# A Note on Asymptotic Equilibrium for Fuzzy Differential Equations

Authors: Yabin Shao; Shengui Zhang; Yanping Xiao
Journal: Journal of Applied Mathematics
 (2014)

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## Abstract

The asymptotic equilibrium results for fuzzy differential systemsGCPx′=f1t,x,y,y′=f2(t,x,y) are investigated, where f1t,x,y satisfies the compactness-type and f2t,x,y satisfies the dissipative-type conditions. It is worth mentioning that the uniformly continuous conditions of ft,x,y are removed in Song et al. (2005). That is to say, the results of Song et al. (2005) are extended. In addition, the global existence and asymptotic equilibrium results of fuzzy differential systems CPx′t=ft,x,x0=x0 are obtained.


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## Body

## 1. Introduction

The Cauchy problems for fuzzy differential equations have been studied by several authors [1–6] on the metric space (En,D) of normal fuzzy convex set with the distance D given by the maximum of the Hausdorff distance between the corresponding level sets. In [4], Nieto proved the Cauchy problems which have a unique solution if f is continuous and bounded. For a general reference to fuzzy differential equations, see a recent book by Lakshmikantham and Mohapatra [7] and references therein. In particular, Wu and Song [8–10] changed the initial value problem of fuzzy differential equations into abstract differential equations on a closed convex cone in a Banach space by the operator j that is the isometric embedding from (En,D) onto its range in the Banach space X. They established the relationships between a solution and its approximate solutions to fuzzy differential equations. Furthermore, they obtained the local existence theorems under the compactness-type and the dissipative-type conditions. Park and Han [11] showed the global existence and uniqueness of fuzzy solutions of fuzzy differential equation using the properties of Hasegawa’s function and successive approximation. Song et al. [12] pointed out a variety of results which assure global existence of solutions to fuzzy differential equations. Song et al. [13] studied the asymptotic equilibrium for fuzzy differential equations:
(CP)x′=f(t,x),x(t0)=x0,
where x0∈En and f is a continuous. Since we are only interested in local solutions to (CP), we assume f:J×B→En and D(f,0~)≤M on J×B for some M, where J=[0,a] and B=B¯(x0,r)={x∈En:D(x,x0)≤r}. The methods used by them are as follows: f satisfied not only the compactness-type conditions but the uniformly continuity condition. Hence, it is of significance to lessen the growth conditions of f.Based on those preceding works, in this paper, we firstly give the existence theorems for(CP) without a uniform continuity assumption on f. In addition, the more general systems than (CP) are considered:
(GCP)x′=(x′y′)=(f1(t,x,y)f2(t,x,y))=f(t,z),
where z=(xy), f∈C[[t0,+∞)×En,En], f=(f1f2), and   E1n and E2n are the fuzzy number space, respectively. fi∈C[[t0,+∞)×En,Ein](i=1,2), f1(t,x,y) satisfies the compactness-type, and f2(t,x,y) satisfies the dissipative-type conditions. In particular, when En=E1n and f2=0, we obtain that Corollary 24 is the promotion of the results of [13]. When En=E2n and f1=0, we obtain that Corollary 26 is the asymptotic equilibrium of fuzzy differential system (CP) under the dissipative-type conditions.As preliminaries we recall some basic results on fuzzy number space(En,D) and list several comparison theorems on classical ordinary differential equations. In Section 3, we will proof the existence theorems for (CP) without a uniform continuity assumption on f. In Section 4, we will show that the asymptotic equilibrium for fuzzy differential system (GCP). Finally, in Section 5, we present some concluding remarks.


## 2. Preliminaries

LetPk(Rn) denote the family of all nonempty compact convex subsets of Rn and define the addition and scalar multiplication in Pk(Rn) as usual. Let A and B be two nonempty bounded subsets of Rn. The distance between A and B is defined by the Hausdorff metric:
(1)dH(A,B)=max⁡{sup⁡a∈Ainf⁡b∈B∥a-b∥,sup⁡b∈Binf⁡a∈A∥b-a∥}.DenoteEn={u:Rn→[0,1]∣u satisfies (1)–(4) below} is a fuzzy number space, where(1)
uis normal; that is, there exists an x0∈Rn such that u(x0)=1;(2)
u is fuzzy convex; that is, u(λx+(1-λ)y)≥min⁡{u(x),u(y)} for any x,y∈Rn and 0≤λ≤1;(3)
u is upper semicontinuous;(4)
[u]0=cl⁡{x∈Rn∣u(x)>0} is compact.For0<α≤1, denote [u]α={x∈Rn∣u(x)≥α}. Then from above (1)–(4), it follows that the α-level set [u]α∈Pk(Rn) for all 0≤α<1. According to Zadeh’s extension principle, we have addition and scalar multiplication in fuzzy number space En as follows:
(2)[u+v]α=[u]α+[v]α,[ku]α=k[u]α,
where u,v∈En and 0≤α≤1.DefineD:En×En→[0,∞):
(3)D(u,v)=sup⁡{dH([u]α,[v]α):α∈[0,1]},
where d is the Hausdorff metric defined in Pk(Rn). Then it is easy to see that D is a metric in En. Using the results in [14, 15], we know that (En,D) is a complete metric space. It is well known that application of fuzzy set theory very often involves the metric space (En,D) of normal fuzzy convex set over Rn, where D is the supremum of the Hausdoff distance between corresponding level sets. This metric has been found very convenient in studying of fuzzy differential equations (see [7]).Definition 1 (see [16, 17]).
LetSn-1 be the unit sphere of Rn; that is, Sn-1={x∈R∣∥x∥=1}, 〈·,·〉 is the inner product in Rn; that is, 〈x,y〉=∑i=1n‍xiyi, where x=〈x1,x2,…,xn〉, y=〈y1,y2,…,yn〉∈Rn. Suppose u∈En, x∈Sn-1,  and r∈[0,1]; then the support function of u is defined by u*(r,x)=sup⁡α∈[u]α〈α,x〉 for all (r,x)∈I×Sn-1.The properties of the support functions can be referred to in [17] for details.Theorem 2 (see [16, 17]).
LetSn-1 be the unit sphere of Rn and 〈·,·〉 the inner product in Rn. Suppose u∈En, u*(r,x) is the support function of u; then(1)
u*(r,x)≤sup⁡a∈[u]r∥a∥; that is,u*(r,x)is bounded onSn-1, for each fixedr∈[u]r;(2)
u*(r,x)is nonincreasing and left continuous inr∈(0,1]and right continuous atr=0, for each fixedx∈Sn-1;(3)
u*(r,x)is Lipschitz continuous inxand(4)∥u*(r,x)-u*(r,y)∥≤(sup⁡a∈[u]r∥a∥)∥x-y∥;(4)
ifu,v∈En,r∈[0,1],
thend([u]r,[v]r)=sup⁡x∈Sn-1|u*(r,x)-v*(r,x)|.Theorem 3 (see [16, 17]).
Define an operatorj on En as j(u)(r,x)=u*(r,x) for any u∈En; then one has(1)
j(u)∈C-(I,C(Sn-1)),(2)
j(su+tv)=sj(u)+tj(v),s,t≥0,(3)
∥j(u)-j(v)∥=D(u,v),(4)
j(En)is closed inC-(I,C(Sn-1)).In the sequel we will recall some integrability and differentiability properties in [18–20] for fuzzy set-valued mappings.LetT=[t0,t0+p]⊂R(p>0) be compact interval. The fuzzy mapping F:T→En is called strong measurable if for all α∈[0,1] the set-valued mapping Fα:T→Pk(Rn) defined by Fα(t)=[F(t)]α is Lebesgue measurable, where Pk(Rn) is endowed with the topology generated by the Hausdorff metric d. A mapping Fα:T→En is called integrable bounded if there exists an integrable function h such that ∥x∥≤h(t) for all x∈F0(t).Definition 4.
LetF:T→En. The integral of F over T, denoted by ∫T‍F(t)dt, is defined level-wise by the equation (see [18]):
(5)[∫T‍F(t)dt]α=∫T‍Fα(t)dt={∫T‍f(t)dt∣f:T→RnhhhhisameasurableselectionforFα∫T‍}.A strongly measurable and integrable mappingFα:T→En is said to be integrable over T if ∫T‍F(t)dt∈En. From [20], we know that if Fα:T→En is continuous, then it is integrable.Letx,y∈En. If there exists a fuzzy number z∈En such that x=y+z, z is called the H-difference of x and y that is denoted by x-Hy. For brevity, we always assume that it satisfies the H-difference when dealing with the operation of subtraction of fuzzy numbers throughout this paper.Definition 5 (see [20]).
A mappingF:T→En is differentiable at t0∈T if there exists a F′(t0)∈En, such that the limits
(6)lim⁡h→0+F(t0+h)-HF(t0)h,lim⁡h→0+F(t0)-HF(t0+h)h
exist and are equal to F′(t0).Here the limits are taken in the metric space(En,D). At the endpoint of T, we consider only one-side fuzzy derivatives. If F:T→En is differentiable at t0∈T, then we say that F′(t0) is the fuzzy derivative of F(t) at the point t0.We note that this definition is fairly strong, because the family of fuzzy-number-valued functionsH-differentiable is very restrictive. For example, the fuzzy-number-valued function f:[a,b]→Rℱ defined by f(x)=C·g(x), where C is a fuzzy number, · is the scalar multiplication (in the fuzzy context), and g:[a,b]→R+, with g′(t0)<0, is not H-differentiable in t0 (see [19, 21]). To avoid the above difficulty, in this paper we consider a more general definition of a derivative for fuzzy-number-valued functions enlarging the class of differentiable fuzzy-number-valued functions, which has been introduced in [19].Definition 6 (see [19]).
Letf:(a,b)→En and x0∈(a,b). One says that f is differentiable at x0 if there exists an element f′(t0)∈En, such that,(1)
for allh>0 sufficiently small, there exist f(x0+h)-Hf(x0), f(x0)-Hf(x0-h) and the limits (in the metric D)
(7)lim⁡h→0f(x0+h)-Hf(x0)h=lim⁡h→0f(x0)-Hf(x0-h)h=f′(x0)(2)
for allh>0 sufficiently small, there exist f(x0)-Hf(x0+h), f(x0-h)-Hf(x0) and the limits
(8)lim⁡h→0f(x0)-Hf(x0+h)-h=lim⁡h→0f(x0-h)-Hf(x0)-h=f′(x0)(3)
for allh>0 sufficiently small, there exist f(x0+h)-Hf(x0), f(x0-h)-Hf(x0) and the limits
(9)lim⁡h→0f(x0+h)-Hf(x0)h=lim⁡h→0f(x0-h)-Hf(x0)-h=f′(x0)(4)
for allh>0 sufficiently small, there exist f(x0)-Hf(x0+h), f(x0)-Hf(x0-h) and the limits
(10)lim⁡h→0f(x0)-Hf(x0+h)-h=lim⁡h→0f(x0)-Hf(x0-h)h=f′(x0) (h and -h at denominators mean 1/h and -1/h, resp.).In addition, we define a continuous fuzzy-valued functionf:T×Ω→En by f∈C[T×Ω,En], where Ω⊂En is an open set.Theorem 7 (see [19]).
Assume thatf∈C[T×Ω,En]. A function x:T→Ω is a solution to the problem x′=f(t,x), x(t0)=x0 if and only if it is continuous and satisfies the integral equation
(11)x(t)=x0+∫t0t‍f(s,x(s))ds,
or
(12)x(t)=x0+(-1)·∫t0t‍f(s,x(s))ds,
for all t∈T.Theorem 8 (see [22]).
LetG⊂R2 be an open set, g∈C[G,R1], (t0,u0)∈G. Suppose that the maximum solution of initial value problem u′(t)=g(t,u), u(t0)=u0 is r(t) and its largest interval of existence of right solution is [t0,t0+a). If m(t)∈C[[t0,t0+a),R1] satisfies (t,m(t))∈G, for all t∈[t0,t0+a),m(t0)≤u0 and
(13)Dm(t)≤g(t,m(t)),∀t∈[t0,t0+a)∖Γ,
where D is one of the four Dini derivatives and Γ at most is a countable set on [t0,t0+a). Then one has m(t)≤r(t), for all t∈[t0,t0+a).Theorem 9 (see [22], (Ascoli-Arzela)).
A setH⊂C(T,X) is a relative compact set if and only if H is equicontinuous and, for any t∈T, H(t) is relative compact set in X.


## 3. On the Cauchy Problem for Fuzzy Differential Equations

Letγ be Kuratowski’s measure of noncompactness (see [22] for details).Lemma 10.
LetS, T are two bounded subsets of Banach space X and t a real number; then Kuratowski’s measure of noncompactness has following properties:(1)
γ(S)=0if and only ifSis a relatively compactness set,(2)
S⊂T⇒γ(S)≤γ(T),(3)
γ(tS)=|t|γ(S), wheretS={tx:x∈S},(4)
γ(S+T)≤γ(S)+γ(T), whereS+T={x+y:x∈S,y∈T},(5)
γ(S×T)=max⁡{γ(S),γ(T)}, whereS×Tis the bounded subsets of Banach spaceE1×E2.It is well known that continuity off is not sufficient for the existence of local solution to (CP) (see [4]). The extra conditions that have been imposed on f are mainly estimates which guarantee that a certain sequence of approximate solutions has at least a uniformly convergent subsequence. In this section, we would like to dispense with the uniform continuity assumption of f.Next, we describe a class of uniqueness functions. LetJ⊂R be an interval. A function ω:J×R+→R is said to satisfy Caratheodory’s condition on J×R+ if ω(t,x) is measurable in t for x∈R+, continuous in x for t∈J and such that to each x0>0 and compact interval J0⊂J there exists a function h0∈L1(J0) with |ω(t,x)|≤h0(t) in J0×[0,x0].Definition 11 (see [23]).
A functionω:(0,a]×R+→R+ with ω∈K((0,a]×R+) is said to be class U, denoted by ω∈U, if ϕ′(t)≤ω(t,ϕ(t)) a.e. on (0,a] and ϕ(t)=o(t) for t→0+ imply ϕ≡0 on [0,a] for each absolutely continuous ϕ:[0,a]→R+.We will consider a sequence of fuzzy-number-valued functions behaving like the sequence of approximate solutions that we will use in next section.Theorem 12.
Letxn be a sequence of continuous fuzzy-number-valued functions from J=[a,b] to En such that there is some function μ∈L1(a,b) with D(xn(t),0~)≤μ(t) on J. Let ϕ(t)=γ(j∘{xn(t):n∈N}). Then ϕ(t) is integrable on J and
(14)γ(j∘{∫ab‍xn(τ)dτ:n∈N})≤∫ab‍ϕ(τ)dτ.Proof.
Fixn,m∈N; we have
(15)dH(∑i=0k-1‍xn(ξi)(τi+1-τi),En)≤∑i=0k-1‍dH(xn(ξi),En)(τi+1-τi)
for every partition (τ0,τ1,…,τk) and every (ξ1,ξ2,…,ξk) with τi≤ξi≤τi+1 for i=0,1,2,…,k-1. With the continuity of xn, we have
(16)dH(∫ab‍xn(τ)dτ,En)≤∫ab‍dH(xn(τ),En)dτ.
An application of Fatou’s Lemma gives
(17)lim⁡n→∞¯dH(∫ab‍xn(τ)dτ,En)≤∫ab‍lim⁡n→∞¯dH(xn(τ),En)dτ
and with Lebesgue’s dominated convergence theorem; since lim⁡n→∞¯dH(xn(t),En)⁡≤μ(t) for t∈J, m∈N, we have
(18)lim⁡m→∞lim⁡n→∞¯dH(∫ab‍xn(τ)dτ,En)≤∫ab‍lim⁡m→∞lim⁡n→∞¯dH(xn(τ),En)dτ.
Therefore, we get
(19)γ(j∘{∫ab‍xn(τ)dτ:n∈N})≤∫ab‍ϕ(τ)dτ.Theorem 13.
Letxn(t):[a,b]→En be a sequence of continuously differentiable functions. Assume that there exists m(t)∈L1[a,b] such that
(20)∥j∘xn(t)∥≤m(t),∥j∘xn′(t)∥≤m(t)
a.e. t∈[a,b], n=1,2,…. Let h(t)=γ(j∘{xn(t):n∈N}); then h(t) is an absolutely continuous function and h′(t)≤γ(j∘{xn(t):n∈N}) a.e. t∈[a,b].Proof.
The absolute continuity ofh follows from
(21)∥h(t)-h(s)∥≤γ({j∘xn(t)-j∘xn(s)})=h(j∘∫st‍xn′(τ)dτ)≤2·sup⁡n∈N∥j∘∫st‍xn′(τ)dτ∥≤2·∫st‍m(τ)dτ.
We conclude by Theorem 12 that
(22)γ(j∘xn(t))-γ(j∘xn(t-l))≤γ(j∘∫t-lt‍xn′(τ)dτ)≤∫t-lt‍γ(j∘xn′(τ))dτ.
Dividing by l>0 and letting l→0+ give
(23)h′(t)≤γ(j∘{xn(t):n∈N}).Theorem 14.
LetB=B¯(x0,r)={x∈En:D(x,x0)≤r} and f:[0,a]×B→En continuous with D(f(t,x),0~)≤μ(t) on [0,a]×B for some μ∈L1[0,a]. Let b<a such that ∫0a‍μ(τ)dτ<r and J=[0,b]. One assumes
(24)γ(j∘f(t,B))≤ω(t,γ(j∘B′))hhhhhhhhfort∈[0,a],B′⊂B,
where ω∈U. Then (CP) has a solution.Proof.
There is a sequence{xn(t)} of approximate solutions to (CP) satisfying
(25)xn′(t)=f(t,xn(t))+yn(t),xn(t)=x0,yn(t)≤1n.
Let h(t)=γ(j∘{xn(t):n∈N}). We have by Theorem 13 that h′(t)≤γ(j∘{xn(t):n∈N}) a.e. on t∈[a,b]. Moreover, by the properties of γ, we have
(26)γ(j∘{xn′(t)})≤γ(j∘{f(t,xn(t))})+γ({j∘yn(t)})≤ω(t,γ(j∘{xn(t)})).
We claim that h(t) is o(t) for t→0+. By the continuity of f and the equicontinuity of the xn(t), we have
(27)xn(t)=x0+∫0t‍(f(τ,xn(τ))-f(0,x0))dτ+tf(0,x0)+∫0t‍yn(τ)dτ=x0+tf(0,x0)+zn(t)
with zn(t)=o(t) for t→0+ uniformly in n. Since ω∈U, we get h(t)=0 on [a,b]. Arzala-Ascoli’s theorem gives a uniformly convergent subsequence of xn(t) and a standard argument shows that the limit of this subsequence is a solution to (CP) on [a,b].Next, consider the terminal value problem(CP)′x′=f(t,x),x(∞)=x0,
where f:[a,∞)×En→En is continuous, a>0, and x0∈En. This problem is equivalent to the following initial value problem:
(28)u′=-t-2f(t-1,u),u(0)=x0,
by means of the transformation t→t-1 on (0,a-1].Definition 15 (see [23]).
A functionω:(0,a]×R+→R+ with ω∈K((0,a]×R+) is said to be of class U¯, denoted by ω∈U¯, if ϕ′(t)≤ω(t,ϕ(t)) a.e. on (0,a] and ϕ(t)=o(1) for t→t-1 imply ϕ(t)=0 on [0,a] for each absolutely continuous ϕ:[0,a]→R+.Theorem 16.
LetEn be a fuzzy number space. B(x0,r)={x∈En:D(x,x0)≤r} and f:[a,∞)×B→En are continuous. Suppose that f satisfies(1)
μ(t)=sup⁡{D(f,0~):x∈B(x0,r)},(2)
γ(j∘f(t,B))≤ω(t,γ(j∘B))fort≥a,where ω:(0,a]×R+→R+ is such that t-2ω(t-1,ϱ)∈U¯. Then there exists b≥a such that the terminal value problem (CP)′ has a solution in [b,∞).Proof.
Consider the following problem:(29)u′=-t-2f(t-1,u),u(n-1)=x0.
By Theorem 14 and condition (1), the problem (29) has a solution un. Let un=x0 for t∈[0,n-1]. As a consequence of Theorem 13 and the choice of ω, we conclude similar to the proof of Theorem 14 that (CP)′ has a solution.


## 4. On the Asymptotic Equilibrium for Fuzzy Differential Equations

In this section, we will consider that the system(GCP) has the asymptotic equilibrium.Let the functions[·,·]+ and [·,·]- be defined as follows (see [11]):
(30)[u,v]+=lim⁡h→01h(∥j∘u+hj∘v∥-∥j∘u∥),[u,v]-=12([u,v]+-[u,-v]+)
for u,v∈En, where j is the embedding operator in Theorem 3.Theorem 17.
For allu,v,w∈En, define 〈u,v〉±=∥j∘u∥[u,v]± and satisfy the following properties:(1)
〈u,v〉±≤∥j∘u∥·∥j∘v∥,(2)
〈u+v,w〉-≤〈u,w〉-+∥j∘v∥·∥jw∥,(3)
ifIis an open interval inRandx:I→Enis a differential function ont∈I, then(31)∥j∘x(t)∥·D-∥j∘x(t)∥≤〈x′(t),x(t)〉-.Proof.
(
1
) and (2) are easy consequences of the definition and (3) follows from Lemma 3.6 in [11].Definition 18.
Assume thatφ:[t*,∞)→R is a real continuous function, φ(t*)=0, φ(t)>0, t∈(t*,∞), φ(t) is differentiable, and φ′(t)≠0 for any t>t*. If ω(t,u) is a continuous and ω(t,0)≡0, the initial value problem u′=ω(t,u) has only solution u≡0 and satisfies
(32)lim⁡t→t*+0u(t)=lim⁡t→t*+0u(t)φ(t)=0.
Then ω:(t*,∞)×R+→R+ is called a function of U1 kind.Definition 19.
Assume thatω(t,u) is a continuous and ω(t,0)≡0; the initial value problem u′=ω(t,u) has only solution u≡0 and satisfies
(33)lim⁡t→t*+0u(t)=0
for any t>t*. Then ω:(t*,∞)×R+→R+ is called a function of U2 kind.Definition 20 (see [13]).
One says that fuzzy equation(CP) has asymptotic equilibrium if every solution of (CP), such that (t0,x0)∈R+×En, exists on [t0,∞) and tends to a limit v∈En as t→∞, and, conversely, to every given vector v∈En there exists a solution for fuzzy system (CP) which tends to v as t→∞.Lemma 21 (see [12]).
Assume that(1)
f(t,x)is locally Lipschitzian inxfor(t,x)∈J×En,(2)
D(f(t,x),0^)≤g(t,D(x,0^)),  for all(t,x)∈J×En,(3)
g∈C[J×[0,∞),[0,∞)],g(t,u)is nondecreasing inu≥0for eacht∈J, and maximal solutionr(t,t0,u0)of the scalar initial value problemu′=g(t,u),u(t0)=u0exists throughoutJ.
Then the largest interval of any solutionr(t,t0,u0) of (1) with D(x,0^)≤u0 is J. In addition, if r(t,t0,u0) is bounded on J, then lim⁡t→∞⁡x(t,t0,x0) exists in (En,D).Lemma 22 (see [13]).
Under the assumptions of Lemma21, given v∈En, there exist a T∈[t0,∞) and a sequence {xn(t)}n=1∞ defined on [T,∞), such that(1)
{xn(t)}n=1∞is equicontinuous on[T,∞);(2)
{xn(t)}n=1∞is uniformly bounded on[T,∞); that is, there existsM∈R+, such thatD(xn(t),0^)≤Mfor allt∈[T,∞)and for alln;(3)
for eachn,xn(t)is a solution ofx′=f(t,x),x(T+n)=v.In the following, we give the main result of this paper.Theorem 23.
Under the assumptions of Lemma21, zn(t)={xn(t),yn(t)} is satisfied Lemma 22, and there exists t*∈[T0,∞) such that {zn(t*)}n=1∞ is relatively compact in En and satisfies the following.(1)
LetA⊂Enis a bounded set; one hasγ(j∘f1(t,A))≤g1(t,γ(j∘A′))for allt∈[t′,∞), whereA′={x:there existsy∈E2nsuch that(x,y)∈A}andg1∈U21.(2)
For allε>0,t∈R+, andr>0there existsδ=δ(t,r)>0such thatD(f2(t,x1,y),f2(t,x2,y))<εwith respect toD(y,0^)≤rwhenD(x1,x2)<δ; we havej∘f2(t,x,y)=h(t)+o[φ′(t)]when(t,x,y)→(t*,x0,y0), whereh(t)is a continuous function andφ(t)∈U2.(3)
For(t,x,y1),(t,x,y2)∈[t*,∞)×E1n×E2n, such that(34)〈f2(t,x,y1)-f(t,x,y2),y1-y2〉-≤ω(t,D(y1,y2))·D(y1,y2),where ω∈U1 and (x0,y0) is the limit point of {xn(t*),yn(t*)}.
Then the fuzzy differential system(GCP) has asymptotic equilibrium.Proof.
Sincezn(t)=(xn(t),yn(t)) is satisfied, xn′=f1(t,xn(t),yn(t)) and yn′=f2(t,xn(t),yn(t)), according to Lemma 10 and Theorem 13, and the following fact: {xn(t):n∈N,t∈[t*,∞)} can produce a subspace of E1n; we let l(t)=γ(j∘{xn(t)}); then we have
(35)l′(t)≤γ(j∘{xn′(t):n∈N})=γ(j∘{f1(t,xn(t),yn(t))}n=1∞)≤g1(t,l(t)).
From the assumption and l(t*)=0, we obtain l(t)=0 on [t*,∞). Since {xn(t)}n=1∞ is equicontinuous and uniformly bounded, from Ascoli-Arzelar’s theorem, then there exists a subsequence of {xn(t)}n=1∞ which uniformly converges to x(t) on any finite closed subset of [t*,∞). Without any loss of generality, let {xn(t)}n=1∞ uniformly converge to x(t) on any finite closed subset of [t*,∞) and let  {zn(t*)}n=1∞ converge too.
Now, we consider the initial problem(36)y′(t)=f2(t,x(t),y),y(t*)=y0.
Next, we will prove that{yn(t)}n=1∞ converges uniformly to y(t) on any finite closed subset of [t*,∞). Let anm=D(yn(t*),ym(t*))=∥j∘yn(t*)-j∘ym(t*)∥; then lim⁡n,m→∞⁡anm=0. And let Δnm(t)=D(yn(t),ym(t))=∥j∘yn(t)-j∘ym(t)∥, according to Theorem 17; we have
(37)ΔnmD-Δnm≤〈yn′(t)-ym′(t),yn(t)-ym(t)〉-=〈f2(t,xn(t),yn(t))hhhhh-f2(t,xm(t),ym(t)),yn(t)-ym(t)〉-≤〈f2(t,x(t),yn(t))-f2(t,x(t),ym(t)),yn(t)-ym(t)〉-+[∥j∘f2(t,xn(t),ym(t))-j∘f2(t,x(t),yn(t))∥+∥j∘f2(t,xm(t),ym(t))-j∘f2(t,x(t),ym(t))∥]Δnm.
Let {xn(t)} converge uniformly to x(t) on [t*,a]  (a>t*) and combine to condition (2); we have
(38)ΔnmD-Δnm≤〈f2(t,x(t),yn(t))-f2(t,x(t),ym(t)),yn(t)-ym(t)(t,x(t),yn(t))〉-+εnmΔnm,
where lim⁡n,m→∞⁡=0,  t∈[t*,a]. Then we have
(39)D-Δnm≤ω(t,Δnm(t))+εnm(t∈[t*,a])
and Δnm(t*)→0 when n,m→∞. Let Δn=sup⁡m>n⁡Δnm; then lim⁡n→∞⁡Δn(t*)=0. According to the definition of Δn, we easily know that {Δn} satisfies the following: there exists M>0 such that |Δn-Δs|≤2M|s-t| and D-Δn(t)≤ω(t,Δn(t))+εn(s,t∈[t*,a]),lim⁡n→∞⁡εn=0. From above, we get that {Δn(t)} is equicontinuous and uniformly bounded on [t*,a], so there exists a subsequence converging uniformly to Δ(t). Since D-Δ(t)≤ω(t,Δ(t)) and Δ(t*)=0, from the hypotheses of the theorem, fixed ε>0, there exist t′∈[t*,a],N>0 such that
(40)D-Δnm(t)≤εφ′(t),
when n,m>N,  t∈[t*,t′], which means that
(41)0≤Δnm(t)≤ε·φ(t),t∈(t*,t′),
implies
(42)lim⁡t→t*Δ(t)φ(t)=0.
Because ω∈U1, we have Δ(t)≡0. Hence {Δn(t)} converge uniformly to Δ(t); that is, {Δn(t)} converge uniformly to 0. From above, we can get that {yn(t)} converge uniformly to y(t) on [t*,a]. Again, from the arbitrariness of a, it is easy to know z(t)=(x(t),y(t))=lim⁡n→∞⁡(xn(t),yn(t)),t∈[t*,∞) and
(43)xn(t)=xn(t*)+∫t*t‍f1(s,xn(s),yn(s))ds,yn(t)=yn(t*)+∫t*t‍f2(s,xn(s),yn(s))ds.
Since zn(t)=(xn(t),yn(t)) converge uniformly on a compact set of [t*,∞), we have
(44)x(t)=x(t*)+∫t*t‍f1(s,x(s),y(s))ds,y(t)=y(t*)+∫t*t‍f2(s,x(s),(s))ds
which means that z(t)=(x(t),y(t)) is a solution of fuzzy system z′=f(t,z),  z(t*)=(x(t*),y*(t)).
Next, we will showlim⁡t→∞⁡z(t)=v. Applying Lemma 22, let u(t,t0,λ) be the largest solution of u′=g(t,u), u(t0)=λ for λ∈R+. According to g(t,u)≥0, we have lim⁡t→∞⁡u(t)=u(∞) and
(45)λ=u(t0)≤u(t)≤u(∞)<∞.
Since g(t,u) is nondecreasing in u for every t∈[t0,∞), we have
(46)∞>u(∞)-u(t0)=∫t0∞‍g(s,u(s))ds≥∫t0∞‍g(s,λ)ds.
So, there exists an integer S>t* which satisfies
(47)∫s∞‍g(t,M)dt≤ε2,
where M=sup⁡n∈N⁡{D(zn(s),0^),s∈[t*,∞)}<∞. In addition, there exists N; when n>N, we have
(48)D(zn(t′),z(t′))=∥j∘zn(t′)-j∘z(t′)∥<ε2.
Hence,
(49)∥j∘z(t′)-j∘v∥=∥j∘z(t′)-j∘zN+S(T+N+S)∥≤∥j∘z(t′)-j∘zN+S(t′)∥+∥j∘zN+S(t′)-j∘zN+S(T+N+S)∥≤ε2+D(∫t′T+N+S‍f(t,zN+S(t)dt),0^)≤ε2+|∫t′T+N+S‍g(t,D(zN+S,0^))dt|≤ε2+∫s∞‍g(t,M)dt<ε.
Hence, we have lim⁡t→∞⁡z(t)=v.WhenEn=E1n and f2=0, we have the following.Corollary 24.
Under the assumptions of Lemma21, {xn(t)}n=1∞ is satisfied Lemma 22, and there exists t*∈[T0,∞) such that {xn(t*)}n=1∞ is relatively compact in En and satisfies condition (1) in Theorem 23. Then the fuzzy differential system (CP) has asymptotic equilibrium.Remark 25.
Compared with Theorem 3.4 in [13], the uniformly continuous conditions are removed.WhenEn=E2n and f1=0, we have the following.Corollary 26.
Under the assumptions of Lemma21, {xn(t)}n=1∞ is satisfied Lemma 22, and there exists t*∈[T0,∞) such that {xn(t*)}n=1∞ is relatively compact in En and satisfies
(50)〈f(t,x)-f(t,y),x-y〉-≤ω(t,D(x,y))·D(x,y),
where t∈[T,∞),  x,y∈En,  ω∈U2. Then the fuzzy differential system (CP) has asymptotic equilibrium.


## 5. Conclusion

In this paper, some global existence and asymptotic equilibrium results for fuzzy differential equations under Bede’s derivative for fuzzy-number-valued functions are proved. We apply our main result to the terminal value problem for ordinary differential equations in fuzzy number spaces, a particular situation where we would like to dispense with the uniform continuity assumption off. Our results improve the results given in [8–10, 12, 13] (where uniform continuity was required), as well as those referred to therein. For future research, we extend the asymptotic stability and global attractivity results for the so-called (GCP) fuzzy systems in our paper.



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*Source: 768457-2014-03-05.xml*