Fuzzy Rate Analysis of Operators and its Applications in Linear Spaces
aa r X i v : . [ m a t h . G M ] D ec Fuzzy Rate Analysis of Operators and its Applications in LinearSpaces
Yijin Zhang , Honggang Li ∗ , Maoming Jin , Zongbing Lin ∗ The corresponding author: Honggang Li
Abstract
In this paper, a new concept, the fuzzy rate of an operator in linear spaces isproposed for the very first time. Some properties and basic principles of it are studied.Fuzzy rate of an operator B which is specific in a plane is discussed. As its application,a new fixed point existence theorem is proved. AMS Subject Classification.
Key Words and Phrases.
Fuzzy rate; Operator; Membership function; Fixed pointexistence theorem.
More and more classical analysis theory are being developed into fuzzy analysis theory.Fuzzy sets, fuzzy logic, fuzzy numbers, fuzzy topologies and so on were introduced andstudied[1-3]. Chang and Huang, Ding and Jong, Jin, Li and others studied several kinds ofvariational inequalities (inclusions) for fuzzy mappings[4-8].Recently, Konwar and Nabanita introduce the notion of continuous linear operators andestablish the uniform continuity theorem and Banach’s contraction principle in an intuition-istic fuzzy n-normed linear space[9].Wang investigates the concepts and some properties ofinterval-valued fuzzy ideals in B-algebras and the homomorphic inverse image of interval-valued intuitionistic fuzzy ideals[10]. Fixed Point Theorems in Partially Ordered FuzzyMetric Spaces and Operator Theory and Fixed Points in Fuzzy Normed Algebras and Ap-plications are studied in [11]. Fuzzy-wavelet-like operators via a real-valued scaling functionare discussed in [12]. A linear fuzzy operator inequality approach is proposed for the first
Yijin Zhang, Honggang Li, Maoming Jin time in [13]. Fuzziness degree’s quantity measure as to fuzzy operator is researched bymeans of fuzzy set theory in [14]. For more details, we reference to the readers [1-15].In this work, we come up with the concept of fuzzy rate of an operator and considerits properties and applications. We also explore fuzzy rate which is produced by an oper-ator effecting an element, as well as some properties and applications of it.These are newextension, attempt and applications to the operator theory in linear space and fuzzy theory.The remainder of this paper is organized as follows. In Section 2, we give an examplewhich helps us introduce the concept of fuzzy rate of an operator. In Section 3, we proposethe concept and prove some basic properties of it. In Section 4, a new Fixed Point ExistenceTheorem with the fuzzy rate of the operator B is obtained as its application. Here, an example is given to introduce a new concept, the fuzzy rate of an operator in linearspaces.
Example 2.1
Let U = R × R be a real plane(Universe), c be a cycle or ellipse whose centeris at (0 , on U , F : U → [0 , be a membership function, and F ( U ) = { F c ( x, y ) | F c ( x, y ) is the membership function for point ( x, y ) belonging to acurve c } .In the plane U , we suppose that the equation of c is c ( λ,r ) : x + λy = r , (2.1) where r > and < λ are two parameters. Define a membership function F c (1 ,r ) ( x, y ) for ( x, y ) belonging to the curve c ( µ,r ) ( µ > , F c ( µ,r ) ( x, y ) = , ( x, y ) / ∈ c ( λ,r ) ,e − ( λ − µ ) , ( x, y ) ∈ c ( λ,r ) (0 < λ = µ ) , , ( x, y ) ∈ c ( λ,r ) (0 < λ = µ ) , (2.2) In the (2.2), let µ = 1 , then the curve c (1 ,r ) is the circle. We have F c (1 ,r ) (0 , r ) = 1 forthe point (0 , r ) ∈ c (1 ,r ) and F c (1 ,r ) ( r √ , r √ ) = e − (3 − = e − for λ = 3 , but F c (1 ,r ) ( r, r ) = 0 for ( r, r ) / ∈ c ( λ,r ) and any λ > .Set B = (cid:18) b (cid:19) (2.3) be an operator on U for b = 0 , then F c (1 ,r ) ( B (0 , r )) = F c (1 ,r ) (0 , br ) = e − ( b − . where the point B (0 , r ) ∈ c ( b ,r ) with the operator B : (0 , r ) B (0 , r ) .The value, F c (1 ,r ) ( B (0 , r )) F c (1 ,r ) (0 , r ) = e − ( b − e − ( b − uzzy Rate Analysis of Operators and its Applications in Linear Spaces expresses a fuzzy rate of operator B at the point (0 , r ) ∈ c (1 ,r ) with F c (1 ,r ) for λ = b . Other examples, if r = 1 , b = √ , then F c (1 , ( B (0 , F c (1 , (0 ,
1) = e − ( b − e − ( − = e − . Furthermore, suppose that X is a linear space, Ø = T ⊆ X , x ∈ X , F T ( x ) : X → [0 ,
1] isa membership function for x belonging to the set T [4, 5, 7, 8], B : X → X is an operator.Then, F T ( B ( x )) reflects the membership degree of the image of x belonging to the set T .It’s clear that the value F T ( B ( x )) F T ( x ) , the ratio of F T ( B ( x )) and F T ( x ), indicates the changingrate which is produced by the mapping B : x → B ( x ). we can consider a special valuesup F ∈F ( X ) F ( B ( y )) F ( y ) to express a fuzzy rate of operator B at a point y ∈ X with F ( X ), it isvery interesting to consider the impact and properties of the operator with respect to F T . In this section, we first give the concept of a fuzzy rate of an operator, then we show somebasic properties of it.
Definition 3.1
Let X be a linear space, B : X → X be an operator, B ( X )= { B | B : X →X } , F : X → [0 , be a membership function over X , P ( X )= { F | F : X → [0 , } be acollection of all membership functions over X . For any ∅ 6 = F ( X ) ⊆ P ( X ) , if k B k y = sup F ∈F ( X ) F ( B ( y )) F ( y ) (3.1) exists, then k B k y is called a fuzzy rate of the operator B at the point y ∈ X on F ( X ) . For Example 2.1,let F ( U ) = { F c ( µ,r ) | µ, r > } by (2.2), we can achieve, obviously k B k (0 ,r ) = sup F c ( µ,r ) ∈F ( U ) F c ( µ,r ) ( B (0 , r )) F c ( µ,r ) ((0 , r ))= sup F c ( µ,r ) ∈F ( U ) F c ( µ,r ) ((0 , br )) F c ( µ,r ) ((0 , r ))= sup F c ( µ,r ) ∈F ( U ) e − ( b − µ ) e − (1 − µ ) = sup µ> e (1 − b ) e µ ( b − = ( + ∞ , | b | < ,e (1 − b ) , | b | ≥ . At the same time, we have the following theorem about the relationship between a fuzzyrate of the operator and fuzzy sets.
Yijin Zhang, Honggang Li, Maoming Jin
Theorem 3.2
Let X be a linear space, B : X → X be an operator, B ( X )= { B | B : X → X } , F : X → [0 , be a membership function over X , P ( X )= { F | F : X → [0 , } be a collectionof all membership functions over X and k B k y be the fuzzy rate of the operator B at the point y ∈ X on F ( X ) . Then for any ∅ 6 = F ( X ) ⊆ P ( X ) , there exist two membership functions F, G ∈ F ( X ) such that, for each y ∈ X , k B k y F ( y ) = G ( B ( y )) . (3.2) Proof.
For any ∅ 6 = F ( X ) ⊆ P ( X ), since k B k y = sup F ∈F ( X ) F ( B ( y )) F ( y ) < + ∞ , then F ( y ) = 0. ∀ n , ∃ G n ∈ F ( X ), we arrive at k B k y ≥ G n ( B ( y )) G n ( y ) > k B k y − n , and k B k y = lim n →∞ G n ( B ( y )) G n ( y ) < + ∞ for each y ∈ X .Since 0 < G n ( y ) , G n ( B ( y )) ≤
1, there exist G n k ( y ) → F ( y ) as k → + ∞ , and G n km ( B ( y )) → G ( B ( y )) as m → + ∞ for any y ∈ X .If 0 < F ( y ) ≤
1, it follows that k B k y = lim n →∞ G n ( B ( y )) G n ( y ) = lim m →∞ G n km ( B ( y )) G n km ( y ) = lim m →∞ G n km ( B ( y ))lim m →∞ G n km ( y ) = G ( B ( y )) F ( y ) , and k B k y F ( y ) = G ( B ( y )) for every y ∈ X .If F ( y ) = 0, it implies that k B k y = lim n →∞ G n ( B ( y )) G n ( y ) = lim m →∞ G n km ( B ( y )) G n km ( y ) = lim m →∞ G n km ( B ( y ))lim m →∞ G n km ( y ) < + ∞ , and lim m →∞ G n km ( B ( y )) = 0 , G ( B ( y )) = lim m →∞ G n km ( B ( y )) = 0[15].Therefore, k B k y F ( y ) = G ( B ( y )) holds for any y ∈ X .It is easy to verify that the converse proposition of Theorem 3.2 holds. We reach Theorem 3.3
Let X be a linear space, B : X → X be an operator, B ( X )= { B | B : X → X } , F : X → [0 , be a membership function over X , P ( X )= { F | F : X → [0 , } be a collectionof all membership functions over X . For any ∅ 6 = F ( X ) ⊆ P ( X ) , if there exist F , G ∈ F ( X ) and F ( y ) = 0 such that k B k y F ( y ) = G ( B ( y )) , then for any y ∈ X , k B k y = sup F ∈F ( X ) F ( B ( y )) F ( y ) = G ( B ( y )) F ( y ) < + ∞ , uzzy Rate Analysis of Operators and its Applications in Linear Spaces B exists.Now, we state some basic properties of the fuzzy rate of the operator B as the nexttheorem. These properties are very useful for further applications. Theorem 3.4
Let X be a linear space, B : X → X be an operator, B ( X )= { B | B : X → X } , F : X → [0 , be a membership function over X , P ( X )= { F | F : X → [0 , } be a collectionof all membership functions over X . For any ∅ 6 = F ( X ) ⊆ P ( X ) , B , B ∈ B ( X ) and theidentity operator I ∈ B ( X ) , then(1) k B k y > for any B ∈ B ( X ) ;(2) k I k y = 1 for any y ∈ X ;(3) If B is a linear operator and a > is a real number, then k aB k y ≤ k B k ( ay ) k aI k y ;(4) If F ( B ( y )) ≥ F ( B ( y )) for any y ∈ X , then k B k y ≥ k B k y ; (5) If F ( B ( y ) + B ( y )) = F ( B ( y )) + F ( B ( y )) for any y ∈ X , then k B + B k y ≤ k B k y + k B k y ; (6) If F ( B ( y ) − B ( y )) = F ( B ( y )) − F ( B ( y )) ≥ for any y ∈ X , then ≤ k B k y − k B k y ≤ k B − B k y ; (7) If ( B B )( y ) = B ( B ( y )) for any y ∈ X , and there exist k B k B ( y ) and k B k y , then k B B k y ≤ k B k B ( y ) k B k y ; (8) If F ( X ) ⊆ F ( X ) ⊆ P ( X ) , k B k y, F ( X ) = sup F ∈F ( X ) F ( B ( y )) F ( y ) and k B k y, F ( X ) = sup F ∈F ( X ) F ( B ( y )) F ( y ) , where k B k y, F ( X ) represents the fuzzy rate of the operator B at the point y ∈ X on F ( X ) and k B k y, F ( X ) on F ( X ) , then for any y ∈ X , k B k y, F ( X ) ≤ k B k y, F ( X ) . Proof. (1) It follows that k B k y ≥ B ∈ B ( X ) from Definition 3.1. On the otherhand, if k B k y = 0, then for any F ∈ F ( X ), F ( B ( y )) = 0. It is false because there existsa membership function F ∈ F ( X ) where F ( z ) = 0 . z = B ( y ) and F ( z ) = 0 when z = B ( y ).(2) Because I is an identity operator, we obtain k I k y = sup F ∈F ( X ) F ( I ( y )) F ( y ) = sup F ∈F ( X ) F ( I ( y )) F ( y ) = 1 Yijin Zhang, Honggang Li, Maoming Jin for all y ∈ X .(3) If B is a linear operator and a > k aB k y = sup F ∈F ( X ) F ( aB ( y )) F ( y ) = sup F ∈F ( X ) ( F ( B ( ay )) F ( ay ) F ( ay ) F ( y ) ) ≤ sup F ∈F ( X ) F ( B ( ay )) F ( ay ) sup F ∈F ( X ) F ( ay ) F ( y ) = k B k ( ay ) k aI k y . (4) If F ( B ( y )) ≥ F ( B ( y )) for any y ∈ X , then k B k y = sup F ∈F ( X ) F ( B ( y )) F ( y ) ≥ sup F ∈F ( X ) F ( B ( y )) F ( y ) = k B k y . (5) If F ( B ( y ) + B ( y )) = F ( B ( y )) + F ( B ( y )) for any y ∈ X , we get k B + B k y = sup F ∈F ( X ) F (( B + B )( y )) F ( y ) ≤ sup F ∈F ( X ) F ( B ( y )) F ( y ) + sup F ∈F ( X ) F ( B ( y )) F ( y ) = k B k y + k B k y . (6) Let F ( B ( y ) − B ( y )) = F ( B ( y )) − F ( B ( y )) ≥ y ∈ X , which means F ( B ( y )) − F ( B ( y )) F ( y ) ≥ k B k y = sup F ∈F ( X ) F ( B ( y )) F ( y ) = sup F ∈F ( X ) [ F ( B ( y )) − F ( B ( y ))] + F ( B ( y )) F ( y ) ≤ sup F ∈F ( X ) F ( B ( y )) − F ( B ( y )) F ( y ) + sup F ∈F ( X ) F ( B ( y )) F ( y ) = k B − B k y + k B k y . Therefore, 0 ≤ k B k y − k B k y ≤ k B − B k y holds by (4).(7) Set ( B B )( y ) = B ( B ( y )) for any y ∈ X . Then there exist k B k B ( y ) and k B k y such that k B B k y = sup F ∈F ( X ) F ( B ( B ( y ))) F ( y ) = sup F ∈F ( X ) F ( B ( B ( y ))) F ( B ( y )) F ( B ( y )) F ( y ) ≤ sup F ∈F ( X ) F ( B ( B ( y ))) F ( B ( y )) sup F ∈F ( X ) F ( B ( y )) F ( y ) = k B k B ( y ) k B k y . (8) It is clear that the result holds by F ( X ) ⊆ F ( X ) ⊆ P ( X ) and Definition 3.1.The following is also a property of the fuzzy rate of the operator B based on its basicproperties. uzzy Rate Analysis of Operators and its Applications in Linear Spaces Corollary 3.5
Let X be a linear space, B : X → X be an operator, B ( X )= { B | B : X → X } , F : X → [0 , be a membership function over X , P ( X )= { F | F : X → [0 , } be a collectionof all membership functions over X . For any ∅ 6 = F ( X ) ⊆ P ( X ) , if B n ( y ) = B n − ( B ( y )) for n = 1 , , · · · , there exist k B k B k − ( y ) for k = 1 , , · · · , n , such that Q ≤ k ≤ n k B k B k − ( y ) ≤ k B n k y , (3.3) where B = I is an identity operator. Proof.
We have k B n k y = sup F ∈F ( X ) F ( B ( B n − ( y ))) F ( y ) = sup F ∈F ( X ) F (( B ( B n − ( y ))) F ( B n − ( y )) F ( B n − ( y )) F ( y ) ≤ sup F ∈F ( X ) F (( B ( B n − ( y )))) F ( B n − ( y )) sup F ∈F ( X ) F ( B n − ( y )) F ( y )= k B k B n − ( y ) sup F ∈F ( X ) F ( B n − ( y )) F ( y ) ≤ · · · ≤ Y ≤ k ≤ n k B k B k − ( y ) . It follows that the result (3.3) holds.In what follows, we will apply the above properties to prove a new Fixed Point ExistenceTheorem.
Fixed Point theory is very important and most generally useful one in classical functionanalysis. In this section, we prove a new Fixed Point Existence Theorem with the fuzzyrate of the operator B as its application. First, we have Lemma 4.1
Let X be a linear space, B : X → X be an operator, B ( X )= { B | B : X → X } , F : X → [0 , be a membership function over X , P ( X )= { F | F : X → [0 , } be a collectionof all membership functions over X . Let k B k y = sup F ∈F ( X ) F ( B ( y )) F ( y ) < + ∞ for ∅ 6 = F ( X ) ⊆ P ( X ) . If for δ ∈ (0 , , there exists a natural number N such that k B n k y ≥ δ as n ≥ N , then there exists a F ∈ F ( X ) such that F ( B n ( y )) = F ( B n − ( y )) , (4.1) or F ( B ( B n − ( y ))) = F ( B n − ( y )) , (4.2) for n ≥ N . Yijin Zhang, Honggang Li, Maoming Jin
Proof.
Note that B n ( y ) = B n − ( B ( y )) for n = 1 , , · · · .If δ ∈ (0 , N such that k B n k y ≥ δ as n ≥ N . Then weknow 1 ≤ Q ≤ k ≤ n k B k B k − ( y ) ≤ k B n k y ≤ δ < + ∞ for n ≥ N by (2.3), and0 ≤ n X k =1 − ln k B k B k − ( y ) ≤ − ln k B n k y ≤ − ln δ < + ∞ , hence lim k → + ∞ ln k B k B k − ( y ) = 0 and lim k → + ∞ k B k B k − ( y ) = 1[15].It follows that for any natural number m there exists a M , as k > max { M, N } such that1 − m < k B k B k − ( y ) < m . Since k B k B k − ( y ) = sup F ∈F ( X ) F ( B ( B k − ( y ))) F ( B k − ( y )) < + ∞ , there exists a membership function F ∈ F ( X ) such that1 − m − m < k B k B k − ( y ) − m < F ( B ( B k − ( y ))) F ( B k − ( y )) < k B k B k − ( y ) + 1 m < m + 1 m . Letting m → + ∞ , we obtain F ( B ( B k − ( y ))) F ( B k − ( y )) = 1 , that’s to say, F ( B ( B k − ( y ))) = F ( B k − ( y )).Then, we give the definition of a quasi-fixed point of the operator B with respect to themembership function F . Definition 4.2
Let X be a linear space, B : X → X be an operator, B ( X )= { B | B : X →X } , F : X → [0 , be a membership function over X , P ( X )= { F | F : X → [0 , } be acollection of all membership functions over X . For y ∈ X , if there exists F ∈ P ( X ) suchthat F ( B ( y )) = F ( y ) , then y is called a quasi-fixed point of the operator B with respect to F . By the proof of Lemma 4.1, B k − ( y ) is a quasi-fixed point of the operator B with respectto F .Now, Fixed Point Existence Theorem with respect to F is presented. Theorem 4.3 (new Fixed Point Existence Theorem) Let X be a linear space, B : X → X be an operator, B ( X )= { B | B : X → X } , F : X → [0 , be a membership function over X , P ( X )= { F | F : X → [0 , } be a collection of all membership functions over X . Presume uzzy Rate Analysis of Operators and its Applications in Linear Spaces k B k y = sup F ∈F ( X ) F ( B ( y )) F ( y ) < + ∞ for ∅ 6 = F ( X ) ⊆ P ( X ) . If δ ∈ (0 , , there exists a natural number N such that k B n k y ≥ δ as n ≥ N . Then if there exists an injection functional F ∈ F ( X ) such that F ( B n ( y )) = F ( B n − ( y )) , (4.3) B n − ( y ) is a fixed point of the operator B with respect to F , and y is a fixed point of theoperator B n with respect to F for n ≥ N . Proof.
It follows directly that the result holds from (4.2) and the injective condition offunctional F ∈ F ( X ).Like the classical fixed point theory applied to differential equations, we believe thatthe Fixed Point Existence Theorem with respect to fuzzy set F might be applied to fuzzyequations or fuzzy differential equations. They are worth further studying in the future. In this work, we have obtained the following results: • The fuzzy rate of an operator in linear spaces is introduced and some properties andbasic principles of the fuzzy rate are studied. • The fuzzy rate of an diagonal matrix B in a plane is discussed. • A new fixed point existence theorem is proved. • This work was supported by Natural Science Foundation Project of Chongqing(Grant No.cstc2019jcyj-msxmX0716). • Competing interests
The authors declare that they have no competing interestsregarding the publication of this article.
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