Fuzzy Rate Analysis of Operators and its Applications in Linear Spaces

In this paper, a new concept, the fuzzy rate of an operator in linear spaces is proposed 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.

Recently, Konwar and Nabanita introduce the notion of continuous linear operators and establish the uniform continuity theorem and Banach's contraction principle in an intuitionistic fuzzy n-normed linear space [9].Wang investigates the concepts and some properties of interval-valued fuzzy ideals in B-algebras and the homomorphic inverse image of intervalvalued intuitionistic fuzzy ideals [10]. Fixed Point Theorems in Partially Ordered Fuzzy Metric Spaces and Operator Theory and Fixed Points in Fuzzy Normed Algebras and Applications are studied in [11]. Fuzzy-wavelet-like operators via a real-valued scaling function are discussed in [12]. A linear fuzzy operator inequality approach is proposed for the first time in [13]. Fuzziness degree's quantity measure as to fuzzy operator is researched by means of fuzzy set theory in [14]. For more details, we reference to the readers [1][2][3][4][5][6][7][8][9][10][11][12][13][14][15].
In this work, we come up with the concept of fuzzy rate of an operator and consider its properties and applications. We also explore fuzzy rate which is produced by an operator effecting an element, as well as some properties and applications of it.These are new extension, 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 example which helps us introduce the concept of fuzzy rate of an operator. In Section 3, we propose the concept and prove some basic properties of it. In Section 4, a new Fixed Point Existence Theorem with the fuzzy rate of the operator B is obtained as its application.

An Example
Here, an example is given to introduce a new concept, the fuzzy rate of an operator in linear spaces.
Example 2.1 Let U = R × R be a real plane(Universe), c be a cycle or ellipse whose center is at (0, 0) on U , F : U → [0, 1] be a membership function, and is the membership function for point (x, y) belonging to a curve c}.
In the plane U , we suppose that the equation of c is where r > 0 and 0 < 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 > 0),

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 for the point (0, r) ∈ c (1,r) and F c (1,r) ( r √ 2 , r √ 6 ) = e −(3−1) 2 = e −4 for λ = 3, but F c (1,r) (r, r) = 0 for (r, r) / ∈ c (λ,r) and any λ > 0. Set be an operator on U for b = 0, then where the point B(0, r) ∈ c ( 1 b 2 ,r) with the operator B : (0, r) → B(0, r). The value, expresses a fuzzy rate of operator B at the point (0, r) ∈ c (1,r) Furthermore, suppose that X is a linear space, Ø = T ⊆ X , x ∈ X , F T (x) : X → [0, 1] is a 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 FT (B(x)) FT (x) , the ratio of F T (B(x)) and F T (x), indicates the changing rate which is produced by the mapping B : x → B(x). we can consider a special value sup F ∈F (X ) to express a fuzzy rate of operator B at a point y ∈ X with F (X ), it is very interesting to consider the impact and properties of the operator with respect to F T .

Fuzzy Rate Analysis of Operators
In this section, we first give the concept of a fuzzy rate of an operator, then we show some basic properties of it.
exists, then B 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 > 0} by (2.2), we can achieve, obviously At the same time, we have the following theorem about the relationship between a fuzzy rate of the operator and fuzzy sets.
Theorem 3.2 Let X be a linear space, B : X → X be an operator, B(X )= {B|B : X → X }, F : X → [0, 1] be a membership function over X , P(X )= {F |F : X → [0, 1]} be a collection of all membership functions over X and B y be the fuzzy rate of the operator B at the point y ∈ X on F (X ). Then for any ∅ = F (X ) ⊆ P(X ), there exist two membership functions F, G ∈ F (X ) such that, for each y ∈ X , and B y F (y) = G(B(y)) for every y ∈ X . If F (y) = 0, it implies that and lim m→∞ G n k m (B(y)) = 0 , G(B(y)) = lim m→∞ G n k m (B(y)) = 0 [15]. Therefore, B 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 then for any y ∈ X , that is, the fuzzy rate of the operator B exists. Now, we state some basic properties of the fuzzy rate of the operator B as the next theorem. These properties are very useful for further applications. (2) I y = 1 for any y ∈ X ; (3) If B 1 is a linear operator and a > 0 is a real number, then aB 1 y ≤ B 1 (ay) aI y ; (4) If F (B 1 (y)) ≥ F (B 2 (y)) for any y ∈ X , then B 1 y ≥ B 2 y ; (5) If F (B 1 (y) + B 2 (y)) = F (B 1 (y)) + F (B 2 (y)) for any y ∈ X , then for any y ∈ X , and there exist B 1 B2(y) and B 1 y , then where B y ,F (X ) 1 represents the fuzzy rate of the operator B at the point y ∈ X on F (X ) 1 and B y ,F (X ) 2 on F (X ) 2 , then for any y ∈ X , Proof.
(1) It follows that B y ≥ 0 for any B ∈ B(X ) from Definition 3.1. On the other hand, if B y = 0, then for any F ∈ F (X ), F (B(y)) = 0. It is false because there exists a membership function F ∈ F (X ) where F (z) = 0.5 when z = B(y) and F (z) = 0 when z = B(y).
(2) Because I is an identity operator, we obtain for all y ∈ X .
The following is also a property of the fuzzy rate of the operator B based on its basic properties.
Proof. We have

It follows that the result (3.3) holds.
In what follows, we will apply the above properties to prove a new Fixed Point Existence Theorem.

Applications-a new Fixed Point Existence Theorem
Fixed Point theory is very important and most generally useful one in classical function analysis. In this section, we prove a new Fixed Point Existence Theorem with the fuzzy rate of the operator B as its application. First, we have for ∅ = F (X ) ⊆ P(X ). If for δ ∈ (0, 1], there exists a natural number N such that B n y ≥ δ as n ≥ N , then there exists a F 0 ∈ F (X ) such that F 0 (B n (y)) = F 0 (B n−1 (y)), (4.1) or F 0 (B(B n−1 (y))) = F 0 (B n−1 (y)), (4.2) for n ≥ N .
Proof. Note that B n (y) = B n−1 (B(y)) for n = 1, 2, · · ·. If δ ∈ (0, 1], there exists a natural number N such that B n y ≥ δ as n ≥ N . Then we know 1 ≤ 1 hence lim k→+∞ ln B B k−1 (y) = 0 and lim k→+∞ B B k−1 (y) = 1 [15]. It follows that for any natural number m there exists a M , as k > max{M, N } such that there exists a membership function F 0 ∈ F (X ) such that Letting m → +∞, we obtain F 0 (B(B k−1 (y))) F 0 (B k−1 (y)) = 1, that's to say, F 0 (B(B k−1 (y))) = F 0 (B k−1 (y)). Then, we give the definition of a quasi-fixed point of the operator B with respect to the membership function F .  F (B(y)) F (y) < +∞ for ∅ = F (X ) ⊆ P(X ). If δ ∈ (0, 1], there exists a natural number N such that B n y ≥ δ as n ≥ N . Then if there exists an injection functional F 0 ∈ F (X ) such that F 0 (B n (y)) = F 0 (B n−1 (y)), (4.3) B n−1 (y) is a fixed point of the operator B with respect to F , and y is a fixed point of the operator B n with respect to F for n ≥ N .
Proof. It follows directly that the result holds from (4.2) and the injective condition of functional F 0 ∈ F (X ).
Like the classical fixed point theory applied to differential equations, we believe that the Fixed Point Existence Theorem with respect to fuzzy set F might be applied to fuzzy equations or fuzzy differential equations. They are worth further studying in the future.

Conclusions
In this work, we have obtained the following results: • The fuzzy rate of an operator in linear spaces is introduced and some properties and basic 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.

Acknowledgments
• 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 interests regarding the publication of this article.