EUROPEAN JOURNAL OF PURE AND APPLIED MATHEMATICS
Vol. 11, No. 2, 2018, 493-504
ISSN 1307-5543 – www.ejpam.com
Published by New York Business Global

On McShane-Stieltjes integrals of interval-valued
functions and fuzzy-number-valued functions on time

scales

Muawya Elsheikh Hamid1,2

1 School of Mathematical Science, Yangzhou University, Yangzhou 225002, China
2 School of Management, Ahfad University for Women, Omdurman, Sudan

Abstract. In this paper, we introduce the notion of the McShane-Stieltjes (MS) integrals of
interval-valued functions and fuzzy-number-valued functions on time scales which are extensions
of the McShane (M) integrals of interval-valued functions and fuzzy-number-valued functions on
time scales [3] and investigate some of their properties.

2010 Mathematics Subject Classifications: 26A39, 26E70

Key Words and Phrases: Fuzzy numbers; (MS) delta integral of interval-valued functions;
(MS) delta integral of fuzzy-number-valued functions.

1. Introduction

The calculus on time scales was introduced for the first time in 1988 by Hilger [2] to
unify the theory of difference equations and the theory of differential equations. In 2016,
Hamid and Elmuiz [4] introduced the concept of the Henstock-Stieltjes (HS) integrals
of interval-valued functions and fuzzy-number-valued functions and discussed a number
of their properties. Very recently, Hamid et al. [5] introduced the thought of the AP-
Henstock integrals of interval-valued functions and fuzzy-number-valued functions and
obtained some of their properties.

In this paper, we introduce the notion of the (MS) delta integrals of interval-valued
functions and fuzzy-number-valued functions on time scales and investigate some of their
properties.

The paper is organized as follows, in Section 2 we provide the preliminary terminology
used in this paper. Section 3 is dedicated to discuss the (MS) delta integral of interval-
valued functions on time scales. In Section 4, we present the (MS) delta integral of
fuzzy-number-valued functions on time scales. The last section provides Conclusions.

Email address: mowia-84@hotmail.com, muawya.ebrahim@gmail.com (M.E. Hamid)

http://www.ejpam.com 493 c© 2018 EJPAM All rights reserved.



M. E. Hamid / Eur. J. Pure Appl. Math, 11 (2) (2018), 493-504 494

2. Preliminaries

A time scale T is a nonempty closed subset of real number R with the subspace
topology inherited from the standard topology of R. For t ∈ T we define the forward
jump operator σ(t) = inf{s ∈ T : s > t} where inf φ = sup{T}, while the backward jump
operator ρ(t) = sup{s ∈ T : s < t} where supφ = inf{T}. If σ(t) > t, we say that t
is right-scattered, while if ρ(t) < t, we say that t is left-scattered. If σ(t) = t, we say
that t is right-dense, while if ρ(t) = t, we say that t is left-dense. The forward graininess
function µ(t) of t ∈ T is defined by µ(t) = σ(t)− t, whlie the backward graininess function
ν(t) of t ∈ T is defined by ν(t) = t − ρ(t). For a, b ∈ T we denote the closed interval
[a, b]T = {t ∈ T : a ≤ t ≤ b}.

Throughout this paper, all considered intervals will be intervals in T. A division P of
[a, b]T is a finite collection of interval-point pairs {([ti−1, ti]T; ξi)}ni=1, where {a = t0 < t1 <
· · · < tn−1 < tn = b} and ξi ∈ [a, b]T for i = 1, 2, · · · , n. By ∆ti = ti − ti−1 we denote the
length of ith subinterval in the division P . δ(ξ) = (δL(ξ), δR(ξ)) is a ∆- gauge for [a, b]T
provided δL(ξ) > 0 on (a, b]T, δR(ξ) > 0 on [a, b)T, δL(a) ≥ 0, δR(b) ≥ 0 and δR(b) ≥ µ(ξ)
for all ξ ∈ [a, b)T. We say that P = {([ti−1, ti]T; ξi)}ni=1 is a δ-fine McShane division of
[a, b]T if [ti−1, ti]T ⊂

(
ξi − δL(ξi), ξi + δR(ξi)

)
T

and ξi ∈ [a, b]T for all i = 1, 2, · · · , n.

Definition 1. [10] Let α : [a, b] → R be an increasing function. A real-valued function
f : [a, b] → R is said to be McShane-Stieltjes (MS) integrable to B with respect to α on
[a, b] if for every ε > 0, there is a function δ(t) > 0 such that for any δ-fine McShane
division P = {[ui, vi]; ξi}ni=1 of [a, b], we have∣∣ n∑

i=1

f(ξi)[α(vi)− α(ui)]−B
∣∣ < ε, (2.1)

we write (MS)
b∫
a
f(t)dα = B , and f ∈MSα[a, b].

Definition 2. Let α : [a, b]T → R be an increasing function. A function f : [a, b]T → R is
McShane-Stieltjes delta integrable (MS ∆-integrable) with respect to α on [a, b]T if there
exists a number A ∈ R such that for each ε > 0 there is a ∆-gauge, δ, on [a, b]T such that∣∣ n∑

i=1

f(ξi)[α(ti)− α(ti−1)]−A
∣∣ < ε (2.2)

for each δ-fine McShane division P = {([ti−1, ti]T; ξi)}ni=1 of [a, b]T. A is called (MS

∆-integral) of f on [a, b]T, and we write A = (MS∆)
b∫
a
f(t)dα.

Theorem 1. Let α : [a, b]T → R be an increasing function. If f(t) and g(t) are (MS)
∆-integrable with respect to α on [a, b]T and f(t) ≤ g(t) almost everywhere on [a, b]T, then

(MS∆)

b∫
a

f(t)dα ≤ (MS∆)

b∫
a

g(t)dα. (2.3)



M. E. Hamid / Eur. J. Pure Appl. Math, 11 (2) (2018), 493-504 495

Proof. The proof is similar to Theorem 2.7 in [10].

3. The MS∆ integral of interval-valued functions on time scales

This section introduces the notion of the MS∆ integral of interval-valued functions on
time scales and investigates some of their properties.

Definition 3. [7] Let IR = {I = [I−, I+] : I is the closed bounded interval on the real line R}.
For A,B ∈ IR, we define A ≤ B iff A− ≤ B− and A+ ≤ B+, A + B = C iff

C− = A− +B− and C+ = A+ +B+, and A ·B = {a · b : a ∈ A, b ∈ B}, where

(A ·B)− = min{A− ·B−, A− ·B+, A+ ·B−, A+ ·B+} (3.1)

and
(A ·B)+ = max{A− ·B−, A− ·B+, A+ ·B−, A+ ·B+}. (3.2)

Define d(A,B) = max(|A− −B−|, |A+ −B+|) as the distance between intervals A and B.

Definition 4. [3] An interval-valued function F : [a, b]T → IR is McShane delta (M∆)
integrable to I0 ∈ IR on [a, b]T if for every ε > 0 there exists a ∆-gauge, δ, on [a, b]T such
that

d
( n∑
i=1

F (ξi)(ti − ti−1), I0

)
< ε, (3.3)

whenever P = {([ti−1, ti]T; ξi)}ni=1 is a δ-fine McShane division of [a, b]T. We write

(IM∆)
b∫
a
F (t)∆t = I0 and F ∈ IM∆[a, b]T.

Definition 5. Let α : [a, b]T → R be an increasing function. An interval-valued function
F : [a, b]T → IR is (MS∆) integrable to I0 ∈ IR with respect to α on [a, b]T if for every
ε > 0 there exists a ∆-gauge, δ, on [a, b]T such that

d
( n∑
i=1

F (ξi)[α(ti)− α(ti−1)], I0

)
< ε, (3.4)

whenever P = {([ti−1, ti]T; ξi)}ni=1 is a δ-fine McShane division of [a, b]T. We write

(IMS∆)
b∫
a
F (t)dα = I0 and F ∈ IMSα∆[a, b]T.

Remark 1. Let α : [a, b]T → R be an increasing function. If F (t) ∈ IMSα∆[a, b]T, then
the integral value is unique.

Theorem 2. Let α : [a, b]T → R be an increasing function. An interval-valued function
F : [a, b]T → IR is (MS∆) integrable with respect to α on [a, b]T if and only if F−, F+ ∈
MSα∆[a, b]T and

(IMS∆)

b∫
a

F (t)dα =

[
(MS∆)

b∫
a

F−(t)dα, (MS∆)

b∫
a

F+(t)dα

]
. (3.5)



M. E. Hamid / Eur. J. Pure Appl. Math, 11 (2) (2018), 493-504 496

Proof. Let F ∈ IMSα∆[a, b]T, then there exists an interval I0 = [I−0 , I
+
0 ] with the

property that for any ε > 0 there exists a ∆-gauge, δ with respect to α on [a, b]T such
that

d
( n∑
i=1

F (ξi)[α(ti)− α(ti−1)], I0

)
< ε, (3.6)

whenever P = {([ti−1, ti]T; ξi)}ni=1 is a δ-fine McShane division of [a, b]T.
Since α(ti)− α(ti−1) ≥ 0 for 1 ≤ i ≤ n, we have

d
( n∑
i=1

F (ξi)[α(ti)− α(ti−1)], I0

)

= max

(∣∣∣∣[ n∑
i=1

F (ξi)[α(ti)− α(ti−1)]

]−
− I−0

∣∣∣∣, ∣∣∣∣[ n∑
i=1

F (ξi)[α(ti)− α(ti−1)]

]+

− I+
0

∣∣∣∣) < ε.

= max

(∣∣∣∣ n∑
i=1

F−(ξi)[α(ti)− α(ti−1)]− I−0

∣∣∣∣, ∣∣∣∣ n∑
i=1

F+(ξi)[α(ti)− α(ti−1)]− I+
0

∣∣∣∣) < ε. (3.7)

Hence

∣∣∣∣ n∑
i=1

F−(ξi)[α(ti)−α(ti−1)]−I−0

∣∣∣∣ < ε,

∣∣∣∣ n∑
i=1

F+(ξi)[α(ti)−α(ti−1)]−I+
0

∣∣∣∣ < ε whenever

P = {([ti−1, ti]T; ξi)}ni=1 is a δ-fine McShane division of [a, b]T. Thus F−, F+ ∈MSα∆[a, b]T
and

(IMS∆)

b∫
a

F (t)dα =

[
(MS∆)

b∫
a

F−(t)dα, (MS∆)

b∫
a

F+(t)dα

]
. (3.8)

Conversely, let F−, F+ ∈MSα∆[a, b]T. Then there exists M1,M2 ∈ R with the property
that given ε > 0 there exists a ∆-gauge, δ with respect to α on [a, b]T such that∣∣∣∣ n∑

i=1

F−(ξi)[α(ti)− α(ti−1)]−M1

∣∣∣∣ < ε,

∣∣∣∣ n∑
i=1

F+(ξi)[α(ti)− α(ti−1)]−M2

∣∣∣∣ < ε

whenever P = {([ti−1, ti]T; ξi)}ni=1 is a δ-fine McShane division of [a, b]T. We define
I0 = [M1,M2], then if P = {([ti−1, ti]T; ξi)}ni=1 is a δ-fine McShane division of [a, b]T, we
have

d
( n∑
i=1

F (ξi)[α(ti)− α(ti−1)], I0

)
< ε. (3.9)

Hence F : [a, b]T → IR is (MS∆) integrable with respect to α on [a, b]T.

Theorem 3. Let α : [a, b]T → R be an increasing function. If F (t), G(t) ∈ IMSα∆[a, b]T
and β, γ ∈ R. Then

[
βF (t) + γG(t)

]
∈ IMSα∆[a, b]T and

(IMS∆)

b∫
a

(βF (t) + γG(t))dα = β(IMS∆)

b∫
a

F (t)dα+ γ(IMS∆)

b∫
a

G(t)dα. (3.10)



M. E. Hamid / Eur. J. Pure Appl. Math, 11 (2) (2018), 493-504 497

Proof. If F (t), G(t) ∈ IMSα∆[a, b]T, then F−(t), F+(t), G−(t), G+(t) ∈ MSα∆[a, b]T by
Theorem 2. Hence βF−(t)+γG−(t), βF−(t)+γG+(t), βF+(t)+γG−(t), βF+(t)+γG+(t) ∈
MSα∆[a, b]T.

(1) If β > 0 and γ > 0, then

(MS∆)

b∫
a

(βF (t) + γG(t))−dα = (MS∆)

b∫
a

(βF−(t) + γG−(t))dα

= β(MS∆)

b∫
a

F−(t)dα+ γ(MS∆)

b∫
a

G−(t)dα

= β

(
(IMS∆)

b∫
a

F (t)dα

)−
+ γ

(
(IMS∆)

b∫
a

G(t)dα

)−

=

(
β(IMS∆)

b∫
a

F (t)dα+ γ(IMS∆)

b∫
a

G(t)dα

)−
.

(2) If β < 0 and γ < 0, then

(MS∆)

b∫
a

(βF (t) + γG(t))−dα = (MS∆)

b∫
a

(βF+(t) + γG+(t))dα

= β(MS∆)

b∫
a

F+(t)dα+ γ(MS∆)

b∫
a

G+(t)dα

= β

(
(IMS∆)

b∫
a

F (t)dα

)+

+ γ

(
(IMS∆)

b∫
a

G(t)dα

)+

=

(
β(IMS∆)

b∫
a

F (t)dα+ γ(IMS∆)

b∫
a

G(t)dα

)−
.

(3) If β > 0 and γ < 0, (or β < 0 and γ > 0), then

(MS∆)

b∫
a

(βF (t) + γG(t))−dα = (MS∆)

b∫
a

(βF−(t) + γG+(t))dα

= β(MS∆)

b∫
a

F−(t)dα+ γ(MS∆)

b∫
a

G+(t)dα

= β

(
(IMS∆)

b∫
a

F (t)dα

)−
+ γ

(
(IMS∆)

b∫
a

G(t)dα

)+



M. E. Hamid / Eur. J. Pure Appl. Math, 11 (2) (2018), 493-504 498

=

(
β(IMS∆)

b∫
a

F (t)dα+ γ(IMS∆)

b∫
a

G(t)dα

)−
.

Similarly, for four cases above we have

(MS∆)

b∫
a

(βF (t) + γG(t))+dα =

(
β(IMS∆)

b∫
a

F (t)dα+ γ(IMS∆)

b∫
a

G(t)dα

)+

. (3.11)

Hence by Theorem 2 βF (t) + γG(t) ∈ IMSα∆[a, b]T and

(IMS∆)

b∫
a

(βF (t) + γG(t))dα = β(IMS∆)

b∫
a

F (t)dα+ γ(IMS∆)

b∫
a

G(t)dα. (3.12)

Theorem 4. Let α : [a, b]T → R be an increasing function. If F (t) ∈ IMSα∆[a, c]T and
F (t) ∈ IMSα∆[c, b]T, then F (t) ∈ IMSα∆[a, b]T and

(IMS∆)

b∫
a

F (t)dα = (IMS∆)

c∫
a

F (t)dα+ (IMS∆)

b∫
c

F (t)dα. (3.13)

Proof. If F (t) ∈ IMSα∆[a, c]T and F (t) ∈ IMSα∆[c, b]T, then by Theorem 2 F−(t), F+(t) ∈
MSα∆[a, c]T and F−(t), F+(t) ∈MSα∆[c, b]T. Hence F−(t), F+(t) ∈MSα∆[a, b]T and

(MS∆)

b∫
a

F−(t)dα = (MS∆)

c∫
a

F−(t)dα+ (MS∆)

b∫
c

F−(t)dα

=

(
(IMS∆)

c∫
a

F (t)dα+ (IMS∆)

b∫
c

F (t)dα

)−
.

Similarly, (MS∆)
b∫
a
F+(t)dα =

(
(IMS∆)

c∫
a
F (t)dα + (IMS∆)

b∫
c
F (t)dα

)+

. Hence by

Theorem 2 F (t) ∈ IMSα∆[a, b]T and

(IMS∆)

b∫
a

F (t)dα = (IMS∆)

c∫
a

F (t)dα+ (IMS∆)

b∫
c

F (t)dα. (3.14)

Theorem 5. Let α : [a, b]T → R be an increasing function. If F (t) ≤ G(t) almost
everywhere with respect to α on [a, b]T and F (t), G(t) ∈ IMSα∆[a, b]T, then

(IMS∆)

b∫
a

F (t)dα ≤ (IMS∆)

b∫
a

G(t)dα. (3.15)



M. E. Hamid / Eur. J. Pure Appl. Math, 11 (2) (2018), 493-504 499

Proof. Let F (t) ≤ G(t) almost everywhere with respect to α on [a, b]T and F (t), G(t) ∈
IMSα∆[a, b]T. Then F−(t), F+(t), G−(t), G+(t) ∈MSα∆[a, b]T and F−(t) ≤ G−(t), F+(t) ≤

G+(t) nearly everywhere with respect to α on [a, b]T. By Theorem 1 (MS∆)
b∫
a
F−(t)dα ≤

(MS∆)
b∫
a
G−(t)dα and (MS∆)

b∫
a
F+(t)dα ≤ (MS∆)

b∫
a
G+(t)dα. Hence

(IMS∆)

b∫
a

F (t)dα ≤ (IMS∆)

b∫
a

G(t)dα, (3.16)

by Theorem 2.

Theorem 6. Let α : [a, b]T → R be an increasing function. Let F (t), G(t) ∈ IMSα∆[a, b]T
and d(F (t), G(t)) is (MS∆) integrable with respect to α on [a, b]T. Then

d
(
(IMS∆)

b∫
a

F (t)dα, (IMS∆)

b∫
a

G(t)dα
)
≤ (MS∆)

b∫
a

d
(
F (t), G(t)

)
dα. (3.17)

Proof. By definition of distance,

d
(
(IMS∆)

b∫
a

F (t)dα, (IMS∆)

b∫
a

G(t)dα
)

= max

(∣∣∣∣((IMS∆)

b∫
a

F (t)dα

)−
−
(

(IMS∆)

b∫
a

G(t)dα

)−∣∣∣∣, ∣∣∣∣((IMS∆)

b∫
a

F (t)dα

)+

−
(

(IMS∆)

b∫
a

G(t)dα

)+∣∣∣∣)

= max

(∣∣∣∣(MS∆)

b∫
a

(
F−(t)−G−(t)

)
dα

∣∣∣∣, ∣∣∣∣(MS∆)

b∫
a

(
F+(t)−G+(t)

)
dα

∣∣∣∣)

≤ max

(
(MS∆)

b∫
a

∣∣∣∣F−(t)−G−(t)

∣∣∣∣dα, (MS∆)

b∫
a

∣∣∣∣F+(t)−G+(t)

∣∣∣∣dα)

≤ (MS∆)

b∫
a

max

(∣∣∣∣F−(t)−G−(t)

∣∣∣∣dα, ∣∣∣∣F+(t)−G+(t)

∣∣∣∣dα)

= (MS∆)

b∫
a

d
(
F (t), G(t)

)
dα. (3.18)



M. E. Hamid / Eur. J. Pure Appl. Math, 11 (2) (2018), 493-504 500

4. The MS∆ integral of fuzzy-number-valued functions on time scales

In this section, we introduce the notion of the (MS∆) integral of fuzzy-number-valued
functions on time scales and discusses some of their properties.

Definition 6. [6, 8, 9] Let à ∈ F (R) be a fuzzy subset on R. If for any λ ∈ [0, 1],
Aλ = [A−λ , A

+
λ ] and A1 6= φ, where Aλ = {t : Ã(t) ≥ λ}, then à is called a fuzzy number.

If à is (1) convex, (2) normal, (3) upper semi-continuous, (4) has the compact support,
we say that à is a compact fuzzy number.

Let R̃ denote the set of all compact.

Definition 7. [6] Let Ã, B̃ ∈ R̃, we define (1) Ã ≤ B̃ iff Aλ ≤ Bλ for all λ ∈ (0, 1], (2)
à + B̃ = C̃ iff Aλ + Bλ = Cλ for any λ ∈ (0, 1], (3) à · B̃ = D̃ iff Aλ · Bλ = Dλ for any
λ ∈ (0, 1].

For Ã, B̃ ∈ R̃C , then
D(Ã, B̃) = sup

λ∈[0,1]
d(Aλ, Bλ), (4.1)

is called the distance between à and B̃.

Lemma 1. [1] If a mapping H : [0, 1]→ IR, λ→ H(λ) = [mλ, nλ], satisfies [mλ1 , nλ1 ] ⊃
[mλ2 , nλ2 ] when λ1 < λ2, then

à :=
⋃

λ∈(0,1]

λH(λ) ∈ R̃ (4.2)

and

Aλ =
∞⋂
n=1

H(λn), (4.3)

where λn = [1− 1
(n+1) ]λ.

Definition 8. Let α : [a, b]T → R be an increasing function and let F̃ : [a, b]T → R̃. If the
interval-valued function Fλ(t) = [F−λ (t), F+

λ (t)] is (MS∆) integrable with respect to α on

[a, b]T for any λ ∈ (0, 1], then F̃ (t) is called (MS∆) integrable with respect to α on [a, b]T
and the integral is defined by (MS∆) integral as follow:

(FMS∆)

b∫
a

F̃ (t)dα :=
⋃

λ∈(0,1]

λ(IMS∆)

b∫
a

Fλ(t)dα

=
⋃

λ∈(0,1]

λ

[
(MS∆)

b∫
a

F−λ (t)dα, (MS∆)

b∫
a

F+
λ (t)dα

]
.

We write F̃ (t) ∈ FMSα∆[a, b]T.



M. E. Hamid / Eur. J. Pure Appl. Math, 11 (2) (2018), 493-504 501

Theorem 7. If F̃ (t) ∈ FMSα∆[a, b]T, then (FMS∆)
b∫
a
F̃ (t)dα ∈ R̃ and

[
(FMS∆)

b∫
a

F̃ (t)dα

]
λ

=
∞⋂
n=1

(IMS∆)

b∫
a

Fλn(t)dα, (4.4)

where λn = [1− 1
(n+1) ]λ.

Proof. LetH : (0, 1]→ IR, be defined byH(λ) =
[
(MS∆)

b∫
a
F−λ (t)dα, (MS∆)

b∫
a
F+
λ (t)dα

]
.

Since F−λ (t) and F+
λ (t) are increasing and decreasing on λ respectively, therefore, when

0 < λ1 ≤ λ2 ≤ 1, we have F−λ1(t) ≤ F−λ2(t), F+
λ1

(t) ≥ F+
λ2

(t), on [a, b]T. From Theorem 5
we have[

(MS∆)

b∫
a

F−λ1(t)dα, (MS∆)

b∫
a

F+
λ1

(t)dα

]
⊃
[
(MS∆)

b∫
a

F−λ2(t)dα, (MS∆)

b∫
a

F+
λ2

(t)dα

]
.

(4.5)
Using Theorem 2 and Lemma 1 we obtain

(FMS∆)

b∫
a

F̃ (t)dα :=
⋃

λ∈(0,1]

λ

[
(MS∆)

b∫
a

F−λ (t)dα, (MS∆)

b∫
a

F+
λ (t)dα

]
∈ R̃ (4.6)

and for all λ ∈ (0, 1],

[
(FMS∆)

b∫
a

F̃ (t)dα
]
λ

=

∞⋂
n=1

(IMS∆)

b∫
a

Fλn(t)dα, (4.7)

where λn = [1− 1
(n+1) ]λ.

Theorem 8. Let α : [a, b]T → R be an increasing function. If F̃ (t), G̃(t) ∈ FMSα∆[a, b]T
and β, γ ∈ R. Then βF̃ (t) + γG̃(t) ∈ FMSα∆[a, b]T and

(FMS∆)

b∫
a

(
βF̃ (t) + γG̃(t)

)
dα = β(FMS∆)

b∫
a

F̃ (t)dα+ γ(FMS∆)

b∫
a

G̃(t)dα. (4.8)

Proof. If F̃ (t), G̃(t) ∈ FMSα∆[a, b]T, then the interval-valued function Fλ(t) = [F−λ (t), F+
λ (t)]

and Gλ(t) = [G−λ (t), G+
λ (t)] are (MS∆) integrable with respect to α on [a, b]T for any

λ ∈ (0, 1] and (FMS∆)
b∫
a
F̃ (t)dα =

⋃
λ∈(0,1]

λ(IMS∆)
b∫
a
Fλ(t)dα and (FMS∆)

b∫
a
G̃(t)dα =



M. E. Hamid / Eur. J. Pure Appl. Math, 11 (2) (2018), 493-504 502

⋃
λ∈(0,1]

λ(IMS∆)
b∫
a
Gλ(t)dα. From Theorem 3 we have βFλ(t) + γGλ(t) ∈ IMSα∆[a, b]T

and (IMS∆)
b∫
a

(
βFλ(t)+γGλ(t)

)
dα = β(IMS∆)

b∫
a
Fλ(t)dα+γ(IMS∆)

b∫
a
Gλ(t)dα for any

λ ∈ (0, 1]. Hence βF̃ (t) + γG̃(t) ∈ FMSα∆[a, b]T and

(FMS∆)

b∫
a

(
βF̃ (t) + γG̃(t)

)
dα =

⋃
λ∈(0,1]

λ(IMS∆)

b∫
a

(
βFλ(t) + γGλ(t)

)
dα

=
⋃

λ∈(0,1]

λ

(
β(IMS∆)

b∫
a

Fλ(t)dα+ γ(IMS∆)

b∫
a

Gλ(t)dα

)

= β
⋃

λ∈(0,1]

λ(IMS∆)

b∫
a

Fλ(t)dα+ γ
⋃

λ∈(0,1]

λ(IMS∆)

b∫
a

Gλ(t)dα

= β(FMS∆)

b∫
a

F̃ (t)dα+ γ(FMS∆)

b∫
a

G̃(t)dα.

Theorem 9. Let α : [a, b]T → R be an increasing function. If F̃ (t) ∈ FMSα∆[a, c]T and
F̃ (t) ∈ FMSα∆[c, b]T, then F̃ (t) ∈ FMSα∆[a, b]T and

(FMS∆)

b∫
a

F̃ (t)dα = (FMS∆)

c∫
a

F̃ (t)dα+ (FMS∆)

b∫
c

F̃ (t)dα. (4.9)

Proof. If F̃ (t) ∈ FMSα∆[a, c]T and F̃ (t) ∈ FMSα∆[c, b]T, then the interval-valued func-
tion Fλ(t) = [F−λ (t), F+

λ (t)] is (MS∆) integrable with respect to α on [a, c]T and [c, b]T for

any λ ∈ (0, 1] and (FMS∆)
c∫
a
F̃ (t)dα =

⋃
λ∈(0,1]

λ(IMS∆)
c∫
a
Fλ(t)dα and (FMS∆)

b∫
c
F̃ (t)dα =

⋃
λ∈(0,1]

λ(IMS∆)
b∫
c
Fλ(t)dα. From Theorem 4 we have Fλ(t) ∈ IMSα∆[a, b]T and (IMS∆)

b∫
a
Fλ(t)dα =

(IMS∆)
c∫
a
Fλ(t)dα + (IMS∆)

b∫
c
Fλ(t)dα for any λ ∈ (0, 1]. Hence F̃ (t) ∈ FMSα∆[a, b]T

and

(FMS∆)

b∫
a

F̃ (t)dα =
⋃

λ∈(0,1]

λ(IMS∆)

b∫
a

Fλ(t)dα

=
⋃

λ∈(0,1]

λ

(
(IMS∆)

c∫
a

Fλ(t)dα+ (IMS∆)

b∫
c

Fλ(t)dα

)



M. E. Hamid / Eur. J. Pure Appl. Math, 11 (2) (2018), 493-504 503

=
⋃

λ∈(0,1]

λ(IMS∆)

c∫
a

Fλ(t)dα+
⋃

λ∈(0,1]

λ(IMS∆)

b∫
c

Fλ(t)dα

= (FMS∆)

c∫
a

F̃ (t)dα+ (FMS∆)

b∫
c

F̃ (t)dα.

Theorem 10. Let α : [a, b]T → R be an increasing function. If F̃ (t) ≤ G̃(t) almost
everywhere with respect to α on [a, b]T and F̃ (t), G̃(t) ∈ FMSα∆[a, b]T, then

(FMS∆)

b∫
a

F̃ (t)dα ≤ (FMS∆)

b∫
a

G̃(t)dα. (4.10)

Proof. If F̃ (t) ≤ G̃(t) almost everywhere with respect to α on [a, b]T and F̃ (t), G̃(t) ∈
FMSα∆[a, b]T, then Fλ(t) ≤ Gλ(t) nearly everywhere with respect to α on [a, b]T for any
λ ∈ (0, 1] and Fλ(t) and Gλ(t) are (MS∆) integrable with respect to α on [a, b]T for any

λ ∈ (0, 1] and (FMS∆)
b∫
a
F̃ (t)dα =

⋃
λ∈(0,1]

λ(IMS∆)
b∫
a
Fλ(t)dα and (FMS∆)

b∫
a
G̃(t)dα =

⋃
λ∈(0,1]

λ(IMS∆)
b∫
a
Gλ(t)dα. From Theorem 5 we have (IMS∆)

b∫
a
Fλ(t)dα ≤ (IMS∆)

b∫
a
Gλ(t)dα

for any λ ∈ (0, 1]. Hence

(FMS∆)

b∫
a

F̃ (t)dα =
⋃

λ∈(0,1]

λ(IMS∆)

b∫
a

Fλ(t)dα

≤
⋃

λ∈(0,1]

λ(IMS∆)

b∫
a

Gλ(t)dα

= (FMS∆)

b∫
a

G̃(t)dα.

5. Conclusions

In this paper, we introduced the concept of the (MS∆) integrals of interval-valued func-
tions and fuzzy number- valued functions on time scales and investigated some properties
of those integrals.



REFERENCES 504

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