Extension of fixed point results in intuitionistic fuzzy b metric space

1 Introduction

In 1965, Zadeh [23] introduced a new branch of concept called fuzzy set theory to explain and represent mathematical structure for the situations where data are imprecise or vague. Simultaneously, in 1986 Atanassaov [2] generalized this concept to a new form which deals with both the degree of membership (belongingness) and non-membership (non-belongingness) of an elements within a set and named it as intuitionistic fuzzy set. Initially the concept of fuzzy metric space was established by Kaleva and Seikkala [14]. After that some mathematician like Kramosil and Michalek [15], George and Veeramani [10] modify the definition of fuzzy metric space. By considering a weaker condition instead of the triangular inequality Bakhtin [4] and Czerwike [7] established the concept of b metric space.

Heilpern [13] initiated the study of fixed point theory in fuzzy metric spaces by establishing an extension of the Banach’s contraction principle in fuzzy setting. This famous principle has been further extended and generalized by many authors in fuzzy settings from different points of view [1, 18–21].

The concept of b-metric space plays a paramount role to signify the existence of fixed point theory. Since existence of fixed point implies the existence of solution of graphs therefore the study of fixed point theory in various generalized spaces place great emphasis in the field of designing and modeling system. However sometimes modeling of systems seem difficult due to the lake of exact distance between two points or elements. In this case, fuzzy sets or fuzzy logic provide a consequential platform in the field of modeling and designing.

As intuitionistic fuzzy set is a generalized form of fuzzy set it deals with more complex situation for higher order fuzzy sets and rescue the complexity of modeling systems. Its more flexible nature in application motives us to work on the current study. The main objective of this work is to extent the concept of existence of fixed point theorems and contraction theorems into generalized intuitionistic fuzzy b metric space and establish some new results which are helpful for further generalization of the current work. In the present work we establish the definition of IFbMS along with some more results. Main motive of this work is to provides some new fixed point theorem and to study the unique fixed point for IFbMS. This work also contain the notion of Cauchy sequence in the setting of IFbMS. Due to the importance of contraction condition our work also study the Ćirić quasi-Contraction theorem for IFbMS along with some examples.

Following definitions are used for the main work.

Definition 1.1. Suppose X is a non-empty set. Consider a three-tuple (X, ℏ , ∗) where ∗ is a continuous t-norm and ℏ are fuzzy set on X² × (0, ∞). Then (X, ℏ , ∗) is called an fuzzy b-metric space if ∀ υ, ξ, z ∈ X and s, t > 0 and a given number b ≥ 1 following conditions are holds:

ℏ (υ, ξ, t) >0,

Definition 1.2. [2] Consider a non-empty set X. Then an intuitionistic fuzzy set (in short “IFS") I in X is defined as E = {(x, ℏ  _E  (x) , ℘  _E  (x)) : x ∈ X, 0 ≤ ℏ  _I  + ℘  _I  ≤ 1} where the functions ℏ I : X → [ 0 , 1 ] ⊆ ℝ and ℘ I : X → [ 0 , 1 ] ⊆ ℝ define the degree of membership and non-membership function of the element x ∈ X respectively.

Definition 1.3. [17] Consider F as a t-norm and for n ∈ ℕ , υ ∈ [0, 1], we have F _n  : [0, 1] → [0, 1] such that F₁ (υ) = F (υ, υ), F_n+1 (υ) = F (F _n  (υ) , υ) then F is called H-type if {F _n  (υ)}, for n ∈ ℕ is equicontinuous at υ = 1.

Proposition 1.4. [12] Consider a sequence of numbers {υ _n } ∈ [0, 1] such that lim n → ∞ υ n = 1 and consider a t-norm F of H-type. Then lim n → ∞ T i = 1 ∞ υ i = lim n → ∞ T i = 1 ∞ υ n + i = 1 .

Main results of the paper is categorized as follows. In Section 2, we establish some propositions related to Convergence and Cauchy sequences in IFbMS. In Section 3, we prove the existence and uniqueness of certain fixed points in this space with respect to some new contractive conditions. Finally, Ciric-quasi contractions are studied in Section 4.

2 Intuitionistic fuzzy b metric space and related propositions

This section contains the concept of intuitionistic fuzzy b metric space(in short, IFbMS). We also established some Lemma and corollary in the settings of IFbMS.

Definition 2.1. Suppose X is a non-empty set. Consider a five-tuple (X, ℏ , ℘ , ∗ , ∘) where ∗ is a continuous t-norm, ∘ is a continuous t-conorm and ℏ, ℘ are fuzzy sets on X² × (0, ∞). Then (X, ℏ , ℘ , ∗ , ∘) is called an intuitionistic fuzzy b metric space(in short, IFbMS) if ∀ υ, ξ, z ∈ X and s, t > 0 and a given number b ≥ 1 following conditions are holds:

ℏ (υ, ξ, t) + ℘ (υ, ξ, t) ≤1,

Example 2.2. Consider that ℏ ( υ , ξ , t ) = e - | υ - ξ | p t and ℘ ( υ , ξ , t ) = 1 - e - | υ - ξ | p t , where p > 1 is a real number. Then (X, ℏ , ℘ , ∗ , ∘) is become a IFbMS with b = 2^p-1.

Lemma 2.3. Consider a sequence {υ _n } in IFbMS (X, ℏ , ℘ , ∗ , ∘). Assume that there exists λ ∈ ( 0 , 1 b ) , t > 0 such that ℏ ( υ n , υ n + 1 , t ) ≥ ℏ ( υ n , υ n + 1 , t λ ) , n ∈ ℕ(1) and ℘ ( υ n , υ n + 1 , t ) ≤ ℘ ( υ n , υ n + 1 , t λ ) , n ∈ ℕ .(2) also there exist υ₀, υ₁ ∈ X, t > 0 and ι ∈ (0, 1) such that lim n → ∞ ∑ i = n ∞ ℏ ( υ 0 , υ 1 , t ι i ) = 1 and lim n → ∞ ∑ i = n ∞ ℘ ( υ 0 , υ 1 , t ι i ) = 0 .(3) Then {υ _n } is a Cauchy sequence.

Proof. Consider ς ∈ (λb, 1). We have ∑ i = 1 ∞ ς i is convergent. Therefore ∃ n 0 ∈ ℕ such that ∑ i = n ∞ ς i < 1 for every n > n₀. Consider n > m > n₀. Since ℏ is b-nondecreasing, and ℘ is b-non-increasing, for every t > 0, we have

ℏ ( υ n , υ n + m , t ) ≥ ℏ ( υ n , υ n + m , t ∑ i = n n + m - 1 ς i b ) ≥ ( ℏ ( υ n , υ n + 1 , t ς n b 2 ) ∗ ℏ ( υ n + 1 , υ n + m , t ∑ i = n + 1 n + m - 1 ς i b 2 ) ) ≥ ( ℏ ( υ n , υ n + 1 , t ς n b 2 ) ∗ ( ℏ ( υ n + 1 , υ n + 2 , t ς n + 1 b 3 ) ∗ … ∗ ℏ ( υ n + m - 1 , υ n + m , t ς n + m - 1 b m ∗ … ) )(4) and

℘ ( υ n , υ n + m , t ) ≤ ℏ ( υ n , υ n + m , t ∑ i = n n + m - 1 ς i b ) ≤ ( ℘ ( υ n , υ n + 1 , t ς n b 2 ) ∘ ℘ ( υ n + 1 , υ n + m , t ∑ i = n + 1 n + m - 1 ς i b 2 ) ) ≤ ( ℘ ( υ n , υ n + 1 , t ς n b 2 ) ∘ ( ℘ ( υ n + 1 , υ n + 2 , t ς n + 1 b 3 ) ∘ … ∘ ℘ ( υ n + m - 1 , υ n + m , t ς n + m - 1 b m ∘ … ) )(5)

By  1 and  2 we have, ℏ ( υ n , υ n + 1 , t ) ≥ ℏ ( υ 0 , υ 1 , t λ n ) , n ∈ ℕ(6) and ℘ ( υ n , υ n + 1 , t ) ≤ ℘ ( υ 0 , υ 1 , t λ n ) , n ∈ ℕ .(7) and as n > m and b > 1, we have

ℏ ( υ n , υ n + m , t ) ≥ ( ℏ ( υ 0 , υ 1 , t ς n b 2 λ n ) ∗ ( ℏ ( υ 0 , υ 1 , t ς n + 1 b 3 λ n + 1 ) ∗ … ∗ ℏ ( υ 0 , υ 1 , t ς n + m - 1 b m + 1 λ n + m - 1 ) ∗ … ) ) ≥ ∑ i = n n + m - 1 ( ℏ ( υ 0 , υ 1 , t ς i b i - n + 2 λ i ) ) ≥ ∑ i = n n + m - 1 ( ℏ ( υ 0 , υ 1 , t ς i b i λ i ) ) ≥ ∑ i = n ∞ ( ℏ ( υ 0 , υ 1 , t ι i ) ) ,(8) and

℘ ( υ n , υ n + m , t ) ≤ ( ℘ ( υ 0 , υ 1 , t ς n b 2 λ n ) ∘ ( ℘ ( υ 0 , υ 1 , t ς n + 1 b 3 λ n + 1 ) ∘ … ∘ ℘ ( υ 0 , υ 1 , t ς n + m - 1 b m + 1 λ n + m - 1 ) ∘ … ) ) ≤ ∑ i = n n + m - 1 ( ℘ ( υ 0 , υ 1 , t ς i b i - n + 2 λ i ) ) ≤ ∑ i = n n + m - 1 ( ℘ ( υ 0 , υ 1 , t ς i b i λ i ) ) ≤ ∑ i = n ∞ ( ℘ ( υ 0 , υ 1 , t ι i ) ) ,(9) where ι = b λ ς . As ι ∈ (0, 1), from 3 we get {υ _n } is a Cauchy sequence.

Corollary 2.4. Consider a sequence υ _n in an IFbMS (X, ℏ , ℘ , ∗ , ∘) and also consider that ℏ and ℘ is of H-type. Then {υ _n } is called a Cauchy sequence if for t > 0 ∃ λ ∈ ( 0 , 1 b ) such that ℏ ( υ n , υ n + 1 , t ) ≥ ℏ ( υ n - 1 , υ n , t λ ) , n ∈ ℕ(10) and ℘ ( υ n , υ n + 1 , t ) ≤ ℘ ( υ n - 1 , υ n , t λ ) , n ∈ ℕ .(11)

Lemma 2.5. Suppose that for some λ ∈ (0, 1), t > 0 and υ, ξ ∈ X ℏ ( υ , ξ , t ) ≥ ℏ ( υ , ξ , t λ ) and ℘ ( υ , ξ , t ) ≤ ℘ ( υ , ξ , t λ ) .(12) then we have υ = ξ.

Proof. Using the given condition 12 we have, for n ∈ ℕ , t > 0

ℏ ( υ , ξ , t ) ≥ ℏ ( υ , ξ , t λ n ) and ℘ ( υ , ξ , t ) ≤ ℘ ( υ , ξ , t λ n ) .(13)

Then from the definition of IFbMS υ = ξ.

Theorem 3.1. Suppose (X, ℏ , ℘ , ∗ , ∘) is a complete IFbMS, υ, ξ ∈ X and f : X ⟶ X. Assume that ∃ λ ∈ ( 0 , 1 b ) , t > 0, ℏ ( f ( υ ) , f ( ξ ) , t ) ≥ ℏ ( υ , ξ , t λ ) and ℘ ( f ( υ ) , f ( ξ ) , t ) ≤ ℘ ( υ , ξ , t λ )(14) and there exists υ₀ ∈ X and ι ∈ (0, 1) such that lim n → ∞ ∑ i = n ∞ ℏ ( υ 0 , f ( υ 0 ) , t ι i ) = 1 and lim n → ∞ ∑ i = n ∞ ℘ ( υ 0 , f ( υ 0 ) , t ι i ) = 0 .(15)

In this case f has a unique fixed point under X.

Proof. Consider υ₀ ∈ X and υ_n+1 = f (υ _n ), ∀ n ∈ ℕ .

Putting υ = υ _n and ξ = υ_n-1 in 14 we have for n ∈ ℕ and t > 0 ℏ ( υ n , υ n + 1 , t ) ≥ ℏ ( υ n - 1 , υ n , t λ ) and(16) ℘ ( υ n , υ n + 1 , t ) ≤ ℘ ( υ n - 1 , υ n , t λ )(17) then from Lemma 2.3 {υ _n } is a Cauchy sequence.

As (X, ℏ , ℘ , ∗ , ∘) is complete therefore ∃ υ ∈ X such that for t > 0

lim n → ∞ υ n = υ and lim n → ∞ ℏ ( υ , υ n , t ) = 1 , lim n → ∞ ℘ ( υ , υ n , t ) = 0 .

From the equation 15 and the definition of IFbMS, ∀ t > 0 we have

ℏ ( f ( υ ) , υ , t ) ≥ ( ℏ ( f ( υ ) , υ n , t 2 b ) ∗ ( ℏ ( υ n , υ , t 2 b ) ) ≥ ( ℏ ( υ , υ n - 1 , t 2 b λ ) ∗ ( ℏ ( υ n , υ , t 2 b ) )(18) and

℘ ( f ( υ ) , υ , t ) ≤ ( ℘ ( f ( υ ) , υ n , t 2 b ) ∘ ( ℘ ( υ n , υ , t 2 b ) ) ≤ ( ℘ ( υ , υ n - 1 , t 2 b λ ) ∘ ( ℘ ( υ n , υ , t 2 b ) )(19)

Now letting n→ ∞, we have

ℏ ( f ( υ ) , υ , t ) ≥ 1 ∗ 1 = 1 and ℘ ( f ( υ ) , υ , t ) ≤ 0 ∘ 0 = 0 .

This implies f (υ) = υ. i.e. υ is a fixed point for f.

Consider two fixed points υ and ξ of f. Then

ℏ ( υ , ξ , t ) = ℏ ( f ( υ ) , f ( ξ ) , t ) ≥ ℏ ( υ , ξ , t λ ) and ℘ ( υ , ξ , t ) = ℘ ( f ( υ ) , f ( ξ ) , t ) ≤ ℘ ( υ , ξ , t λ )

From the Lemma 2.5 we have υ = ξ.

Example 3.2. Suppose X = [0, 1]. Consider for all υ, ξ ∈ X and t > 0, ℏ ( υ , ξ , t ) = e - ( υ - y ) 2 t and ℘ ( υ , ξ , t ) = 1 - e - ( υ - y ) 2 t . Then (X, ℏ , ℘ , ∗ , ∘) is a complete IFbMS with b = 2.

Consider f (υ) = kυ, k < 2 2 , υ ∈ X. Then for υ, ξ ∈ X and t > 0,

ℏ ( f ( υ ) , f ( ξ ) , t ) = e k 2 ( υ - y ) 2 t ≥ e λ ( υ - y ) 2 t = ℏ ( υ , ξ , t λ )(20) and

℘ ( f ( υ ) , f ( ξ ) , t ) = 1 - e k 2 ( υ - ξ ) 2 t ≤ 1 - e λ ( υ - ξ ) 2 t = ℏ ( υ , ξ , t λ )(21) for 1 b > λ > k 2 .

Therefore, equation 15 of the Theorem 3.1 is satisfied and f has a unique fixed point under X.

Theorem 3.3. Suppose (X, ℏ , ℘ , ∗ , ∘) is a complete IFbMS, υ, ξ ∈ X and f : X ⟶ X. Assume that there exists λ ∈ ( 0 , 1 b ) , t > 0, ℏ ( f ( υ ) , f ( ξ ) , t ) ≥ min { ℏ ( υ , ξ , t λ ) , ℏ ( f ( υ ) , υ , t λ ) , ℏ ( f ( y ) , y , t λ ) }(22) and ℘ ( f ( υ ) , f ( ξ ) , t ) ≤ max { ℘ ( υ , ξ , t λ ) , ℘ ( f ( υ ) , υ , t λ ) , ℘ ( f ( y ) , y , t λ ) }(23) and there exists υ₀ ∈ X and ι ∈ (0, 1) such that ∀ t > 0 lim n → ∞ ∑ i = n ∞ ℏ ( υ 0 , f ( υ 0 ) , t ι i ) = 1 and lim n → ∞ ∑ i = n ∞ ℘ ( υ , t ι i ) = 0 .(24)

In this case f has a unique fixed point under X.

Proof. Consider υ₀ ∈ X and υ_n+1 = f (υ _n ), ∀ n ∈ ℕ .

Putting υ = υ _n and ξ = υ_n-1 in 22 and  23 we have for n ∈ ℕ and t > 0

ℏ ( υ n + 1 , υ n , t ) ≥ min { ℏ ( υ n , υ n - 1 , t λ ) , ℏ ( υ n + 1 , υ n , t λ ) , ℏ ( υ n , υ n - 1 , , t λ ) } ≥ min { ℏ ( υ n , υ n - 1 , t λ ) , ℏ ( υ n + 1 , υ n , t λ ) }(25) and

℘ ( υ n + 1 , υ n , t ) ≤ max { ℘ ( υ n , υ n - 1 , , t λ ) , ℘ ( υ n + 1 , υ n , t λ ) , ℘ ( υ n , υ n - 1 , , t λ ) } ≤ max { ℘ ( υ n , υ n - 1 , , t λ ) , ℘ ( υ n + 1 , υ n , t λ ) }(26)

If we consider ℏ ( υ n + 1 , υ n , t ) ≥ ℏ ( υ n + 1 , υ n , t λ ) and ℘ ( υ n + 1 , υ n , t ) ≤ ℘ ( υ n + 1 , υ n , t λ ) then from the Lemma 2.5 υ _n  = υ_n+1, n ∈ ℕ .

Therefore, for n ∈ ℕ , t > 0

ℏ ( υ n + 1 , υ n , t ) ≥ ℏ ( υ n , υ n - 1 , t λ ) and ℘ ( υ n + 1 , υ n , t ) ≤ ℘ ( υ n , υ n - 1 , t λ ) .(27) then from Lemma 2.3 {υ _n } is a Cauchy sequence.

Therefore, ∃ υ ∈ X such that for t > 0

lim n → ∞ υ n = υ and lim n → ∞ ℏ ( υ , υ n , t ) = 1 , lim n → ∞ ℘ ( υ , υ n , t ) = 0 .(28)

Next suppose ς₁ ∈ (λb, 1) and ς₂ = 1 - ς₁. From  22 and  23 we have,

ℏ ( f ( υ ) , υ , t ) ≥ ( ℏ ( f ( υ ) , f ( υ n ) , t ς 1 b ) ∗ ( ℏ ( υ n + 1 , υ , t ς 2 b ) ) ≥ ( min { ℏ ( υ , υ n , t ς 1 b λ ) , ℏ ( υ , f ( υ ) , t ς 1 b λ ) , ℏ ( υ n , υ n + 1 , t ς 1 b λ ) } ∗ ℏ ( υ n + 1 , υ , t ς 2 b ) )(29) and

℘ ( f ( υ ) , υ , t ) ≤ ( ℘ ( f ( υ ) , f ( υ n ) , t ς 1 b ) ∘ ( ℘ ( υ n + 1 , υ , t ς 2 b ) ) ≤ ( max { ℘ ( υ , υ n , t ς 1 b λ ) , ℘ ( υ , f ( υ ) , t ς 1 b λ ) , ℘ ( υ n , υ n + 1 , t ς 1 b λ ) } ∘ ℘ ( υ n + 1 , υ , t ς 2 b ) )(30)

Now letting n→ ∞, we have for t > 0

ℏ ( f ( υ ) , υ , t ) ≥ ( min 1 , ℏ ( υ , f ( υ ) , t ς 1 b λ ) , 1 ∗ 1 ) = ( ℏ ( υ , f ( υ ) , t ς 1 b λ ) ∗ 1 ) = ℏ ( υ , f ( υ ) , t ι ) and ℘ ( f ( υ ) , υ , t ) ≤ ( max 0 , ℘ ( υ , f ( υ ) , t ς 1 b λ ) , 0 ∘ 0 ) = ( ℘ ( υ , f ( υ ) , t ς 1 b λ ) ∘ 0 ) = ℘ ( υ , f ( υ ) , t ι )

where ι = b λ ς 1 ∈ ( 0 , 1 ) .

Using Lemma 2.5 we have f (υ) = υ. i.e. υ is a fixed point for f.

Consider two fixed points υ and ξ of f, i.e. f (υ) = υ and f (ξ) = ξ. Then from 22 and  23, for t > 0

ℏ ( f ( υ ) , f ( ξ ) , t ) ≥ min { ℏ ( υ , ξ , t λ ) , ℏ ( υ , f ( υ ) , t λ ) , ℏ ( ξ , f ( ξ ) , t λ ) } = min { ℏ ( υ , ξ , t λ ) , 1 , 1 } = ℏ ( υ , ξ , t λ ) = ℏ ( f ( υ ) , f ( ξ ) , t λ ) and ℘ ( f ( υ ) , f ( ξ ) , t ) ≤ max { ℘ ( υ , ξ , t λ ) , ℘ ( υ , f ( υ ) , t λ ) , ℘ ( ξ , f ( ξ ) , t λ ) } = max { ℘ ( υ , ξ , t λ ) , 0 , 0 } = ℘ ( υ , ξ , t λ ) = ℘ ( f ( υ ) , f ( ξ ) , t λ )

Hence from the Lemma 2.5 we have f (υ) = f (ξ), i.e. υ = ξ.

Example 3.4. Consider X = (0, 2) and f : X ⟶ X be a function such that for all υ, ξ ∈ X,

f ( υ ) = { 2 - υ , υ ∈ ( 0 , 1 ) 1 , υ ∈ [ 1 , 2 ) ,

Consider ℏ ( υ , ξ , t ) = e - ( υ - ξ ) 2 t and ℘ ( υ , ξ , t ) = 1 - e - ( υ - ξ ) 2 t then (X, ℏ , ℘ , ∗ , ∘) is a complete IFbMS with b = 2.

Case 1: For t > 0 first assume υ, ξ ∈ [1, 2), then ℏ (f (υ) , f (ξ) , t) =1 and ℘ (f (υ) , f (ξ) , t) =0.

Therefore equation 22 and  23 are trivially satisfied.

Case 2: Consider υ ∈ [1, 2) and y ∈ (0, 1), then for λ ∈ ( 1 4 , 1 2 ) and t > 0 we have

ℏ ( f ( υ ) , f ( ξ ) , t ) = e - ( 1 - ξ ) 2 t ≥ e - 4 λ ( 1 - ξ ) 2 t = ℏ ( f ( ξ ) , y , t λ ) , and ℘ ( f ( υ ) , f ( ξ ) , t ) = 1 - e - ( 1 - ξ ) 2 t ≤ 1 - e - 4 λ ( 1 - ξ ) 2 t = ℘ ( f ( ξ ) , y , t λ ) Case 3: Consider y ∈ [1, 2) and υ ∈ (0, 1), then for λ ∈ ( 1 4 , 1 2 ) and t > 0 we have

ℏ ( f ( υ ) , f ( ξ ) , t ) = e - ( υ - 1 ) 2 t ≥ e - 4 λ ( υ - 1 ) 2 t = ℏ ( f ( υ ) , υ , t λ ) , and ℘ ( f ( υ ) , f ( ξ ) , t ) = 1 - e - ( υ - 1 ) 2 t ≤ 1 - e - 4 λ ( υ - 1 ) 2 t = ℘ ( f ( υ ) , υ , t λ )

Case 4: Finally consider υ, ξ ∈ (0, 1) then for λ ∈ ( 1 4 , 1 2 ) and t > 0 ℏ ( f ( υ ) , f ( ξ ) , t ) = e - ( υ - ξ ) 2 t ≥ e - ( 1 - ξ ) 2 t ≥ e - 4 λ ( 1 - ξ ) 2 t = ℏ ( f ( ξ ) , y , t λ ) , and ℘ ( f ( υ ) , f ( ξ ) , t ) = 1 - e - ( υ - ξ ) 2 t ≤ 1 - e - ( 1 - ξ ) 2 t ≤ 1 - e - 4 λ ( 1 - ξ ) 2 t = ℘ ( f ( ξ ) , y , t λ ) ,

In this case if we consider υ < y then we have ℏ ( f ( υ ) , f ( ξ ) , t ) ≥ ℏ ( f ( υ ) , υ , t λ ) and ℘ ( f ( υ ) , f ( ξ ) , t ) ≤ ℘ ( f ( υ ) , υ , t λ )

Therefore the equation 22 and  23 of the theorem 3.3 are satisfied for all υ, ξ ∈ X and t > 0. Hence f has a fixed point in X and υ = 1 is a unique fixed point for f.

This section contains study of Ćirić quasi-contraction as an extension work on fixed point theorem in IFbMS. We also establish a weaker contraction condition for existence of a fixed point under t-norm and t-conorm.

Theorem 4.1. Suppose (X, ℏ , ℘ , ∗ , ∘) is a complete IFbMS, υ, ξ ∈ X and f : X ⟶ X. Assume that for some λ ∈ ( 0 , 1 b 2 ) , t > 0, ℏ ( f ( υ ) , f ( ξ ) , t ) ≥ min { ℏ ( υ , ξ , t λ ) , ℏ ( f ( υ ) , υ , t λ ) , ℏ ( f ( ξ ) , ξ , t λ ) , ℏ ( f ( υ ) , ξ , 2 t λ ) , ℏ ( υ , f ( ξ ) , t λ ) }(31) and ℘ ( f ( υ ) , f ( ξ ) , t ) ≤ max { ℘ ( υ , ξ , t λ ) , ℘ ( f ( υ ) , υ , t λ ) , ℘ ( f ( ξ ) , ξ , t λ ) , ℘ ( f ( υ ) , ξ , 2 t λ ) , ℘ ( υ , f ( ξ ) , t λ ) }(32)

In this case f has a unique fixed point under X.

Proof. Consider υ₀ ∈ X and υ_n+1 = f (υ _n ), for all n ∈ ℕ .

Now putting υ = υ _n and ξ = υ_n-1 in 39 and  40 and using the definition of IFbMS we have for n ∈ ℕ and t > 0

ℏ ( υ n + 1 , υ n , t ) ≥ min { ℏ ( υ n , υ n - 1 , t λ ) , ℏ ( υ n + 1 , υ n , t λ ) , ℏ ( υ n , υ n - 1 , t λ ) , min { ℏ ( υ n + 1 , υ n , t b λ ) , ℏ ( υ n , υ n - 1 , , t b λ ) } , ℏ ( υ n , υ n , t λ ) } ≥ min { ℏ ( υ n , υ n - 1 , , t b λ ) , ℏ ( υ n + 1 , υ n , t b λ ) }(33) and

℘ ( υ n + 1 , υ n , t ) ≥ max { ℘ ( υ n , υ n - 1 , , t λ ) , ℘ ( υ n + 1 , υ n , t λ ) , ℘ ( υ n , υ n - 1 , t λ ) , min { ℘ ( υ n + 1 , υ n , t b λ ) , ℘ ( υ n , υ n - 1 , , t b λ ) } , ℘ ( υ n , υ n , t λ ) } ≥ max { ℘ ( υ n , υ n - 1 , , t b λ ) , ℘ ( υ n + 1 , υ n , t b λ ) }(34)

Now from Theorem 3.3, Lemma 2.5 and Corollary 2.4 we have for n ∈ ℕ , t > 0,

ℏ ( υ n + 1 , υ n , t ) ≥ ℏ ( υ n , υ n - 1 , t b λ ) and ℘ ( υ n + 1 , υ n , t ) ≤ ℘ ( υ n , υ n - 1 , t b λ ) .(35) and hence {υ _n } is a Cauchy sequence.

Therefore ∃ υ ∈ X such that for t > 0

lim n → ∞ υ n = υ and lim n → ∞ ℏ ( υ , υ n , t ) = 1 , lim n → ∞ ℘ ( υ , υ n , t ) = 0 .(36)

Next suppose ς₁ ∈ (λb², 1) and ς₂ = 1 - ς₁. From  39 and  40 we have, ∀ n ∈ ℕ and t > 0,

ℏ ( f ( υ ) , υ , t ) ≥ min { ℏ ( f ( υ ) , f ( υ n ) , t ς 1 b ) , ( ℏ ( f ( υ n ) , υ , t ς 2 b ) } ≥ min { min { ℏ ( υ , υ n , t ς 1 b λ ) , ℏ ( υ , f ( υ ) , t ς 1 b λ ) , ℏ ( υ n , υ n + 1 , t ς 1 b λ ) , min { ℏ ( f ( υ ) , υ , t ς 2 b 2 λ ) , ℏ ( υ , υ n , t ς 2 b 2 λ ) } , ℏ ( υ , υ n + 1 , t ς 1 b λ ) } , ℏ ( υ n + 1 , υ , t ς 2 b ) }(37) and

℘ ( f ( υ ) , υ , t ) ≤ max { ℘ ( f ( υ ) , f ( υ n ) , t ς 1 b ) , ( ℘ ( f ( υ n ) , υ , t ς 2 b ) } ≤ max { max { ℘ ( υ , υ n , t ς 1 b λ ) , ℘ ( υ , f ( υ ) , t ς 1 b λ ) , ℘ ( υ n , υ n + 1 , t ς 1 b λ ) , max { ℘ ( f ( υ ) , υ , t ς 2 b 2 λ ) , ℘ ( υ , υ n , t ς 2 b 2 λ ) } , ℘ ( υ , υ n + 1 , t ς 1 b λ ) } , ℘ ( υ n + 1 , υ , t ς 2 b ) }(38)

Now letting n→ ∞, we have for t > 0 ℏ ( f ( υ ) , υ , t ) ≥ min { min { 1 , ℏ ( υ , f ( υ ) , t ς 1 b λ ) , 1 , min { ℏ ( f ( υ ) , υ , t ς 2 b 2 λ ) , 1 } , 1 } , 1 } = ℏ ( f ( υ ) , υ , t ς 1 b 2 λ ) = ℏ ( f ( υ ) , υ , t ι ) and ℘ ( f ( υ ) , υ , t ) ≤ max { max { 0 , ℏ ( υ , f ( υ ) , t ς 1 b λ ) , 0 , min { ℏ ( f ( υ ) , υ , t ς 2 b 2 λ ) , 0 } , 0 } , 0 } = ℘ ( f ( υ ) , υ , t ς 1 b 2 λ ) = ℘ ( f ( υ ) , υ , t ι ) where ι = b λ ς 1 ∈ ( 0 , 1 ) .

Using Lemma 2.5 we have f (υ) = υ. i.e. υ is a fixed point for f.

Consider two fixed points υ and ξ of f, i.e. f (υ) = υ and f (ξ) = ξ. Then from 39 and  40, for t > 0 ℏ ( f ( υ ) , f ( ξ ) , t ) ≥ min { ℏ ( υ , ξ , t λ ) , ℏ ( f ( υ ) , υ , t λ ) , ℏ ( f ( ξ ) , ξ , t λ ) , min { ℏ ( f ( υ ) , υ , t b λ ) , ℏ ( υ , ξ , t b λ ) } , ℏ ( υ , f ( ξ ) , t λ ) } = min { ℏ ( υ , ξ , t λ ) , 1 , 1 , min { 1 , ℏ ( υ , ξ , t b λ ) } , ℏ ( υ , ξ , t λ ) } = ℏ ( υ , ξ , t b λ ) = ℏ ( f ( υ ) , f ( ξ ) , t b λ ) and ℘ ( f ( υ ) , f ( ξ ) , t ) ≤ max { ℘ ( υ , ξ , t λ ) , ℘ ( f ( υ ) , υ , t λ ) , ℘ ( f ( ξ ) , ξ , t λ ) , max { ℘ ( f ( υ ) , υ , t b λ ) , ℘ ( υ , ξ , t b λ ) } , ℘ ( υ , f ( ξ ) , t λ ) } = max { ℘ ( υ , ξ , t λ ) , 0 , 0 , max { 0 , ℏ ( υ , ξ , t b λ ) } , ℘ ( υ , ξ , t λ ) } = ℘ ( υ , ξ , t b λ ) = ℘ ( f ( υ ) , f ( ξ ) , t b λ )

Hence from the Lemma 2.5 we have f (υ) = f (ξ), i.e. υ = ξ.

Theorem 4.2. Suppose (X, ℏ , ℘ , ∗ , ∘) is a complete IFbMS, υ, ξ ∈ X and f : X ⟶ X. Assume that for some λ ∈ ( 0 , 1 b 2 ) , t > 0, ℏ ( f ( υ ) , f ( ξ ) , t ) ≥ min { ℏ ( υ , ξ , t λ ) , ℏ ( f ( υ ) , υ , t λ ) , ℏ ( f ( ξ ) , ξ , t λ ) , ℏ ( f ( υ ) , y , 2 t λ ) , ℏ ( υ , f ( ξ ) , t λ ) }(39) and ℘ ( f ( υ ) , f ( ξ ) , t ) ≤ max { ℘ ( υ , ξ , t λ ) , ℘ ( f ( υ ) , υ , t λ ) , ℘ ( f ( ξ ) , ξ , t λ ) , ℘ ( f ( υ ) , y , 2 t λ ) , ℘ ( υ , f ( ξ ) , t λ ) }(40)

Also ∃ υ₀ ∈ X and ι ∈ (0, 1) such that lim n → ∞ ∑ i = n ∞ ℏ ( υ 0 , f ( υ 0 ) , t ι i ) = 1 and(41) lim n → ∞ ∑ i = n ∞ ℘ ( υ 0 , f ( υ 0 ) , t ι i ) = 0 .(42) in this case f has a unique fixed point under X.

Proof. Consider υ₀ ∈ X and υ_n+1 = f (υ _n ), ∀ n ∈ ℕ .

Now putting υ = υ _n and ξ = υ_n-1 in 39 and  40 and using the definition of IFbMS we have for n ∈ ℕ and t > 0

ℏ ( υ n + 1 , υ n , t ) ≥ min { ℏ ( υ n , υ n - 1 , t λ ) , ℏ ( υ n + 1 , υ n , t λ ) , ℏ ( υ n , υ n - 1 , , t λ ) , ℏ ( υ n + 1 , υ n , t b λ ) ∗ ℏ ( υ n , υ n - 1 , , t b λ ) , ℏ ( υ n , υ n , t λ ) }(43) and

℘ ( υ n + 1 , υ n , t ) ≥ max { ℘ ( υ n , υ n - 1 , t λ ) , ℘ ( υ n + 1 , υ n , t λ ) , ℘ ( υ n , υ n - 1 , , t λ ) , ℘ ( υ n + 1 , υ n , t b λ ) ∘ ℘ ( υ n , υ n - 1 , t b λ ) , ℘ ( υ n , υ n , t λ ) }(44)