Download Double sequences of interval numbers defined by Orlicz functions

ACTA ET COMMENTATIONES UNIVERSITATIS TARTUENSIS DE MATHEMATICA
Volume 17, Number 1, June 2013
Available online at www.math.ut.ee/acta/
Double sequences of interval numbers defined by
Orlicz functions
Ayhan Esi
Abstract. We define and study λ2 -convergence of double sequences of
interval numbers defined by Orlicz function and λ2 -statistical convergence of double sequences of interval numbers. We also establish some
inclusion relations between them.
1. Introduction
The idea of statistical convergence for ordinary sequences was introduced
by Fast [7] in 1951. Schoenberg [17] studied statistical convergence as a
summability method and listed some elementary properties of statistical
convergence. Both of these authors noted that if a bounded sequence is
statistically convergent, then it is Cesàro summable. Recently Mursaleen
[15] defined and studied λ-statistical convergence for sequences as follows.
Let λ = (λi ) be a non-decreasing sequence of positive numbers tending to
infinity such that λi+1 ≤ λi + 1, λ1 = 1. Then a sequence x = (xk ) is said
to be λ-statistically convergent to a number L if for every ε > 0,
1
lim
|{k ∈ Ii : d (xk , L) ≥ ε }| = 0,
i→∞ λi
where Ii = [i − λi + 1, i].
Recall (see [10]) that an Orlicz function M is a continuous, convex, nondecreasing function, defined for u ≥ 0, such that M (0) = 0 and M (u) > 0 if
u > 0. An Orlicz function M is said to satisfy ∆2 -condition for all values of
u if there exists a number K > 0 such that M (2u) ≤ KM (u), u ≥ 0.
Interval arithmetic was first suggested by Dwyer [2] in 1951. Development
of interval arithmetic as a formal system and evidence of its value as a
computational device was provided by Moore [12] in 1959 and Moore and
Received February 17, 2012.
2010 Mathematics Subject Classification. 40A05, 40A35, 40C05, 46A45.
Key words and phrases. Double sequence space, interval numbers, Orlicz function, statistical convergence.
http://dx.doi.org/10.12697/ACUTM.2013.17.04
57
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AYHAN ESI
Yang [13] in 1962. Furthermore, Moore and others (see [3], [8], [12] and [11])
have developed applications to differential equations.
Chiao [1] introduced sequences of interval numbers and defined the usual
convergence of sequences of interval numbers. Şengönül and Eryılmaz [18]
introduced and studied bounded and convergent sequence spaces of interval
numbers and showed that these spaces are complete metric spaces. Recently,
Esi in [4] and [5] defined and studied λ-statistical and lacunary statistical
convergence of interval numbers, respectively.
We denote the set of all real valued closed intervals by IR. Any element of
IR is called an interval number and is denoted by A = [xl , xr ]. Let xl and xr
be the smallest and the greatest points of an interval number A, respectively.
For interval numbers A1 = [x1l , x1r ], A2 = [x2l , x2r ] and a number α ∈ R we
have
A1 = A2 ⇐⇒ x1l = x2l , x1r = x2r ,
A1 + A2 = {x ∈ R : x1l + x2l ≤ x ≤ x1r + x2r }
and
(
{x ∈ R : αx1l ≤ x ≤ αx1r } if α ≥ 0,
αA =
{x ∈ R : αx1r ≤ x ≤ αx1l } if α < 0.
The set of all interval numbers IR is a complete metric space with the
distance
d A1 , A2 = max {|x1l − x2l | , |x1r − x2r |}
(see [14]). In the special case A1 = [a, a] and A2 = [b, b] we obtain the usual
metric of R.
Now we give the definition of a convergent sequence of interval numbers
(see [1]).
Definition 1.1. A sequence Ak of interval numbers is said to be converA0 if for each ε > 0 there exists a positive integer
gent to an interval number
k0 such that d Ak , A0 < ε for all k ≥ k0 . We denote it by limk Ak = A0 .
Thus, limk Ak = A0 if and only if limk xkl = x0l and limk xkr = x0r .
In this paper, we introduce and study the concepts of λ2 -summable and
statistically λ2 -convergent double sequences of interval numbers, and relations between them.
2. Definitions
Recall that a double sequence (ak,i ) of real numbers is said to be convergent
in the Pringsheim sense to a number L if for every ε > 0 there exists an
index n such that |ak,i − L| < ε whenever k, i > n (see [16]). We transfer
DOUBLE SEQUENCES OF INTERVAL NUMBERS
59
this definition to the double sequences of interval numbers in the following
way.
Definition 2.1. An interval valued double sequence Ak,l is said to be
convergent in the Pringsheim sense to an interval number A0 if for every
ε > 0 there exists n ∈ N such that
d Ak,l , A0 < ε for k, l > n.
In this case we write P -lim Ak,l = A0 .
We denote by c2 the set of all convergent in the Pringsheim sense double
sequences of interval numbers.
Definition 2.2. An interval valued double sequence Ak,l is said to be
bounded if there
exist an interval number A0 and a positive number B such
that d Ak,l , 0 ≤ B for all k, l ∈ N.
We will denote the set of all bounded double sequences of interval number
2
by the symbol `∞ . It should be noted that, similarly to the case of double
2
number sequences, c2 is not the subset of `∞ .
Let a double sequence λ2 = (λi,j ) of positive real numbers tend to infinity
and satisfy
λi+1,j ≤ λi,j + 1, λi,j+1 ≤ λi,j + 1,
λi,j − λi+1,j ≤ λi,j+1 − λi+1,j+1 , λ1,1 = 1.
Put
Ii,j = {(k, l) : i − λi,j + 1 ≤ k ≤ i, j − λi,j + 1 ≤ l ≤ j} .
Definition 2.3. An interval valued double sequence Ak,l is said to be
λ2 -summable if there exists an interval number A0 such that
P - lim
i,j
1
λi,j
X
d Ak,l , A0 = 0.
(k,l)∈Ii,j
Definition 2.4. Let M be an Orlicz function, let Ak,l be an interval
valued double sequence, and let p = (pk,l ) be a double sequence of positive
real numbers. Let λ2 = (λi,j ) be the double sequence defined above. We
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AYHAN ESI
define
h

i 
1
2
V λ2 , M, p = Ak,l : P - lim
i,j λi,j

"
X
M
(k,l)∈Ii,j
d Ak,l , A0
r
!#pk,l
= 0,
o
for some r > 0 and A0 ∈ IR ,

"
!#pk,l
h
i 
X
d Ak,l , 0
1
2
= 0,
V λ2 , M, p = Ak,l : P - lim
M
i,j λi,j
r
0 
(k,l)∈Ii,j
o
for some r > 0
and


h
i
1
2
V λ2 , M, p = Ak,l : sup

∞
i,j λi,j
"
X
M
(k,l)∈Ii,j
!#pk,l
d Ak,l , 0
< ∞,
r
o
for some r > 0 ,
where 0 = [0, 0].
If we consider various assignments of M , λ2 and p in Definition 2.4, then
we obtain different special sets of sequences. For example, if pk,l = 1 for
sets
hall k, l ∈i N,
h then these
i
h reduce
i to the sets denoted, respectively, by
2
2
2
V λ2 , M , V λ2 , M and V λ2 , M . In the special case λi,j = ij (i, j ∈
∞ i
h
0
2
N), we write c2 , M instead of V λ2 , M .
For M (x) = x we obtain

i 
h
X p
1
2
V λ2 , p = Ak,l : P - lim
d Ak,l , A0 k,l = 0
i,j λi,j

(k,l)∈Ii,j
o
for some A0 ∈ IR ,
h
i
h
i
2
2
and similarly, V λ2 , p and V λ2 , p .
0
∞
3. Main theorems
Theorem 3.1. If 0 < pk,l < qk,l and
h
i
2
⊂ V λ2 , M, q .
qk,l
pk,l
h
i
2
is bounded, then V λ2 , M, p
DOUBLE SEQUENCES OF INTERVAL NUMBERS
Proof. If we take M
d(Ak,l ,A0 )
r
61
pk,l
= wk,l for all k, l ∈ N, then using
the same technique employed in the proof of Theorem 2.9 from [9] we get
the result.
Corollary 3.2. The following statements are valid.
h
i h
i
2
2
(i) If 0 < inf k,l pk,l ≤ 1 for all k, l ∈ N, then V λ2 , M, p ⊂ V λ2 , M .
h
i
2
(ii) If 1 ≤ pk,l ≤ supk,l pk,l = H < ∞ for all k, l ∈ N, then V λ2 , M
h
i
2
⊂ V λ2 , M, p .
Proof. (i) follows from Theorem 3.1 with qk,l = 1 for all k, l ∈ N and (ii)
follows from Theorem 3.1 with pk,l = 1 for all k, l ∈ N.
The proof of the following result is a routine work, so we omit it.
Proposition 3.3. Let M
that
h be an
i Orlicz
h function
i such
h
i ∆h2 -conditioni
2
2
2
2
is satisfied. Then we have V λ2 , p ⊂ V λ2 , M, p , V λ2 , p ⊂ V λ2 , M, p
0
0
h
i
h
i
2
2
and V λ2 , p
⊂ V λ2 , M, p .
∞
∞
The following definition was presented by Esi [6] for a single
sequence
of interval numbers. A sequence of interval numbers Ak is said to be
statistically λ-convergent to an interval number A0 if for every ε > 0,
1 lim
k ∈ In : d Ak , A0 ≥ ε = 0,
n→∞ λn
where the vertical bars indicate the number of elements in the enclosed set.
Now, we will give definitions of statistical convergence and statistical
λ2 -convergence for double sequences of interval numbers.
Definition 3.1. A double sequence Ak,l of interval numbers is said to
be statistically convergent to an interval number A0 provided that for each
ε > 0,
1 P - lim
(k, l) ∈ N × N; k ≤ i, l ≤ j : d Ak,l , A0 ≥ ε = 0.
i,j→∞ ij
We denote the set of all statistically convergent double sequences of interval
numbers by s2 .
Definition 3.2. A double sequence Ak,l of interval numbers is said to
be statistically λ2 -convergent to an interval number A0 if for each ε > 0,
1 P - lim
(k, l) ∈ Ii,j : d Ak,l , A0 ≥ ε = 0.
i,j→∞ λi,j
We denote the set of all statistically λ2 -convergent double sequences of interval numbers by s2λ2 .
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AYHAN ESI
Theorem 3.4. Let M hbe an Orlicz
i function. If 0 < h ≤ inf k,l pk,l ≤
2
supk,l pk,l = H < ∞, then V λ2 , M, p ⊂ s2λ2 .
i
h 2
Proof. Let Ak,l ∈ V λ2 , M, p . Then there exists r > 0 such that
1
λi,j
"
X
M
(k,l)∈Ii,j
d Ak,l , A0
r
!#pk,l
→0
in the Pringsheim sense if i, j → ∞. If ε > 0, then we obtain
1
λi,j
≥
"
X
M
(k,l)∈Ii,j
1
λi,j
X
(k,l)∈Ii,j
(
!#pk,l
d Ak,l , A0
r
"
!#pk,l
d Ak,l , A0
M
r
)
d Ak,l ,A0 ≥ε
≥
1
λi,j
h
X
(k,l)∈Ii,j
M
ε ipk,l
r
)
(
d Ak,l ,A0 ≥ε
1
≥
λi,j
X
(k,l)∈Ii,j
min M
ε h
r
,M
ε H r
(
)
ε H 1 ε h
≥
(k, l) ∈ Ii,j : d Ak,l , A0 ≥ ε min M
,M
.
λi,j
r
r
d Ak,l ,A0 ≥ε
Hence Ak,l ∈ s2λ2 , which completes the proof.
Theorem 3.5. Let M be an Orlicz hfunction and
i let 0 < h ≤ inf k,l pk,l ≤
2
2
2
supk,l pk,l = H < ∞. Then sλ2 ∩ `∞ ⊂ V λ2 , M, p .
2
Proof. Let Ak,l ∈ s2λ2 ∩ `∞ . Then there is a constant N > 0 such that
d Ak,l , A0 ≤ N for all k, l ∈ N. Given ε > 0, for an arbitrarily fixed r > 0
DOUBLE SEQUENCES OF INTERVAL NUMBERS
63
we have
1
λi,j
=
!#pk,l
d Ak,l , A0
r
"
!#pk,l
d Ak,l , A0
M
r
"
X
M
(k,l)∈Ii,j
1
λi,j
X
(k,l)∈Ii,j
(
)
1
+
λi,j
X
d Ak,l ,A0 <ε
"
(k,l)∈Ii,j
(
M
d Ak,l , A0
r
)
h ε ipk,l
1
M
+
r
λi,j
!#pk,l
d Ak,l ,A0 ≥ε
1
≤
λi,j
X
(k,l)∈Ii,j
(
)
ε H ε h
,M
≤ max M
r
r
d Ak,l ,A0 <ε
X
(k,l)∈Ii,j
(
( H )
N h
N
max M
,M
r
r
)
d Ak,l ,A0 ≥ε
( H )
h
N
1 N
+
,M
(k, l) ∈ Ii,j : d Ak,l , A0 ≥ ε max M
.
λi,j
r
r
i
h 2
Hence Ak,l ∈ V λ2 , M, p . This completes the proof.
The following corollary follows directly from Theorems 3.4 and 3.5.
Corollary 3.6. If 0 < h ≤ inf k,l pk,l ≤ supk,l pk,l = H < ∞, then s2λ2 ∩
h
i
2
2
2
`∞ = V λ2 , M, p ∩ `∞ .
If we take λi,j = ij and pk,l = 1 for all k, l ∈ N in Theorems 3.4, 3.5 and
Corollary 3.6, then we get
Corollary 3.7. Let M be an Orlicz function. Then the following statements hold.
(i) c2 , M ⊂ s2 .
2
(ii) s2 ∩ `∞ ⊂ c2 , M .
2
2
(iii) c2 , M ∩ `∞ = s2 ∩ `∞ .
Acknowledgement
The author is extremely grateful to the referee for his/her many valuable
comments and suggestions.
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AYHAN ESI
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Adiyaman University, Science and Art Faculty, Department of Mathematics, 02040, Adıyaman, Turkey
E-mail address: [email protected].