The fuzzy Henstock–Kurzweil delta integral
on time scales
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.
Keywords:
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 that it is hard to understand without substantial mathematical maturity. Also, the Lebesgue integral does not inherit the naturalness of the Riemann integral. Henstock H1 and Kurzweil K1 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 BPM ; H1 ; S1 ; S2 ; Y ; ZY and monographs G ; L2 ; SY . As an important branch of the HK integration theory, the fuzzy Henstock–Kurzweil (FHK) integral has been extensively studied in BPM12 ; D16 ; G04 ; GS09 ; HG15 ; M15 ; SH13 ; SH014 ; WG00 ; WG01 .
In 1988, Hilger introduced the theory of time scales in his Ph.D. thesis H4 . A time scale is an arbitrary nonempty closed subset of . The aim is to unify and generalize discrete and continuous dynamical systems, see, e.g., BT16 ; BCT2 ; BCT1 ; BP1 ; BP2 ; MR3393831 ; FB16 ; G1 ; OTT . In PT , 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 realvalued and vectorvalued functions on time scales has been developed rather intensively, see, e.g., the papers A ; C ; FMS ; MS2 ; MS1 ; ST1 ; S3 ; T ; MyID:375 and references cited therein.
In 2015, Fard and Bidgoli introduced the FHK delta integral and presented some of its basic properties FB15 . 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 3.3 and 3.7). Moreover, we obtain some convergence theorems for the FHK delta integral, in particular Theorem 3.9 of dominated convergence and Theorem 3.10 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 is called a fuzzy number provided that

is normal: there exists with ;

is fuzzy convex: for all and all ;

is upper semicontinuous;

is compact.
Denote by the space of fuzzy numbers. We define the level set by
From conditions –, is denoted by . For and , we define
for all . The Hausdorff distance between and is defined by
Then, the metric space is complete. Let . We define the halfopen interval by
The open and closed intervals are defined similarly. For , we denote by the forward jump operator, i.e., , and by the backward jump operator, i.e., . Here, we put and , where and are finite. In this situation, and , otherwise, and . If , then we say that is rightscattered, while if , then we say that is leftscattered. If and , then is called rightdense, and if and , then is leftdense. The graininess functions and are defined by and , respectively.
In what follows, all considered intervals are intervals in . A division of is a finite set of intervalpoint pairs such that
and for each . We write . We say that
is a gauge on if on , on , , and for any . The symbol stands for the set of gauge on . Let and be gauges such that
for any and for any . Then we call finer than and write . We say that is a fine HK division of if for each . Let be the set of all fine HK divisions of . Given an arbitrary , , we write
for integral sums over , whenever .
Lemma 1 (See K87 )
Suppose that . Then,

the interval is closed for ;

for ;

for any sequence satisfying and , we have .
Conversely, if a collection of subsets verify (1)–(3), then there exists a unique such that for and .
Lemma 2 (See Gv86 )
Suppose that . Then,

is bounded and nondecreasing;

is bounded and nonincreasing;

;

for , and ;

and .
Conversely, if and satisfy items (1)–(5), then there exists such that
3 The fuzzy Henstock–Kurzweil delta integral
We introduce the concept of fuzzy Henstock–Kurzweil (FHK) delta integrability.
Definition 1
A function is called FHK integrable on with the FHK integral , if for each there exists a such that for each . The family of all FHK integrable functions on is denoted by .
Remark 1
Theorem 3.1
The FHK integral of is unique.
Theorem 3.2
If and , then
with
Follows a Cauchy–Bolzano condition for the FHK integral.
Theorem 3.3 (The Cauchy–Bolzano condition)
Function if and only if for each there exists a such that
for any .
Proof
(Necessity) Let . By hypothesis, there exists such that
for any . Let . Then,
(Sufficiency) For each , choose a such that
for any . Replacing by , we may assume that . For each , fix a . For , we have , so . Thus, and it follows that is a Cauchy sequence. We denote the limit of by and let . Choose and let . Then,
Hence, .
Theorem 3.4
Let . If , then
with
Proof
Let . By assumption, there exist gauges
such that
respectively for any , , and for any , . We define on by setting
and
Now, let , . It follows that

either and ;

or and .
The case (ii) is straightforward. For (i), one has
The intended result follows.
Corollary 1
If , then for any .
Definition 2 (See Pt )
Let . We say has delta measure zero if it has Lebesgue measure zero and contains no rightscattered points. A property is said to hold a.e. on if there exists of measure zero such that holds for every .
Theorem 3.5
Let a.e. on . If , then so . Moreover,
Proof
Let . Then there exists a such that
for any , . Set , where
For each , there exists consisting of a collection of open intervals with total length less than , such that . Define
Then, for any , , one has
Therefore,
The proof is complete.
Theorem 3.6 (See Pt )
Let be given. Assume

holds a.e.;

holds a.e.;

.
Then . Moreover,
Theorem 3.7
Function if and only if , for all uniformly, i.e., the gauge in Definition 1 is independent of .
Proof
(Necessity) Let . Given , there exists a
such that for any . Then,
and
for any and for any .
Thus, ,
uniformly
for any .
(Sufficiency) Let . By assumption, there exists a such that
for any and for any , where
To prove that represents a fuzzy number, it is enough to check that satisfies items (1)–(3) of Lemma 1:

for , if , then , i.e., the interval is closed.

and are, respectively, nondecreasing and nonincreasing functions on . For any one has
This implies .

For any satisfying and , we have
that is,
and . Moreover,
Thanks to Theorem 3.6, we have and
Consequently,
Define by . Thus,
for each .
Definition 3
A sequence of HK integrable functions is called uniformly FHK integrable on if for each there exists a such that
for any and for any .
Theorem 3.8
Let , , satisfy:

on ;

are uniformly FHK integrable on .
Then and
Proof
Let . By assumption, there exists a