The fuzzy Henstock-Kurzweil delta integral on time scales

The fuzzy Henstock–Kurzweil delta integral on time scales

arXiv:1711.11089v1 [math.CA] 29 Nov 2017

Dafang Zhao, Guoju Ye, Wei Liu and Delfim F. M. Torres

Abstract We investigate properties of the fuzzy Henstock–Kurzweil delta integral (shortly, FHK ∆ -integral) on time scales, and obtain two necessary and sufficient conditions for FHK ∆ -integrability. The concept of uniformly FHK ∆ -integrability is introduced. Under this concept, we obtain a uniformly integrability convergence theorem. Finally, we prove monotone and dominated convergence theorems for the FHK ∆ -integral. Key words: fuzzy Henstock–Kurzwei integral, convergence theorems, time scales. Mathematics Subject Classification (2010 ): 26A42; 26E50; 26E70.

1 Introduction The Lebesgue integral, with its convergence properties, is superior to the Riemann integral. However, a disadvantage with respect to Lebesgue’s integral, is Dafang Zhao College of Science, Hohai University, Nanjing, 210098, P. R. China School of Mathematics and Statistics, Hubei Normal University, Huangshi, 435002, P. R. China e-mail: [email protected] Guoju Ye College of Science, Hohai University, Nanjing, 210098, P. R. China e-mail: [email protected] Wei Liu College of Science, Hohai University, Nanjing, 210098, P. R. China e-mail: [email protected] Delfim F. M. Torres ( ) Center for Research and Development in Mathematics and Applications (CIDMA), Department of Mathematics, University of Aveiro, 3810–193 Aveiro, Portugal e-mail: [email protected]

1

2


that it is hard to understand without substantial mathematical maturity. Also, the Lebesgue integral does not inherit the naturalness of the Riemann integral. Henstock [23] and Kurzweil [26] gave, independently, a slight, yet powerful, modification of the Riemann integral to get the now called Henstock–Kurzweil (HK) integral, which possesses all the convergence properties of the Lebesgue integral. For the fundamental results of HK integral, we refer to the papers [7, 23, 38, 39, 43, 45] and monographs [20, 27, 34]. As an important branch of the HK integration theory, the fuzzy Henstock–Kurzweil (FHK) integral has been extensively studied in [8, 13, 18, 19, 22, 30, 35, 36, 41, 42]. In 1988, Hilger introduced the theory of time scales in his Ph.D. thesis [24]. A time scale T is an arbitrary nonempty closed subset of R. The aim is to unify and generalize discrete and continuous dynamical systems, see, e.g., [2–6, 9, 15, 21, 31]. In [32], Peterson and Thompson introduced a more general concept of integral, i.e., the HK ∆ -integral, which gives a common generalization of the Riemann ∆ and Lebesgue ∆ -integral. The theory of HK integration for real-valued and vectorvalued functions on time scales has been developed rather intensively, see, e.g., the papers [1, 11, 16, 28, 29, 33, 37, 40, 44] and references cited therein. In 2015, Fard and Bidgoli introduced the FHK delta integral and presented some of its basic properties [14]. Nonetheless, to our best knowledge, there is no systematic theory for the FHK delta integral on time scales. In this work, in order to complete the FHK delta integration theory, we give two necessary and sufficient conditions of FHK delta integrability (see Theorems 3 and 7). Moreover, we obtain some convergence theorems for the FHK delta integral, in particular Theorem 9 of dominated convergence and Theorem 10 of monotone convergence. After Section 2 of preliminaries, in Section 3 the definition of FHK delta integral is introduced, and our necessary and sufficient conditions of FHK delta integrability proved. We also obtain some convergence theorems. Finally, in Section 4, we give conclusions and point out some directions that deserve further study.

2 Preliminaries A fuzzy subset of the real axis u : R → [0, 1] is called a fuzzy number provided that (1) u is normal: there exists x0 ∈ R with u(x0 ) = 1; (2) u is fuzzy convex: u(λ x1 + (1 − λ )x2) ≥ min{u(x1 ), u(x2 )} for all x1 , x2 ∈ R and all λ ∈ (0, 1); (3) u is upper semi-continuous; (4) [u]0 = {x ∈ R : u(x) > 0} is compact. Denote by RF the space of fuzzy numbers. We define the α -level set [u]α by [u]α = {x ∈ R : u(x) ≥ α } ,

α ∈ (0, 1]. From conditions (1)–(4), [u]α is denoted by [u]α = uα , uα . For u1 , u2 ∈ RF and λ ∈ R, we define


3

[u1 + u2]α = [u1 ]α + [u2 ]α and [λ ⊙ u1]α = λ [u1 ]α for all α ∈ [0, 1]. The Hausdorff distance between u1 and u2 is defined by o n D(u1 , u2 ) = sup max |uα1 − uα2 |, |uα1 − uα2 | . α ∈[0,1]

Then, the metric space (RF , D) is complete. Let a, b ∈ T. We define the half-open interval [a, b)T by [a, b)T = {x ∈ T : a ≤ x < b} . The open and closed intervals are defined similarly. For x ∈ T, we denote by σ the forward jump operator, i.e., σ (x) := inf{y > x : y ∈ T}, and by ρ the backward jump operator, i.e., ρ (x) := sup{y < x : y ∈ T}. Here, we put σ (sup T) = sup T and ρ (inf T) = inf T, where sup T and inf T are finite. In this situation, Tκ := T\{sup T} and Tκ := T\{inf T}, otherwise, Tκ := T and Tκ := T. If σ (x) > x, then we say that x is right-scattered, while if ρ (x) < x, then we say that x is left-scattered. If σ (x) = x and x < sup T, then x is called right-dense, and if ρ (x) = x and x > inf T, then x is left-dense. The graininess functions µ and η are defined by µ (x) := σ (x) − x and η (x) := x − ρ (x), respectively. In what follows, all considered intervals are intervals in T. A division D of [a, b]T is a finite set of interval-point pairs {([xi−1 , xi ]T , ξi )}ni=1 such that n [

[xi−1 , xi ]T = [a, b]T

i=1

and ξi ∈ [a, b]T for each i. We write ∆ xi = xi − xi−1 . We say that

δ (ξ ) = (δL (ξ ), δR (ξ )) is a ∆ -gauge on [a, b]T if δL (ξ ) > 0 on (a, b]T , δR (ξ ) > 0 on [a, b)T , δL (a) ≥ 0, δR (b) ≥ 0 and δR (ξ ) ≥ µ (ξ ) for any ξ ∈ [a, b)T . The symbol Γ (∆ , [a, b]T ) stands for the set of ∆ -gauge on [a, b]T . Let δ 1 (ξ ) and δ 2 (ξ ) be ∆ -gauges such that 0 < δL1 (ξ ) < δL2 (ξ ) for any ξ ∈ (a, b]T and 0 < δR1 (ξ ) < δR2 (ξ ) for any ξ ∈ [a, b)T . Then we call δ 1 (ξ ) finer than δ 2 (ξ ) and write δ 1 (ξ ) < δ 2 (ξ ). We say that D = {([xi−1 , xi ]T , ξi )}ni=1 is a δ -fine HK division of [a, b]T if ξi ∈ [xi−1 , xi ]T ⊂ (ξi − δL (ξi ), ξi + δR (ξi ))T for each i. Let D(δ , [a, b]T ) be the set of all δ -fine HK divisions of [a, b]T . Given an arbitrary D ∈ D(δ , [a, b]T ), D = {([xi−1 , xi ]T , ξi )}ni=1 , we write n

S( f , D, δ ) = ∑ f (ξi )∆ xi i=1

for integral sums over D, whenever f : [a, b]T → RF .

4


Lemma 1 (See [25]). Suppose that u ∈ RF . Then, (1) the interval [u]α is closed for α ∈ [0, 1]; (2) [u]α1 ⊃ [u]α2 for 0 ≤ α1 ≤ α2 ≤ 1; (3) for any sequence {αn } satisfying αn ≤ αn+1 and αn → α ∈ (0, 1], we have T∞ αn α n=1 [u] = [u] . Conversely, if a collection of subsets {uα : α ∈ [0, 1]} verify (1)–(3), then there exists S a unique u ∈ RF such that [u]α = uα for α ∈ (0, 1] and [u]0 = α ∈(0,1] uα ⊂ u0 . Lemma 2 (See [17]). Suppose that u ∈ RF . Then,

(1) uα is bounded and nondecreasing; (2) uα is bounded and nonincreasing; (3) u1 ≤ u1 ; (4) for c ∈ (0, 1], limα →c− uα = uc and limα →c− uα = uc ; (5) limα →0+ uα = u0 and limα →0+ uα = u0 . Conversely, if uα and uα satisfy items (1)–(5), then there exists u ∈ RF such that [u]α = uα , uα = uα , uα .

3 The fuzzy Henstock–Kurzweil delta integral We introduce the concept of fuzzy Henstock–Kurzweil (FHK) delta integrability. Definition 1. A function f : [a, b]T →ZRF is called FHK ∆ -integrable on [a, b]T with the FHK ∆ -integral A˜ = (FHK) f (x)∆ x, if for each ε > 0 there exists [a,b]T a δ ∈ Γ (∆ , [a, b]T ) such that D S( f , D, δ ), A˜ < ε for each D ∈ D(δ , [a, b]T ). The family of all FHK ∆ -integrable functions on [a, b]T is denoted by F H K [a,b]T .

Remark 1. It is clear that Definition 1 is more general than the HK ∆ -integral introduced by Peterson and Thompson in [32] and more general than the FH integral introduced by Wu and Gong in [41, 42]. The proofs of Theorems 1 and 2 are straightforward and are left to the reader. Theorem 1. The FHK ∆ -integral of f (x) is unique. Theorem 2. If f (x), g(x) ∈ F H K

[a,b]T

and α , β ∈ R, then

α f (x) + β g(x) ∈ F H K

[a,b]T

with (FHK)

Z

[a,b]T

(α f (x) + β g(x))∆ x = α (FHK)

Z

[a,b]T

f (x)∆ x + β (FHK)

Z

[a,b]T

g(x)∆ x.


5

Follows a Cauchy–Bolzano condition for the FHK ∆ -integral. Theorem 3 (The Cauchy–Bolzano condition). Function f (x) ∈ F H K and only if for each ε > 0 there exists a δ ∈ Γ (∆ , [a, b]T ) such that

[a,b]T

if

D (S( f , D1 , δ ), S( f , D2 , δ )) < ε for any D1 , D2 ∈ D(δ , [a, b]T ). Proof. (Necessity) Let ε > 0. By hypothesis, there exists δ ∈ Γ (∆ , [a, b]T ) such that Z ε D S( f , D, δ ), (F HK) f (x)∆ x < 2 [a,b]T for any D ∈ D(δ , [a, b]T ). Let D1 , D2 ∈ D(δ , [a, b]T ). Then, D (S( f , D1 , δ ), S( f , D2 , δ )) Z ≤ D S( f , D1 , δ ), (FHK)

[a,b]T

ε ε < + = ε. 2 2

Z f (x)∆ x + D S( f , D2 , δ ), (FHK)

[a,b]T

f (x)∆ x

(Sufficiency) For each n, choose a δn ∈ Γ (∆n , [a, b]T ) such that D (S( f , D1 , δn ), S( f , D2 , δn ))
n, we have δ j ⊂ δn , so D j ∈ D(δn , [a, b]T ). Thus, D (S( f , Dn , δn ), S( f , D j , δn )) < 1n and it follows that {S( f , Dn , δn )} is a Cauchy sequence. We denote the limit of {S( f , Dn , δn )} by A˜ and let ε > 0. Choose N > ε2 and let D ∈ D(δN , [a, b]T ). Then, T

D S( f , D, δN ), A˜ ≤ D (S( f , D, δN ), S( f , DN , δN )) + D S( f , DN , δN ), A˜ 1 1 < + N N < ε. Hence, f (x) ∈ F H K

[a,b]T .

Theorem 4. Let c ∈ (a, b)T . If f (x) ∈ F H K f (x) ∈ F H K

[a,c]T

T

FH K

[c,b]T ,

then

Z

f (x)∆ x.

[a,b]T

with (FHK)

Z

[a,b]T

f (x)∆ x = (FHK)

Z

[a,c]T

f (x)∆ x + (FHK)

[c,b]T

6


Proof. Let ε > 0. By assumption, there exist ∆ -gauges

δ i (ξ ) = (δLi (ξ ), δRi (ξ )),

i = 1, 2,

such that Z D S( f , D1 , δ 1 ), (FHK) f (x)∆ x < ε , [a,c]T Z f (x)∆ x < ε , D S( f , D2 , δ 2 ), (FHK) [c,b]T

n

respectively for any D1 ∈ D(δ 1 , [a, c]T ), D1 = {([x1k−1 , xκ1 ]T , ξκ1 )}k=1 , and for any m D2 ∈ D(δ 2 , [c, b]T ), D2 = {([x2k−1 , x2κ ]T , ξκ2 )}k=1 . We define δ (ξ ) = (δL (ξ ), δR (ξ )) on [a, b]T by setting

δL (ξ ) =

and

δR (ξ ) =

(

 1 δL (ξ ),    δ 1 (ξ ),  L n

if ξ ∈ [a, c)T , if ξ = c = ρ (c),

o 1 (ξ ), η (c) , if ξ = c > ρ (c), δ min  L 2  n o   min δ 2 (ξ ), ξ −c , if ξ ∈ (c, b] , T L 2

n n oo min δR1 (ξ ), max µ (ξ ), c−2 ξ , if ξ ∈ [a, c)T , min{δR2 (ξ )},

if ξ ∈ [c, b]T .

p . It follows that Now, let D ∈ D(δ , [a, b]T ), D = {([xk−1 , xk ]T , ξk )}k=1

(i) either c = ξq and tq > c; (ii) or ξq = ρ (c) < c and tq = c. The case (ii) is straightforward. For (i), one has Z Z D S( f , D, δ ), (FHK) f (x)∆ x + (FHK) [a,c]T

p

=D

∑ f (ξk )∆ xk , (FHK) k=1

[c,b]T

Z

[a,c]T

f (x)∆ x + (FHK)

q−1

≤D

∑ f (ξk )∆ xk + f (c)(c − tq−1), (FHK) k=1

Z

[a,c]T

p

+D

∑

f (ξk )∆ xk + f (c)(tq − c), (FHK)

k=q+1

< ε + ε = 2ε . The intended result follows.

f (x)∆ x

Z

Z

[c,b]T

f (x)∆ x

f (x)∆ x

[c,b]T

!

f (x)∆ x

!

!


Corollary 1. If f ∈ F H K

[a,b]T ,

then f ∈ F H K

7 [r,s]T

for any [r, s]T ⊂ [a, b]T .

Definition 2 (See [32]). Let T′ ⊂ T. We say T′ has delta measure zero if it has Lebesgue measure zero and contains no right-scattered points. A property P is said to hold ∆ a.e. on T if there exists T′ of measure zero such that P holds for every t ∈ T \ T′ . Theorem 5. Let f (x) = g(x) ∆ a.e. on [a, b]T . If f (x) ∈ F H K [a,b]T , then so g(x). Moreover, Z Z f (x)∆ x = (FHK) (FHK) g(x)∆ x. [a,b]T

[a,b]T

Proof. Let ε > 0. Then there exists a δ ∈ Γ (∆ , [a, b]T ) such that Z f (x)∆ x < ε D S( f , D, δ ), (F HK) [a,b]T

for any D ∈ D(δ , [a, b]T ), D = {([xi−1 , xi ]T , ξi )}. Set E = ∑∞j=1 E j , where o∞ n E j = x : j − 1 < D( f (x), g(x)) ≤ j, t ∈ [a, b]T . j=1

For each j, there exists Fj consisting of a collection of open intervals with total length less than ε · 2− j · j−1 , such that E j ⊂ Fj . Define ( (δL0 (ξ ), δR0 (ξ )), if ξ ∈ [a, b]T \E, δ (ξ ) = (δL1 (ξ ), δR1 (ξ )), if ξ ∈ E j satisfies (ξ − δL1 (ξ ), ξ + δR1 (ξ ))T ⊂ Fj . Then, for any D ∈ D(δ , [a, b]T ), D = {([xi−1 , xi ]T , ξi )}, one has Z D S(g, D, δ ), (FHK) f (x)∆ x [a,b]T ! Z =D

∑

g(ξi )∆ xi , (FHK)

[a,b]T

ξi ∈[a,b]T

=D

∑ g(ξi )∆ xi +

ξi ∈E

=D

∑ g(ξi )∆ xi +

ξi ∈E

(FHK) Therefore,

Z

[a,b]T

f (x)∆ x

∑

g(ξi )∆ xi , (FHK)

∑

f (ξi )∆ xi +

ξi ∈[a,b]T \E

ξi ∈[a,b]T \E

f (x)∆ x +

∑

ξi ∈E

∑

ξi ∈E

!

f (ξi )∆ xi .

Z

[a,b]T

f (x)∆ x

f (ξi )∆ xi ,

!

8


Z D S(g, D, δ ), (FHK)

∑

≤D

[a,b]T

f (x)∆ x

f (ξi )∆ xi , (FHK)

ξi ∈[a,b]T

∑ g(ξi )∆ xi , ∑

+D

ξi ∈E ∞

≤ε+∑

Z

[a,b]T

f (ξi )∆ xi

ξi ∈E

∑

f (x)∆ x

!

!

D ( f (ξi ), g(ξi )) ∆ xi

j=1 ξi ∈E j

≤ 2ε . The proof is complete. Theorem 6 (See [32]). Let [a, b]T be given. Assume (1) limn→∞ fn (x) = f (x) holds ∆ a.e.; (2) G(x) ≤ fn (x) ≤ H(x) holds ∆ a.e.; (3) fn (x), G(x), H(x) ∈ H K [a,b]T . Then f (x) ∈ H K

[a,b]T .

Moreover,

lim (HK)

n→∞

Z

[a,b]T

fn (x)∆ x = (HK)

Z

[a,b]T

f (x)∆ x.

Theorem 7. Function f (x) ∈ F H K [a,b]T if and only if f (x)α , f (x)α ∈ H K [a,b]T for all α ∈ [0, 1] uniformly, i.e., the ∆ -gauge in Definition 1 is independent of α . Proof. (Necessity) Let A˜ = (FHK)

Z

[a,b]T

f (x)∆ x. Given ε > 0, there exists a

δ ∈ Γ (∆ , [a, b]T ) such that D S( f , D, δ ), A˜ < ε for any D ∈ D(δ , [a, b]T ). Then, o n sup max [S( f , D, δ )]α − A˜ α , [S( f , D, δ )]α − A˜ α

α ∈[0,1]

o n = sup max S( f α , D, δ ) − A˜ α , S( f α , D, δ ) − A˜ α α ∈[0,1]

0. By assumption, there exists a δ ∈ Γ (∆ , [a, b]T ) such that α S( f , D, δ ) − A˜ α < ε , S( f α , D, δ ) − A˜ α < ε for any D ∈ D(δ , [a, b]T ) and for any α ∈ [0, 1], where A˜ α = (FHK)

Z

[a,b]T

f α ∆ x, A˜ α = (FHK)

Z

f α ∆ x. [a,b]T

nh i o A˜ α , A˜ α , α ∈ [0, 1] represents a fuzzy number, it is enough to To prove that h i check that A˜ α , A˜ α satisfies items (1)–(3) of Lemma 1: i h (1) for α ∈ [0, 1], if f α ≤ f α , then A˜ α ≤ A˜ α , i.e., the interval A˜ α , A˜ α is closed.

(2) f α and f α are, respectively, nondecreasing and nonincreasing functions on [0, 1]. For any 0 ≤ α1 ≤ α2 ≤ 1 one has (FHK)

Z

[a,b]T

f α1 ∆ x ≤ (FHK) ≤ (FHK) ≤ (FHK)

Z

Z Z

f α2 ∆ x [a,b]T

f α2 ∆ x [a,b]T

f α1 ∆ x. [a,b]T

i h i h This implies A˜ α1 , A˜ α1 ⊃ A˜ α2 , A˜ α2 . (3) For any {αn } satisfying αn ≤ αn+1 and αn → α ∈ (0, 1], we have ∞ \

[ f ]αn = [ f ]α ,

n=1

that is,

∞ \ f αn , f αn = f α , f α ,

n=1

limn→∞ f αn = f α and limn→∞ f αn = f α . Moreover, f 0 ≤ f αn ≤ f 1 ,

f 1 ≤ f αn ≤ f 0 .

Thanks to Theorem 6, we have f α , f α ∈ H K lim (HK)

n→∞

lim (HK)

n→∞

Z

Z

[a,b]T [a,b]T

[a,b]T

f αn ∆ x = (HK) f αn ∆ x = (HK)

Z Z

and f α ∆ x,

[a,b]T

f α ∆ x. [a,b]T

10


Consequently,

∞ h \

n=1

i h i A˜ αn , A˜ αn = A˜ α , A˜ α .

i o nh Define A˜ by A˜ α , A˜ α , α ∈ [0, 1] . Thus,

D S( f , D, δ ), A˜ < ε

for each D ∈ D(δ , [a, b]T ).

Definition 3. A sequence { fn (x)} of HK ∆ -integrable functions is called uniformly FHK ∆ -integrable on [a, b]T if for each ε > 0 there exists a δ ∈ Γ (∆ , [a, b]T ) such that Z fn (x)∆ x < ε D S( fn , D, δ ), (FHK) [a,b]T

for any D ∈ D(δ , [a, b]T ) and for any n. Theorem 8. Let fn (x) ∈ F H K

[a,b]T ,

n = 1, 2, . . ., satisfy:

(1) limn→∞ fn (x) = f (x) on [a, b]T ; (2) fn (x) are uniformly FHK ∆ -integrable on [a, b]T . Then f (x) ∈ F H K

[a,b]T

and

lim (FHK)

n→∞

Z

[a,b]T

fn (x)∆ x = (FHK)

Z

[a,b]T

f (x)∆ x.

Proof. Let ε > 0. By assumption, there exists a δ ∈ Γ (∆ , [a, b]T ) such that Z D S( fn , D, δ ), (FHK) fn (x)∆ x < ε [a,b]T

for any D ∈ D(δ , [a, b]T ) and for every n. Fix a D0 ∈ D(δ , [a, b]T ). From (1) of Theorem 8, there exists N such that D (S( fn , D0 , δ ), S( fm , D0 , δ )) < ε for arbitray n, m > N. Then, Z Z fm (x)∆ x fn (x)∆ x, (FHK) D (FHK) [a,b]T [a,b]T Z ≤ D S( fn , D0 , δ ), (FHK) fn (x)∆ x + D (S( fn , D0 , δ ), S( fm , D0 , δ )) [a,b]T Z + D S( fm , D0 , δ ), (FHK) fm (x)∆ x [a,b]T

< 3ε


11

n o R for any n, m > N and, hence, (FHK) [a,b]T fn (x)∆ x is a Cauchy sequence. Let lim (FHK)

n→∞

Z

[a,b]T

˜ fn (x)∆ x = A.

We now prove that A˜ = (FHK)

Z

[a,b]T

f (x)∆ x.

Let ε > 0. By hypothesis, there exists a δ ∈ Γ (∆ , [a, b]T ) such that Z D S( fn , D, δ ), (FHK) fn (x)∆ x < ε [a,b]T

for any D ∈ D(δ , [a, b]T ) and for all n. Choose N that satisfies Z ˜ fn (x)∆ x, A < ε D (FHK) [a,b]T

for all n > N. For the above D and N, there exists N0 > N satisfying D S( fN0 , D, δ ), S( f , D, δ ) < ε . Therefore,

D S( f , D, δ ), A˜

Z fN0 (x)∆ x ≤ D S( f , D, δ ), S( fN0 , D, δ ) + D S( fN0 , D, δ ), (FHK) [a,b]T Z + D (FHK) fN0 (x)∆ x, A˜ [a,b]T

< 3ε

and the result follows. Definition 4 (See [10]). A function f : [a, b]T → R is called absolutely continuous on [a, b]T , if for each ε > 0 there exists γ > 0 such that n

∑ | f (xi ) − f (xi−1)| < ε

i=1

whenever

Sn

i=1 [xi−1 , xi ]T

⊂ [a, b]T and ∑ni=1 ∆ xi < γ .

Theorem 9 (Dominated convergence theorem). Let the time scale interval [a, b]T be given. If fn (x) ∈ F H K [a,b]T , n = 1, 2, . . ., satisfy (1) limn→∞ fn (x) = f (x) ∆ a.e.; (2) G(x) ≤ fn (t) ≤ H(x) ∆ a.e. and G(x), H(x) ∈ F H K

[a,b]T ;

12


then sequence { fn (x)} is uniformly FHK ∆ -integrable. Thus, f (x) ∈ F H K and Z Z f (x)∆ x. lim (FHK) fn (x)∆ x = (FHK) n→∞

[a,b]T

[a,b]T

[a,b]T

Proof. By hypothesis, one has n o D ( f p (x), fq (x)) = sup max | f p (x)α − fq (x)α |, | f p (x)α − fq (x)α | α ∈[0,1]

n o ≤ sup max |H(x)α − G(x)α |, |H(x)α − G(x)α | α ∈[0,1]

= D (H(x), G(x)) .

Then, D (H(x), G(x)) is Lebesgue ∆ -integrable. Let D(x) =

Z

[a,x]T

D (H(s), G(s)) ∆ s.

From [10], D(x) is absolutely continuous on [a, b]T . Let ε > 0. Then there exists γ > 0 such that n ε ∑ |D(xi ) − D(xi−1)| < b − a i=1 whenever ni=1 [xi−1 , xi ]T ⊂ [a, b]T and ∑ni=1 ∆ xi < γ . The limit limn→∞ fn (x) = f (x) holds ∆ a.e. on [a, b]T and {D( fn (x), f (x))} is a sequence of ∆ -measurable functions. Thanks to the Egorov’s theorem, there exists an open set Ω with m(Ω ) < δ such that limn→∞ fn (x) = f (x) uniformly for x ∈ [a, b]T \Ω . Thus, there exists N such ε that D( f p (x), fq (x)) < b−a for any p, q > N and for any x ∈ [a, b]T \Ω . Suppose that δ1 ∈ Γ (∆ , [a, b]T ) such that Z S(D (H(x), G(x)) , D, δ1 ) − D (H(x), G(x)) ∆ x < ε S

[a,b]T

and

Z D S( fn , D, δ1 ), (FHK)

[a,b]T

fn (x)∆ x < ε

for 1 ≤ n ≤ N and for any D ∈ D(δ1 , [a, b]T ). Define δ ∈ Γ (∆ , [a, b]T ) by ( δ1 (ξ ) if ξ ∈ [a, b]T \Ω , δ (ξ ) = min{δ1 (ξ ), ρ (ξ , Ω )}, if ξ ∈ Ω , where ρ (ξ , Ω ) = inf{|ξ − ξ ′| : ξ ′ ∈ Ω }. Fix n > N. One has D (S( fn , D, δ ), S( fN , D, δ )) = D

∑

ξi ∈[a,b]T

fn (ξi )∆ xi ,

∑

ξi ∈[a,b]T

fN (ξi )∆ xi

!


∑

≤D

∑

+D

∑

fN (ξi )∆ xi

ξi ∈[a,b]T \Ω

fn (ξi )∆ xi ,

ξi ∈Ω

≤ε+

∑

fn (ξi )∆ xi ,

ξi ∈[a,b]T \Ω

∑

fN (ξi )∆ xi

ξi ∈Ω

13

!

!

D( fn (ξi ), fN (ξi ))∆ xi

ξi ∈Ω

Z Z ≤ ε + ∑ D(H(ξi ), G(ξi ))∆ xi − D(H(x), G(x))∆ x + D(H(x), G(x))∆ x ξ ∈Ω Ω Ω i

≤ 3ε

for any D ∈ D(δ , [a, b]T ). Hence, Z D S( fn , D, δ ), (FHK) fn (x)∆ x [a,b]T Z fN (x)∆ x ≤ D (S( fn , D, δ ), S( fN , D, δ )) + D S( fN , D, δ ), (FHK) [a,b]T Z Z fn (x)∆ x + D (FHK) fN (x)∆ x, (FHK) [a,b]T

[a,b]T

≤ 5ε .

Our dominated convergence theorem is proved. As a consequence of Theorem 9, we get the following monotone convergence theorem. Theorem 10 (Monotone convergence theorem). Let the time scale interval [a, b]T be given. If fn (x) ∈ F H K [a,b]T , n = 1, 2, . . ., satisfy (1) limn→∞ fn (x) = f (x) ∆ a.e.; (2) { fn (x)} is a monotone sequence and fn (x) ∈ F H K

[a,b]T ;

then { fn (x)} is uniformly FHK ∆ -integrable. Consequently, f (x) ∈ F H K Moreover, Z Z fn (x)∆ x = (FHK) lim (FHK) f (x)∆ x. n→∞

[a,b]T

[a,b]T .

[a,b]T

4 Conclusion We investigated the fuzzy Henstock–Kurzweil (FHK) delta integral on time scales. Our results give a common generalization of the classical FHK and HK integrals. For future researches, we will investigate the characterization of FHK delta integrable functions. Another interesting line of research consists to study the concept of fuzzy Henstock–Stieltjes integral on time scales.

14


Acknowledgements This research is supported by Chinese Fundamental Research Funds for the Central Universities, grant 2017B19714 (Ye, Liu and Zhao); by the Educational Commission of Hubei Province, grant B2016160 (Zhao); and by Portuguese funds through FCT and CIDMA, within project UID/MAT/04106/2013 (Torres).

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