Approximate Fixed Point Theorems in Fuzzy Norm Spaces for an Operator

Abstract

We define approximate fixed point and fuzzy diameter in fuzzy norm spaces. We prove theorems for various types of well-known generalized contractions on fuzzy norm spaces with the use of two general lemmas that are given regarding approximate fixed points of operators on fuzzy norm spaces.

1. Introduction

In this paper, starting from the article of Berinde [1], we study some well-known types of operators on fuzzy norm spaces, and we give some fuzzy approximate fixed points of such operators.

Fuzzy set was defined by Zadeh [2]. Katsaras [3], while studying fuzzy topological vector spaces, was the first to introduce in 1984 the idea of fuzzy norm on a linear space. In 1992, Felbin [4] defined a fuzzy norm on a linear space with an associated metric of the Kaleva and Seikkala type [5]. A further development along this line of inquiry took place when, in 1994, Cheng and Mordeson [6] evolved the definition of a further type of fuzzy norm having a corresponding metric of the Kramosil and Michálek type [7].

Chitra and Mordeson [8] introduce a definition of norm fuzzy, and thereafter the concept of fuzzy norm space has been introduced and generalized in different ways by Bag and Samanta in [9–11].

Throughout this paper, the symbols ∧ and ∨ mean the min and the max, respectively.

2. Some Preliminary Results

We start our work with the following definitions.

Definition 1. Let 𝑈 be a linear space on 𝐑. A function 𝑁∶𝑈×𝐑→[0,1] is called fuzzy norm if and only if for every 𝑥,𝑢∈𝑈 and for every 𝑐∈𝐑, the following properties are satisfied:(𝐹_𝑁1): 𝑁(𝑥,𝑡)=0 for every 𝑡∈𝐑⁻∪{0}, (𝐹_𝑁2): 𝑁(𝑥,𝑡)=1 if and only if 𝑥=0 for every 𝑡∈𝐑⁺, (𝐹_𝑁3): 𝑁(𝑐𝑥,𝑡)=𝑁(𝑥,𝑡/|𝑐|) for every 𝑐≠0 and 𝑡∈𝐑⁺, (𝐹_𝑁4): 𝑁(𝑥+𝑢,𝑠+𝑡)≥min{𝑁(𝑥,𝑠),𝑁(𝑢,𝑡)} for every 𝑠,𝑡∈𝐑⁺, (𝐹_𝑁5): the function 𝑁(𝑥,⋅) is nondecreasing on 𝐑, and lim_𝑡→∞𝑁(𝑥,𝑡)=1.A pair (𝑈,𝑁) is called a fuzzy norm space. Sometimes, we need two additional conditions as follows: (𝐹_𝑁6): for all 𝑡∈𝐑⁺𝑁(𝑥,𝑡)>0⇒𝑥=0. (𝐹_𝑁7): function 𝑁(𝑥,⋅) is continuous for every 𝑥≠0 and on subset {𝑡∶0<𝑁(𝑥,𝑡)<1} (1)is strictly increasing.
Let (𝑈,𝑁) be a fuzzy norm space. For all 𝛼∈(0,1), we define 𝛼 norm on 𝑈 as follows: ‖𝑥‖_𝛼=∧{𝑡>0∶𝑁(𝑥,𝑡)≥𝛼}forevery𝑥∈𝑈. (2)
Then {‖𝑥‖_𝛼∶𝛼∈(0,1]} is an ascending family of normed on 𝑈 and they are called 𝛼-norm on 𝑈 corresponding to the fuzzy norm 𝑁 on 𝑈. Some notation, lemmas, and examples which will be used in this paper are given in the following.

Lemma 2 (see [9]). Let (𝑈,𝑁) be a fuzzy norm space such that it satisfies conditions (𝐹_𝑁6) and (𝐹_𝑁7). Define the function 𝑁^′∶𝑈×𝐑→[0,1] as follows: 𝑁^′()=∨()∶‖𝑥‖𝑥,𝑡𝛼∈0,1_𝛼)≠(),0()=().≤𝑡(𝑥,𝑡0,0𝑥,𝑡0,0 (3) Then(a)𝑁^′ is a fuzzy norm on 𝑈.(b)𝑁=𝑁^′.

Lemma 3 (see [9]). Let (𝑈,𝑁) be a fuzzy norm space such that it satisfies conditions (𝐹_𝑁6) and (𝐹_𝑁7), and {𝑥_𝑛}⊆𝑈. Then lim_𝑛→∞𝑁(𝑥_𝑛−𝑥,𝑡)=1 if and only if lim_𝑛→∞𝑥_𝑛−𝑥_𝛼=0 (4) for every 𝛼∈(0,1).
Note that the sequence {𝑥_𝑛}⊆𝑈 converges if there exists a 𝑥∈𝑈 such that lim_𝑛→∞𝑁𝑥_𝑛−𝑥,𝑡=1𝑓𝑜𝑟𝑒𝑣𝑒𝑟𝑦𝑡∈𝐑⁺. (5) In this case, 𝑥 is called the limit of {𝑥_𝑛}.

Example 4 (see [9]). Let 𝑉 be the real or complex vector space and let 𝑁 be defined on 𝑉×𝑅 as follows: 𝑁()=|𝑥||𝑥|,𝑥,𝑡1𝑡>0𝑡≤ (6) for all 𝑥∈𝑉 and 𝑡∈𝑅. Then (𝑉,𝑁) is a fuzzy norm space, and the function 𝑁 satisfies conditions (𝐹_𝑁6) and ‖𝑥‖_𝛼=|𝑥| for every 𝛼∈(0,1).

3. Fuzzy Approximate Fixed Point

In the section, we begin with two lemmas which will be used in order to prove all the results given in the same section.

Definition 5. Let (𝑈,𝑁) be a fuzzy norm space, 𝑇∶𝑈→𝑈, 𝜖>0, and 𝑢₀∈𝑈. Then 𝑢₀ is an 𝐹^𝑧-approximate fixed point (fuzzy approximate fixed point) of 𝑇 if for some 𝛼∈(0,1), ∧𝑡>0∶𝑁𝑢₀−𝑇𝑢₀,𝑡≥𝛼≤𝜖. (7)

Remark 6. In the rest of the paper we will denote the set of all 𝐹^𝑧-approximate fixed points of 𝑇, for a given 𝜖>0, by 𝐹^𝑧_𝜖(𝑇)={{()}(.𝑢∈𝑈∶∧𝑡>0∶𝑁𝑢−𝑇𝑢,𝑡≥𝛼≤𝜖forsome𝛼∈0,1)} (8)

Definition 7. Let 𝑇∶𝑈→𝑈. Then 𝑇 has the fuzzy approximate fixed point property (f.a.f.p.p.) if ∀𝜖>0𝐹^𝑧_𝜖(𝑇)≠∅. (9)

Lemma 8. Let (𝑈,𝑁) be a fuzzy norm space such that it satisfies conditions (𝐹_𝑁6) and (𝐹_𝑁7), 𝑇∶𝑈→𝑈 such that 𝑇 is asymptotic regular for some 𝛼∈(0,1); that is, lim_𝑛→∞∧𝑡>0∶𝑁𝑇^𝑛𝑢−𝑇^𝑛+1𝑢,𝑡≥𝛼=0,∀𝑢∈𝑈. (10) Then 𝑇 has the fuzzy approximate fixed point property.

Proof. Suppose 𝑢∈𝑈 and 𝜖>0. Since lim_𝑛→∞∧𝑡>0∶𝑁𝑇^𝑛𝑢−𝑇^𝑛+1𝑢,𝑡≥𝛼=0, (11)∃𝑁₀>0s.t. for all 𝑛≥𝑁₀∶∧{𝑡>0∶𝑁(𝑇^𝑛𝑢−𝑇^𝑛+1𝑢,𝑡)≥𝛼}<𝜖 for some  𝛼∈(0,1).
If 𝑛=𝑁₀, then for some 𝛼∈(0,1), ∧𝑡>0∶𝑁𝑇^𝑁0𝑢−𝑇𝑇^𝑁0𝑢,𝑡≥𝛼<𝜖. (12) Therefore, 𝑇^𝑁0𝑢∈𝐹^𝑧_𝜖(𝑇) and 𝐹^𝑧_𝜖(𝑇)≠∅. Hence there exists a fuzzy approximate fixed point in 𝑈.

Definition 9. Let (𝑈,𝑁) be a fuzzy norm space such that satisfies conditions (𝐹_𝑁6) and (𝐹_𝑁7). We define fuzzy diameter of 𝑈≠∅ for some 𝛼∈(0,1) as 𝛿^𝛼(𝑈)=∨[∧{𝑡>0∶𝑁(𝑢−𝑣,𝑡)≥𝛼∀𝑢,𝑣∈𝑈}]. (13)

Lemma 10. Let (𝑈,𝑁) be a fuzzy norm space such that satisfy conditions (𝐹_𝑁6) and (𝐹_𝑁7), 𝑇∶𝑈→𝑈, and 𝜖>0. One assumes that for some 𝛼∈(0,1), (1)𝐹^𝑧_𝜖(𝑇)≠∅, (2)for all 𝜇>0 there exists 𝜑(𝜇)>0 such that ∧{()}{()}{()}𝜇𝑡>0∶𝑁𝑢−𝑣,𝑡≥𝛼−∧𝑡>0∶𝑁𝑇𝑢−𝑇𝑣,𝑡≥𝛼≤𝜇⟹∧𝑡>0∶𝑁𝑢−𝑣,𝑡≥𝛼≤𝜑∀𝑢,𝑣∈𝐹^𝑧_𝜖(𝑇). (14)Then 𝛿^𝛼𝐹^𝑧_𝜖(𝑇)≤𝜑(2𝜖). (15)

Proof. Let 𝜖>0 and 𝑢,𝑣∈𝐹^𝑧_𝜖(𝑇). Then ∧{()}∧{()}𝑡>0∶𝑁𝑢−𝑇𝑢,𝑡≥𝛼<𝜖,𝑡>0∶𝑁𝑣−𝑇𝑣,𝑡≥𝛼<𝜖. (16) We can write ∧{()}{()}{()}{()}{()}{()}{()}𝑡>0∶𝑁𝑢−𝑣,𝑡≥𝛼≤∧𝑡>0∶𝑁𝑢−𝑇𝑢,𝑡≥𝛼+∧𝑡>0∶𝑁𝑇𝑢−𝑇𝑣,𝑡≥𝛼+∧𝑡>0∶𝑁𝑣−𝑇𝑣,𝑡≥𝛼≤∧𝑡>0∶𝑁𝑇𝑢−𝑇𝑣,𝑡≥𝛼+2𝜖⟹∧𝑡>0∶𝑁𝑢−𝑣,𝑡≥𝛼−∧𝑡>0∶𝑁𝑇𝑢−𝑇𝑣,𝑡≥𝛼≤2𝜖. (17) Now by (44), it follows that ∧{𝑡>0∶𝑁(𝑢−𝑣,𝑡)≥𝛼}≤𝜑(2𝜖). (18) So 𝛿^𝛼𝐹^𝑧_𝜖(𝑇)≤𝜑(2𝜖) (19) for some 𝛼∈(0,1).

Lemma 11. Let (𝑈,𝑁) be a fuzzy norm space such that it satisfies conditions (𝐹_𝑁6) and (𝐹_𝑁7), 𝑇∶𝑈→𝑈, and 𝜖>0. One assumes that for some 𝛼∈(0,1),(1)lim_𝑛→∞∧{𝑡>0∶𝑁(𝑇^𝑛𝑢−𝑇^𝑛+1𝑥,𝑡)≥𝛼}=0, for all 𝑢∈𝑈,(2)for all 𝜇>0 there exists 𝜑(𝜇)>0 such that ∧{()}{()}{()}𝜇𝑡>0∶𝑁𝑢−𝑣,𝑡≥𝛼−∧𝑡>0∶𝑁𝑇𝑢−𝑇𝑣,𝑡≥𝛼≤𝜇⟹∧𝑡>0∶𝑁𝑢−𝑣,𝑡≥𝛼≤𝜑∀𝑢,𝑣∈𝐹^𝑧_𝜖(𝑇). (20)Then 𝛿^𝛼𝐹^𝑧_𝜖(𝑇)≤𝜑(2𝜖). (21)

Definition 12. A mapping 𝑇∶𝑈→𝑈 is an 𝑎-contraction if there exists 𝑎∈(0,1) such that ∧{()}{()}𝑡>0∶𝑁𝑇𝑢−𝑇𝑣,𝑡≥𝛼≤𝑎∧𝑡>0∶𝑁𝑢−𝑣,𝑡≥𝛼,∀𝑢,𝑣∈𝑈. (22)

Proposition 13. Let (𝑈,𝑁) be a fuzzy norm space and 𝑇∶𝑈→𝑈 an 𝑎-contraction. Then ∀𝜖>0,𝐹^𝑧_𝜖(𝑇)≠∅. (23)

Proof. Let 𝜖>0 and 𝑢∈𝑈. Consider ∧𝑡>0∶𝑁𝑇^𝑛𝑢−𝑇^𝑛+1𝑇𝑇𝑢,𝑡≥𝛼=∧𝑡>0∶𝑁^𝑛−1𝑢−𝑇𝑇^𝑛𝑢𝑇,𝑡≥𝛼≤𝑎∧𝑡>0∶𝑁^𝑛−1𝑢−𝑇^𝑛⋮𝑢,𝑡≥𝛼≤𝑎^𝑛∧{𝑡>0∶𝑁(𝑢−𝑇𝑢,𝑡)≥𝛼}. (24) But 𝑎∈(0,1). Therefore lim_𝑛→∞∧𝑡>0∶𝑁𝑇^𝑛𝑢−𝑇^𝑛+1𝑥,𝑡≥𝛼=0,∀𝑢∈𝑈. (25) Now by Lemma 8 it follows that 𝐹^𝑧_𝜖(𝑇)≠∅, for all 𝜖>0.

In 1968, Kannan (see [12, 13]) proved a fixed point theorem for operators which needs not be continuous. We apply it to fuzzy norm space for 𝐹^𝑧-approximate fixed points.

Definition 14. A mapping 𝑇∶𝑈→𝑈 is an 𝐹-Kannan operator if there exists 𝑎∈(0,1/2) such that ∧{()}[∧{()}{(),𝑡>0∶𝑁𝑇𝑢−𝑇𝑣,𝑡≥𝛼≤𝑎𝑡>0∶𝑁𝑢−𝑇𝑢,𝑡≥𝛼+∧𝑡>0∶𝑁𝑣−𝑇𝑣,𝑡≥𝛼}] (26) for all 𝑢,𝑣∈𝑈.

Proposition 15. Let (𝑈,𝑁) be a fuzzy norm space and 𝑇∶𝑈→𝑈 a Kannan operator. Then ∀𝜖>0,𝐹^𝑧_𝜖(𝑇)≠∅. (27)

Proof. Let 𝜖>0 and 𝑢∈𝑈. Consider ∧𝑡>0∶𝑁𝑇^𝑛𝑢−𝑇^𝑛+1𝑇𝑇𝑢,𝑡≥𝛼=∧𝑡>0∶𝑁^𝑛−1𝑢−𝑇𝑇^𝑛𝑢∧𝑇,𝑡≥𝛼≤𝑎𝑡>0∶𝑁^𝑛−1𝑢−𝑇𝑇^𝑛−1𝑢𝑇,𝑡≥𝛼+∧𝑡>0∶𝑁^𝑛𝑢−𝑇𝑇^𝑛𝑢𝑇,𝑡≥𝛼=𝑎∧𝑡>0∶𝑁^𝑛−1𝑢−𝑇𝑇^𝑛−1𝑢𝑇,𝑡≥𝛼+𝑎∧𝑡>0∶𝑁^𝑛𝑢−𝑇𝑇^𝑛𝑢,𝑡≥𝛼. (28) Therefore (1−𝑎)∧𝑡>0∶𝑁𝑇^𝑛𝑢−𝑇^𝑛+1𝑇𝑢,𝑡≥𝛼≤𝑎∧𝑡>0∶𝑁^𝑛−1𝑢−𝑇^𝑛𝑢,𝑡≥𝛼. (29) Then ∧𝑡>0∶𝑁𝑇^𝑛𝑢−𝑇^𝑛+1≤𝑎𝑢,𝑡≥𝛼∧𝑇1−𝑎𝑡>0∶𝑁^𝑛−1𝑢−𝑇^𝑛⋮≤𝑎𝑢,𝑡≥𝛼1−𝑎^𝑛∧{𝑡>0∶𝑁(𝑢−𝑇𝑢,𝑡)≥𝛼}. (30) But 𝑎∈(0,1/2); hence 𝑎/(1−𝑎)∈(0,1). Therefore lim_𝑛→∞∧𝑡>0∶𝑁𝑇^𝑛𝑢−𝑇^𝑛+1𝑥,𝑡≥𝛼=0,∀𝑢∈𝑈. (31) Now by Lemma 8 it follows that 𝐹^𝑧_𝜖(𝑇)≠∅, for all 𝜖>0.

In 1972, Chatterjea (see [14]) considered another one which again does not impose the continuity of the operator. We apply it to fuzzy norm space for 𝐹^𝑧-approximate fixed points.

Definition 16. A mapping 𝑇∶𝑈→𝑈 is an 𝐹-Chatterjea operator if there exists 𝑎∈(0,1/2) such that ∧{()}[∧{()}{(),𝑡>0∶𝑁𝑇𝑢−𝑇𝑣,𝑡≥𝛼≤𝑎𝑡>0∶𝑁𝑢−𝑇𝑣,𝑡≥𝛼+∧𝑡>0∶𝑁𝑣−𝑇𝑢,𝑡≥𝛼}] (32) for all 𝑢,𝑣∈𝑈.

Proposition 17. Let (𝑈,𝑁) be a fuzzy norm space and 𝑇∶𝑈→𝑈 a Chatterjea operator. Then ∀𝜖>0,𝐹^𝑧_𝜖(𝑇)≠∅. (33)

Proof. Let 𝜖>0 and 𝑢∈𝑈. Consider ∧𝑡>0∶𝑁𝑇^𝑛𝑢−𝑇^𝑛+1𝑇𝑇𝑢,𝑡≥𝛼=∧𝑡>0∶𝑁^𝑛−1𝑢−𝑇𝑇^𝑛𝑢∧𝑇,𝑡≥𝛼≤𝑎𝑡>0∶𝑁^𝑛−1𝑢−𝑇𝑇^𝑛𝑢𝑇,𝑡≥𝛼+∧𝑡>0∶𝑁^𝑛𝑢−𝑇𝑇^𝑛−1𝑢𝑇,𝑡≥𝛼=𝑎∧𝑡>0∶𝑁^𝑛−1𝑢−𝑇^𝑛+1𝑇𝑢,𝑡≥𝛼+𝑎∧𝑡>0∶𝑁^𝑛𝑢−𝑇^𝑛∧𝑇𝑢,𝑡≥𝛼=𝑎𝑡>0∶𝑁^𝑛−1𝑢−𝑇^𝑛+1𝑢,𝑡≥𝛼. (34) On the other hand, ∧𝑡>0∶𝑁𝑇^𝑛−1𝑢−𝑇^𝑛+1𝑇𝑢,𝑡≥𝛼≤∧𝑡>0∶𝑁^𝑛−1𝑢−𝑇^𝑛𝑇𝑢,𝑡≥𝛼+∧𝑡>0∶𝑁^𝑛𝑢−𝑇^𝑛+1𝑢,𝑡≥𝛼. (35) Then (1−𝑎)∧𝑡>0∶𝑁𝑇^𝑛𝑢−𝑇^𝑛+1𝑇𝑢,𝑡≥𝛼≤𝑎∧𝑡>0∶𝑁^𝑛−1𝑢−𝑇^𝑛𝑢,𝑡≥𝛼. (36) Hence ∧𝑡>0∶𝑁𝑇^𝑛𝑢−𝑇^𝑛+1≤𝑎𝑢,𝑡≥𝛼∧𝑇1−𝑎𝑡>0∶𝑁^𝑛−1𝑢−𝑇^𝑛⋮≤𝑎𝑢,𝑡≥𝛼1−𝑎^𝑛∧{𝑡>0∶𝑁(𝑢−𝑇𝑢,𝑡)≥𝛼}. (37) But 𝑎∈(0,1/2); hence 𝑎/(1−𝑎)∈(0,1). Therefore lim_𝑛→∞∧𝑡>0∶𝑁𝑇^𝑛𝑢−𝑇^𝑛+1𝑥,𝑡≥𝛼=0,∀𝑢∈𝑈. (38) Now by Lemma 8 it follows that 𝐹^𝑧_𝜖(𝑇)≠∅, for all 𝜖>0.

We, by combining the three independent contraction conditions above, obtain another 𝐹^𝑧-approximation fixed points result for operators which satisfies the following.

Definition 18. A mapping 𝑇∶𝑈→𝑈 is an 𝐹-Zamfirescu operator if there exist 𝛼,𝛽,𝛾∈𝐑,  𝛼∈[0,1[, 𝛽∈[0,1/2[, 𝛾∈[0,1/2[ such that for all 𝑥,𝑦∈𝑈, at least one of the following is true: 𝐹_𝑍1∶∧{𝑡>0∶𝑁(𝑇𝑢−𝑇𝑣,𝑡)≥𝛼}≤𝛼₁∧{()},𝐹𝑡>0∶𝑁𝑢−𝑣,𝑡≥𝛼_𝑍2{()}[∧{()}{(),𝐹∶∧𝑡>0∶𝑁𝑇𝑢−𝑇𝑣,𝑡≥𝛼≤𝛽𝑡>0∶𝑁𝑢−𝑇𝑢,𝑡≥𝛼+∧𝑡>0∶𝑁𝑣−𝑇𝑣,𝑡≥𝛼}]_𝑍3{()}[∧{()}{().∶∧𝑡>0∶𝑁𝑇𝑢−𝑇𝑣,𝑡≥𝛼≤𝛾𝑡>0∶𝑁𝑢−𝑇𝑣,𝑡≥𝛼+∧𝑡>0∶𝑁𝑣−𝑇𝑢,𝑡≥𝛼}] (39)

Proposition 19. Let (𝑈,𝑁) be a fuzzy norm space and 𝑇∶𝑈→𝑈 an 𝐹-Zamfirescu operator. Then ∀𝜖>0,𝐹^𝑧_𝜖(𝑇)≠∅. (40)

Proof. Let 𝑥,𝑦∈𝑈.
Supposing that (𝐹_𝑍2) holds, we have ∧{()}[∧{()}{()[∧{()}{()}{(){()}{()}{()}.𝑡>0∶𝑁𝑇𝑢−𝑇𝑣,𝑡≥𝛼≤𝛽𝑡>0∶𝑁𝑢−𝑇𝑢,𝑡≥𝛼+∧𝑡>0∶𝑁𝑣−𝑇𝑣,𝑡≥𝛼}]≤𝛽𝑡>0∶𝑁𝑣−𝑢,𝑡≥𝛼+∧𝑡>0∶𝑁𝑢−𝑇𝑢,𝑡≥𝛼+∧𝑡>0∶𝑁𝑇𝑢−𝑇𝑣,𝑡≥𝛼}]=2𝛽∧𝑡>0∶𝑁𝑢−𝑇𝑢,𝑡≥𝛼+𝛽∧𝑡>0∶𝑁𝑢−𝑣,𝑡≥𝛼+𝛽∧𝑡>0∶𝑁𝑇𝑢−𝑇𝑣,𝑡≥𝛼 (41) Thus ∧{()}≤𝑡>0∶𝑁𝑇𝑢−𝑇𝑣,𝑡≥𝛼2𝛽∧{()}+𝛽1−𝛽𝑡>0∶𝑁𝑢−𝑇𝑢,𝑡≥𝛼∧{()}.1−𝛽𝑡>0∶𝑁𝑢−𝑣,𝑡≥𝛼 (42)
Supposing (𝐹_𝑍3) holds, we have ∧{()}[∧{()}{()[∧{()}{()[∧{()[∧{()}{(){()}{()}{()}.𝑡>0∶𝑁𝑇𝑢−𝑇𝑣,𝑡≥𝛼≤𝛾𝑡>0∶𝑁𝑢−𝑇𝑣,𝑡≥𝛼+∧𝑡>0∶𝑁𝑣−𝑇𝑢,𝑡≥𝛼}]≤𝛾𝑡>0∶𝑁𝑢−𝑣,𝑡≥𝛼+∧𝑡>0∶𝑁𝑣−𝑇𝑣,𝑡≥𝛼}]+𝛾𝑡>0∶𝑁𝑣−𝑇𝑣,𝑡≥𝛼}]+𝛾𝑡>0∶𝑁𝑣−𝑇𝑣,𝑡≥𝛼+∧𝑡>0∶𝑁𝑇𝑣−𝑇𝑢,𝑡≥𝛼}]=𝛾∧𝑡>0∶𝑁𝑇𝑢−𝑇𝑣,𝑡≥𝛼+2𝛾∧𝑡>0∶𝑁𝑣−𝑇𝑣,𝑡≥𝛼+𝛾∧𝑡>0∶𝑁𝑢−𝑣,𝑡≥𝛼 (43) Thus ∧{()}≤𝑡>0∶𝑁𝑇𝑢−𝑇𝑣,𝑡≥𝛼2𝛾∧{()}+𝛾1−𝛾𝑡>0∶𝑁𝑣−𝑇𝑣,𝑡≥𝛼∧{()}.1−𝛾𝑡>0∶𝑁𝑢−𝑣,𝑡≥𝛼 (44)
Similarly, ∧{()}[∧{()}{()[∧{()}{()[∧{()}{(){()}{()}{()}.𝑡>0∶𝑁𝑇𝑢−𝑇𝑣,𝑡≥𝛼≤𝛾𝑡>0∶𝑁𝑢−𝑇𝑣,𝑡≥𝛼+∧𝑡>0∶𝑁𝑣−𝑇𝑢,𝑡≥𝛼}]≤𝛾𝑡>0∶𝑁𝑢−𝑇𝑢,𝑡≥𝛼+∧𝑡>0∶𝑁𝑇𝑢−𝑇𝑣,𝑡≥𝛼}]+𝛾𝑡>0∶𝑁𝑣−𝑢,𝑡≥𝛼+∧𝑡>0∶𝑁𝑢−𝑇𝑢,𝑡≥𝛼}]=𝛾∧𝑡>0∶𝑁𝑇𝑢−𝑇𝑣,𝑡≥𝛼+2𝛾∧𝑡>0∶𝑁𝑢−𝑇𝑢,𝑡≥𝛼+𝛾∧𝑡>0∶𝑁𝑢−𝑣,𝑡≥𝛼 (45)
Then ∧{()}≤𝑡>0∶𝑁𝑇𝑢−𝑇𝑣,𝑡≥𝛼2𝛾∧{()}+𝛾1−𝛾𝑡>0∶𝑁𝑢−𝑇𝑢,𝑡≥𝛼∧{()}.1−𝛾𝑡>0∶𝑁𝑢−𝑣,𝑡≥𝛼 (46)
Now looking at (𝐹_𝑍1), (42), (44), and (46), we can denote 𝜂=max𝛼₁,𝛽,𝛾1−𝛽,1−𝛾 (47) and it is easy to see that 𝜂∈[0,1[.
For 𝑇 satisfying at least one of the conditions (𝐹_𝑍1), (𝐹_𝑍2), and (𝐹_𝑍3), we have that ∧{()}{()}{()},∧{()}{()}{()}𝑡>0∶𝑁𝑇𝑢−𝑇𝑣,𝑡≥𝛼≤2𝜂∧𝑡>0∶𝑁𝑢−𝑇𝑢,𝑡≥𝛼+𝜂∧𝑡>0∶𝑁𝑢−𝑣,𝑡≥𝛼𝑡>0∶𝑁𝑇𝑢−𝑇𝑣,𝑡≥𝛼≤2𝜂∧𝑡>0∶𝑁𝑣−𝑇𝑣,𝑡≥𝛼+𝜂∧𝑡>0∶𝑁𝑢−𝑣,𝑡≥𝛼 (48) hold.
Using these conditions implied by (𝐹_𝑍1)--(𝐹_𝑍3) and taking 𝑥∈𝑈, we have ∧𝑡>0∶𝑁𝑇^𝑛𝑢−𝑇^𝑛+1𝑇𝑇𝑢,𝑡≥𝛼=∧𝑡>0∶𝑁^𝑛−1𝑢−𝑇𝑇^𝑛𝑢,𝑡≥𝛼₍₄₈₎≤2𝜂∧𝑡>0∶𝑁𝑇^𝑛−1𝑢−𝑇𝑇^𝑛−1𝑢𝑇,𝑡≥𝛼+𝜂∧𝑡>0∶𝑁^𝑛−1𝑢−𝑇𝑇^𝑛𝑢𝑇,𝑡≥𝛼=3𝜂∧𝑡>0∶𝑁^𝑛−1𝑢−𝑇^𝑛𝑢,𝑡≥𝛼. (49) Then ∧𝑡>0∶𝑁𝑇^𝑛𝑢−𝑇^𝑛+1𝑢,𝑡≥𝛼≤⋯≤3𝜂³∧{𝑡>0∶𝑁((𝑢−𝑇𝑢),𝑡)≥𝛼}. (50)
Therefore lim_𝑛→∞∧𝑡>0∶𝑁𝑇^𝑛𝑢−𝑇^𝑛+1𝑥,𝑡≥𝛼=0,∀𝑢∈𝑈. (51) Now by Lemma 8 it follows that 𝐹^𝑧_𝜖(𝑇)≠∅, for all 𝜖>0.

Definition 20. A mapping 𝑇∶𝑈→𝑈 is an 𝐹-weak contraction operator if there exist 𝛼₁∈[0,1[ and 𝐿≥0 such that∧{𝑡>0∶𝑁(𝑇𝑢−𝑇𝑣,𝑡)≥𝛼}≤𝛼₁∧{()}{()}𝑡>0∶𝑁𝑢−𝑣,𝑡≥𝛼+𝐿∧𝑡>0∶𝑁𝑣−𝑇𝑢,𝑡≥𝛼 (52) for all 𝑥,𝑦∈𝑈.

Proposition 21. Let (𝑈,𝑁) be a fuzzy norm space and 𝑇∶𝑈→𝑈 an 𝐹-weak contraction operator. Then ∀𝜖>0,𝐹^𝑧_𝜖(𝑇)≠∅. (53)

Proof. Let 𝑥∈𝑈. Consider ∧𝑡>0∶𝑁𝑇^𝑛𝑢−𝑇^𝑛+1𝑇𝑇𝑢,𝑡≥𝛼=∧𝑡>0∶𝑁^𝑛−1𝑢−𝑇𝑇^𝑛𝑢,𝑡≥𝛼≤𝛼₁∧𝑡>0∶𝑁𝑇^𝑛−1𝑢−𝑇^𝑛𝑢𝑇,𝑡≥𝛼+𝐿∧𝑡>0∶𝑁^𝑛𝑢−𝑇^𝑛𝑇𝑢,𝑡≥𝛼=∧𝑡>0∶𝑁^𝑛−1𝑢−𝑇^𝑛𝑢,𝑡≥𝛼≤⋯≤𝛼^𝑛₁∧{𝑡>0∶𝑁(𝑢−𝑇𝑢,𝑡)≥𝛼}. (54) But 𝛼₁∈[0,1[. Therefore lim_𝑛→∞∧𝑡>0∶𝑁𝑇^𝑛𝑢−𝑇^𝑛+1𝑥,𝑡≥𝛼=0,∀𝑢∈𝑈. (55) Now by Lemma 8, it follows that 𝐹^𝑧_𝜖(𝑇)≠∅, for all 𝜖>0.

For the same operators, we will now use Lemma 10 in order to obtain new results.

Proposition 22. Let (𝑈,𝑁) be a fuzzy norm space and 𝑇∶𝑈→𝑈 an 𝑎-contraction operator.
Then for some 𝛼∈(0,1), 𝛿^𝛼𝐹^𝑧_𝜖(𝑇)≤2𝜖1−𝑎,∀𝜖>0. (56)

Proof. Let 𝜖>0. Condition (1) in Lemma 10 is satisfied, as one can see in the proof of Proposition 13.
We will show now that (2) also holds for 𝑎-contractions.
Let 𝜇>0 and 𝑥,𝑦∈𝐹^𝑧_𝜖(𝑇) and assume that ∧{()}{()}𝑡>0∶𝑁𝑢−𝑣,𝑡≥𝛼−∧𝑡>0∶𝑁𝑇𝑢−𝑇𝑣,𝑡≥𝛼≤𝜇 (57) and aim to show that there exists an 𝜑(𝜇)>0 such that ∧{𝑡>0∶𝑁(𝑢−𝑣,𝑡)≥𝛼}≤𝜑𝜇. (58) We have ∧{()}{()}{()}𝑡>0∶𝑁𝑢−𝑣,𝑡≥𝛼≤∧𝑡>0∶𝑁𝑇𝑢−𝑇𝑣,𝑡≥𝛼+𝜇≤𝑎∧𝑡>0∶𝑁𝑢−𝑣,𝑡≥𝛼+𝜇. (59)
Therefore (1−𝑎)∧{𝑡>0∶𝑁(𝑢−𝑣,𝑡)≥𝛼}≤𝜇, (60) which implies ∧{𝑡>0∶𝑁(𝑢−𝑣,𝑡)≥𝛼}≤𝜇/(1−𝑎).
So for every 𝜇>0, there exist 𝜑(𝜇)=𝜇/(1−𝑎)>0 such that ∧{()}{()}𝑡>0∶𝑁𝑢−𝑣,𝑡≥𝛼−∧𝑡>0∶𝑁𝑇𝑢−𝑇𝑣,𝑡≥𝛼≤𝜇. (61) Therefore ∧{𝑡>0∶𝑁(𝑇𝑢−𝑇𝑣,𝑡)≥𝛼}≤𝜑(𝜇).
Now by Lemma 10 it follows that 𝛿^𝛼𝐹^𝑧_𝜖(𝑇)≤𝜑(2𝜖),∀𝜖>0, (62) for some 𝛼∈(0,1), which means exactly that 𝛿^𝛼𝐹^𝑧_𝜖(𝑇)≤2𝜖1−𝑎,∀𝜖>0, (63) for some 𝛼∈(0,1).

Proposition 23. Let (𝑈,𝑁) be a fuzzy norm space and 𝑇∶𝑈→𝑈 an 𝐹-Kannan operator.
Then for some 𝛼∈(0,1), 𝛿^𝛼𝐹^𝑧_𝜖(𝑇)≤2𝜖(1+𝑎),∀𝜖>0. (64)

Proof. Let 𝜖>0. As in the proof of Proposition 22 we only verify that condition (2) in Lemma 10 holds.
Let 𝜇>0 and 𝑥,𝑦∈𝐹^𝑧_𝜖(𝑇) and assume that ∧{()}{()}𝑡>0∶𝑁𝑢−𝑣,𝑡≥𝛼−∧𝑡>0∶𝑁𝑇𝑢−𝑇𝑣,𝑡≥𝛼≤𝜇. (65) Then ∧{()}[∧{()}{()𝑡>0∶𝑁𝑢−𝑣,𝑡≥𝛼≤𝑎𝑡>0∶𝑁𝑢−𝑇𝑢,𝑡≥𝛼+∧𝑡>0∶𝑁𝑣−𝑇𝑣,𝑡≥𝛼}]+𝜇. (66) As 𝑥,𝑦∈𝐹^𝑧_𝜖(𝑇), we know that ∧{𝑡>0∶𝑁(𝑢−𝑇𝑢,𝑡)≥𝛼}<𝜖 and ∧{𝑡>0∶𝑁(𝑣−𝑇𝑣,𝑡)≥𝛼}<𝜖.
Then ∧{𝑡>0∶𝑁(𝑢−𝑣,𝑡)≥𝛼}≤2𝑎𝜖+𝜇. So for every 𝜇>0 there exist 𝜑(𝜇)=𝜇+2𝑎𝜖>0 such that ∧{()}{()}𝑡>0∶𝑁𝑢−𝑣,𝑡≥𝛼−𝑎∧𝑡>0∶𝑁𝑇𝑢−𝑇𝑣,𝑡≥𝛼≤𝜇. (67) Therefore ∧{𝑡>0∶𝑁(𝑢−𝑣,𝑡)≥𝛼}≤𝜇. Now by Lemma 10 it follows that 𝛿^𝛼𝐹^𝑧_𝜖(𝑇)≤𝜑(2𝜖),∀𝜖>0, (68) for some 𝛼∈(0,1), which means exactly that 𝛿^𝛼𝐹^𝑧_𝜖(𝑇)≤2𝜖(1+𝑎),∀𝜖>0, (69) for some 𝛼∈(0,1).

Proposition 24. Let (𝑈,𝑁) be a fuzzy norm space and 𝑇∶𝑈→𝑈 an 𝐹-Chatterjea operator.
Then for some 𝛼∈(0,1), 𝛿^𝛼𝐹^𝑧_𝜖(𝑇)≤2𝜖(1+𝑎)1−2𝑎,∀𝜖>0. (70)

Proof. Let 𝜖>0. We will only verify that condition (2) in Lemma 10 holds.
Let 𝜇>0 and 𝑥,𝑦∈𝐹^𝑧_𝜖(𝑇) and assume that ∧{()}{()}𝑡>0∶𝑁𝑢−𝑣,𝑡≥𝛼−∧𝑡>0∶𝑁𝑇𝑢−𝑇𝑣,𝑡≥𝛼≤𝜇. (71) Then ∧{()}[∧{()}{(){()}{()}[∧{()}{()[∧{()}{()𝑡>0∶𝑁𝑢−𝑣,𝑡≥𝛼≤𝑎𝑡>0∶𝑁𝑢−𝑇𝑣,𝑡≥𝛼+∧𝑡>0∶𝑁𝑣−𝑇𝑢,𝑡≥𝛼}]+𝜇≤𝑎∧𝑡>0∶𝑁𝑢−𝑇𝑣,𝑡≥𝛼+𝑎∧𝑡>0∶𝑁𝑣−𝑇𝑢,𝑡≥𝛼+𝜇≤𝑎𝑡>0∶𝑁𝑢−𝑣,𝑡≥𝛼−∧𝑡>0∶𝑁𝑣−𝑇𝑣,𝑡≥𝛼}]+𝑎𝑡>0∶𝑁𝑣−𝑢,𝑡≥𝛼−∧𝑡>0∶𝑁𝑢−𝑇𝑢,𝑡≥𝛼}]+𝜇. (72) As 𝑥,𝑦∈𝐹^𝑧_𝜖(𝑇), it follows that ∧{()}{()}𝑡>0∶𝑁𝑢−𝑣,𝑡≥𝛼≤2𝑎∧𝑡>0∶𝑁𝑢−𝑣,𝑡≥𝛼2𝑎𝜖+𝜇. (73) Then (1−2𝑎)∧{𝑡>0∶𝑁(𝑢−𝑣,𝑡)≥𝛼}≤2𝑎𝜖+𝜇⇒∧{𝑡>0∶𝑁(𝑢−𝑣,𝑡)≥𝛼}≤2𝑎𝜖+𝜇.1−2𝑎 (74)
So for every 𝜇>0 there exist 𝜑(𝜇)=(2𝑎𝜖+𝜇)/(1−2𝑎)>0 such that ∧{()}{()}𝑡>0∶𝑁𝑢−𝑣,𝑡≥𝛼−∧𝑡>0∶𝑁𝑇𝑢−𝑇𝑣,𝑡≥𝛼≤𝜇. (75) Then ∧{𝑡>0∶𝑁(𝑢−𝑣,𝑡)≥𝛼}≤𝜑(𝜇). Now by Lemma 10 it follows that 𝛿^𝛼𝐹^𝑧_𝜖(𝑇)≤𝜑(2𝜖),∀𝜖>0, (76) for some 𝛼∈(0,1), which means exactly that 𝛿^𝛼𝐹^𝑧_𝜖(𝑇)≤2𝑎𝜖+𝜇1−2𝑎,∀𝜖>0, (77) for some 𝛼∈(0,1).

Proposition 25. Let (𝑈,𝑁) be a fuzzy norm space and 𝑇∶𝑈→𝑈 an 𝐹-Zamfirescu operator. Then for some 𝛼∈(0,1), 𝛿^𝛼𝐹^𝑧_𝜖(𝑇)≤2𝜖1+𝜂1−𝜂,∀𝜖>0, (78) where 𝜂=max{𝛼₁,𝛽/(1−𝛽),𝛾/(1−𝛾)}, and 𝛼, 𝛽, 𝛾 as in Definition 18.

Proof. In the proof of Proposition 19 we have already shown that if 𝑓 satisfies at least one of the conditions (𝐹_𝑍1), (𝐹_𝑍2), and (𝐹_𝑍3) from Definition 18, then∧{()}{()}{()},∧{()}{()}{()}𝑡>0∶𝑁𝑇𝑢−𝑇𝑣,𝑡≥𝛼≤2𝜂∧𝑡>0∶𝑁𝑢−𝑇𝑢,𝑡≥𝛼+𝜂∧𝑡>0∶𝑁𝑢−𝑣,𝑡≥𝛼𝑡>0∶𝑁𝑇𝑢−𝑇𝑣,𝑡≥𝛼≤2𝜂∧𝑡>0∶𝑁𝑣−𝑇𝑣,𝑡≥𝛼+𝜂∧𝑡>0∶𝑁𝑢−𝑣,𝑡≥𝛼 (79) hold.
Let 𝜖>0. We will only verify that condition (2) in Lemma 10 is satisfied, as (1) holds; see the Proof of Proposition 19.
Let 𝜇>0 and 𝑥,𝑦∈𝐹^𝑧_𝜖(𝑇) and assume that ∧{()}{()}𝑡>0∶𝑁𝑢−𝑣,𝑡≥𝛼−∧𝑡>0∶𝑁𝑇𝑢−𝑇𝑣,𝑡≥𝛼≤𝜇. (80)
Let 𝜇>0 and 𝑥,𝑦∈𝐹^𝑧_𝜖(𝑇) and assume that ∧{()}{()}𝑡>0∶𝑁𝑢−𝑣,𝑡≥𝛼−∧𝑡>0∶𝑁𝑇𝑢−𝑇𝑣,𝑡≥𝛼≤𝜇. (81) Then ∧{()}{()}{()}{()}⟹∧{()}{()}≤𝑡>0∶𝑁𝑢−𝑣,𝑡≥𝛼≤∧𝑡>0∶𝑁𝑇𝑢−𝑇𝑣,𝑡≥𝛼+𝜇≤2𝜂∧𝑡>0∶𝑁𝑢−𝑇𝑢,𝑡≥𝛼+𝜂∧𝑡>0∶𝑁𝑢−𝑣,𝑡≥𝛼+𝜇1−𝜂𝑡>0∶𝑁𝑢−𝑣,𝑡≥𝛼≤2𝜂𝜖+𝜇⟹∧𝑡>0∶𝑁𝑣−𝑢,𝑡≥𝛼2𝜂𝜖+𝜇.1−𝜂 (82)
So for every 𝜇>0 there exist 𝜑(𝜇)=(2𝜂𝜖+𝜇)/(1−𝜂)>0 such that ∧{()}{()}𝑡>0∶𝑁𝑢−𝑣,𝑡≥𝛼−∧𝑡>0∶𝑁𝑇𝑢−𝑇𝑣,𝑡≥𝛼≤𝜇. (83) Then ∧{𝑡>0∶𝑁(𝑢−𝑣,𝑡)≥𝛼}≤𝜑(𝜇). Now by Lemma 10 it follows that 𝛿^𝛼𝐹^𝑧_𝜖(𝑇)≤𝜑(2𝜖),∀𝜖>0, (84) for some 𝛼∈(0,1), which means exactly that 𝛿^𝛼𝐹^𝑧_𝜖(𝑇)≤2𝜖1+𝜂1−𝜂,∀𝜖>0, (85) for some 𝛼∈(0,1).

Example 26. Let us consider the linear space 𝑈=𝐶[0,1], the space of all continuous real valued functions on [0,1] with the usual linear operations.
Consider the two norms on 𝐶[0,1] defined by ‖𝑥‖₀=¹_𝑥0(𝑥(𝑡))²𝑑𝑡^1/2,‖𝑥‖₁=sup_0≤𝑡≤1|𝑥(𝑡)|. (86) Define two fuzzy norms 𝑁₁ and 𝑁₂ by 𝑁₁()=‖𝑥‖𝑥,𝑡1𝑡>₀,0𝑡≤‖𝑥‖₀,𝑁₂()=‖𝑥‖𝑥,𝑡1𝑡>₁,12‖𝑥‖₀≤𝑡≤‖𝑥‖₁,0𝑡≤‖𝑥‖₀. (87) For the fuzzy norm 𝑁₁, its 𝛼-norm ‖⋅‖¹_𝛼 is given by ‖⋅‖¹_𝛼=‖𝑥‖₀ for all 𝛼∈(0,1), and for the fuzzy norm 𝑁₂, its 𝛼-norm ‖⋅‖²_𝛼 is given by ‖𝑥‖_𝛼=‖𝑥‖₀10<𝛼≤2,‖𝑥‖₁12<𝛼<1. (88) Suppose that 𝑇∶𝑈→𝑈 with 𝑇(𝑥)=𝑥+1 for all 𝑥∈𝑈. It can easily be seen that for 1/2<𝛼<1 and 𝑥,𝑦∈𝑈, ∧≤1𝑡>0∶𝑁𝑇𝑥−𝑇𝑦,𝑡≥𝛼2∧𝑡>0∶𝑁𝑥−𝑦,𝑡≥𝛼. (89) However, 𝑇 has no any fixed point of course. But by the Proposition 13 for some 𝜖>0, 𝑇 has a fuzzy approximate fixed point in 𝑈. That is, there exists 𝑢₀∈𝑈 such that ∧𝑡>0∶𝑁𝑢₀−𝑇𝑢₀,𝑡≥𝛼≤𝜖. (90)

4. Conclusions

Nowadays, fixed point and operator theory play an important role in different areas of mathematics and its applications, particularly in mathematics, physics, differential equation, game theory, and dynamic programming. Since fuzzy mathematics and fuzzy physics along with the classical ones are constantly developing, the fuzzy type of the fixed point and operator theory can also play an important role in the new fuzzy area and fuzzy mathematical physics.

We think that this paper could be of interest to the researchers working in the field fuzzy functional analysis in particular, fuzzy approximate fixed point theory. We proved results about fuzzy approximate fixed points and fuzzy diameter on fuzzy norm spaces, starting from a result presented in [1], but the study may go further to other classes of operators, which will be the subject of future papers.

Acknowledgment

The authors are extremely grateful to the referees for their helpful suggestions for the improvement of the paper.