Relations on continuities and boundedness in intuitionistic fuzzy pseudo normed linear spaces
aa r X i v : . [ m a t h . G M ] F e b RELATIONS ON CONTINUITIES AND BOUNDEDNESSIN INTUITIONISTIC FUZZY PSEUDO NORMEDLINEAR SPACES
BIVAS DINDA, SANTANU KUMAR GHOSH, T.K. SAMANTA
Abstract.
In this study different types of intuitionistic fuzzy con-tinuities (IFCs) and intuitionistic fuzzy boundedness (IFBs) in intu-itionistic fuzzy pseudo normed linear spaces are studied. Relations(intra and inter) on intuitionistic fuzzy continuities and intuition-istic fuzzy boundedness are investigated.
Key Words:
Strongly intuitionistic fuzzy continuity, weakly intuitionistic fuzzy continu-ity, sequentially intuitionistic fuzzy continuity, strongly intuitionistic fuzzy bounded, weaklyintuitionistic fuzzy bounded, uniformly intuitionistic fuzzy bounded. Introduction
The fuzzy norm concept was originated by A. Katsaras [12, 13]. Sub-sequently, this notion swotted by multiple researchers, viz. C. Falbin[9], S.C. Cheng and J. N. Moderson [3], I. Golet [10] and many others.Chasing the conviction of Cheng-Moderson[3], Bag-Samanta[1] consid-ered another definition of fuzzy norm, it became most acceptable amongresearchers. Motivated by the work of Bag-Samanta [1, 2], S. N˘ad˘aban[14] introduce the idea of fuzzy pseudo norm.Study of intuitionistic fuzzy normed spaces attracted lots of interest inrecent years [4, 5, 6, 7, 16, 17]. In particular, Dinda et al. [4] swotted theconcept of intuitionistic fuzzy pseudo normed linear spaces and deducedthat the concept of of intuitionistic fuzzy pseudo norm is more generalset up than intuitionistic fuzzy norm.In this paper, intuitionistic fuzzy continuities and intuitionistic fuzzyboundedness of linear operator are studied in intuitionistic fuzzy pseudonormed spaces, a more general concept of intuitionistic fuzzy normedspaces. In section 3, the concept of intuitionistic fuzzy continuities andintra relation on various types of intuitionistic fuzzy continuities are emphasized. In section 4,various types of intuitionistic fuzzy bounded-ness are studied. First intra relations on different types of intuitionisticfuzzy boundedness is obtained, then the interrelations on different typesof continuities and boundedness are gone into.2.
Preliminaries
Definition 2.1. [15] Let X be a linear space. A function k · k : X → R is said to be a pseudo norm on X if the following conditions hold:( P. k x k ≥ , ∀ x ∈ X ;( P. k x k = 0 if and only if x = θ, ∀ x ∈ X ;( P. k c x k ≤ k x k , ∀ x ∈ X, ∀ c ∈ K with | c | ≤ P. k x + y k ≤ k x k + k y k , ∀ x, y ∈ X . Definition 2.2. [4] Let X be linear space over the field K (= R / C ). Aintuitionistic fuzzy subset ( µ, ν ) of ( X × R X × R ) is said to be an IFPNon X if ∀ x, y ∈ X (IFP.1) µ ( x , t ) + ν ( x , t ) ≤ . (IFP.2) ∀ t ∈ R with t ≤ , µ ( x , t ) = 0 ; (IFP.3) ∀ t ∈ R + , µ ( x , t ) = 1 if and only if x = θ ; (IFP.4) ∀ t ∈ R + , µ ( cx , t ) ≥ µ ( x , t ) if | c | ≤ , ∀ c ∈ K ; (IFP.5) µ ( x + y , s + t ) ≥ min { µ ( x , s ) , µ ( y , t ) } , ∀ s, t ∈ R + ; (IFP.6) lim t → ∞ µ ( x , t ) = 1 . (IFP.7) if there exists α ∈ (0 ,
1) such that µ ( x , t ) > α , ∀ t ∈ R + then x = θ . (IFP.8) ∀ x ∈ X, µ ( x , · ) is left continuous on R . (IFP.9) ∀ t ∈ R with t ≤ , ν ( x , t ) = 1 ; (IFP.10) ∀ t ∈ R + , ν ( x , t ) = 0 if and only if x = θ ; (IFP.11) ∀ t ∈ R + , ν ( cx , t ) ≤ ν ( x , t ) if | c | ≤ , ∀ c ∈ K ; (IFP.12) ν ( x + y , s + t ) ≤ max { ν ( x , s ) , ν ( y , t ) } , ∀ s, t ∈ R + ; (IFP.13) lim t → ∞ ν ( x , t ) = 0 . (IFP.14) if there exists α ∈ (0 ,
1) such that ν ( x , t ) < α , ∀ t ∈ R + then x = θ . (IFP.15) ∀ x ∈ X , ν ( x , · ) is left continuous on R .Here ( X, µ, ν ) is called intuitionistic fuzzy pseudo normed linear space.
Note . [11] a ∗ a = a and a ⋄ a = a, ∀ a ∈ [0 ,
1] is satisfied only when a ∗ b = min { a, b } and a ⋄ b = max { a, b } . elations on continuities and boundedness in IFPNLS 3 Definition 2.4. [4] Let (
X, µ, ν ) be an intuitionistic fuzzy pseudo normlinear space. A sequence { a n } n ∈ N converges to a ∈ X if and only iflim n →∞ µ ( a n − a, t ) = 1 and lim n →∞ ν ( a n − a, t ) = 0. Theorem 2.5. [4]
Let ( X, µ, ν ) be a intuitionistic fuzzy pseudo normedlinear space. Then for any α ∈ (0 , the functions k x k α , k x k ∗ α : X → [0 , ∞ ) defined as k x k α = ^ { t > µ ( x, t ) ≥ α } is a ascending family of pseudo norm on X . k x k ∗ α = ^ { t > ν ( x, t ) ≤ α } is a descending family of pseudo norm on X . Theorem 2.6. [4]
Let ( X, µ, ν ) be an intuitionistic fuzzy pseudo normedlinear space and let µ ′ , ν ′ : X × R → [ 0 , be defined by µ ′ ( x , t ) = ( W { α ∈ (0 ,
1) : k x k α ≤ t } , if t > , if t ≤ ν ′ ( x , t ) = ( V { α ∈ (0 ,
1) : k x k ∗ α ≤ t } , if t > , if t ≤ then ( i ) ( µ ′ , ν ′ ) is an intuitionistic fuzzy pseudo norm on X . ( ii ) µ ′ = µ and ν ′ = ν, where k · k α is a ascending family of pseudonorms and k · k ∗ α is a descending family of pseudo norms defined in The-orem 2.5. Intuitionistic Fuzzy continuities of operators
This section deals with the study of different types of continuities ofbounded linear operators and their intra relations in intuitionistic fuzzypseudo normed linear spaces.
Definition 3.1.
Let (
X, µ , ν ) , ( Y, µ , ν ) be intuitionistic fuzzy pseudonormed linear spaces. A mapping T : ( X, µ , ν ) → ( Y, µ , ν ) is saidto be IFC at x ∈ X if for any given ǫ > α ∈ (0 ,
1) there exists δ = δ ( α, ǫ ) > β = β ( α, ǫ ) ∈ (0 ,
1) such that for all x ∈ X , µ ( x − x , δ ) > − β ⇒ µ ( T ( x ) − T ( x ) , ǫ ) > − αν ( x − x , δ ) < β ⇒ ν ( T ( x ) − T ( x ) , ǫ ) < α. Bivas Dinda, S.K. Ghosh and T.K. Samanta
Definition 3.2.
Let (
X, µ , ν ) , ( Y, µ , ν ) be intuitionistic fuzzy pseudonormed linear spaces. A mapping T : ( X, µ , ν ) → ( Y, µ , ν ) is said tobe sequentially IFC at x ∈ X if for any sequence { x n } n , x n ∈ X and t >
0, lim t → ∞ µ ( x n − x , t ) = 1 ⇒ lim t → ∞ µ ( T ( x n ) − T ( x ) , t ) = 1lim t → ∞ ν ( x n − x , t ) = 0 ⇒ lim t → ∞ ν ( T ( x n ) − T ( x ) , t ) = 0 . Theorem 3.3.
If a linear operator T : ( X, µ , ν ) → ( Y, µ , ν ) is se-quentially IFC at a point a ∈ X then it is sequentially IFC on X , where ( X, µ , ν ) and ( Y, µ , ν ) are intuitionistic fuzzy pseudo normed linearspaces.Proof. Let { a n } n be a sequence in X and a n → a, ( a ∈ X ). Then ∀ t > , lim n → ∞ µ ( a n − a, t ) = 1 and lim n → ∞ ν ( a n − a, t ) = 0. Therefore,lim n → ∞ µ (( a n − a + a ) − a , t ) = 1 and lim n → ∞ ν (( a n − a + a ) − a , t ) = 0.Since T is sequentially IFC at x , ∀ t > n → ∞ µ ( T ( a n − a + a ) − T ( a ) , t ) = 1 and lim n → ∞ ν ( T ( a n − a + a ) − T ( a ) , t ) = 0. ⇒ lim n → ∞ µ ( T ( a n ) − T ( a ) + T ( a ) − T ( a ) , t ) = 1 and lim n → ∞ ν ( T ( a n ) − T ( a ) + T ( a ) − T ( a ) , t ) = 0, since T is linear. ⇒ lim n → ∞ µ ( T ( a n ) − T ( a ) , t ) = 1 and lim n → ∞ ν ( T ( a n ) − T ( a ) , t ) = 0.Since a ∈ X is arbitrary, T is sequentially IFC on X . (cid:3) Theorem 3.4.
A linear operator T : ( X, µ , ν ) → ( Y, µ , ν ) is sequen-tially IFC if and only if it is IFC, where ( X, µ , ν ) and ( Y, µ , ν ) areintuitionistic fuzzy pseudo normed linear spaces.Proof. First suppose T be IFC at a ∈ X . Let { a n } n be a sequencein X converges to a . Then for any given ǫ > , α ∈ (0 ,
1) there exist δ = δ ( ǫ, α ) > β = β ( ǫ, α ) ∈ (0 ,
1) such that ∀ a ∈ X , µ ( a − a , δ ) > − β ⇒ µ ( T ( a ) − T ( a ) , ǫ ) > − αν ( a − a , δ ) < β ⇒ ν ( T ( a ) − T ( a ) , ǫ ) < α Since a n converges to a there exists n ∈ N such that ∀ n ≥ n , µ ( a n − a , δ ) > − β, ν ( a n − a , δ ) < β and since T is IFC at a ∈ X ,we have µ ( T ( a n ) − T ( a ) , ǫ ) > − α and ν ( T ( a n ) − T ( a ) , ǫ ) < α ⇒ T ( a n ) → T ( a ) i.e., T is sequentially IFC at a ∈ X .Conversely, suppose T be not IFC at a ∈ X . Then there exist b ∈ X elations on continuities and boundedness in IFPNLS 5 such that for any given ǫ > , α ∈ (0 ,
1) there exist δ > , β ∈ (0 , µ ( b − a , δ ) > − β ⇒ µ ( T ( y ) − T ( x ) , ǫ ) ≤ − α and ν ( b − a , δ ) < β ⇒ ν ( T ( y ) − T ( x ) , ǫ ) ≥ − α .Hence for δ = β = n +1 there exist b n for n = 1 , , · · · , such that µ ( b n − a , δ ) = µ ( b n − a , n +1 ) > − n +1 ⇒ µ ( T ( b n ) − T ( a ) , ǫ ) ≤ − α and ν ( b n − a , δ ) = ν ( b n − a , n +1 ) < n +1 ⇒ ν ( T ( b n ) − T ( a ) , ǫ ) ≥ α .Therefore, lim n → ∞ µ ( b n − a , δ ) = 1 ⇒ lim n → ∞ µ ( T ( b n ) − T ( a ) , ǫ ) = 1 and lim n → ∞ ν ( b n − a , δ ) = 0 ⇒ lim n → ∞ ν ( T ( b n ) − T ( a ) , ǫ ) = 0. Hence T is not sequentially IFC at a . (cid:3) Definition 3.5.
Let (
X, µ , ν ) , ( Y, µ , ν ) be intuitionistic fuzzy pseudonormed linear spaces. A mapping T : ( X, µ , ν ) → ( Y, µ , ν ) is said tobe strongly IFC at x ∈ X if for any given ǫ > δ ( ǫ ) > x ∈ X , µ ( T ( x ) − T ( x ) , ǫ ) ≥ µ ( x − x , δ ) , ν ( T ( x ) − T ( x ) , ǫ ) ≤ ν ( x − x , δ ) . Theorem 3.6.
If a linear operator T : ( X, µ , ν ) → ( Y, µ , ν ) isstrongly IFC at a point a ∈ X then it is strongly IFC on X , where ( X, µ , ν ) and ( Y, µ , ν ) are intuitionistic fuzzy pseudo normed linearspaces.Proof. Since T is strongly IFC at a , for given ǫ > δ ( ǫ ) > ∀ a ∈ X , µ ( T ( a ) − T ( a ) , ǫ ) ≥ µ ( a − a , δ ) and ν ( T ( a ) − T ( a ) , ǫ ) ≤ ν ( a − a , δ ).Taking b ∈ X we have a + a − b ∈ X . Therefore replacing a by a + a − b we have, µ ( T ( a + a − b ) − T ( a ) , ǫ ) ≥ µ ( a + a − b − a , δ ) and ν ( T ( a + a − b ) − T ( a ) , ǫ ) ≤ ν ( a + a − b − a , δ ). ⇒ µ ( T ( a ) + T ( a ) − T ( b ) − T ( a ) , ǫ ) ≥ µ ( a − b, δ ) and ν ( T ( a ) + T ( a ) − T ( b ) − T ( a ) , ǫ ) ≤ ν ( a − b, δ ). ⇒ µ ( T ( a ) − T ( b ) , ǫ ) ≥ µ ( a − b, δ ) and ν ( T ( a ) − T ( b ) , ǫ ) ≤ ν ( a − b, δ ).Hence T is strongly IFC at b . Since b ∈ X is arbitrary, T is stronglyIFC on X . (cid:3) Definition 3.7.
Let (
X, µ , ν ) , ( Y, µ , ν ) be intuitionistic fuzzy pseudonormed linear spaces. A mapping T : ( X, µ , ν ) → ( Y, µ , ν ) is said to Bivas Dinda, S.K. Ghosh and T.K. Samanta be weakly IFC at x ∈ X if for any given ǫ > α ∈ (0 ,
1) thereexists δ = δ ( α, ǫ ) > x ∈ X , µ ( x − x , δ ) ≥ α ⇒ µ ( T ( x ) − T ( x ) , ǫ ) ≥ αν ( x − x , δ ) ≤ α ⇒ ν ( T ( x ) − T ( x ) , ǫ ) ≤ α. Theorem 3.8.
If a linear operator T : ( X, µ , ν ) → ( Y, µ , ν ) is weaklyIFC at a point a ∈ X then it is weakly IFC on X , where ( X, µ , ν ) and ( Y, µ , ν ) are intuitionistic fuzzy pseudo normed linear spaces.Proof. Since T is weakly IFC on a ∈ X then for given ǫ > , α ∈ (0 , δ ( α, ǫ ) > ∀ a ∈ X , µ ( a − a , δ ) ≥ α ⇒ µ ( T ( a ) − T ( a ) , ǫ ) ≥ α and ν ( a − a , δ ) ≤ α ⇒ ν ( T ( a ) − T ( a ) , ǫ ) ≤ α .Taking b ∈ X we have a + a − b ∈ X . Therefore replacing a by a + a − b we have, µ ( a − b, δ ) ≥ α ⇒ µ ( T ( a + a − b ) − T ( a ) , ǫ ) ≥ α ⇒ µ ( T ( a ) + T ( a ) − T ( b ) − T ( a ) , ǫ ) ≥ α ⇒ µ ( T ( a ) − T ( b ) , ǫ ) ≥ α and ν ( a − b, δ ) ≤ α ⇒ ν ( T ( a + a − b ) − T ( a ) , ǫ ) ≤ α ⇒ ν ( T ( a ) + T ( a ) − T ( b ) − T ( a ) , ǫ ) ≤ α ⇒ ν ( T ( a ) − T ( b ) , ǫ ) ≤ α .Since y ( ∈ X ) is arbitrary, T is weakly IFC on X . (cid:3) Theorem 3.9.
If a linear operator T : ( X, µ , ν ) → ( Y, µ , ν ) isstrongly IFC then it is weakly IFC, where ( X, µ , ν ) and ( Y, µ , ν ) areintuitionistic fuzzy pseudo normed linear spaces.Proof. From the definitions of strongly IFC and weakly IFC it follows. (cid:3)
The next example shows that in an intuitionistic fuzzy pseudo normedlinear space weakly intuitionistic fuzzy continuity may not imply stronglyintuitionistic fuzzy continuity.
Example . Let ( X, k · k ) be a pseudo normed linear space and µ, ν : X × R → [0 ,
1] be defined by: µ ( x, t ) = if t > , k x k < t. tt + k x k if t > , k x k ≥ t . if t ≤ .ν ( x, t ) = if t > , k x k < t. k x k t + k x k if t > , k x k ≥ t . if t ≤ . elations on continuities and boundedness in IFPNLS 7 then by Example 3.2 of [4], ( X, µ, ν ) is an IFPNLS.Let T : ( X, µ, ν ) → ( X, µ, ν ) be a linear operator defined by T ( x ) = x x .Let x ∈ X then for each x ∈ X, ǫ > α ∈ (0 , µ ( T ( x ) − T ( x ) , ǫ ) ≥ α ⇐ ǫǫ + k T ( x ) − T ( x ) k ≥ α ⇐ ǫǫ + k x x − x x k ≥ α ⇐ ǫ k (1 + x )(1 + x ) k ǫ k (1 + x )(1 + x ) k + k x + x x − x − xx k ≥ α ⇐ ǫ k x + x + xx ) ǫ k x + x + xx ) k + k ( x − x )( x + xx + x ) + xx ( x + x )( x − x ) k ≥ α ⇐ ǫ k x + x + xx ) k ǫ k x + x + xx ) k + k ( x − x ) kk ( x + xx + x + x x + xx ) k ≥ α ⇐ ǫ k x + x + xx ) kk ( x + xx + x + x x + xx ) k ǫ k x + x + xx ) kk ( x + xx + x + x x + xx ) k + k ( x − x ) k ≥ α ⇐ ǫ k x + x + xx ) kk ( x + xx + x + x x + xx ) k ≥ α.ǫ k x + x + xx ) kk ( x + xx + x + x x + xx ) k + α k ( x − x ) k⇐ ǫ ≥ α.ǫ + α k ( x − x ) k . k ( x + xx + x + x x + xx ) kk x + x + xx ) k ≥ α.ǫ + α k ( x − x ) k ,since k ( x + xx + x + x x + xx ) kk x + x + xx ) k ≥ ⇐ δ ≥ α.δ + α k ( x − x ) k , by taking ǫ = δ . ⇐ δδ + k ( x − x ) k ≥ α ⇐ µ ( x − x , δ ) ≥ α ; and ν ( T ( x ) − T ( x ) , ǫ ) ≤ α ⇐ k T ( x ) − T ( x ) k ǫ + k T ( x ) − T ( x ) k ≤ α ⇐ k x x − x x k ǫ + k x x − x x k ≤ α ⇐ k x + x x − x − xx k ǫ k (1 + x )(1 + x ) k + k x + x x − x − xx k ≤ α ⇐ k ( x − x )( x + xx + x ) + xx ( x + x )( x − x ) k ǫ k x + x + xx ) k + k ( x − x )( x + xx + x ) + xx ( x + x )( x − x ) k ≤ Bivas Dinda, S.K. Ghosh and T.K. Samanta α ⇐ k ( x − x )( x + xx + x + x x + xx ) k ǫ k x + x + xx ) k + k ( x − x )( x + xx + x + x x + xx ) k ≤ α ⇐ k ( x − x ) kk ( x + xx + x + x x + xx ) k ǫ k x + x + xx ) k + k ( x − x ) kk ( x + xx + x + x x + xx ) k ≤ α ⇐ k ( x − x ) k ǫ k x + x + xx ) kk ( x + xx + x + x x + xx ) k + k ( x − x ) k ≤ α ⇐ α k ( x − x ) k + α.ǫ k x + x + xx ) kk ( x + xx + x + x x + xx ) k ≥ k ( x − x ) k⇐ (1 − α ) k ( x − x ) k ≤ α.ǫ k x + x + xx ) kk ( x + xx + x + x x + xx ) k ≤ α.ǫ ,since k x + x + xx ) kk ( x + xx + x + x x + xx ) k ≤ ⇐ k ( x − x ) k − α k ( x − x ) k ≤ α.δ , by taking δ = ǫ ⇐ k ( x − x ) k ≤ α ( δ + k ( x − x ) k ) ⇐ k ( x − x ) k δ + k ( x − x ) k ≤ α ⇐ ν ( x − x , δ ) ≤ α .Thus for every ǫ > α ∈ (0 ,
1) there exists δ = δ ( α, ǫ ) > x ∈ X and x ∈ X µ ( x − x , δ ) ≥ α ⇒ µ ( T ( x ) − T ( x ) , ǫ ) ≥ α, ν ( x − x , δ ) ≤ α ⇒ ν ( T ( x ) − T ( x ) , ǫ ) ≤ α. Hence T is weakly intuitionistic fuzzy continuous at x ∈ X and henceon X .To show T is not strongly intuitionistic fuzzy continuous, it is enoughto for any given ǫ > δ > µ ( T ( x ) − T ( x ) , ǫ ) ≥ µ ( x − x , δ ) or ν ( T ( x ) − T ( x ) , ǫ ) ≤ ν ( x − x , δ ) . Let ǫ >
0, then ∀ x ∈ X and x ∈ X , µ ( T ( x ) − T ( x ) , ǫ ) ≥ µ ( x − x , δ ) ⇒ µ ( x x − x x , ǫ ) ≥ δδ + k x − x k ⇒ ǫǫ + k x + x x − x − xx kk (1+ x )(1+ x ) k ≥ δδ + k x − x k ⇒ ǫ k (1 + x + x + xx ) k ǫ k (1 + x + x + xx ) k + k x − x kk x + xx + x + x x + xx k ≥ δδ + k x − x k ⇒ ǫ k x − x kk (1+ x + x + xx ) k ≥ δ k x − x kk x + xx + x + x x + xx k⇒ δ ≤ ǫ k (1 + x + x + xx ) kk x + xx + x + x x + xx k . elations on continuities and boundedness in IFPNLS 9 Now inf { k (1 + x + x + xx ) kk x + xx + x + x x + xx k } = 0 , ∀ x ∈ X .Therefore, δ = 0, which is not possible. This shows that T is not stronglyintuitionistic fuzzy continuous. Theorem 3.11.
If a linear operator T : ( X, µ , ν ) → ( Y, µ , ν ) isstrongly IFC then it is sequentially IFC, where ( X, µ , ν ) and ( Y, µ , ν ) are intuitionistic fuzzy pseudo normed linear spaces.Proof. Let { a n } n be a sequence in X such that a n → a .i.e., lim n → ∞ µ ( a n − a ) = 1 and lim n → ∞ ν ( a n − a ) = 0 , ∀ t > T is strongly IFC at a ∈ X . Then for ǫ > , ∃ δ ( ǫ ) > ∀ a ∈ X, µ ( T ( a ) − T ( a ) , ǫ ) ≥ µ ( a − a , δ ) and ν ( T ( a ) − T ( a ) , ǫ ) ≤ ν ( a − a , δ ).Now, lim n → ∞ µ ( T ( a n ) − T ( a ) , ǫ ) ≥ lim n → ∞ µ ( a n − a , δ ) = 1 ⇒ lim n → ∞ µ ( T ( a n ) − T ( a ) , ǫ ) = 1, andlim n → ∞ ν ( T ( a n ) − T ( a ) , ǫ ) ≤ lim n → ∞ ν ( a n − a , δ ) = 0 ⇒ lim n → ∞ ν ( T ( a n ) − T ( a ) , ǫ ) = 0.Since ǫ is arbitrary small positive number, T is sequentially IFC. (cid:3) The next example shows that in an intuitionistic fuzzy pseudo normedlinear space sequentially intuitionistic fuzzy continuity may not implystrongly intuitionistic fuzzy continuity.
Example . Consider the intuitionistic fuzzy pseudo normed linearspace (
X, µ, ν ) as Example 3.10 and the linear operator T is defined by T ( x ) = x x .Let { x n } n be a sequence in X such that x n → x in X . Now ∀ t > n → ∞ µ ( x n − x , t ) = 1 , lim n → ∞ ν ( x n − x , t ) = 0. ⇒ lim n → ∞ tt + k x n − x k = 1 , lim n → ∞ k x n − x k t + k x n − x k = 0.(3.1) ⇒ lim n → ∞ k x n − x k = 0Now µ ( T ( x n ) − T ( x ) , t ) = tt + k x n x n − x x k = tt + k x n + x x n − x − x n x (1+ x n )(1+ x ) k = t k (1 + x n )(1 + x ) k t k (1 + x n )(1 + x ) k + k x n − x kk x n + x n x + x + x x n + x n x k =1 as n → ∞ by Equation 3.1. Also, µ ( T ( x n ) − T ( x ) , t ) = x n x n − x x t + k x n x n − x x k = k x n + x x n − x − x n x (1+ x n )(1+ x ) k t + k x n + x x n − x − x n x (1+ x n )(1+ x ) k = k x n − x kk x n + x n x + x + x x n + x n x k t k (1 + x n )(1 + x ) k + k x n − x kk x n + x n x + x + x x n + x n x k =0 as n → ∞ by Equation 3.1.Thus T is sequentially IFC at x ∈ X and hence on X . From Example3.10 it is apparent that T is not strongly IFC. Corollary 3.13.
If a linear operator T : ( X, µ , ν ) → ( Y, µ , ν ) isstrongly IFC then it is IFC, where ( X, µ , ν ) and ( Y, µ , ν ) are intu-itionistic fuzzy pseudo normed linear spaces.Proof. From Theorem 3.4 and Theorem 3.11 the corollary follows. (cid:3) Intuitionistic Fuzzy boundedness of operators
Definition 4.1.
Let (
X, µ , ν ) , ( Y, µ , ν ) be intuitionistic fuzzy pseudonormed linear spaces. A linear operator T : ( X, µ , ν ) → ( Y, µ , ν ) issaid to be strongly IFB if ∀ x ∈ X and ∀ t ∈ R + , µ ( T ( x ) , t ) ≥ µ ( x, t ) , ν ( T ( x ) , t ) ≤ ν ( x, t ) . Definition 4.2.
Let (
X, µ , ν ) , ( Y, µ , ν ) be intuitionistic fuzzy pseudonormed linear spaces. A mapping T : ( X, µ , ν ) → ( Y, µ , ν ) is said tobe weakly IFB if for any α ∈ (0 , ∀ x ∈ X and ∀ t ∈ R + , µ ( x, t ) ≥ α ⇒ µ ( T ( x ) , t ) ≥ α , ν ( x, t ) ≤ − α ⇒ ν ( T ( x ) , t ) ≤ − α. Theorem 4.3.
If a linear operator T : ( X, µ , ν ) → ( Y, µ , ν ) isstrongly IFB then it is weakly IFB, where ( X, µ , ν ) and ( Y, µ , ν ) areintuitionistic fuzzy pseudo normed linear spaces.Proof. This theorem easily perceived from the definition of strongly IFBand weakly IFB of linear operators. (cid:3)
Definition 4.4.
Let (
X, µ , ν ) , ( Y, µ , ν ) be intuitionistic fuzzy pseudonormed linear spaces. A mapping T : ( X, µ , ν ) → ( Y, µ , ν ) is said tobe uniformly IFB if there exist c > , α ∈ (0 ,
1) such that k T ( x ) k α ≤ k x k α , k T ( x ) k ∗ α ≤ k x k ∗ α . Where k · k α , k · k α are ascending family of pseudo norms and k · k ∗ α , k · k ∗ α descending family of pseudo norms defined by k x k α = ^ { t > µ ( x, t ) ≥ α } , k T ( x ) k α = ^ { t > µ ( T ( x ) , t ) ≥ α } elations on continuities and boundedness in IFPNLS 11 k x k ∗ α = ^ { t > ν ( x, t ) ≤ α } , k T ( x ) k ∗ α = ^ { t > ν ( T ( x ) , t ) ≤ α } Theorem 4.5.
A linear operator T : ( X, µ , ν ) → ( Y, µ , ν ) is stronglyIFB if and only if it is uniformly IFB with respect to corresponding α -norms, α ∈ (0 , , where ( X, µ , ν ) and ( Y, µ , ν ) are intuitionisticfuzzy pseudo normed linear spaces.Proof. First we suppose T is strongly IFB. Then ∀ x ∈ X and ∀ t ∈ R + ,(4.1) µ ( T ( x ) , t ) ≥ µ ( x, t ) , ν ( T ( x ) , t ) ≤ ν ( x, t ) . Let k x k α < t ⇒ V { s ( >
0) : µ ( x, s ) ≥ α } < t. ⇒ ∃ s > t such that µ ( x, s ) ≥ α. ⇒ ∃ s > t such that µ ( T ( x ) , s ) ≥ α. (by Equation 4.1) ⇒ k T ( x ) k α ≤ s < t .Thus, k T ( x ) k α ≤ k x k α . Also, let k x k ∗ α > t ⇒ V { s ( >
0) : ν ( x, s ) ≤ α } > t. ⇒ ∃ s > t such that ν ( x, s ) ≤ α. ⇒ ∃ s > t such that ν ( T ( x ) , s ) ≤ α. (by Equation 4.1) ⇒ k T ( x ) k α ≥ s > t .Thus, k T ( x ) k ∗ α ≥ k x k ∗ α . Hence T is uniformly IFB.Conversely, suppose T is uniformly IFB with respect to corresponding α -norms. Then for α ∈ (0 , k T ( x ) k α ≤ k x k α , k T ( x ) k ∗ α ≤ k x k ∗ α Let µ ( x, t ) > a ⇒ W { α ∈ (0 ,
1) : k x k α ≤ t } > a. ⇒ ∃ α ∈ (0 ,
1) such that α > a and k x k α ≤ t ⇒ ∃ α ∈ (0 ,
1) such that α > a and k T ( x ) k α ≤ t (by Equation 4.2) ⇒ µ ( T ( x ) , t ) ≥ α > a Therefore, µ ( T ( x ) , t ) ≥ µ ( x, t ).Also, let ν ( x, t ) < b ⇒ V { α ∈ (0 ,
1) : k x k ∗ α ≤ t } < b. ⇒ ∃ α ∈ (0 ,
1) such that α < b and k x k ∗ α ≤ t ⇒ ∃ α ∈ (0 ,
1) such that α < b and k T ( x ) k ∗ α ≤ t (by Equation 4.2) ⇒ ν ( T ( x ) , t ) ≤ α < b Therefore, ν ( T ( x ) , t ) ≤ ν ( x, t ).Hence T is strongly IFB. (cid:3) Theorem 4.6.
A linear operator T : ( X, µ , ν ) → ( Y, µ , ν ) is stronglyIFC if and only if it is strongly IFB, where ( X, µ , ν ) and ( Y, µ , ν ) areintuitionistic fuzzy pseudo normed linear spaces.Proof. First suppose T is strongly IFB then ∀ x ∈ X and ∀ ǫ ∈ R + , µ ( T ( x ) , ǫ ) ≥ µ ( x, ǫ ) , ν ( T ( x ) , ǫ ) ≤ ν ( x, ǫ ). ⇒ µ ( T ( x − θ ) , ǫ ) ≥ µ ( x − θ, ǫ ) , ν ( T ( x − θ ) , ǫ ) ≤ ν ( x − θ, ǫ ) ⇒ µ ( T ( x ) − T ( θ ) , ǫ ) ≥ µ ( x − θ, δ ) , ν ( T ( x − θ ) , ǫ ) ≤ ν ( x − θ, δ ),where δ = ǫ .Therefore T is strongly IFC at θ and hence by Theorem 3.6 T is stronglyIFC on X .Conversely, suppose T is strongly IFC on X. Then T is strongly IFC atany point of X , say θ . ∀ x ∈ X take ǫ = t = δ , then µ ( T ( x ) − T ( θ ) , t ) ≥ µ ( x − θ, t ) , ν ( T ( x ) − T ( θ ) , t ) ≤ ν ( x − θ, t ). ⇒ µ ( T ( x ) , t ) ≥ µ ( x, t ) , ν ( T ( x ) , t ) ≤ ν ( x, t ).If x = θ, t > µ ( T ( θ ) , t ) = µ ( θ Y , t ) = 1 = µ ( θ, t ) and ν ( T ( θ ) , t ) = ν ( θ Y , t ) = 0 = ν ( θ, t ).For any x, t ≤ µ ( T ( x ) , t ) = 0 = µ ( x, t ) and ν ( T ( x ) , t ) = 1 = ν ( x, t ).Hence ∀ x ∈ X, ∀ t ∈ R , T is strongly IFB. (cid:3) Corollary 4.7.
A linear operator T : ( X, µ , ν ) → ( Y, µ , ν ) is stronglyIFB then it is sequentially IFC, where ( X, µ , ν ) and ( Y, µ , ν ) are in-tuitionistic fuzzy pseudo normed linear spaces.Proof. From Theorem 3.11 and Theorem 4.6 the corollary follows. (cid:3)
Corollary 4.8.
A linear operator T : ( X, µ , ν ) → ( Y, µ , ν ) is stronglyIFB then it is IFC, where ( X, µ , ν ) and ( Y, µ , ν ) are intuitionisticfuzzy pseudo normed linear spaces.Proof. From Corollary 3.13 and Theorem 4.6 the corollary follows. (cid:3)
Theorem 4.9.
A linear operator T : ( X, µ , ν ) → ( Y, µ , ν ) is weaklyIFC if and only if it is weakly IFB, where ( X, µ , ν ) and ( Y, µ , ν ) areintuitionistic fuzzy pseudo normed linear spaces.Proof. First suppose T is weakly IFB. Then for any α ∈ (0 , , ∀ x ∈ X, ∀ t ∈ R + , µ ( x, t ) ≥ α ⇒ µ ( T ( x ) , t ) ≥ α and ν ( x, t ) ≤ α ⇒ ν ( T ( x ) , t ) ≤ α . µ ( x − θ, t ) ≥ α ⇒ µ ( T ( x − θ ) , t ) ≥ α and ν ( x − θ, t ) ≤ α ⇒ ν ( T ( x − θ ) , t ) ≤ α . µ ( x − θ, δ ) ≥ α ⇒ µ ( T ( x ) − T ( θ ) , ǫ ) ≥ α and ν ( x − θ, δ ) ≤ α ⇒ elations on continuities and boundedness in IFPNLS 13 ν ( T ( x ) − T ( θ ) , ǫ ) ≤ α , where ǫ = t = δ . Therefore, T is weakly IFC at θ and hence by Theorem 3.8, T is weakly IFC.Conversely suppose T is weakly IFC on X. Then T is weakly IFC at anypoint of X , say θ . ∀ x ∈ X take ǫ = t = δ , then µ ( x − θ, t ) ≥ α ⇒ µ ( T ( x ) − T ( θ ) , t ) ≥ α and ν ( x − θ, t ) ≤ α ⇒ ν ( T ( x ) − T ( θ ) , t ) ≤ αµ ( x, t ) ≥ α ⇒ µ ( T ( x ) , t ) ≥ α and ν ( x, t ) ≤ α ⇒ ν ( T ( x ) , t ) ≤ α If x = θ, t > µ ( x, t ) = 1 = µ ( T ( x ) , t ) and ν ( x, t ) = 0 = ν ( T ( x ) , t ).For any x, t ≤ µ ( x, t ) = 0 = µ ( T ( x ) , t ) and ν ( x, t ) = 1 = ν ( T ( x ) , t ).Hence for any α ∈ (0 , , ∀ x ∈ X, ∀ t ∈ R , T is weakly IFB. (cid:3) References [1] T. Bag, S.K. Samanta,
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Department of Mathematics, Mahishamuri Ramkrishna Vidyapith, Howrah 711401,West Bengal, India
Email: [email protected]
Santanu Kumar Ghosh
Department of Mathematics, Kazi Nazrul University, Asansol 713340, West Bengal,India
Email:santanu − T.K. Samanta
Department of Mathematics, Uluberia College, Howrah 711315, West Bengal, India
Email: mumpu −−