Rough I -convergence in cone metric spaces
aa r X i v : . [ m a t h . M G ] A ug ROUGH I -CONVERGENCE IN CONE METRIC SPACES. AMAR KUMAR BANERJEE AND ANIRBAN PAULA
BSTRACT . Here we have studied the notion of rough I -convergence as an extension of the idea ofrough convergence in a cone metric space using ideals. We have further introduced the notion of rough I ∗ -convergence of sequences in a cone metric space to find the relationship between rough I and I ∗ -convergence of sequences. Key words and phrases:
Cone metric spaces, Rough convergence, Rough I -convergence, Rough I ∗ -convergence, (AP) condition. (2010) AMS subject classification : NTRODUCTION
The concept of statistical convergence was given independently by Fast[10] and Steinhaus[27] asa generalization of ordinary convergence of real sequences. Lot of devolopments have been made inthis area after the remakable works done in [11, 21, 26]. The idea of I -convergence was introducedby Kostyrko et al.[12] as a generalization of statistical convergence using the idea of the ideal I ofsubsets of the set of natural numbers. Later many more works have been carried out in this direction[3, 4, 5, 6, 8, 14, 16, 17].In 2001, Phu [23] introduced the idea of rough convergence and rough Cauchyness of sequences in afinite dimensional normed space. Later in 2003, he studied the same in an infinite dimensional normedspace [24]. Using these ideas Aytar[1], in 2008 gave the concept of rough statistical convergenceof a sequence. In 2014, using the concepts of I -convergence and rough convergence D ¨undar etal.[9] introduced the notion of rough I -convergence. Many more works have been done in differentdirection [2, 19, 20] by several authors using this idea given by Phu [23].The idea of cone metric spaces was given by Huang and Xian [18]. In their paper they have replacedthe distance between two points by the elements of a real Banach space. Obviously such space isa generalization of the notion of an ordinary metric space. Since then many more works have beencarried out specially in the field of summability theory.In [7] Banerjee and Mondal have studied the idea of rough convergence of sequences in a cone metricspace. In our paper we have studied the notion of rough I -convergence of sequences in a cone metricspace and examine how far several results as valid in [7] are affected. Also we have introduced herethe idea of rough I ∗ -convergence of sequence in a cone metric space and obtain the relations betweenrough I and I ∗ -convergence of sequences.2. P RELIMINARIES
We give the basic ideas of statistical convergence and then ideal convergence of real sequencesand few definitions and notions related to these ideas are also furnished which will be needed in thesequel.
Let K be a subset of the set of positive integers N . Then the natural density of K is given by δ ( K ) =lim n →∞ | K n | n , where K n = { k ∈ K : k ≤ n } and | K n | denotes the number of elements in K n . Definition 2.1. [10] A sequence x = { x n } of real numbers is said to be statistically convergent to x if for any ε > the set { n ∈ N : | x n − x | ≥ ε } have natural density zero.Note that the idea of statistical convergence of sequences is a generalization ordinary convergence.A family I ⊂ N of subsets of X is said to be an ideal [13] in N if the following conditions holds: ( i ) A, B ∈ I ⇒ A ∪ B ∈ I ( ii ) A ∈ I, B ⊆ A ⇒ B ∈ II is called a non-trivial ideal in N if I = { φ } or N / ∈ I . A non-trivial ideal in N is said to beadmissible if { n } ∈ I for each n ∈ N .Clearly if I is a nontrivial ideal in N then the family of sets F ( I ) = { M ⊂ N : there exists A ∈ I, M = N \ A } is a filter in N . It is called the filter associatedwith the ideal I .An admissible ideal I ⊂ N is said to satisfy the condition (AP) [12] if for any sequence { A , A , · · · } of mutually disjoint sets in I , there is a sequence { B , B , · · · } of subsets of N such that A i ∆ B i ( i = 1 , , · · · ) is finite and B = ∪ j ∈ N B j ∈ I . Definition 2.2. [12] A sequence x = { x n } of real numbers is said to be I -convergent to x if for any ε > the set A ( ε ) = { n ∈ N : | x n − x | ≥ ε } ∈ I . Definition 2.3. [9] A sequence x = { x n } in a normed linear space is said to be I -bounded if thereexists a M ( > ∈ R such that the set { n ∈ N : k x n k ≥ M } ∈ I .We now give the idea of a cone metric space [18] as follows:Let E be a real Banach space and P ⊂ E . Then P is called a cone if and only if the following aresatisfied: ( i ) P is closed, non-empty and P = { } ( be the zero element of E ). ( ii ) a, b ∈ R and a, b ≥ then x, y ∈ P implies ax + by ∈ P . ( iii ) x ∈ P and − x ∈ P implies that x = 0 (zero element of E ).Let E be a real Banach space and P be a cone in E . Then a partial ordering ≤ with respect to P can be defined by x ≤ y if and only if y − x ∈ P , whereas x < y indicates x ≤ y and x = y , also x << y stands for y − x ∈ intP , intP denotes the interior P .The cone P is called normal if there exists a positive real number K such that for all x, y ∈ E , ≤ x ≤ y implies k x k ≤ K k y k .The least positive number satisfying above is called the normal constant of P . In [28], it is knownthat any cone metric space is a first countable Hausdorff topological space with topology induced bythe open balls defined naturally for each element z ∈ X and for every (0 << ) c ∈ E . Definition 2.4. [18] Let ( X, d ) be a cone metric space. A sequence x = { x n } in X is said to beconvergent to x if for any c ∈ E with << c there is N ∈ N such that d ( x n , x ) << c for all n > N . Lemma 2.1. [18]
Let ( X, d ) be a cone metric space and P be a normal cone with normal constant K . Let { x n } be a sequence in X . Then { x n } converges to x if and only if d ( x n , x ) → as n → ∞ Lemma 2.2. [15]
Let ( X, E ) be a cone space with x ∈ P and y ∈ intP . Then one can find n ∈ N such that x << ny . Now we recall some useful results from [7].
Theorem 2.3. [7]
Let E be a real Banach space with cone P . If x ∈ intP and α ( > ∈ R then αx ∈ intP OUGH I -CONVERGENCE IN CONE METRIC SPACES. 3 Theorem 2.4. [7]
Let E be a real Banach space and P be a cone in E . If x ∈ P and y ∈ intP then x + y ∈ intP . Corollary 1. [7] If x , y ∈ intP then x + y ∈ intP . Theorem 2.5. [7]
Let E be a real Banach space with cone P , then / ∈ intP ( be the zero elementof E ). Definition 2.5. [7] Let { x n } be a sequence in a cone metric space ( X, d ) . A point c ∈ X is said tobe a cluster point of { x n } if for any (0 << ) ε in E and for any k ∈ N , there exists a k ∈ N such that k > k with d ( x k , c ) << ε .The definition of I -convergent and I ∗ -convergent of a sequence in a cone metric is as follows: Definition 2.6. [25] Let ( X, d ) be a cone metric space. A sequence x = { x n } in X is said to be I -convergent to x if for any c ∈ E with << c the set { n ∈ N : c − d ( x n , x ) / ∈ intP } ∈ I . Definition 2.7. [25] Let ( X, d ) be a cone metric space. A sequence x = { x n } in X is said to be I ∗ -convergent to x if and only if there exists a set M ∈ F ( I ) , M = { m < m < · · · < m k < · · · } such that { x n } n ∈ M is convergent to x i.e., for any c ∈ E with << c , there exists p ∈ N such that c − d ( x m k , x ) ∈ intP for all k ≥ p . Definition 2.8. [23] Let x = { x n } be a sequence in a normed linear space ( X, k . k ) and r ( ≥ ∈ R .Then { x n } is said to be rough convergent of roughness degree r to x if for any ε > there exists anatural number N such that k x n − x k < r + ε for all n ≥ k .For r = 0 we obtain the ordinary convergence of sequences. Definition 2.9. [9] A sequence x = { x n } in a normed linear space is said to be rough I -convergent ofroughness degree r to x ∗ for some r ≥ if for any ε > the set { n ∈ N : k x n − x ∗ k ≥ r + ε } ∈ I .We denote this by x n r − I −−→ x ∗ .The definition of I -bounded sequence in a normed linear space has been given in [9] as follows: Definition 2.10. [9] A sequence { x n } is said to be I -bounded if there exists a positive real number M such that { n ∈ N : || x n || ≥ M } ∈ I . Definition 2.11. [7] A sequence { x n } in a cone metric space is said to be bounded if for any fixed x ∈ X there exists a (0 << ) M ∈ E such that d ( x n , x ) << M for all n ∈ N .Now we recall the definition of rough convergence in cone metric space from [7]. Definition 2.12. [7] Let ( X, d ) be a cone metric space. A sequence x = { x n } in X is said to berough convergent of roughness degree r to x for some r ∈ E with << r or r = 0 if for any ε with (0 << ) ε there exists a m ∈ N such that d ( x n , x ) << r + ε for all n ≥ m .We denote this by x n r −→ x . 3. M AIN R ESULTS
Throughout our discussion ( X, d ) will always stands for a cone metric space where d : X × X E is the cone metric and E being a real Banach space. I be a admissible ideal, N and R stands for theset of natural numbers and the set of real numbers respectively. A ∁ denotes the complement of the set A unless otherwise stated. Definition 3.1.
Let ( X, d ) be a cone metric space. A sequence x = { x n } in X is said to be rough I -convergent of roughness degree r to x ∗ ∈ X for some r ∈ E with << r or r = 0 if for any (0 << ) ε ∈ E the set A ( ε ) = { n ∈ N : ( r + ε − d ( x n , x ∗ )) / ∈ intP } ∈ I . AMAR KUMAR BANERJEE AND ANIRBAN PAUL
We denote this by x n r − I −−→ x ∗ . For r = 0 the definition reduces to the definition of I -convergenceof sequence in a cone metric space. If a sequences x = { x n } is rough I -convergent of roughnessdegree r to x ∗ ∈ X then x ∗ is called the rough I -limit of x = { x n } . In general, the rough I -limit ofa sequence x = { x n } is not unique which can be seen from the next example. So the set of all rough I -limits of a sequence x = { x n } denoted by I − LIM r x is called the rough I -limit set of a sequence x = { x n } i.e., I − LIM r x := n x ∗ ∈ X : x n r − I −−→ x ∗ o .Therefore, a sequence x = { x n } is said to be rough I -convergent in a cone metric space if I − LIM r x = φ Example 3.1.
Let X = R , E = R , P = { ( x, y ) ∈ E : x, y ≥ } ⊂ E and d : X × X E besuch that d ( x, y ) = ( | x − y | , | x − y | ) . Then ( X, d ) is a cone metric space. Now let us consider theideal in N which consists of sets whose natural density are zero i.e., I = I d . Now, let us consider thesequence x = { x n } in X defined by x n = ( ( − n , if n = k ( where k ∈ N ) n, otherwise . Now we can seethat for any r = ( r , r ) ∈ E with << r , if min( r , r ) = r ∗ and r ∗ ≥ then I − LIM r x =[ − ( r ∗ − , ( r ∗ − if r ∗ ≥ , since for any x ∈ [ − ( r ∗ − , ( r ∗ − and r ∗ ≥ we have { n ∈ N :( r + ε − d ( x n , x ) / ∈ intP } ⊂ { , , , · · · } , therefore { n ∈ N : ( r + ε − d ( x n , x ) / ∈ intP } ∈ I and if r ∗ < or r = 0 then I − LIM r x = φ . Also since the sequence is unbounded therefore LIM r x = φ , for any r . Note 1.
From the above example we can see that in general I − LIM r x = φ does not imply that LIM r x = φ . But since I is an admissible ideal therefore LIM r x = φ implies I − LIM r x = φ .That is, if a sequence x = { x n } in ( X, d ) is rough convergent of roughness degree r , where r ∈ E with << r or r = 0 , then it is also rough I -convergent of same roughness degree r . Therefore ifwe denote all rough convergence sequences in a cone metric space ( X, d ) by LIM r and the set of allrough I -convergent sequences by I − LIM r , then we have LIM r ⊆ I − LIM r .It is seen in [7] that if a sequence x = { x n } in a cone metric space ( X, d ) is bounded then LIM r x = φ for some (0 << ) r ∈ E . So in view of note 1, the following theorem is evident. Theorem 3.1.
If a sequence x = { x n } in a cone metric space ( X, d ) is bounded, then there existssome r ∈ E with << r such that I − LIM r x = φ . We recall that a sequence { x n } in a metric space ( X, d ) is said to be bounded if there exists x ∈ X and r > satisfying d ( x n , x ) < r for all n ∈ N . Using this idea we define I -bounded sequence in acone metric space as follows: Definition 3.2.
A sequence x = { x n } in a cone metric space ( X, d ) is said to be I -bounded if thereexists a y ∈ X and M ∈ E with << M such that { n ∈ N : M − d ( x n , y ) intP } ∈ I .Let { x n } be bounded sequence in a cone metric space ( X, d ) , then there exists x ∈ X and M ∈ E with << M such that d ( x, x n ) << M for all n ∈ N . This implies that M − d ( x, x n ) ∈ intP for all n ∈ N . So { n ∈ N : M − d ( , x n ) / ∈ intP } = φ ∈ I . Hence { x n } is I -bounded. But the converse may not be true as seen in the example 3.1. For, if we choose y = 2 and (0 << ) M = (5 , then we get { n ∈ N : ( M − d ( x n , y )) / ∈ intP } ⊂ { , , , · · · } , whichimplies that { n ∈ N : M − d ( x n , y ) / ∈ intP } ∈ I . So the sequence considered here is I -boundedalthough the sequence is not bounded.From the example 3.1 it follows that the reverse implication of the theorem 3.1 is not valid, howeverthe reverse implication is true in case of I -boundedness as seen in the following theorem. OUGH I -CONVERGENCE IN CONE METRIC SPACES. 5 Theorem 3.2.
Let I be an admissible ideal of N . Then a sequence x = { x n } in ( X, d ) is I -boundedif and only if there exists some r ∈ E with << r or r = 0 such that I − LIM r x = φ .Proof. Let the sequence x = { x n } be I -bounded. Then there exists a y ∈ x and (0 << ) r ∈ E such that the set { n ∈ N : r − d ( x n , y ) intP } ∈ I . Let (0 << ) ε ∈ E ( i.e., ε ∈ intP ). Then { n ∈ N : r + ε − d ( x n , y ) / ∈ intP } ⊂ { n ∈ N : r − d ( x n , y ) / ∈ intP } ∈ I ( For, let n ∈ { n ∈ N : r + ε − d ( x n , y ) / ∈ intP } ⇒ r + ε − d ( x n , y ) / ∈ intP . So r − d ( x n , y ) / ∈ intP ⇒ n ∈{ r − d ( x n , y ) / ∈ intP } ). Therefore y ∈ I − LIM r x .Conversely, let I − LIM r x = φ for some r ∈ E with << r or r = 0 and x ∗ ∈ I − LIM r x .Therefore for any (0 << ) ε ∈ E (i.e., ε ∈ intP ) the set { n ∈ N : r + ε − d ( x n , x ∗ ) / ∈ intP } ∈ I .Now r + ε ∈ intP for any ε ∈ intP . So taking M = r + ε ∈ intP ( i.e., << M ), we have { n ∈ N : M − d ( x n , x ∗ ) / ∈ intP } ∈ I . So the sequence x = { x n } is I -bounded. (cid:3) Theorem 3.3. An I -bounded sequence x = { x n } in a cone metric space ( X, d ) always contains asubsequence which is rough I -convergent of roughness degree r for some (0 << ) r ∈ E .Proof. Let a sequence x = { x n } in a cone metric space ( X, d ) be I -bounded. Therefore there existsa z ∈ X and (0 << ) M ∈ E such that the set { n ∈ N : M − d ( x n , z ) / ∈ intP } ∈ I . Therefore theset L = { n ∈ N : M − d ( x n , z ) ∈ intP } ∈ F ( I ) . Now if we consider the subsequence { x n } n ∈ L then this subsequence is bounded. Now as for any bounded sequence x = { x n } , LIM r x = φ for some (0 << ) r ∈ E , so the subsequence { x n } n ∈ L is rough convergent of roughness degree r ( (0 << ) r ∈ E ). Hence in view of note 1 { x n } n ∈ L is also rough I -convergent of same roughnessdegree (0 << ) r ∈ E . (cid:3) Theorem 3.4.
Let { x n } be a sequence in a cone metric space ( X, d ) which is I -convergent to x . If { y n } is another sequence in ( X, d ) such that d ( x n , y n ) ≤ r for some (0 << ) r ∈ E and for all n ∈ N .Then { y n } is rough I -convergent of roughness degree r to x .Proof. Let { x n } be a sequence in a cone metric space ( X, d ) which is I -convergent to x . Therefore foran (0 << ) ε ∈ E , the set { n ∈ N : ε − d ( x n , x ) / ∈ intP } ∈ I . So { n ∈ N : ε − d ( x n , x ) ∈ intP } ∈ F ( I ) . Now d ( y n , x ) ≤ d ( y n , x n ) + d ( x, x n ) ≤ r + d ( x n , x ) . This implies that r + d ( x n , x ) − d ( y n , x ) ∈ P . Hence if ε − d ( x n , x ) ∈ intP then ( r + d ( x n , x ) − d ( y n , x )) + ( ε − d ( x n , x )) = r + ε − d ( y n , x ) ∈ intP . Therefore the set { n ∈ N : r + ε − d ( y n , x ) ∈ intP } ∈ F ( I ) . Thus { n ∈ N : r + ε − d ( y n , x ) / ∈ intP } ∈ I . Hence the results follows. (cid:3) Theorem 3.5.
Let x = { x n } be a sequence in a cone metric space ( X, d ) which is rough I -convergentof roughness degree r for some (0 << ) r ∈ E . Then there does not exists y, z ∈ I − LIM r x suchthat mr < d ( y, z ) , where m is a real number grater than .Proof. Suppose on contrary that there exists such y, z ∈ I − LIM r x for which mr < d ( y, z ) and m ( ∈ R ) > . Let (0 << ) ε be arbitrarily chosen in E . Now as y, z ∈ I − LIM r x , so each of the sets M = (cid:8) n ∈ N : r + ε − d ( x n , y ) / ∈ intP (cid:9) and M = (cid:8) n ∈ N : r + ε − d ( x n , z ) / ∈ intP (cid:9) belongsto I . Then both of M ∁ and M ∁ belongs to F ( I ) . Let p ∈ M ∁ ∩ M ∁ . Then r + ε − d ( x p , y ) ∈ intP and r + ε − d ( x p , z ) ∈ intP . Hence ( r + ε − d ( x p , y )) + ( r + ε − d ( x p , z )) = 2 r + ε − ( d ( x p , y ) + d ( x p , z )) ∈ intP . Now d ( y, z ) ≤ d ( x p , y )+ d ( x p , z ) . So d ( x p , y )+ d ( x p , z ) − d ( y, z ) ∈ P . Therefore (2 r + ε − ( d ( x p , y ) + d ( x p , z ))) + ( d ( x p , y ) + d ( x p , z ) − d ( y, z )) = 2 r + ε − d ( y, z ) ∈ intP . Againby our assumption d ( y, z ) − mr ∈ P . So (2 r + ε − d ( y, z )) + ( d ( y, z ) − mr ) = 2 r + ε − mr ∈ intP .That is ε − r ( m − ∈ intP . But choosing ε = r ( m − we get ∈ intP , which is a contradiction.Hence the result follows. (cid:3) Theorem 3.6.
Let { x n } be a sequence in ( X, d ) which is rough I -convergent of roughness degree r .Then { x n } is also rough I -convergent of roughness degree r for any r with r < r . AMAR KUMAR BANERJEE AND ANIRBAN PAUL
Proof.
Proof is trivial and so is omitted. (cid:3)
In view of the theorem 3.6 we have the following corollary.
Corollary 2.
Let x = { x n } be a rough I -convergent sequence in ( X, d ) of roughness degree r . Thenfor a (0 << ) r with r < r , LIM r x ⊂ LIM r x . Definition 3.3. (cf. [17] ) A point c ∈ X is said to be a I -cluster point of a sequence { x n } in ( X, d ) if for any (0 << ) ε the set { k ∈ N : ε − d ( x k , c ) ∈ intP } / ∈ I .For << r and a fixed y ∈ X , the closed spheres B r ( y ) and open spheres B r ( y ) centred at y withradius r is defined in [7] as follows: B r ( y ) = { x ∈ X : d ( x, y ) ≤ r } and B r ( y ) = { x ∈ X : d ( x, y ) << r } .Now we have the following theorems. Theorem 3.7.
Let ( X, d ) be a cone metric space. c ∈ X and (0 << ) r be such that for any x ∈ X either d ( x, c ) ≤ r or r << d ( x, c ) . If c is a I -cluster point of a sequence { x n } then I − LIM r x ⊂ B r ( c ) .Proof. If possible assume that there exists a y ∈ I − LIM r x but y / ∈ B r ( c ) . Now by our assump-tion r << d ( y, c ) . Let (0 << ) ε = d ( y, c ) − r and hence d ( y, c ) = r + ε . Let (0 << ) ε = ε .Then we have d ( y, c ) = r + 2 ε . Also B r + ε ( y ) ∩ B ε ( c ) = φ . For, if l ∈ B r + ε ( y ) ∩ B ε ( c ) then d ( l, y ) << r + ε and d ( l, c ) << ε . Thus r + ε − d ( l, y ) ∈ intP and ε − d ( l, c ) ∈ intP .Therefore ( r + ε − d ( l, y )) + ( ε − d ( l, c )) = r + 2 ε − ( d ( l, y ) + d ( l, c )) ∈ intP → ( i ) . Nowas d ( y, c ) ≤ d ( y, l ) + d ( l, c ) , therefore d ( y, l ) + d ( l, c ) − d ( y, c ) ∈ P → ( ii ) . Hence from ( i ) and ( ii ) we get r + 2 ε − ( d ( l, y ) + d ( l, c )) + d ( y, l ) + d ( l, c ) − d ( y, c ) = r + 2 ε − d ( y, c ) =0 ∈ intP , a contradiction. Hence B r + ε ( y ) ∩ B ε ( c ) = φ . As y ∈ I − LIM r x , so the set A = { n ∈ N : r + ε − d ( x n , y ) / ∈ intP } ∈ I . So the set A ∁ = N \ A ∈ F ( I ) . Again since c is a I -cluster point of { x n } , so for << ε the set { k ∈ N : ε − d ( x k , c ) ∈ intP } / ∈ I . Therefore the set { k ∈ N : ε − d ( x k , c ) ∈ intP } can not be a subset of A . For, if { k ∈ N : ε − d ( x k , c ) ∈ intP } ⊂ A then we have { k ∈ N : ε − d ( x k , c ) ∈ intP } ∈ I , which contradicts to the fact that c is a I -clusterpoint of { x n } . We consider an element m ∈ A ∁ . So m ∈ { k ∈ N : ε − d ( x k , c ) ∈ intP } . Now m ∈ A ∁ implies r + ε − d ( x m , y ) ∈ intP . Hence d ( x m , y ) << r + ε , which implies x m ∈ B r + ε ( y ) .Also m ∈ { k ∈ N : ε − d ( x k , c ) ∈ intP } implies ε − d ( x m , c ) ∈ intP . Therefore d ( x m , c ) << ε ,which further implies that x m ∈ B ε ( c ) . Thus we see that x m ∈ B r + ε ( y ) ∩ B ε ( c ) , which is a contra-diction. Hence we can conclude that our assumption is wrong and y ∈ B r ( c ) . (cid:3) Theorem 3.8.
Let x = { x n } be a rough I -convergent sequence of roughness degree r in a conemetric space ( X, d ) and { y n } be a I -convergent sequence in I − LIM r x which is I -convergent to y .Then y ∈ I − LIM r x .Proof. Let (0 << ) ε be given. Since the sequence { y n } is I -convergent to y , for (0 << ) ε the set A = (cid:8) n ∈ N : ε − d ( y n , y ) / ∈ intP (cid:9) ∈ I . So the set A ∁ = N \ A ∈ F ( I ) . Choose a p ∈ A ∁ . Then ε − d ( y p , y ) ∈ intP and so d ( y p , y ) << ε → ( i ) . Also since { y n } is a sequence in I − LIM r x ,let y p ∈ I − LIM r . Therefore the set B = (cid:8) n ∈ N : r + ε − d ( x n , y p ) / ∈ intP (cid:9) ∈ I . Henceit’s complement B ∁ = N \ B ∈ F ( I ) . Let us choose an element l ∈ B ∁ ( ∈ F ( I )) . Therefore r + ε − d ( x l , y p ) ∈ intP and so d ( x l , y p ) << r + ε → ( ii ) . Also for all n ∈ N we have d ( x n , y ) ≤ d ( x n , y p ) + d ( y p , y ) . So d ( x n , y p ) + d ( y p , y ) − d ( x n , y ) ∈ P , for all n ∈ N . In particular d ( x l , y p ) + d ( y p , y ) − d ( x l , y ) ∈ P → ( iii ) . Now by ( i ) and ( ii ) using the theorem 2.4 we get ( ε − d ( y p , y )) + ( r + ε − d ( x l , y p )) = r + ε − ( d ( y p , y ) + d ( x l , y p )) ∈ intP → ( iv ) . Applying againthe theorem 2.4 we get from ( iii ) and ( iv ) , ( d ( x l , y p ) + d ( y p , y ) − d ( x l , y )) + ( r + ε − ( d ( y p , y ) + OUGH I -CONVERGENCE IN CONE METRIC SPACES. 7 d ( x l , y p ))) = r + ε − d ( x l , y ) ∈ intP . Now since l is chosen arbitrarily from B ∁ , therefore the set { l ∈ N : r + ε − d ( x l , y ) / ∈ intP } ⊂ B and so { l ∈ N : r + ε − d ( x l , y ) / ∈ intP } ∈ I . Hence y ∈ I − LIM r x . (cid:3) Theorem 3.9. If { x n } and { y n } are two sequence in a cone metric space ( X, d ) such that for any (0 << ) ε the set { n ∈ N : d ( x n , y n ) > ε } ∈ I . Then { x n } is rough I -convergent of roughness degree r to x if and only if { y n } is rough I -convergent of same roughness degree r to x .Proof. Let { x n } be rough I -convergent of roughness degree r to x . Let (0 << ) ε be given. Thenthe set (cid:8) n ∈ N : r + ε − d ( x n , x ) / ∈ intP (cid:9) ∈ I → ( i ) . Also according to our assumption the set (cid:8) n ∈ N : d ( x n , y n ) > ε (cid:9) ∈ I → ( ii ) . Now complement of the sets in ( i ) and ( ii ) belong to F ( I ) and hence their intersection belong to F ( I ) . Let us choose an element k ∈ N in that intersection.Therefore r + ε − d ( x k , x ) ∈ intP and d ( x k , y k ) ≤ ε i.e., ε − d ( x k , y k ) ∈ P . So, ( r + ε − d ( x k , x )) + ( ε − d ( x k , y k )) = r + ε − ( d ( x k , x ) + d ( x k , y k )) ∈ intP → ( iii ) . Also for all n d ( y n , x ) ≤ d ( x n , y n ) + d ( x n , x ) . That is d ( x n , y n ) + d ( x n , x ) − d ( y n , x ) ∈ P . In particular d ( x k , y k ) + d ( x k , x ) − d ( y k , x ) ∈ P → ( iv ) . Hence from ( iii ) and ( iv ) we get ( r + ε − ( d ( x k , x ) + d ( x k , y k ))) + ( d ( x k , y k ) + d ( x k , x ) − d ( y k , x )) = r + ε − d ( y k , x ) ∈ intP . Therefore the set { n ∈ N : r + ε − d ( y k , x ) / ∈ intP } ∈ I , which implies that { y n } is rough I -convergent of roughnessdegree r to x .Converse part can be proved by interchanging the role of { x n } and { y n } . (cid:3) Theorem 3.10.
Let C be the set of all I -cluster points of a sequence { x n } . Also let (0 << ) r ∈ E besuch that for any x ∈ X and for each c ∈ C either d ( x, c ) ≤ r or r << d ( x, c ) . Then I − LIM r x ⊂ T c ∈C B r ( c ) ⊂ n y ∈ X : C ⊂ B r ( y ) o .Proof. From the theorem 3.7 we can say that I − LIM r x ⊂ T c ∈C B r ( c )( ⊂ B r ( c )) . To prove thepart T c ∈C B r ( c ) ⊂ n y ∈ X : C ⊂ B r ( y ) o , let us take a z ∈ T c ∈C B r ( c ) . So z ∈ B r ( c ) for each c ∈ C and therefore d ( z, c ) ≤ r for every c ∈ C . This implies that c ∈ B r ( z ) for each c ∈ C . Thuswe get C ⊂ B r ( z ) . Hence T c ∈C B r ( c ) ⊂ n y ∈ X : C ⊂ B r ( y ) o . Hence the results follows. (cid:3) Definition 3.4.
A sequence { x n } in a cone metric space ( X, d ) is said to be rough I ∗ -convergent ofroughness degree r to x if there exists a set M = { m < m < · · · < m k < · · · } ∈ F ( I ) such thatthe subsequence { x n } n ∈ M is rough convergent of roughness degree r to x for some (0 << ) r ∈ E or r = 0 . That is for any ε with (0 << ) ε there exists a k ∈ N such that d ( x m p , x ) << r + ε for all p ≥ k . Here x is called the rough I ∗ -limit of the sequence { x n } .We denote this by x n r − I ∗ −−−→ x . Note 2.
For r = 0 we have the ordinary I ∗ -convergence of sequences in a cone metric space. Clearlythe rough I ∗ -limit of a sequence in general not unique. We shall denote the set of all rough I ∗ -limitsof a sequence { x n } by I ∗ − LIM r = n x ∈ X : x n r − I ∗ −−−→ x o of roughness degree r . Theorem 3.11.
If a sequence x = { x n } is rough I ∗ -convergent of roughness degree r to x then it isalso rough I -convergent of same roughness degree r to x .Proof. Let us assume that the sequence { x n } is rough I ∗ -convergent of roughness degree r to x .Therefore by the definition, there exists a set M = { m < m < · · · < m k < · · · } ∈ F ( I ) suchthat { x n } n ∈ M is rough convergent of roughness degree r to x . That is for any (0 << ) ε there exists a p ∈ N such that d ( x m k , x ) << r + ε for all k ≥ p . Now the set { n ∈ N : r + ε − d ( x n , x ) / ∈ intP } ⊂ N \ M ∪ { m , m , · · · , m p − } . As N \ M ∪ { m , m , · · · , m p − } ∈ I therefore the set AMAR KUMAR BANERJEE AND ANIRBAN PAUL { n ∈ N : r + ε − d ( x n , x ) / ∈ intP } ∈ I . Hence the sequence { x n } is rough I -convergent of rough-ness degree r to x . This proves our theorem. (cid:3) It may happen that a sequence { x n } in a cone metric space ( X, d ) is rough I -convergent of rough-ness degree r to x ∈ X without being rough I ∗ -convergent of same roughness degree r to x . Follow-ing example is such one in support of our claim. Example 3.2.
Let N = ∞ [ j =1 D j be a decomposition of N such that D j = (cid:8) j − (2 s −
1) : s = 1 , , · · · (cid:9) . Then each D j is infinite and D i ∩ D j = φ for i = j . Put I be the class of all A ⊂ N such that A intersects with only a finite numbers of D j ’s. Then it is easy tosee that I is an admissible ideal in N . Let X = R , E = R and P = { ( x, y ) : x, y ≥ } ⊂ R be acone. Define d : X × X → E be such that d ( x, y ) = ( | x − y | , | x − y | ) . Then ( X, d ) is a cone metricspace. Define a sequence x = { x n } in ( X, d ) such that x n = j if n ∈ D j . Let r = ( r , r ) ∈ intP and min( r , r ) = r ∗ . Let (0 << ) ε = ( ε , ε ) be arbitrary and min( ε , ε ) = ε ∗ . Then byArchimedean property of R , there exists a l ∈ N such that ε ∗ > l . Then it is easy to see that [ − r ∗ , r ∗ ] ⊂ I − LIM r x , as { n ∈ N : r + ε − d ( x n , x ∗ ) / ∈ intP } ⊂ D ∪ D ∪ · · · ∪ D l for any x ∗ ∈ [ − r ∗ , r ∗ ] . Therefore the sequence defined above is rough I -convergent.If possible let this sequence be rough I ∗ -convergent of roughness degree r = ( r , r ) to x ∗ = r ∗ ,where r ∗ = min( r , r ) . Then there exists a set M = { m < m < · · · < m k < · · · } ∈ F ( I ) suchthat { x m k } is rough convergent of roughness degree r . Now obviously N \ M = H ∈ I . So thereexists a p ∈ N such that H ⊂ D ∪ D ∪· · ·∪ D p and D p +1 ⊂ N \ H = M . Hence we have x m k = p +1 for infinitely many k ’s. Let us take (0 << ) r ∈ E in such a way that r ∗ = p +1) . Let (0 << ) ε =( ε , ε ) ∈ E be chosen such that ( ε , ε ) = ( p +1) , p +1) ) . Then r + ε − d ( x m k , x ∗ ) / ∈ intP forinfinitely many k ’s. Therefore the sequence is not rough I ∗ -convergent to x ∗ = r ∗ for this chosen r although the sequence is rough I -convergent to x ∗ = r ∗ for the same r . Remark 1.
However the sequence in the example 3.2 may be rough I ∗ -convergent of different rough-ness degree to different limit with respect to the same ideal defined in that example. For, if we chosesay r = (5 , and l = p +1 , then d ( x m k , l ) << r + ε for all k ’s. Therefore { x n } n ∈ M is roughconvergent and so { x n } is rough I ∗ -convergent.Rough I -limit and rough I ∗ -limit are same for a sequence { x n } in a cone metric space if the idealhas the property (AP). To prove this we need the following lemma. Lemma 3.12. [22]
Let { A n } n ∈ N be a countable family of subsets of N such that A n ∈ F ( I ) for each n , where F ( I ) is the filter associated with an admissible ideal I with the property (AP). Then thereexists a set B ⊂ N such that B ∈ F ( I ) and the sets B \ A n is finite for all n . Theorem 3.13.
If an ideal I has the property (AP) then a sequence x = { x n } in a cone metric space ( X, d ) which is rough I -convergent of roughness degree r to x ∗ ∈ X is also rough I ∗ -convergent ofsame roughness degree r to x ∗ .Proof. Let I be a ideal in N which satisfy the property (AP). Let the sequence x = { x n } be rough I -convergent of the roughness degree r to x ∗ . Then for any (0 << ) ε the set { n ∈ N : r + ε − d ( x n , x ∗ ) / ∈ intP } ∈ I . Therefore the set { n ∈ N : r + ε − d ( x n , x ∗ ) ∈ intP } ∈ F ( I ) . Let (0 << ) l ∈ E . Now define A i = (cid:8) n ∈ N : d ( x n , x ∗ ) << r + li (cid:9) , where i = 1 , , · · · . It isclear that A i ∈ F ( I ) for all i = 1 , , · · · . Since I has the property (AP), therefore there exists a set B ⊂ N such that B ∈ F ( I ) and B \ A i is finite for i = 1 , , · · · . Now let (0 << ) ε ∈ E , then by OUGH I -CONVERGENCE IN CONE METRIC SPACES. 9 lemma 2.2 there exists j ∈ N such that lj << ε . As B \ A j is finite, so there exists a k = k ( j ) ∈ N such that n ∈ B ∩ A j for all n ≥ k . Therefore d ( x n , x ∗ ) << r + lj << r + ε for all n ∈ B and n ≥ k . Thus the sequence { x n } n ∈ B is rough convergent of roughness degree r to x ∗ . Hence thesequence { x n } is rough I ∗ -convergent of roughness degree r to x ∗ . Hence the theorem. (cid:3) Corollary 3.
Let { x n } be a sequence in a cone metric space ( X, d ) . Then rough I -limit set of { x n } equals with rough I ∗ -limit set of { x n } of roughness degree r if and only if I has the property (AP).Proof. In view of theorem 3.11 and theorem 3.13 the result follows. (cid:3)
Theorem 3.14. If y = { x n k } be a subsequence of the sequence x = { x n } , then I − LIM r x ⊂ I − LIM r y .Proof. If possible let x ∗ ∈ I − LIM r x . Then for any (0 << ) ε ∈ E the set { n ∈ N : r + ε − d ( x n , x ∗ ) / ∈ intP } ∈ I . Now for the subsequence y = { x n k } , as { n k ∈ N : r + ε − d ( x n k , x ∗ ) / ∈ intP } ⊂ { n ∈ N : r + ε − d ( x n , x ∗ ) / ∈ intP } and { n ∈ N : r + ε − d ( x n , x ∗ ) / ∈ intP } ∈ I , so { n k ∈ N : r + ε − d ( x n k , x ∗ ) / ∈ intP } ∈ I . Hence the set L = { n k ∈ N : r + ε − d ( x n k , x ∗ ) ∈ intP } ∈ F ( I ) . Let us write L = { m < m < m · · · } . Then { x m k } m k ∈ L is a subsequence of y . So for the sequence { x m k } m k ∈ L we have d ( x m k , x ∗ ) << r + ε and hence { x m k } m k ∈ L is roughconvergent of roughness degree r to x ∗ . Therefore the sequence y = { x n k } is rough I ∗ -convergenceof roughness degree r to x ∗ . So y = { x n k } is also rough I -convergent of roughness degree r to x ∗ .Hence x ∗ ∈ I − LIM r y . (cid:3) We now recall following two lemmas from [7].
Lemma 3.15. [7]
Let ( X, d ) be a cone metric space with normal cone P and normal constant K .Then for any ε ( > ∈ R , we can choose c ∈ E with c ∈ intP and K || c || < ε . Lemma 3.16. [7]
Let ( X, d ) be a cone metric space with normal cone P and normal constant K .Then for each c ∈ E with << c , there is a δ > such that || x || < δ implies c − x ∈ intP . Theorem 3.17.
Let ( X, d ) be a cone metric space with normal cone P and normal constant K . Let I be an ideal in N which has the property (AP). Then a sequence x = { x n } in ( X, d ) is rough I -convergent of roughness degree r to x if and only if { d ( x n , x ) − r } is I -convergent to , providedthat { d ( x n , x ) − r } is a sequence in P .Proof. Firstly let us assume that x = { x n } is rough I -convergent of roughness degree r to x . Since I has the property (AP) by theorem 3.13, the sequence x = { x n } is also rough I ∗ -convergent ofroughness degree r to x . So there exists a set M = { m < m < · · · < m k < · · · } ∈ F ( I ) such thatthe subsequence { x n } n ∈ M is rough convergent of roughness degree r to x . Let (0 < ) ε ∈ R be given.Then according to lemma 3.15 we have an element (0 << ) c ∈ E with K k c k < ε . Now since thesequence { x n } n ∈ M is rough convergent of roughness degree r to X , so for this (0 << ) c we have anelement l ∈ N such that d ( x m k , x ) << r + c for all k ≥ l . That is d ( x m k , x ) − r << c for all k ≥ l .Now as P is normal cone with normal constant K , therefore we have k d ( x m k , x ) − r k ≤ K k c k < ε for all k ≥ l . Since this is true for any arbitrary (0 < ) ε ∈ R , by lemma 2.1 we see that the sequence { d ( x n , x ) − r } n ∈ M converges to . This implies that the sequence { d ( x n , x ) − r } is I ∗ -convergentto and hence it is also I -convergent to .Conversely suppose that the sequence { d ( x n , x ) − r } is I -convergent to . Since the ideal I hasthe property (AP) and any cone metric space is first countable, therefore the sequence { d ( x n , x ) − r } is also I ∗ -convergent to . Thus there exists a set M = { m < m < · · · < m k < · · · } ∈ F ( I ) such that { d ( x n , x ) − r } n ∈ M is convergent to . Let c ∈ E with << c . Then by lemma 3.16, thereexists a δ > , such that k x k < δ implies c − x ∈ intP → ( i ) . Now since { d ( x n , x ) − r } n ∈ M is convergent to , so for this δ there exists a k ∈ N such that (cid:13)(cid:13) d ( x m p , x ) − r (cid:13)(cid:13) < δ for all p ≥ k .So by ( i ) , c − ( d ( x m p , x ) − r ) ∈ intP for all p ≥ k . Thus d ( x m p , x ) << r + c for all p ≥ k .Therefore { n ∈ N : r + c − d ( x n , x ) / ∈ intP } ⊂ N \ M ∪ { m < m < · · · < m k − } and hence { n ∈ N : r + c − d ( x n , x ) / ∈ intP } ∈ I . Therefore { x n } is rough I -convergent of roughness degree r to x . (cid:3) Theorem 3.18.
Let ( X, d ) be a cone metric space with normal cone P and normal constant K . Alsolet { x n } and { y n } be two sequence in ( X, d ) rough I -convergent of roughness degree K +2 r to x and y respectively. Then the sequence { z n } in E is rough I -convergent to d ( x, y ) of roughness degree || r || where z n = d ( x n , y n ) for all n ∈ N .Proof. Let ε > be given and x ∈ intP . Then c = εx k x k (4 K +2) ∈ intP . Obviously k c k < ε K +2) .Now as { x n } and { y n } both are rough I -convergent of same roughness degree K +2 r to x and y re-spectively, therefore for (0 << ) c ∈ E the sets A = n n ∈ N : c + K +2 r − d ( x n , x ) / ∈ intP o and A = n n ∈ N : c + K +2 r − d ( y n , y ) / ∈ intP o both belongs to I . Therefore the set A ∁ = N \ A and A ∁ = N \ A both belongs to F ( I ) . Therefore M = A ∁ ∩ A ∁ ∈ F ( I ) . Let us choose an element m ∈ A ∁ ∩ A ∁ . So ( c + K +2 r ) − d ( x m , x ) ∈ intP → ( i ) and ( c + K +2 r ) − d ( y m , y ) ∈ intP → ( ii ) .From ( i ) and ( ii ) we get ( c + K +2 r ) − d ( x m , x ) + ( c + K +2 r ) − d ( y m , y ) = 2( c + K +2 r ) − ( d ( x m , x ) + d ( y m , y )) ∈ intP → ( iii ) .Again d ( x, y ) ≤ d ( x m , x ) + d ( x m , y ) i.e., d ( x m , x ) + d ( x m , y ) − d ( x, y ) ∈ P → ( iv ) . Alsoas d ( x m , y ) ≤ d ( x m , y m ) + d ( y m , y ) , so d ( x m , y m ) + d ( y m , y ) − d ( x m , y ) ∈ P → ( v ) . Thusfrom ( iv ) and ( v ) we get d ( x m , x ) + d ( y m , y ) + d ( x m , y m ) − d ( x, y ) ∈ P → ( vi ) . Also as d ( x, y m ) ≤ d ( x, y ) + d ( y, y m ) i.e., d ( x, y ) + d ( y, y m ) − d ( x, y m ) ∈ P and as d ( x m , y m ) ≤ d ( x m , x ) + d ( x, y m ) i.e., d ( x m , x ) + d ( x, y m ) − d ( x m , y m ) ∈ P so their sum also belongs to P .That is d ( x, y ) + d ( y, y m ) + d ( x m , x ) − d ( x m , y m ) ∈ P → ( vii ) . Now from ( iii ) and ( vii ) weget c + k +2 r ) + d ( x, y ) − d ( x m , y m ) ∈ intP → ( viii ) . Again from ( iii ) and ( vi ) we have c + K +2 r ) + d ( x m , y m ) − d ( x, y ) ∈ intP , i.e., c + K +2 r ) − (2( c + K +2 r ) + d ( x, y ) − d ( x m , y m )) ∈ intP . This implies that c + K +2 r ) + d ( x, y ) − d ( x m , y m ) << c + K +2 r ) . Alsofrom ( viii ) we have c + K +2 r )+ d ( x, y ) − d ( x m , y m ) ∈ intP . Again we have << c + K +2 r ) .Now as P is normal therefore || c + K +2 r ) + d ( x, y ) − d ( x m , y m ) || ≤ K || c + K +2 r ) || → ( ix ) .Also || d ( x, y ) − d ( x m , y m ) || = || d ( x, y ) − d ( x m , y m ) + 2( c + K +2 r ) − c + K +2 r ) || ≤|| d ( x, y ) − d ( x m , y m ) + 2( c + K +2 r ) || + 2 || ( c + K +2 r ) || → ( x ) . Thus from ( x ) using ( ix ) weget || d ( x, y ) − d ( x m , y m ) || ≤ K || ( c + K +2 r ) || + 2 || ( c + K +2 r ) || = (4 K + 2) || ( c + K +2 r ) || ≤ (4 K + 2) || c || + (4 K + 2) K +2 || r || = ε + || r || < ε + || r || → ( xi ) . Since the inequality in ( xi ) holdsfor any arbitrary m ∈ M , therefore { n ∈ N : || d ( x n , y n ) − d ( x, y ) || ≥ ε + || r ||} ⊂ N \ M . Hencethe set { n ∈ N : || d ( x n , y n ) − d ( x, y ) || ≥ ε + || r ||} ∈ I . Hence the sequence { z n } in E is rough I -convergent to d ( x, y ) ∈ E of roughness degree || r || . (cid:3) R EFERENCES [1] S. Aytar,
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