ISSN: 1821-1291, URL: http://www.bmathaa.org Volume 10 Issue 3(2018), Pages 1-9.
ROUGH CONVERGENCE IN 2-NORMED SPACES
MUKADDES ARSLAN AND ERD˙INC¸ D ¨UNDAR
Abstract. In this work, we introduced the notions of rough convergence, rough Cauchy sequence and the set of rough limit points of a sequence and obtained rough convergence criteria associated with this set in 2-normed space. Later, we proved that this set is closed and convex. Finally, we examined the relations between rough convergence and rough Cauchy sequence in 2-normed space.
1. Introduction
The concept of 2-normed spaces was initially introduced by G¨ahler [8, 9] in the 1960’s. Since then, this concept has been studied by many authors. G¨urdal and Pehlivan [13] studied statistical convergence, statistical Cauchy sequence and investigated some properties of statistical convergence in 2-normed spaces. S¸ahiner et al. [24] and G¨urdal [15] studied I-convergence in 2-normed spaces. G¨urdal and A¸cık [14] investigatedI-Cauchy andI∗-Cauchy sequences in 2-normed spaces. Sarabadan and Talebi [20] studied statistical convergence and ideal convergence of sequences of functions in 2-normed spaces. Arslan and D¨undar [1] investigated the concepts ofI-convergence,I∗-convergence,I-Cauchy andI∗-Cauchy sequences of functions in 2-normed spaces. Also, Yeg¨ul and D¨undar [25] studied statistical convergence of sequence of functions in 2-normed spaces. Futhermore, a lot of development have been made in this area (see [5, 16, 21, 23]).
The idea of rough convergence was first introduced by Phu [17] in finite - dimen-sional normed spaces. In [17], he showed that the set LIMrxi is bounded, closed, and convex; and he introduced the notion of rough Cauchy sequence. He also inves-tigated the relations between rough convergence and other convergence types and the dependence ofLIMrxi on the roughness degreer. In another paper [18] related to this subject, he defined the rough continuity of linear operators and showed that every linear operator f :X →Y isr -continuous at every pointx∈X under the assumption dimY < ∞ and r > 0 where X and Y are normed spaces. In [19], he extended the results given in [17] to infinite-dimensional normed spaces. Aytar [3] studied of rough statistical convergence and defined the set of rough statistical limit points of a sequence and obtained two statistical convergence criteria asso-ciated with this set and prove that this set is closed and convex. Also, Aytar [4]
2000Mathematics Subject Classification. 40A05, 40A35.
Key words and phrases. Rough convergence, 2-normed space, Rough Cauchy sequence. c
2018 Universiteti i Prishtin¨es, Prishtin¨e, Kosov¨e. Submitted May 11, 2018. Published July 27, 2018. Communicated by N.L. Braha.
studied that the r-limit set of the sequence is equal to the intersection of these sets and thatr-core of the sequence is equal to the union of these sets. Recently, D¨undar and C¸ akan [6, 7] introduced the notion of roughI-convergence and the set of roughI-limit points of a sequence and studied the notion of rough convergence and the set of rough limit points of a double sequence.
In this paper, using the concept of rough convergence and concept of 2-normed spaces, we introduce the notion of rough convergence in 2-normed spaces. Defining the set of rough limit points of a sequence, we obtain two convergence criteria asso-ciated with this set. Later, we prove that this set is closed and convex. Finally, we examine the relations between the set of cluster points and the set of rough limit points of a sequence. We note that our results and proof techniques presented in this paper are analogues of those in Phu’s [17] paper. Namely, the actual origin of most of these results and proof techniques is them papers. The following our theorems and results are the extension of theorems and results in [17].
Now, we recall the concept of 2-normed space, ideal convergence and some fun-damental definitions and notations (See [10, 11, 12, 13, 14, 15, 20, 24]).
LetX be a real vector space of dimensiond, where 2≤d <∞. A 2-norm onX is a functionk·,·k:X×X →Rwhich satisfies the following statements:
(i) kx, yk= 0 if and only if xandy are linearly dependent. (ii) kx, yk=ky, xk.
(iii) kαx, yk=|α|kx, yk, α∈R.
(iv) kx, y+zk ≤ kx, yk+kx, zk.
As an example of a 2-normed space we may takeX =R2 being equipped with
the 2-norm
kx, yk:= the area of the parallelogram based on the vectors xandy which may be given explicitly by the formula
kx, yk=|x1y2−x2y1|; x= (x1, x2), y= (y1, y2)∈R2.
In this study, we supposeX to be a 2-normed space having dimensiond; where 2≤d <∞. The pair (X,k·,·k) is then called a 2-normed space.
A sequence (xn) in 2-normed space (X,k·,·k) is said to be convergent toLinX if
lim
n→∞kxn−L, yk= 0, for everyy∈X. In such a case, we write lim
n→∞xn=Land callLthe limit of (xn).
Example 1.1. Let x= (xn) = (n+1n ,n1), L = (1,0) and z = (z1, z2). It is clear
that (xn)convergent toL= (1,0) in 2-normed space.
Let (xn) be a sequence in (X,k., .k) 2-normed space. If for∀ε >0, there is an N =N(ε)∈Nsuch that for∀m, n≥N and for every z∈X,
kxm−xn, zk< ε
Letrbe a nonnegative real number andRndenotes the realn-dimensional space
with the normk.k. Consider a sequencex= (xi)⊂Rn.
The sequencex= (xi) is said to ber-convergent tox∗, denoted by xi r −→ x∗ provided that
∀ε >0 ∃iε∈N: i≥iε⇒ kxi−x∗k< r+ε. The set
LIMrxi:={x∗∈Rn:xi r −→x∗} is called ther-limit set of the sequencex= (xi).
A sequence x = (xi) is said to be r-convergent if LIMrx 6= ∅. In this case, r is called the convergence degree of the sequence x= (xi). For r = 0, we get the ordinary convergence.
The sequence (xn) is said to be a rough Cauchy sequence satisfying
∀ε >0,∃nε:m, n≥nε⇒ kxm−xnk< ρ+ε
for ρ > 0. ρ is roughness degree of (xn). Shortly (xn) is called a rough Cauchy sequence. ρis also a Cauchy degree of (xn).
Lemma 1.1. [19] Let (xi) be r-convergent, i.e., LIMrxi 6=∅. Then, (xi) is a ρ
-Cauchy sequence for every ρ ≥ 2r. This bound for the Cauchy degree cannot be generally decreased.
2. MAIN RESULTS
In this work, we introduced the notions of rough convergence, rough Cauchy sequence and the set of rough limit points of a sequence and obtained rough con-vergence criteria associated with this set in 2-normed space. Later, we proved that this set is closed and convex. Finally, we examined the relations between rough convergence and rough Cauchy sequence in 2-normed space.
Definition 2.1. Let (xn)be a sequence in (X,k., .k) 2-normed linear space and r
be a non-negative real number. (xn) is said to be rough convergent (r-convergent)
toLdenoted by xn k.,.k −→rL if
∀ε >0,∃nε∈N:n≥nε⇒ kxn−L, zk< r+ε (2.1)
or equivalently, if
lim supkxn−L, zk ≤r,
for everyz∈X.
If (2.1) holdsL is anr-limit point of (xn), which is usually no more unique (for r >0). So, we have to consider the so-calledr-limit set (or shortlyr-limit) of (xn) defined by
LIMr2xn:={L∈X :xn k.,.k
−→rL}. (2.2)
The sequence (xn) is said to be rough convergent if
LIMr2xn6=∅.
measured or calculated) exactly, one has to do with an approximated sequence (xn) satisfying
kxn−yn, zk ≤r
for all n and for every z ∈X, where r >0 is an upper bound of approximation error. Then, (xn) is no more convergent in the classical sense, but for everyz∈X,
kxn−L, zk ≤ kxn−yn, zk+kyn−L, zk ≤ r+kyn−L, zk
implies that isr-convergent in the sense of (2.1).
Example 2.1. The sequence x = (xn) = ((−1)n,(−1)n) is not convergent in
2-normed space(X,k., .k)but it is rough convergent toL= (0,0) for everyz∈X. It is clear that
LIMr2xn =
∅ , if r <1 [(−r,−r),(r, r)] , otherwise.
Sometimes we are interested in the set of r-limit points lying in a given subset D⊂X,which is calledr-limit inD and denoted by
LIMD,r2 xn :={L∈D:xn k.,.k
−→rL}. (2.3)
It is clearly
LIMX,r2 xn= LIMr2xn and LIM D,r
2 xn=D∩LIMr2xn.
First, let us transform some properties of classical convergence to rough con-vergence. It is well known if a sequence converges then its limit is unique. This property is nor maintained for rough convergence with roughness degree. r >0, but only has the following analogy.
Theorem 2.1. Let (X,k., .k) be a 2-normed space and consider a sequence x= (xn)∈X.We havediam(LIMr2xn)≤2r. In general,diam(LIMr2xn)has no smaller
bound.
Proof 2.1. We have to show that
diam(LIMr2xn) =sup{ky−t, zk:y, t∈LIMr2xn ≤2r}, (2.4)
where(X,k., .k)is a 2-normed space and for everyz∈X.Assume the contrary that
diam(LIMr2xn)>2r
then, there existy, t∈LIMr2xn satisfying
d:=ky−t, zk>2r,
for every z ∈X. For an arbitrary ε∈(0, d/2−r), it follows from (2.1) and (2.2) that there is annε∈Nsuch that forn≥nε,
kxn−y, zk< r+ε and kxn−t, zk< r+ε,
for everyz∈X.This implies
ky−t, zk ≤ kxn−y, zk+kxn−t, zk
< 2(r+ε)
< 2r+ 2(d/2−r)
for every z ∈ X, which conflicts with d= ky−t, zk. Hence, (2.4) must be true. Consider a convergent sequence(xn)with lim
n→∞xn=L.Then, for
Br(L) :={y∈X :ky−L, zk ≤r},
it follows from
kxn−y, zk ≤ kxn−L, zk+kL−y, zk ≤ kxn−L, zk+r
for everyz∈X and fory∈Br(L),(2.1) and (2.2) that
LIMr2xn =Br(L).
Since diam(Br(L)) = 2r, this shows that in general the upper bound 2r of the diameter of an r-limit set cannot be decreased anymore.
Obviously the uniqueness of limit (of classical convergence) can be regarded as a special case of latter property, because ifr= 0 thendiam(LIMr2xn) = 2r= 0,that is, LIMr2xn is either empty or a singleton. A further property of classical concept is the boundedness of convergent sequences. Its analogy for rough convergence is:
Theorem 2.2. Let (X,k., .k) be a 2-normed space and consider a sequence x= (xn) ∈ X. The sequence (xn) is bounded if and only if there exist an r ≥ 0 such that LIMr2xn 6= ∅. For all r > 0, a bounded sequence (xn) is always contains a
subsequence xnk with
LIM(xnk),r
2 xnk6=∅.
Proof 2.2. For every z∈X if
s:=sup{kxn, zk:n∈N}<∞
then, LIMs2xn contains the origin ofX.On the other hand LIMr2xn 6=∅, for some r ≥0 then, all but finite elements xn are contained in some ball with any radius
greater thanr. Therefore, the sequence (xn) is bounded in2-normed space X. As (xn) is bounded sequence in 2-normed space X, it certainly contains a convergent
subsequence (xnk).Let Lbe its limit point, then
LIMr2xnk=Br(L)
and forr >0,
LIM(xnk),r
2 xnk={xnk :kL−xnk, zk ≤r} 6=∅, for everyz∈X.
Note that the second part of the previous proposition is concerned withr-limit points lying in the subsequence (xnk) in 2-normed spaceX. It is straightforward
that a sequence contained in some bounded set D always possesses a subsequence beingr-convergent (for an arbitraryr >0) to some point ofD.Here, the closedness ofDis not needed as for classical convergence. Corresponding to the property that each subsequence of a convergent sequence also converges to the same limit point, we now have the following one whose proof is rather simple.
Proposition 2.1. Let (X,k., .k)be a 2-normed space and consider a sequencex= (xn)∈X.If(x0n)is a subsequence of(xn)then,
LIMr2xn⊆LIMr2x0n
Theorem 2.3. Let (X,k., .k) be a 2-normed space and consider a sequence x= (xn) ∈ X. For all r≥ 0, the r-limit setLIMr2xn of an arbitrary sequence (xn) is
closed.
Proof 2.3. Let(ym)be an arbitrary sequence in LIMr2xn which converges to some
point L.For each ε >0and every z∈X,by definition there aremε/2 and an nε/2
such that
kymε/2−L, zk< ε/2 and kxn−ymε/2, zk< r+ε/2
whenevern≥nε/2. Consequently for everyz∈X,
kxn−L, zk ≤ kxn−ymε/2, zk+kymε/2−L, zk < r+ε,
if n≥nε/2.That meansL∈LIMr2xn too. Hence,LIMr2xn closed.
Theorem 2.4. Let (X,k., .k) be a 2-normed space and consider a sequence x= (xn)∈X.If
y0∈LIMr20xn and y1∈LIMr21xn
then,
yα:= (1−α)y0+αy1∈LIM
(1−α)r0+αr1
2 xn, f or α∈[0,1].
Proof 2.4. By definition, for everyε >0,r0, r1>0 and everyz∈X there exists
annε such that n > nεimplies
kxn−yo, zk< r0+ε and kxn−y1, zk< r1+ε,
which yields also, for everyz∈X,
kxn−yα, zk ≤ (1−α)kxn−yo, zk+αkxn−y1, zk < (1−α)(r0+ε) +α(r1+ε)
= (1−α)r0+αr1+ε.
Hence, we have
yα∈LIM
(1−α)r0+αr1
2 xn.
Theorem 2.5. Let (X,k., .k) be a 2-normed space and consider a sequence x= (xn)∈X.LIMr2xn is convex.
Proof 2.5. In particular, for r =r0 =r1, Theorem 2.4 yields immediately that LIMr
2xn is convex.
Theorem 2.6. If xn k.,.k
−→rL1 andyn k.,.k
−→rL2. Then,
(i) (xn+yn) k.,.k
−→r(L1+L2)and
(ii)c.(xn)−→k.,.krc.L1,(c∈R).
Proof 2.6. (i)By definition for every z∈X,
∀ε >0,∃nε∈N:n≥nε⇒ kxn−L1, zk ≤r1+ ε 2
and
Let j=max(nε, jε)andr1+r2=r. For everyn > j and every z∈X we have
k(xn+yn)−(L1+L2), zk = kxn−L1, zk+kyn−L2, zk
≤ r1+ ε 2 +r2+
ε 2 = r+ε
and so
(xn+yn) k.,.k
−→r(L1+L2).
(ii) It is obvious for c= 0.Let c6= 0.Since
xn k.,.k −→rL
for everyε >0 and every z∈X,∃nε∈Nsuch taht for everyn≥nεwe have
kxn−L, zk ≤ r+ε
|c| .
According to this, for ∀n≥nε and everyz∈X,we can write
kc.xn−c.L, zk = |c|.|xn−L, zk
≤ |c|.r+ε
|c| = r+ε.
So,
(c.xn) k.,.k −→rc.L.
Definition 2.2. Let (xn)be a sequence in(X,k., .k) 2-normed space. (xn)is said to be a rough Cauchy sequence with roughness degreeρ, if
∀ε >0,∃kε:m, n≥kε⇒ kxm−xn, zk ≤ρ+ε
is hold forρ >0,L∈X and everyz∈X. ρis also called a Cauchy degree of(xn).
Proposition 2.2. (i) Monotonicity: Assume ρ0 > ρ. If ρ is a Cauchy degree of a given sequence(xn)in(X,k., .k)2-normed space, soρ0is a Cauchy degree of(xn).
(ii) Boundedness: A sequence(xn)is bounded if and only if there exists aρ≥0
such that (xn)is aρ-Cauchy sequence in(X,k., .k)2-normed space.
Theorem 2.7. If (xn) is rough convergent in 2-normed space (X,k., .k), i.e., LIMr2xn 6= ∅ if and only if (xn) is a ρ-Cauchy sequence for every ρ ≥ 2r. This
bound for the Cauchy degree cannot be generally decreased.
Proof 2.7. Let L be any point in LIMr2xn Then, for all ε > 0, there exists an kε∈N such thatm, n≥kεimplies
kxm−L, zk ≤r+ ε
2 and kxn−L, zk ≤r+ ε 2,
for everyz∈X. Therefore,m, n≥kε, we have
kxm−xn, zk = kxm−L+L−xn, zk ≤ kxm−L, zk+kL−xn, zk
≤ r+ε 2 +r+
for everyz ∈X. Hence, (xn) is a ρ-Cauchy sequence for ρ≥2r. By proposition 2.2, everyρ≥2r is also a Cauchy degree of(xn).
Let(xn)be a rough Cauchy sequence in 2-normed space(X,k., .k).Since(xn)be a rough Cauchy sequence, then it is bounded and consequently it is rough convergent forρ >0 and for everyz ∈X. It is clear that this bound2r can not be generally decreased by Lemma 1.1.
Acknowledgments. The authors would like to thank the anonymous referee for his/her comments that helped us improve this article.
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Mukaddes Arslan
˙Ihsaniye Anadolu ˙Imam Hatip Lisesi, 03370 Afyonkarahisar, Turkey
E-mail address:[email protected]
Erdinc¸ D¨undar
Department of Mathematics, Afyon Kocatepe University, 03200 Afyonkarahisar, Turkey.
