on two-valued measure and double statistical convergence in 2-normed spaces

Savas Journal of Inequalities and Applications 2013, 2013:347http://www.journalofinequalitiesandapplications.com/content/2013/1/347

RESEARCH Open Access

On two-valued measure and doublestatistical convergence in -normed spacesEkrem Savas*

Dedicated to Professor Hari M Srivastava

*Correspondence:[email protected];[email protected] of Mathematics,Istanbul Ticaret University, Üsküdar,Istanbul, Turkey

AbstractIn this paper we introduce some new double difference lacunary sequence spacesusing Orlicz functions, generalized double difference sequences and a two-valuedmeasure μ in 2-normed spaces, and we also examine some of their properties.MSC: Primary 40H05; secondary 40C05

Keywords: statistical convergence; μ-statistical convergence; 2-normed space;paranormed space; sequence spaces; double lacunary sequence

1 IntroductionThe notion of summability of single sequences with respect to a two-valued measurewas introduced by Connor [, ] as a very interesting generalization of statistical con-vergence which was defined by Fast []. Over the years, and under different names, sta-tistical convergence was discussed in the theory of Fourier analysis, ergodic theory andnumber theory. Later on, it was further investigated from the sequence spaces point ofview and linked with summability theory by Fridy [], Salat []. The notion of statisti-cal convergence was further extended to double sequences independently by Móricz []and Mursaleen et al. []. Savaş [] studied statistical convergence in random -normedspace. Formore recent developments on double sequences one can consult the papers (see[–]) where more references can be found. In particular, very recently Das and Bhuniainvestigated the summability of double sequences of real numbers with respect to a two-valued measure and made many interesting observations []. In [], Das and Savaş et al.introduced some generalized double difference sequence spaces using summability withrespect to a two-valued measure and an Orlicz function in -normed spaces which havea unique non-linear structure. The study of Orlicz sequence spaces was initiated with acertain specific purpose in Banach space theory. Lindenstrauss and Tzafriri [] investi-gated Orlicz sequence spaces in more detail and they proved that every Orlicz sequencespace lM contains a subspace isomorphic to lp ( ≤ p < ∞). The Orlicz sequence spacesare the special cases of Orlicz spaces studied in []. Orlicz spaces find a number of usefulapplications in the theory of nonlinear integral equations. Whereas the Orlicz sequencespaces are the generalization of lp-spaces, the lp-spaces find themselves enveloped in Or-licz spaces [].

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Savas Journal of Inequalities and Applications 2013, 2013:347 of 11http://www.journalofinequalitiesandapplications.com/content/2013/1/347

The concept of -normed spaces was initially introduced by Gahler [, ] as a veryinteresting non-linear extension of the idea of usual normed linear spaces. Some initialstudies on this structure can be seen in [–]. Recently a lot of interesting develop-ments have occurred in -normed spaces in summability theory and related topics (see[, –]).In this paper, in a natural way, we first define statistical convergence for double sequences

in -normed spaces using a two-valued measure and also prove some interesting theo-rems. Furthermore, we introduce some new sequence spaces in -normed spaces usingOrlicz functions, generalized double difference sequences and a two-valued measure μ.

2 PreliminariesThroughout the paper N denotes the set of all natural numbers, χA represents the char-acteristic function of A⊆N and R represents the set of all real numbers.Recall that a set A⊆N is said to have the asymptotic density d(A) if

d(A) = limn→∞

(n

) n∑j=

χA(j)

exists.

Definition . [] A sequence {xn}n∈N of real numbers is said to be statistically convergentto ξ ∈ R if for any ε > we have d(A(ε)) = , where A(ε) = {n ∈N : |xn – ξ | ≥ ε}.

In [] the notion of convergence for double sequences was presented by Pringsheim.A double sequence x = {xkl} of real numbers is said to be convergent to L ∈ R if, for any

ε > , there exists Nε ∈ N such that |xkl – L| < ε whenever k, l ≥ Nε . In this case, we writelimk,l→∞ xkl = L.A double sequence x = {xkl} of real numbers is said to be bounded if there exists a positive

real numberM such that |xkl| <M for all k, l ∈N. That is, ‖x‖(∞,) = supk,l∈N |xkl| < ∞.Let K ⊆ N × N and let K(k, l) be the cardinality of the set {(m,n) ∈ K : m ≤ k,n ≤ l}.

If the sequence {K (k,l)k.l }k,l∈N has a limit in the Pringsheim’s sense, then we say that K has

double natural density and is denoted by d(K) = limk,l→∞ K (k,l)k.l .

A statistically convergent double sequence of elements of ametric space (X,ρ) is definedessentially in the same way (ρ(xkl, ξ ) ≥ ε instead of |xkl – ξ | ≥ ε).Throughout the paper μ will denote a complete {, }-valued finite additive measure

defined on algebra � of subsets of N×N that contains all subsets of N ×N that are con-tained in the union of a finite number of rows and columns of N×N and μ(A) = if A iscontained in the union of a finite number of rows and columns of N×N (see []).

Definition. [] A double sequence x = {xkl} of real numbers is said to beμ-statisticallyconvergent to L ∈ R if and only if for any ε > , μ({(k, l) ∈N×N : |xkl – L| ≥ ε}) = .

Definition . [] A double sequence x = {xkl} of real numbers is said to be convergent toL ∈R in μ-density if there exists an A ∈ � with μ(A) = such that {xkl}(k,l)∈A is convergentto L.



Definition . [] Let X be a real vector space of dimension d, where ≤ d < ∞.A -norm on X is a function ‖·, ·‖ : X × X → R which satisfies (i) ‖x, y‖ = if and onlyif x and y are linearly independent; (ii) ‖x, y‖ = ‖y,x‖; (iii) ‖αx, y‖ = |α|‖x, y‖, α ∈ R;(iv) ‖x, y+ z‖ ≤ ‖x, y‖+ ‖x, z‖. The ordered pair (X,‖·, ·‖) is then called a -normed space.

Let (X,‖·, ·‖) be a finite dimensional -normed space and let u = {u,u, . . . ,ud} be thebasis of X. We can define the norm ‖·, ·‖∞ on X by ‖·, ·‖∞ = max{‖x,ui‖ : i = , . . . ,d}. Also,‖x – y‖∞ = max{‖x – y,uj‖ : j = , . . . ,d}.Let (X,‖·, ·‖) be any -normed space and let S′′( –X) be the set of all double sequences

defined over the -normed space (X,‖·, ·‖). Clearly S′′( –X) is a linear space under addi-tion and scalar multiplication.Recall in [] that an Orlicz function M : [,∞) → [,∞) is a continuous, convex and

non-decreasing function such that M() = and M(x) > for x > , and M(x) → ∞ asx → ∞.Note that ifM is an Orlicz function, thenM(λx)≤ λ for all λ with < λ < .In the later stage, different classes ofOrlicz sequence spaceswere introduced and studied

by Parashar and Choudhary [], Savaş [–, –] and many others.

Definition . [] A double sequence x = {xkl} in a -normed space (X,‖·, ·‖) is said tobe convergent to L in (X,‖·, ·‖) if for each ε > and each z ∈ X, there exists nε ∈ N suchthat ‖xkl – L, z‖ < ε for all i, j ≥ nε .

Definition . [] Let μ be a two-valued measure on N × N. A double sequence {xkl}in a -normed space (X,‖·, ·‖) is said to be μ-statistically convergent to some point x inX if for each pre-assigned ε > and for each z ∈ X, μ(A(z, ε)) = , where A(z, ε) = {(k, l) ∈N×N : ‖xkl – x, z‖ ≥ ε}.

Definition . Let μ be a two-valued measure on N × N. A double sequence {xkl} in a-normed space (X,‖·, ·‖) is said to be μ-statistically Cauchy if for each pre-assigned ε > and for each z ∈ X, there exist integersN =N(ε, z) andM =M(ε, z) such thatμ(A(z, ε)) = ,where A(z, ε) = {(k, l) ∈N×N : ‖xkl – xN(ε,z)M(ε,z), z‖ ≥ ε}.

We first give the following theorem.

Theorem . Let μ be a two-valued measure on N×N and two sequences (xkl) and (ykl)in -normed space (X,‖·, ·‖). If (ykl) is a μ-convergent sequence such that xkl = ykl a.a. (k, l),then (xkl) is μ-statistically convergent.

Proof Suppose μ({(k, l) ∈ N × N : xkl = ykl}) = and limm,n→∞‖ykl, z‖ = ‖L, z‖. Then, forevery ε > and z ∈ X,

{(k, l) ∈N×N : ‖xkl – L, z‖ ≥ ε

} ⊆ {(k, l) ∈ N×N : ‖ykl – L, z‖ ≥ ε

}∪ {

(k, l) ∈N×N : xkl = ykl}.

Therefore

μ{(k, l) ∈N×N : ‖xkl – L, z‖ ≥ ε

} ≤ μ{(k, l) ∈N×N : ‖ykl – L, z‖ ≥ ε

}+μ

{(k, l) ∈N×N : xkl = ykl

}.



Since

limmn→∞‖ykl, z‖ = ‖L, z‖

for every z ∈ X, the set

{(k, l) ∈N×N : ‖ykl – L, z‖ ≥ ε

}contains a finite number of integers. Hence,

μ{(k, l) ∈N×N : ‖ykl – L, z‖ ≥ ε

}= .

We get

μ{(k, l) ∈N×N : ‖xkl – L, z‖ ≥ ε

}=

for every ε > and z ∈ X. Consequently,μ-st-limk,l→∞‖xkl, z‖ = ‖L, z‖. This completes theproof. �

Theorem . Let μ be a two-valued measure on N × N and let {xkl}k,l≥, be a μ-statistically Cauchy sequence in a finite dimensional -normed space (X,‖·, ·‖). Then thereexits a μ-convergent sequence {ykl}k,l≥ in (X,‖·, ·‖) such that xkl = ykl for a.a. (k, l).

Proof First suppose that {xkl}k,l≥ is a statistically Cauchy sequence in (X,‖·, ·‖∞). Choosenatural numbers M and N such that the closed ball B,

u = Bu(xMN , ) contains xklfor a.a. (k, l). Then choose natural numbers M and N such that the closed ball B, =Bu(xMN ,

(.) ) contains xkl for a.a. (k, l). Note that B,u = B,

u ∪ B, also contains xkl fora.a. (k, l). Thus, by continuing this process, we can obtain a sequence {Bp,q

u }p,q≥, of nestedclosed balls such that diam{Bp,q

u } ≤ pq . Therefore

⋂∞,∞p,q=, B

p,qu = {A}. Since each Bp,q

u con-tains xkl for a.a. (k, l), we can choose a sequence of strictly increasing natural numbers{Tp,q}p,q≥ such that

kl

∣∣{(k, l) ∈N×N : xkl /∈ Bp,qu

}∣∣ < p,q

if k, l > Tp,q.

Hence we have

μ{(k, l) ∈N×N : xkl /∈ Bp,q

u}= if k, l > Tp,q.

Put Wp;q = {(k, l) ∈ N × N : (k, l) > Tp,q,xkl /∈ Bp,qu } for all p,q ≥ , and W =

⋃∞,∞p,q=,Wp,q.

Now define the sequence {ykl}kl≥ as follows:

ykl ={A if (k, l) ∈W ,xkl otherwise.

Note that limk,l→∞,∞ykl = A. In fact, for each ε > , choose a natural number p, q suchthat ε >

p,q > . Then, for each kl > Tpq, or ykl = A or ykl = xkl ∈ Bp,qu and so in each case



‖ykl – A‖∞ ≤ diam{Bp,qu } ≤

p–q– . Since {(k, l) ∈ N×N : ykl = xkl} ⊆ {(k, l) ∈ N×N : xkl /∈Bp,qu }, we have

μ{(k, l) ∈N×N : ykl = xkl

} ⊆ μ{(k, l) ∈N×N : xkl /∈ Bp,q

u}.

Henceμ({(k, l) ∈N×N : ykl = xkl}) = . Thus, in the space (X,‖·, ·‖∞), ykl = xkl for a.a. (k, l).Suppose that {u,u, . . . ,ud} is the basis for (X,‖·, ·‖). Since limmn→∞‖ykl – A‖∞ = and‖ykl –A,ui‖ ≤ ‖ykl –A‖∞ for all ≤ i≤ d, limmn→∞‖ykl –A, z‖∞ = for every z ∈ X. Thiscompletes the proof. �

In order to prove the equivalence of Definitions . and ., we shall find it helpful touse Theorems . and ..

Theorem . Let μ be a two-valued measure on N×N and let {xkl}kl≥ be a sequence ina -normed space (X,‖·, ·‖). The sequence {xkl} is μ-statistically convergent if and only if{xkl} is a μ-statistically Cauchy sequence.

3 New double sequence spacesWe first state an inequality which will be used throughout this paper : If {pkl} is a boundeddouble sequence of non-negative real numbers and supk,l∈N pkl =H and D = Max{, H–},then

|akl + bkl|pkl ≤ D{|akl|pkl + |bkl|pkl

}for all k, l and akl,bkl ∈C, the set of all complex numbers. Also,

|a|pkl ≤ Max{, |a|H}

for all a ∈C.By a lacunary sequence θ = (kr); r = , , , . . . , where k = , we shall mean an increasing

sequence of non-negative integers with kr – kr– as r → ∞. The intervals determined by θ

will be denoted by Ir = (kr–,kr] and hr = kr – kr–. The ratio krkr– will be denoted by qr .

Definition . The double sequence θr,s = {(kr , ls)} is called double lacunary if there existtwo increasing sequences of integers such that

k = , hr = kr – kk– → ∞ as r → ∞

and

l = , hs = ls – ls– → ∞ as s→ ∞.

Let us denote kr,s = krls, hr,s = hrhs and θr,s is determined by Ir,s = {(k, l) : kr– < k ≤kr & ls– < l ≤ ls}, qr = kr

kr– , qs =lsls– , and qr,s = qrqs.

Definition . Suppose that as before μ is a two-valued measure on N×N and letM bean Orlicz function and (X,‖·, ·‖) be a -normed space. Further, let p = {pkl} be a bounded



sequence of positive real numbers. Now we introduce the following different types of se-quence spaces, for all ε > ,

Wμ(θ,M,�m,p,‖·, ·‖)

={x ∈ S′′( –X) : μ

((k, l) ∈N×N :

hrs

∑(k,l)∈Irs

[M

(∥∥∥∥�m xkl – Lρ

, z∥∥∥∥)]pkl

≥ ε

)=

for some ρ > and L ∈ X and each z ∈ X},

Wμ(θ,M,�m,p,‖·, ·‖)

={x ∈ S′′( –X) : μ

((k, l) ∈N×N :

hrs

∑(k,l)∈Irs

[M

(∥∥∥∥�m xklρ

, z∥∥∥∥)]pkl

≥ ε

)=

for some ρ > and each z ∈ X},

Wμ∞

(θ,M,�m,p,‖·, ·‖)

={x ∈ S′′( –X) : ∃k > ,μ

({(k, l) ∈N×N :

hrs

∑(k,l)∈Irs

[M


, z∥∥∥∥)]pkl

≥ k})

= for some ρ > and each z ∈ X},

where �m xkl =�m– xkl–�m– xk+,l–�m– xk,l++�m– xk+,l+.

We now prove the following theorem.

Theorem . Wμ(θ ,M,�m,p,‖·, ·‖), Wμ (θ,M,�m,p,‖·, ·‖) and Wμ∞(θ,M,�m,p,‖·, ·‖)

are linear spaces.

Proof We shall prove the theorem for Wμ (θ,M,�m,p,‖·, ·‖) and others can be proved

similarly. Let ε > be given. Assume that x, y ∈ Wμ (θ ,M,�m,p,‖·, ·‖) and α,β ∈R, where

x = {xkl} and y = {ykl}. Further let z ∈ X. Then

μ

({(k, l) ∈N×N :

hrs

∑(k,l)∈Irs

[M


, z∥∥∥∥)]pkl

≥ ε

})= (.)

for some ρ > and

μ

({(k, l) ∈N×N :

hrs

∑(k,l)∈Irs

[M

(∥∥∥∥�m yklρ

, z∥∥∥∥)]pkl

≥ ε

})= (.)

for some ρ > .Since ‖·, ·‖ is -normed, �m is linear, therefore the following inequality holds:

hrs

∑(k,l;)∈Irs

[M

(∥∥∥∥�m (αxkl + βykl)|α|ρ + |β|ρ

, z∥∥∥∥)]pkl

≤ D hrs

∑(k,l)∈Irs

[ |α||α|ρ + |β|ρ

M(∥∥∥∥�m xkl

ρ, z

∥∥∥∥)]pst



+D hrs

∑(k,l)∈Ikl

[ |β||α|ρ + |β|ρ

M(∥∥∥∥�m ykl

ρ, z

∥∥∥∥)]pkl

≤ DF hrs

∑(k,l)∈Irs

[M


, z∥∥∥∥)]pkl

+DF hrs

∑(k,l)∈Irs

[M


, z∥∥∥∥)]pkl

,

where

F =Max{,

[ |α||α|ρ + |β|ρ

]H,[ |β|

|α|ρ + |β|ρ

]H}.

From the above inequality we get{(k, l) ∈N×N :

hrs

∑(k,l)∈Irs

[M

(∥∥∥∥�m (αxkl + βykl)|α|ρ + |β|ρ

, z∥∥∥∥)]pkl

≥ ε

}

⊆{(k, l) ∈ N×N :DF

hrs

∑(k,l)∈Irs

[M


, z∥∥∥∥)]pkl

≥ ε

}

∪{(k, l) ∈N×N :DF

hrs

∑(k,l)∈Irs

[M


, z∥∥∥∥)]pkl

≥ ε

}.

Hence from (.) and (.) the required result is proved. �

Theorem . For any fixed (k, l) ∈N×N,Wμ∞(θ,M,�m,p,‖·, ·‖) is a paranormed spacewith respect to the paranorm gkl : X →R, defined by

gkl(x) = infz∈X

∑(k,l)∈Irs

‖xkl, z‖

+ inf

{ρpkl/H : ρ > s.t. sup

(k,l)∈N×N

[M


, z∥∥∥∥)]pkl

≤ ,∀z ∈ X}.

Proof This can be proved by using the techniques similar to those used in Theorem .in []. �

Theorem . Let M,M,M be Orlicz functions. Then(i) Wμ

(θ,M,�m,p,‖·, ·‖) ⊆Wμ (θ,MoM,�m,p,‖·, ·‖), provided {pkl}k,l∈N×N are

such that H = infpkl > ;(ii) Wμ

(θ,M,�m,p,‖·, ·‖)∩Wμ (θ,M,�m,p,‖·, ·‖) ⊆Wμ

(θ,M +M,�m,p,‖·, ·‖).

Proof (i) Let ε > be given. Choose ε > such that max{εH , εH } < ε. Now, using the

continuity of M, choose < δ < such that < t < δ implies that M(t) < ε. Let {xkl} ∈Wμ

(θ,M,�m,p,‖·, ·‖). Now, from the definition μ(A(δ)) = , where

A(δ) ={(k, l) ∈N×N :

hrs

∑(k,l)∈Irs

[M


, z∥∥∥∥)]pkl

≥ δH}.

Thus if (k, l) /∈ A(δ), then

hrs

∑(k,l)∈Irs

[M


, z∥∥∥∥)]pkl

< δH ,



i.e.,

∑(r,s)∈Irs

[M


, z∥∥∥∥)]pkl

< hrsδH ,

i.e.,

[M


, z∥∥∥∥)]pkl

< δH

for all (k, l) ∈ Irs. Hence

[M


, z∥∥∥∥)]

< δ

for all (k, l) ∈ Irs.Hence, from the above, using the continuity ofM, we must have

M([

M


, z∥∥∥∥)])

< ε

for all (k, l) ∈ Irs. This implies that

[MoM


, z∥∥∥∥)]pkl

< max{εH , εH

}

for all (k, l) ∈ Irs, i.e.,

∑(k,l)∈Irs

[MoM


, z∥∥∥∥)]pkl

< hrs max{εH , εH

}< hrsε,

which again implies that

hrs

∑(r,s)∈Irs

[MoM


, z∥∥∥∥)]pkl

< ε.

This shows that{(k, l) ∈N×N :

hrs

∑(k,l)∈Irs

[MoM


, z∥∥∥∥)]pkl

≥ ε

}⊆ A(δ).

Therefore

μ

({(k, l) ∈N×N :

hrs

∑(k,l)∈Irs

[MoM


, z∥∥∥∥)]pkl

≥ ε

})= .

Thus

{xkl} ∈ Wμ(θ,M,�m,p,‖·, ·‖).



(ii) Let {xkl} ∈ Wμ (θ,M,�m,p,‖·, ·‖)∩Wμ

(θ,M,�m,p,‖·, ·‖). Then the fact that

hrs

[(M +M)


, z∥∥∥∥)]pkl

≤ Dhrs

[M


, z∥∥∥∥)]pkl

+Dh rs

[M


, z∥∥∥∥)]pkl

gives us the result. Hence this completes the proof of the theorem. �

Finally we conclude this paper by stating the following theorem.

Theorem . Let X(�m–), m ≥ , stand for Wμ(θ,M,�m–,p,‖·, ·‖) or Wμ (θ,M,�m–,

p,‖·, ·‖) or Wμ∞(θ,M,�m–,p,‖·, ·‖). Then X(�m–) � X(�m). In general, X(�i) � X(�m)for all i = , , , . . . ,m – .

Proof We shall give the proof forWμ (θ,M,�m–,p,‖·, ·‖) only. It can be proved in a sim-

ilar way forWμ(θ,M,�m–,p,‖·, ·‖) and Wμ∞(θ ,M,�m–,p,‖·, ·‖).Let x = {xkl}k,l∈N ∈Wμ

(θ,M,�m–,p,‖·, ·‖). Let also ε > be given. Then

μ

({(k, l) ∈N×N :

hrs

∑(k,l)∈Irs

[M

(∥∥∥∥�m– xklρ

, z∥∥∥∥)]pkl

≥ ε

})= (.)

for some ρ > . SinceM is non-decreasing and convex, it follows that

hrs

∑(k,l)∈Irs

[M


, z∥∥∥∥)]pkl

=hrs

∑(k,l)∈Irs

[M

(∥∥∥∥�m– xk+,l+–�m– xk+,l–�m– xk,l++�m– xklρ

, z∥∥∥∥)]pkl

≤ D

hrs

∑(k,l)∈Irs

([M

(∥∥∥∥�m– xk+,l+ρ

, z∥∥∥∥)]pkl

+[M

(∥∥∥∥�m– xk+,lρ

, z∥∥∥∥)]pkl

+[M

(∥∥∥∥�m– xk,l+ρ

, z∥∥∥∥)]pkl

+[M

(∥∥∥∥�m– xk,lρ

, z∥∥∥∥)]pkl)

≤ DGhrs

∑(k,l)∈Irs

([M

(∥∥∥∥�m– xk+,l+ρ

, z∥∥∥∥)]pkl

+[M

(∥∥∥∥�m– xk+,lρ

, z∥∥∥∥)]pkl

+[M

(∥∥∥∥�m– xk,l+ρ

, z∥∥∥∥)]pkl

+[M

(∥∥∥∥�m– xk,lρ

, z∥∥∥∥)]pkl)

,

where G =Max{, ( )H}. Hence we have

{(k, l) ∈N×N :

hrs

∑(k,l)∈Irs

[M


, z∥∥∥∥)]pkl

≥ ε

}

⊆{(k, l) ∈ N×N :

DGhrs

∑(k,l)∈Irs

[M

(∥∥∥∥�m– xk+,l+ρ

, z∥∥∥∥)]pkl

≥ ε

}

∪{(k, l) ∈N×N :

DGhrs

∑(k,l)∈Irs

[M

(∥∥∥∥�m– xk+,lρ

, z∥∥∥∥)]pkl

≥ ε

}



∪{(k, l) ∈N×N :

DGhrs

∑(k,l)∈Irs

[M

(∥∥∥∥�m– xk,l+ρ

, z∥∥∥∥)]pkl

≥ ε

}

∪{(k, l) ∈N×N :

DGhrs

∑(k,l)∈Irs

[M

(∥∥∥∥�m– xk,lρ

, z∥∥∥∥)]pkl

≥ ε

}.

Using (.) we get

μ

({(k, l) ∈N×N :

hrs

∑(k,l)∈Irs

[M


, z∥∥∥∥)]pkl

≥ ε

})= .

Therefore x = {xkl} ∈Wμ (θ,M,�m,p,‖·, ·‖). This completes the proof. �

Competing interestsThe author declares that they have no competing interests.

Received: 16 November 2012 Accepted: 3 July 2013 Published: 26 July 2013

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doi:10.1186/1029-242X-2013-347Cite this article as: Savas: On two-valued measure and double statistical convergence in 2-normed spaces. Journal ofInequalities and Applications 2013 2013:347.


on two-valued measure and double statistical convergence in 2-normed spaces

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