Ideal Statistically Pre-Cauchy Triple Sequences of Fuzzy Number and Orlicz Functions ()
1. Introduction
The notion of statistical convergence was introduced by Fast [1] and also independently by Buck [2] and Schoenberg [3] for real and complex sequences.Over the years and under different names statistical convergence has been discussed in the theory of Fourier analysis, Ergodic theory and Number theory. Later on it was further investigated from the sequence spaces point of view and linked with summability theory by Altinok and Et [4], Connor [5], Et et al. ( [6] [7] [8] ), Fridy [9], Fridy and Orhan [10], Mursaleen [11] and many others.
Matloka [12] defined the notion of fuzzy sequence and introduced bounded and convergent sequences of fuzzy real numbers and studied their some properties. After then, Nuray and Savas [13] defined the notion of statistical convergence for sequences of fuzzy numbers. Since then, there has been increasing interest in the study of statistical convergence of fuzzy sequences (see [14] - [19] ).
Lindesstrauss and Tzafriri [20] used the idea of Orlicz sequence space,
, which is Banachi space with the norm:
. The space
is closely related to
the space
, which is an Orlicz sequence space with
for
.
Connor, Fridy and Kline [21] proved that statistical convergent sequences are statistically pre-Cauchy and any bounded statistically pre-Cauchy sequence with nowhere dense set of limit points is statistically convergent. They also gave an example showing statistically pre-Cauchy sequences are not necessarily statistically convergent.
In this paper, we extend the notions of ideal statistically convergence for sequence of fuzzy number. We introduce the notions ideal statistically pre-Cauchy triple sequences of fuzzy number about Orlicz function, and give some correlation theorem. Also, some properties of these new sequence spaces are investigated. It popularized the work of predecessors.
2. Definitions and Preliminaries
In this section, we give some basic notions which will be used throughout the paper.
Let
be a fuzzy subset on R. If
is convex, normal, upper semi-continuous and has compact support, we say that
is a fuzzy number. Let
denote the set of all fuzzy numbers.
For
, we write the level set of
as
and
. Let
, we define
iff
,
iff
and
for any
.
, where
Define
where d is the Hausdorff metric.
is called the distance between
and
.
Using the results of [22] [23], we see that
1)
is a complete metric space,
2)
,
3)
,
4)
,
5)
,
6)
,
Where
,
represents zero fuzzy number.
A sequence
of fuzzy numbers is said to be statistically convergent to a fuzzy number
if for each
the set
has natural density zero. The fuzzy number
is called the statistical limit of the sequence
and we write st-
[24].
The concept of Orlicz function was introduced by Parashar and Choudhary [25], A mapping
is said to be an Orlicz funtion [26]
1)
iff
,
2)
for
,
3)
as
,
4) M is continuous, nondecreasing and convex.
An Orlicz function may be bounded or unbounded. For example,
is bounded.
A triple sequence can be defined as a function
where N, R nad C denote the set of natural numbers, real numbers and complec numbers respectively. A triple sequence
is said to be Cauchy sequence if for every
, there exist
such that
whenever
,
,
[27].
A triple sequence
is called statistically pre-Cauchy if for every
there exist
and
such that
where the vertical bars indicate the number of elements in the set [28].
3. Main Results
Definition 3.1. A triple sequence of fuzzy numbers is said to be ideal statistically pre-Cauchy if for every
there exist
and
such that
where the I denote the nontrivival ideal of N.
Theorem 3.1. Let
be a triple sequence of fuzzy number and let M be a bounded Oricz function. Then x is ideal statistically pre-Cauchy if and only if
Proof. Suppose that
For each
and
,
, we have
Now suppose that x is ideal statistically pre-Cauchy and that
has been given.
Let
be such that
.
Since M is bounded Orlicz function, there exist an integer G such that
for all
.
Not that, for each
Hence
Theorem 3.2. Let
be a triple sequence of fuzzy numbers and let M be a bounded Orlicz function. Then x is ideal statistically convergent to
if and only if
Proof. Suppose that
For each
and
,
, we have
We have x is ideal statistically convergent to
.
Now suppose that x is ideal statistically convergent to
, let
be such that
.
Since M is bounded Orlicz function, there exist an integer G such that
for all
.
Note that, for each
Hence
Corollary 3.3. Let
be a bound triple sequence of fuzzy number. Then x is ideal statistically pre-Cauchy if and only if
Proof. Let
and defined
, then
, and
Hence
, if and only if
, and an immediate application of Theorem 3.1 completes the proof.
Corollary 3.4. Let
be a bound triple sequence of fuzzy number. Then x is ideal statistically convergent
if and only if
Proof. Let
and defined
, then
, and
Hence
, if and only if
, and an immediate application of Theorem 3.1 completes the proof.
4. Conclusion
In this article, we introduced ideal statistically pre-Cauchy triple sequences of fuzzy numbers about Orlicz function. At the same time, we have proved some properties and relationships.
Fund
This work is supported by National Natural Science Fund of China (11761056); the Natural Science Foundation of Qinghai Province (2020-ZJ-920); University level planning project of Qinghai Minzu University (2021XJGH24).