# On Fuzzy Fixed-Point Results in Complex Valued Extendedb-Metric Spaces with Application

Authors: Amnah Essa Shammaky; Jamshaid Ahmad; Abdelhamied Farrag Sayed
Journal: Journal of Mathematics
 (2021)

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## Abstract

The aim of this paper is to define fuzzy contraction in the context of complex valued extendedb-metric space and prove fuzzy fixed-point results. Our results improve and extend certain recent results in literature. Moreover, we discuss an illustrative example to highlight the realized improvements. As application, we derive fixed-point results for multivalued mappings in the setting 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 2011, Azam et al. [7] introduced the concept of complex valued metric space and obtained some common fixed-point results for rational contraction which consist of a pair of single valued mappings. Later on, many researchers [8–15] worked on this generalized metric space. Ahmad et al. [16] and Azam et al. [17] defined the generalized Housdorff metric function in the setting of complex valued metric space and obtained common fixed-point results for multivalued mappings. In [18], Mukheimer generalized the concept of complex valued metric space to complex valued b-metric space. Recently, Naimatullah et al. [19] 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 contrary, Heilpern [20] introduced the concept of fuzzy mappings in the setting of metric linear spaces and extended Banach Contraction Principle [21]. In 2014, Kutbi et al. [22] 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. [23] 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. [7], Rouzkard and Imdad [9], Ahmad et al. [16], and Kutbi et al. [22] for fuzzy and multivalued mappings in complex valued metric spaces.


## 2. Preliminaries

In 2011, Azam et al. [7] introduced the complex valued metric space as follows.Definition 1.
(see [7]). Let ℂ be the set of complex numbers and ℓ1,ℓ2∈ℂ. A partial order ≾ on ℂ is defined in this way:(1)ℓ1≾ℓ2⇔Reℓ1≤Reℓ2,Imℓ1≤Imℓ2.It follows that(2)ℓ1≾ℓ2if one of these assertions is satisfied:(3)aReℓ1=Reℓ2,Imℓ1<Imℓ2,bReℓ1<Reℓ2,Imℓ1=Imℓ2,cReℓ1<Reℓ2,Imℓ1<Imℓ2,dReℓ1=Reℓ2,Imℓ1=Imℓ2.(CV1)0≾℘ℓ,ℏ, for all ℓ,ℏ∈Q and ℘ℓ,ℏ=0 if and only if ℓ=ℏ(CV2)℘ℓ,ℏ=℘ℏ,ℓ, for all ℓ,ℏ∈Q(CV3)℘ℓ,ℏ≾℘ℓ,ν+℘ν,ℏ, for all ℓ,ℏ,ν∈Q.Definition 2.
(see [7]). Let Q≠∅. A mapping ℘:Q×Q⟶ℂ is said to be a complex valued metric if the following assertions hold.Then,Q,℘ is called a complex valued metric space (CVMS).In 2014, Mukheimer [18] introduced the notion of complex valued b-metric space as follows.(CVB1)0≾℘ℓ,ℏ, for all ℓ,ℏ∈Q and ℘ℓ,ℏ=0 if and only if ℓ=ℏ(CVB2)℘ℓ,ℏ=℘ℏ,ℓ, for all ℓ,ℏ∈Q(CVB3)℘ℓ,ℏ≾π℘ℓ,ν+℘ν,ℏ, for all ℓ,ℏ,ν∈QDefinition 3 (see [18]).
LetQ≠∅ and π≥1 be a real number. A mapping ℘:Q×Q⟶ℂ is said to be a complex valued b-metric space if the following assertions hold.Then,Q,℘ is called a complex valued b- metric space (CVbMS).Recently, Naimatullah et al. [19] defined the notion of complex valued extended b-metric space in the following way.(ECVB1)0≾℘ℓ,ℏ, for all ℓ,ℏ∈Q and ℘ℓ,ℏ=0 if and only if ℓ=ℏ(ECVB2)℘ℓ,ℏ=℘ℏ,ℓ, for all ℓ,ℏ∈Q(ECVB3)℘ℓ,ℏ≾φℓ,ℏ℘ℓ,ν+℘ν,ℏ, for all ℓ,ℏ,ν∈QDefinition 4.
(see [19]). Let Q≠∅ and φ:Q×Q⟶1,∞. A mapping ℘:Q×Q⟶ℂ is called a complex valued extended b-metric if following conditions hold:Then,Q,℘ is called a complex valued extended b- metric space (CVEbMS).Lemma 1 (see [19]).
LetQ,℘ be a CVEbMS and let ℓn⊆Q. Then, ℓn converges to ℓ⇔℘ℓn,ℓ⟶0asn⟶∞.Lemma 2 (see [19]).
LetQ,℘ be a CVEbMS and let ℓn⊆Q. Then, ℓn is a Cauchy sequence ⇔℘ℓn,ℓn+m⟶0asn⟶∞, where m∈ℕ.LetQ,℘ be a CVEbMS; then, ℭBQ denotes the family of all nonempty, closed, and bounded subsets of Q.From now on, we denotesℓ1=ℓ2∈ℂ:ℓ1≺¯ℓ2 for ℓ1∈ℂ, and (4)sℓ1,ℜ2=∪ℓ2∈ℜ2s℘ℓ1,ℓ2=∪ℓ2∈ℜ2ℓ∈ℂ:℘ℓ1,ℓ2≺¯ℓ,for a∈Q and ℜ2∈ℭBQ.Forℜ1,ℜ2∈ℭBQ, we denote(5)sℜ1,ℜ2=∩ℓ1∈ℜ1sℓ1,ℜ2∩∩ℓ2∈ℜ2sℓ2,ℜ1.(i)
Letℓ1,ℓ2∈ℂ. If ℓ1≺¯ℓ2, then sℓ2⊂sℓ1.(ii)
Letℓ∈Q and ℜ∈NQ. If θ∈sℓ,ℜ, then ℓ∈ℜ.(iii)
Letℓ∈ℂ and let ℜ1,ℜ2∈ℭBQ and ℓ1∈ℜ1. If ℓ∈sℜ1,ℜ2, then ℓ∈sℓ1,ℜ2, for all ℓ1∈ℜ1, or ℓ∈sℜ1,ℓ2, for all ℓ2∈ℜ2.Lemma 3 (see [19]).
LetQ,℘ be a CVEbMS.LetQ,℘ be a complex valued extended b-metric space and ℭQ be a collection of nonempty closed subsets of Q. Let ℑ:Q⟶ℭBQbe a multivalued mapping. For ℓ∈Q and ℜ∈ℭBQ, we define(6)Wℓℜ=℘ℓ,ℓ1:ℓ1∈ℜ.Thus, forℓ,y∈Q,(7)Wℓℑy=℘ℓ,ℓ1:ℓ1∈ℑy.Definition 5.
(see [19]). Let Q,℘ be a complex valued metric space. A subset ℜ of Q is called bounded below if ∃ℓ∈Q, such that ℓ≺¯ℓ1, for all ℓ1∈ℜ.Definition 6.
(see [19]). Let Q,℘ be a complex valued metric space. A multivalued mapping ℑ:Q⟶2ℂ is called bounded below if, for each ℓ∈Q,∃ℓx∈ℂ,(8)ℓx≺¯u,for all u∈ℑℓ.In 1981, Heilpern [20] utilized the concept of fuzzy set and introduced the notion of fuzzy mappings in metric spaces (MS). A fuzzy set in Q is a function with domain Q and values in 0,1, and IQ is the collection of all fuzzy sets in Q. If ℜ is a fuzzy set and x∈Q, then the function values ℜℓ is called the grade of membership of ℓ in ℜ. The α-level set of ℜ is denoted by ℜα and is defined as follows:(9)ℜα=ℓ:ℜℓ≥αifα∈0,1,ℜ0=x:ℜℓ>0¯.Here,ℜ¯ denotes the closure of the set ℜ. Let ℱQ be the collection of all fuzzy sets in a metric space Q.Definition 7.
(see [20]). Let Q1 be a nonempty set and Q2,℘ be a MS. A mapping ℑ is called fuzzy mapping if ℑ is a mapping from Q1 into ℱQ2. A fuzzy mapping ℑ is a fuzzy subset on Q1×Q2 with membership function ℑxy. The function ℑxyis the grade of membership of y in ℑx.Definition 8.
(see [20]). Let Q,℘ be a MS and ℑ1,ℑ2: Q⟶ℱQ. A point ℓ∈Q is said to be a fuzzy fixed point of ℑ2 if ℓ∈ℑ2ℓα, for some α∈0,1. The point ℓ∈Q is said to be a common fuzzy fixed point of ℑ1 and ℑ2 if ℓ∈ℑ1ℓα∩ℑ2ℓα, for some α∈0,1.In 2014, Kutbi et al. [22] used the above notion of fuzzy mappings in complex valued metric space (CVMS) and established the result for these mappings.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. [22] for fuzzy mappings and some fixed-point result for multivalued mappings in CVMS.


## 3. Main Result

Definition 9.
LetQ,℘ be a CVEbMS. The fuzzy mapping ℑ:Q⟶ℱQ is said to have g.l.b. property on Q,℘ if, for any ℓ∈Q and any α∈0,1, greatest lower bound of Wℓℑℏα exists in ℂ, ∀ℏ∈Q. We denote ℘ℓ,ℑℏα by the g.l.b of Wℓℑℏα. That is,(10)℘ℓ,ℑℏα=inf℘ℓ,ν:ν∈ℑℏα.Now, we state our main result in this way.Theorem 1.
LetQ,℘ be a complete CVEbMS, φ:Q×Q⟶1,∞, and let ℑ1,ℑ2: Q⟶ℱQ satisfy g.l.b property. Assume that ∃α∈0,1, such that, for each ℓ∈Q, ℑ1ℓα,ℑ2ℓα∈CBQ and there exist nonnegative real numbers ζ,κ,μ with ζ+κ+μ<1 and λ1−κ=ζ, where λ∈0,1 such that(11)ζ℘ℓ,ℏ+κ℘ℓ,ℑ1ℓα℘ℏ,ℑ2ℏα+μ℘ℏ,ℑ1ℓα℘ℓ,ℑ2ℏα1+℘ℓ,ℏ∈sℑ1ℓα,ℑ2ℏα,for all ℓ,ℏ∈Q. If, for each ℓ0∈Q, limn,m⟶∞φℓn,ℓmλ<1, then ∃ℓ∗∈Q such that ℓ∗∈ℑ1ℓ∗α∩ℑ2ℓ∗α.Proof.
Letℓ0 be an arbitrary point in Q. By assumption, we can find ℓ1∈ℑ1ℓ0α. So, we have(12)ζ℘ℓ0,ℓ1+κ℘ℓ0,ℑ1ℓ0α℘ℓ1,ℑ2ℓ1αμ℘ℓ1,ℑ1ℓ0α℘ℓ0,ℑ2ℓ1α1+℘ℓ0,ℓ1∈sℑ1ℓ0α,ℑ2ℓ1α,ζ℘ℓ0,ℓ1+κ℘ℓ0,ℑ1ℓ0α℘ℓ1,ℑ2ℓ1α+μ℘ℓ1,ℑ1ℓ0α℘ℓ0,ℑ2ℓ1α1+℘ℓ0,ℓ1∈∩ω∈ℑ1ℓ0αsω,ℑ2ℓ1α.
Sinceℓ1∈ℑ1ℓ0α, so, we have(13)ζ℘ℓ0,ℓ1+κ℘ℓ0,ℑ1ℓ0α℘ℓ1,ℑ2ℓ1α+μ℘ℓ1,ℑ1ℓ0α℘ℓ0,ℑ2ℓ1α1+℘ℓ0,ℓ1∈sℓ1,ℑ2ℓ1α.
By definition,(14)ζ℘ℓ0,ℓ1+κ℘ℓ0,ℑ1ℓ0α℘ℓ1,ℑ2ℓ1α+μ℘ℓ1,ℑ1ℓ0α℘ℓ0,ℑ2ℓ1α1+℘ℓ0,ℓ1∈∪ϱ∈ℑ2ℓ1αs℘ℓ1,ϱ.
This implies that∃ℓ2∈ℑ2ℓ1α such that(15)ζ℘ℓ0,ℓ1+κ℘ℓ0,ℑ1ℓ0α℘ℓ1,ℑ2ℓ1α+μ℘ℓ1,ℑ1ℓ0α℘ℓ0,ℑ2ℓ1α1+℘ℓ0,ℓ1∈s℘ℓ1,ℓ2.
That is,(16)℘ℓ1,ℓ2⪯ζ℘ℓ0,ℓ1+κ℘ℓ0,ℑ1ℓ0α℘ℓ1,ℑ2ℓ1α+μ℘ℓ1,ℑ1ℓ0α℘ℓ0,ℑ2ℓ1α1+℘ℓ0,ℓ1.
By the meaning ofWℓℑ2ℏα and Wℓℑ1ℏα for ℓ,ℏ∈Q, we obtain(17)℘ℓ1,ℓ2⪯ζ℘ℓ0,ℓ1+κ℘ℓ0,ℓ1℘ℓ1,ℓ2+μ℘ℓ1,ℓ1℘ℓ0,ℓ21+℘ℓ0,ℓ1=ζ℘ℓ0,ℓ1+κ℘ℓ0,ℓ1℘ℓ1,ℓ21+℘ℓ0,ℓ1=ζ℘ℓ0,ℓ1+κ℘ℓ1,ℓ2℘ℓ0,ℓ11+℘ℓ0,ℓ1.
This implies(18)℘ℓ1,ℓ2≤ζ℘ℓ0,ℓ1+κ℘ℓ1,ℓ2℘ℓ0,ℓ11+℘ℓ0,ℓ1≤ζ℘ℓ0,ℓ1+κ℘ℓ1,ℓ2,such that(19)℘ℓ1,ℓ2≤ζ1−κ℘ℓ0,ℓ1=λ℘ℓ0,ℓ1.
Similarly, forℓ2∈ℑ2ℓ1α, we have(20)ζ℘ℓ2,ℓ1+κ℘ℓ1,ℑ2ℓ1α℘ℓ2,ℑ1ℓ2α+μ℘ℓ2,ℑ2ℓ1α℘ℓ1,ℑ1ℓ2α1+℘ℓ2,ℓ1∈sℑ2ℓ1α,ℑ1ℓ2α,ζ℘ℓ2,ℓ1+κ℘ℓ1,ℑ2ℓ1α℘ℓ2,ℑ1ℓ2α+μ℘ℓ2,ℑ2ℓ1α℘ℓ1,ℑ1ℓ2α1+℘ℓ2,ℓ1∈∩ω∈ℑ2ℓ1αsω,ℑ1ℓ2α.
Sinceℓ2∈ℑ2ℓ1α, so, we have(21)ζ℘ℓ2,ℓ1+κ℘ℓ1,ℑ2ℓ1α℘ℓ2,ℑ1ℓ2α+μ℘ℓ2,ℑ2ℓ1α℘ℓ1,ℑ1ℓ2α1+℘ℓ2,ℓ1∈sℓ2,ℑ1ℓ2α.
By definition of “s” function, we have(22)ζ℘ℓ2,ℓ1+κ℘ℓ1,ℑ2ℓ1α℘ℓ2,ℑ1ℓ2α+μ℘ℓ2,ℑ2ℓ1α℘ℓ1,ℑ1ℓ2α1+℘ℓ2,ℓ1∈∪∈ℑ1ℓ2αs℘ℓ2,.
By definition of “s” function, there exists some ℓ3∈ℑ1ℓ2α, such that(23)ζ℘ℓ2,ℓ1+κ℘ℓ1,ℑ2ℓ1α℘ℓ2,ℑ1ℓ2α+μ℘ℓ2,ℑ2ℓ1α℘ℓ1,ℑ1ℓ2α1+℘ℓ2,ℓ1∈s℘ℓ2,ℓ3.
That is,(24)℘ℓ2,ℓ3⪯ζ℘ℓ2,ℓ1+κ℘ℓ1,ℑ2ℓ1α℘ℓ2,ℑ1ℓ2α+μ℘ℓ2,ℑ2ℓ1α℘ℓ1,ℑ1ℓ2α1+℘ℓ2,ℓ1.
By the meaning ofWℓℑ2ℏα and Wℓℑ1ℏα, for ℓ,ℏ∈Q, we obtain(25)℘ℓ2,ℓ3≺¯ζ℘ℓ2,ℓ1+κ℘ℓ1,ℓ2℘ℓ2,ℓ3+μ℘ℓ2,ℓ2,℘ℓ1,ℓ31+℘ℓ2,ℓ1=ζ℘ℓ1,ℓ2+κ℘ℓ1,ℓ2℘ℓ2,ℓ31+℘ℓ1,ℓ2,which implies that(26)℘ℓ2,ℓ3≤ζ℘ℓ1,ℓ2+κ℘ℓ2,ℓ3℘ℓ1,ℓ21+℘ℓ1,ℓ2,which implies(27)℘ℓ2,ℓ3≤ζ1−κ℘ℓ1,ℓ2=λ|℘ℓ1,ℓ2|
Inductively, we can construct a sequenceℓn in Q such that(28)℘ℓ1,ℓ2≤λ℘ℓ0,ℓ1,℘ℓ2,ℓ3≤λ2℘ℓ0,ℓ1,···℘ℓn,ℓn+1≤λn℘ℓ0,ℓ1,for all n∈ℕ. Now, by triangular inequality, for m>n, we have(29)℘ℓn,ℓm⪯φℓn,ℓmλn℘ℓ0,ℓ1+φℓn,ℓmφℓn+1,ℓmλn+1℘ℓ0,ℓ1+···+φℓn,ℓmφℓn+1,ℓm···φℓm−2,ℓmφℓm−1,ℓmλm−1℘ℓ0,ℓ1≺¯wpℓ0,ℓ1φℓn,ℓmλn+φℓn,ℓmφℓn+1,ℓmλn+1+···+φℓn,ℓmφℓn+1,ℓm···φℓm−2,ℓmφℓm−1,ℓmλm−1.
Sincelimn,m⟶∞φℓn,ℓmλ<1, so the series ∑n=1∞λn∏i=1pφℓi,ℓm converges by ratio test for each m∈ℕ. Let(30)S=∑n=1∞λn∏i=1pφℓi,ℓm,Sn=∑j=1nλj∏i=1pφℓi,ℓm.
Thus, form>n, the above inequality can be written as(31)℘ℓn,ℓm≺¯℘ℓ0,ℓ1Sm−1−Sn.
Now, by takingn⟶∞, we obtain(32)℘ℓn,ℓm⟶0.
By Lemma2, we conclude that ℓn is a Cauchy sequence. Since Q is complete, then there exists an element ℓ∗ such that ℓn⟶ϱ∗∈Q as n⟶∞. Now, to show ϱ∗∈ℑ1ϱ∗ and ϱ∗∈ℑ2ϱ∗, from (1), we have(33)ζ℘ℓ2n,ϱ∗+κ℘ℓ2n,ℑ1ℓ2nα℘ϱ∗,ℑ2ϱ∗α+μ℘ϱ∗,ℑ1ℓ2nα℘ℓ2n,ℑ2ϱ∗α1+℘ℓ2n,ϱ∗∈sℑ1ℓ2nα,ℑ2ϱ∗α,ζ℘ℓ2n,ϱ∗+κ℘ℓ2n,ℑ1ℓ2nα℘ϱ∗,ℑ2ϱ∗α+μ℘ϱ∗,ℑ1ℓ2nα℘ℓ2n,ℑ2ϱ∗α1+℘ℓ2n,ϱ∗∈∩ω∈ℑ1ℓ2nαsω,ℑ2ϱ∗α.
Sinceℓ2n+1∈ℑ1ℓ2nα, we have(34)ζ℘ℓ2n,ϱ∗+κ℘ℓ2n,ℑ1ℓ2nα℘ϱ∗,ℑ2ϱ∗α+μ℘ϱ∗,ℑ1ℓ2nα℘ℓ2n,ℑ2ϱ∗α1+℘ℓ2n,ϱ∗∈sℓ2n+1,ℑ2ϱ∗α,ζ℘ℓ2n,ϱ∗+κ℘ℓ2n,ℑ1ℓ2nα℘ϱ∗,ℑ2ϱ∗α+μ℘ϱ∗,ℑ1ℓ2nα℘ℓ2n,ℑ2ϱ∗α1+℘ℓ2n,ϱ∗∈∪ϱ/∈ℑ2ϱ∗αs℘ℓ2n+1,ϱ/.
This implies that∃ϱn∈ℑ2ϱ∗α such that(35)ζ℘ℓ2n,ϱ∗+κ℘ℓ2n,ℑ1ℓ2nα℘ϱ∗,ℑ2ϱ∗α+μ℘ϱ∗,ℑ1ℓ2nα℘ℓ2n,ℑ2ϱ∗α1+℘ℓ2n,ϱ∗∈s℘ℓ2n+1,ϱn.
That is,(36)℘ℓ2n+1,ϱn⪯ζ℘ℓ2n,ϱ∗+κ℘ℓ2n,ℑ1ℓ2nα℘ϱ∗,ℑ2ϱ∗α+μ℘ϱ∗,ℑ1ℓ2nα℘ℓ2n,ℑ2ϱ∗α1+℘ℓ2n,ϱ∗.
The g.l.b property ofℑ2 yields(37)℘ℓ2n+1,ϱn⪯ζ℘ℓ2n,ϱ∗+κ℘ℓ2n,ℓ2n+1℘ϱ∗,ϱn+μ℘ϱ∗,ℓ2n+1℘ℓ2n,ϱn1+℘ℓ2n,ϱ∗.
We know that(38)℘ϱ∗,ϱn⪯θϱ∗,ϱn℘ϱ∗,ℓ2n+1+℘ℓ2n+1,ϱn.
Hence,(39)℘ϱ∗,ϱn⪯θϱ∗,ϱn℘ϱ∗,ℓ2n+1+ζθϱ∗,ϱn℘ℓ2n,ϱ∗+κθϱ∗,ϱn℘ℓ2n,ℓ2n+1℘ϱ∗,ϱn1+℘ℓ2n,ϱ∗+μθϱ∗,ϱn℘ϱ∗,ℓ2n+1℘ℓ2n,ϱn1+℘ℓ2n,ϱ∗.
It follows that(40)℘ϱ∗,ϱn≤θϱ∗,ϱn℘ϱ∗,ℓ2n+1+ζθϱ∗,ϱn℘ℓ2n,ϱ∗+κθϱ∗,ϱn℘ℓ2n,ℓ2n+1℘ϱ∗,ϱn1+℘ℓ2n,ϱ∗+μθϱ∗,ϱn℘ϱ∗,ℓ2n+1℘ℓ2n,ϱn1+℘ℓ2n,ϱ∗.
Lettingn⟶∞, we get ℘ϱ∗,ϱn⟶0. By using Lemma 1, 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∗∈Q such that ϱ∗∈ℑ1ϱ∗α∩ℑ2ϱ∗α.
By settingμ=0 in Theorem 1, we get the following Corollary.Corollary 1.
LetQ,℘ be a complete CVEbMS, φ:Q×Q⟶1,∞, and let ℑ1,ℑ2: Q⟶ℱQ satisfy g.l.b property. Assume that ∃α∈0,1, such that, for each ℓ∈Q, ℑ1ℓα,ℑ2ℓα∈CBQ and there exist nonnegative real numbers ζ,κ with ζ+κ<1 and λ1−κ=ζ, where λ∈0,1 such that(41)ζ℘ℓ,ℏ+κ℘ℓ,ℑ1ℓα℘ℏ,ℑ2ℏα1+℘ℓ,ℏ∈sℑ1ℓα,ℑ2ℏα.for all ℓ,ℏ∈Q. If, for each ℓ0∈Q, limn,m⟶∞φℓn,ℓmλ<1, then ∃ℓ∗∈Q such that ℓ∗∈ℑ1ℓ∗α∩ℑ2ℓ∗α.By settingℑ1=ℑ2 in Theorem 1, we get the following corollary.Corollary 2.
LetQ,℘ be a complete CVEbMS, φ:Q×Q⟶1,∞, and let ℑ: Q⟶ℱQ satisfy g.l.b property. Assume that ∃α∈0,1, such that, for each ℓ∈Q, ℑℓ∈CBQ and there exist nonnegative real numbers ζ,κ,μ with ζ+κ+μ<1 and λ1−κ=ζ, where λ∈0,1 such that(42)ζ℘ℓ,ℏ+κ℘ℓ,ℑℓα℘ℏ,ℑℏα+μ℘ℏ,ℑℓα℘ℓ,ℑℏα1+℘ℓ,ℏ∈sℑℓα,ℑℏα,for all ℓ,ℏ∈Q. If, for each ℓ0∈Q, limn,m⟶∞φℓn,ℓmλ<1, then ∃ℓ∗∈Q such that ℓ∗∈ℑℓ∗α.Remark 1.
If we takeφℓ,ℏ=1 in Theorem 1, then we get main result of Kutbi et al. [22].


## 4. Application

In this section, we derive some fixed-point results for multivalued mappings as application of our main result.Theorem 2.
LetQ,℘ be a complete CVEbMS, φ:Q×Q⟶1,∞, and let ℵ1,ℵ2: Q⟶CBQ satisfy g.l.b property. If there exist nonnegative real numbers ζ,κ,μ with ζ+κ+μ<1 and λ1−κ=ζ, where λ∈0,1 such that(43)ζ℘ℓ,ℏ+κ℘ℓ,ℵ1ℓ℘ℏ,ℵ2ℏ+μ℘ℏ,ℵ1ℓ℘ℓ,ℵ2ℏ1+℘ℓ,ℏ∈sℵ1ℓ,ℵ2ℏ,for all ℓ,ℏ∈Q. If, for each ℓ0∈Q, limn,m⟶∞φℓn,ℓmλ<1, then ∃ℓ∗∈Q such that ℓ∗∈ℵ1ℓ∗∩ℵ2ℓ∗.Proof.
Considerℑ1,ℑ2:Q⟶ℱQ defined by(44)ℑ1ℓt=α,t∈ℵ1ℓ0,t∉ℵ1ℓ,ℑ2ℓt=α,t∈ℵ2ℓ0,t∉ℵ2ℓ,where α∈0,1. Then,(45)ℑ1ℓα=t:ℑ1ℓt≥α=ℵ1ℓ,ℑ2ℓα=ℵ2ℓ.
Thus, Theorem1 can be applied to obtain ℓ∗∈Q such that(46)ℓ∗∈ℑ1ℓ∗α∩ℑ2ℓ∗α=ℵ1ℓ∗∩ℵ2ℓ∗.
If we consider just one multivalued mapping, then we get the following result.Corollary 3.
LetQ,℘ be a complete CVEbMS, φ:Q×Q⟶1,∞, and let ℵ: Q⟶CBQ satisfy g.l.b property. If there exist nonnegative real numbers ζ,κ,μ with ζ+κ+μ<1 and λ1−κ=ζ, where λ∈0,1, then(47)ζ℘ℓ,ℏ+κ℘ℓ,ℵℓ℘ℏ,ℵℏ+μ℘ℏ,ℵℓ℘ℓ,ℵℏ1+℘ℓ,ℏ∈sℵℓ,ℵℏ,for all ℓ,ℏ∈Q. If for each ℓ0∈Q, limn,m⟶∞φℓn,ℓmλ<1, then ∃ℓ∗∈Q such that ℓ∗∈ℵℓ∗.Corollary 4.
LetQ,℘ be a complete CVEbMS, φ:Q×Q⟶1,∞, and let ℵ1,ℵ2: Q⟶CBQ satisfy g.l.b property. If there exist nonnegative real numbers ζ,κ with ζ+κ<1 and λ1−κ=ζ, where λ∈0,1 such that(48)ζ℘ℓ,ℏ+κ℘ℓ,ℵ1ℓ℘ℏ,ℵ2ℏ1+℘ℓ,ℏ∈sℵ1ℓ,ℵ2ℏ,for all ℓ,ℏ∈Q. If, for each ℓ0∈Q, limn,m⟶∞φℓn,ℓmλ<1, then ∃ℓ∗∈Q such that ℓ∗∈ℵ1ℓ∗∩ℵ2ℓ∗.Proof.
Takeμ=0 in Theorem 2.Corollary 5.
LetQ,℘ be a complete CVEbMS, φ:Q×Q⟶1,∞, and let ℵ1,ℵ2: Q⟶CBQ satisfy g.l.b property. If there exists nonnegative real number ζ∈0,1, then(49)ζ℘ℓ,ℏ∈sℵ1ℓ,ℵ2ℏ,for all ℓ,ℏ∈Q. If, for each ℓ0∈Q, limn,m⟶∞φℓn,ℓmζ<1, then ∃ℓ∗∈Q such that ℓ∗∈ℵ1ℓ∗∩ℵ2ℓ∗.Proof.
Takeκ=μ=0 in Theorem 2.
If we takeφℓ,ℏ=1 in Theorem 2, then we get main result of Ahmad et al. [16] as follows.Corollary 6 (see [16]).
LetQ,℘ be a complete CVMS, and let ℵ1,ℵ2 :Q⟶CBQ satisfy g.l.b property. If there exist nonnegative real numbers ζ,κ,μ with ζ+κ+μ<1 such that(50)ζ℘ℓ,ℏ+κ℘ℓ,ℵ1ℓ℘ℏ,ℵ2ℏ+μ℘ℏ,ℵ1ℓ℘ℓ,ℵ2ℏ1+℘ℓ,ℏ∈sℵ1ℓ,ℵ2ℏ,for all ℓ,ℏ∈Q, then ∃ℓ∗∈Q such that ℓ∗∈ℵ1ℓ∗∩ℵ2ℓ∗.The following result is a direct consequence of Theorem2 if we replace multivalued mappings with single valued mappings.Theorem 3.
LetQ,℘ be a complete CVEbMS, φ:Q×Q⟶1,∞, and let ℵ1,ℵ2: Q⟶Q. If there exist nonnegative real numbers ζ,κ,μ with ζ+κ+μ<1 and λ1−κ=ζ, where λ∈0,1 such that(51)℘ℵ1ℓ,ℵ2ℏ≺¯ζ℘ℓ,ℏ+κ℘ℓ,ℵ1ℓ℘ℏ,ℵ2ℏ+μ℘ℏ,ℵ1ℓ℘ℓ,ℵ2ℏ1+℘ℓ,ℏ,for all ℓ,ℏ∈Q. If, for each ℓ0∈Q, limn,m⟶∞φℓn,ℓmλ<1, then ∃ℓ∗∈Q such that ℓ∗=ℵ1ℓ∗∩ℵ2ℓ∗.If we takeφℓ,ℏ=1 in Theorem 3, then we get main result of Rouzkard and Imdad [9] as follows.Corollary 7 (see [9]).
LetQ,℘ be a complete CVMS and let ℵ1,ℵ2: Q⟶Q. If there exist nonnegative real numbers ζ,κ,μ with ζ+κ+μ<1 such that(52)℘ℵ1ℓ,ℵ2ℏ≺¯ζ℘ℓ,ℏ+κ℘ℓ,ℵ1ℓ℘ℏ,ℵ2ℏ+μ℘ℏ,ℵ1ℓ℘ℓ,ℵ2ℏ1+℘ℓ,ℏ,for all ℓ,ℏ∈Q, then ∃ℓ∗∈Q such that ℓ∗=ℵ1ℓ∗∩ℵ2ℓ∗.If we takeφℓ,ℏ=1 and μ=0 in Theorem 3, then we get the main result of Azam et al. [7] as follows.Corollary 8 (see [7]).
LetQ,℘ be a complete CVMS, and let ℵ1,ℵ2: Q⟶Q. If there exist nonnegative real numbers ζ,κ with ζ+κ<1 such that(53)℘ℵ1ℓ,ℵ2ℏ≺¯ζ℘ℓ,ℏ+κ℘ℓ,ℵ1ℓ℘ℏ,ℵ2ℏ1+℘ℓ,ℏ,for all ℓ,ℏ∈Q, then ∃ℓ∗∈Q such that ℓ∗=ℵ1ℓ∗∩ℵ2ℓ∗.


## 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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