C om mun.Fac.Sci.U niv.A nk.Series A 1 Volum e 66, N umb er 2, Pages 80–90 (2017) D O I: 10.1501/C om mua1_ 0000000803 ISSN 1303–5991
http://com munications.science.ankara.edu.tr/index.php?series= A 1
SOME CESÀRO-TYPE SUMMABILITY SPACES DEFINED BY A
MODULUS FUNCTION OF ORDER ( ; )
HACER ¸SENGÜL
Abstract. In this article, we introduce strong w [ ; f; p] summability of or-der ( ; ) for sequences of complex (or real) numbers and give some inclusion relations between the sets of lacunary statistical convergence of order ( ; ), strong w [ ; f; p] summability and strong w (p) summability.
1. Introduction
In 1951, Steinhaus [15] and Fast [9] introduced the concept of statistical con-vergence and later in 1959, Schoenberg [13] reintroduced independently. Caserta et al. [2], Çakall¬[3], Connor [8], Çolak [7], Et [4], Fridy [10], Gadjiev and Orhan [5], Kolk [6], Salat [14] and many others investigated some arguments related to this notion.
Çolak [7] studied statistical convergence order by giving the de…nition as fol-lows:
We say that the sequence x = (xk) is statistically convergent of order to ` if
there is a complex number ` such that lim
n!1
1
n jfk n : jxk `j "gj = 0:
Let 0 < 1. We de…ne the ( ; ) density of the subset E of N by (E) = lim
n
1
n jfk n : k 2 Egj
provided the limit exists (…nite or in…nite), where jfk n : k 2 Egj denotes the th power of number of elements of E not exceeding n:
If a sequence x = (xk) satis…es property P (k) for all k except a set of ( ; ) density
zero, then we say that xk satis…es P (k) for "almost all k according to " and we
abbreviate this by "a:a:k ( ; )".
Received by the editors: August 04, 2016; Accepted: November 01, 2016.
2010 Mathematics Subject Classi…cation. Primary 40A05, 40C05; Secondary 46A45. Key words and phrases. Lacunary sequence, modulus function, statistical convergence.
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Throughout this paper w indicate the space of sequences of real number. Let 0 < 1; 0 < 1; and x = (xk) 2 w: The sequence x = (xk)
is said to be statistically convergent of order ( ; ) if there is a complex number L such that
lim
n!1
1
n jfk n : jxk Lj "gj = 0
i.e. for a:a:k( ; ) jxk Lj < " for every " > 0, in that case a sequence x is said to
be statistically convergent of order ( ; ) ; to L: This convergence is indicated by S lim xk = L ([16]).
By a lacunary sequence we mean an increasing integer sequence = (kr) such
that hr = (kr kr 1) ! 1 as r ! 1 and 2 (0; 1] : Throughout this paper the
intervals determined by will be denoted by Ir= (kr 1; kr] and the ratio kkrr1 will
be abbreviated by qr: Lacunary sequence spaces were studied in ([11], [12], [17],
[18]).
First of all, the notion of a modulus was given by Nakano [20]. Maddox [25] and Ruckle [28] used a modulus function to construct some sequence spaces. Afterwards di¤erent sequence spaces de…ned by modulus have been studied by Alt¬n [1], Et ([26], [27]) , Gaur and Mursaleen [21], I¸s¬k [23], Nuray and Sava¸s [22], Pehlivan and Fisher [29] and everybody else.
We recall that a modulus f is a function from [0; 1) to [0; 1) such that i) f (x) = 0 if and only if x = 0;
ii) f (x + y) f (x) + f (y) for x; y 0; iii) f is increasing,
iv) f is continuous from the right at 0.
It follows that f must be continuous everywhere on [0; 1).
The following inequality will be used frequently throughout the paper:
jak+ bkjpk D (jakjpk+ jbkjpk) (1)
where ak; bk 2 C; 0 < pk sup pk= H; D = max 1; 2H 1 ([24]).
2. Main Results
In this part we will describe the sets of strongly w (p) summable sequences and strongly w [ ; f; p] summable sequences with respect to the modulus function f: We will examine these spaces and we give some inclusion relations between the S ( ) statistical convergent, strong w [ ; f; p] summability and strong
w (p) summability.
De…nition 1. Let = (kr) be a lacunary sequence and 0 < 1 be given.
We say that the sequence x = (xk) is S ( ) statistically convergent (or lacunary
that
lim
r!1
1
hr jfk 2 Ir: jxk Lj "gj = 0;
where Ir = (kr 1; kr] and hr denotes the th power (hr) of hr; that is h =
(hr) = (h1; h2; :::; hr; :::) and jfk n : k 2 Egj denotes the th power of number of elements of E not exceeding n: In the present case this convergence is indicated by S ( ) lim xk= L. S ( ) will indicate the set of all S ( ) statistically convergent
sequences. If = (2r) ; then we will write S in the place of S ( ). If = = 1
and = (2r) ; then we will write S in the place of S ( ) :
De…nition 2. Let = (kr) be a lacunary sequence, 0 < 1 and p be a
posi-tive real number. We say that the sequence x = (xk) is strongly N ( ; p) summable
(or strongly N ( ; p) summable of order ( ; )) if there is a real number L such that lim r!1 1 hr X k2Ir jxk Ljp ! = 0.
In the present case we denote N ( ; p) lim xk = L. N ( ; p) will denote the set
of all strongly N ( ; p) summable of order ( ; ). If = = 1; then we will write N ( ; p) in the place of N ( ; p). If = (2r) ; then we will write w (p) in the place
of N ( ; p) : If L = 0; then we will write w ;0(p) in the place of w (p). N ;0( ; p) will denote the set of all strongly N (p) summable of order ( ; ) to 0.
De…nition 3. Let f be a modulus function, p = (pk) be a sequence of strictly
positive real numbers and 0 < 1 be real numbers. We say that the sequence x = (xk) is strongly w [ ; f; p] summable to L (a real number) such that
w [ ; f; p] = 8 < :x = (xk) : limr!1 1 hr X k2Ir [f (jxk Lj)]pk ! = 0; for some L 9 = ;: In the present case, we denote w [ ; f; p] lim xk= L: In the special case pk= 1;
for all k 2 N and f (x) = x we will denote N ( ; p) in the place of w [ ; f; p] : w ;0[ ; f; p] will denote the set of all strongly w [ ; f; p] summable of order ( ; ) to 0:
In the following theorems we shall assume that the sequence p = (pk) is bounded
and 0 < h = infkpk pk supkpk= H < 1.
Theorem 1. The class of sequences w ;0[ ; f; p] is linear space. Proof. Omitted.
g (x) = sup r 8 < : 1 hr X k2Ir [f (jxkj)]pk ! 9= ; 1 M
where 0 < 1 and M = max (1; H) :
Proof. Clearly g (0) = 0 and g (x) = g ( x) : Take any x; y 2 w ;0[ ; f; p] : Since pk
M 1 and M 1; using the Minkowski’s inequality and de…nition of f; we can
write 8 < : 1 hr X k2Ir [f (jxk+ ykj)]pk ! 9= ; 1 M 8< : 1 hr X k2Ir [f (jxkj) + f (jykj)]pk ! 9= ; 1 M = 1 hM r X k2Ir [f (jxkj) + f (jykj)]pk ! 1 M 1 hM r 8 < : X k2Ir [f (jxkj)]pk ! 9= ; 1 M + 1 hM r 8 < : X k2Ir [f (jykj)]pk ! 9= ; 1 M :
Therefore g (x + y) g (x)+g (y) for x; y 2 w ;0[ ; f; p] : Let be complex number. By de…nition of f we have g ( x) = sup r 8 < : 1 hr X k2Ir [f (j xkj)]pk ! 9= ; 1 M K H M g (x)
where [ ] denotes the integer part of ; and K = 1 + [j j] : Now, let ! 0 for any …xed x with g (x) 6= 0: By de…nition of f, for j j < 1 and 0 < 1; we have
1 hr X k2Ir [f (j xkj)]pk ! < " for n > N (") : (2) Also, for 1 n N; taking small enough, since f is continuous we have
1 hr X k2Ir [f (j xkj)]pk ! < " (3)
Proposition 1. ([19]) Let f be a modulus and 0 < < 1: Then for each kuk ; we have f (kuk) 2f (1) 1kuk :
Theorem 3. If 0 < = 1, p > 1 and lim infu!1f (u)u > 0; then w [ ; f; p] =
w (p) :
Proof. Let pk = p be a positive real number: If lim infu!1f (u)u > 0 then there
exists a number c > 0 such that f (u) > cu for u > 0. We have x 2 w [ ; f; p] : Clearly 1 hr X k2Ir [f (jxk Lj)]p ! 1 hr X k2Ir [c jxk Lj]p ! =c p hr X k2Ir jxk Ljp ! ; therefore w [ ; f; p] w (p) :
Let x 2 w (p) : Then we have 1 hr X k2Ir jxk Ljp ! ! 0 as r ! 1:
Let " > 0; = and choose with 0 < < 1 such that cu < f (u) < " for every u with 0 u : We can write
1 hr X k2Ir [f (jxk Lj)]p ! = 1 hr 0 B B @ X k2Ir jxk Lj [f (jxk Lj)]p 1 C C A + 1 hr 0 B B @ X k2Ir jxk Lj> [f (jxk Lj)]p 1 C C A 1 hr "p hr + 1 hr 0 B B @ X k2Ir jxk Lj> 2f (1) 1jxk Lj p 1 C C A 1 hr" p h r + 2p f (1)p hr p X k2Ir jxk Ljp ! by Proposition 1. Therefore x 2 w [ ; f; p] :
Example 1. We now give an example to show that w [ ; f; p] 6= w (p) in this case when lim infu!1f (u)u = 0: Consider the sequence f (x) =
p
De…ne x = (xk) by
xk =
hr; if k = kr
0; if otherwise: We have, for L = 0; p = 32 and =
1 hr X k2Ir [f (jxkj)]p ! = 1 hr p hr 3 2 ! 0 as r ! 1 and so x 2 w [ ; f; p] : But 1 hr X k2Ir jxkjp ! =(hr) 3 2 hr ! 1 as r ! 1 and so x =2 w (p) :
Theorem 4. Let 0 < 1 and lim inf pk > 0: Then xk ! L implies
w [ ; f; p] lim xk = L:
Proof. Let xk! L: By de…nition of f we have f (jxk Lj) ! 0: Since lim inf pk > 0;
we have [f (jxk Lj)]pk! 0: Therefore w [ ; f; p] lim xk= L:
Theorem 5. Let 1; 2; 1; 2 2 (0; 1] be real numbers such that 0 < 1 2 1 2 1; f be a modulus function and let = (kr) be a lacunary sequence;
then w 2
1[ ; f; p] S 1 2( ) :
Proof. Let x 2 w 2
1[ ; f; p] and let " > 0 be given and
P
1and
P
2denote the sums
over k 2 Ir; jxk Lj " and k 2 Ir; jxk Lj < " respectively. Since hr1 hr2 for
each r we may write 1 h 1 r X k2Ir [f (jxk Lj)]pk ! 2 = 1 h 1 r hX 1[f (jxk Lj)] pk +X 2[f (jxk Lj)] pki 2 1 h 2 r hX 1[f (jxk Lj)] pk +X 2[f (jxk Lj)] pki 2 1 h 2 r hX 1[f (")] pki 2 1 h 2 r hX 1min([f (")] h ; [f (")]H)i 2 1 h 2 r jfk 2 Ir: jxk Lj "gj 1 h min([f (")]h; [f (")]H)i 1: Hence x 2 S 1 2( ) :
Theorem 6. If the modulus f is bounded and limr!1h 2 r h 1 r = 1 then S 2 1( ) w 1 2[ ; f; p] : Proof. Let x 2 S 2
1( ). Assume that f is bounded. Therefore f (x) K; for a
positive integer K and all x 0: Then for each r 2 N and " > 0 we can write 1 h 2 r X k2Ir [f (jxk Lj)]pk ! 1 1 h 1 r X k2Ir [f (jxk Lj)]pk ! 1 = 1 h 1 r X 1[f (jxk Lj)] pk+X 2[f (jxk Lj)] pk 1 1 h 1 r X 1max K h; KH +X 2[f (")] pk 1 max Kh; KH 2 1 h 1 r jfk 2 Ir: f (jxk Lj) "gj 2 +h 2 r h 1 r max f (")h; f (")H 2: Hence x 2 w 1 2[ ; f; p] :
Theorem 7. Let f be a modulus function. If lim pk > 0, then w [ ; f; p] lim xk=
L uniquely.
Proof. Let lim pk = s > 0: Assume that w [ ; f; p] lim xk= L1 and w [ ; f; p]
lim xk= L2: Then lim r 1 hr X k2Ir [f (jxk L1j)]pk ! = 0; and lim r 1 hr X k2Ir [f (jxk L2j)]pk ! = 0: By de…nition of f and using (1), we have
1 hr X k2Ir [f (jL1 L2j)]pk ! D hr X k2Ir [f (jxk L1j)]pk+ X k2Ir [f (jxk L2j)]pk ! D hr X k2Ir [f (jxk L1j)]pk ! + D hr X k2Ir [f (jxk L2j)]pk !
where supkpk = H; 0 < 1 and D = max 1; 2H 1 : Hence lim r 1 hr X k2Ir [f (jL1 L2j)]pk ! = 0:
Since limk!1pk= s we have L1 L2= 0: Thus the limit is unique.
Theorem 8. Let = (kr) and 0= (sr) be two lacunary sequences such that Ir Jr
for all r 2 N and let 1; 2; 1 and 2 be such that 0 < 1 2 1 2 1;
(i) If lim inf r!1 h 1 r ` 2 r > 0 (4) then w 2 2 h 0 ; f; pi w 1 1[ ; f; p] ;
(ii) If the modulus f is bounded and lim r!1 `r h 2 r = 1 (5) then w 2 1[ ; f; p] w 1 2 h 0 ; f; p i : Proof. (i) Let x 2 w 2
2 h 0 ; f; pi: We can write 1 ` 2 r X k2Jr [f (jxk Lj)]pk ! 2 = 1 ` 2 r X k2Jr Ir [f (jxk Lj)]pk ! 2 + 1 ` 2 r X k2Ir [f (jxk Lj)]pk ! 2 1 ` 2 r X k2Ir [f (jxk Lj)]pk ! 2 h 1 r ` 2 r 1 h 1 r X k2Ir [f (jxk Lj)]pk ! 1 : Thus if x 2 w 2 2 h 0 ; f; p i ; then x 2 w 1 1[ ; f; p] :
(ii) Let x = (xk) 2 w 21[ ; f; p] and (2) holds. Assume that f is bounded.
and hr `r for all r 2 N; we can write 1 ` 2 r X k2Jr [f (jxk Lj)]pk ! 1 = 1 ` 2 r X k2Jr Ir [f (jxk Lj)]pk ! 1 + 1 ` 2 r X k2Ir [f (jxk Lj)]pk ! 1 `r hr ` 2 r 1 Kpk 1+ 1 ` 2 r X k2Ir [f (jxk Lj)]pk ! 1 `r hr2 h 2 r KH 1+ 1 h 2 r X k2Ir [f (jxk Lj)]pk ! 2 `r h 2 r 1 KH 1+ 1 h 1 r X k2Ir [f (jxk Lj)]pk ! 2
for every r 2 N: Therefore w 2
1[ ; f; p] w 1 2
h 0
; f; pi:
Now as a result of Theorem 8 we have the following Corollary 1.
Corollary 1. Let = (kr) and 0 = (sr) be two lacunary sequences such that
Ir Jr for all r 2 N:
If (4) holds then, for 0 < 1 2 1 2 1
(i) If 0 < 1 2 1 1 and 2= 1; then w 2
h 0 ; f; p
i w 1
1[ ; f; p] ;
(ii) If 0 < 1 2 1 and 1= 2= 1; then w 2
h 0 ; f; p
i
w 1[ ; f; p] ;
(iii) If 0 < 1 1 and 2= 1= 2= 1; then w
h 0
; f; pi w 1[ ; f; p] ;
(iv) If 0 < 1 2 1 and 1= 2= ; then w 2
h 0 ; f; p i w 1[ ; f; p] ; (v) If 1= 2= and 0 < 1 2 1; then w 2 h 0 ; f; pi w 1[ ; f; p] ;
(vi) If 1= 2= 1 and 1= 2= 1; then w
h 0 ; f; p
i
w [ ; f; p] : If (5) holds then, for 0 < 1 2 1 2 1
(i) If 0 < 1 2 1 1 and 2= 1; then w 1[ ; f; p] w 1 2
h 0 ; f; pi; (ii) If 0 < 1 2 1 and 1= 2= 1; then w 1[ ; f; p] w 2
h 0
; f; pi; (iii) If 0 < 1 1 and 2= 1= 2= 1; then w 1[ ; f; p] w
h 0
; f; pi; (iv) If 0 < 1 2 1 and 1= 2= ; then w 1[ ; f; p] w 2
h 0 ; f; p i ; (v) If 1= 2= and 0 < 1 2 1; then w 2[ ; f; p] w 1 h 0 ; f; pi;
(vi) If 1= 2= 1 and 1= 2= 1; then w [ ; f; p] w h 0 ; f; p i : References
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Current address : Hacer ¸Sengül: Department of Mathematics ; Siirt University 56100; Siirt; TURKEY.
