‖ eL K (t−τ) for all t ∈ [τ, T ), (11)

where T = min{T1, T2} and K > 0 is a constant such that 1 + K (ω) ≤ K for allω ∈ Ω .

Proof On [τ, T ) solution X (ϕ) satisfies the relation

X (t) ={ϕ(t − τ), t ∈ [τ − q, τ ]ϕ(0)+ ∫ t

τF(s, Xs(ϕ)ds + ∫ t

τG (s, Xs) dC(s), t ∈ [τ, T )

and solution X (ψ) satisfies the same relation but with ψ in place of ϕ. Then, fort ∈ [τ, T ), we have

|X (ϕ)(t)− X (ψ)(t)| ≤ |ϕ(0)− ψ(0)| +t∫

τ

|F(s, Xs(ϕ)− F(s, Xs(ψ)|ds

+K (ω)

t∫

τ

|G(s, Xs(ϕ)− G(s, Xs(ψ)|ds

≤ ‖ϕ − ψ‖ + L[1 + K (ω)]t∫

τ

‖Xs(ϕ)− Xs(ψ)‖ ds

≤ ‖ϕ − ψ‖ + L K

t∫

τ

maxθ∈[τ−q,s] |X (ϕ)(θ)− X (ψ)(θ)|ds.

123

Fuzzy delay differential equations 111

If we let w(t) = supθ∈[τ−q,s] |X (ϕ)(θ)− X (ψ)(θ)|, τ ≤ s ≤ t , then we have

w(t) ≤ ‖ϕ − ψ‖ + L K

t∫

τ

w(s)ds , τ ≤ t < T .

Then the Gronwall’s inequality gives

w(t) ≤ ‖ϕ − ψ‖ eL K (t−τ) , τ ≤ t < T,

implying that (11) holds. �

4 Conclusion

Based on the concept of Liu process, we prove the existence and uniqueness theoremfor fuzzy delay differential equations. We also establish continuous dependence ofsolutions with respect to initial data.

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