Gen. Math. Notes, Vol. 23, No. 1, July 2014, pp.89-108
c ICSRS Publication, 2014
ISSN 2219-7184; Copyright www.i-csrs.org
Available free online at http://www.geman.in
On the Classical Sets of Sequences
with Fuzzy
b-Metric
U§ur Kadak1,2
1
Department of Mathematics, Faculty of Science
Gazi University, Teknikokullar, 06500-Ankara, Turkey
2
Department of Mathematics, Faculty of Science
Bozok University, Teknikokullar, 66100-Yozgat, Turkey
E-mail: ugurkadak@gmail.com; ugurkadak@gazi.edu.tr
(Received: 18-2-14 / Accepted: 11-4-14)
Abstract
Since the utilization of Zadeh's Extension Principle is quite dicult in practice, we prefer the idea of using the sum of the series of level sets. In this paper
we present some classical sets of sequences of fuzzy numbers with respect to the
notion of fuzzy b-metric. Also, we introduce the completeness of such spaces
and derive the relationships between these sets and their classical forms. In
addition, we use our results corresponding with series not only directly improve
and generalize some results in metric spaces and b-metric spaces, and also expand and complement some previous results in fuzzy metric spaces with the
level sets.
Keywords: Set of the sequences of fuzzy numbers, fuzzy level sets, bmetric, complete metric space.
1
Introduction
Some problems, particularly the problem of the convergence of measurable
functions with respect to a measure, lead to a generalization of notion of
a metric.
As a continuation of metric notion Banach proved a very impor-
tant result regarding a contraction mapping, known as the Banach contraction
principle in metric spaces which is a very popular tool for solving problems
90
U§ur Kadak
in nonlinear analysis.
In [7], Czerwik introduced
alization of metric spaces.
in
b-metric
b-metric
spaces as a gener-
Also he proved the contraction mapping principle
spaces which generalizes the famous Banach contraction principle.
Later, several papers have devoted to the xed point theory or the variational
principle for single-valued and multi-valued operators in
b-metric
spaces have
been obtained (see [2, 4, 5, 15, 16, 17, 23, 24, 30, 31, 36, 39, 43]).
In recent
investigations, the xed point in nonconvex analysis, especially in an ordered
normed space, occupies a prominent place in many aspects.
Especially, many researchers have focused on dierent contractive conditions in complete metric spaces endowed with a partial order and studied
xed point theory in the so-called bistructural spaces.
For more details on
xed point results, its applications, comparison of dierent contractive conditions and related results in ordered metric spaces, the reader may refer to
[1, 3, 6, 9, 10, 11, 12, 13, 21, 27, 29, 40, 42, 49] and the references mentioned
therein.
By
ω(F ),
we denote the set of all sequences of fuzzy numbers. We dene
the classical sets
`∞ (b, λ), c(b, λ), c0 (b, λ) and `q (b, λ) with respect to the bb-bounded, b-convergent, b-null and absolutely q -th
metric consisting of the
b-summable sequences of fuzzy numbers, as follows:
b
`∞ (b, λ) :=
u = (uk ) ∈ ω(F ) : sup D (uk , 0) < ∞ ,
k∈N
n
o
c(b, λ) :=
u = (uk ) ∈ ω(F ) : ∃l ∈ E 1 3 b lim Db (uk , l) = 0 ,
k→∞
n
o
b
b
c0 (b, λ) :=
u = (uk ) ∈ ω(F ) : lim D (uk , 0) = 0 ,
k→∞
)
(
X
Db (uk , 0)q < ∞ , (1 ≤ q < ∞)
`q (b, λ) :=
u = (uk ) ∈ ω(F ) :
order of
k
where the distance function
the
b-metric ρ
D
b
on the space
E1
of fuzzy numbers by means of
dened by
+ + p
− p
Db (u, v) := sup ρ([u]λ , [v]λ ) := sup max{(d(u−
λ , vλ ) , (d(uλ , vλ ) }
λ∈[0,1]
with
s = 2p−1
denotes fuzzy
where
λ∈[0,1]
u, v ∈ E 1
b-metric,
p ∈ R with p > 1.
of ρ based on λ and
and
by means
Since
Db
Hausdor
b-metric,
metric d.
is a
For simplicity in notation, here and in what follows, the summation without
limits runs from
c0 (b, λ)
0
to
∞.
It can be shown that the sets
`∞ (b, λ), c(b, λ)
and
b,λ
are b-complete with the b-metric D∞ dened by
b,λ
(u, v) := sup Db (uk , vk )
D∞
k∈N
p−1
s = 2
and u = (uk ), v = (vk )
c0 (b, λ) or `∞ (b, λ).
with
are the elements of the sets
c(b, λ),
91
On the Classical Sets of Sequences...
On the other hand, we can introduce and prove that the set
`q (b, λ)
is
b-complete with Dqb,λ dened by
(
)1/q
X
q
Dqb,λ (u, v) :=
Db (uk , vk )
, (1 ≤ q < ∞)
k
with
s = 2q−1
where
u = (uk ), v = (vk )
are the points of
`p (b, λ).
Mursaleen and Ba³arr [28] have recently introduced some new sets of se-
A some
`∞ (p, F ), c(p, F ), c0 (p, F ) and
special cases of that matrix A.
quences of fuzzy numbers generated by a non-negative regular matrix
of which are reduced to the Maddox spaces
`(p, F )
of sequences of fuzzy numbers for the
Altn, Et and Çolak [51] have recently dened the concepts of lacunary statistical convergence and lacunary strongly convergence of generalized dierence
sequences of fuzzy numbers.
They have also given some relations related to
these concepts and showed that lacunary
4m -statistical convergence and la4m -bounded sequences of
m
cunary strongly 4
(p) -convergence are equivalent for
fuzzy numbers.
Quite recently; Talo and Ba³ar [33] have extended the main
results of Ba³ar and Altay [8] to the fuzzy numbers. Also, Talo and Ba³ar [35]
have recently studied the normed quasilinearity of the classical sets
`∞ (F ),
c(F ), c0 (F ) and `p (F ) of sequences of fuzzy numbers and derived some related
results. Furthermore, Talo and Ba³ar [34] have introduced the sets `∞ (F, f ),
c(F, f ), c0 (F, f ) and `p (F, f ) of sequences of fuzzy numbers dened by a modulus function and given some topological properties of the sets together with
some inclusion relations. Finally, Kadak and Ba³ar [44, 45, 46, 47] have presented some new notions about the power series and Fourier series of fuzzy
numbers on fuzzy level sets.
The main purpose of the present paper is to
`∞ (b, λ), c(b, λ), c0 (b, λ) and `q (b, λ) of sequences
b-metric to the classical spaces `∞ , c, c0 and `p of se-
study the corresponding sets
of fuzzy numbers via
quences with real or complex terms.
The rest of this paper is organized, as follows:
In section 2, some required denitions and consequences related with the
b-metric,
sequences and series of fuzzy numbers are given. The most relevant
and recent literature is also reported in Section 2.
Section 3 is terminated
with the condensation of the results on the sum of the series of the fuzzy sets
given by M. Stojakovi¢ and Z. Stojakovi¢ in [26].
on
b-convergence
Additionally, an example
of series of fuzzy numbers is also presented in this section.
Furthermore, some notions i.e uniformly convergent, continuity and boundedness are established via fuzzy
spaces of fuzzy numbers via
b-metric
b-metric
and the completeness of such sequence
are presented.
92
U§ur Kadak
2
Preliminaries, Background and Notation
2.1
A
Fuzzy Level Sets
fuzzy number
is a fuzzy set on the real axis, i.e.
a mapping
u : R → [0, 1]
which satises the following four conditions:
(i) u is normal, i.e.
there exists an
x0 ∈ R
such that
u(x0 ) = 1.
(ii) u is fuzzy convex, i.e. u[λx + (1 − λ)y] ≥ min{u(x), u(y)} for all x, y ∈ R
and for all
λ ∈ [0, 1].
(iii) u is upper semi-continuous.
(iv)
[u]0 = {x ∈ R : u(x) > 0} is compact, (cf. Zadeh [19]), where
{x ∈ R : u(x) > 0} denotes the closure of the set {x ∈ R : u(x) > 0} in
the usual topology of R.
The set
λ-level set [u]λ
u ∈ E 1 is dened by
{t ∈ R : u(t) ≥ λ} , (0 < λ ≤ 1),
[u]λ :=
{t ∈ R : u(t) > λ} ,
(λ = 0).
The set
[u]λ
of
which is dened by
each
r∈R
−
+
[u]λ := [u (λ), u (λ)]. R
can be regarded as a fuzzy number
u, v, w ∈ E 1
and
k ∈ R.
cation and product dened on
can be embedded
r
dened by
1 , (x = r),
0 , (x 6= r).
r(x) :=
Let
λ ∈ [0, 1]
1
in E , since
is closed, bounded and non-empty interval for each
Then the operations addition, scalar multipli-
E1
by
u + v = w ⇔ [w]λ = [u]λ + [v]λ for all λ ∈ [0, 1]
⇔ w− (λ) = u− (λ) + v − (λ), w+ (λ) = u+ (λ) + v + (λ).
Further
[ku]λ = k[u]λ
for all
λ ∈ [0, 1]
and
uv = w ⇔ [w]λ = [u]λ [v]λ
for all
λ ∈ [0, 1],
where it is immediate that
w− (λ) = min{u− (λ)v − (λ), u− (λ)v + (λ), u+ (λ)v − (λ), u+ (λ)v + (λ)}
93
On the Classical Sets of Sequences...
and
w+ (λ) = max{u− (λ)v − (λ), u− (λ)v + (λ), u+ (λ)v − (λ), u+ (λ)v + (λ)}
for all
λ ∈ [0, 1].
Let
W
numbers with endpoints
W
be the set of all closed bounded intervals
A
and
A,
A := [A, A].
i.e.
A
of real
Dene the relation
d
on
by
d(A, B) := max{|A − B|, |A − B|}.
Then it can easily be observed that
Kloeden [37]) and
(W, d)
AB
d
is a metric on
W
(cf.
Diamond and
is a complete metric space, (cf. Nanda [41]).
if and only if
The partial ordering relation
on
A≤B
E1
u v ⇔ [u]λ [v]λ ⇔ u− (λ) ≤ v − (λ)
and
A ≤ B.
is dened as follows:
and
u+ (λ) ≤ v + (λ)
for all
λ ∈ [0, 1].
Denition 2.1 (Triangular Fuzzy Number) [18, Denition, p. 137] We can
dene the triangular fuzzy number u as u = (u1 , u2 , u3 ) whose membership
function µ is interpreted as follows;
µ(x) =


x−u1
u2 −u1
u3 −x
u3 −u2

0
, u1 ≤ x ≤ u2 ,
, u2 ≤ x ≤ u3 ,
, x < u1 , x > u3 .
Then, the result [u]λ := [u− (λ), u+ (λ)] = [(u2 − u1 )λ + u1 , −(u3 − u2 )λ + u3 ]
holds for each λ ∈ [0, 1].
2.2
b-Metric
Consistent with [7] and [43], the following denitions and results will be needed
in the sequel.
Denition 2.2 [14] Let X be a (nonempty) set and s ≥ 1 be a given real
number. A function d : X × X → [0, ∞) is a b-metric if, for all x, y, z ∈ X ,
the following conditions are satised:
(b1 ) d(x, y) = 0 i x = y,
(b2 ) d(x, y) = d(y, x),
(b3 ) d(x, z) ≤ s[d(x, y) + d(y, z)].
The pair (X, d) is called a b-metric space.
94
U§ur Kadak
It should be noted that the class of
than that of metric spaces, since a
b-metric
b-metric
spaces is eectively larger
is a metric if (and only if )
We present an easy example to show that in general a
b-metric
s = 1.
need not be a
metric.
Example 2.3 [2] Let (X, d) be a metric space, and db (x, y) = (d(x, y)p ), where
p ≥ 1 is a real number. Then db is a b-metric with s = 2p−1 .
However, (X, d) is not necessarily a metric space. For example, if X = R
is the set of real numbers and db (x, y) = |x − y| is the usual Euclidean metric,
then db (x, y) = (x − y)2 is a b-metric on R with s = 2, but it is not a metric
on R.
Denition 2.4 [22] Let (X, ρ) be a b-metric space. Then a sequence {xn } in
X is called:
(a) b-convergent if and only if there exists x ∈ X such that ρ(xn , x) → 0, as
n → ∞. In this case, we write b limn→∞ xn = x.
(b) b-Cauchy if and only if ρ(xn , xm ) → 0, as n, m → ∞.
Denition 2.5 [22] In a b-metric space (X, ρ) the following assertions hold:
(a) A b-convergent sequence has a unique limit.
(b) Each b-convergent sequence is b-Cauchy.
(c) In general, a b-metric is not continuous.
Also very recently N. Hussain et al. have presented an example of a b-metric
which is not continuous (see Example 3 in [30]).
Denition 2.6 [22]
(a) The b-metric space (X, ρ) is b-complete i every b-Cauchy sequence in X
is b-convergent.
(b) Let (X, ρ) be a b-metric space. If Y is a nonempty subset of X , then the
closure Y of Y is the set of limits of all b-convergent sequences of points
in Y , i.e.,
Y := x ∈ X : there exists a sequence {xn } in Y so that
b
lim xn = x .
n→∞
Taking into account of the above denition, we have the following concepts.
95
On the Classical Sets of Sequences...
(c) Let (X, ρ) be a b-metric space. Then a subset Y ⊂ X is called closed if
and only if for each sequence {xn } ∈ Y which b-converges to an element
x, we have x ∈ Y .
(d) Let (X, ρ) and (X 0 , ρ0 ) be two b-metric spaces. Then a function f : X →
X 0 is b-continuous at a point x ∈ X if and only if it is b-sequentially
continuous at x, that is, whenever {xn } is b-convergent to x;{f (xn )} is
b-convergent to f (x).
Lemma 2.7 [2] Let (X, d) be a b-metric space with s ≥ 1, and suppose that
{xn }, {yn } are b-convergent to x, y , respectively. Then we have,
1
d(x, y) ≤ b lim inf d(xn , yn ) ≤ b lim sup d(xn , yn ) ≤ s2 d(x, y).
2
n→∞
s
n→∞
In particular, if x = y, then we have b limn→∞ d(xn , yn ) = 0. Moreover, for
each z ∈ X , we have,
1
d(x, z) ≤ b lim inf d(xn , z) ≤ b lim sup d(xn , z) ≤ s2 d(x, z).
n→∞
s
n→∞
Further, we have the following basic denitions with respect to an arbitrary
b-metric
on real which is essential in the text.
Denition 2.8 (Uniform b-convergence) Let {fn (x)} be a sequence of realvalued functions dened on a set A ⊆ R and ρ be arbitrary b-metric on R. If
{fn (x)} b-converges pointwise on a set A, then we can dene f : A → R by
b
lim fn (x) = f (x) or fn (x) →
− f (x)
ρ
n→∞
for each x ∈ A.
In other words, {fn (x)} b-converges to f on A if and only if for each x ∈ A
and for an arbitrary > 0, there exists an integer N = N (, x) such that
ρ(fn (x), f (x)) < whenever n > N . The integer N in the denition of pointwise convergence may, in general, depend on both > 0 and x ∈ A. If,
however, one integer can be found that works for all points in A, then the bconvergence is said to be uniform. That is, a sequence of functions {fn (x)}
uniformly b-converges to f on a set A if for each > 0, there exists an integer
N () such that
ρ(fn (x), f (x)) < or fn (x) −→ f (x) whenever n > N () and for all x ∈ A.
uρ
Denition 2.9 Let ρ be arbitrary b-metric on R with s ≥ 1 then the following
basic denitions can be given as follows:
96
U§ur Kadak
(i) (b-Limit) Suppose f : R → R is dened on the real line and c, L ∈ R. It is
said the b-limit of f , as x approaches c with respect to b-metric ρ, is L
and written b limx→c f (x) = L. On the other hand
b
lim f (x) = L ⇔ ∀ > 0, ∃δ > 0 3 ρ(f (x), L) ≤ for all ρ(x, c) < δ.
x→c
(ii) (b-Continuity) A function f : R → R is said to be b-continuous at c if it is
both dened at c and its value at c equals the b-limit of f as x approaches
c with respect to b-metric ρ, denoted by b limx→c f (x) = f (c).
(iii) (b-Boundedness) A sequence (xn ) ∈ ω is called b-bounded if and only if
the set which consists the terms of the sequence (xn ) is a bounded set.
That is to say that a sequence (xn ) ∈ ω is said to be b-bounded if and
only if there exist a number m > 0 such that ρ(xn , 0) ≤ m for all n ∈ N.
By taking the value
s=1
for an arbitrary
b-metric ρ
the notions in De-
nition 2.9 are reduced to those of classical limit, continuity and boundedness.
3
Main Results
Lemma 3.1 Let d be usual metric. Then the distance function Db dened by
− p
+ + p
Db (u, v) := sup ρ([u]λ , [v]λ ) := sup max{(d(u−
λ , vλ ) , (d(uλ , vλ ) }
λ∈[0,1]
(1)
λ∈[0,1]
is a fuzzy b-metric for u = [u]λ , v = [v]λ ∈ E 1 . Furthermore the function ρ is
also b-metric denoted by fuzzy usual b-metric with s = 2p−1 where p > 1 is a
real number.
Proof 3.2 One can easily show by a routine verication that ρ and Db satisfy
b-metric axioms in Denition 2.2. So, we prove only the axiom for Db .
Let u = [u]λ , v = [v]λ and w = [w]λ ∈ E 1 . By taking into account the
conditions max{a + b, c + d} ≤ max{a, c} + max{b, d} for all a, b, c, d > 0 and
(a + b)p ≤ 2p−1 (ap + bp ) from convexity of the function xp for p ≥ 1. We
immediately deduce that
On the Classical Sets of Sequences...
97
(b3 ) Db (u, w)
=
− p
+
+ p
sup max{(d(u−
λ , wλ ) , (d(uλ , wλ ) }
λ∈[0,1]
≤
−
−
− p
+ +
+
+ p
sup max [d(u−
λ , vλ ) + d(vλ , wλ )] , [d(uλ , vλ ) + d(vλ , wλ )] )
λ∈[0,1]
− p
−
− p
+ + p
+
+ p
≤ 2p−1 sup max d(u−
λ , vλ ) + d(vλ , wλ ) , d(uλ , vλ ) + d(vλ , wλ )
λ∈[0,1]
p−1
≤ 2
− p
+ + p
sup max d(u−
λ , vλ ) , d(uλ , vλ )
λ∈[0,1]
+
sup max d(vλ− , wλ− )p , d(vλ+ , wλ+ )p
λ∈[0,1]
= 2p−1 (Db (u, v) + Db (v, w)).
Therefore, (E 1 , Db ) is b-metric space on E 1 with s = 2p−1 where p > 1 is a
real number.
As a result of Bolzano-Weierstrass Theorem for every bounded innite sequence of fuzzy numbers in [48] we may give the next corollary based on fuzzy
level set completeness.
Corollary 3.3 The set (E 1 , Db ) is b-complete metric space on E 1 with s =
2p−1 where p > 1 is a real number.
Example 3.4 Let ρ be b-metric in Lemma 3.1. Consider the membership
functions u(x) and v(x) dened by triangular form as


 2x − 1 , 1/2 ≤ x ≤ 2/3,
 12x − 1 , 1/12 ≤ x ≤ 1/6,
3 − 2x , 2/3 < x ≤ 1,
3 − 12x , 1/6 ≤ x ≤ 1/4,
u(x) =
v(x) =


0
, otherwise,
0
, otherwise,
for all k ∈ N. It is trivial that u−λ = λ+1
, vλ− = λ+1
and u+λ = 3−λ
, vλ+ = 3−λ
2
12
2
12
for all λ ∈ [0, 1]. Therefore we can calculate the distance between two fuzzy
numbers u and v with respect to ρ
Db (u, v) =
− p
+ + p
sup max{(d(u−
λ , vλ ) , (d(uλ , vλ ) }
λ∈[0,1]
+ p
− p
+
sup max{|u−
λ − vλ | , |uλ − vλ | }
λ∈[0,1]
5(λ + 1) p 5(3 − λ) p
= (5/4)p .
= sup max ,
12 12 λ∈[0,1]
=
If we choose p = 1, then the distance Db (u, v) = 5/4 which is equal to the
usual fuzzy distance. If we take p = 3 then s = 4 and Db (u, v) = (5/4)3 .
Furthermore one can conclude that the fuzzy distance based on b-metric ρ of
two fuzzy numbers depends on the choice of s.
98
U§ur Kadak
To avoid undue repetition in the statements, we give the next proposition
without proof which is concerning Lemma 3.1. Now, we may give:
+ p
p
− p
+ p
Db (u, 0) = sup max{|u−
λ | , |uλ | } = max{|u0 | , |u0 | }.
(2)
λ∈[0,1]
Proposition 3.5 Let u, v, w, z ∈ E 1 and k ∈ R. Then,
(i) Db (ku, kv) = |k|p Db (u, v) with p ≥ 1.
(ii) Db (u + v, w + v) = Db (u, w).
(iii) Db (u + v, w + z) ≤ s[Db (u, w) + Db (v, z)] with s ≥ 1.
(iv) s|Db (u, 0) − Db (v, 0)| ≤ Db (u, v) ≤ s[Db (u, 0) + Db (v, 0)] for all s ≥ 1.
Following Matloka [27], we give some denitions concerning the sequences
of fuzzy numbers with respect to the fuzzy
b-metric
below, which are needed
in the text.
Lemma 3.6 (cf. [32]) The following statements hold:
(i) Db (uv, 0) ≤ Db (u, 0)Db (v, 0) for all u, v ∈ E 1 .
(ii) If uk → u, as k → ∞ then Db (uk , 0) → Db (u, 0), as k → ∞; where
(uk ) ∈ ω(F ).
Proof 3.7 (i) It is clear that the inequalities |u−λ |p ≤ Db (u, 0) and |u+λ |p ≤
Db (u, 0) hold for all λ ∈ [0, 1]. By considering these facts, one can see that the
distance Db (uv, 0)
=
sup max{|(uv)− (λ)|p , |(uv)+ (λ)|p }
λ∈[0,1]
≤
+ p + p
− p + p
+ p − p
p − p
sup max{|u−
λ | |vλ | , |uλ | |vλ | , |uλ | |vλ | , |uλ | |vλ | }
λ∈[0,1]
≤
sup max{Db (u, 0)|vλ− |p , Db (u, 0)|vλ+ |p , Db (u, 0)|vλ− |p , Db (u, 0)|vλ+ |p }
λ∈[0,1]
= Db (u, 0) sup max{|vλ− |p , |vλ+ |p }
λ∈[0,1]
b
= D (u, 0)Db (v, 0),
which completes the proof of part (i).
(ii) This is trivial by using the fact given by (iv) of Proposition 3.5 and
Denition 2.4(a).
Representation Theorem 1 Let [u]λ = [u− (λ), u+ (λ)] for u ∈ E 1 and for
each λ ∈ [0, 1]. Then the following statements hold:
99
On the Classical Sets of Sequences...
(i) u− (λ) is a b-bounded and non-decreasing left b-continuous and u+ (λ) is a
b-bounded and non-increasing left b-continuous function on (0, 1].
(ii) The functions u− (λ) and u+ (λ) are right b-continuous at the point λ = 0.
Proof 3.8 The proof can be obtained in a similar way from [38] which consists
as a generalization of this theroem. We omit the detail.
Remark 3.9 If the number u is the uniform b-limit
in Denition 2.8 of the
ρ
1
sequence {un } of fuzzy numbers on E , write un →
− u, as n → ∞ then the real
uρ
uρ
−
−
+
sequences of functions un (λ) −→ uλ and un (λ) −→ u+λ in [0, 1], respectively.
Theorem 3.10 Let (uk ), (vk ) ∈ ω(F ) with b limk uk = a , b limk vk = b. Then,
(i) b limk (uk + vk ) = a + b as k → ∞,
(ii) b limk (uk − vk ) = a − b as k → ∞,
(iii) b limk uk vk = ab as k → ∞,
(iv) b limk uk /vk = a/b as k → ∞ where 0 ∈ [uk ]0 for all k ∈ N and 0 ∈ [u]0 .
Denition 3.11 A sequence (uk ) ∈ ω(F ) is called fuzzy b-bounded if and only
if the set of fuzzy numbers consisting of the terms of the sequence (uk ) is a
b-bounded set. That is to say that a sequence (uk ) ∈ ω(F ) is said to be fuzzy
b-bounded if and only if there exist two fuzzy numbers m and M such that
−
−
m uk M for all k ∈ N. This means that m−
λ ≤ uk (λ) ≤ Mλ and
+
+
m+
λ ≤ uk (λ) ≤ Mλ for all λ ∈ [0, 1].
b-boundedness of the sequence (uk ) ∈ ω(F ) is
+
b-boundedness of the functions u−
k (λ) and uk (λ)
on [0, 1]. Therefore, one can see by using the relation (2) that the fuzzy bboundedness of the sequence (uk ) ∈ ω(F ) is equivalent to the fact that
The fact that the fuzzy
equivalent to the uniform
+
p
p
sup Db (uk , 0) = sup sup max{|u−
k (λ)| , |uk (λ)| } < ∞ (p ≥ 1)
k∈N
k∈N λ∈[0,1]
Now, prior to stating and proving the lemma concerning the sum of a
convergent series of fuzzy numbers we give the following denition (cf.
b-
Kim
and Ghil [50]):
b
1
Denition 3.12
. Then
P Let (uk ) ∈ ω(F ) and D be a fuzzy b-metric on E P
the expression uk is called a series of fuzzy numbers. Denote sk = nk=0 uk
for all n ∈ N, if the sequence (sk ) b-converges to a fuzzy number u then we say
100
U§ur Kadak
that the series uk of fuzzy numbers b-converges to u and write
which implies as n → ∞ that
P
n
X
uρ
u−
→
k (λ) −
n
X
u (λ) and
−
k=0
P∞
k=0
uk = u
uρ
u+
→ u+ (λ),
k (λ) −
k=0
uniformly in λ ∈ [0, 1]. On the other hand by taking into account ρ is a b-metric
on R, there exists an integer N () such that
!
ρ
X
±
u±
k (λ), u (λ)
< ε whenever n > N () and for all λ ∈ [0, 1].
k
Conversely,
if the fuzzy numbers uk = {(u− (λ), u+ (λ)) : λ ∈ [0, 1]}, k u−k (λ) =
P
−
+
+
u− (λ) and k u+
k (λ) = u (λ) uρ-converge
P in λ, then u = {[uλ , uλ ] : λ ∈ [0, 1]}
denes a fuzzy number such that u = uk .
P
Lemma 3.13 (cf. [32])
If the fuzzy number uk = {[u−k (λ), u+k (λ)] : λ ∈ [0, 1]},
P
P
−
−
+
+
−
+
k uk (λ) = uλ and
k uk (λ) = uλ uρ-converge
P in λ, then u = {[uλ , uλ ] :
λ ∈ [0, 1]} denes a fuzzy number such that u =
uk .
Proof 3.14 To prove the lemma, we must show that the pair of functions u−
and u+ satises the conditions of Representation Theorem. For this, we prove
that u− is a b-bounded, non-decreasing, left b-continuous function on (0, 1] and
right b-continuous at the point λ = 0. u−k 's are the b-bounded, non-decreasing,
left b-continuous functions on (0, 1] and right b-continuous at the point λ = 0
for each k ∈ N.
−
(i) LetPλ1 < λ2 . Then,
u−
k (λ1 ) ≤ uk (λ2 ) for each k ∈ N. Therefore, we have
P
−
−
−
−
k uk (λ1 ) ≤
k uk (λ2 ) which yields the fact that u (λ1 ) ≤ u (λ2 ).
−
Hence, u is non-decreasing.
(ii) By taking into account the uniform b-convergence in λ of limλ→λ−0 u−k (λ) =
P −
−
u−
k (λ0 ),
k uk (λ) = u (λ) for each k ∈ N we obtain for λ0 ∈ (0, 1] that
lim− u− (λ) = lim−
λ→λ0
λ→λ0
X
u−
k (λ) =
k
X
k
lim− u−
k (λ) =
λ→λ0
X
−
u−
k (λ0 ) = u (λ0 )
k
which shows that u− is a left b-continuous function on (0, 1].
(iii) By using the uniform b-convergence in λ in the expressions lim+ u−k (λ) =
λ→0
P −
−
u−
(0)
for
each
k
∈
N
and
u
(λ)
=
u
(λ)
,
we
see
that
k
k k
lim+ u− (λ) = lim+
λ→0
λ→0
X
k
u−
k (λ) =
X
k
lim+ u−
k (λ) =
λ→0
X
−
u−
k (0) = u (0).
(3)
k
This means that u− is a right b-continuous function at the point λ = 0.
101
On the Classical Sets of Sequences...
(iv) There exists Mk > 0 such thatPρ(u−k (λ), 0) ≤ Mk for all λ ∈ [0, 1] and for
−
all k ∈ N. Since the series k u−k (λ) = uP
(λ) uρ-converges
in λ there
−
exists n0 ∈ N for all ε > 0 such that ρ ∞
u
(λ),
0
< ε for all
k=n+1 k
n ≥ n0 and for all λ ∈ [0, 1]. Therefore, we have
ρ(u−
λ , 0) = ρ
∞
X
!
u−
k (λ), 0
n
X
= ρ
k=0
k=0
≤
≤
n
X
k=0
n
X
!
∞
X
u−
k (λ) +
u−
k (λ), 0
k=n+1
ρ(u−
k (λ), 0) + ρ
∞
X
!
u−
k (λ), 0
k=n+1
Mk + ε ≤ Kε
k=0
This leads us to the fact that u− is a b-bounded function.
Since one can establish in the similar way that u+ is a b-bounded, nonincreasing, left b-continuous function on (0, 1], and right b-continuous at the
point λ = 0, we omit the detail.
−
+
Therefore, it is deduced
fuzzy number.
P that [u]λ = [uk (λ), uk (λ)] denes a P
Finally, we show that uk = u. Since the series of functions k u−k (λ) and
P
+
−
+
k uk (λ) uρ-converge in λ to u (λ) and u (λ), respectively, for all ε > 0 and
s = 2p−1 there exists n0 ∈ N such that
Db
n
X
k=0
!
uk , u
p p )
( n
n
X
X
−
+
−
+
= sup max uk (λ) − u (λ) , uk (λ) − u (λ)
λ∈[0,1]
k=0
k=0
(
)
n
n
X
X
p
p
−
−
+
+
≤ max sup uk (λ) − uλ , sup uk (λ) − uλ < ε
λ λ k=0
for
P all n ≥ n0 , the sequence (
uk = u.
Pn
k=0
k=0
uk ) b-converges to the fuzzy number u, i.e.
Example 3.15 As an example for b-convergent
P series corresponding fuzzy bmetric Db with s = 2p−1 , consider the series uk with
 k
2 t−1



1,
uk (t) =
2
−
2k−2 t



0
, 21k ≤ t ≤ 22k ,
, 22k < t ≤ 24k ,
, 24k < t ≤ 28k ,
, otherwise,
for all k ∈ N. It is obvious that u−k (λ) = λ+1
and u+k (λ) = 4(2−λ)
for all
2k
2k
λ ∈ [0, 1]. Therefore by using the uniform b-convergence of b-metric we see
102
that
P
U§ur Kadak
P
−
k (uk )λ
= 2p (λ + 1) and
+
k (uk )λ
P
uk = u, where




t
2p
−1
1
u(t) :=
2 − 8tp



0
3.1
= 8p (2 − λ). Then, it is conclude that
, (2 ≤ t ≤ 4),
, (4 ≤ t ≤ 8),
, (8 ≤ t ≤ 16),
, (otherwise).
Completeness of Some Fuzzy Sequence Spaces
Lemma 3.16 Dene the relation Db,λ on the space γ by
b,λ
: γ × γ −→ R+
D∞
b,λ
(u, v) 7−→ D∞
(u, v) = sup Db (uk , vk ) = sup sup ρ(uk , vk )
k∈N
k∈N λ∈[0,1]
where γ denotes any of the spaces `∞ (b, λ), c(b, λ), c0 (b, λ) and u = (uk ), v =
b,λ
) is a b-complete metric space.
(vk ) ∈ γ . Then, (γ, D∞
Proof 3.17 Since the proof is similar for the spaces c(b, λ) and c0 (b, λ), we
prove the theorem only for the space `∞ (b, λ).
b,λ
satises (b1 ) and
One can easily show by a routine verication that D∞
(b2 ). So, we prove only (b3 ). Let u = (uk ), v = (vk ) and w = (wk ) ∈ `∞ (b, λ).
Then,
(b3 ) By using the axiom (b3 ) in Denition 2.2 we get
b,λ
D∞
(uk , wk ) = sup Db (uk , wk ) ≤ sup s[Db (uk , vk ) + Db (vk , wk )]
k∈N
k∈N
b
≤ s sup D (uk , vk ) + s sup Db (vk , wk )
k∈N
k∈N
b
= s sup D (uk , vk ) + sup Db (vk , wk )
k∈N
k∈N
b,λ
b,λ
= s D∞ (uk , vk ) + D∞ (vk , wk )
b,λ
for all s ≥ 1. Therefore (`∞ (b, λ), D∞
) is b-metric space on `∞ (b, λ). It
remains to prove the b-completeness of the space `∞ (b, λ).
(m)
Let xm = {x(m)
1 , x2 , . . .} be a b-Cauchy sequence in `∞ (b, λ). Then,
b,λ
for any > 0 there exists a positive integer m0 such that D∞
(xm , xr ) =
(m)
(r)
b
supk∈N D (xk , xk ) < for all m, r > m0 . A fortiori, for every xed k ∈ N
and for m, r > m0 then
n
o
(m)
(r)
Db xk , xk : k ∈ N < .
(4)
In this case for any xed k ∈ N, by using completeness of (E 1 , Db ) in Corollary
3.3 , we say that x(m)
is a b-Cauchy sequence and is b-convergent. That is to
k
103
On the Classical Sets of Sequences...
say that (x−λ )(m)
and (x+λ )(m)
are uρ-convergent for all λ ∈ [0, 1]. Now, we
k
k
(m)
b
suppose that limm→∞ xk = xk and x = (x1 , x2 , . . .). We must show that
b
b,λ
limm→∞ D∞
(xm , x) = 0 and x ∈ `∞ (b, λ).
The constant m0 ∈ N for all m > m0 , taking the b-limit as r → ∞ in (4),
(m)
we obtain Db (x(m)
∈ `∞ (b, λ), there exists
k , xk ) < for all k ∈ N. Since xk
b (m)
a positive number δ > 0 such that D (xk , 0) ≤ δ . By taking into account
b-metric axiom (b3 ) we get
(m)
(m)
Db (xk , 0) ≤ s[Db (xk , xk ) + Db (xk , 0)] < s( + δ)
(5)
for all s ≥ 1. It is clear that (5) holds for every k ∈ N whose right-hand side
does not involve k. This leads us to the consequence that x = (xk ) ∈ `∞ (b, λ).
Also, we immediately deduce that the inequality
(m)
b,λ
D∞
(xm , x) = sup Db xk , xk < k∈N
b,λ
holds for m > m0 . This shows that D∞
(xm , x) → 0 as m → ∞. Since (xm )
is an arbitrary b-Cauchy sequence, `∞ (b, λ) is b-complete.
Lemma 3.18 Dene the distance function Dqb,λ by
Dqb,λ : `q (b, λ) × `q (b, λ) −→ R+
P b
q 1/q
(u, v)
7−→ Dqb,λ (u, v) :=
k D (uk , vk )
where x = (xk ), y = (yk ) ∈ `q (b, λ) and 1 ≤ q < ∞. Then, (`q (b, λ), Dqb,λ ) is a
b-complete metric space.
Proof 3.19 It is obvious that Dqb,λ satises (b1 ) and (b2 ). Let u = (uk ), v =
(vk ) and w = (wk ) ∈ `q (b, λ). Then, we derive by applying the Minkowski's
inequality Dqb,λ (u, w) can be evaluated as
=
X
b
q
D (uk , wk )
1/q
≤
X
k
≤
q 1/q
(s )
q
s[D (uk , vk ) + D (vk , wk )]
b
= s
1/q
k
X
b
q
D (uk , vk )
1/q
+
k
= s
b
X
b
q
D (vk , wk )
1/q k
b
q
D (uk , vk )
k
b,λ
[Dq (u, v)
X
1/q
+
X
b
q
D (vk , wk )
1/q k
+
Dqb,λ (v, w)].
This shows that (b3 ) also holds. Therefore (`q (b, λ), Dqb,λ ) is b-metric space with
s = 2q−1 . Since the remaining part of this proof is analogous Lemma 3.16 we
omit the detail. Hence (`q (b, λ), Dqb,λ ) is a b-complete metric space.
104
U§ur Kadak
Concluding Remarks
In this research some new sequence spaces of fuzzy numbers are introduced by
b-metric with respect to the usual metric, and shown that
the given spaces are b-complete. In addition, this work presents a new tool for
using the notion of
the description of xed point theorems for fuzzy contractive mappings.
The
potential applications of the obtained results include the establishment of new
sequence spaces which are interesting topics for the future works. Of course, it
will be meaningful to determine the alpha-, beta- and gamma-duals of
b-metric
sequence spaces. We should record that one can study on the domain of some
triangle matrices in
b-metric sequence spaces `∞ (b, λ), c(b, λ) and `q (b, λ) which
is a new development on the theory of sequence spaces and matrix transformations.
Finally, we should note from now on that our next papers will be
devoted to Kothe-Toeplitzs duals and matrix transformations between some
classical sets of sequences of fuzzy numbers with respect to
Acknowledgements:
b-metric.
We record our pleasure to the anonymous referee for
his/her constructive report and many helpful suggestions on the main results
of the earlier version of the manuscript which improved the presentation of the
paper.
The authors declare that they have no competing interests.
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