The Strong Fuzzy Henstock Integrals and Discontinuous Fuzzy Differential Equations

Abstract

We generalized the existence theorems and the continuous dependence of a solution on parameters for initial problems of fuzzy discontinuous differential equation by the strong fuzzy Henstock integral and its controlled convergence theorem.

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 that the Cauchy problem has a uniqueness result if 𝑓 is continuous and bounded. In [1, 3, 7–9], the authors presented a uniqueness result when 𝑓 satisfies a Lipschitz condition. For a general reference to fuzzy differential equations, see a recent book by Lakshmikantham and Mohapatra [10] and references therein. In 2002, Xue and Fu [11] established the solutions to fuzzy differential equations with right-hand side functions satisfying Carathéodory conditions on a class of Lipschitz fuzzy sets. However, there are discontinuous systems in which the right-hand side functions 𝑓∶[𝑎,𝑏]×𝐸^𝑛→𝐸^𝑛 are not integrable in the sense of Kaleva [1] on certain intervals, and their solutions are not absolute continuous functions. To illustrate, we consider the following example.

Example 1. Consider the following discontinuous system: 𝑥^′(𝑡)(𝑡)(0)=𝑔(𝑡)=1=ℎ,𝑥𝐴,2𝑡sin𝑡²−2𝑡1cos𝑡²𝐴(𝑠)=ℎ(𝑡),𝑡≠0,0,𝑡=0,𝑠,0≤𝑠≤1,2−𝑠,1<𝑠≤2,0,others,=𝜒_|𝑔(𝑡)|+𝐴. (1) Then, ℎ(𝑡)=𝜒_|𝑔(𝑡)|+𝐴 is not integrable in the sense of Kaleva. However, the above system has the following solution: 𝑥(𝑡)=𝜒_|𝐺(𝑡)|+𝐴𝑡, (2) where 𝐺(𝑡)=𝑡²1sin𝑡²,𝑡≠0,0,𝑡=0. (3)
It is well known that the Henstock integral is designed to integrate highly oscillatory functions which the Lebesgue integral fails to do. It is known as nonabsolute integration and is a powerful tool. It is well known that the Henstock integral includes the Riemann, improper Riemann, Lebesgue, and Newton integrals [12, 13]. Though such an integral was defined by Denjoy in 1912 and also by Perron in 1914, it was difficult to handle using their definitions. But with the Riemann-type definition introduced more recently by Henstock [12] in 1963 and also independently by Kurzweil [13], the definition is now simple, and furthermore the proof involving the integral also turns out to be easy. For more detailed results about the Henstock integral, we refer to [14]. Recently, Wu and Gong [15, 16] have combined the fuzzy set theory and nonabsolute integration theory, and they discussed the fuzzy Henstock integrals of fuzzy-number-valued functions which extended Kaleva [1] integration. In order to complete the theory of fuzzy calculus and to meet the solving need of transferring a fuzzy differential equation into a fuzzy integral equation, Gong and Shao [17, 18] defined the strong fuzzy Henstock integrals and discussed some of their properties and the controlled convergence theorem.
In this paper, according to the idea of [19] and using the concept of generalized differentiability [20], we will prove other controlled convergence theorems for the strong fuzzy Henstock integrals, which will be of foundational significance for studying the existence and uniqueness of solutions to the fuzzy discontinuous systems. As we know, we inevitably use the controlled convergence theorems for solving the numerical solutions of differential equations. As the main outcomes, we will deal with the Cauchy problem of discontinuous fuzzy systems as follows: 𝑥^′(𝑡)=𝑓(),𝑥(𝜏)𝑡,𝑥=𝜉∈𝐸^𝑛, (4) where 𝑓∶𝑈→𝐸^𝑛 is a strong fuzzy Henstock integrable function and 𝑈=(𝑡,𝑥)∶|𝑡−𝜏|≤𝑎,𝑥∈𝐸^𝑛,𝐷(𝑥,𝜉)≤𝑏. (5)
To make our analysis possible, we will first recall some basic results of fuzzy numbers and give some definitions of absolutely continuous fuzzy-number-valued function. In addition, we present the concept of generalized differentiability. In Section 3, we present the concept of strong fuzzy Henstock integrals, and we prove a controlled convergence theorem for the strong fuzzy Henstock integrals. In Section 4, we deal with the Cauchy problem of discontinuous fuzzy systems. And in Section 5, we present some concluding remarks.

2. Preliminaries

Let 𝑃_𝑘(𝑅^𝑛) denote the family of all nonempty compact convex subset 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 [21] as follows: 𝑑_𝐻(𝐴,𝐵)=maxsup_𝑎∈𝐴inf_𝑏∈𝐵‖𝑎−𝑏‖,sup_𝑏∈𝐵inf_𝑎∈𝐴‖𝑏−𝑎‖. (6)

Denote that 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 that [𝑢]^𝛼={𝑥∈𝑅^𝑛∣𝑢(𝑥)≥𝛼}. Then from the 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 [21]: [𝑢+𝑣]^𝛼=[𝑢]^𝛼+[𝑣]^𝛼,[𝑘𝑢]^𝛼=𝑘[𝑢]^𝛼, (7) 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], (8) 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 [22], we know that(1)(𝐸^𝑛,𝐷) is a complete metric space,(2)𝐷(𝑢+𝑤,𝑣+𝑤)=𝐷(𝑢,𝑣) for all 𝑢,𝑣,𝑤∈𝐸^𝑛,(3)𝐷(𝜆𝑢,𝜆𝑣)=|𝜆|𝐷(𝑢,𝑣) for all 𝑢,𝑣,𝑤∈𝐸^𝑛 and 𝜆∈𝑅.

Let 𝑥,𝑦∈𝐸^𝑛. If there exist 𝑧∈𝐸^𝑛 such that 𝑥=𝑦+𝑧, then 𝑧 is called the 𝐻-difference of 𝑥 and 𝑦 and is denoted by 𝑥−_𝐻𝑦. As mentioned above which always is called the condition (𝐻). It is well known that the 𝐻-derivative for fuzzy-number-functions was initially introduced by Puri et al. [5, 23] and it is based on the condition (𝐻) of sets. 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 [20, 24]). 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 [20].

Definition 2 (see [20]). 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 exists 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 (9) or(2)for all ℎ>0 sufficiently small, there exists 𝑓(𝑥₀)−_𝐻𝑓(𝑥₀+ℎ),𝑓(𝑥₀−ℎ)−_𝐻𝑓(𝑥₀) and the limits lim_ℎ→0𝑓𝑥₀−_𝐻𝑓𝑥₀+ℎ−ℎ=lim_ℎ→0𝑓𝑥₀−ℎ−_𝐻𝑓𝑥₀=𝑓−ℎ_′𝑥₀ (10) or(3)for all ℎ>0 sufficiently small, there exists 𝑓(𝑥₀+ℎ)−_𝐻𝑓(𝑥₀),𝑓(𝑥₀−ℎ)−_𝐻𝑓(𝑥₀) and the limits lim_ℎ→0𝑓𝑥₀+ℎ−_𝐻𝑓𝑥₀ℎ=lim_ℎ→0𝑓𝑥₀−ℎ−_𝐻𝑓𝑥₀=𝑓−ℎ_′𝑥₀ (11) or(4)for all ℎ>0 sufficiently small, there exists 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 (12) (ℎ and −ℎ at denominators mean (1/ℎ)⋅ and −(1/ℎ)⋅, resp.).

3. The Convergence Theorem of Strong Fuzzy Henstock Integral

In this section, we define the strong Henstock integrals of fuzzy-number-valued functions in fuzzy number space 𝐸^𝑛, and we give some properties and controlled convergence theorem of this integral by using new conditions.

Definition 3 (see [18]). A fuzzy-number-valued function 𝑓 is said to be termed additive on [𝑎,𝑏] if, for any division 𝑇∶𝑎≤𝑥₁≤𝑥₂≤⋯≤𝑥_𝑛≤𝑏, one has 𝑓([𝑥_𝑖,𝑥_𝑗])(1≤𝑖<𝑗≤𝑛) that exists and 𝑓([𝑥_𝑖,𝑥_𝑗])=∑^𝑗−1_𝑘=𝑖𝑓([𝑥_𝑘,𝑥_𝑘+1]) or 𝑓([𝑥_𝑗,𝑥_𝑖])(1≤𝑖<𝑗≤𝑛) that exists and (−1)⋅𝑓([𝑥_𝑗,𝑥_𝑖])=(−1)⋅∑^𝑗−1_𝑘=𝑖𝑓([𝑥_𝑘+1,𝑥_𝑘]). For convenience, 𝑓([𝑠,𝑡]) denotes 𝑓(𝑡)−_𝐻𝑓(𝑠).

Definition 4 (see [17, 18]). A fuzzy-number-valued function 𝑓 is said to be strong Henstock integrable on [𝑎,𝑏] if there exists a piecewise additive fuzzy-number-valued function 𝐹 on [𝑎,𝑏] such that for every 𝜀>0 there is a function 𝛿(𝜉)>0 and for any 𝛿-fine division 𝑃={([𝑢,𝑣],𝜉)} of [𝑎,𝑏] one has _{𝑖∈𝐾𝑛}𝐷𝑓𝜉_𝑖𝑣_𝑖−𝑢_𝑖,𝐹𝑢_𝑖,𝑣_𝑖+_{𝑗∈𝐼𝑛}𝐷𝑓𝜉_𝑗𝑣_𝑗−𝑢_𝑗,(−1)⋅𝐹𝑢_𝑗,𝑣_𝑗−1<𝜀, (13) where 𝐾_𝑛={𝑖∈{1,2,…,𝑛} such that 𝐹([𝑥_𝑖−1,𝑥_𝑖]) is a fuzzy number and 𝐼_𝑛={𝑗∈{1,2,…,𝑛} such that 𝐹([𝑥_𝑗,𝑥_𝑗−1]) is a fuzzy number. One writes 𝑓∈SFH[𝑎,𝑏].

Definition 5. A fuzzy-number-valued function 𝐹 defined on 𝑋⊂[𝑎,𝑏] is said to be 𝐴𝐶^∗_𝛿(𝑋) if for every 𝜀>0 there exists 𝜂>0 and 𝛿(𝜉)>0 such that for any 𝛿-fine partial division 𝑃={([𝑢,𝑣],𝜉)} with 𝜉∈𝑋_𝑖 satisfying ∑^𝑛_𝑖=1|𝑣−𝑢|<𝜂 one has ∑𝐷(𝐹[𝑢,𝑣])<𝜀.

Definition 6. A fuzzy-number-valued function 𝐹 is said to be 𝐴𝐶𝐺^∗_𝛿 on 𝑋⊂[𝑎,𝑏] if 𝑋 is the union of a sequence of closed sets {𝑋_𝑖} such that on each 𝑋_𝑖, 𝐹 is 𝐴𝐶^∗_𝛿(𝑋_𝑖).

Definition 7. The sequence of fuzzy-number-function {𝐹_𝑛} is 𝑈𝐴𝐶𝐺^∗_𝛿 on 𝑋⊂[𝑎,𝑏] if 𝑋 is the sequence of subsets 𝑋_𝑖 such that {𝐹_𝑛} is 𝑈𝐴𝐶^∗_𝛿 for each 𝑖, independent of 𝑛.

Definition 8. Let {𝐹_𝑛} be a sequence of fuzzy-number-function defined on [𝑎,𝑏], and let 𝑥⊂[𝑎,𝑏] be measurable.(i)The sequence of fuzzy-number-function {𝐹_𝑛} is 𝒫-Cauchy on 𝐸^𝑛 if {𝐹_𝑛} converges pointwise on 𝑋 and if for each 𝜀>0 there exist 𝛿(𝜉)>0 on 𝑋 and a positive integer 𝑁 such that 𝐷(𝐹_𝑚(𝑃),𝐹_𝑛(𝑃))<𝜀 for all 𝑚,𝑛≥𝑁 whenever 𝑃 is 𝑋-subordinate to 𝛿(𝜉).(ii)The sequence of fuzzy-number-function {𝐹_𝑛} is generalized 𝒫-Cauchy on 𝑋 if 𝑋 can be written as a countable union of measurable sets on each of which {𝐹_𝑛} is 𝒫-Cauchy.

Theorem 9. Let the following conditions be satisfied:(i)𝑓_𝑛,𝑋(𝑥)→𝑓_𝑥 a.e. on [𝑎,𝑏] as 𝑛→∞ where each 𝑓_𝑛,𝑋 is strong Fuzzy Henstock integrable on [𝑎,𝑏];(ii)the primitives 𝐹_𝑛,𝑋 of 𝑓_𝑛,𝑋 are 𝑈𝐴𝐶^∗_𝛿 with closed set 𝑋 in [𝑎,𝑏].Then 𝑓_𝑋(𝑥) is strong fuzzy Henstock integrable on [𝑎,𝑏] with the primitive 𝐹_𝑋(𝑥).

Proof. By (ii), for every 𝜀>0 there exist a 𝛿(𝜉)>0 and 𝜂>0, such that for any 𝛿-fine partial division 𝑃 of 𝑋 satisfying ∑|𝑣−𝑢|<𝜂 we have ∑𝐷(𝐹_𝑛,𝑋(𝑣,𝑢),0)<𝜀. By Egoroff’s theorem [18, Theorem 3.4], there is an open set 𝐺 with |𝐺|<𝜂 such that 𝐷(𝑓_𝑛(𝜉),𝑓_𝑚(𝜉))<𝜀 for 𝑛,𝑚≥𝑁 and 𝜉∉𝐺. Consider the following, in which 𝑃 is a 𝛿-fine division of [𝑥,𝑦] and 𝑃=𝑃₁∪𝑃₂ so that 𝑃₁ contains the intervals with the associated points 𝜉∉𝐺 and 𝑃₂ otherwise: 𝐷𝐹_𝑛,𝑋𝑥,𝑦,𝐹_𝑚,𝑋=𝑃𝑥,𝑦₂𝐷𝐹_𝑛,𝑋(𝑢,𝑣),𝐹_𝑚,𝑋()𝐹𝑢,𝑣≤𝐷_𝑛,𝑋(𝑢,𝑣),𝑓_𝑛,𝑋(𝜉)𝐹)(𝑣−𝑢+𝐷_𝑚,𝑋(𝑢,𝑣),𝑓_𝑚,𝑋(𝜉)+𝑓)(𝑣−𝑢𝐷_𝑚,𝑋(𝜉)(𝑣−𝑢),𝑓_𝑚,𝑋(𝜉)𝐹)(𝑣−𝑢+𝐷_𝑛,𝑋(𝑣),𝐹_𝑛,𝑋(𝑢)+𝐷𝐹_𝑚,𝑋(𝑣),𝐹_𝑚,𝑋(𝑢)<𝜀(4+𝑏−𝑎). (14) Hence, for any 𝛿-fine partial division 𝑃 of [𝑎,𝑏] we have 𝐷𝐹_𝑛,𝑋(𝑢,𝑣),𝐹_𝑚,𝑋(𝑢,𝑣)<𝜀, (15) for 𝑚,𝑛≥𝑁. Therefore the fuzzy sequence {𝐹_𝑛,𝑋} is generalized 𝒫-Cauchy on [𝑎,𝑏]. Then, by (i), we have that 𝑓_𝑋 is strong fuzzy Henstock integrable on [𝑎,𝑏] with primitive 𝐹_𝑋.

Definition 10. (a) A sequence {𝐹_𝑛} of fuzzy-number-valued function is uniformly 𝐴𝐶^∇ on 𝑋 whenever to each 𝜀>0 there exist 𝜂>0 and 𝛿(𝑥)>0 such that(1)sup_𝑛𝐷(∑_{𝐽𝑘∈𝑃1}𝐹_𝑛(𝐽_𝑘),∑_{𝐿ℎ∈𝑃2}𝐹_𝑛(𝐿_ℎ))<𝜀, for each 𝑃₁,𝑃₂∈∏(𝑋,𝛿), (2)with |(∪𝑃₁)Δ(∩𝑃₂)|<𝜂.
(b) A sequence {𝐹_𝑛} of fuzzy-number-valued function is uniformly 𝐴𝐶𝐺^∇ on [𝑎,𝑏] if [𝑎,𝑏]=∪_𝑖𝑋_𝑖, where 𝑋_𝑖 are measurable sets and {𝐹} is uniformly 𝐴𝐶^∇ on each 𝑋_𝑖.

Theorem 11. If {𝐹} is uniformly 𝐴𝐶𝐺^∇, then {𝐹} is uniformly 𝐴𝐶𝐺^∗_𝛿.

Proof. Let [𝑎,𝑏]=∪_𝑖𝑋_𝑖 be such that {𝐹} is uniformly 𝐴𝐶𝐺^∇ on each 𝑋_𝑖. So, for each 𝜀>0 there exist 𝜂>0 and 𝛿(𝑥)>0 such that (1) holds in Definition 10 for each 𝑃₁,𝑃₂∈∏(𝑋,𝛿) satisfying condition (2). We take 𝑃={([𝑐_𝑘,𝑑_𝑘],𝑥_𝑘)}^𝑝_𝑖=1 with ∑_𝑘|𝑑_𝑘−𝑐_𝑘|<𝜂 and put 𝑃₁={([𝑐_𝑘,𝑑_𝑘],𝑥_𝑘)∶𝐹_𝑛(𝐶_𝑘,𝑑_𝑘≥0)} and 𝑃₂={([𝑐_𝑘,𝑑_𝑘],𝑥_𝑘)∶𝐹_𝑛(𝑐_𝑘,𝑑_𝑘<0)}. So, we have |(∪𝑃₁)Δ(∩𝑃₂)|=∑^𝑝_𝑘=1|𝑑_𝑘−𝑐_𝑘|<𝜂. Then by condition (1) in Definition 10, we have sup_𝑛^𝑝_𝑘=1𝐷𝐹_𝑛𝑐_𝑘,𝐹_𝑛𝑑_𝑘=sup𝐷_{𝑐𝑘,𝑑𝑘∈𝑃1}𝐹_𝑛𝑐_𝑘,𝑑_𝑘,_{(𝑐𝑘,𝑑𝑘)∈𝑃2}𝐹_𝑛𝑐_𝑘,𝑑_𝑘<𝜀. (16) Hence, we have that {𝐹} is uniformly 𝐴𝐶𝐺^∗_𝛿.

We get the following theorem by Theorems 9 and 11.

Theorem 12. Let the following conditions be satisfied:(i)𝑓_𝑛,𝑋→𝑓_𝑋 a.e. in [𝑎,𝑏] where each 𝑓_𝑛,𝑋 is strong fuzzy Henstock integrable on [𝑎,𝑏];(ii)the primitives 𝐹_𝑛,𝑋 of 𝑓_𝑛,𝑋 are 𝑈𝐴𝐶^∇(𝑋) with closed set 𝑋 in [𝑎,𝑏].Then 𝑓_𝑋 is strong fuzzy Henstock integrable on [𝑎,𝑏] with primitive 𝐹_𝑋.

Next, we give the controlled convergence theorem for the strong fuzzy Henstock integrals by the definition of the 𝑈𝐴𝐶𝐺_𝛿 for a fuzzy-number-valued function.

Definition 13. Let 𝐹∶[𝑎,𝑏]→𝐸^𝑛, and let 𝑋⊂[𝑎,𝑏]. A fuzzy-number-valued function 𝐹 is 𝐴𝐶_𝛿 on 𝑋 if for each 𝜀>0 there exist 𝜂>0 and 𝛿(𝑥)>0 on 𝑋 such that ∑^𝑁_𝑖=1𝐷(𝐹(𝑐_𝑖),𝐹(𝑑_𝑖))<𝜀 for ∑^𝑁_𝑖=1(𝑑_𝑖−𝑐_𝑖)<𝜂. A fuzzy-number-valued function 𝐹 is 𝐴𝐶𝐺_𝛿 on [𝑎,𝑏] if [𝑎,𝑏] is the union of a sequence of set {𝑋_𝑖} such that the function 𝐹 is 𝐴𝐶_𝛿(𝑋_𝑖) for each 𝑖.

Definition 14 (see [18]). A fuzzy-number-valued function 𝐹 defined on 𝑋⊂[𝑎,𝑏] is said to be 𝐴𝐶^∗(𝑋) if for every 𝜀>0 there exists 𝜂>0 such that for every finite sequence of non-overlapping intervals {[𝑎_𝑖,𝑏_𝑖]}, satisfying ∑^𝑛_𝑖=1|𝑏_𝑖−𝑎_𝑖|<𝜂 where 𝑎_𝑖,𝑏_𝑖∈𝑋 for all 𝑖, one has ∑𝜔(𝐹,[𝑎_𝑖,𝑏_𝑖])<𝜀, where 𝜔 denotes the oscillation of 𝐹 over [𝑎_𝑖,𝑏_𝑖]; that is, 𝜔(𝐹,[𝑎_𝑖,𝑏_𝑖])=sup{𝐷(𝐹(𝑦),𝐹(𝑥));𝑥,𝑦∈[𝑎_𝑖,𝑏_𝑖]}. A fuzzy-number-valued function 𝐹 is said to be 𝐴𝐶𝐺^∗ on 𝑋 if 𝑋 is the union of a sequence of closed sets {𝑋_𝑖} such that, on each 𝑋_𝑖, 𝐹 is 𝐴𝐶^∗(𝑋_𝑖).

Theorem 15. A fuzzy-number-valued function 𝐹 is 𝐴𝐶𝐺_𝛿 if and only if it is 𝐴𝐶𝐺^∗ on [𝑎,𝑏].

Theorem 16 (controlled convergence theorem). Let the following conditions be satisfied:(1)𝑓_𝑛(𝑥)→𝑓(𝑥) almost everywhere in [𝑎,𝑏] as 𝑛→∞ where each 𝑓_𝑛 is strong fuzzy Henstock integrable on [𝑎,𝑏];(2)the primitives 𝐹_𝑛(𝑥)=(𝑆𝐹𝐻)∫^𝑥_𝑎𝑓_𝑛(𝑠)d𝑥 of 𝑓_𝑛 are 𝑈𝐴𝐶𝐺_𝛿 uniformly in 𝑛;(3)the sequence {𝐹_𝑛(𝑥)} converges uniformly to a continuous function on [𝑎,𝑏]. Then 𝑓(𝑥) is strong fuzzy Henstock integrable on [𝑎,𝑏] and one has lim_𝑛→∞(𝑆𝐹𝐻)^𝑏_𝑎𝑓_𝑛(𝑥)d𝑥=(𝑆𝐹𝐻)^𝑏_𝑎𝑓(𝑥)d𝑥. (17) If conditions (1) and (2) are replaced by condition (4):(4)𝑔(𝑥)≤𝑓(𝑥)≤ℎ(𝑥) almost everywhere on [𝑎,𝑏], where 𝑔(𝑥) and ℎ(𝑥) are strong fuzzy Henstock integrable.

Proof. By condition (2) there exists a sequence {𝑋_𝑖} such that 𝐹_𝑛∈𝑈𝐴𝐶_𝛿 in 𝑋_𝑖 a bounded, closed set with bounds 𝑎 and 𝑏, and put 𝑋=𝑋_𝑖. We note that 𝐹_𝑛(𝑥)→𝐹(𝑥); we have 𝐹∈𝐴𝐶_𝛿 on 𝑋, and hence 𝐹∈𝐴𝐶𝐺_𝛿 on [𝑎,𝑏]. By Theorem 15, 𝐹∈𝐴𝐶^∗ on 𝑋 and hence 𝐹∈𝐴𝐶𝐺^∗ on [𝑎,𝑏] and also 𝐹∈𝐴𝐶 on 𝑋 and hence 𝐹∈𝐴𝐶𝐺 on [𝑎,𝑏].
Now we prove that 𝐹^′(𝑥)=𝑓(𝑥) a.e. on [𝑎,𝑏]. In fact, let 𝐺∶[𝑎,𝑏]→𝐸^𝑛 equal 𝐹_𝑛 on 𝑋, and extend 𝐺_𝑛 linearly to the closed interval contiguous to 𝑋. Likewise, we define 𝐺 from 𝐹. We see that 𝐺_𝑛 and 𝐺 are 𝑈𝐴𝐶 on [𝑎,𝑏]. By condition (3), we have 𝐺_𝑛→𝐺 on [𝑎,𝑏]. Let [𝑐_𝑘,𝑑_𝑘] be the intervals contiguous to 𝑋. Then we have 𝐷(𝐺^′_𝑛)≤𝑀_𝑘. We define a fuzzy-number-valued function as follows: 𝐺^′(𝑥)=𝐺_𝑛𝑑_𝑘−_𝐻𝐺_𝑛𝑐_𝑘𝑑_𝑘−𝑐_𝑘,𝑥∈𝑐_𝑘,𝑑_𝑘. (18) Consequently, 𝐺^′_𝑛(𝑥) converges on (𝑐_𝑘,𝑑_𝑘). Hence 𝐺^′_𝑛 converges on [𝑎,𝑏] a.e. Since {𝐺_𝑛}∈𝐴𝐶 on [𝑎,𝑏], then 𝐺^′_𝑛(𝑥)=𝑓_𝑛→𝑓 on 𝑋. Therefore, we have 𝐺^′_𝑛(𝑥)=𝑔(𝑥)=𝑓(𝑥)=𝐹^′(𝑥) a.e. on 𝑋. Thus 𝐹^′(𝑥)=𝑓(𝑥) a.e. on [𝑎,𝑏] by Theorem 15. Therefore, there exists an 𝐴𝐶𝐺_𝛿 function on [𝑎,𝑏] such that 𝐹^′(𝑥)=𝑓(𝑥) a.e. on [𝑎,𝑏]. Hence 𝑓 is strong fuzzy Henstock integrable on [𝑎,𝑏], and we have lim_𝑛→∞(SFH)^𝑏_𝑎𝑓_𝑛(𝑥)d𝑥=(SFH)^𝑏_𝑎𝑓(𝑥)d𝑥. (19)

4. The Generalized Solutions of Discontinuous Fuzzy Differential Equations

In this section, a generalized fuzzy differential equation of form (4) is defined by using strong fuzzy Henstock integral. The main results of this section are existence theorems for the generalized solution to the discontinuous fuzzy differential equation.

Definition 17 (see [11]). Let 𝜏 and 𝜉 be fixed, and let a fuzzy-number-valued function 𝑓(𝑡,𝑥) be a Carathéodory function defined on a rectangle 𝑈∶|𝑡−𝜏|≤𝑎,𝐷(𝑥,𝜉)≤𝑏; that is, 𝑓 is continuous in 𝑥 for almost all 𝑡 and measurable in 𝑡 for each fixed 𝑥.

Theorem 18. Let a fuzzy-number-valued function 𝑓 be a function as given in Definition 17; then there exist two strong fuzzy Henstock integrable functions ℎ and 𝑔 defined on |𝑡−𝜏|≤𝑎 such that 𝑔(𝑡)≤𝑓(𝑡,𝑥)≤ℎ(𝑡) for all (𝑡,𝑥)∈𝑈.

Proof. Note that 𝑓 is a Carathéodory function. Thus, there exist two measurable functions 𝑢(𝑡) and 𝑣(𝑡) defined on |𝑡−𝜏|≤𝑎 with values in 𝐷(𝑥,𝜉)≤𝑏 such that 𝑓(𝑡,𝑢(𝑡))≤𝑓(𝑡,𝑥)≤𝑓(𝑡,𝑣(𝑡)) for all (𝑡,𝑥)∈𝑈. Next, we will show that 𝑓(𝑡,𝑢(𝑡)) and 𝑓(𝑡,𝑣(𝑡)) are fuzzy Henstock integrable by using controlled convergence Theorem 16. First, there exists a sequence {𝑘_𝑛(𝑡)} of step functions defined on |𝑡−𝜏|≤𝑎 with values in 𝐷(𝑥,𝜉)≤𝑏 such that 𝑘_𝑛(𝑡)→𝑢(𝑡) almost everywhere as 𝑛→∞. Let 𝐹_𝑛(𝑡)=∫^𝑡_𝜏𝑓(𝑠,𝑘_𝑛(𝑠))d𝑠. Then {𝐹_𝑛(𝑡)} is 𝑈𝐴𝐶𝐺_𝛿 uniformly in 𝑛 and equicontinuous. By controlled convergence Theorem 16, 𝑓(𝑡,𝑢(𝑡)) is strong fuzzy Henstock integrable. Similarly, 𝑓(𝑡,𝑣(𝑡)) is strong fuzzy Henstock integrable.

Definition 19. A fuzzy-number-valued function 𝑥(𝑡)∶𝐼→𝐸^𝑛 is said to be a solution of the discontinuous fuzzy differential equation (4) if 𝑥(𝑡) satisfies the following conditions:(i)𝑥(𝑡) is 𝐴𝐶𝐺_𝛿 on each compact subinterval of 𝐼;(ii)(𝑡,𝑥)∈𝑈 for 𝑡∈𝐼;(iii)𝑥^′(𝑡) for almost everywhere 𝑡∈𝐼.

Now we will state the existence theorem for the generalized solution of discontinuous fuzzy differential equation (4).

Theorem 20. Suppose that 𝑓 satisfies the condition of Theorem 18; then there exists a generalized solution Φ of the discontinuous fuzzy differential equation (4) on some interval |𝑡−𝜏|≤𝑎 which satisfies Φ(𝜏)=𝜉.

Proof. Given 𝑔(𝑡)≤𝑓(𝑡,𝑥)≤ℎ(𝑡) for all 𝑥 and almost all 𝑡 with (𝑡,𝑥)∈𝑈, we get 0≤𝑓(𝑡,𝑥)−_𝐻𝑔(𝑡)≤ℎ(𝑡)−_𝐻𝑔(𝑡). Let 𝐹(𝑡,𝑥)=𝑓𝑡,𝑥+^𝑡_𝜏𝑔(𝑠)d𝑠−_𝐻𝑔(𝑡), (20) or 𝐹(𝑡,𝑥)=𝑓𝑡,𝑥+(−1)⋅^𝑡_𝜏𝑔(𝑠)d𝑠−_𝐻𝑔(𝑡). (21) Then 𝐹 is a Carathéodory function. Furthermore, 0≤𝐹(𝑡,𝑥)≤ℎ(𝑡)−_𝐻𝑔(𝑡) for all (𝑡,𝑥)∈𝑈₀, where 𝑈₀⊂𝑈, such that 𝐷𝑥+^𝑡_𝜏𝑔(𝑠)d𝑠,𝜉≤𝑏∀(𝑡,𝑥)∈𝑈₀. (22) By Carathéodory existence theorem (see Theorem 7 in [11]), there is a fuzzy-number-valued function 𝜓 on some interval |𝑡−𝜏|≤𝑎 such that 𝜓^′(𝑡)=𝐹(𝑡,𝜓(𝑡)) almost everywhere in this interval and 𝜓(𝜏)=𝜉. Let 𝜙(𝑡)=𝜓(𝑡)+^𝑡_𝜏𝑔(𝑠)d𝑠or𝜙(𝑡)=𝜓(𝑡)+(−1)⋅^𝑡_𝜏𝑔(𝑠)d𝑠. (23) Then, for almost all 𝑡, we have the following.
Case  1. Consider 𝜙^′(𝑡)=𝜓^′(𝑡)+𝑔(𝑡)=𝐹(𝑡+𝑔(𝑡)=𝑓(𝑡)+𝑡,𝜓)𝑡,𝜓^𝑡_𝜏𝑔(𝑠)d𝑠−_𝐻𝑔(𝑡)+𝑔(𝑡)=𝑓(𝑡),𝜙(𝑡)(𝜏)+𝑡,𝜙=𝜓^𝑡_𝜏𝑔(𝑠)d𝑠=𝜉. (24)
Case  2. Consider 𝜙^′(𝑡)=𝜓^′(𝑡)+𝑔(𝑡)=𝐹(𝑡+𝑔(𝑡)=𝑓(𝑡)+()⋅𝑡,𝜓)𝑡,𝜓−1^𝑡_𝜏𝑔(𝑠)d𝑠−_𝐻𝑔(𝑡)+𝑔(𝑡)=𝑓(𝑡),𝜙(𝑡)(𝜏)+()⋅𝑡,𝜙=𝜓−1^𝑡_𝜏𝑔(𝑠)d𝑠=𝜉. (25) The proof is complete.

Example 21. Consider fuzzy differential equation 𝑥^′=𝑓(𝑡,𝑥)=𝑔(𝑡,𝑥)+ℎ(𝑡), where 𝐷(𝑔(𝑡,𝑥),0)≤𝐷(𝑔₁(𝑡),0) for all |𝑡|≤1,𝐷(𝑥,0)≤1 and 𝑔₁(𝑡) is Kaleva integrable on |𝑡|≤1 and ℎ(𝑡)=𝐴⋅(𝑑/𝑑𝑡)(𝑡²sin𝑡⁻²) if 𝑡≠0 and ℎ(0)=0. Here 𝐴 is defined in Example 1. Note that ℎ is strong fuzzy Henstock integrable but not Kaleva integrable, and ℎ(𝑡)−_𝐻𝑔₁(𝑡)≤𝑓(𝑡,𝑥)≤ℎ(𝑡)+𝑔₁(𝑡),for|𝑡|≤1,𝐷𝑥,0≤1. (26) Thus, by Theorem 20, there exists a solution of 𝑥^′=𝑓(𝑡,𝑥) with 𝑥(0)=0. For instance, if 𝑔(𝑡,𝑥)=𝑡²𝑥, then 𝜙(𝑡)=𝑒^𝑡3/3⋅^𝑡₀𝑒^−𝑠3/3ℎ(𝑠)d𝑠 (27) is a solution by using integrating factor.

We get the following existence theorem by Theorems 18 and 20.

Theorem 22. Let a fuzzy-number-valued function 𝑓 be a Carathéodory function defined on a rectangle 𝑈∶|𝑡−𝜏|≤𝑎,𝐷(𝑥,𝜉)≤𝑏. Let 𝑓(𝑡,𝑢(𝑡)) be strong fuzzy Henstock integrable on |𝑡−𝜏|≤𝑎 for any step function 𝑢(𝑡) defined on |𝑡−𝜏|≤𝑎 with values in 𝐷(𝑥,𝜉)≤𝑏. Denote that 𝐹_𝑢(𝑡)=∫^𝑡_𝜏𝑓(𝑠,𝑢(𝑠))d𝑠. If {𝐹_𝑢∶𝑢 is a step function} is 𝑈𝐴𝐶𝐺_𝛿 uniformly in 𝑢 and equicontinuous on |𝑡−𝜏|≤𝑎, then there exists a solution 𝜙 of 𝑥^′=𝑓(𝑡,𝑥) on some interval |𝑡−𝜏|≤𝛽 with 𝜙(𝜏)=𝜉.

Finally, in this paper, we will show the continuous dependence of a solution on parameters by using Theorems 16, 18, and 22.

Let 𝑈_𝑝 be a connected set in 𝑈. Let 𝑐>0 and let 𝜇₀ be fixed: 𝐼_𝜇=𝜇;𝜇−𝜇₀,𝑈<𝑐_𝜇=𝑡,𝑥,𝜇;(𝑡,𝑥)∈𝑈_𝑝,𝜇∈𝐼_𝜇. (28) Let 𝑓(𝑡,𝑥,𝜇) be a fuzzy-number-valued function defined on 𝑈_𝜇 such that, for each fixed 𝜇, the function 𝑓 is a Carathéodory function defined on 𝑈_𝑝 for each fixed 𝑡 and continuous at (𝑥,𝜇₀) for every 𝑥. For 𝜇=𝜇₀, let 𝑥^′(𝑡)=𝑓,𝑥(𝜏)𝑡,𝑥,𝜇=𝜉∈𝐸^𝑛 (29) have a solution 𝜙₀ on [𝑎,𝑏], where 𝜏∈[𝑎,𝑏]. Let 𝑓(𝑡,𝑢(𝑡),𝜇) be strong fuzzy Henstock integrable on [𝑎,𝑏] for any step function 𝑢(𝑡) with (𝑡,𝑢(𝑡))∈𝑈_𝑝 for 𝑡∈[𝑎,𝑏].

Definition 23. Denote 𝐹_𝜇,𝑢(𝑡)=^𝑡_𝑎𝑓𝑠,𝑢(𝑠),𝜇d𝑠. (30) The family {𝐹_𝜇,𝑢} is said to be equicontinuous in 𝑢 and near 𝜇₀ if, for each 𝑡₀∈[𝑎,𝑏], there exists an interval |𝜇−𝜇₀|<𝑟₀ such that the family {𝐹_𝜇,𝑢(𝑡)∶𝑢and𝜇with|𝜇−𝜇₀|<𝑟₀} is equicontinuous at 𝑡₀.

Theorem 24. Let 𝑓 be a fuzzy-number-valued function as given above. If the primitive 𝐹_𝜇,𝑢(𝑡) of 𝑓(𝑡,𝑥,𝜇) is 𝐴𝐶𝐺_𝛿 uniformly in 𝑢 and 𝜇 and equicontinuous in 𝑢 and near 𝜇₀, then there exists 𝛿>0 such that, for any fixed 𝜇 with |𝜇−𝜇₀|<𝛿, a solution 𝜙_𝜇 of discontinuous fuzzy differential equation (29) exists over [𝑎,𝑏] and as 𝜇→𝜇₀,𝜙_𝜇→𝜙₀ uniformly over [𝑎,𝑏].

Proof. Firstly, we will consider the case 𝜏∈(𝑎,𝑏). By Theorem 22 and the equicontinuity of 𝐹_𝜇,𝑢(𝑡), all solutions 𝜙_𝜇 of problem (29) with |𝜇−𝜇₀|≤𝛿₁ for some 𝛿₁>0 exist over some interval |𝑡−𝜏|≤𝛼 with 𝛼>0. Then 𝐺={𝜙_𝜇∶|𝜇−𝜇₀|≤𝛿₁} is equicontinuous on |𝑡−𝜏|≤𝛼. Because 𝜙_𝜇(𝜏)=𝜉, 𝐺 is uniformly bounded. Hence, for all sequence of 𝐺 with 𝜇→𝜇₀, there exists a subsequence which converges uniformly. That is to say, 𝜙_𝜇(𝑘)→𝜓 uniformly on |𝑡−𝜏|≤𝛼 as 𝜇(𝑘)→𝜇₀. Since we have 𝜙_𝜇(𝑘)(𝑡)=𝜉+^𝑡_𝜏𝑓𝑠,𝜙_𝜇(𝑘),𝜇(𝑘)d𝑠 (31) or 𝜙_𝜇(𝑘)(𝑡)=𝜉+(−1)⋅^𝑡_𝜏𝑓𝑠,𝜙_𝜇(𝑘),𝜇(𝑘)d𝑠, (32) by Theorem 16 we get 𝜓(𝑡)=𝜉+^𝑡_𝜏𝑓𝑠,𝜓(𝑠),𝜇₀d𝑠 (33) or 𝜓(𝑡)=𝜉+(−1)⋅^𝑡_𝜏𝑓𝑠,𝜓(𝑠),𝜇₀d𝑠. (34) Thus, 𝜓 is a solution of (29) for 𝜇=𝜇₀. Hence, 𝜓=𝜙₀ on |𝑡−𝜏|≤𝛼 because the solution 𝜙₀ is unique. Consequently, 𝜙_𝑢(𝑘)→𝜙₀ uniformly on |𝑡−𝜏|≤𝛼 as 𝜇(𝑘)→𝜇₀. By Reductio ad absurdum, 𝜙_𝜇→𝜙₀ uniformly there as 𝜇→𝜇₀.
Secondly, we will extend the result over [𝑎,𝑏]. We consider the case [𝜏,𝑏]. Assume that there exists 𝑡₀∈[𝜏,𝑏) such that the result is valid over [𝜏,𝑡₀−ℎ] but not [𝜏,𝑡₀+ℎ]. Obviously, 𝑡₀≥𝜏+𝛼. Let 𝛾,𝛽 be such that 𝐻=(𝑡,𝑥)∶𝑡−𝑡₀≤𝛼,𝐷𝑥,𝜙₀𝑡−𝛾≤𝛽 (35) is contained in 𝑈. Since {𝐹_𝜇,𝑢(𝑡)} is equicontinuous in 𝑢 and near 𝜇₀, we may choose 𝛾>0 small enough such that 𝐷^𝑡_𝑡0𝑓(𝑠)0<𝛽𝑠,𝑢,𝜇d𝑠,3 (36) for all step functions 𝑢 whenever |𝑡−𝑡₀|≤𝛾 and |𝜇−𝜇₀|<𝛿₂. Since 𝜙_𝜇→𝜙₀ uniformly on [𝜏,𝑡₀−ℎ], there exists a 𝛿 such that 𝐷𝜙_𝜇𝑡₀−𝛾,𝜙₀𝑡₀<𝛽−𝛾3 (37) for |𝜇−𝜇₀|<𝛿. Hence, we have 𝐷^𝑡_𝑡0−𝛾𝑓𝑠,𝑢(𝑠),𝜇d𝑠,𝜙_𝑚𝑢𝑡₀−𝛾−_𝐻𝜙₀𝑡₀−𝛾≤𝐷^𝑡0_𝑡0−𝛾𝑓𝑠,𝑢(𝑠),𝜇d𝑠,0+𝐷^𝑡_𝑡0𝑓(𝑠)0𝜙𝑠,𝑢,𝜇d𝑠,+𝐷_𝑚𝑢𝑡₀−𝛾,𝜙₀𝑡₀𝛽−𝛾<3⋅3. (38) Thus, for each 𝑢 with |𝜇−𝜇₀|<𝛿, a solution of (29) with 𝑥(𝑡₀−𝛾)=𝜙_𝜇(𝑡₀−𝛾) exists on |𝑡−𝑡₀|≤𝛾. Hence, 𝜙_𝜇 can be continued to 𝑡₀+𝛾. So, in the case [𝜏−𝛼,𝜏+𝛼],𝜙_𝜇→𝜙₀ uniformly on [𝜏,𝑡₀+𝛾]. It leads to a contradiction, similarly, for the cases 𝑡₀=𝑏 and [𝑎,𝜏]. Therefore, the theorem holds over [𝑎,𝑏].

Example 25. Let 𝐻(𝑡)=𝐴⋅𝑡²sin𝑡⁻²𝑡≠0,0,𝑡=0, (39) where fuzzy number 𝐴 is defined in Example 1. Let ℎ(𝑡)=𝐻^′(𝑡). We define ℎ_𝜇(𝑡)=ℎ(𝑡)[)()𝜇∈0,1,𝑡∈−2,2,−𝜇,𝜇0,otherwise, (40) and ℎ_𝜇(𝑡)=ℎ_−𝜇(𝑡) for 𝜇∈(−1,0]. Let 𝑓(𝑡,𝑥,𝜇)=𝑡²𝑥+ℎ_𝜇(𝑡) defined on (−2,2)×(−1,1). Then, we have 𝐷𝐹_𝜇,𝑢𝑡₁,𝐹_𝜇,𝑢𝑡₂=𝐷^𝑡2_𝑡1𝑓(𝑠)0≤𝑠,𝑢,𝜇d𝑠,_𝑡2𝑡1𝑠²d𝑠+𝐷^𝑡2_𝑡1ℎ_𝜇0,𝐷d𝑠,_𝑡2𝑡1ℎ_𝜇d𝑠,0≤^𝑡2_𝑡1𝐷ℎ(𝑠)d𝑠,0, (41) where 𝑡₁,𝑡₂∈(0,1] or 𝑡₁,𝑡₂∈[−1,0), and 𝐷^𝑡₀ℎ_𝜇(𝑠)0=𝐷d𝑠,𝐴⋅𝑡²sin𝑡⁻²,𝐴⋅𝜇²sin𝜇⁻²𝐷,𝜇≠0,𝐴⋅𝑡²sin𝑡⁻²,0,𝜇=0. (42) Note that ℎ is Kaleva integrable on every subinterval of [−1,0) and (0,1]. Therefore, we have that 𝐹_𝜇,𝑢(𝑡) is equicontinuous on [−1,1] in 𝑢 and near 𝜇₀=0. Furthermore, 𝐹_𝜇,𝑢(𝑡) is 𝐴𝐶_𝛿(𝑋_𝑛) uniformly in 𝜇 and 𝑢, where 𝑋_𝑛=[1/𝑛,1],𝑛=1,2,…. On the other hand, by using integrating factor, for 𝑡∈[−1,1], we have 𝜙_𝜇(𝑡)=𝑒^𝑡2/3𝑡₀𝑒^−𝑠3/3𝐴⋅ℎ_𝜇(𝑠)d𝑠, (43) with 𝜙_𝜇(0)=0. Obviously, 𝜙₀ is unique. Thus, by Theorem 24, 𝜙_𝜇→𝜙₀ uniformly on [−1,1].

5. Conclusion

In this paper, we give the definition of the 𝑈𝐴𝐶𝐺_𝛿 for a fuzzy-number-valued function and the nonabsolute fuzzy integral and its controlled convergence theorem. In addition, we deal with the Cauchy problem and the continuous dependence of a solution on parameters of discontinuous fuzzy differential equations involving the strong fuzzy Henstock integral in fuzzy number space. The function governing the equations is supposed to be discontinuous with respect to some variables and satisfy nonabsolute fuzzy integrability. Our result improves the result given in [1, 11, 19, 20] (where uniform continuity was required), as well as those referred therein.

Acknowledgments

The authors are very grateful to the anonymous referees and Professor Mehmet Sezer for many valuable comments and suggestions which helped to improve the presentation of the paper. The authors would like to thank the National Natural Science Foundation of China (no. 11161041) and the Fundamental Research Fund for the Central Universities (no. 31920130010).