A Note on Asymptotic Equilibrium for Fuzzy Differential Equations

Abstract

The asymptotic equilibrium results for fuzzy differential systems (𝐺𝐶𝑃)𝑥^′=𝑓₁(𝑡,𝑥,𝑦),𝑦^′=𝑓₂(𝑡,𝑥,𝑦) are investigated, where 𝑓₁(𝑡,𝑥,𝑦) satisfies the compactness-type and 𝑓₂(𝑡,𝑥,𝑦) satisfies the dissipative-type conditions. It is worth mentioning that the uniformly continuous conditions of 𝑓(𝑡,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 (𝐶𝑃)𝑥^′(𝑡)=𝑓(𝑡,𝑥),𝑥(0)=𝑥₀ are obtained.

1. Introduction

The Cauchy problems for fuzzy differential equations have been studied by several authors [1–6] on the metric space (𝐸^𝑛,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 (𝐸^𝑛,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: 𝑥^′=𝑓(𝑡,𝑥),𝑥𝑡₀=𝑥₀,(𝐶𝑃) where 𝑥₀∈𝐸^𝑛 and 𝑓 is a continuous. Since we are only interested in local solutions to (𝐶𝑃), we assume 𝑓∶𝐽×𝐵→𝐸^𝑛 and 𝐷(𝑓,0)≤𝑀 on 𝐽×𝐵 for some 𝑀, where 𝐽=[0,𝑎] and 𝐵=𝐵(𝑥₀,𝑟)={𝑥∈𝐸^𝑛∶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: 𝑥^′=𝑥^′𝑦^′=𝑓₁𝑓𝑡,𝑥,𝑦₂(),()𝑡,𝑥,𝑦=𝑓𝑡,𝑧𝐺𝐶𝑃 where 𝑧=^𝑥_𝑦, 𝑓∈𝐶[[𝑡₀,+∞)×𝐸^𝑛,𝐸^𝑛], 𝑓=^𝑓1_𝑓2, and   𝐸^𝑛₁ and 𝐸^𝑛₂ are the fuzzy number space, respectively. 𝑓_𝑖∈𝐶[[𝑡₀,+∞)×𝐸^𝑛,𝐸^𝑛_𝑖] (𝑖=1,2), 𝑓₁(𝑡,𝑥,𝑦) satisfies the compactness-type, and 𝑓₂(𝑡,𝑥,𝑦) satisfies the dissipative-type conditions. In particular, when 𝐸^𝑛=𝐸^𝑛₁ and 𝑓₂=0, we obtain that Corollary 24 is the promotion of the results of [13]. When 𝐸^𝑛=𝐸^𝑛₂ 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 (𝐸^𝑛,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

Let 𝑃_𝑘(𝑅^𝑛) denote the family of all nonempty compact convex subsets of 𝑅^𝑛 and define the addition and scalar multiplication in 𝑃_𝑘(𝑅^𝑛) as usual. Let 𝐴 and 𝐵 be two nonempty bounded subsets of 𝑅^𝑛. The distance between 𝐴 and 𝐵 is defined by the Hausdorff metric: 𝑑_𝐻(𝐴,𝐵)=maxsup_𝑎∈𝐴inf_𝑏∈𝐵‖𝑎−𝑏‖,sup_𝑏∈𝐵inf_𝑎∈𝐴‖𝑏−𝑎‖.(1)

Denote 𝐸^𝑛={𝑢∶𝑅^𝑛→[0,1]∣𝑢 satisfies (1)–(4) below} is a fuzzy number space, where(1)𝑢is normal; that is, there exists an 𝑥₀∈𝑅^𝑛 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∈𝑅^𝑛 and 0≤𝜆≤1;(3)𝑢 is upper semicontinuous;(4)[𝑢]⁰=cl{𝑥∈𝑅^𝑛∣𝑢(𝑥)>0} is compact.

For 0<𝛼≤1, denote [𝑢]^𝛼={𝑥∈𝑅^𝑛∣𝑢(𝑥)≥𝛼}. Then from above (1)–(4), it follows that the 𝛼-level set [𝑢]^𝛼∈𝑃_𝑘(𝑅^𝑛) for all 0≤𝛼<1. According to Zadeh’s extension principle, we have addition and scalar multiplication in fuzzy number space 𝐸^𝑛 as follows: [𝑢+𝑣]^𝛼=[𝑢]^𝛼+[𝑣]^𝛼,[𝑘𝑢]^𝛼=𝑘[𝑢]^𝛼,(2) where 𝑢,𝑣∈𝐸^𝑛 and 0≤𝛼≤1.

Define 𝐷∶𝐸^𝑛×𝐸^𝑛→[0,∞): 𝐷(𝑢,𝑣)=sup𝑑_𝐻[𝑢]^𝛼,[𝑣]^𝛼∶𝛼∈[0,1],(3) where 𝑑 is the Hausdorff metric defined in 𝑃_𝑘(𝑅^𝑛). Then it is easy to see that 𝐷 is a metric in 𝐸^𝑛. Using the results in [14, 15], we know that (𝐸^𝑛,𝐷) is a complete metric space. It is well known that application of fuzzy set theory very often involves the metric space (𝐸^𝑛,𝐷) of normal fuzzy convex set over 𝑅^𝑛, 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]). Let 𝑆^𝑛−1 be the unit sphere of 𝑅^𝑛; that is, 𝑆^𝑛−1={𝑥∈𝑅∣‖𝑥‖=1}, ⟨⋅,⋅⟩ is the inner product in 𝑅^𝑛; that is, ⟨𝑥,𝑦⟩=∑^𝑛_𝑖=1𝑥_𝑖𝑦_𝑖, where 𝑥=⟨𝑥₁,𝑥₂,…,𝑥_𝑛⟩, 𝑦=⟨𝑦₁,𝑦₂,…,𝑦_𝑛⟩∈𝑅^𝑛. Suppose 𝑢∈𝐸^𝑛, 𝑥∈𝑆^𝑛−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×𝑆^𝑛−1.

The properties of the support functions can be referred to in [17] for details.

Theorem 2 (see [16, 17]). Let 𝑆^𝑛−1 be the unit sphere of 𝑅^𝑛 and ⟨⋅,⋅⟩ the inner product in 𝑅^𝑛. Suppose 𝑢∈𝐸^𝑛, 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 on 𝑆^𝑛−1, for each fixed 𝑟∈[𝑢]^𝑟;(2)𝑢^∗(𝑟,𝑥) is nonincreasing and left continuous in 𝑟∈(0,1] and right continuous at 𝑟=0, for each fixed 𝑥∈𝑆^𝑛−1;(3)𝑢^∗(𝑟,𝑥) is Lipschitz continuous in 𝑥 and 𝑢^∗(𝑟,𝑥)−𝑢^∗𝑟,𝑦≤sup_{[𝑢]𝑎∈𝑟}‖𝑎‖𝑥−𝑦;(4)(4)if 𝑢,𝑣∈𝐸^𝑛, 𝑟∈[0,1], then 𝑑([𝑢]^𝑟,[𝑣]^𝑟)=sup_𝑥∈𝑆^𝑛−1|𝑢^∗(𝑟,𝑥)−𝑣^∗(𝑟,𝑥)|.

Theorem 3 (see [16, 17]). Define an operator 𝑗 on 𝐸^𝑛 as 𝑗(𝑢)(𝑟,𝑥)=𝑢^∗(𝑟,𝑥) for any 𝑢∈𝐸^𝑛; then one has(1)𝑗(𝑢)∈𝐶(𝐼,𝐶(𝑆^𝑛−1)),(2)𝑗(𝑠𝑢+𝑡𝑣)=𝑠𝑗(𝑢)+𝑡𝑗(𝑣), 𝑠,𝑡≥0,(3)‖𝑗(𝑢)−𝑗(𝑣)‖=𝐷(𝑢,𝑣),(4)𝑗(𝐸^𝑛) is closed in 𝐶(𝐼,𝐶(𝑆^𝑛−1)).

In the sequel we will recall some integrability and differentiability properties in [18–20] for fuzzy set-valued mappings.

Let 𝑇=[𝑡₀,𝑡₀+𝑝]⊂𝑅(𝑝>0) be compact interval. The fuzzy mapping 𝐹∶𝑇→𝐸^𝑛 is called strong measurable if for all 𝛼∈[0,1] the set-valued mapping 𝐹_𝛼∶𝑇→𝑃_𝑘(𝑅^𝑛) defined by 𝐹_𝛼(𝑡)=[𝐹(𝑡)]^𝛼 is Lebesgue measurable, where 𝑃_𝑘(𝑅^𝑛) is endowed with the topology generated by the Hausdorff metric 𝑑. A mapping 𝐹_𝛼∶𝑇→𝐸^𝑛 is called integrable bounded if there exists an integrable function ℎ such that ‖𝑥‖≤ℎ(𝑡) for all 𝑥∈𝐹₀(𝑡).

Definition 4. Let 𝐹∶𝑇→𝐸^𝑛. The integral of 𝐹 over 𝑇, denoted by ∫_𝑇𝐹(𝑡)𝑑𝑡, is defined level-wise by the equation (see [18]): _𝑇𝐹(𝑡)𝑑𝑡^𝛼=_𝑇𝐹_𝛼(𝑡)=𝑑𝑡_𝑇𝑓(𝑡)𝑑𝑡∣𝑓∶𝑇→𝑅^𝑛isameasurableselectionfor𝐹_𝛼.(5)

A strongly measurable and integrable mapping 𝐹_𝛼∶𝑇→𝐸^𝑛 is said to be integrable over 𝑇 if ∫_𝑇𝐹(𝑡)𝑑𝑡∈𝐸^𝑛. From [20], we know that if 𝐹_𝛼∶𝑇→𝐸^𝑛 is continuous, then it is integrable.

Let 𝑥,𝑦∈𝐸^𝑛. If there exists a fuzzy number 𝑧∈𝐸^𝑛 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 mapping 𝐹∶𝑇→𝐸^𝑛 is differentiable at 𝑡₀∈𝑇 if there exists a 𝐹^′(𝑡₀)∈𝐸^𝑛, such that the limits lim_ℎ→0+𝐹𝑡₀+ℎ−_𝐻𝐹𝑡₀ℎ,lim_ℎ→0+𝐹𝑡₀−_𝐻𝐹𝑡₀+ℎℎ(6) exist and are equal to 𝐹^′(𝑡₀).

Here the limits are taken in the metric space (𝐸^𝑛,𝐷). At the endpoint of 𝑇, we consider only one-side fuzzy derivatives. If 𝐹∶𝑇→𝐸^𝑛 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 functions 𝐻-differentiable is very restrictive. For example, the fuzzy-number-valued function 𝑓∶[𝑎,𝑏]→𝑅_ℱ 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]). Let 𝑓∶(𝑎,𝑏)→𝐸^𝑛 and 𝑥₀∈(𝑎,𝑏). One says that 𝑓 is differentiable at 𝑥₀ if there exists an element 𝑓^′(𝑡₀)∈𝐸^𝑛, such that,(1)for all ℎ>0 sufficiently small, there exist 𝑓(𝑥₀+ℎ)−_𝐻𝑓(𝑥₀), 𝑓(𝑥₀)−_𝐻𝑓(𝑥₀−ℎ) and the limits (in the metric 𝐷) lim_ℎ→0𝑓𝑥₀+ℎ−_𝐻𝑓𝑥₀ℎ=lim_ℎ→0𝑓𝑥₀−_𝐻𝑓𝑥₀−ℎℎ=𝑓^′𝑥₀(7)(2)for all ℎ>0 sufficiently small, there exist 𝑓(𝑥₀)−_𝐻𝑓(𝑥₀+ℎ), 𝑓(𝑥₀−ℎ)−_𝐻𝑓(𝑥₀) and the limits lim_ℎ→0𝑓𝑥₀−_𝐻𝑓𝑥₀+ℎ−ℎ=lim_ℎ→0𝑓𝑥₀−ℎ−_𝐻𝑓𝑥₀−ℎ=𝑓^′𝑥₀(8)(3)for all ℎ>0 sufficiently small, there exist 𝑓(𝑥₀+ℎ)−_𝐻𝑓(𝑥₀), 𝑓(𝑥₀−ℎ)−_𝐻𝑓(𝑥₀) and the limits lim_ℎ→0𝑓𝑥₀+ℎ−_𝐻𝑓𝑥₀ℎ=lim_ℎ→0𝑓𝑥₀−ℎ−_𝐻𝑓𝑥₀−ℎ=𝑓^′𝑥₀(9)(4)for all ℎ>0 sufficiently small, there exist 𝑓(𝑥₀)−_𝐻𝑓(𝑥₀+ℎ), 𝑓(𝑥₀)−_𝐻𝑓(𝑥₀−ℎ) and the limits lim_ℎ→0𝑓𝑥₀−_𝐻𝑓𝑥₀+ℎ−ℎ=lim_ℎ→0𝑓𝑥₀−_𝐻𝑓𝑥₀−ℎℎ=𝑓^′𝑥₀(10) (ℎ and −ℎ at denominators mean 1/ℎ and −1/ℎ, resp.).

In addition, we define a continuous fuzzy-valued function 𝑓∶𝑇×Ω→𝐸^𝑛 by 𝑓∈𝐶[𝑇×Ω,𝐸^𝑛], where Ω⊂𝐸^𝑛 is an open set.

Theorem 7 (see [19]). Assume that 𝑓∈𝐶[𝑇×Ω,𝐸^𝑛]. 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 𝑥(𝑡)=𝑥₀+^𝑡_𝑡0𝑓(𝑠,𝑥(𝑠))𝑑𝑠,(11) or 𝑥(𝑡)=𝑥₀+(−1)⋅^𝑡_𝑡0𝑓(𝑠,𝑥(𝑠))𝑑𝑠,(12) for all 𝑡∈𝑇.

Theorem 8 (see [22]). Let 𝐺⊂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 𝐷𝑚(𝑡)≤𝑔(𝑡,𝑚(𝑡)),∀𝑡∈𝑡₀,𝑡₀+𝑎⧵Γ,(13) 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 set 𝐻⊂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

Let 𝛾 be Kuratowski’s measure of noncompactness (see [22] for details).

Lemma 10. Let 𝑆, 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)=0 if and only if 𝑆 is a relatively compactness set,(2)𝑆⊂𝑇⇒𝛾(𝑆)≤𝛾(𝑇),(3)𝛾(𝑡𝑆)=|𝑡|𝛾(𝑆), where 𝑡𝑆={𝑡𝑥∶𝑥∈𝑆},(4)𝛾(𝑆+𝑇)≤𝛾(𝑆)+𝛾(𝑇), where 𝑆+𝑇={𝑥+𝑦∶𝑥∈𝑆,𝑦∈𝑇},(5)𝛾(𝑆×𝑇)=max{𝛾(𝑆),𝛾(𝑇)}, where 𝑆×𝑇 is the bounded subsets of Banach space 𝐸₁×𝐸₂.

It is well known that continuity of 𝑓 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. Let 𝐽⊂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 function 𝜔∶(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. Let 𝑥_𝑛 be a sequence of continuous fuzzy-number-valued functions from 𝐽=[𝑎,𝑏] to 𝐸^𝑛 such that there is some function 𝜇∈𝐿¹(𝑎,𝑏) with 𝐷(𝑥_𝑛(𝑡),0)≤𝜇(𝑡) on 𝐽. Let 𝜙(𝑡)=𝛾(𝑗∘{𝑥_𝑛(𝑡)∶𝑛∈𝑁}). Then 𝜙(𝑡) is integrable on 𝐽 and 𝛾𝑗∘^𝑏_𝑎𝑥_𝑛(𝜏)𝑑𝜏∶𝑛∈𝑁≤^𝑏_𝑎𝜙(𝜏)𝑑𝜏.(14)

Proof. Fix 𝑛,𝑚∈𝑁; we have 𝑑_𝐻^𝑘−1_𝑖=0𝑥_𝑛𝜉_𝑖𝜏_𝑖+1−𝜏_𝑖,𝐸^𝑛≤^𝑘−1_𝑖=0𝑑_𝐻𝑥_𝑛𝜉_𝑖,𝐸^𝑛𝜏_𝑖+1−𝜏_𝑖(15) for every partition (𝜏₀,𝜏₁,…,𝜏_𝑘) and every (𝜉₁,𝜉₂,…,𝜉_𝑘) with 𝜏_𝑖≤𝜉_𝑖≤𝜏_𝑖+1 for 𝑖=0,1,2,…,𝑘−1. With the continuity of 𝑥_𝑛, we have 𝑑_𝐻^𝑏_𝑎𝑥_𝑛(𝜏)𝑑𝜏,𝐸^𝑛≤^𝑏_𝑎𝑑_𝐻𝑥_𝑛(𝜏),𝐸^𝑛𝑑𝜏.(16) An application of Fatou’s Lemma gives lim_𝑛→∞𝑑_𝐻^𝑏_𝑎𝑥_𝑛(𝜏)𝑑𝜏,𝐸^𝑛≤^𝑏_𝑎lim_𝑛→∞𝑑_𝐻𝑥_𝑛(𝜏),𝐸^𝑛𝑑𝜏(17) and with Lebesgue’s dominated convergence theorem; since lim_𝑛→∞𝑑_𝐻(𝑥_𝑛(𝑡),𝐸^𝑛)≤𝜇(𝑡) for 𝑡∈𝐽, 𝑚∈𝑁, we have lim_𝑚→∞lim_𝑛→∞𝑑_𝐻^𝑏_𝑎𝑥_𝑛(𝜏)𝑑𝜏,𝐸^𝑛≤^𝑏_𝑎lim_𝑚→∞lim_𝑛→∞𝑑_𝐻𝑥_𝑛(𝜏),𝐸^𝑛𝑑𝜏.(18) Therefore, we get 𝛾𝑗∘^𝑏_𝑎𝑥_𝑛(𝜏)𝑑𝜏∶𝑛∈𝑁≤^𝑏_𝑎𝜙(𝜏)𝑑𝜏.(19)

Theorem 13. Let 𝑥_𝑛(𝑡)∶[𝑎,𝑏]→𝐸^𝑛 be a sequence of continuously differentiable functions. Assume that there exists 𝑚(𝑡)∈𝐿¹[𝑎,𝑏] such that 𝑗∘𝑥_𝑛(𝑡)≤𝑚(𝑡),𝑗∘𝑥^′_𝑛(𝑡)≤𝑚(𝑡)(20) a.e. 𝑡∈[𝑎,𝑏], 𝑛=1,2,…. Let ℎ(𝑡)=𝛾(𝑗∘{𝑥_𝑛(𝑡)∶𝑛∈𝑁}); then ℎ(𝑡) is an absolutely continuous function and ℎ^′(𝑡)≤𝛾(𝑗∘{𝑥_𝑛(𝑡)∶𝑛∈𝑁}) a.e. 𝑡∈[𝑎,𝑏].

Proof. The absolute continuity of ℎ follows from ‖ℎ(𝑡)−ℎ(𝑠)‖≤𝛾𝑗∘𝑥_𝑛(𝑡)−𝑗∘𝑥_𝑛(𝑠)=ℎ𝑗∘^𝑡_𝑠𝑥^′_𝑛(𝜏)𝑑𝜏≤2⋅sup_𝑛∈𝑁𝑗∘^𝑡_𝑠𝑥^′_𝑛(𝜏)𝑑𝜏≤2⋅^𝑡_𝑠𝑚(𝜏)𝑑𝜏.(21) We conclude by Theorem 12 that 𝛾𝑗∘𝑥_𝑛(𝑡)−𝛾𝑗∘𝑥_𝑛(𝑡−𝑙)≤𝛾𝑗∘^𝑡_𝑡−𝑙𝑥^′_𝑛(𝜏)≤𝑑𝜏_{𝑡𝑡−𝑙}𝛾𝑗∘𝑥^′_𝑛(𝜏)𝑑𝜏.(22) Dividing by 𝑙>0 and letting 𝑙→0⁺ give ℎ^′(𝑡)≤𝛾𝑗∘𝑥_𝑛(𝑡)∶𝑛∈𝑁.(23)

Theorem 14. Let 𝐵=𝐵(𝑥₀,𝑟)={𝑥∈𝐸^𝑛∶𝐷(𝑥,𝑥₀)≤𝑟} and 𝑓∶[0,𝑎]×𝐵→𝐸^𝑛 continuous with 𝐷(𝑓(𝑡,𝑥),0)≤𝜇(𝑡) on [0,𝑎]×𝐵 for some 𝜇∈𝐿¹[0,𝑎]. Let 𝑏<𝑎 such that ∫^𝑎₀𝜇(𝜏)𝑑𝜏<𝑟 and 𝐽=[0,𝑏]. One assumes 𝛾𝑗∘𝑓(𝑡,𝐵)≤𝜔𝑡,𝛾𝑗∘𝐵^′𝑓𝑜𝑟𝑡∈[0,𝑎],𝐵^′⊂𝐵,(24) where 𝜔∈𝑈. Then (𝐶𝑃) has a solution.

Proof. There is a sequence {𝑥_𝑛(𝑡)} of approximate solutions to (𝐶𝑃) satisfying 𝑥^′_𝑛(𝑡)=𝑓𝑡,𝑥_𝑛(𝑡)+𝑦_𝑛(𝑡),𝑥_𝑛(𝑡)=𝑥₀,𝑦_𝑛(𝑡)≤1𝑛.(25) Let ℎ(𝑡)=𝛾(𝑗∘{𝑥_𝑛(𝑡)∶𝑛∈𝑁}). We have by Theorem 13 that ℎ^′(𝑡)≤𝛾(𝑗∘{𝑥_𝑛(𝑡)∶𝑛∈𝑁}) a.e. on 𝑡∈[𝑎,𝑏]. Moreover, by the properties of 𝛾, we have 𝛾𝑗∘𝑥^′_𝑛(𝑡𝑓)≤𝛾𝑗∘𝑡,𝑥_𝑛(𝑡)+𝛾𝑗∘𝑦_𝑛(𝑡𝑥)≤𝜔𝑡,𝛾𝑗∘_𝑛(𝑡).(26) We claim that ℎ(𝑡) is 𝑜(𝑡) for 𝑡→0⁺. By the continuity of 𝑓 and the equicontinuity of the 𝑥_𝑛(𝑡), we have 𝑥_𝑛(𝑡)=𝑥₀+^𝑡₀𝑓𝜏,𝑥_𝑛(𝜏)−𝑓0,𝑥₀𝑑𝜏+𝑡𝑓0,𝑥₀+^𝑡₀𝑦_𝑛(𝜏)𝑑𝜏=𝑥₀+𝑡𝑓0,𝑥₀+𝑧_𝑛(𝑡)(27) with 𝑧_𝑛(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 𝑥_𝑛(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 𝑥^′=𝑓(𝑡,𝑥),𝑥(∞)=𝑥₀,(𝐶𝑃)^′ where 𝑓∶[𝑎,∞)×𝐸^𝑛→𝐸^𝑛 is continuous, 𝑎>0, and 𝑥₀∈𝐸^𝑛. This problem is equivalent to the following initial value problem: 𝑢^′=−𝑡⁻²𝑓𝑡⁻¹,𝑢,𝑢(0)=𝑥₀,(28) by means of the transformation 𝑡→𝑡⁻¹ on (0,𝑎⁻¹].

Definition 15 (see [23]). A function 𝜔∶(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. Let 𝐸^𝑛 be a fuzzy number space. 𝐵(𝑥₀,𝑟)={𝑥∈𝐸^𝑛∶𝐷(𝑥,𝑥₀)≤𝑟} and 𝑓∶[𝑎,∞)×𝐵→𝐸^𝑛 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)) for 𝑡≥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: 𝑢^′=−𝑡⁻²𝑓𝑡⁻¹,𝑢,𝑢𝑛⁻¹=𝑥₀.(29) By Theorem 14 and condition (1), the problem (29) has a solution 𝑢_𝑛. Let 𝑢_𝑛=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]): [𝑢,𝑣]₊=lim_ℎ→01ℎ𝑗∘𝑢+ℎ𝑗∘𝑣−𝑗∘𝑢,[𝑢,𝑣]₋=12[𝑢,𝑣]₊−[𝑢,−𝑣]₊(30) for 𝑢,𝑣∈𝐸^𝑛, where 𝑗 is the embedding operator in Theorem 3.

Theorem 17. For all 𝑢,𝑣,𝑤∈𝐸^𝑛, 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)if 𝐼 is an open interval in 𝑅 and 𝑥∶𝐼→𝐸^𝑛 is a differential function on 𝑡∈𝐼, then 𝑗∘𝑥(𝑡)⋅𝐷⁻𝑗∘𝑥(𝑡)≤𝑥^′(𝑡),𝑥(𝑡)₋.(31)

Proof. (1) and (2) are easy consequences of the definition and (3) follows from Lemma 3.6 in [11].

Definition 18. Assume that 𝜑∶[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 lim_{𝑡→𝑡∗+0}𝑢(𝑡)=lim_{𝑡→𝑡∗+0}𝑢(𝑡)𝜑(𝑡)=0.(32) Then 𝜔∶(𝑡^∗,∞)×𝑅⁺→𝑅⁺ is called a function of 𝑈₁ kind.

Definition 19. Assume that 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 lim_{𝑡→𝑡∗+0}𝑢(𝑡)=0(33) 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₊×𝐸^𝑛, exists on [𝑡₀,∞) and tends to a limit 𝑣∈𝐸^𝑛 as 𝑡→∞, and, conversely, to every given vector 𝑣∈𝐸^𝑛 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 in 𝑥 for (𝑡,𝑥)∈𝐽×𝐸^𝑛,(2)𝐷(𝑓(𝑡,𝑥),0)≤𝑔(𝑡,𝐷(𝑥,0)),  for all (𝑡,𝑥)∈𝐽×𝐸^𝑛,(3)𝑔∈𝐶[𝐽×[0,∞),[0,∞)], 𝑔(𝑡,𝑢) is nondecreasing in 𝑢≥0 for each 𝑡∈𝐽, and maximal solution 𝑟(𝑡,𝑡₀,𝑢₀) of the scalar initial value problem 𝑢^′=𝑔(𝑡,𝑢), 𝑢(𝑡₀)=𝑢₀ exists throughout 𝐽.
Then the largest interval of any solution 𝑟(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 (𝐸^𝑛,𝐷).

Lemma 22 (see [13]). Under the assumptions of Lemma 21, given 𝑣∈𝐸^𝑛, there exist a 𝑇∈[𝑡₀,∞) and a sequence {𝑥_𝑛(𝑡)}^∞_𝑛=1 defined on [𝑇,∞), such that(1){𝑥_𝑛(𝑡)}^∞_𝑛=1 is equicontinuous on [𝑇,∞);(2){𝑥_𝑛(𝑡)}^∞_𝑛=1 is uniformly bounded on [𝑇,∞); that is, there exists 𝑀∈𝑅₊, such that 𝐷(𝑥_𝑛(𝑡),0)≤𝑀 for all 𝑡∈[𝑇,∞) and for all 𝑛;(3)for each 𝑛,𝑥_𝑛(𝑡) is a solution of 𝑥^′=𝑓(𝑡,𝑥), 𝑥(𝑇+𝑛)=𝑣.

In the following, we give the main result of this paper.

Theorem 23. Under the assumptions of Lemma 21, 𝑧_𝑛(𝑡)={𝑥_𝑛(𝑡),𝑦_𝑛(𝑡)} is satisfied Lemma 22, and there exists 𝑡^∗∈[𝑇₀,∞) such that {𝑧_𝑛(𝑡^∗)}^∞_𝑛=1 is relatively compact in 𝐸^𝑛 and satisfies the following.(1)Let 𝐴⊂𝐸^𝑛 is a bounded set; one has 𝛾(𝑗∘𝑓₁(𝑡,𝐴))≤𝑔₁(𝑡,𝛾(𝑗∘𝐴^′)) for all 𝑡∈[𝑡^′,∞), where 𝐴^′={𝑥∶ there exists 𝑦∈𝐸^𝑛₂ such that (𝑥,𝑦)∈𝐴} and 𝑔₁∈𝑈₂1.(2)For all 𝜀>0, 𝑡∈𝑅⁺, and 𝑟>0 there exists 𝛿=𝛿(𝑡,𝑟)>0 such that 𝐷(𝑓₂(𝑡,𝑥₁,𝑦),𝑓₂(𝑡,𝑥₂,𝑦))<𝜀 with respect to 𝐷(𝑦,0)≤𝑟 when 𝐷(𝑥₁,𝑥₂)<𝛿; we have 𝑗∘𝑓₂(𝑡,𝑥,𝑦)=ℎ(𝑡)+𝑜[𝜑^′(𝑡)] when (𝑡,𝑥,𝑦)→(𝑡^∗,𝑥₀,𝑦₀), where ℎ(𝑡) is a continuous function and 𝜑(𝑡)∈𝑈₂.(3)For (𝑡,𝑥,𝑦₁), (𝑡,𝑥,𝑦₂)∈[𝑡^∗,∞)×𝐸^𝑛₁×𝐸^𝑛₂, such that⟨𝑓₂𝑡,𝑥,𝑦₁−𝑓𝑡,𝑥,𝑦₂,𝑦₁−𝑦₂⟩₋≤𝜔𝑡,𝐷𝑦₁,𝑦₂⋅𝐷𝑦₁,𝑦₂,(34)where 𝜔∈𝑈₁ and (𝑥₀,𝑦₀) is the limit point of {𝑥_𝑛(𝑡^∗),𝑦_𝑛(𝑡^∗)}.
Then the fuzzy differential system (𝐺𝐶𝑃) has asymptotic equilibrium.

Proof. Since 𝑧_𝑛(𝑡)=(𝑥_𝑛(𝑡),𝑦_𝑛(𝑡)) is satisfied, 𝑥^′_𝑛=𝑓₁(𝑡,𝑥_𝑛(𝑡),𝑦_𝑛(𝑡)) and 𝑦^′_𝑛=𝑓₂(𝑡,𝑥_𝑛(𝑡),𝑦_𝑛(𝑡)), according to Lemma 10 and Theorem 13, and the following fact: {𝑥_𝑛(𝑡)∶𝑛∈𝑁,𝑡∈[𝑡^∗,∞)} can produce a subspace of 𝐸^𝑛₁; we let 𝑙(𝑡)=𝛾(𝑗∘{𝑥_𝑛(𝑡)}); then we have 𝑙^′(𝑡)≤𝛾𝑗∘𝑥^′_𝑛(𝑡)𝑓∶𝑛∈𝑁=𝛾𝑗∘₁𝑡,𝑥_𝑛(𝑡),𝑦_𝑛(𝑡)^∞_𝑛=1≤𝑔₁(𝑡,𝑙(𝑡)).(35) From the assumption and 𝑙(𝑡^∗)=0, we obtain 𝑙(𝑡)=0 on [𝑡^∗,∞). Since {𝑥_𝑛(𝑡)}^∞_𝑛=1 is equicontinuous and uniformly bounded, from Ascoli-Arzelar’s theorem, then there exists a subsequence of {𝑥_𝑛(𝑡)}^∞_𝑛=1 which uniformly converges to 𝑥(𝑡) on any finite closed subset of [𝑡^∗,∞). Without any loss of generality, let {𝑥_𝑛(𝑡)}^∞_𝑛=1 uniformly converge to 𝑥(𝑡) on any finite closed subset of [𝑡^∗,∞) and let  {𝑧_𝑛(𝑡^∗)}^∞_𝑛=1 converge too.
Now, we consider the initial problem 𝑦^′(𝑡)=𝑓₂(𝑡),𝑦𝑡𝑡,𝑥,𝑦_∗=𝑦₀.(36)
Next, we will prove that {𝑦_𝑛(𝑡)}^∞_𝑛=1 converges uniformly to 𝑦(𝑡) on any finite closed subset of [𝑡^∗,∞). Let 𝑎_𝑛𝑚=𝐷(𝑦_𝑛(𝑡^∗),𝑦_𝑚(𝑡^∗))=‖𝑗∘𝑦_𝑛(𝑡^∗)−𝑗∘𝑦_𝑚(𝑡^∗)‖; then lim_{𝑛,𝑚→∞}𝑎_𝑛𝑚=0. And let Δ_𝑛𝑚(𝑡)=𝐷(𝑦_𝑛(𝑡),𝑦_𝑚(𝑡))=‖𝑗∘𝑦_𝑛(𝑡)−𝑗∘𝑦_𝑚(𝑡)‖, according to Theorem 17; we have Δ_𝑛𝑚𝐷⁻Δ_𝑛𝑚≤𝑦^′_𝑛(𝑡)−𝑦^′_𝑚(𝑡),𝑦_𝑛(𝑡)−𝑦_𝑚(𝑡)₋=𝑓₂𝑡,𝑥_𝑛(𝑡),𝑦_𝑛(𝑡)−𝑓₂𝑡,𝑥_𝑚(𝑡),𝑦_𝑚(𝑡),𝑦_𝑛(𝑡)−𝑦_𝑚(𝑡)₋≤𝑓₂𝑡,𝑥(𝑡),𝑦_𝑛(𝑡)−𝑓₂𝑡,𝑥(𝑡),𝑦_𝑚(𝑡),𝑦_𝑛(𝑡)−𝑦_𝑚(𝑡)₋+𝑗∘𝑓₂𝑡,𝑥_𝑛(𝑡),𝑦_𝑚(𝑡)−𝑗∘𝑓₂𝑡,𝑥(𝑡),𝑦_𝑛(𝑡)+𝑗∘𝑓₂𝑡,𝑥_𝑚(𝑡),𝑦_𝑚(𝑡)−𝑗∘𝑓₂𝑡,𝑥(𝑡),𝑦_𝑚(𝑡)Δ_𝑛𝑚.(37) Let {𝑥_𝑛(𝑡)} converge uniformly to 𝑥(𝑡) on [𝑡^∗,𝑎]  (𝑎>𝑡^∗) and combine to condition (2); we have Δ_𝑛𝑚𝐷⁻Δ_𝑛𝑚≤𝑓₂𝑡,𝑥(𝑡),𝑦_𝑛(𝑡)−𝑓₂𝑡,𝑥(𝑡),𝑦_𝑚(𝑡),𝑦_𝑛(𝑡)−𝑦_𝑚(𝑡)₋+𝜀_𝑛𝑚Δ_𝑛𝑚,(38) where lim_{𝑛,𝑚→∞}=0,  𝑡∈[𝑡^∗,𝑎]. Then we have 𝐷⁻Δ_𝑛𝑚≤𝜔𝑡,Δ_𝑛𝑚(𝑡)+𝜀_𝑛𝑚𝑡∈𝑡^∗,𝑎(39) and Δ_𝑛𝑚(𝑡^∗)→0 when 𝑛,𝑚→∞. Let Δ_𝑛=sup_𝑚>𝑛Δ_𝑛𝑚; then lim_𝑛→∞Δ_𝑛(𝑡^∗)=0. According to the definition of Δ_𝑛, we easily know that {Δ_𝑛} satisfies the following: there exists 𝑀>0 such that |Δ_𝑛−Δ_𝑠|≤2𝑀|𝑠−𝑡| and 𝐷⁻Δ_𝑛(𝑡)≤𝜔(𝑡,Δ_𝑛(𝑡))+𝜀_𝑛(𝑠,𝑡∈[𝑡^∗,𝑎]),lim_𝑛→∞𝜀_𝑛=0. From above, we get that {Δ_𝑛(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 𝐷⁻Δ_𝑛𝑚(𝑡)≤𝜀𝜑^′(𝑡),(40) when 𝑛,𝑚>𝑁,  𝑡∈[𝑡^∗,𝑡^′], which means that 0≤Δ_𝑛𝑚(𝑡)≤𝜀⋅𝜑(𝑡),𝑡∈𝑡^∗,𝑡^′,(41) implies lim_{𝑡→𝑡∗}Δ(𝑡)𝜑(𝑡)=0.(42) Because 𝜔∈𝑈₁, we have Δ(𝑡)≡0. Hence {Δ_𝑛(𝑡)} converge uniformly to Δ(𝑡); that is, {Δ_𝑛(𝑡)} converge uniformly to 0. From above, we can get that {𝑦_𝑛(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→∞(𝑥_𝑛(𝑡),𝑦_𝑛(𝑡)),𝑡∈[𝑡^∗,∞) and 𝑥_𝑛(𝑡)=𝑥_𝑛𝑡^∗+^𝑡_𝑡∗𝑓₁𝑠,𝑥_𝑛(𝑠),𝑦_𝑛(𝑠)𝑦𝑑𝑠,_𝑛(𝑡)=𝑦_𝑛𝑡^∗+^𝑡_𝑡∗𝑓₂𝑠,𝑥_𝑛(𝑠),𝑦_𝑛(𝑠)𝑑𝑠.(43) Since 𝑧_𝑛(𝑡)=(𝑥_𝑛(𝑡),𝑦_𝑛(𝑡)) converge uniformly on a compact set of [𝑡^∗,∞), we have 𝑥(𝑡)=𝑥𝑡^∗+^𝑡_𝑡∗𝑓₁(𝑠)(𝑠)𝑦(𝑡)𝑡𝑠,𝑥,𝑦𝑑𝑠,=𝑦^∗+^𝑡_𝑡∗𝑓₂(𝑠,𝑥(𝑠),(𝑠))𝑑𝑠(44) 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 show lim_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 𝜆=𝑢𝑡₀≤𝑢(𝑡)≤𝑢(∞)<∞.(45) Since 𝑔(𝑡,𝑢) is nondecreasing in 𝑢 for every 𝑡∈[𝑡₀,∞), we have ∞>𝑢(∞)−𝑢𝑡₀=^∞_𝑡0𝑔(𝑠,𝑢(𝑠))𝑑𝑠≥^∞_𝑡0𝑔(𝑠,𝜆)𝑑𝑠.(46) So, there exists an integer 𝑆>𝑡^∗ which satisfies ^∞_𝑠𝑔()𝜀𝑡,𝑀𝑑𝑡≤2,(47) where 𝑀=sup_𝑛∈𝑁{𝐷(𝑧_𝑛(𝑠),0),𝑠∈[𝑡^∗,∞)}<∞. In addition, there exists 𝑁; when 𝑛>𝑁, we have 𝐷𝑧_𝑛𝑡^′,𝑧𝑡^′=𝑗∘𝑧_𝑛𝑡^′−𝑗∘𝑧𝑡^′<𝜀2.(48) Hence, 𝑗∘𝑧𝑡^′=𝑡−𝑗∘𝑣𝑗∘𝑧^′−𝑗∘𝑧_𝑁+𝑆(≤𝑡𝑇+𝑁+𝑆)𝑗∘𝑧^′−𝑗∘𝑧_𝑁+𝑆𝑡^′+𝑗∘𝑧_𝑁+𝑆𝑡^′−𝑗∘𝑧_𝑁+𝑆(≤𝜀𝑇+𝑁+𝑆)2+𝐷_𝑡^{𝑇+𝑁+𝑆}_′𝑓𝑡,𝑧_𝑁+𝑆(𝑡),0≤𝜀𝑑𝑡2+_𝑡^{𝑇+𝑁+𝑆}_′𝑔𝑡,𝐷𝑧_𝑁+𝑆,0≤𝜀𝑑𝑡2+^∞_𝑠𝑔(𝑡,𝑀)𝑑𝑡<𝜀.(49) Hence, we have lim_𝑡→∞𝑧(𝑡)=𝑣.

When 𝐸^𝑛=𝐸^𝑛₁ and 𝑓₂=0, we have the following.

Corollary 24. Under the assumptions of Lemma 21, {𝑥_𝑛(𝑡)}^∞_𝑛=1 is satisfied Lemma 22, and there exists 𝑡^∗∈[𝑇₀,∞) such that {𝑥_𝑛(𝑡^∗)}^∞_𝑛=1 is relatively compact in 𝐸^𝑛 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.

When 𝐸^𝑛=𝐸^𝑛₂ and 𝑓₁=0, we have the following.

Corollary 26. Under the assumptions of Lemma 21, {𝑥_𝑛(𝑡)}^∞_𝑛=1 is satisfied Lemma 22, and there exists 𝑡^∗∈[𝑇₀,∞) such that {𝑥_𝑛(𝑡^∗)}^∞_𝑛=1 is relatively compact in 𝐸^𝑛 and satisfies 𝑓(𝑡,𝑥)−𝑓𝑡,𝑦,𝑥−𝑦₋≤𝜔𝑡,𝐷𝑥,𝑦⋅𝐷𝑥,𝑦,(50) where 𝑡∈[𝑇,∞),  𝑥,𝑦∈𝐸^𝑛,  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 of 𝑓. 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.

Conflict of Interests

The authors declare that there is no conflict of interests regarding the publication of this paper.

Acknowledgments

The authors are very grateful to the anonymous referees and Professor Junjie Wei for many valuable comments and suggestions which helped in improving the presentation of the paper. The authors would like to thank National Natural Science Foundation of China (no. 11161041) and Fundamental Research Fund for the Central Universities (no. 31920130004 and no. zyz2012079).