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DENSITY BY MODULI AND LACUNARY STATISTICAL CONVERGENCE OF DOUBLE SEQUENCES
1 A. G. K. Ali, 2A. M. Brono and 3A. Masha
^Department of Mathematics and Computer Science, Borno State University Maiduguri, Nigeria ^Department of Mathematical Sciences, University of Maiduguri, Borno State, Nigeria Email: [email protected], [email protected], [email protected]
Abstract
In this paper, we introduced and studied the concept of lacunary statistical convergence of double sequence with respect to modulus function where the modulus function is an unbounded double sequence. We also introduced the concept of lacunary strong convergence of double sequence via modulus function. We further characterized those lacunary convergence of double sequence for which the lacunary statistically convergent of double sequence with respect to modulus function equals statistically convergent of double sequence with respect to modulus function. Finally, we established some inclusion relations between these two lacunary methods and proved some essential analogue for double sequence.
Keywords: modulus function, statistical convergence, lacunary strong convergence, lacunary statistical convergence, double sequence.
1. Introduction
The concept of statistical convergence was formally introduced by [1] and [2] independently. Although statistical convergence was introduced over fifty years ago, it has become an active area of research in recent years. It has been applied in various areas such as summability theory [3] and [4], topological groups [5] and [6], topological spaces [7], locally convex spaces [8], measure theory [9], [10 and [11], Fuzzy Mathematics [12] and [13]. In recent years generalization of statistical convergence has appeared in the study of strong summability and the structure of ideals of bounded functions, [14]. Extension of the notion of statistical convergence of single sequence to double sequences by proposed by [15]. The concept of lacunary statistical convergence of single sequence was introduced by [16]. The extension of the concept of lacunary statistical of single sequence to double sequences was proposed by [17]. The notion of modulus function was introduced by [18]. Following [19] and [20], we recall that a function /: [0, &>) ^ [0, ro) is said to be a modulus function if it satisfies the following properties
(1) /(x) = 0 if and only if x = 0
(2) /(x + y) < /(x) + /(y) for x > 0,y > 0,
(3) / is increasing,
(4) / is continuous from the right at 0.
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DENSITY BY MODULI_Volume 18, December 2023
It follows that f is continuous on [0, ro). The modulus function may be bounded or unbounded. For example, if we take f(x) =~~, then f(x) is bounded. But, 0< p < 1, f(x) = xp is not bounded. The definition of a new concept of density with help of an unbounded modulus function was proposed by [21], as a consequence, they obtained a new concept of non-matrix convergence, namely, /-statistical convergence, which is intermediate between the ordinary convergence and statistical and agrees with the statistical convergence when the modulus function is the identity mapping.
Quite recently, [22] and [23] have introduced and studied the concepts of /-statistical convergence of order a and /-statistical boundedness, respectively, by using approach of [21]. Quite recently, [24] introduced and studied the concept of /-lacunary statistical convergence and the concept of strong lacunary statistical convergence with respect to modulus function. We further extended and introduced some analogues results of double in line with that of [24].
Definition 1.1: ([15]): A real double sequence x = (xjk) is statistically convergent to a number l if for each e > 0, the set
{(j,k),j < n and k <m: \xjk — Z\ > e} (1)
has double natural density zero. In this case we write st2 — lim xjk = I and we denote the set of all statistically convergent double sequences by st2.
Definition 1.2 ([17]).The double sequence 9rs = (jr , ks) is called double lacunary if there exist two increasing sequences of integers such that j0 = 0,hr = jr — jr-1 ^ ro as r ^ ro and k0 = 0,hs =
ks—ks-1 ^ ro as s ^ ro. Let jrjS = jrks,hrs = hrhs and 9rs is determined by IrjS = {(j,k): jr-1 < j <
j fe
jr and ks-1 < k < ks}, qr qs =—— and qrs = qr(fs.
jr-1
Definition 1.3 ([17]): Let 9r s be a double lacunary sequence, the double number sequence x is double lacunary statistical convergent to L provided that for every e > 0,
1
lim—\{(j,k) E Ir/: \xjjk — L\> e}\ = 0. (2)
r,s hr s J
Throughout this paper s, L™ and c will denote the spaces of all, bounded and convergent double sequences of real numbers, respectively.
Now in this paper we introduce the concept of -lacunary statistical convergence of double sequence, where fjk is an unbounded modulus functions of double sequence. Definition 1.4: Let fjk be an unbounded modulus functions of double sequence. Let 9r s= (jr, ks) be double lacunary sequence. A double sequence x = (xjk) is said to be /,fc-lacunary statistically convergent of double sequence to L or SgJ,k- convergent to L, if, for each £ > 0,
1
lim f (h JiA\{(j,k) E Ir,s: \xjk —L\> e}\) = 0. (3)
In this case we write
Sk* — limX* =L or XJk ^ L d*).
For a given double lacunary sequence 9r s = (jr , ks) and unbounded modulus function fj k, by S^ we denote the set of all fjk -lacunary statistically convergent of double sequences.
2. Methods
2.1 /y,fc-Lacunary Statistical Convergence of Double Sequence
We begin by establishing elementary connections between convergence of double sequence, /¿fc-lacunary statistical convergence of double sequence and double lacunary statistical convergence. Theorem 2.1: Every convergent double sequence is /¿fc-lacunary statistically convergent of sequence, that is c c S^ for any unbounded modulus functions / of double sequence and double lacunary statistical convergence sequence 0rs.
Proof: Let x = (x7-fc) be any convergent double sequence. Then, for each e > 0, the set
(0', |x,-fc - i| > e) is finite. Suppose {(/', fc) E N x N: |x_,-fc - i| > e) = g0.
Now, since {(/', fc) E /r s: |xjfc — i| > e) c {(/, fc) E N x N: |xjfc — i| > e) and is modulus increasing, therefore
/m(|(0', ft) E /r,s: |x7-fc — i| > e)|) /(5o)
<
Taking limit as r, s ^ ro, on both sides, we get
/■,fc E ^r,s: |x7-fc - L| > e)
lim —---^ = o,
as /)-,fc(ftr,s) ^ ro as r,s ^ ro.
Theorem 2.2: Every /,fc-lacunary statistical convergent double sequence is double lacunary statistical convergent.
Proof: Suppose x = (x7-fc) is /,fc-lacunary statistically convergent double sequence to L. Then by the definition of limit and the fact that /,fc being modulus is subadditive, for every p£M, there exist r0, s0EM such that, for r, s > r0, s0, we have
1 . /Ka r (hrA
/J.fc(|(0',fc) e /r,s: |x7-fc - L| > £}|) < < = fo (^t)
Since /-,fc is increasing, we have
-H((/,fc)E/r,s:|x/fc-L| >e}| <1.
Hence, x= (x7-fc) is a double lacunary statistically convergent to L.
Remark 2.1: It seems that the inclusion Sg7,ft c Sgrs is strict. But right now we are not in a position to give an example of a double sequence which is S6r s-convergent but not S^-convergent. So it is left as an open problem.
Remark 2.2: From theorem 2.1 and 2.2, we can say that the concept of /,fc-lacunary statistical convergence is intermediate between the usual notion of convergence of double sequence and the double lacunary statistical convergence of double sequences.
We now establish a relationship between /,fc -lacunary statistical convergence of double sequences and double lacunary strong convergence with respect to modulus functions /,fc of double sequence.
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Theorem 2.3 Let 9r s = (jr, ks) be a double lacunary sequence, then consider the following:
(a) For any unbounded modulus functions f for which lim > 0 and there is a positive
constant c such that f(xy) > cf(x)f(y), for all x>0,y> 0, (i) xjk ^ L imphes xjk ^ L (Sg'*"),
(ii) Ng'ris a proper subset of .
(b) x EL2m and Xjk ^ L (n^*) imply Xjk ^ L (Sq^), for any unbounded modulus functions fjik of double sequence.
(c) Ng^ nL22 = n L22 for any unbounded modulus function fjk of double sequence for which lim ^^ > 0 and there is a positive constant c such that f(xy) > cf(x)f(y), for all x >0,y > 0.
Proof: (a) (i) For any double sequence x = (xjk) and e >0, by the definition of a modulus function (1) and (3) we have
if \ i ( X
rJ]JkElr,s r,S \jJkElr|S J r,s jJkElr'S
\\x]k-L\>£
1
>^fi,k(\{(j,k) E Ir,s: \x]k —L\> e}\e) >^fj,k(\{(j,k) E Ir,s: \xjk —L\> e}\)f(e) c fj,k(\{(j,k) E Ir/: \xjk —L\> e^
hr,s fj,k(hr,s)
-fj,k(K,s)f(£)
From where it follows that x £ Sfj'k as x £ Nfjk and lim (Jj,k(hr,s)/ ) > o
^r,s &r,s r,s^œ \ / h.
(fj,k(hr,s)l \ ( /h r, )
(ii) To show the strictness of inclusion, consider the double sequence x = (xjk) such that xjk is to be 1,2, ...,[^hrs\ at the first [jhrs\ integers in lr s, and xjk =0 otherwise. Note that (xjk) is not bounded. Also, for every e> 0,
fik(\{(j, k) £ Irs: \xik -L\> 0}|) = ^^ = x
Because lim „A lim (fik(hr^/h ) are positive and
I (\jhr s\) I '' / hr,s '
rSs.li 'hi-jKxr0
Thus, xjk ^ 0 (sfjk). On other hand,
i v v* n , ^ _ fj,k(i) + fj,k(2) + -+fjAUKJiK fjk(i + 2 + - + Lfjk(\xjk-Ll)= h~s > h~s
j ,k£Ir,s
fk ([« ^ + %)) flM)f,k (« + %)
r,
= c x r JL^— x-, .-——— x--—---> 0
([JK2 +1)/ hr,s
h
r,
A. G. K. Ali, A. M. Brono and A. Masha RT&A, No 4 (76) DENSITY BY MODULI_Volume 18, December 2023
As c, rlim„(/-,k([VO/[VO, ^(/^([V^II + !)/2)/((№3 + 1)/2), and r, J)(([V"r,sJ + 1)/2)/"r,s) are positive. Hence xjfc ^ 0
lim ([V^J)(([V^TJ + 1)/2)/^r,s) are positive. Hence x_,fc ^ 0 i^e/fl
, s*—^^O V r /
(b) Suppose that x7-k — L (sf7,,c) and x E L^, say |xjfc — L | < H for all y, k E M. Given e > 0, we have
Z I L|)+ I I^--L|)<^|((/,k)
r,i _/',fcE/r,s r,i ./',fcE/r,s ./',fcE/r,s r,i
|X7-ft-L|>£ |X7-ft-L|<£ 1
E Zr,s: |x7-fc — L| > e}|/J-,fc(H) ^ —ftr,s/;,fc(e).
^r.s
Taking limit on both sides as r,s — ro, we get lim (-MH ;kE/ H/;k(|x,k — L|) = 0, in view of
theorem 2.2 and the fact that /,k is increasing.
(c) This is an immediate consequence of (a) and (b)
Remark 2.3 The example given in part (a) of the above theorem shows that the boundedness condition cannot be omitted from the hypothesis of part (b).
3. Results
3.1 /-,k-Lacunary Statistical Convergence of Double Sequence Versus /-,k-Statistical Convergence of Double Sequence
In this section we study the inclusion Sf7,ft c SfJ,fc and SfJ,fc c Sf,fc under certain restrictions on 0r s and /-,fc.
Lemma 3.1.1: For any double lacunary sequence 0r s and unbounded modulus function /-,k for which lim(/( t)/ t) > 0 and there is a positive constant c such that /(xy) > c/(x)/(y), for all x > 0 ,y > 0,
t—ro
f f k
one has S—c Se7, if and only if lim inf qr,s > 1.
Proof: Sufficiency: If lim inf qr,s > 1 then there exists S > 0 such that <7r s >1 + 5 for sufficiently large r,s. Since s = kr s — kr-l s-1, we have
ftr,s > / S \2 kr s > (1 + S)
For sufficiently large , . If xjk — L(Sf ¿,fc), then, for given e > 0 and sufficiently large r, s we have
/■fc(|{/ < y'r and k < ks: |x,fc — L| > e}|) > ^O^M^^MD = /¿^
■'.M'VIU — Jr — s \ ]H \ — ¡\J — f,-,,ii' k,-) f:,,(i-k.-)
■ X
r k5)
( Vs ) (f7',ftOrks)) Vrks1 /¿,ftOrks) > ( hr,s ) V/fcOrksV (1+5)
'¿,fc(|(q,fc)E/r,s:|^fc-L|>6)|)
This proves the sufficiency.
Necessity: Assume that lim infqrs = 1. We can select a subsequence (/r(i)kSQ-)) of 0rs satisfying
7r( i)ks(j) ^ 1 + £ Jr(t)ksQ') >
7r(i)-1ks0')-1 y 7r(i)-1ks0')-1
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DENSITY BY MODULI_Volume 18, December 2023
Where r(i) > r(i — 1) + 2 and s(/) > s(/ — 1) + 2. Define a bounded double sequence by
_ f 1 i/y, k E /r,s(y),/or some i,y = 1,2,3,... )k 10 otherwise.
It is shown that x g Vgr? but x E w. Thus, we have x g Sf,fc. Hence S-7'^ £ Sf7,fc. But this is a
Ji*.
r,5 ' ^r 5 #r
contradiction to the assumption that Sf7',fc c Se7,ft. This contradiction shows that our assumption is wrong. Hence liminfqrs > 1.
r,
Remark 3.1.1: The double sequence x = (x)k), constructed in the necessity part of the above lemma, is an example of /,k-statistically convergent double sequence which is not /,k -lacunary statistically convergent of double sequence.
Lemma 3.1.2: For any double lacunary sequence 0rs and unbounded modulus functions /,k for which lim(/( t)/ t) > 0 and there is a positive constant c such that /(xy) > c/(x)/(y), for all x >
t—ro
0,y > 0 one has Sf,fc c S-7^ if and only if limsup qrs > 1.
r,s r,s
Proof: Sufficiency: If Iim sup qr s, then there is H > 0 such that qr s < H for all r, s. Now, suppose that
r,
x,k — L (Sf7,ft) and lim (/(hrs)/hrs) = L'. Therefore, for given e > 0, there exist r0,s0 E M such that
J V er,5 / r,S—ro '
for all , > 0, 0
/)',k (^r,s)
hr,s
< L' + ,
:/l,k(|(0',k) E /r,s: |x7-k — L| > e}|) < e.
/),k (^r,s)
Let Vrs = |{(/, k) E /r s: |xjk — L| > e}|. Using this notion, we have
fj,fc(Wr,s) f7,ft(^r,s)
< £ V r, S > r0,s0.
Now, let M =max(/1,k k (V2,2) . , //,k(^r0,S0)} and let m n be integers such that 7r-1 < m < 7r
and ks-1 < n < ks, then we can write
1
/),k(|(; < m,k < n: |x,k —L| > e}|) < ^/),k(|(/ < m,k < n: |x,k —L| > e}|)
/',k(/r-1ks-1)
/1, k , S 0 + ^ro + 1,So + 1
< f n\ X (/^1,1) + /,^2,2) + - + /),k(Vro,So) + /kOW+1) + • + /),k(^r,s))
/1, k( r-1 k -1)
0 0 M
< - - ---+[/),k(Vro+1 ,So + 1) + - + /),k(Vr,s)J
/)',k0r-1ks-1)
r0s0M
/',k(/r-1ks-1)
/),k (hro + 1,so + 1) /),k (Vro + 1,sp + 1) ,
h f (h \ "ro+1,so+1 + '
"ro + 1,So + 1 //,k(hro + 1,so + 1)
/,k (h r, ) /1, k( Vr, )
hr,s /),k(hr,s) r'S.
1
+ TT7-t—^[(L' + e)ehro+1,so+1 + •"+ (L' + e)ehr,sJ
/)',k(/r-1Ks-1)
1
1
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DENSITY BY MODULI_Volume 18, December 2023
r0s0M 1 r ,
' + TT;-1-\£(l + £)\hr0+i,s0+i + ■■■+ KA
fj,k(Jr-lks-l) fj,k(jr-lks-l)
0 0 M 1
■ + TTi—T^e(l + £)Urks - Jr0kS0\
^ -0-0-- f., jrk.
<7—T--i-7 + £(l + £)
fj, k ( r- -lks- l)
0 oM
fj,k 0r- -lks- l)
0 oM
fj,k (r- -lks- l)
0 oM
fj,k (r- -lks- l)
0 oM
r— • —
fj,k(jr-lks-l)J
+ £(l + £)f r -irlS
Jj,k\Jr-lks-l)/]r-lks-l]r-lks-l 1
+ £(ll + e)qr,s -¿—r.-t—TT-1—
Jj,k\]r-lks-l)/]r-lks-l
< '0'°'-~ + £(l'+£)H 1
fj,k(]r-lks-l) fj,k(jr-lks-l)/jr-lks-l
From where the sufficiency follows immediately, in view of the above fact that
lim (fj,k(]'r-lks-l)/jr-lks-l) > 0
r,s^m
Necessity: Suppose that lim sup qrs = m. We can select a subsequence (jr(i)kSj) of double lacunary
r,
sequence 9rs such that qr(i),S(j) > ij. Define a bounded double sequence x = (Xjk) by
x. = [1 if jr(i)K(j) < Jk ^ 2jr(i)-lks(j)-l, for some i,j = 1,2,3,..., jk I 0 otherwise.
7fj,k
It is shown that x £ Ng but x £ w. We conclude that x £ Sfj' , but x for every f,k-statistically
^ Jjk
convergent of double sequence is statistically convergent double sequence. Sgj'k £ Sfjk. But this is a
gr,s
f
contradiction to the assumption that S This contradiction shows that lim sup qrs < m.
r,s r,s
Remark 3.2.1: The double sequence x = (xjk), constructed in the necessity part of the above lemma, is an example of fjkk -lacunary statistically convergent double sequence which is not fjkk-statistically convergent double sequence.
Combining lemma 3.1.1 and 3.2.1 we have the following.
Theorem 3.1.1: For any double lacunary sequence 9rs and unbounded modulus functions fjkk for which lim (f(t)/t) > 0 and there is positive constant c such that f(xy) > c f(x)f(y), for all x > 0,y >
t^m
f
0, one has Sgh = Sfjk if and only if 1 < liminfqrs < lim sup qrs < m.
T,s r,s ' r,s
Theorem 3.2.1: For any double lacunary sequence 9rs and unbounded modulus functions fjkk for which lim (
t^m
0, one has
which lim (f(t)/1) > 0 and there is positive constant c such that f(xy) > c f(x)f(y), for all x > 0,y >
t^m
Sfjk = n Sfj,k = U Sfj,k. (4)
lim infqr s>1 gT.s lim sup qr s<TC gr,s
n S
™ infqr,s>i
h- rfi* i_____r.. \ r- cfj,k .
Proof: In view lemma 3.1, we have c n SgJ' . Suppose if possible x = (xjk) £
n Stlk but x £ Sfj,k. We have (xjk) £ Sij,k for all 9r s = (jr, ks) for which lim infqr s > 1. If
if - grS grS ' ■v c '
lim infqr,s>i gr,s
we
take 9rfS = (2r+s), then, in view theorem 3.1, we have Sgh ■■ = Sfj,k and so x £ Sfj,k, contrary to our assumption. Hence Sfjk = n Sgj . The remaining part can be proved similarly and hence is
lim irI1sfqr,s>1 r,S
omitted.
Remark 3.3.1: The double sequence x = (xjk) constructed in part (a) of theorem 2.1 belongs to Sfj,k
r.s
A. G. K. Ali, A. M. Brono and A. Masha RT&A, No 4 (76) DENSITY BY MODULI_Volume 18, December 2023
for every double lacunary sequence 0rs, as well unbounded modulus functions /-,fc for which
lim(/( t)/ t) > 0 and there is a positive constant c such that /(xy) > c/(x)/(y) for all x > 0,y > 0.
Hence n lim in/ qr s ^ 0.
r,s ' r,s
3.2 Inclusion Between two Lacunary Methods of /-,fc-Statistical Convergence.
Our first results shows that, for certain modulus function /¿fc, if $r,s is a lacunary refinement of the
double lacunary sequence 0rs S , c S^ . To establish this result, we first recall the definition of
pr,s "r,s
double lacunary refinement of double sequence.
Definition 3.2.1: The double lacunary sequence $r,s = Or- ks) is called a double lacunary refinement of double lacunary sequence 0rs = (/r, fcs) if (/r, fcs) c (/r, fc^). Theorem 3.2.1: If 0r,s = (#,
is a double lacunary refinement of 0rs = (/r, fcs) and is an unbounded modulus functions of double sequence such that
|/},fc(x) - /-,fc(y)| = /y,fc(|x - y|),vx > 0,y > 0, Then x £ sf/,,c implies x £ sf7,,c.
(5)
Proof: Suppose each /rs of 0rs contains the points (/^ o^so'))-^ of #r,s so that
7r-1'ks-1 <7r,1'ks,1 <7r,2'ks,2 < "■ < 7r,v(r)'ks,v(s) = 7r>
where /r,s = {(/, fc):yr-1 < / < y*r-1 and
< fc' <
Note that, for all (r, s), v(r, s) > 1 because 0r-ks) c 0r-1-ks-1). Let x7-fc ^ ¿(s^). Therefore, for each e > 0, we have
1
Ä ^"¡v-T/S,fc(|{0"'£ /r©,s0'): |x/'fc - L| > e}|) =
,/7,fc("r(i),s0'))
1< u<v(r,s) /7',fc("r(i),s0'))
where ftr(i),sO') = kr(i),sO') kr(i-1),s0'-1) and fyr,1),(s,1) = k(r,1),(s,1) kr-1,s-1, whence
lim
y X "T^-:/',*(l{(/'fc) £ Wo'V |x,'fc - > e}|) = 0.
' ' ' ' / I, i (.' 1
7r( i),s(m)c/r(i) 1< ij'<v(r,s)
For each e > 0, we have
/)',fc (^r,s)
(|{0'.fc) £/r,s:|x,fc-L| >e}|)
/)',fc (^r,s)
0',fc)£, u /^(0,s0.):|x7-fc-L| >
/)',fc (^r,s)
1<ij'<v(r,s)
/ \ y y|{0'-k) £ ^r(i),sO'): |x7fc - L| > e}|
\ 'r(i),sO)C'r,s \1<ij'<v(r,s)
/
1
1
1
- f. (h \ Z Z fjk ({(' k) £ ■\xjk-l\> £1)
J-J^ , Z Zfj,k W'k) £ 'r(i),S(j): lxjk
, , ^r(V),s(j') clr,s l< i,j<v(r,s)
= TTh' ) Z Zfj,k(h'r(i),sU))
r( i),s(j)c'r,s l< i,j<v(r,s)
1
fj,k ({('k) £ K(.i),sO)'- lxjk -l\> £}).
fj,k(h'r(i),s(j))
Also, in view of the choice of unbounded modulus functions f and using the fact that 9'jS = (¡r, k's) is increasing, we have
Z ^Zfj,k(h(r,i),(s,j)) = fj,k(h'(r,l),(s,l)) + fj,k(h(r,2),(s,2)) + + fj,k(h(r,s),v(r,s))
l< i,j<v(r,s)
= fj,k(j(r,l)'kls,l)) + fj,k(j'(r,2)'k'(s,2)) + + fj,k (0(r,v(r))' k(s,v(s))) — Q(r,v(r)-l)' k(s,v(s)-l)))
= fj,k(IQr,l'Kl) - (Jr-l'ks-l)l) + fj,k(IQr,2'ks,2) - Qr,l'ks,l)l) + ■
+ fj,k(\(Jr,v(r)'ks,v(s)) - (J(r,v(r)-1'k(s,v(s)-1)\)
= \fj,k(J^,l'k's,l) - fj,k(J^-l'k's-l)l + \fj,k(J^,2'k's,2) - fj,k(j',l'ks,l)l + ■■■ + \ fj,k (Jr,v(r)' ks,v(s)) — fj,k(j"r,v(r)-l' It'svis)-^1
= fjjkOr.l'Kjl) — fjjkOr-l'K-D + fj,k(J^,2'k's,2) - fj,k (jr,l' ks,l) +
+ fj,k(Jr,v(r)'ks,v(s)) = fj,k(j(r,v(r)-1'k(s,v(s)-1) = \fj,k(J(r,v(r)-1'k(s,v(s)-1)\ = fj.kQj'l'r.vW-l'Ks.v^-l1) = fj,k(\hr,s\) = fj,k(hr,s).
Thus, we have
)fj,k(\{(j'k) £ ¡r,s■ \xjk -l\> ^D < ^-V fl „(h.......) Vl'm V fjk (h'r(i),s(j))tr(i),sU)'
fj,kihr,s)",k^- " -^r().i.c,rsVfj,kih'r(i),sa)^'r(0,s(j)<
rW,sO) l<i,j<v(r,s)
1< i,j<v(r,s)
(6)
where
trUXsij) = (fj,k(h'r(i),sU)))-lfj,k(\{(j'k) £ rr(i),suy \xjk -l\> £}\).
Since the term on the right hand of (6) is regular weighted mean transformation of the double sequence tr(¿),s(j), which tend to zero as r.s ^ m, therefore the term on the right hand side of (6) also tends to zero as r.s ^ m. Thus,
fj,k(hr,s)
fj,k(\{(j, k) £ lr s: \xjk - l\> e}\) ^0asr,s^m. Hence x £ Sfj,k.
Theorem 3.2.2: Let fj k be an unbounded modulus functions and = (j^, k'n) is a double lacunary refinement of double lacunary sequence
9r,s = (jr,ks). Let lrs = {(j'k): jr-l < j < jr and ks-l < k < ks}, hr = jr -jr-l and hs = ks-ks-l, where
hr,s = hrhs, r.s = 1,2,3' .... and 1'n = {(j, k): < j' < j' and k'n-l < k' < k'n}, h'm = j'm - j'm-l and h'n = kn — k'n, where h'mn = h'mh'n, m.n = 1,2,3, ..if there exists 5 > 0, such that
'r( i)Mj)c'r,s
1
1
//,fc(^m,n) ^ o c r' _ r
; fh , > $ for every /m,„ c / ,
Then x £ sf7,fc and x £ sf/,ft
Proof: For any e > 0, and for /m,n c /rs, we can find /r s such that /m,n c /rs, then we have
1 1 -TTT-^/mGÎO'^) £ /m,n= Ix-fc — l| > £}|) < £ /r,s: |x7-fc — l| > £}|)
= /^f- 1 /^(Ifr,£ ^ - L| > £)I) 11
" £ ^ ^ - L| > e}|)
From where it follows that sf7,ic c sf/,fc.
In the next theorem we deal with a more general situation.
Theorem 3.2.3: Let / and g be any two modulus functions of double sequence such that /(x) < ^(x), for all x £ [0, œ), and = 0m, is a double lacunary refinement of the double sequence 0r,s = 0'r,ks). Let /r,s = {0',fc):;'r-i < 7 < 7r and < ^ < M, ^r = 7r — 7r-i and = fes-fes-i, where ftr s = r, s = 1,2,3,..., and /m,n = {(/, 7m-1 < < 7m ^n-1 < < ^n), ^m = 7m — ;m-1 and ^n = ^n — ^, where = ^m^n, n = 1,2,3,..., if there exists 0 < 5 < 1, such that
-7—— > 5 for every / C ^s, Then x £ sf7,ic and x £ sf/,fc.
Proof: For any e > 0, and every /m,n, we can find /r s such that /m,n c /r s, then we have 11
'—т/>.fc(|{0",£ 4,n: |x7-fc — L| > e}|) < —r(|{0,£ |x,-fc — i| > e}|)
f (h' N-'VAVKV--' - -m^- |--/k "I — ~JIJ — f (h, \ y^kv"^^ J j.k\lt"m.nJ
1
< TT^^kM^ £ /r,s: |x,k — L| > e}|)
7 7,k v tm,n)
1, n.V
№ M^) e 7r,s: |x,k - L| > £}|)
/¿ft ("m,n) O/./c ("r,s) 1 1
- 0a (h )gj,fc(|{0',fe) £ /r,s: ^ - l| > e}|)
$ 5/,/c (hr,s)
From where it follows that sf7,ic c sf,7,\
In the next theorem we show that the inclusion sf7,,c c sf/,fc is possible if even if none of 0r s and 0,1 s is refinement of the other.
Theorem 3.2.4: Let / be an unbounded modulus functions such that
l/(x)-/(y)| = /(|x-y|),Vx> 0,y> 0. (7)
Suppose 0r,s = (&, fcn) and 0r s = (/V, ks). Let /r s = {(/, k): 7r-1 < 7 — 7r and ks-1 < k — ks}, hr = y'r -yr-1 and hs = ks—ks-1, where hr s = hrhi.;, r, s = 1,2,3,..., and = {(/, k):< 7' — and kn-1 < k' — kn}, = and hn = kn - kn, where h^,n = h^hn, m,n = 1,2,3,..., and /p,q,m,n = n
/m n, p, q, m, n = 1,2,3, if there exists $ > 0 such that
———-> o for every p, q, m, n = 1,2,3,...,
S/,fc(frr,s)
provided > 0, where denotes the length of the interval /p,q,m,n then x £ implies
rfj,k
Remark 3.2.1: If the condition in theorem 4.4 is replaced by f{op,q,m,n)/f(h'^n) > 5 for every r,s,m,n = 1,2,3,..., provided op,q,m,n > 0, where op,q,m,n denotes the length of the interval Ipqmn = Ipqq n Imn, p, q,m,n = 1,2,3,..., it can be seen that x £ S^J,,k implies x £ Sgj,k.
Combining remark 4.1 and theorem 4.4, we get the following. Theorem 3.2.5: Let f be an unbounded modulus functions such that
lf(x)-f(y)l=f(lx-yl),Vx>0,y>0. (8)
Suppose 9's = (j'm, k'n) and 9rs = (jr, ks) are two double lacunary sequences. Let lTiS = {(j,k): jr-1 < j < jr and ks-1 < k < ks}, hr = jr — jr-1 and hs = ks-ks-1, where hr s = hrhs, r,s = 1,2,3,..., l!m n = {Q, k): j'm-i < j' < jm and k'n-i < k' < k'n}, h'm = j'm — j'm-i and h'n = k'n — k'n, where h'm,n = h'mh'n, m,n = 1,2,3,..., and Ip,q,m,n = IpA n Im n, p, q,m,n = 1,2,3,..., if there exists 5 > 0 such that
> 5 for every p, q,m,n = 1,2,3,...,
aj,k(hr,s+h'mn)
provided ap q m n > 0, where ap q m n denotes the length of the interval Ip,q,m,n then Sgj* = S^,,k.
4. Discussion
The concept of modulus lacunary statistical convergence of double sequence was introduced via modulus functions where the modulus function is bounded or unbounded. We have also introduced the concept of lacunary strong convergence of double sequence with respect modulus function. We established some inclusion relations between these two lacunary methods and proved some essential analogues results for double sequence. This concept can be further extended in the direction of fuzzy numbers of double sequence.
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