GENERALIZED DIFFERENCE SEQUENCE SPACES OF FRACTIONAL ORDER DEFINED BY ORLICZ FUNCTIONS

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C om mun.Fac.Sci.U niv.A nk.Ser. A 1 M ath. Stat. Volum e 69, N umb er 1, Pages 941–951 (2020) D O I: 10.31801/cfsuasm as.628863

ISSN 1303–5991 E-ISSN 2618-6470

http://com munications.science.ankara.edu.tr

NAZLIM DENIZ ARAL AND MIKAIL ET

Abstract. The main purpose of this paper is to introduce the concepts of lacunary statistical convergence of order (0 < 1)with the fractional order of and lacunary strongly convergence of order (0 < 1)with the fractional order of . We establish some connections between lacunary strongly convergence of order and lacunary statistical convergence of order .

1. Introduction

The idea of statistical convergence was given by Zygmund [45] in the …rst edition of his monograph published in Warsaw in 1935. The concept of statistical conver-gence was introduced by Steinhaus [42] and Fast [20] and later reintroduced by Schoenberg [38]. Over the years and under di¤erent names statistical convergence was discussed in the theory of Fourier analysis, Ergodic theory, Number theory, Measure theory, Trigonometric series, Turnpike theory and Banach spaces. Later on it was further investigated from the sequence space point of view and linked with summability theory by Çakall¬ et al. ([7],[8],[9]). Caserta et al. [10], Ç¬nar et al. [11], Connor [13], Et et al. ([15],[17]), Fridy [22], Fridy and Orhan [23], Mursaleen [33], Salat [41], Mohiuddine et al. ([5], [31]) and many others.

The idea of statistical convergence depends upon the density of subsets of the set N of natural numbers. The density of a subset E of N is de…ned by

(E) = lim_n !1 1 n n X k=1

E(k); provided that the limit exists.

A sequence x = (xk) is said to be statistically convergent to L if for every " > 0;

(fk 2 N : jxk Lj "g) = 0:

Received by the editors: October 03, 2019; Accepted: March 11, 2020. 2010 Mathematics Subject Classi…cation. 40A05, 40C05, 46A45.

Key words and phrases. Di¤erence sequence, statistical convergence, lacunary sequence. c 2 0 2 0 A n ka ra U n ive rsity C o m m u n ic a tio n s Fa c u lty o f S c ie n c e s U n ive rs ity o f A n ka ra -S e rie s A 1 M a t h e m a t ic s a n d S ta t is t ic s

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Recently, Çolak [12] generalized the statistical convergence by ordering the in-terval (0; 1] and de…ned the statistical convergence of order and strong p Cesàro summability of order , where 0 < 1 and p is a positive real number. ¸Sengül and Et ([19],[39]) generalized the concepts such as lacunary statistical convergence of order and lacunary strong p Cesàro summability of order for sequences of real numbers.

Di¤erence sequence spaces was de…ned by K¬zmaz [27] and the concept was generalized by Et et al. ([14],[18]) as follows:

m

(X) = fx = (xk) : ( mxk) 2 Xg ;

where X is any sequence space, m 2 N; 0_{x = (x}

k) ; x = (xk xk+1) ; mx =

( m_x

k) = m 1xk m 1xk+1 and so mxk =Pm_v=0( 1)v m_v xk+v.

If x 2 m_{(X) then there exists one and only one sequence y = (y}

k) 2 X such that yk= mxk and xk = k m_X v=1 ( 1)m k v 1 m 1 yv= k X v=1 ( 1)m k + m v 1 m 1 yv m; (1) y1 m= y2 m= = y0= 0

for su¢ ciently large k, for instance k > 2m. After then some properties of di¤erence sequence spaces have been studied in ([1],[2],[16],[18], [25], [26], [32], [36]).

By (r), we denote the Gamma function of a real number r and r =_{2 f0; 1; 2; 3; :::g.} By the de…nition, it can be expressed as an improper integral as:

(r) = Z ₁

0

e ttr 1dt: From the de…nition, it is observed that:

(i) For any natural number n; (n + 1) = n!;

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Without loss of generality, we assume throughout that the series de…ned in (2) is convergent. Moreover, if is a positive integer, then the in…nite sum de…ned in (2) reduces to a …nite sum i.e.,P_i=0( 1)i ( +1)

i! ( i+1)xk+i. In fact, this operator

generalized the di¤erence operator introduced by Et and Çolak [14].

Recently, using fractional operator (fractional order of ; _{2 R )} Baliars-ingh et al. ([3],[4],[35]) de…ned the sequence space (X) such as: (X) = fx = (xk) : ( xk) 2 Xg ; where X is any sequence space.

By a lacunary sequence we mean an increasing integer sequence = (kr) of

non-negative integers such that k0= 0 and hr= (kr kr 1) ! 1 as r ! 1: The

intervals determined by will be denoted by Ir = (kr 1; kr] and the ratio _kk_rr₁

will be abbreviated by qr, and q1 = k1 for convenience: In recent years, lacunary

sequences have been studied in ([7],[8],[9],[21],[23], [24], [40]).

An Orlicz function is a function M : [0; 1) ! [0; 1), which is continuous, non-decreasing and convex with M (0) = 0, M (x) > 0 for x > 0 and M (x) ! 1 as x ! 1.

The study of Orlicz sequence spaces was initiated with a certain speci…c purpose in Banach space theory. Indeed, Lindberg [29] got interested in Orlicz sequence spaces in connection with …nding Banach spaces with symmetric Schauder bases having complementary subspaces isomorphic to c0or `p(0 p < 1). Subsequently,

Lindenstrauss and Tzafriri [30] used the idea of Orlicz function to construct the sequence space `M = ( x 2 w : 1 X k=1 M jxkj _{< 1 for some} > 0 ) : The space `M with the norm

kxk = infn > 0 :

1

X

k=1

M jxkj 1o

becomes a Banach space, called an Orlicz sequence space. The space `M is closely

related to the space `p which is an Orlicz sequence space with M (x) = jxjp for

1 _{p < 1. Lindenstrauss and Tzafriri [30] proved that every Orlicz sequence} space `M contains a subspace isomorphic to lp (1 p < 1). The Orlicz sequence

spaces are the special cases of Orlicz spaces studied in [28].

It is well known that if M is a convex function and M (0) = 0, then M ( x) M (x) for all with 0 < < 1.

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2. Main Results and proofs

De…nition 1. Let = (kr) be a lacunary sequence, 2 (0; 1] and be a proper

fraction. The sequence x = (xk) is said to be lacunary statistically convergent

of order of fractional order of (or (S ) convergent to L ) to the number L, if there is a real number L such that

lim

r!1

1 hr

jfk 2 Ir: j xk Lj "gj = 0

for all " > 0: In this case, we write xk! L( (S ):

The set of all (S ) convergent sequences will be denoted by (S ). If = (2r) ; then we write (S ) instead of (S ). In the special cases = (2r) and = 1; we write (S) instead of (S ):

In particular, (S ) convergence includes many special cases; for example, in case of _{= m 2 N;} = 1; lacunary statistical convergence of order reduces to the m lacunary statistical convergence which was de…ned and studied by Tripathy and Et [44].

De…nition 2. Let M be an Orlicz function, = (kr) be a lacunary sequence,

2 (0; 1]; be a proper fraction and p = (pk) be a sequence of strictly positive real

numbers. The sequence x = (xk) is said to be strongly (N ; (p)) summable to

L with respect to the Orlicz function M (or strongly (N ; M; (p)) summable to L), if there is a real number L such that

lim r!1 1 hr X k2Ir M j xk Lj pk = 0;

for all " > 0 and some > 0: In this case, we write xk ! L( (N ; M; (p))):

The set of all (N ; M; (p)) summable sequences will be denoted by (N ; M; (p)). In the special cases M (x) = x; pk= p for each k 2 N, we obtain the set (N ; p).

If = (2r_{) ; M (x) = x; p}

k = 1 for each k 2 N and = 1; then we write (j 1j)

instead of (N ; p) and say that x = (xk) is strongly Cesàro summable to

L:

The proof the following theorems are straightforward, so we choose to state these results without proof.

Theorem 3. Let = (kr) be a lacunary sequence; 2 (0; 1], be a proper fraction

and x = (xk); y = (yk) are sequences of real numbers, then

i) If xk ! L( (S )) and c 2 C, then cxk ! cL( (S )).

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Theorem 4. Let the sequence (pk) be bounded, then the sequence space (N ; M; (p))

is a linear space over the set of complex numbers.

Theorem 5. If a _{bounded sequence (that is x 2} (`₁)) is statistically convergent to L then it is strongly Cesàro summable to L:

Proof. Suppose that x 2 (`₁_{) \} (S) with xk ! L( (S): Without loss of

generality we may assume that L = 0: Set K = k xk1: Let " > 0 be given and

choose N" such that 1_njfk n : j xkj "₂gj < _2K" for all n > N". Now, we get

1 n n X k=1 j xkj = 1 n X 1 k n j xkj 2" j xkj + 1 n X 1 k n j xkj<"2 j xkj 1 n n" 2KK + n n " 2 = ": for all n > N": Thus lim_n1

Pn

k=1j xkj = 0 which means that x 2 (j 1j) :

Converse of Theorem 5 does not holds. For this choose = 1; then the sequence x = (0; 1; 1; 2; 2; 3 ; 3; 4; 4; :::) belongs to _(j 1j) and does not belong

to (S) :

Theorem 6. (N ; p) is a Banach space normed by kxk 1 = 1 X i=1 jxij + sup r 1 hr X k2Ir j xkjp !1=p ; 1 _{p < 1} (3)

and a complete p-normed space for 0 < p < 1 by kxk 2 = 1 X i=1 jxij + sup r 1 hr X k2Ir j xkjp= 0 (4)

Proof. Proof follows from Theorem 3 [4] and Theorem 2.4 [39].

Theorem 7. (N ; p) is a BK space normed by (3) : Proof. Omitted.

Theorem 8. Let = (kr) be a lacunary sequence, 2 (0; 1]; be a proper fraction

and p be a …xed positive real number, then

i) If xk ! L( (N ; p)), then xk ! L( (S )) and the inclusion is strict,

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Proof. The inclusion part of the proof is easy. In order to establish "the inclusion is strict ", let be given, choose = m, = 1; p = 1 and de…ne a sequence x = (xk)

by m_{x to be 1; 2; :::;} p_h

r at the …rst phr integers in Ir; and mxk = 0

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It is clear that x is not m _{bounded. Since}

1 hrjfk 2 I r: j mxk 0j "gj = p hr hr ! 0; as r ! 1 and 1 hr X k2Ir j mxk 0j = p hr phr + 1 2hr ! 1 2; as r ! 1: From (1) we have x 2 m(S ), xk 2= m(N ) :

Theorem 9. Let 0 < 1 and = (kr) be a lacunary sequence. If lim infrqr> 1;

then S S :

Proof. Suppose that lim infrqr> 1; then there exists a > 0 such that qr 1 +

for su¢ ciently large r; which implies that hr kr 1 + =) hr kr 1 + =) 1 kr (1 + ) 1 hr : If xk ! L S ; then for every " > 0 and for su¢ ciently large r; we have

1 kr jfk kr: j xk Lj "gj 1 kr jfk 2 I r: j xk Lj "gj (1 + ) 1 hr jfk 2 Ir: j xk Lj "gj ; So x 2 S .

Theorem 10. Let 0 < 1 and = (kr) be a lacunary sequence. If lim suprqr<

1; then S S :

Proof. Omitted.

In the following theorems, assume that the sequence p = (pk) is bounded and

0 < h = infkpk pk supkpk = H < 1.

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Proof. Let x 2 (N (M; (p)), " > 0 be given andP₁ and P₂ denote the sums over k 2 Ir; j xk Lj " and j xk Lj < " respectively. As hr hr for each

r, we have 1 hr X k2Ir M j xk Lj pk 1 hr 2 4 P 1 h M j xk Lj ipk +P₂ h M j xk Lj ipk 3 5 1 hr " X 1 M " #pk 1 hr X 1 min [M ("1)]h; [M ("1)]H ; "1= " 1 hrjfk 2 Ir: j xk Lj "gj min [M ("1)]h; [M ("1)]H :

As x 2 (N (M; (p)), the left hand side of the above inequality tends to zero as r ! 1. Therefore, the right hand side of the above inequality tends to zero as r ! 1, hence x 2 (S ):

Corollary 12. Let 0 < 1, M be an Orlicz function and = (kr) be a lacunary

sequence, then (N (M; (p)) (S ):

Theorem 13. Let M be an Orlicz function, x = (xk) be a bounded sequence

and = (kr) be a lacunary sequence. If limr!1

hr

hr = 1, then x 2 (S ) ) x 2

(N (M; (p)).

Proof. Suppose that x = (xk) be a bounded sequence, that is x 2 (`1)

and xk ! L( (S )). As x 2 (`1), then there is a constant T > 0 such that

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+ 1 hr X 2 max M " pk maxn[M (K)]h; [M (K)]Ho 1 hr jfk 2 I r: j xk Lj "gj +hr hr max n [M ("1)]h; [M ("1)]H o ; T = K; " = "1: Hence x 2 (N (M; (p)).

Theorem 14. If lim pk > 0 and x = (xk) is strongly (N (M; (p)) summable

to L with respect to the Orlicz function M , then that limit L is unique.

Proof. Let lim pk= s > 0. Suppose that xk! L( (N ; p)) and xk! L1( (N ; p)).

Then we have lim r!1 1 hr X k2Ir M j xk Lj 1 pk = 0; for some ₁> 0 and lim r!1 1 hr X k2Ir M j xk L1j 2 pk = 0; for some ₂> 0: We de…ne the = max(2 ₁; 2 ₂). As M is nondecreasing and convex, we have

1 hr X k2Ir M L L1 pk D hr X k2Ir 1 2pk M j xk Lj 1 pk + M j xk L1j 2 pk D hr X k2Ir M j xk Lj 1 pk +D hr X k2Ir M j xk L1j 2 pk ! 0; (r ! 1); where sup_kpk = H and D = max(1; 2H 1). Hence,

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As limk!1pk = s, we have lim k!1 M jL L1j pk = M jL L1j s

and so L = L1. Thus, the limit is unique.

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Current address : Nazl¬m Deniz ARAL: Department of Mathematics, Faculty of Arts and Sciences, Bitlis Eren University, Bitlis, 13000, Turkey.

E-mail address : [email protected]

ORCID Address: https://orcid.org/0000-0002-8984-2620

Current address : Mikail ET: Department of Mathematics, Faculty of Science, F¬rat University, Elaz¬¼g, 23119, Turkey.

E-mail address : [email protected]