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In 2000, Wu and Gong [1] introduced the thought of the Henstock integrals of inter-valvalued functions and fuzzy-number-valued functions and obtained a number of their properties. The aim of this paper is to introduce the thought of the AP- Henstock integrals of interval-valued functions and fuzzy-number-valued functions which are extensions of [1] and investigate a number of their properties.

As it is well known, the Henstock integral for a real function was first defined by Henstock [

In this paper, we introduce the concept of the AP-Henstock integrals of interval-valued functions and fuzzy-number-valued functions and discuss some of their properties.

The paper is organized as follows. In Section 2, we have a tendency to provide the preliminary terminology used in this paper. Section 3 is dedicated to discussing the AP-Henstock integral of interval-valued functions. In Section 4, we introduce the AP- Henstock integral of fuzzy-number-valued functions. The last section provides conclusions.

Let

provided the limit exists. The point

A measurable set

A division

1) a division of

2)

Definition 2.1. [

where the sum

Definition 2.2. [

for each

Theorem 2.1. If

Proof. The proof is similar to the Theorem 3.6 in [

In this section, we shall give the definition of the AP-Henstock integrals of interval-valued functions and discuss some of their properties.

Definition 3.1. [

For

and

Define

Definition 3.2. [

then

Definition 3.3. A interval-valued function

whenever

Theorem 3.1. If

Proof. Let integral value is not unique and let

whenever

Whence it follows from the Triangle Inequality that:

Since for

Theorem 3.2. An interval-valued function

Proof. Let

whenever

Hence

Conversely, let

whenever

Hence

Theorem 3.3. If

Proof. If

(1) If

(2) If

(3) If

Similarly, for four cases above we have

Hence by Theorem 3.2

W

Theorem 3.4. If

Proof. If

Similarly,

W

Theorem 3.5. If

Proof. Let

by Theorem 3.2. W

Theorem 3.6. Let

Proof. By definition of distance,

W

This section introduces the concept of the AP-Henstock integral of fuzzy-number- valued functions and investigates some of their properties.

Definition 4.1. [

Let

Definition 4.2. [

For

Lemma 4.1. [

and

where

Definition 4.3. [

For brevity, we write

Definition 4.4. Let

We write

Theorem 4.1.

where

Proof. Let

Since

From Theorem 3.2 and Lemma 4.1 we have

and for all

Theorem 4.2. If

Proof. If

Hence

W

Theorem 4.3. If

Proof. If

W

Theorem 4.4. If

Proof. If

In this paper, we have a tendency to introduce the concept of the AP-Henstock integrals of interval-valued functions and fuzzy number-valued functions and investigate some properties of those integrals.

Hamid, M.E., Elmuiz, A.H. and Sheima, M.E. (2016) On AP-Henstock Integrals of Interval-Valued Functions and Fuzzy-Number-Valued Func- tions. Applied Mathematics, 7, 2285-2295. http://dx.doi.org/10.4236/am.2016.718180