# Commonα-Fuzzy Fixed Point Results for Kannan Type Contractions with Application

Authors: Amer Hassan Albargi; Jamshaid Ahmad
Journal: Journal of Function Spaces
 (2022)

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## Abstract

The purpose of this paper is to utilize complex valued extendedb-metric space and obtain common α-fuzzy fixed point theorems under Kannan type contractions. As application, we derive common α-fixed point results for multivalued mappings in the context of complex valued extended b-metric space.


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## Body

## 1. Introduction

In the theory of fixed points, there is vital role of metric spaces which have useful applications in mathematics as well as in computer science, medicine, physics, and biology (see [1–3]). Many mathematicians generalized, improved, and extended the notion of metric spaces to vector-valued metric spaces of Perov ([4]), b-metric space of Czerwik [5], cone metric spaces of Huang and Zhang [6], and others.In 1969, Kannan [7] gave an analogue sort of contractive condition that demonstrated the existence of fixed point. The basic distinction between Banach fixed point theorem (BFT) and that of Kannan contraction is that continuity of contraction is not required in the latter.In 2011, Azam et al. [8] introduced the concept of complex valued metric space and obtained some common fixed points results for rational contraction which consist of a pair of single valued mappings. Later, on many researchers, [9–17] worked on this generalized metric space. Ahmad et al. [18] and Azam et al. [19] defined the generalized Hausdorff metric function in the setting of complex valued metric space and obtained common fixed point results for multivalued mappings. In [20], Mukheimer generalized the concept of complex valued metric space to complex valued b-metric space. Recently, Naimatullah et al. [21] introduced the notion of complex valued extended b-metric space as extension of complex valued b-metric space and established some results for rational contractions in this generalized space.On the other hand, Heilpern [22] introduced the concept of fuzzy mappings in the setting of metric linear spaces and extended Banach contraction principle [23]. In 2014, Kutbi et al. [24] established fuzzy fixed point results in complex valued metric spaces and generalized the results in metric spaces. Owing to the notion of a complex valued metric space, Humaira et al. [25] proved some common fixed point results under contractive condition for rational expressions.In this paper, we define the generalized fuzzy contraction in the setting of complex valued extendedb-metric space and obtain some fuzzy fixed point results. As application, we derive the main results of Azam et al. [8], Rouzkard et al. [10], Ahmad et al. [18], and Kutbi et al. [24] for fuzzy and multivalued mappings in complex valued metric spaces.


## 2. Preliminaries

In 2011, Azam et al. [8] introduced the complex valued metric space as follows:Definition 1. (see [8]).
Letℂ be the set of complex number and σ1,σ2∈ℂ. A partial order ≾ on ℂ is defined in this way:
(1)σ1≾σ2⇔Reσ1≤Reσ2,Imσ1≤Imσ2.
It follows that(2)σ1≾σ2,if one of these assertions is satisfied:
(3)aReσ1=Reσ2,Imσ1<Imσ2,bReσ1<Reσ2,Imσ1=Imσ2,cReσ1<Reσ2,Imσ1<Imσ2,dReσ1=Reσ2,Imσ1=Imσ2.Definition 2. (see [8]).
Letℵ=∅. A mapping ℏ:ℵ×ℵ⟶ℂ is said to be a complex valued metric if the following assertions hold:
(CV1)0≾ℏσ,ρ,for all σ,ρ∈ℵ and ℏσ,ρ=0 if and only if σ=ρ
(CV2)ℏσ,ρ=ℏρ,σ for all σ,ρ∈ℵ
(CV3)ℏσ,ρ≾ℏσ,ν+ℏν,ρ, for all σ,ρ,ν∈ℵ
Then,ℵ,ℏ is called a complex valued metric space.Example 3. (see [8]).
Letℵ=0,1 and let σ,ρ∈ℵ. Define ℏ:ℵ×ℵ⟶ℂ by
(4)ℏσ,ρ=0,ifσ=ρ,i2,ifσ=ρ.
Then,ℵ,ℏ is a complex valued metric space (CVMS).In 2014, Mukheimer [20] introduced the notion of complex valued b-metric space as follows:Definition 4. (see [20]).
Letℵ=∅ and π≥1 be a real number. A mapping ℏ:ℵ×ℵ⟶ℂ is said to be a complex valued b-metric space if the following assertions hold:
(CVB1)0≾ℏσ,ρ, for all σ,ρ∈ℵ and ℏσ,ρ=0 if and only if σ=ρ
(CVB2)ℏσ,ρ=ℏρ,σ for all σ,ρ∈ℵ
(CVB3)ℏσ,ρ≾πℏσ,ν+ℏν,ρ, for all σ,ρ,ν∈ℵ
Then,ℵ,ℏ is called a complex valued b-metric space (CVbMS).Example 5. (see [20]).
Letℵ=0,1. Define ℏ:ℵ×ℵ⟶ℂ by
(5)ℏσ,ρ=σ−ρ2+iσ−ρ2,for all σ,ρ∈ℵ. Then, ℵ,ℏ is a complex valued b-metric space with π=2.Recently, Naimatullah et al. [21] defined the notion of complex valued extended b-metric space in this way.Definition 6. (see [21]).
Letℵ=∅ and φ:ℵ×ℵ⟶1,∞. A mapping ℏ:ℵ×ℵ⟶ℂ is called a complex valued extended b-metric if following conditions hold:
(ECVB1)0≾ℏσ,ρ, for all σ,ρ∈ℵ and ℏσ,ρ=0 if and only if σ=ρ
(ECVB2)ℏσ,ρ=ℏρ,σ for all σ,ρ∈ℵ
(ECVB3)ℏσ,ρ≾φσ,ρℏσ,ν+ℏν,ρ, for all σ,ρ,ν∈ℵ
Then,ℵ,ℏ is called a complex valued extended b-metric space (CVEbMS).Example 7. (see [21]).
Letℵ=∅ and φ:ℵ×ℵ⟶1,∞ be defined by
(6)φσ,ρ=1+σ+ρσ+ρ,and ℏ:ℵ×ℵ⟶ℂ by
(i)
ℏσ,ρ=i/σρ,∀0<σ,ρ≤1(ii)
ℏσ,ρ=0⇔σ=ρ,∀0≤σ,ρ≤1(7)ℏσ,0=ℏ0,σ=iσ,∀0<σ≤1.
Then,ℵ,ℏ is a CVEbMS.Example 8. (see [21]).
Letℵ=0,∞ and φ:ℵ×ℵ⟶1,∞ be a function defined by φσ,ρ=1+σ+ρ and ℏ:ℵ×ℵ⟶ℂ by
(8)ℏσ,ρ=0,ifσ=ρ,i,ifσ=ρ.
Then,ℵ,ℏ is a CVEbMS.Lemma 9. (see [21]).
Letℵ,ℏ be a CVEbMS, and let σn⊆ℵ. Then, σn converges to σ⇔ℏσn,σ⟶0 as n⟶∞.Lemma 10. (see [21]).
Letℵ,ℏ be a CVEbMS, and let σn⊆ℵ. Then, σn is a Cauchy sequence ⇔ℏσn,σn+m⟶0asn⟶∞, where m∈ℕ.
Letℵ,ℏ be a CVEbMS, and then CBℵ denotes the family of all nonempty, closed, and bounded subsets of ℵ.
From now on, we denotesσ1=σ2∈ℂ:≾σ1σ2 for σ1∈ℂ, and
(9)sσ1,R2=∪σ2∈R2sℏσ1,σ2=∪σ2∈R2σ∈ℂ:ℏσ1,σ2≾σ,for a∈ℵ and R2∈CBℵ.
ForR1,R2∈CBℵ, we denote
(10)sR1,R2=∩σ1∈R1sσ1,R2∩∩σ2∈R2sσ2,R1.Lemma 11. (see [21]).
Letℵ,ℏ be a CVEbMS:
(i)
Letσ1,σ2∈ℂ. If σ1≾σ2, then sσ2⊂sσ1(ii)
Letσ∈ℵ and R∈Nℵ. If θ∈sσ,R,then σ∈R(iii)
Letσ∈ℂ and let R1,R2∈CBℵ and σ1∈R1. If σ∈sR1,R2, then σ∈sσ1,R2 for all σ1∈R1 or σ∈sR1,σ2 for all σ2∈R2
Letℵ,ℏ be a complex valued extended b-metric space and Cℵ be a collection of nonempty closed subsets of ℵ. Let I:ℵ⟶CBℵ be a multivalued mapping. For σ∈ℵ and R∈CBℵ, define
(11)WσR=ℏσ,σ1:σ1∈R.
Thus, forσ,y∈ℵ,
(12)WσIy=ℏσ,σ1:σ1∈Iy.Definition 12. (see [21]).
Letℵ,ℏ be a complex valued metric space. A subset R of ℵ is called bounded below if there exists σ∈ℵ, such that σ°σ1 for all σ1∈R.Definition 13. (see [21]).
Letℵ,ℏ be a complex valued metric space. A multivalued mapping I:ℵ⟶2ℂ is called bounded below if for each σ∈ℵ, there exists σx∈ℂ such that σx°u for all u∈Iσ.
In 1981, Heilpern [22] utilized the concept of fuzzy set and introduced the notion of fuzzy mappings in metric spaces (MS). A fuzzy set in ℵ is a function with domain ℵ and values in 0,1, and Iℵ is the collection of all fuzzy sets in ℵ. If R is a fuzzy set and x∈ℵ, then the function values Rσ are called the grade of membership of σ in R. The α-level set of R is denoted by Rα and is defined as follows:
(13)Rα=σ:Rσ≥αifα∈0,1,R0=x:Rσ>0.¯
Here,R¯ denotes the closure of the set R. Let ℵ be the collection of all fuzzy sets in a metric space ℵ.Definition 14. (see[22]).
Letℵ1 be a nonempty set and ℵ2,ℏ be a MS. A mapping I is called fuzzy mapping if I is a mapping from ℵ1 into ℵ2. A fuzzy mapping I is a fuzzy subset on ℵ1×ℵ2 with membership function Ixy. The function Ixy is the grade of membership of y in Ix.Definition 15. (see[22]).
Letℵ,ℏ be a MS and I1,I2: ℵ⟶ℵ. A point σ∈ℵ is said to be a fuzzy fixed point of I2 if σ∈I2σα, for some α∈0,1. The point σ∈ℵ is said to be a common fuzzy fixed point of I1 and I2 if σ∈I1σα∩I2σα for some α∈0,1.
In 2014, Kutbi et al. [24] used the above notion of fuzzy mappings in complex valued metric space (CVMS) and established the following result.Theorem 16. (see [24]).
Letℵ,ℏ be a complete CVMS and let I1,I2: ℵ⟶ℵ satisfying g.l.b property. Assume that there exists α∈0,1, such that for each σ∈ℵ,I1σα,I2σα∈CBℵ, and there exists nonnegative real numbers ℓ1,ℓ2,ℓ3 with ℓ1+ℓ2+ℓ3<1 such that
(14)ℓ1ℏσ,ρ+ℓ2ℏσ,I1σαℏρ,I2ρα1+ℏσ,ρ+ℓ3ℏρ,I1σαℏσ,I2ρα1+ℏσ,ρ∈sI1σα,I2ρα,for all σ,ρ∈ℵ, and then ∃σ∗∈ℵ such that σ∗∈I1σ∗α∩I2σ∗α.
In this paper, we establish fuzzy fixed point results in the setting of complex valued extendedb-metric spaces (CVEbMS) and derive the above result of Kutbi et al. [24] for fuzzy mappings and some fixed point result for multivalued mappings in CVMS.


## 3. Main Result

Definition 17.
Letℵ,ℏ be a CVEbMS. The fuzzy mapping I:ℵ⟶ℵ is said to have g.l.b. property on ℵ,ℏ, if for any σ∈ℵ and any α∈0,1, the greatest lower bound of WσIρα exists in ℂ, for all ρ∈ℵ. We denote ℏσ,Iρα by the g.l.b of WσIρα. That is,
(15)ℏσ,Iρα=infℏσ,ν:ν∈Iρα.Now, we state our main result in this way.Theorem 18.
Letℵ,ℏ be a complete CVEbMS, φ:ℵ×ℵ⟶1,∞, and let I1,I2: ℵ⟶ℵ satisfying g.l.b property. Assume that there exists α∈0,1s, such that for each σ∈ℵ,I1σα,I2σα, ∈CBℵ, and there exists nonnegative real numbers ℓ1,ℓ2,ℓ3 with 2ℓ1+ℓ2+ℓ3<1 and λ1−ℓ1−ℓ2=ℓ1, where λ∈0,1 such that
(16)ℓ1ℏσ,I1σα+ℏρ,I2ρα+ℓ2ℏσ,I1σαℏρ,I2ρα1+ℏσ,ρ+ℓ3ℏρ,I1σαℏσ,I2ρα1+ℏσ,ρ∈sI1σα,I2ρα,
for allσ,ρ∈ℵ. If for each σ0∈ℵ,limn,m⟶∞φσn,σmλ<1, then there exists σ∗∈ℵ such that σ∗∈I1σ∗α∩I2σ∗α.Proof.
Letσ0 be an arbitrary point in ℵ. By assumption, we can find σ1∈I1σ0α. So, we have
(17)ℓ1ℏσ0,I1σ0α+ℏσ1,I2σ1α+ℓ2ℏσ0,I1σ0αℏσ1,I2σ1α1+ℏσ0,σ1+ℓ3ℏσ1,I1σ0αℏσ0,I2σ1α1+ℏσ0,σ1∈sI1σ0α,I2σ1α.That is,
(18)ℓ1ℏσ0,I1σ0α+ℏσ1,I2σ1α+ℓ2ℏσ0,I1σ0αℏσ1,I2σ1α1+ℏσ0,σ1+ℓ3ℏσ1,I1σ0αℏσ0,I2σ1α1+ℏσ0,σ1∈ω∈I1σ0αsω,I2σ1α.
Sinceσ1∈I1σ0α, so, we have
(19)ℓ1ℏσ0,I1σ0α+ℏσ1,I2σ1α+ℓ2ℏσ0,I1σ0αℏσ1,I2σ1α1+ℏσ0,σ1+ℓ3ℏσ1,I1σ0αℏσ0,I2σ1α1+ℏσ0,σ1∈sσ1,I2σ1α.
By definition,(20)ℓ1ℏσ0,I1σ0α+ℏσ1,I2σ1α+ℓ2ℏσ0,I1σ0αℏσ1,I2σ1α1+ℏσ0,σ1+ℓ3ℏσ1,I1σ0αℏσ0,I2σ1α1+ℏσ0,σ1∈ρ∈I2σ1αsℏσ1,ρ.
This implies that∃σ2∈I2σ1α such that
(21)ℓ1ℏσ0,I1σ0α+ℏσ1,I2σ1α+ℓ2ℏσ0,I1σ0αℏσ1,I2σ1α1+ℏσ0,σ1+ℓ3ℏσ1,I1σ0αℏσ0,I2σ1α1+ℏσ0,σ1∈sℏσ1,σ2.
That is,(22)ℏσ1,σ2°ℓ1ℏσ0,I1σ0α+ℏσ1,I2σ1α+ℓ2ℏσ0,I1σ0αℏσ1,I2σ1α1+ℏσ0,σ1+ℓ3ℏσ1,I1σ0αℏσ0,I2σ1α1+ℏσ0,σ1.
By the meaning ofWσI2ρα and WσI1ρα for σ,ρ∈ℵ, we get
(23)ℏσ1,σ2°ℓ1ℏσ0,σ1+ℏσ1,σ2+ℓ2ℏσ0,σ1ℏσ1,σ2+ℓ3ℏσ1,σ1ℏσ0,σ21+ℏσ0,σ1=ℓ1ℏσ0,σ1+ℏσ1,σ2+ℓ2ℏσ0,σ1ℏσ1,σ21+ℏσ0,σ1=ℓ1ℏσ0,σ1+ℏσ1,σ2+ℓ2ℏσ1,σ2ℏσ0,σ11+ℏσ0,σ1.
This implies(24)ℏσ1,σ2≤ℓ1ℏσ0,σ1+ℓ1ℏσ1,σ2+ℓ2ℏσ1,σ2ℏσ0,σ11+ℏσ0,σ1≤ℓ1ℏσ0,σ1+ℓ1ℏσ1,σ2+ℓ2ℏσ1,σ2,which implies that
(25)ℏσ1,σ2≤ℓ11−ℓ1−ℓ2ℏσ0,σ1=λℏσ0,σ1.
Similarly, forσ2∈I2σ1α, we have
(26)ℓ1ℏσ1,I2σ1α+ℏσ2,I1σ2α+ℓ2ℏσ1,I2σ1αℏσ2,I1σ2α1+ℏσ2,σ1+ℓ3ℏσ2,I2σ1αℏσ1,I1σ2α1+ℏσ2,σ1∈sI2σ1α,I1σ2α.That is,
(27)ℓ1ℏσ1,I2σ1α+ℏσ2,I1σ2α+ℓ2ℏσ1,I2σ1αℏσ2,I1σ2α1+ℏσ2,σ1+ℓ3ℏσ2,I2σ1αℏσ1,I1σ2α1+ℏσ2,σ1∈ω∈I2σ1αsω,I1σ2α.
Sinceσ2∈I2σ1α, so, we have
(28)ℓ1ℏσ1,I2σ1α+ℏσ2,I1σ2α+ℓ2ℏσ1,I2σ1αℏσ2,I1σ2α1+ℏσ2,σ1+ℓ3ℏσ2,I2σ1αℏσ1,I1σ2α1+ℏσ2,σ1∈sσ2,I1σ2α.
By definition of “s” function, we have
(29)ℓ1ℏσ1,I2σ1α+ℏσ2,I1σ2α+ℓ2ℏσ1,I2σ1αℏσ2,I1σ2α1+ℏσ2,σ1+ℓ3ℏσ2,I2σ1αℏσ1,I1σ2α1+ℏσ2,σ1∈ρ∈I1σ2αsℏσ2,ρ.
By definition of “s” function, there exists some σ3∈I1σ2α, such that
(30)ℓ1ℏσ1,I2σ1α+ℏσ2,I1σ2α+ℓ2ℏσ1,I2σ1αℏσ2,I1σ2α1+ℏσ2,σ1+ℓ3ℏσ2,I2σ1αℏσ1,I1σ2α1+ℏσ2,σ1∈sℏσ2,σ3.That is,
(31)ℏσ2,σ3°ℓ1ℏσ1,I2σ1α+ℏσ2,I1σ2α+ℓ2ℏσ1,I2σ1αℏσ2,I1σ2α1+ℏσ2,σ1+ℓ3ℏσ2,I2σ1αℏσ1,I1σ2α1+ℏσ2,σ1.
By the meaning ofWσI2ρα and WσI1ρα for σ,ρ∈ℵ, we get
(32)ℏσ2,σ3°ℓ1ℏσ1,σ2+ℏσ2,σ3+ℓ2ℏσ1,σ2ℏσ2,σ3+ℓ3ℏσ2,σ2,ℏσ1,σ31+ℏσ2,σ1=ℓ1ℏσ1,σ2+ℏσ2,σ3+ℓ2ℏσ1,σ2ℏσ2,σ31+ℏσ1,σ2,which implies that
(33)ℏσ2,σ3≤ℓ1ℏσ1,σ2+ℓ1ℏσ2,σ3+ℓ2ℏσ2,σ3ℏσ1,σ21+ℏσ1,σ2,which implies
(34)ℏσ2,σ3≤ℓ11−ℓ1−ℓ2ℏσ1,σ2=λℏσ1,σ2.
Inductively, we can construct a sequenceσn in ℵ such that
(35)ℏσ1,σ2≤λℏσ0,σ1,ℏσ2,σ3≤λ2ℏσ0,σ1,···ℏσn,σn+1≤λnℏσ0,σ1,for all n∈ℕ. Now, by triangular inequality, for m>n, we have
(36)ℏσn,σm°φσn,σmλnℏσ0,σ1+φσn,σmφσn+1,σmλn+1ℏσ0,σ1+⋯+,φσn,σmφσn+1,σm⋯φσm−2,σmφσm−1,σmλm−1ℏσ0,σ1,°ℏσ0,σ1φσn,σmλn+φσn,σmφσn+1,σmλn+1+⋯+φσn,σmφσn+1,σm⋯φσm−2,σmφσm−1,σmλm−1.
Sincelimn,m⟶∞φσn,σmλ<1, so the series ∑n=1∞λn∏i=1pφσi,σm converges by ratio test for each m∈ℕ. Let
(37)S=∑n=1∞λn∏i=1pφσi,σm,Sn=∑j=1nλj∏i=1pφσi,σm.
Thus, form>n, the above inequality can be written as
(38)ℏσn,σm°ℏσ0,σ1Sm−1−Sn.
Now, by takingn⟶∞, we get
(39)ℏσn,σm⟶0.
By Lemma10, we conclude that σn is a Cauchy sequence. Since ℵ is complete, then there exists an element σ∗ such that σn⟶ρ∗∈ℵ as n⟶∞. Now, to show ρ∗∈I1ρ∗ and ρ∗∈I2ρ∗ from (16), we have
(40)ℓ1ℏσ2n,I1σ2nα+ℏρ∗,I2ρ∗α+ℓ2ℏσ2n,I1σ2nαℏρ∗,I2ρ∗α1+ℏσ2n,ρ∗+ℓ3ℏρ∗,I1σ2nαℏσ2n,I2ρ∗α1+ℏσ2n,ρ∗∈sI1σ2nα,I2ρ∗α.That is,
(41)ℓ1ℏσ2n,I1σ2nα+ℏρ∗,I2ρ∗α+ℓ2ℏσ2n,I1σ2nαℏρ∗,I2ρ∗α1+ℏσ2n,ρ∗+ℓ3ℏρ∗,I1σ2nαℏσ2n,I2ρ∗α1+ℏσ2n,ρ∗∈ω∈I1σ2nαsω,I2ρ∗α.
Sinceσ2n+1∈I1σ2nα, we have
(42)ℓ1ℏσ2n,I1σ2nα+ℏρ∗,I2ρ∗α+ℓ2ℏσ2n,I1σ2nαℏρ∗,I2ρ∗α1+ℏσ2n,ρ∗+ℓ3ℏρ∗,I1σ2nαℏσ2n,I2ρ∗α1+ℏσ2n,ρ∗∈sσ2n+1,I2ρ∗αℓ1ℏσ2n,I1σ2nα+ℏρ∗,I2ρ∗α+ℓ2ℏσ2n,I1σ2nαℏρ∗,I2ρ∗α1+ℏσ2n,ρ∗+ℓ3ℏρ∗,I1σ2nαℏσ2n,I2ρ∗α1+ℏσ2n,ρ∗∈ρ/∈I2ρ∗αsℏσ2n+1,ρ/.
This implies that there existsρn∈I2ρ∗α such that
(43)ℓ1ℏσ2n,I1σ2nα+ℏρ∗,I2ρ∗α+ℓ2ℏσ2n,I1σ2nαℏρ∗,I2ρ∗α1+ℏσ2n,ρ∗+ℓ3ℏρ∗,I1σ2nαℏσ2n,I2ρ∗α1+ℏσ2n,ρ∗∈sℏσ2n+1,ρn.
That is,(44)ℏσ2n+1,ρn°ℓ1ℏσ2n,I1σ2nα+ℏρ∗,I2ρ∗α+ℓ2ℏσ2n,I1σ2nαℏρ∗,I2ρ∗α1+ℏσ2n,ρ∗+ℓ3ℏρ∗,I1σ2nαℏσ2n,I2ρ∗α1+ℏσ2n,ρ∗.
The g.l.b property ofI2 yields
(45)ℏσ2n+1,ρn°ℓ1ℏσ2n,σ2n+1+ℏρ∗,ρn+ℓ2ℏσ2n,σ2n+1ℏρ∗,ρn1+ℏσ2n,ρ∗+ℓ3ℏρ∗,σ2n+1ℏσ2n,ρn1+ℏσ2n,ρ∗.
We know that(46)ℏρ∗,ρn°θρ∗,ρnℏρ∗,σ2n+1+ℏσ2n+1,ρn.Hence,
(47)ℏρ∗,ρn°θρ∗,ρnℏρ∗,σ2n+1+ℓ1θρ∗,ρnℏσ2n,σ2n+1+ℏρ∗,ρn+ℓ2θρ∗,ρnℏσ2n,σ2n+1ℏρ∗,ρn1+ℏσ2n,ρ∗+ℓ3θρ∗,ρnℏρ∗,σ2n+1ℏσ2n,ρn1+ℏσ2n,ρ∗.
It follows that(48)ℏρ∗,ρn≤θρ∗,ρnℏρ∗,σ2n+1+ℓ1θρ∗,ρnℏσ2n,σ2n+1+ℓ1θρ∗,ρnℏρ∗,ρn+ℓ2θρ∗,ρnℏσ2n,σ2n+1ℏρ∗,ρn1+ℏσ2n,ρ∗+ℓ3θρ∗,ρnℏρ∗,σ2n+1ℏσ2n,ρn1+ℏσ2n,ρ∗.
Lettingn⟶∞, we get ℏρ∗,ρn⟶0. By using Lemma 9, we have ρn⟶ρ∗. Since I2ρ∗α is closed, so ρ∗∈I2ρ∗α. Following the similar steps, we can prove that ρ∗∈I1ρ∗α. Hence, there exists ρ∗∈ℵ such that ρ∗∈I1ρ∗α∩I2ρ∗α.
By settingℓ3=0 in Theorem 18, we get the following Corollary:Corollary 19.
Letℵ,ℏ be a complete CVEbMS, φ:ℵ×ℵ⟶1,∞, and let I1,I2: ℵ⟶ℵ satisfying g.l.b property. Assume that there exists α∈0,1, such that for each σ∈ℵ,I1σα,I2σα∈CBℵ, and there exists nonnegative real numbers ℓ1,ℓ2 with 2ℓ1+ℓ2<1 and λ1−ℓ1−ℓ2=ℓ1, where λ∈0,1 such that
(49)ℓ1ℏσ,I1σα+ℏρ,I2ρα+ℓ2ℏσ,I1σαℏρ,I2ρα1+ℏσ,ρ∈sI1σα,I2ρα,for all σ,ρ∈ℵ. If for each σ0∈ℵ,limn,m⟶∞φσn,σmλ<1, then there exists σ∗∈ℵ such that σ∗∈I1σ∗α∩I2σ∗α.By settingI1=I2 in Theorem 18, we get the following Corollary:Corollary 20.
Letℵ,ℏ be a complete CVEbMS, φ:ℵ×ℵ⟶1,∞, and let I: ℵ⟶ℵ satisfying g.l.b property. Assume that there exists α∈0,1, such that for each σ∈ℵ,Iσ∈CBℵ, and there exists nonnegative real numbers ℓ1,ℓ2,ℓ3 with 2ℓ1+ℓ2+ℓ3<1 and λ1−ℓ1−ℓ2=ℓ1, where λ∈0,1 such that
(50)ℓ1ℏσ,Iσα+ℏρ,Iρα+ℓ2ℏσ,Iσαℏρ,Iρα1+ℏσ,ρ+ℓ3ℏρ,Iσαℏσ,Iρα1+ℏσ,ρ∈sIσα,Iρα,for all σ,ρ∈ℵ. If for each σ0∈ℵ,limn,m⟶∞φσn,σmλ<1, then there exists σ∗∈ℵ such that σ∗∈Iσ∗α.Remark 21.
If we takeφσ,ρ=1 in the above Theorem 18, then we get the main result of Kutbi et al. [24].


## 4. Application

In this section, we derive some fixed point results for multivalued mappings as application of our main result.Theorem 22.
Letℵ,ℏ be a complete CVEbMS, φ:ℵ×ℵ⟶1,∞, and let Ə1,Ə2:ℵ⟶CBℵ satisfying g.l.b property. If there exists nonnegative real numbers ℓ1,ℓ2,ℓ3 with 2ℓ1+ℓ2+ℓ3<1 and λ1−ℓ1−ℓ2=ℓ1, where λ∈0,1 such that
(51)ℓ1ℏσ,Ə1σ+ℏρ,Ə2ρ+ℓ2ℏσ,Ə1σℏρ,Ə2ρ+ℓ3ℏρ,Ə1σℏσ,Ə2ρ1+ℏσ,ρ∈sƏ1σ,Ə2ρ,for all σ,ρ∈ℵ. If for each σ0∈ℵ,limn,m⟶∞φσn,σmλ<1, then there exists σ∗∈ℵ such that σ∗∈Ə1σ∗∩Ə2σ∗.Proof.
ConsiderI1,I2:ℵ⟶ℵ defined by
(52)I1σt=α,t∈Ə1σ,0,t∈Ə1σ,I2σt=α,t∈Ə2σ,0,t∈Ə2σ,where α∈0,1. Then,
(53)I1σα=t:I1σt≥α=Ə1σ,I2σα=Ə2σ.
Thus, Theorem18 can be applied to obtain σ∗∈ℵ such that
(54)σ∗∈I1σ∗α∩I2σ∗α=Ə1σ∗∩Ə2σ∗.
If we consider just one multivalued mapping, then we get the following result.Corollary 23.
Letℵ,ℏ be a complete CVEbMS, φ:ℵ×ℵ⟶1,∞, and let Ə:ℵ⟶CBℵ satisfying g.l.b property. If there exists nonnegative real numbers ℓ1,ℓ2,ℓ3 with 2ℓ1+ℓ2+ℓ3<1 and λ1−ℓ1−ℓ2=ℓ1, where λ∈0,1 such that
(55)ℓ1ℏσ,Əσ+ℏρ,öρ+ℓ2ℏσ,Əσℏρ,Əρ+ℓ3ℏρ,Əσℏσ,Əρ1+ℏσ,ρ∈sƏσ,Əρ,for all σ,ρ∈ℵ. If for each σ0∈ℵ,limn,m⟶∞φσn,σmλ<1, then there exists σ∗∈ℵ such that σ∗∈öσ∗.Corollary 24.
Letℵ,ℏ be a complete CVEbMS, φ:ℵ×ℵ⟶1,∞, and let Ə1,Ə2:ℵ⟶CBℵ satisfying g.l.b property. If there exists nonnegative real numbers ℓ1,ℓ2 with 2ℓ1+ℓ2<1 and λ1−ℓ1−ℓ2=ℓ1, where λ∈0,1 such that
(56)ℓ1ℏσ,Ə1σ+ℏρ,Ə2ρ+ℓ2ℏσ,Ə1σℏρ,Ə2ρ1+ℏσ,ρ∈sƏ1σ,Ə2ρ,for all σ,ρ∈ℵ. If for each σ0∈ℵ,limn,m⟶∞φσn,σmλ<1, then there exists σ∗∈ℵ such that σ∗∈ö1σ∗∩ö2σ∗.Proof.
Takingℓ3=0 in Theorem 22.Corollary 25.
Letℵ,ℏ be a complete CVEbMS, φ:ℵ×ℵ⟶1,∞ and let Ə1,Ə2:ℵ⟶CBℵ satisfying g.l.b property. If there exists nonnegative real number 2ℓ1∈0,1 such that
(57)ℓ1ℏσ,Ə1σ+ℏρ,Ə2ρ∈sƏ1σ,Ə2ρfor all σ,ρ∈ℵ. If for each σ0∈ℵ,limn,m⟶∞φσn,σmℓ1<1, then there exists σ∗∈ℵ such that σ∗∈Ə1σ∗∩Ə2σ∗.Proof.
Takingℓ2=ℓ3=0 in Theorem 22, if we take φσ,ρ=1 in the above Theorem 22, then we get main result of Ahmad et al. [18] in this way.Corollary 26. (see [18]).
Letℵ,ℏ be a complete CVMS, and let Ə1,Ə2:ℵ⟶CBℵ satisfying g.l.b property. If there exists nonnegative real numbers ℓ1,ℓ2,ℓ3 with 2ℓ1+ℓ2+ℓ3<1 such that
(58)ℓ1ℏσ,Ə1σ+ℏρ,Ə2ρ+ℓ2ℏσ,Ə1σℏρ,Ə2ρ+ℓ3ℏρ,Ə1σℏσ,Ə2ρ1+ℏσ,ρ∈sƏ1σ,Ə2ρ,for all σ,ρ∈ℵ, then there exists σ∗∈ℵ such that σ∗∈Ə1σ∗∩Ə2σ∗.The following result is a direct consequence of Theorem22 if we replace multivalued mappings with single valued mappings.Theorem 27.
Letℵ,ℏ be a complete CVEbMS, φ:ℵ×ℵ⟶1,∞, and let Ə1,Ə2:ℵ⟶ℵ. If there exists nonnegative real numbers ℓ1,ℓ2,ℓ3 with 2ℓ1+ℓ2+ℓ3<1 and λ1−ℓ1−ℓ2=ℓ1, where λ∈0,1 such that
(59)ℏƏ1σ,Ə2ρ°ℓ1ℏσ,Ə1σ+ℏρ,Ə2ρ+ℓ2ℏσ,Ə1σℏρ,Ə2ρ+ℓ3ℏρ,Ə1σℏσ,Ə2ρ1+ℏσ,ρ,for all σ,ρ∈ℵ. If for each σ0∈ℵ,limn,m⟶∞φσn,σmλ<1, then there exists σ∗∈ℵ such that σ∗=Ə1σ∗∩Ə2σ∗.If we takeφσ,ρ=1 in the above Theorem 27, then we get main result of Rouzkard et al. [10] in this way.Corollary 28. (see [10]).
Letℵ,ℏ be a complete CVMS, and let Ə1,Ə2:ℵ⟶ℵ. If there exists nonnegative real numbers ℓ1,ℓ2,ℓ3 with 2ℓ1+ℓ2+ℓ3<1 such that
(60)ℏƏ1σ,Ə2ρ≾ℓ1ℏσ,Ə1σ+ℏρ,Ə2ρ+ℓ2ℏσ,Ə1σℏρ,Ə2ρ+ℓ3ℏρ,Ə1σℏσ,Ə2ρ1+ℏσ,ρ,for all σ,ρ∈ℵ, then there exists σ∗∈ℵ such that σ∗=Ə1σ∗∩Ə2σ∗.
If we takeφσ,ρ=1 and ℓ3=0 in the above Theorem 27, then we get the main result of Azam et al. [8] in this way.Corollary 29. (see [8]).
Letℵ,ℏ be a complete CVMS, and let Ə1,Ə2:ℵ⟶ℵ. If there exists nonnegative real numbers ℓ1,ℓ2 with 2ℓ1+ℓ2<1 such that
(61)ℏƏ1σ,Ə2ρ≾ℓ1ℏσ,Ə1σ+ℏρ,Ə2ρ+ℓ2ℏσ,Ə1σℏρ,Ə2ρ1+ℏσ,ρ,for all σ,ρ∈ℵ, then there exists σ∗∈ℵ such that σ∗=Ə1σ∗∩Ə2σ∗.


## 5. Conclusion

In this paper, we have defined fuzzy contraction in the setting of complex valued extendedb-metric space and proved fuzzy fixed point results. As application of our main theorems, we have derived fixed point results for multivalued mappings. We hope that the results proved in this paper will form new connections for those who are working in complex valued extended b-metric space.In this direction, the future work will focus on studying the fixed points ofL-fuzzy mappings in complex valued extended b-metric spaces. Also, coincidence point results for single valued mappings and left total relations can be established as future work. As applications, one can investigate the fractional differential inclusion problems.



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*Source: 5632119-2022-05-17.xml*