Generalized difference Cesaro sequence spaces of fuzzy real numbers defined by Orlicz functions
aa r X i v : . [ m a t h . F A ] J un GENERALIZED DIFFERENCE CES ` A RO SEQUENCE SPACES OFFUZZY REAL NUMBERS DEFINED BY ORLICZ FUNCTIONB inod C handra T ripathy ∗ and S tuti B orgohain ∗∗∗ Mathematical Sciences DivisionInstitute of Advanced Study in Science and TechnologyPaschim Boragaon; Garchuk; Guwahati-781035; Assam,
INDIA .E-mail: ∗ [email protected]; tripathybc@rediffmail.com ∗∗ Department of MathematicsIndian Institute of Technology, BombayPowai:400076, Mumbai, Maharashtra,
INDIA .Email: ∗∗ [email protected] Abstract:
In this paper, we introduced different types of generalized difference Ces` a rosequnecs spaces of fuzzy real numbers defined by Orlicz function. We study some topo-logical properties of these spaces. We obtain some inclusion relations involving thesesequence spaces. These notions generalize many notions on difference Ces` a ro sequencespaces. Keywords:
Orlicz function; Ces` a ro sequence; fuzzy set; metric space; completeness. The concept of fuzzy set theory was introduced by L.A.Zadeh in 1965. Later on sequencesof fuzzy numbers have been discussed by Esi (2006), Tripathy and Dutta (2007, 2010), The work of the authors was carried under University Grants Commision of India project No.- F . No.30 − / Definition 1.1
Kizmaz (1981) defined the difference sequence spaces ℓ ∞ (∆) , c (∆) and c (∆) of crisp sets as follows: Z (∆) = { ( x k ) ∈ w : (∆ x k ) ∈ Z } , for Z = ℓ ∞ , c and c and ∆ x = (∆ x k ) = ( x k − x k +1 ) for all k ∈ N .The above spaces are Banach spaces, normed by, k ( x k ) k ∆ = | x | + sup k ≥ | ∆ x k | . The idea of Kizmaz (1981) was apllied to introduce different types of difference se-quence spaces and study their different properties by Tripathy (2004), Tripathy and Esi(2006), Tripathy and Mahanta (2004), Tripathy and Sarma (2008b), Tripathy, Altin andEt (2008) and many others.Tripathy and Esi (2006) introduced the new type of difference sequence spaces, forfixed m ∈ N , as follows: Z (∆ m ) = { x = ( x k ) : (∆ m x k ) ∈ Z } , for Z = ℓ ∞ , c and c and ∆ m x = (∆ m x k ) = ( x k − x k + m ) for all k ∈ N .This generalizes the notion of difference sequence spaces studied by Kizmaz (1981).The above spaces are Banach spaces, normed by, k ( x k ) k ∆ m = m X r =1 | x r | + sup k ≥ | ∆ m x k | . Tripathy, Esi and Tripathy (2005) further generalized this notion and introduced thefollowing notion. For m ≥ n ≥ Z (∆ nm ) = { x = ( x k ) : (∆ nm x k ) ∈ Z } , for Z = ℓ ∞ , c and c . where (∆ nm x k ) = (∆ n − m x k − ∆ n − m x k + m ), for all k ∈ N .This generalized difference has the following binomial representation,2 nm x k = n X r =0 ( − r nr ! x k + rm . (1) Definition 1.2
Ng and Lee (1978) defined the Ces` a ro sequence spaces X p of non-absolutetype as follows: x = ( x k ) ∈ X p if and only if σ ( x ) ∈ ℓ p , ≤ p < ∞ , where σ ( x ) = n n X k =1 x k ! ∞ n =1 . Orhan (1983) defined the Ces` a ro difference sequence spaces X p (Λ), for 1 ≤ p < ∞ and studied their different properties and proved some inclusion results. He also obtainedthe duals of these sequence spaces.Mursaleen, Gaur and Saifi (1996) defined the second difference Ces` a ro sequence spaces X p (∆ ), for 1 ≤ p < ∞ and studied their different topological properties and proved someinclusion results. They also calculated their dual sequence spaces.Later on, Tripathy, Esi and Tripathy (2005) further introduced new types of dif-ference Ces` a ro sequence spaces as C ∞ (∆ nm ) , O ∞ (∆ nm ) , C p (∆ nm ) , O p (∆ nm ) , and ℓ p (∆ nm ), for1 ≤ p < ∞ .For m = 1, the spaces C p (∆ n ) and C ∞ (∆ n ) are studied by Et (1996-1997). Definition 1.3
An Orlicz function is a function M : [0 , ∞ ) → [0 , ∞ ), which is continu-ous, non-decreasing and convex with M (0) = 0 , M ( x ) >
0, for x > M ( x ) → ∞ , as x → ∞ .An Orlicz function M is said to satisfy ∆ -condition for all values of x , if there existsa constant K > , M ( Lx ) ≤ KLM ( x ), for all x > L > M ( x + y ) ≤ M ( x ) + M ( y ), then this function is called as modulus function. Remark 1.1
An Orlicz function satisfies the inequality M ( λx ) ≤ λM ( x ), for all λ with0 < λ <
1. 3hroughout the article w F , ℓ F , ℓ F ∞ represent the classes of all ; absolutely summable and bounded sequences of fuzzy real numbers respectively. A fuzzy real number X is a fuzzy set on R i.e. a mapping X : R → I (=[0 , t with its grade of membership X ( t ). Definition 2.2
A fuzzy real number X is called convex if X ( t ) ≥ X ( s ) ∧ X ( r ) =min( X ( s ) , X ( r )), where s < t < r . Definition 2.3
If there exists t ∈ R such that X ( t ) = 1, then the fuzzy real number X is called normal . Definition 2.4
A fuzzy real number X is said to be upper semi continuous if for each ε > , X − ([0 , a + ε )), for all a ∈ I , is open in the usual topology of R .The class of all upper semi-continuous, normal, convex fuzzy real numbers is denotedby R ( I ). Definition 2.5
For X ∈ R ( I ), the α -level set X α , for 0 < α ≤ X α = { t ∈ R : X ( t ) ≥ α } . The 0-level i.e. X is the closure of strong 0-cut, i.e. X = cl { t ∈ R : X ( t ) > } . Definition 2.6
The absolute value of X ∈ R ( I ) i.e. | X | is defined by, | X | ( t ) = max { X ( t ) , X ( − t ) } , for t ≥
00 otherwise
Definition 2.7
For r ∈ R, r ∈ R ( I ) is defined as, r ( t ) = t = r t = r Definition 2.8
The additive identity and multiplicative identity of R ( I ) are denoted by0 and 1 respectively. The zero sequence of fuzzy real numbers is denoted by θ .4 efinition 2.9 Let D be the set of all closed bounded intervals X = [ X L , X R ].Define d : D × D → R by d ( X, Y ) = max {| X L − Y L | , | X R − Y R |} . Then clearly ( D, d )is a complete metric space.Define by d : R ( I ) × R ( I ) → R ( I ) by d ( X, Y ) = sup <α ≤ d ( X α , Y α ), for X, Y ∈ R ( I ).Then it is well known that ( R ( I ) , d ) is a complete metric space. Definition 2.10
A sequence X = ( X k ) of fuzzy real numbers is said to converge to thefuzzy number X , if for every ε >
0, there exists k ∈ N such that d ( X k , X ) < ε , for all k ≥ k . Definition 2.11
A sequence space E is said to be solid if ( Y n ) ∈ E , whenever ( X n ) ∈ E and | Y n | ≤ | X n | , for all n ∈ N . Definition 2.12
Let X = ( X n ) be a sequence, then S ( X ) denotes the set of all per-mutations of the elements of ( X n ) i.e. S ( X ) = { ( X π ( n ) ) : π is a permutation of N } . Asequence space E is said to be symmetric if S ( X ) ⊂ E for all X ∈ E . Definition 2.13
A sequence space E is said to be convergence-free if ( Y n ) ∈ E whenever( X n ) ∈ E and X n = 0 implies Y n = 0. Definition 2.14
A sequence space E is said to be monotone if E contains the canonicalpre-images of all its step spaces. Lemma 2.1
A sequence space E is solid implies that E is monotone. Definition 2.15
Lindenstrauss and Tzafriri (1971) used the notion of Orlicz function andintroduced the sequence space: ℓ M = ( x ∈ w : ∞ X k =1 M | x k | ρ ! , for some ρ > ) The space ℓ M with the norm, k x k = inf ( ρ > ∞ X k =1 M | x k | ρ ! ≤ ) ℓ M closely relatedto the space ℓ p which is an Orlicz sequence space with M ( x ) = x p , for 1 ≤ p < ∞ .Later on different classes of Orlicz sequence spaces were introduced and studied byTripathy and Mahanta (2004), Et, Altin, Choudhary and Tripathy (2006), Tripathy, Altinand Et (2008), Tripathy and Sarma (2008a,2009,2011)many others.Let m, n ≥ ≤ p < ∞ . In this article we introduced thefollowing new types of generalized difference Ces` a ro sequence spaces of fuzzy real numbers: C Fp ( M, ∆ nm ) = ( ( X k ) : ∞ X i =1 i i X k =1 M d (∆ nm X k , ρ !!! p < ∞ , for some ρ > ) .C F ∞ ( M, ∆ nm ) = ( ( X k ) : sup i i i X k =1 M d (∆ nm X k , ρ !! < ∞ , for some ρ > ) .ℓ Fp ( M, ∆ nm ) = ( ( X k ) : ∞ X k =1 M d (∆ nm X k , ρ !! p < ∞ , for some ρ > ) .O Fp ( M, ∆ nm ) = ( ( X k ) : ∞ X i =1 i i X k =1 M d (∆ nm X k , ρ !!! p < ∞ , for some ρ > ) .O F ∞ ( M, ∆ nm ) = ( ( X k ) : sup i i i X k =1 M d (∆ nm X k , ρ ! < ∞ , for some ρ > ) . Lemma 2.2
Let ≤ p < ∞ . Then ,( i ) The space C Fp ( M ) is a complete metric space with the metric , η ( X, Y ) = inf ρ > ∞ X i =1 i i X k =1 M d ( X k , Y k ) ρ !! p ! p ≤ . ( ii ) The space C F ∞ ( M ) is a complete metric space with respect to the metric , η ( X, Y ) = inf ( ρ > i i i X k =1 M d ( X k , Y k ) ρ ! ≤ ) . iii ) The space ℓ Fp ( M ) is a complete metric space with the metric , η ( X, Y ) = inf ρ > ∞ X i =1 M d ( X k , Y k ) ρ !! p ! p ≤ . ( iv ) The space O Fp ( M ) is a complete metric space with the metric , η ( X, Y ) = inf ρ > ∞ X i =1 i i X k =1 M d ( X k , Y k ) ρ !! p ! p ≤ . ( v ) The space O F ∞ ( M ) is a complete metric space with respect to the metric , η ( X, Y ) = inf ( ρ > i i i X k =1 M d ( X k , Y k ) ρ ! ≤ ) . Proof of lemma 2.2(i)
Let ( X ( u ) ) be a Cauchy sequence in C Fp ( M ) such that X ( u ) =( X ( u ) n ) ∞ n =1 , for i ∈ N .Let ε > x >
0, choose r > M (cid:16) rx (cid:17) ≥
1. Thenthere exits a positive integer n = n ( ε ) such that η ( X ( u ) , X ( v ) ) < εrx , for all u, v ≥ n . By the definition of η , we get:inf ρ > ∞ X i =1 i i X k =1 M d ( X ( u ) k , X ( v ) k ) ρ p p ≤ < ε, for all u, v ≥ n . (2)Which implies that, M d ( X ( u ) k , X ( v ) k ) ρ ≤ , for all u, v ≥ n . (3) ⇒ M (cid:18) d ( X ( u ) k ,X ( v ) k ) η ( X ( i ) ,X ( j ) ) (cid:19) ≤ ≤ M (cid:16) rx (cid:17) , for all u, v ≥ n . Since M is continuous, we get, d ( X ( u ) k , X ( v ) k ) ≤ rx .η ( X ( u ) , X ( v ) ) , for all u, v ≥ n . d ( X ( u ) k , X ( v ) k ) < rx . εrx = ε , for all u, v ≥ n . ⇒ d ( X ( u ) k , X ( v ) k ) < ε , for all u, v ≥ n . Which implies that ( X ( u ) k ) is a Cauchy sequence in R ( I ) and so it is convergent in R ( I ) by the completeness property of R ( I ).Also, lim u X ( u ) k = X k , for each k ∈ N .Now, taking v → ∞ and fixing u and using the continuity of M , it follows from (3), M d ( X ( u ) k , X k ) ρ ≤ , for some ρ > . Now on taking the infimum of such ρ s, we get,inf ρ > ∞ X i =1 i i X k =1 M d ( X ( u ) k , X k ) ρ p p ≤ < ε, for all u ≥ n (by (2)) . Which implies that, η ( X ( u ) , X ) < ε, for all u ≥ n . i.e. lim u X ( u ) = X .Now, we show that X ∈ C Fp ( M ).We know that, d ( X k , ≤ d ( X ( u ) k , X k ) + d ( X ( u ) k , M is continuous and non-decreasing, so we get, ∞ X i =1 i i X k =1 M d ( X k , ρ !! p ≤ ∞ X i =1 i i X k =1 M d ( X ( u ) k , X k ) ρ p + ∞ X i =1 i i X k =1 M d ( X ( u ) k , ρ p ∞ . (finite)Which implies that X ∈ C Fp ( M ).Hence C Fp ( M ) is a complete metric space.This completes the proof. In this section, we prove the results relating to the introduced sequence spaces. The proofof the following result is a routine verification.
Proposition 3.1
The classes of sequences C F ∞ ( M, ∆ nm ) , O F ∞ ( M, ∆ nm ) , C Fp ( M, ∆ nm ) , O Fp ( M, ∆ nm ) and ℓ Fp ( M, ∆ nm ) , for ≤ p < ∞ , are metric spaces with respect to the metric, f ( X, Y ) = mn X k =1 d ( X k ,
0) + η (∆ nm X k , ∆ nm Y k ) where Z = C F ∞ , C Fp , O F ∞ , O Fp , ℓ Fp . Theorem 3.1
Let Z ( M ) be a complete metric space with respect to the metric η , thespace Z ( M, ∆ nm ) is a complete metric space with respect to the metric, f ( X, Y ) = mn X k =1 d ( X k ,
0) + η (∆ nm X k , ∆ nm Y k ) where Z = C F ∞ , C Fp , O F ∞ , O Fp , ℓ Fp .Proof Let ( X ( u ) ) be a Cauchy sequence in Z ( M, ∆ nm ) such that X ( u ) = ( X ( u ) n ) ∞ n =1 .We have for ε >
0, there exists a positive integer n = n ( ε ) such that, f ( X ( u ) , X ( v ) ) < ε, for all u, v ≥ n . By the definition of f , we get: mn X r =1 d ( X ( u ) r , X ( v ) r ) + η (∆ nm X ( u ) k , ∆ nm X ( v ) k ) < ε, for all u, v ≥ n . (4)Which implies that, 9 n X r =1 d ( X ( u ) r , X ( v ) r ) < ε, for all u, v ≥ n . ⇒ d ( X ( u ) r , X ( v ) r ) < ε, for all u, v ≥ n , r = 1 , , ...mn. Hence ( X ( u ) r ) is a Cauchy sequence in R ( I ), so it is convergent in R ( I ), by the com-pleteness property of R ( I ), for r = 1 , , .....mn .Let, lim u →∞ X ( u ) r = X r , for r = 1 , , ....mn (5)Next we have, η (∆ nm X ( u ) k , ∆ nm X ( v ) k ) < ε, for all u, v ≥ n (6)Which implies that (∆ nm X ( u ) k ) is a Cauchy sequence in Z ( M ), since M is a continuousfunction and so it is convergent in Z ( M ) by the completeness property of Z ( M ).Let, lim u ∆ nm X ( u ) k = Y k (say), in Z ( M ), for each k ∈ N .We have to prove that,lim u X ( u ) = X and X ∈ Z ( M, ∆ nm ) . For k = 1, we have, from (1) and (5),lim u X ( u ) mn +1 = X mn +1 , for m ≥ , n ≥ . Proceeding in this way of induction, we get,lim u X ( u ) k = X k , for each k ∈ N. Also, lim u ∆ nm X ( u ) k = ∆ nm X k , for each k ∈ N. Now, taking v → ∞ and fixing u it follows from (4), mn X r =1 d ( X ( u ) r , X r ) + η (∆ nm X ( u ) k , ∆ nm X k ) < ε, for all u, v ≥ n . Which implies that, f ( X ( u ) , X ) < ε, for all u ≥ n . u X ( u ) = X. Now, it is to show that X ∈ Z ( M, ∆ nm ) . We know that, f (∆ nm X k , ≤ f (∆ nm X ( i ) k , ∆ nm X k ) + f (∆ nm X ( i ) k , < ∞ . Which implies that X ∈ Z ( M, ∆ nm ) . Hence Z ( M, ∆ nm ) is a complete metric space.This completes the proof of the theorem.The proof of the following results is a consequence of the above result and lemma. Proposition 3.2
Let ≤ p < ∞ .Then ,( i ) The space C Fp ( M, ∆ nm ) is a complete metric space with the metric , f ( X, Y ) = mn X r =1 d ( X r , Y r ) + inf ρ > ∞ X i =1 i i X k =1 M d (∆ nm X k , ∆ nm Y k ) ρ !! p ! p ≤ . ( ii ) The space C F ∞ ( M, ∆ nm ) is a complete metric space with respect to the metric , f ( X, Y ) = mn X r =1 d ( X r , Y r ) + inf ( ρ > i i M d (∆ nm X k , ∆ nm Y k ) ρ ! ≤ ) . ( iii ) The space ℓ Fp ( M, ∆ nm ) is a complete metric space with respect to the metric , f ( X, Y ) = mn X r =1 d ( X r , Y r ) + inf ρ > ∞ X i =1 M d (∆ nm X k , ∆ nm Y k ) ρ !! p ! p ≤ . ( iv ) The space O Fp ( M, ∆ nm ) is a complete metric space with the metric ,11 ( X, Y ) = mn X r =1 d ( X r , Y r ) + inf ρ > ∞ X i =1 i i X k =1 M d (∆ nm X k , ∆ nm Y k ) ρ !! p ! p ≤ . ( v ) O F ∞ ( M, ∆ nm ) is a complete metric space with respect to the metric , f ( X, Y ) = mn X r =1 d ( X r , Y r ) + inf ( ρ > i i i X k =1 M d (∆ nm X k , ∆ nm Y k ) ρ ! ≤ ) . Theorem 3.2
The classes of spaces Z ( M, ∆ nm ) , where Z = C F ∞ , O F ∞ , C Fp , O Fp and ℓ Fp , for ≤ p < ∞ , are not monotone and as such are not solid for m, n ≥ .Proof. Let us consider the proof for C Fp ( M, ∆ nm ). The proof follows from the followingexample: Example 3.1
Let X k = k , for all k ∈ N .Let m = 3 and n = 2. Let M ( x ) = | x | , for all x ∈ [0 , ∞ ).Then, we have, d (∆ X k ,
0) = 0, for all k ∈ N .Hence, we get, for 1 ≤ p < ∞ , ∞ X i =1 i i X k =1 M d (∆ X k , ρ !! p ! p < ∞ , for some ρ > X k ) ∈ C Fp ( M, ∆ ).Let J = { k : k is even } ⊆ N . Let ( Y k ) be the canonical pre-image of ( X k ) J for thesubsequence J of N . Then, Y k = ( X k , for k odd , , for k even . But ( Y k ) / ∈ C Fp ( M, ∆ ).Hence the spaces are not monotone as such not solid.12his completes the proof. Remark 3.1
For m = 0 or n = 0, the spaces C Fp ( M ) and C F ∞ ( M ) are neither solidnor monotone, where as the spaces ℓ Fp ( M ) , O Fp ( M ) and O F ∞ ( M ) are solid and hence aremonotone. Theorem 3.3
The classes of spaces Z ( M, ∆ nm ) , where Z = C F ∞ , O F ∞ , C Fp , O Fp and ℓ Fp , for ≤ p < ∞ , are not symmetric, for m, n ≥ Proof
Let us consider the proof for C F ∞ ( M, ∆ nm ). The proof follows from the followingexample: Example 3.2
Let X k = 1, for all k ∈ N .Let m = 4 and n = 1. Let M ( x ) = | x | , for all x ∈ [0 , ∞ ).Then, we have, d (∆ X k ,
0) = 0, for all k ∈ N .Hence, we get, sup i i i X k =1 M d (∆ X k , ρ !! < ∞ , for some ρ > . Which implies that, ( X k ) ∈ C F ∞ ( M, ∆ ).Consider the rearranged sequence ( Y k ) of ( X k ) such that ( Y k ) = ( X , X , X , X , X , X , X , X , X , ... )such that d (∆ Y k , ≈ k − ( k − ≈ k , for all k ∈ N .Which shows,sup i i i X k =1 M d (∆ Y k , ρ !! = ∞ , for some fixed ρ > . Hence, ( Y k ) / ∈ C F ∞ ( M, ∆ ) . It follows that the spaces are not symmetric.This completes the proof. 13 heorem 3.4 The classes of spaces Z ( M, ∆ nm ) , where Z = C F ∞ , O F ∞ , C Fp , O Fp and ℓ Fp , for ≤ p < ∞ , are not convergence-free, for m, n ≥ Proof
Let us consider the proof for C F ∞ ( M, ∆ nm ). The proof follows from the followingexample: Example 3.3
Let m = 3 and n = 1. Let M ( x ) = x , for all x ∈ [0 , ∞ ).Consider the sequence ( X k ) defined as follows: X k ( t ) = k t, for t ∈ [ − k , , − k t, for t ∈ [0 , k ] , , otherwiseThen, ∆ X k ( t ) = k ( k +3) k +6 k +9 t, for t ∈ h − k +6 k +9 k ( k +3) , i , − k ( k +3) k +6 k +9 t, for t ∈ h , k +6 k +9 k ( k +3) i , , otherwiseSuch that, d (∆ X k ,
0) = k +6 k +9 k ( k +3) = k + k +3) .We have,sup i i i X k =1 M d (∆ X k , ρ !! < ∞ , for some fixed ρ > . Thus ( X k ) ∈ C F ∞ ( M, ∆ ).Now, let us take another sequence ( Y k ) such that, Y k ( t ) = tk , for t ∈ [ − k , , − tk , for t ∈ [0 , k ] , , otherwiseSo that, ∆ Y k ( t ) = t k +6 k +9 , for t ∈ [ − (2 k + 6 k + 9) , , − t k +6 k +9 , for t ∈ [0 , (2 k + 6 k + 9)] , , otherwiseBut, d (∆ Y k ,
0) = (2 k + 6 k + 9) , for all k ∈ N .14hich implies that,sup i i i X k =1 M d (∆ X k , ρ !! = ∞ , for some fixed ρ > . Thus, ( Y k ) / ∈ C F ∞ ( M, ∆ ).Hence C F ∞ ( M, ∆ nm ) is not convergence-free, in general.This completes the proof. Theorem 3.5 ( a ) ℓ Fp ( M, ∆ nm ) ⊂ O Fp ( M, ∆ nm ) ⊂ C Fp ( M, ∆ nm ) and the inclusions are strict .( b ) Z ( M, ∆ n − m ) ⊂ Z ( M, ∆ nm ) (in general Z ( M, ∆ im ) ⊂ Z ( M, ∆ nm ) , for i = 1 , , , ...n − ,for Z = C F ∞ , O F ∞ , C Fp , O Fp and ℓ Fp , for ≤ p < ∞ .( c ) O F ∞ ( M, ∆ nm ) ⊂ C F ∞ ( M, ∆ nm ) and the inclusion is strict . Proof ( b ) Let ( X k ) ∈ C F ∞ ( M, ∆ n − m ) . Then we have,sup i i i X k =1 M d (∆ n − m X k , ρ !! < ∞ , for some ρ > . Now we have,sup i i i X k =1 M d (∆ nm X k , ρ !! = sup i i i X k =1 M d (∆ n − m X k − ∆ n − m X k +1 , ρ !! ≤ sup i i i X k =1 M d (∆ n − m X k , ρ !!! + sup i i i X k =1 M d (∆ n − m X k +1 , ρ !!! .< ∞ .Proceeding in this way, we have, Z ( M, ∆ im ) ⊂ Z ( M, ∆ nm ), for 0 ≤ i < n , for Z = C F ∞ , O F ∞ , C Fp , O Fp and ℓ Fp , for1 ≤ p < ∞ .This completes the proof. 15 heorem 3.6 ( a ) If ≤ p < q , then ,( i ) C Fp ( M, ∆ nm ) ⊂ C Fq ( M, ∆ nm )( ii ) ℓ Fp ( M, ∆ nm ) ⊂ ℓ Fq ( M, ∆ nm )( b ) C Fp ( M ) ⊂ C Fp ( M, ∆ nm ) , for all m ≥ and n ≥ . Theorem 3.7
Let
M, M and M be Orlicz functions satisfying ∆ - condition. Then, for Z = C F ∞ , O F ∞ , C Fp , O Fp and ℓ Fp , for ≤ p < ∞ ,( i ) Z ( M , ∆ nm ) ⊆ Z ( M ◦ M , ∆ nm )( ii ) Z ( M , ∆ nm ) ∩ Z ( M , ∆ nm ) ⊆ Z ( M + M , ∆ nm ). Proof ( i ) Let ( X k ) ∈ Z ( M , ∆ nm ). For ε >
0, there exists η > ε = M ( η ).Then, M (cid:18) d (∆ nm X k ,L ) ρ (cid:19) < η , for some ρ > , L ∈ R ( I ).Let Y k = M (cid:18) d (∆ nm X k ,L ) ρ (cid:19) , for some ρ > , L ∈ R ( I ).Since M is continuous and non-decreasing, we get, M ( Y k ) = M M d (∆ nm X k , L ) ρ !! < M ( η ) = ε, for some ρ > . Which implies that, ( X k ) ∈ Z ( M ◦ M , ∆ nm ).This completes the proof.( ii ) Let ( X k ) ∈ Z ( M , ∆ nm ) ∩ Z ( M , ∆ nm ).Then, M (cid:18) d (∆ nm X k ,L ) ρ (cid:19) < ε , for some ρ > , L ∈ R ( I ).and M (cid:18) d (∆ nm X k ,L ) ρ (cid:19) < ε , for some ρ > , L ∈ R ( I ).The rest of the proof follows from the equality,16 M + M ) d (∆ nm X k , L ) ρ ! = M d (∆ nm X k , L ) ρ ! + M d (∆ nm X k , L ) ρ ! < ε + ε = 2 ε, for some ρ > . Which implies that ( X k ) ∈ Z ( M + M , ∆ nm ).This completes the proof. References
Altin Y, Et M, Tripathy BC (2004) The sequence space | N p | ( M, r, q, s ) on seminormedspaces. Applied Math & Computation 154: 423-430.Esi A (2006) On some new paranormed sequence spaces of fuzzy numbers defined by Orliczfunctions and statistical convergence. Mathematical Modelling and Analysis 11(4):379-388.Et M (1996-1997) On some generalized Ces` a ro difference sequence spaces. Istanbul Univfen fak Mat Dergisi 55-56: 221-229.Et M, Altin Y, Choudhary B, Tripathy BC (2006) Some classes of sequences defined bysequences of Orlicz functions. Math Ineq Appl 9(2):335- 342.Kizmaz H (1981) On certain sequence spaces. Canad Math Bull 24(2) :169-176.Lindenstrauss J, Tzafriri L (1971) On Orlicz sequence spaces. Israel J Math 10: 379-390.Mursaleen M, Gaur, Saifi AH (1996) Some new sequence spaces their duals and matrixtransformations. Bull Cal Math Soc 88 :207-212.Ng PN, Lee PY (1978) Ces` a ro sequence spaces of non absolute type. Comment Math 20:429-433.Orhan C (1983) Ces` a ro difference sequence spaces and related matrix transformations.Comm Fac Univ Ankara ser A 32 :55-63. 17hiue JS (1970) On the Ces` a ro sequence spaces. Tamkang Journal Math 1:19-25.Tripathy BC (2004) On generalized difference paranormed statistically convergent se-quences. Indian J Pure Appl Math 35(5) :655-663.Tripathy BC, Altin Y, Et M (2008) Generalized difference sequences spaces on semi-normed spaces defined by Orlicz functions. Math. Slovaca 58(3): 315-324.Tripathy BC, Baruah A (2009), New type of difference sequence spaces of fuzzy real num-bers. Mathematical Modelling and Analysis, 14(3) : 391-397.Tripathy BC, Borgohain S (2008) The sequence space m ( M, φ, ∆ nm , p ) F . MathematicalModelling and Analysis 13(4): 577-586.Tripathy BC, Borgogain S (2011), Some classes of difference sequence spaces of fuzzy realnumbers defined by Orlicz function, Advances in Fuzzy Systems Article ID216414, 6 pages.Tripathy BC, Dutta AJ (2007) On fuzzy real-valued double sequence spaces ....Mathe-matical and Computer Modelling 46(9-10):1294-1299.Tripathy BC, Dutta AJ (2010), Bounded variation double sequence space of fuzzy realnumbers. Computers and Mathematics with Applications 59(2) : 1031-1037.Tripathy BC, Esi A (2006) A new type of difference sequence spaces. Inter Jour Sci Tech1(1) :11-14.Tripathy BC, Esi A, Tripathy BK (2005) On a new type of generalized difference Ces` a rosequence spaces. Soochow J Math 31: 333-340.Tripathy BC, Hazarika B (2011), I -convergent sequence spaces defined by Orlicz func-tions. Acta Mathematica Applicata Sinica 27(1) : 149-154.Tripathy BC, Mahanta S (2004) On a class of generalized lacunary difference sequencespaces defined by Orlicz function. Acta Math Applicata Sinica 20(2):231-238.18ripathy BC, Sarma B (2008a) Sequence spaces of fuzzy real numbers defined by Orliczfunctions. Math Slovaca 58(5): 621-628.Tripathy BC, Sarma B, (2008b)Statistically convergent difference double sequence spaces.Acta Mathematica Sinica 24(5): 737-742.Tripathy BC, Sarma B (2009) Vector valued double sequence spaces defined by Orliczfunction. Mathematica Slovaca 59(6) : 767-776.Tripathy BC, Sarma B (2011), Double sequence spaces of fuzzy numbers defined by Orliczfunctions. Acta Mathematicae Scientia 31B(1) : 134-140.Tripathy BC, Sen M, Nath S (2011), I -convergence in probabilistic nn