Tauberian conditions under which convergence follows from the weighted mean summability and its statistical extension for sequences of fuzzy number
UDC 517.5 Let $(p_n)$ be a sequence of nonnegative numbers such that $p_0>0$ and$$P_n:=\sum_{k=0}^{n}p_k\to\infty\qquad \text{as}\qquad n\to\infty.$$Let $(u_n)$ be a sequence of fuzzy numbers.The weighted mean of $(u_n)$ is defined by$$t_n:=\frac{1}{P_n}\sum_{k=0}^{n}p_ku_k\qquad \text{fo...
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Institute of Mathematics, NAS of Ukraine
2021
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| author | Önder, Z. Çanak, İ. Önder, Zerrin Çanak, İbrahim Önder, Z. Çanak, İ. |
| author_facet | Önder, Z. Çanak, İ. Önder, Zerrin Çanak, İbrahim Önder, Z. Çanak, İ. |
| author_sort | Önder, Z. |
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| description | UDC 517.5
Let $(p_n)$ be a sequence of nonnegative numbers such that $p_0>0$ and$$P_n:=\sum_{k=0}^{n}p_k\to\infty\qquad \text{as}\qquad n\to\infty.$$Let $(u_n)$ be a sequence of fuzzy numbers.The weighted mean of $(u_n)$ is defined by$$t_n:=\frac{1}{P_n}\sum_{k=0}^{n}p_ku_k\qquad \text{for}\qquad n =0,1,2,\ldots \,. $$It is known that the existence of the limit $\lim u_n=\mu_{0}$ implies that of $\lim t_n=\mu_{0}.$ For the the existence of the limit $st$-$\lim t_n=\mu_{0},$ we require the boundedness of $(u_n)$ in addition to the existence of the limit $\lim u_n=\mu_{0}.$ But, in general, the converse of this implication is not true. In this paper, we obtain Tauberian conditions, under which the existence of the limit $\lim u_n=\mu_{0}$ follows from that of $\lim t_n=\mu_{0}$ or $st$-$\lim t_n=\mu_{0}.$ These Tauberian conditions are satisfied if $(u_n)$ satisfies the two-sided condition of Hardy type relative to $(P_n).$ |
| doi_str_mv | 10.37863/umzh.v73i8.584 |
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DOI: 10.37863/umzh.v73i8.584
UDC 517.5
Z. Önder, İ. Çanak (Ege Univ., Izmir, Turkey)
TAUBERIAN CONDITIONS UNDER WHICH CONVERGENCE
FOLLOWS FROM THE WEIGHTED MEAN SUMMABILITY
AND ITS STATISTICAL EXTENSION FOR SEQUENCES OF FUZZY NUMBER
ТАУБЕРОВI УМОВИ, ЗА ЯКИХ ЗБIЖНIСТЬ ВИПЛИВАЄ
З СЕРЕДНЬОВАГОВОЇ СУМОВНОСТI, ТА ЇХ СТАТИСТИЧНЕ
РОЗШИРЕННЯ НА ПОСЛIДОВНОСТI НЕЧIТКИХ ЧИСЕЛ
Let (pn) be a sequence of nonnegative numbers such that p0 > 0 and
Pn :=
n\sum
k=0
pk \rightarrow \infty as n \rightarrow \infty .
Let (un) be a sequence of fuzzy numbers. The weighted mean of (un) is defined by
tn :=
1
Pn
n\sum
k=0
pkuk for n = 0, 1, 2, . . . .
It is known that the existence of the limit \mathrm{l}\mathrm{i}\mathrm{m}un = \mu 0 implies that of \mathrm{l}\mathrm{i}\mathrm{m} tn = \mu 0. For the the existence of the limit
st-\mathrm{l}\mathrm{i}\mathrm{m} tn = \mu 0, we require the boundedness of (un) in addition to the existence of the limit \mathrm{l}\mathrm{i}\mathrm{m}un = \mu 0. But, in general,
the converse of this implication is not true. In this paper, we obtain Tauberian conditions, under which the existence of the
limit \mathrm{l}\mathrm{i}\mathrm{m}un = \mu 0 follows from that of \mathrm{l}\mathrm{i}\mathrm{m} tn = \mu 0 or st-\mathrm{l}\mathrm{i}\mathrm{m} tn = \mu 0. These Tauberian conditions are satisfied if (un)
satisfies the two-sided condition of Hardy type relative to (Pn).
Нехай (pn) — послiдовнiсть невiд’ємних чисел таких, що p0 > 0 i
Pn :=
n\sum
k=0
pk \rightarrow \infty при n \rightarrow \infty .
Нехай (un) — послiдовнiсть нечiтких чисел. Вагове середнє для (un) визначається як
tn :=
1
Pn
n\sum
k=0
pkuk для n = 0, 1, 2, . . . .
Вiдомо, що з iснування границi \mathrm{l}\mathrm{i}\mathrm{m}un = \mu 0 випливає \mathrm{l}\mathrm{i}\mathrm{m} tn = \mu 0. Для iснування границi st-\mathrm{l}\mathrm{i}\mathrm{m} tn = \mu 0
вимагається обмеженiсть (un) як додаткова умова до iснування границi \mathrm{l}\mathrm{i}\mathrm{m}un = \mu 0. Але обернена iмплiкацiя
взагалi не є правильною. У цiй роботi запропоновано тауберовi умови, за яких iснування границi \mathrm{l}\mathrm{i}\mathrm{m}un = \mu 0
випливає з того, що \mathrm{l}\mathrm{i}\mathrm{m} tn = \mu 0 або st-\mathrm{l}\mathrm{i}\mathrm{m} tn = \mu 0. Цi тауберовi умови виконуються, якщо (un) задовольняє
двостороннi умови типу Гардi вiдносно (Pn).
1. Introduction. In this section, we begin with some remarks about history of fuzzy set theory and its
applications to (N, p) summability method, that is about the history from almost fifty years ago until
these days. We shortly mention emergence of the concept of statistical convergence for sequences
of fuzzy numbers and advancement of that in Tauberian theory. In the sequel, bringing together the
concepts of (N, p) summability and statistical convergence for sequences of fuzzy numbers under the
same roof, we refer certain results obtained by several researchers concerning these concepts. After
dwelling on studies that encourages us to do this research, we complete this section summarizing
theorems and corollaries attained in this paper.
c\bigcirc Z. ÖNDER, İ. ÇANAK, 2021
ISSN 1027-3190. Укр. мат. журн., 2021, т. 73, № 8 1085
1086 Z. ÖNDER, İ. ÇANAK
Improved based upon the fuzzy sets and fuzzy set operations which was introduced by Zadeh [1],
fuzzy set theory has increasingly received attention from researchers in a diverse range of disciplines
in the last few years. Aspiring to apply concept of fuzziness to individual works with a broad
viewpoint from theoretical to practical in almost all sciences and technology, researchers have reached
numerous and varied applications of its in fields such as statistics, nuclear science, biomedicine,
agriculture, geography, weather prediction, finance and stock market, engineering, computer science,
artificial intelligence, pattern recognition, handwriting analysis, decision theory, robotics etc. In
addition to these, one of areas which the concept of fuzziness was carried out is also pure mathematics
and there have been several authors discussing many important properties and applications of fuzzy
sets. Dubois and Prade [2] introduced the fuzzy numbers and defined basic operations of addition,
subtraction, multiplication and division. In [3], Goetschel and Voxman presented a less restrictive
definition of fuzzy numbers. Matloka [4] introduced bounded and convergent sequences of fuzzy
numbers, studied some of their properties and showed that every convergent sequence of fuzzy
numbers is bounded. Nanda [5] studied the spaces of bounded and convergent sequence of fuzzy
numbers and proved that they are complete metric spaces.
In recent years, there has been an increasing interest on summability methods of sequences of
fuzzy numbers. One of these summability methods which has attracted the attention of many re-
searchers is (N, p) summability method. Tripathy and Baruah [6] introduced (N, p) method for
sequences of fuzzy numbers and obtained fuzzy analogues of classical Tauberian theorems for this
method. Çanak [7] investigated some conditions needed for the (N, p) summable sequences to
be convergent. Later, Önder et al. [8] established the Tauberian condition controlling one-sided
oscillatory behavior of a sequence of fuzzy numbers for the (N, p) summability method. Con-
trary to the common belief that the concept of statistical convergence, which is a natural gene-
ralization of that of ordinary convergence, was introduced by Fast [9] and Schoenberg [10], this
concept was firstly came up with by Zygmund [11] who used the term almost convergence in
place of statistical convergence and proved some theorems related to it. After the definition of
statistical convergence was put into the final form by Fast [9] and Schoenberg [10], Nuray and
Savaş [12] extended this concept to sequences of fuzzy numbers and discussed some properties
related to its. Savaş [13] obtained some equivalent conditions for a sequence of fuzzy numbers
to be statistically convergent and statistically Cauchy. Aytar and Pehlivan [14] indicated that sta-
tistical convergence of a sequence of fuzzy numbers is equivalent to uniform statistical conver-
gence of the sequence of functions which are defined via the endpoints of a-cuts of same se-
quence. In [15], Başar presented some results on statistical convergence of sequences of fuzzy
numbers. In the sequel, this concept was associated with Tauberian conditions given by several
researchers from past to present. Kwon [16] established the Tauberian theorem and a decompo-
sition theorem for statistical convergence of sequences of fuzzy numbers. Considering statistical
convergence as a regular summability method, Talo and Başar [17] found out that necessary condi-
tions for convergence of sequences of fuzzy numbers which are statistically convergent are slow
decrease and one-sided condition nun \succeq nun - 1 - \=H for some H > 0. After the results ob-
tained related to concept of statistical convergence were published, it was combined with (N, p)
summability method. In relation to that, Talo and Bal [18] presented Tauberian conditions under
which statistical convergence of sequence of fuzzy numbers follows from its statistically (N, p)
ISSN 1027-3190. Укр. мат. журн., 2021, т. 73, № 8
TAUBERIAN CONDITIONS UNDER WHICH CONVERGENCE FOLLOWS . . . 1087
summability. For some interesting results on statistical convergence of fuzzy numbers, we refer to
[6, 19].
Besides the studies mentioned up to now, the studies that encourage us to do this research is in
fact those including some results obtained by Móricz [20, 21] for Cesàro (or (C, 1)) and Harmonic
(or (H, 1)) summability methods for sequence of real numbers. Móricz formulated these results as
follows.
Theorem 1 [20]. If the real (or complex) sequence (sn) is statistically (C, 1) summable to \mu
and slowly decreasing (or slowly oscillating), then (sn) is convergent to \mu .
Theorem 2 [21]. If the real (or complex) sequence (sn) is statistically (H, 1) summable to \mu
and slowly decreasing (or slowly oscillating) with respect to the (H, 1) summability, then (sn) is
convergent to \mu .
In case that pn = 1 and pn =
1
n
for all nonnegative integers n, (N, p) summability method
reduces to Cesàro and Harmonic summability methods, respectively. Here, our aim extend the
theorems presented by Móricz for Cesàro and Harmonic summability methods for sequence of real
(or complex) numbers to the (N, p) summability method for sequence of fuzzy numbers.
In this paper, we indicate that some conditions under which convergence follows from (N, p)
summability and its statistical extension for sequences of fuzzy numbers. In Section 2, we recall
some notations, basic definitions and theorems for fuzzy numbers. In Section 3, we present some
lemmas which will be benefited in the proofs of main results for sequences of fuzzy numbers. In
Section 4, we prove the Tauberian theorem for (N, p) summable sequences of fuzzy numbers. In
the sequel, replacing (N, p) summable sequences of fuzzy numbers by statistically (N, p) summable
sequences of fuzzy numbers, we establish the Tauberian theorem which convergence follows from
statistically (N, p) summability under condition of slow oscillation relative to (Pn) and additional
conditions on (pn) and we present a corollary related to this theorem. We end this section by giving
an extension of the obtained theorem and corollary from statistically (N, p) summability method to
statistically (N, p, \alpha ) summability method.
2. Preliminaries. In this section, we present background needed to make easier readability of
our study. For this, we begin with basic definitions and notations with respect to fuzzy numbers that
will be used throughout this paper. In the sequel, we mention its linear structure, set operations on
the space of fuzzy numbers and some algebraic properties related to its. We recall metric on the
space of fuzzy numbers and exhibit a list of fundamental properties of its. We end this section by
giving some definitions concerning the sequences of fuzzy numbers. For the sake of completeness of
the paper, we give our study in Section 4.
In [3], Goetschel and Voxman introduced concept of fuzzy numbers as follows.
Definition 1. Consider a fuzzy subset of real line u : \BbbR \rightarrow [0, 1]. Then the mapping u is a fuzzy
number if it satisfies following additional properties:
(i) u is normal, i.e., there exists t0 \in \BbbR such that u(t0) = 1;
(ii) u is fuzzy convex, i.e., for any t0, t1 \in \BbbR and for any \alpha \in [0, 1], u(\alpha t0 + (1 - \alpha )t1) \geq
\geq \mathrm{m}\mathrm{i}\mathrm{n}\{ u(t0), u(t1)\} ;
(iii) u is upper semicontinuous on \BbbR ;
(iv) the support of u, [u]0 := \{ t \in \BbbR : u(t) > 0\} is compact, where \{ t \in \BbbR : u(t) > 0\} denotes
the closure of the set \{ t \in \BbbR : u(t) > 0\} in usual topology of \BbbR .
ISSN 1027-3190. Укр. мат. журн., 2021, т. 73, № 8
1088 Z. ÖNDER, İ. ÇANAK
We denote the set of all fuzzy numbers on \BbbR by E1 and call it the space of fuzzy numbers.
We recall the linear structure of E1 as follows. For u \in E1, the \lambda -level set of u is defined by
[u]\lambda :=
\Biggl\{
\{ x \in \BbbR : u(x) \geq \lambda \} , 0 < \lambda \leq 1,
\{ x \in \BbbR : u(x) > \lambda \} , \lambda = 0.
Then, it is easily established (see [22]) that u is a fuzzy number if and only if [u]\lambda is a closed,
bounded and nonempty interval for each \lambda \in [0, 1] with [u]\beta \subseteq [u]\lambda if 0 \leq \lambda \leq \beta \leq 1. From this
characterization of fuzzy numbers, it follows that a fuzzy number u is completely determined by the
end points of the intervals [u]\lambda = [u - (\lambda ), u+(\lambda )] where u - (\lambda ) \leq u+(\lambda ) and u - (\lambda ), u+(\lambda ) \in \BbbR for
each \lambda \in [0, 1].
In the sequel, Goetschel and Voxman [3] presented another representation of a fuzzy number as
a pair of functions that satisfy some properties.
Theorem 3 [3]. Let u \in E1 and [u]\lambda = [u - (\lambda ), u+(\lambda )]. Then the functions u - , u+ : [0, 1] \rightarrow
\rightarrow \BbbR , defining the endpoints of the \lambda -level sets, satisfy following conditions:
(i) u - (\lambda ) \in \BbbR is a bounded, non-decreasing and left continuous function on (0, 1];
(ii) u+(\lambda ) \in \BbbR is a bounded, non-increasing and left continuous function on (0, 1];
(iii) The functions u - (\lambda ) and u+(\lambda ) are right continuous at \lambda = 0;
(iv) u - (1) \leq u+(1).
Conversely, if the pair of functions f and g satisfies the above conditions (i) – (iv), then there
exists a unique fuzzy number u such that [u]\lambda := [f(\lambda ), g(\lambda )] for each \lambda \in [0, 1] and u(x) :=
:= \mathrm{s}\mathrm{u}\mathrm{p}\lambda \in [0,1] \{ \lambda : f(\lambda ) \leq x \leq g(\lambda )\} .
Suppose that u, v \in E1 are represented by [u - (\lambda ), u+(\lambda )] and [v - (\lambda ), v+(\lambda )] for each \lambda \in
\in [0, 1], respectively. Then the operations addition, subtraction and scalar multiplication on the set
of fuzzy numbers are defined as follows:
[u+ v]\lambda :=
\bigl[
u - (\lambda ) + v - (\lambda ), u+(\lambda ) + v+(\lambda )
\bigr]
,
[u - v]\lambda :=
\bigl[
u - (\lambda ) - v+(\lambda ), u+(\lambda ) - v - (\lambda )
\bigr]
,
[ku]\lambda = k[u]\lambda :=
\Biggl\{
[ku - (\lambda ), ku+(\lambda )] , k \geq 0,
[ku+(\lambda ), ku - (\lambda )] , k < 0.
The set of all real numbers can be embedded in E1. For r \in \BbbR , \=r \in E1 is defined by
\=r(x) :=
\Biggl\{
1, x = r,
0, x \not = r.
The following lemma deals with the algebraic properties of fuzzy numbers.
Lemma 1 [23]. On the set of fuzzy numbers there are two binary operations, denoted by +, .
and called addition, scalar multiplication, respectively. These operations satisfy following properties:
(i) the addition of fuzzy numbers is associative and commutative, i.e., u + v = v + u and
u+ (v + w) = (u+ v) + w for any u, v, w \in E1;
(ii) \=0 \in E1 is neutral element with respect to +, i.e., u+ \=0 = \=0 + u = u for any u \in E1;
(iii) with respect to +, none of u \in E1 \setminus \BbbR has opposite in E1;
(iv) \=1 \in E1 is neutral element with respect to, i.e., u\=1 = \=1u = u for any u \in E1;
ISSN 1027-3190. Укр. мат. журн., 2021, т. 73, № 8
TAUBERIAN CONDITIONS UNDER WHICH CONVERGENCE FOLLOWS . . . 1089
(v) for any a, b \in \BbbR with ab \geq 0 and any u \in E1, we have (a + b)u = au + bu; for general
a, b \in \BbbR , this property does not holds;
(vi) for any a \in \BbbR and u, v \in E1, we have a(u+ v) = au+ av;
(vii) for any a, b \in \BbbR and u \in E1, we have (ab)u = a(bu).
As a conclusion, we attain by Lemma 1 that the space of fuzzy numbers is not a linear space.
Concept of metric space may be defined as an arbitrary fuzzy set which a distance between all
elements of the set are described. It is possible to define several different metrics on the space of
fuzzy numbers; however, the most well-known and preferential metric among these metrics is the
Hausdorff distance for fuzzy numbers based on the classical Hausdorff distance between compact
convex subsets of \BbbR n. Let W denote the set of all closed and bounded intervals. For the case when
A = [a - , a+], B = [b - , b+] are the two intervals, the Hausdorff distance on W is defined by
d(A,B) := \mathrm{m}\mathrm{a}\mathrm{x}
\bigl\{ \bigm| \bigm| a - - b -
\bigm| \bigm| , \bigm| \bigm| a+ - b+
\bigm| \bigm| \bigr\} .
It can be noted that W is a complete separable metric space on the basis of the Hausdorff distance
(cf. Nanda [5]). Now, we may define the metric D on the space of fuzzy numbers with the help of
the Hausdorff metric d.
Definition 2 [23]. Let D : E1 \times E1 \rightarrow \BbbR + and let u, v \in E1 represented, respectively, by
[u - (\lambda ), u+(\lambda )] and [v - (\lambda ), v+(\lambda )] for each \lambda \in [0, 1]
D(u, v) = \mathrm{s}\mathrm{u}\mathrm{p}
\lambda \in [0,1]
d([u]\lambda , [v]\lambda ).
Then D is called the Hausdorff distance between fuzzy numbers u and v.
It is easy to see that
D(u, \=0) = \mathrm{s}\mathrm{u}\mathrm{p}
\lambda \in [0,1]
\mathrm{m}\mathrm{a}\mathrm{x}
\bigl\{ \bigm| \bigm| u - (\lambda )\bigm| \bigm| , \bigm| \bigm| u+(\lambda )\bigm| \bigm| \bigr\} = \mathrm{m}\mathrm{a}\mathrm{x}
\bigl\{ \bigm| \bigm| u - (0)\bigm| \bigm| , \bigm| \bigm| u+(0)\bigm| \bigm| \bigr\} .
The following lemma presents some fundamental properties of the Hausdorff distance between
fuzzy numbers.
Lemma 2 [23]. Let u, v, w, z \in E1 and k \in \BbbR . Then following statements hold true:
(i) (E1, D) is a complete metric space;
(ii) D(u+ w, v + w) = D(u, v), i.e., D is translation invariant;
(iii) D(ku, kv) = | k| D(u, v);
(iv) D(u+ v, w + z) \leq D(u,w) +D(v, z);
(v) | D(u, \=0) - D(v, \=0)| \leq D(u, v) \leq D(u, \=0) +D(v, \=0).
We now refer following definitions concerning sequences of fuzzy numbers which will be needed
in the sequel.
Definition 3 [4]. A sequence u = (un) of fuzzy numbers is a function u from the set \BbbN of all
positive integers into the set E1. The fuzzy number un denotes the value of the function at a point
n \in \BbbN and is called the nth term of the sequence.
We denote the set of all sequences of fuzzy numbers by \omega (F ).
Definition 4 [4]. A sequence (un) of fuzzy numbers is said to be convergent to the fuzzy number
\mu 0, written as \mathrm{l}\mathrm{i}\mathrm{m}n\rightarrow \infty un = \mu 0, if for every \epsilon > 0 there exists a positive integer n0 such that
D(un, \mu 0) < \epsilon whenever n \geq n0. (1)
The number \mu 0 is called the limit of (un).
ISSN 1027-3190. Укр. мат. журн., 2021, т. 73, № 8
1090 Z. ÖNDER, İ. ÇANAK
We denote the set of all convergent sequences of fuzzy numbers by c(F ).
Definition 5 [6]. Let (un) be a sequence of fuzzy numbers and p = (pn) be a sequence of
nonnegative numbers such that
p0 > 0 and Pn :=
n\sum
k=0
pk \rightarrow \infty as n \rightarrow \infty . (2)
Weighted means of (un) is defined by
t(1)n :=
1
Pn
n\sum
k=0
pkuk for n \in \BbbN .
A sequence (un) is said to be summable by weighted mean method determined by the sequence p to
the fuzzy number \mu 0 if for every \epsilon > 0 there exists a positive integer n0 such that
D(t(1)n , \mu 0) < \epsilon for n > n0.
Weighted mean methods are also called Riesz methods or (N, p) methods in the literature.
(N, p) summability method is regular if and only if condition (2) is satisfied. In other words,
every convergent sequence of fuzzy numbers is also (N, p) summable to the same number under
condition (2). However, converse of this statement is not true in general. Truth of that is possible
under some suitable condition which is so-called the Tauberian condition on the sequence. Any
theorem stating that convergence of a sequence follows from its (N, p) summability and some
Tauberian condition is said to be the Tauberian theorem for (N, p) summability method.
If pn = 1 for all n \in \BbbN , then (N, p) summability method reduces to Cesàro summability method.
In addition, weighted means of integer order \alpha \geq 0 of a sequence (un) of fuzzy numbers is
defined by
t(\alpha )n :=
\left\{
1
Pn
n\sum
k=0
pkt
(\alpha - 1)
k , if \alpha \geq 1,
un, if \alpha = 0.
As similar to first order, a sequence (un) is said to be summable by weighted mean method of integer
order \alpha \geq 0 determined by the sequence p to the fuzzy number \mu 0 if, for every \epsilon > 0, there exists
a positive integer n0 such that
D(t(\alpha )n , \mu 0) < \epsilon for n > n0.
We present definition of natural density of K \subset \BbbN and generate statistically convergent sequences
of fuzzy numbers by using this concept. Let K \subset \BbbN be a subset of positive integers and Kn = \{ k \in
\in K : k \leq n\} . Then the set K has a natural density if sequence
\biggl(
| Kn|
n+ 1
\biggr)
has a limit. In this case,
we write \delta (K) = \mathrm{l}\mathrm{i}\mathrm{m}n\rightarrow \infty
| Kn|
n+ 1
, where the vertical bar denotes the cardinality of the enclosed set.
In [12], Nuray and Savaş introduced concept of statistical convergence for sequences of fuzzy
numbers as follows.
ISSN 1027-3190. Укр. мат. журн., 2021, т. 73, № 8
TAUBERIAN CONDITIONS UNDER WHICH CONVERGENCE FOLLOWS . . . 1091
Definition 6 [12]. A sequence (un) of fuzzy numbers is said to be statistically convergent to the
fuzzy number \mu 0 if for every \epsilon > 0 the set K\epsilon := \{ k \in \BbbN : D(uk, \mu 0) \geq \epsilon , k \leq n\} has natural
density zero, i.e., for each \epsilon > 0,
\mathrm{l}\mathrm{i}\mathrm{m}
n\rightarrow \infty
1
n+ 1
\bigm| \bigm| \bigm| \{ k \in \BbbN : D(uk, \mu 0) \geq \epsilon , k \leq n\}
\bigm| \bigm| \bigm| = 0. (3)
We denote the set of all statistically convergent sequences of fuzzy numbers by st(F ). In this
case, we write st - \mathrm{l}\mathrm{i}\mathrm{m}n\rightarrow \infty un = \mu 0 if the limit (3) exists.
We write down that every convergent sequence of fuzzy numbers is statistically convergent to
same number since all finite subsets of the natural numbers have density zero. Accordingly, statistical
convergence may be considered as a regular summability method. However, converse of that is not
always true. For example, sequence (un) of fuzzy numbers defined by
un(t) =
\left\{
n+ 3
8
t - 3n+ 1
8
, if t \in
\biggl[
3n+ 1
n+ 3
, 3
\biggr]
1, if t \in [3, 5]
11n+ 17
6n+ 2
- n+ 3
6n+ 2
t, if t \in
\biggl[
5,
11n+ 17
n+ 3
\biggr]
0, otherwise
\right\}
if n = k3 and k \in \BbbN
0, otherwise
is statistically convergent to \=0, since
\mathrm{l}\mathrm{i}\mathrm{m}
n\rightarrow \infty
1
n+ 1
\bigm| \bigm| \bigm| \{ k \in \BbbN : D(uk, \=0) \geq \epsilon , k \leq n\}
\bigm| \bigm| \bigm| \leq \mathrm{l}\mathrm{i}\mathrm{m}
n\rightarrow \infty
3
\surd
n+ 1
n+ 1
= 0
for every \epsilon > 0, but not convergent in the ordinary sense.
Recall that a sequence (un) of fuzzy numbers is called statistically (N, p, \alpha ) summable to the
fuzzy number \mu 0 for each nonnegative integer \alpha if st - \mathrm{l}\mathrm{i}\mathrm{m}n\rightarrow \infty t
(\alpha )
n = \mu 0.
If \alpha = 1, then statistically (N, p, \alpha ) summability method reduces to statistically (N, p) summa-
bility method.
At present, we define concepts of slow oscillation relative to (Pn) and two-sided condition of
Hardy type relative to (Pn), respectively. In pursuit of defining of these concepts, we mention about
how a transition exists between them.
Definition 7. A sequence (un) of fuzzy numbers is said to be slowly oscillating relative to (Pn) if
\mathrm{l}\mathrm{i}\mathrm{m} \mathrm{s}\mathrm{u}\mathrm{p}
n\rightarrow \infty
D(um, un) = 0 as m \geq n,
Pm
Pn
\rightarrow 1 (n \rightarrow \infty ). (4)
Using \epsilon and \delta , (4) is equivalent to following statement: for every \epsilon > 0 there exist \delta > 0 and
n0 \in \BbbN 0 such that
D(um, un) \leq \epsilon whenever m \geq n \geq n0 and 1 \leq Pm
Pn
\leq 1 + \delta .
We emphasize that if pn = 1 for all n \in \BbbN in (4), then concept of slow oscillation relative to
(Pn) correspond to concept of slow oscillation (cf. [24]).
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1092 Z. ÖNDER, İ. ÇANAK
Definition 8. A sequence (un) of fuzzy numbers satisfies two-sided condition of Hardy type
relative to (Pn) if there exist positive constants n0 and C such that
D(un, un - 1) \leq C
pn
Pn
for n > n0. (5)
We emphasize that if pn = 1 for all n \in \BbbN in (5), then two-sided condition of Hardy type relative
to (Pn) correspond to two-sided condition of Hardy type.
Additionally, it is easy to see that if the sequence (pn) satisfies condition (2), then two-sided
condition of Hardy type relative to (Pn) implies condition of slow oscillation relative to (Pn).
As a matter of fact, we assume that (pn) satisfies condition (2). Since two-sided condition
of Hardy type relative to (Pn) is satisfied, there exist positive constants n0 and C such that
D(un, un - 1) \leq C
pn
Pn
for n > n0.
Let m \geq n \geq n0 and 1 \leq Pm
Pn
\leq 1 + \delta . Then, for a given \epsilon > 0, we have
D(um, un) = D(um, um - 1) +D(um - 1, um - 2) + . . .+D(un+1, un) =
=
m\sum
k=n+1
D(uk, uk - 1) \leq C
m\sum
k=n+1
pk
Pk
\leq C
\biggl(
Pm
Pn
- 1
\biggr)
< C\delta < \epsilon
in case we choose 0 < \delta <
\epsilon
C
. Therefore, we obtain that (un) \in \omega (F ) is slowly oscillating relative
to (Pn).
3. Lemmas. In this section, we express and prove following assertions which will be benefited
in proofs of our main results for sequences of fuzzy numbers. The following lemma which we prove
following a procedure resembling proof done for sequences of real numbers by Mikhalin [25] plays
a crucial role in proofs of subsequent two lemmas which are necessary to achieve our main results
for sequences of fuzzy numbers.
Lemma 3. Let (pn) satisfy conditions (2) and
pn
Pn
\rightarrow 0 as n \rightarrow \infty . If (un) \in \omega (F ) satisfies
condition
\mathrm{l}\mathrm{i}\mathrm{m} \mathrm{s}\mathrm{u}\mathrm{p}
n\rightarrow \infty
D(um, un) \leq r (0 \leq r < \infty ) as m \geq n,
Pm
Pn
\rightarrow 1 (n \rightarrow \infty ), (6)
then there exist positive numbers a and b such that D(um, un) \leq a \mathrm{l}\mathrm{o}\mathrm{g}
Pm
Pn
+ b for all m \geq n \geq 0.
Proof. Assume that (pn) satisfies conditions (2),
pn
Pn
\rightarrow 0 as n \rightarrow \infty and (un) satisfies
condition (6). We can say from condition (6) that for every r + 1 > 0 there exist \delta > 0 and n0 \in \BbbN
such that D(um, un) < r + 1 whenever n > n0 and 1 \leq Pm
Pn
\leq 1 + \delta .
Let n \leq n0 and Pm \leq Pn0(1 + \delta ). In this case, we find that D(um, un) has a maximum
depending only on n0, and so there exist \delta > 0 and \ell \geq r + 1 such that D(um, un) < \ell for all
m,n \in \BbbN related by 0 \leq Pm - Pn \leq \delta Pn. Choose n1 such that 1 \leq Pn+1
Pn
\leq (1 + \delta ) for all
n \geq n1. We investigate chosen n1 in three cases such that q \geq w \geq n1, 0 \leq w < n1 \leq q and
0 \leq w \leq q < n1 for arbitrary fixed q, w \in \BbbN .
We firstly take the case q \geq w \geq n1 into consideration. For this, we define subsequence (wi+1)
ISSN 1027-3190. Укр. мат. журн., 2021, т. 73, № 8
TAUBERIAN CONDITIONS UNDER WHICH CONVERGENCE FOLLOWS . . . 1093
where w0 = w and wi+1 is the largest natural number for which inequality Pn \leq Pwi(1 + \delta ) holds
for all i \in \BbbN . Therefore, we attain from this defining that inequalities Pwi+1 \leq Pwi(1 + \delta ) and
Pwi+1+1 > Pwi(1 + \delta ) is valid. Now, let wk \leq q - 1 < wk+1. Then we obtain that inequalities
Pq \leq Pwk+1
and 0 \leq Pwi+1 - Pwi \leq \delta Pwi hold for all k \in \BbbN . So, we get
D (uq, uw) \leq D (uq, uwk
) +D (uwk
, uw0) \leq
\leq D (uq, uwk
) +D
\bigl(
uwk
, uwk - 1
\bigr)
+D
\bigl(
uwk - 1
, uwk - 2
\bigr)
+ . . .+D (uw1 , uw0) =
=
k - 1\sum
j=0
D
\bigl(
uwj+1 , uwj
\bigr)
+D (uq, uwk
) \leq
k - 1\sum
j=0
\ell +D (uq, uwk
) \leq (k + 1)\ell .
Due to the fact that we have also inequalities
Pq \geq Pwk+1 > Pwk - 1
(1 + \delta ) \geq Pwk - 2+1(1 + \delta ) >
> Pwk - 3
(1 + \delta )2 \geq . . . \geq Pw0(1 + \delta )[
k
2 ] > Pw(1 + \delta )
k
2
- 1,
we reach inequality \mathrm{l}\mathrm{o}\mathrm{g}Pq \geq \mathrm{l}\mathrm{o}\mathrm{g}Pw +
k - 2
2
\mathrm{l}\mathrm{o}\mathrm{g}(1 + \delta ). This implies that inequality (1 + k) \leq
\leq \mathrm{l}\mathrm{o}\mathrm{g}Pq - \mathrm{l}\mathrm{o}\mathrm{g}Pw
\mathrm{l}\mathrm{o}\mathrm{g}(1 + \delta )1/2
+ 3 holds and, hence, we get inequality
D (uq, uw) \leq (k + 1)\ell <
\ell
\mathrm{l}\mathrm{o}\mathrm{g}(1 + \delta )1/2
(\mathrm{l}\mathrm{o}\mathrm{g}Pq - \mathrm{l}\mathrm{o}\mathrm{g}Pw) + 3\ell (7)
for any q \geq w \geq n1.
On the other hand, if we take the case 0 \leq w < n1 \leq q into consideration, then we obtain
D (uq, uw) \leq D (uq, un1) + \mathrm{m}\mathrm{a}\mathrm{x}
0\leq w<n1
D (un1 , uw) \leq
\leq \ell
\mathrm{l}\mathrm{o}\mathrm{g}(1 + \delta )1/2
(\mathrm{l}\mathrm{o}\mathrm{g}Pq - \mathrm{l}\mathrm{o}\mathrm{g}Pn1) + 3\ell + \mathrm{m}\mathrm{a}\mathrm{x}
0\leq w<n1
D (un1 , uw) \leq
\leq \ell
\mathrm{l}\mathrm{o}\mathrm{g}(1 + \delta )1/2
(\mathrm{l}\mathrm{o}\mathrm{g}Pq - \mathrm{l}\mathrm{o}\mathrm{g}Pw) + 3\ell + \mathrm{m}\mathrm{a}\mathrm{x}
0\leq w<n1
D (un1 , uw) . (8)
Finally, if we consider the case 0 \leq w \leq q < n1, then we have
D (uq, uw) \leq \mathrm{m}\mathrm{a}\mathrm{x}
0\leq w\leq q\leq n1 - 1
D (uq, uw) \leq
\ell
\mathrm{l}\mathrm{o}\mathrm{g}(1 + \delta )1/2
(\mathrm{l}\mathrm{o}\mathrm{g}Pq - \mathrm{l}\mathrm{o}\mathrm{g}Pw)+3\ell + \mathrm{m}\mathrm{a}\mathrm{x}
0\leq w\leq q\leq n1
D (uq, uw) .
(9)
If we define positive numbers a, b as a =
\ell
\mathrm{l}\mathrm{o}\mathrm{g}(1 + \delta )1/2
and b = \mathrm{m}\mathrm{a}\mathrm{x}\{ 3\ell , 3\ell + \mathrm{m}\mathrm{a}\mathrm{x}
0\leq w\leq q\leq n1
D (uq, uw)\} ,
then we conclude by (7) – (9) that
D (uq, uw) \leq a \mathrm{l}\mathrm{o}\mathrm{g}
Pq
Pw
+ b
for all q \geq w \geq 0.
Lemma 3 is proved.
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1094 Z. ÖNDER, İ. ÇANAK
Due to the fact that condition (6) corresponds to condition of slow oscillation relative to (Pn)
in case of r = 0 in Lemma 3, we prove in following lemma that the below-mentioned sequence is
bounded under condition of slow oscillation relative to (Pn) which is restrictive in comparison with
condition (6) and some additional condition on (pn) with the help of Lemma 3.
Lemma 4. Let (pn) satisfies conditions (2) and
pn
Pn
\rightarrow 0 as n \rightarrow \infty . If (un) \in \omega (F ) is slowly
oscillating relative to (Pn), then \Biggl(
1
Pm
m\sum
n=0
pnD(um, un)
\Biggr)
(10)
is bounded.
Proof. Assume that (pn) satisfies conditions (2),
pn
Pn
\rightarrow 0 as n \rightarrow \infty and (un) is slowly
oscillating relative to (Pn). Then, by taking these hypotheses into consideration, we conclude by the
help of Lemma 3 that there exist positive numbers a and b such that D(um, un) \leq a \mathrm{l}\mathrm{o}\mathrm{g}
Pm
Pn
+ b for
all m \geq n \geq 0. In addition to this, since (pn) satisfies condition
pn
Pn
\rightarrow 0 as n \rightarrow \infty , we have
Pn
Pn+1
= 1 - pn+1
Pn+1
\rightarrow 1 as n \rightarrow \infty . (11)
By the fact that tn \rightarrow \ell implies
1
tn
\rightarrow 1
\ell
whenever \ell \not = 0 as n \rightarrow \infty , we find by (11) that
Pn+1
Pn
\rightarrow 1 as n \rightarrow \infty
and, so,
1 \leq Pm
Pn
=
Pm
Pm - 1
Pm - 1
Pm - 2
. . .
Pn+1
Pn
\rightarrow 1 as m \geq n \rightarrow \infty . (12)
This means that for every \delta > 0 there exists n0 \in \BbbN 0 such that 1 \leq Pm
Pn
\leq 1 + \delta whenever
m \geq n \geq n0. Therefore, from condition of slow oscillation relative to (Pn) we declare that for
every \epsilon > 0 there exist \delta > 0 and n0 \in \BbbN 0 such that D(um, un) \leq \epsilon whenever m \geq n \geq n0 and
1 \leq Pm
Pn
\leq 1 + \delta . With reference to above inequalities, we obtain that for all m \geq 0 and given any
\epsilon > 0
1
Pm
m\sum
n=0
pnD(um, un) =
1
Pm
n0\sum
n=0
pnD(um, un) +
1
Pm
m\sum
n=n0+1
pnD(um, un) \leq
\leq 1
Pm
n0\sum
n=0
pn
\biggl(
a \mathrm{l}\mathrm{o}\mathrm{g}
Pm
Pn
+ b
\biggr)
+
1
Pm
m\sum
n=n0+1
pn\epsilon \leq
\leq 1
Pm
n0\sum
n=0
pn
\biggl(
a \mathrm{l}\mathrm{o}\mathrm{g}
Pm
P0
+ b
\biggr)
+
1
Pm
m\sum
n=n0+1
pn\epsilon =
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TAUBERIAN CONDITIONS UNDER WHICH CONVERGENCE FOLLOWS . . . 1095
=
Pn0 - P0
Pm
\biggl(
a \mathrm{l}\mathrm{o}\mathrm{g}
Pm
P0
+ b
\biggr)
+
Pm - Pn0
Pm
\epsilon =
= (Pn0 - P0)
\biggl(
a
Pm
\mathrm{l}\mathrm{o}\mathrm{g}
Pm
P0
\biggr)
+
Pn0 - P0
Pm
b+
\biggl(
1 - Pn0
Pm
\biggr)
\epsilon \leq
\leq (Pn0 - P0)
\biggl(
a
P0
\biggr)
+
Pn0 - P0
P0
b+
\biggl(
1 - Pn0
Pm
\biggr)
\epsilon =
=
\biggl(
Pn0
P0
- 1
\biggr)
(a+ b) +
\biggl(
1 - Pn0
Pm
\biggr)
\epsilon .
In conjunction with the information obtained up to now if we consider that
\biggl(
Pn0
Pm
\biggr)
is convergent
to 0 by condition (2) and every convergent sequence is also bounded, then there exists a positive
constant H such that
1
Pm
m\sum
n=0
pnD(um, un) \leq
\biggl(
Pn0
P0
- 1
\biggr)
(a+ b) +
\biggl(
1 - Pn0
Pm
\biggr)
\epsilon \leq
\biggl(
Pn0
P0
- 1
\biggr)
(a+ b) +H := M
for all m \geq 0 and some constant M > 0. In conclusion, we reach that the sequence in (10) is
bounded.
Lemma 4 is proved.
Lemma 5. Let (pn) satisfies conditions (2) and
pn
Pn
\rightarrow 0 as n \rightarrow \infty . If (un) \in \omega (F ) is slowly
oscillating relative to (Pn), then (t
(1)
n ) \in \omega (F ) is also slowly oscillating relative to (Pn).
Proof. Assume that (pn) satisfies conditions (2),
pn
Pn
\rightarrow 0 as n \rightarrow \infty and (un) is slowly
oscillating relative to (Pn). Given \epsilon > 0. By the definition of slow oscillation relative to (Pn), this
means that there exist \delta > 0 and n0 \in \BbbN 0 such that D(um, un) \leq \epsilon whenever m \geq n \geq n0 and
1 \leq Pm
Pn
\leq 1 + \delta .
Let m \geq n \geq n0 and 1 \leq Pm
Pn
\leq 1 + \delta \prime . By the definition of the weighted means of first order
of (un) and Lemma 4, we obtain
D
\Bigl(
t(1)m , t(1)n
\Bigr)
= D
\Biggl(
1
Pm
\Biggl\{
n\sum
k=0
+
m\sum
k=n+1
\Biggr\}
pkuk,
1
Pn
n\sum
k=0
pkuk
\Biggr)
=
= D
\Biggl(
1
Pm
n\sum
k=0
pkuk +
1
Pm
m\sum
k=n+1
pkuk +
Pm - Pn
PmPn
n\sum
k=0
pkun,
Pm - Pn
PmPn
n\sum
k=0
pkun +
1
Pn
n\sum
k=0
pkuk
\Biggr)
=
= D
\Biggl(
1
Pm
n\sum
k=0
pkuk +
1
Pm
m\sum
k=n+1
pkuk +
Pm - Pn
PmPn
n\sum
k=0
pkun,
Pm - Pn
PmPn
n\sum
k=0
pkun +
+
Pm - Pn
PmPn
n\sum
k=0
pkuk +
1
Pm
n\sum
k=0
pkuk
\Biggr)
=
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1096 Z. ÖNDER, İ. ÇANAK
= D
\Biggl(
1
Pm
m\sum
k=n+1
pkuk +
Pm - Pn
PmPn
n\sum
k=0
pkun,
Pm - Pn
PmPn
n\sum
k=0
pkun +
Pm - Pn
PmPn
n\sum
k=0
pkuk
\Biggr)
\leq
\leq D
\Biggl(
1
Pm
m\sum
k=n+1
pkuk,
Pm - Pn
PmPn
n\sum
k=0
pkun
\Biggr)
+D
\Biggl(
Pm - Pn
PmPn
n\sum
k=0
pkun,
Pm - Pn
PmPn
n\sum
k=0
pkuk
\Biggr)
\leq
\leq 1
Pm
m\sum
k=n+1
pkD(uk, un) +
Pm - Pn
Pm
1
Pn
n\sum
k=0
pkD(un, uk) \leq
\leq 1
Pm
m\sum
k=n+1
pk\epsilon +
\biggl(
1 - Pn
Pm
\biggr)
M =
=
\biggl(
1 - Pn
Pm
\biggr)
(M + \epsilon )
whenever m \geq k > n \geq n0, 1 <
Pk
Pn
\leq Pm
Pn
\leq 1 + \delta \prime and for some constant M > 0. Since we have
that for m \geq n \geq n0 and 1 \leq Pm
Pn
\leq 1 + \delta \prime
0 \leq 1 - Pn
Pm
\leq \delta \prime
1 + \delta \prime
,
if we choose 0 < \delta \prime \leq \epsilon
M
, then we arrive
D
\Bigl(
t(1)m , t(1)n
\Bigr)
\leq
\biggl(
1 - Pn
Pm
\biggr)
(M + \epsilon ) \leq \delta \prime
1 + \delta \prime
(M + \epsilon ) \leq \epsilon .
Therefore, we obtain that (t(1)n ) is slowly oscillating relative to (Pn), as well.
Lemma 5 is proved.
The following lemma can be given as a corollary of theorem proved by Talo and Başar [17].
Lemma 6 [17]. If (un) \in \omega (F ) is statistically convergent to \mu 0 \in E1 and slowly oscillating,
then (un) is convergent to \mu 0.
4. Main results. In this section, we prove the Tauberian theorem for (N, p) summable sequences
of fuzzy numbers following a procedure resembling proof done for sequences of real numbers by
Boos [26] at first. In the sequel, replacing (N, p) summable sequences of fuzzy numbers by statis-
tically (N, p) summable sequences of fuzzy numbers, we establish some Tauberian theorems which
convergence follows from statistically (N, p) summability under some Tauberian conditions and ad-
ditional conditions on (pn) and we present some corollaries related to these theorems. We end this
section by giving an extension of obtained theorems from statistically (N, p) summability method to
statistically (N, p, s) summability method.
Theorem 4. Let (pn) satisfies conditions (2) and
Pn
Pn+1
\rightarrow 1 as n \rightarrow \infty . (13)
If (un) \in \omega (F ) is (N, p) summable to \mu 0 \in E1 and slowly oscillating relative to (Pn), then (un) is
convergent to \mu 0.
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TAUBERIAN CONDITIONS UNDER WHICH CONVERGENCE FOLLOWS . . . 1097
Proof. Assume that (pn) satisfies conditions (2), (13) and that (un) being (N, p) summable
to \mu 0 is slowly oscillating relative to (Pn). We may suppose that \mu 0 = \=0 without loss of generality
and that (un) does not converge. In this case, because (un) cannot converge to \=0 as well, we may
consider 0 < \alpha := \mathrm{l}\mathrm{i}\mathrm{m} \mathrm{s}\mathrm{u}\mathrm{p}D(un, \=0) \leq \infty . Thus, there exists an index sequence (ni) such that
uni \rightarrow \alpha as i \rightarrow \infty . We examine defined \alpha in two cases such that 0 < \alpha < \infty and \alpha = \infty .
If we firstly take the case 0 < \alpha < \infty into account, then for every \epsilon 1 := \alpha /2 > 0 and i \in \BbbN
there exists N1 \in \BbbN such that \alpha /2 \leq D(uni , \=0) whenever ni \geq i \geq N1. On the other hand, if we
consider the case \alpha = \infty , then for every \epsilon 2 := 1 > 0 and i \in \BbbN there exists N2 \in \BbbN such that
1 \leq D(uni , \=0) whenever ni \geq i \geq N2. Taking account of both cases, for every i \in \BbbN we may choose
\vargamma := \mathrm{m}\mathrm{i}\mathrm{n}\{ 1, \alpha /2\} and N \in \BbbN such that \vargamma \leq D(uni , \=0) whenever ni \geq i \geq N = \mathrm{m}\mathrm{a}\mathrm{x}\{ N1, N2\} .
In addition to these, because (un) is slowly oscillating relative to (Pn), the subsequence (uni) \subset
\subset (un) is also slowly oscillating relative to (Pn). This means that for every \epsilon := \vargamma /2 > 0 there exist
\delta > 0 and N \in \BbbN such that D(um, uni) \leq \vargamma /2 whenever m \geq ni \geq N and Pni \leq Pm \leq (1+\delta )Pni .
At the same time, inequality
D(\=0, uni) \leq D(\=0, um) +D(um, uni)
is verified, we obtain by taking slow oscillation relative to (Pn) of (uni) and assumption into account
inequality
D(\=0, um) \geq D(\=0, uni) - D(um, uni) \geq \vargamma - \vargamma
2
=
\vargamma
2
for all such m. We now define index sequence
mi := \mathrm{m}\mathrm{i}\mathrm{n}
\biggl\{
n \in \BbbN : Pn \geq Pni
\biggl(
1 +
\delta
2
\biggr) \biggr\}
for all i \in \BbbN .
It follows from the definition of the index sequence (mi) that mi \geq ni \geq N and Pmi - 1 <
< Pni
\biggl(
1 +
\delta
2
\biggr)
. On the other hand, by the fact that tn \rightarrow \ell implies
1
tn
\rightarrow 1
\ell
whenever \ell \not = 0
as n \rightarrow \infty , we find by (13) that
Pn+1
Pn
\rightarrow 1 as n \rightarrow \infty
and, so,
Pn
Pn - 1
\leq 1 +
\delta
4
for sufficiently large n. Thus, we get
Pmi =
Pmi
Pmi - 1
Pmi - 1 \leq
\biggl(
1 +
\delta
4
\biggr)
Pni
\biggl(
1 +
\delta
2
\biggr)
\leq Pni(1 + \delta )
for sufficiently large i. From this point of view, we can say that (mi) satisfies condition Pni
\biggl(
1 +
\delta
2
\biggr)
\leq
\leq Pmi \leq Pni(1 + \delta ) for sufficiently large i. By the definition of the weighted means of first order
of (un), the assumption and Lemma 2, we obtain
D
\Bigl(
t(1)mi
, t(1)ni
\Bigr)
+D
\biggl(
t(1)ni
,
Pni
Pmi
t(1)ni
\biggr)
\geq D
\biggl(
t(1)mi
,
Pni
Pmi
t(1)ni
\biggr)
=
= D
\left( 1
Pmi
\left( ni\sum
k=0
pkuk +
mi\sum
k=ni+1
pkuk
\right) ,
1
Pmi
ni\sum
k=0
pkuk
\right) =
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1098 Z. ÖNDER, İ. ÇANAK
= D
\left( 1
Pmi
mi\sum
k=ni+1
pkuk, \=0
\right) \geq
\geq D
\left( 1
Pmi
mi\sum
k=ni+1
pkuni , \=0
\right) - D
\left( 1
Pmi
mi\sum
k=ni+1
pkuk,
1
Pmi
mi\sum
k=ni+1
pkuni
\right) \geq
\geq D
\biggl(
Pmi - Pni
Pmi
uni , \=0
\biggr)
- 1
Pmi
mi\sum
k=ni+1
pkD(uk, uni) \geq
\geq Pmi - Pni
Pmi
D(uni , \=0) -
1
Pmi
mi\sum
k=ni+1
pk
\vargamma
2
\geq
\geq
\biggl(
Pmi - Pni
Pmi
\biggr)
\vargamma -
\biggl(
Pmi - Pni
Pmi
\biggr)
\vargamma
2
=
=
\biggl(
1 - Pni
Pmi
\biggr)
\vargamma
2
\geq
\biggl(
\delta
2 + \delta
\biggr)
\vargamma
2
> 0 (14)
whenever mi \geq k \geq ni \geq i \geq N and Pni
\biggl(
1 +
\delta
2
\biggr)
\leq Pmi \leq Pni(1 + \delta ). Additionally, we get
inequality
D
\biggl(
t(1)mi
,
Pni
Pmi
t(1)ni
\biggr)
\leq D
\Bigl(
t(1)mi
, t(1)ni
\Bigr)
+D
\biggl(
t(1)ni
,
Pni
Pmi
t(1)ni
\biggr)
=
= D
\Bigl(
t(1)mi
, t(1)ni
\Bigr)
+D
\Biggl(
Pmi
PniPmi
ni\sum
k=0
pkuk,
Pni
PniPmi
ni\sum
k=0
pkuk
\Biggr)
=
= D
\Bigl(
t(1)mi
, t(1)ni
\Bigr)
+
1
PniPmi
D
\Biggl(
Pni + (Pmi - Pni)
ni\sum
k=0
pkuk, Pni
ni\sum
k=0
pkuk
\Biggr)
\leq
\leq D
\Bigl(
t(1)mi
, t(1)ni
\Bigr)
+
Pmi - Pni
Pmi
D
\Bigl(
t(1)ni
, \=0
\Bigr)
\leq
\leq D
\Bigl(
t(1)mi
, \=0
\Bigr)
+D
\Bigl(
t(1)ni
, \=0
\Bigr)
+
\delta
2 + \delta
D
\Bigl(
t(1)ni
, \=0
\Bigr)
(15)
by the help of Lemma 2. If we take limit of both sides of inequality (15) as i \rightarrow \infty , because (un) is
(N, p) summable to \=0, we reach inequality
\mathrm{l}\mathrm{i}\mathrm{m}
i\rightarrow \infty
D
\biggl(
t(1)mi
,
Pni
Pmi
t(1)ni
\biggr)
\leq \mathrm{l}\mathrm{i}\mathrm{m}
i\rightarrow \infty
D(t(1)mi
, \=0) + \mathrm{l}\mathrm{i}\mathrm{m}
i\rightarrow \infty
D(t(1)ni
, \=0) +
\biggl(
\delta
2 + \delta
\biggr)
D
\Bigl(
t(1)ni
, \=0
\Bigr)
= 0. (16)
Therefore, from inequalities (14) and (16) we attain
0 <
\biggl(
\delta
2 + \delta
\biggr)
\vargamma
2
\leq \mathrm{l}\mathrm{i}\mathrm{m}
i\rightarrow \infty
D
\biggl(
t(1)mi
,
Pni
Pmi
t(1)ni
\biggr)
\leq 0.
ISSN 1027-3190. Укр. мат. журн., 2021, т. 73, № 8
TAUBERIAN CONDITIONS UNDER WHICH CONVERGENCE FOLLOWS . . . 1099
Because this contradicts that (t(1)n ) converges to \=0, we conclude that (un) converges to \=0.
Theorem 4 is proved.
Theorem 5. Let (pn) satisfies conditions (2),
pn
Pn
\rightarrow 0 as n \rightarrow \infty and
1 \leq Pm
Pn
\rightarrow 1 whenever 1 <
m
n
\rightarrow 1 as n \rightarrow \infty . (17)
If (un) \in \omega (F ) is statistically (N, p) summable to \mu 0 \in E1 and slowly oscillating relative to (Pn),
then (un) is convergent to \mu 0.
Proof. Assume that (pn) satisfies conditions (2), (17) and
pn
Pn
\rightarrow 0 as n \rightarrow \infty and (un) being
statistically (N, p) summable to \mu 0 is slowly oscillating relative to (Pn). In the circumstances, we
arrive by the help of Lemma 5 that (t(1)n ) is also slowly oscillating relative to (Pn). In other words,
we can say by the definition of slow oscillation relative to (Pn) that condition
\mathrm{l}\mathrm{i}\mathrm{m} \mathrm{s}\mathrm{u}\mathrm{p}
n\rightarrow \infty
D(t(1)m , t(1)n ) = 0 as m \geq n,
Pm
Pn
\rightarrow 1 (n \rightarrow \infty ),
holds and, so, by condition (17) we obtain
\mathrm{l}\mathrm{i}\mathrm{m} \mathrm{s}\mathrm{u}\mathrm{p}
n\rightarrow \infty
D(t(1)m , t(1)n ) = 0 as m \geq n,
m
n
\rightarrow 1 (n \rightarrow \infty ).
This statement implies slow oscillation of (t
(1)
n ). Since (t
(1)
n ) is slowly oscillating and statistically
convergent to \mu 0, we reach by the help of Lemma 6 that (t(1)n ) is convergent to \mu 0 which means
that (un) is (N, p) summable to \mu 0. In addition to this, since (pn) satisfies condition
pn
Pn
\rightarrow 0 as
n \rightarrow \infty , we have
Pn
Pn+1
= 1 - pn+1
Pn+1
\rightarrow 1 as n \rightarrow \infty .
If we consider that condition of slowly oscillating relative to (Pn) is the Tauberian condition for
(N, p) summable sequence under additional conditions on (pn) as a result of Theorem 4, then we
conclude that (un) is convergent to \mu 0.
Theorem 5 is proved.
Corollary 1. Let (pn) satisfies conditions (2), (17) and
pn
Pn
\rightarrow 0 as n \rightarrow \infty . If (un) \in \omega (F ) is
statistically (N, p) summable to \mu 0 \in E1 and satisfies two-sided condition of Hardy type relative to
(Pn), then (un) is convergent to \mu 0.
As a generalization of Theorem 5 and Corollary 1, we can present Theorem 6 and Corollary 2,
respectively.
Theorem 6. Let (pn) satisfies conditions (2), (17) and
pn
Pn
\rightarrow 0 as n \rightarrow \infty . If (un) \in \omega (F ) is
statistically (N, p, \alpha ) summable to \mu 0 \in E1 for some integer \alpha \geq 0 and slowly oscillating relative
to (Pn), then (un) is convergent to \mu 0.
Proof. Assume that (pn) satisfies conditions (2), (17) and
pn
Pn
\rightarrow 0 as n \rightarrow \infty and (un) being
statistically (N, p, \alpha ) summable to \mu 0 for some integer \alpha \geq 0 is slowly oscillating relative to (Pn).
In the circumstances, we arrive by the help of Lemma 5 that (t(\beta )n ) is also slowly oscillating relative
ISSN 1027-3190. Укр. мат. журн., 2021, т. 73, № 8
1100 Z. ÖNDER, İ. ÇANAK
to (Pn) for each integer \beta \geq 1. In other words, we can say by the definition of slow oscillation
relative to (Pn) that condition
\mathrm{l}\mathrm{i}\mathrm{m} \mathrm{s}\mathrm{u}\mathrm{p}
n\rightarrow \infty
D(t(\beta )m , t(\beta )n ) = 0 as m \geq n,
Pm
Pn
\rightarrow 1 (n \rightarrow \infty ),
holds and, so, by condition (17) we obtain
\mathrm{l}\mathrm{i}\mathrm{m} \mathrm{s}\mathrm{u}\mathrm{p}
n\rightarrow \infty
D(t(\beta )m , t(\beta )n ) = 0 as m \geq n,
m
n
\rightarrow 1 (n \rightarrow \infty ).
This statement implies slow oscillation of (t
(\beta )
n ) for each integer \beta \geq 1. Since (t
(\alpha )
n ) is statistically
convergent to \mu 0 from the assumption, we reach by taking \beta = \alpha with the help of Lemma 6 that
(t
(\alpha )
n ) is convergent to \mu 0 which means that (t(\alpha - 1)
n ) is (N, p) summable to \mu 0. In the case when
\beta = \alpha - 1, we have that (t(\alpha - 1)
n ) is slowly oscillating relative to (Pn). In addition to this, since (pn)
satisfies condition
pn
Pn
\rightarrow 0 as n \rightarrow \infty , we find
Pn
Pn+1
= 1 - pn+1
Pn+1
\rightarrow 1 as n \rightarrow \infty .
If we consider that condition of slowly oscillating relative to (Pn) is the Tauberian condition for
(N, p) summable sequence under additional conditions on (pn) as a result of Theorem 4, then we
conclude that (t
(\alpha - 1)
n ) is convergent to \mu 0. Continuing in a similar way, it follows that (t
(1)
n ) is
convergent to \mu 0 which means that (un) is (N, p) summable to \mu 0. Therefore, we accomplish by
the help of Theorem 4 that (un) is convergent to \mu 0.
Theorem 6 is proved.
Corollary 2. Let (pn) satisfies conditions (2), (17) and
pn
Pn
\rightarrow 0 as n \rightarrow \infty . If (un) \in \omega (F ) is
statistically (N, p, \alpha ) summable to \mu 0 \in E1 for some integer \alpha \geq 0 and satisfies two-sided condition
of Hardy type relative to (Pn), then (un) is convergent to \mu 0.
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ISSN 1027-3190. Укр. мат. журн., 2021, т. 73, № 8
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| id | umjimathkievua-article-584 |
| institution | Ukrains’kyi Matematychnyi Zhurnal |
| keywords_txt_mv | keywords |
| language | English |
| last_indexed | 2026-03-24T02:03:10Z |
| publishDate | 2021 |
| publisher | Institute of Mathematics, NAS of Ukraine |
| record_format | ojs |
| resource_txt_mv | umjimathkievua/6f/e9debc64be6d15b13d9f2d1d44108a6f.pdf |
| spelling | umjimathkievua-article-5842025-03-31T08:47:35Z Tauberian conditions under which convergence follows from the weighted mean summability and its statistical extension for sequences of fuzzy number Tauberian conditions under which convergence follows from the weighted mean summability and its statistical extension for sequences of fuzzy number Tauberian conditions under which convergence follows from the weighted mean summability and its statistical extension for sequences of fuzzy number Önder, Z. Çanak, İ. Önder, Zerrin Çanak, İbrahim Önder, Z. Çanak, İ. Sequences of fuzzy numbers, slow oscillation, slow oscillation relative to $(P_n)$, statistical convergence, two-sided conditions of Hardy type, Tauberian theorems, weighted mean summability method UDC 517.5 Let $(p_n)$ be a sequence of nonnegative numbers such that $p_0&gt;0$ and$$P_n:=\sum_{k=0}^{n}p_k\to\infty\qquad \text{as}\qquad n\to\infty.$$Let $(u_n)$ be a sequence of fuzzy numbers.The weighted mean of $(u_n)$ is defined by$$t_n:=\frac{1}{P_n}\sum_{k=0}^{n}p_ku_k\qquad \text{for}\qquad n =0,1,2,\ldots \,. $$It is known that the existence of the limit $\lim u_n=\mu_{0}$ implies that of $\lim t_n=\mu_{0}.$ For the the existence of the limit $st$-$\lim t_n=\mu_{0},$ we require the boundedness of $(u_n)$ in addition to the existence of the limit $\lim u_n=\mu_{0}.$ But, in general, the converse of this implication is not true. In this paper, we obtain Tauberian conditions, under which the existence of the limit $\lim u_n=\mu_{0}$ follows from that of $\lim t_n=\mu_{0}$ or $st$-$\lim t_n=\mu_{0}.$ These Tauberian conditions are satisfied if $(u_n)$ satisfies the two-sided condition of Hardy type relative to $(P_n).$ УДК 517.5 Тауберові умови, за яких збіжність випливає з середньовагової сумовності, та їх статистичне розширення на послідовності нечітких чисел Нехай $(p_n)$ - послідовність невід'ємних чисел таких, що $p_0&gt;0$ і$$P_n:=\sum_{k=0}^{n}p_k\to\infty\qquad \text{при}\qquad n\to\infty.$$Нехай $(u_n)$ - послідовність нечітких чисел.Вагове середнє для $(u_n)$ визначається як$$t_n:=\frac{1}{P_n}\sum_{k=0}^{n}p_k u_k\qquad \text{для}\qquad n =0,1,2,\ldots \,. $$Відомо, що з існування границі $\lim u_n=\mu_{0}$ випливає $\lim t_n=\mu_{0}.$ Для існування границі $st$-$\lim t_n=\mu_{0}$ вимагається обмеженість $(u_n)$ як додаткова умова до існування границі $\lim u_n=\mu_{0}.$ Але обернена імплікація взагалі не є правильною. У цій роботі запропоновано тауберові умови, за яких існування границі $\lim u_n=\mu_{0}$ випливає з того, що $\lim t_n=\mu_{0}$ або $st$-$\lim t_n=\mu_{0}.$ Ці тауберові умови виконуються, якщо $(u_n)$ задовольняє двосторонні умови типу Гарді відносно $(P_n).$ Institute of Mathematics, NAS of Ukraine 2021-08-18 Article Article application/pdf https://umj.imath.kiev.ua/index.php/umj/article/view/584 10.37863/umzh.v73i8.584 Ukrains’kyi Matematychnyi Zhurnal; Vol. 73 No. 8 (2021); 1085 - 1101 Український математичний журнал; Том 73 № 8 (2021); 1085 - 1101 1027-3190 en https://umj.imath.kiev.ua/index.php/umj/article/view/584/9096 Copyright (c) 2021 İbrahim Çanak |
| spellingShingle | Önder, Z. Çanak, İ. Önder, Zerrin Çanak, İbrahim Önder, Z. Çanak, İ. Tauberian conditions under which convergence follows from the weighted mean summability and its statistical extension for sequences of fuzzy number |
| title | Tauberian conditions under which convergence follows from the weighted mean summability and its statistical extension for sequences of fuzzy number |
| title_alt | Tauberian conditions under which convergence follows from the weighted mean summability and its statistical extension for sequences of fuzzy number Tauberian conditions under which convergence follows from the weighted mean summability and its statistical extension for sequences of fuzzy number |
| title_full | Tauberian conditions under which convergence follows from the weighted mean summability and its statistical extension for sequences of fuzzy number |
| title_fullStr | Tauberian conditions under which convergence follows from the weighted mean summability and its statistical extension for sequences of fuzzy number |
| title_full_unstemmed | Tauberian conditions under which convergence follows from the weighted mean summability and its statistical extension for sequences of fuzzy number |
| title_short | Tauberian conditions under which convergence follows from the weighted mean summability and its statistical extension for sequences of fuzzy number |
| title_sort | tauberian conditions under which convergence follows from the weighted mean summability and its statistical extension for sequences of fuzzy number |
| topic_facet | Sequences of fuzzy numbers slow oscillation slow oscillation relative to $(P_n)$ statistical convergence two-sided conditions of Hardy type Tauberian theorems weighted mean summability method |
| url | https://umj.imath.kiev.ua/index.php/umj/article/view/584 |
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