# Fuzzy Fixed Point Results in ConvexC∗-Algebra-Valued Metric Spaces

Authors: Mohammed Shehu Shagari; Shazia Kanwal; Hassen Aydi; Yaé Ulrich Gaba
Journal: Journal of Function Spaces
 (2022)

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## Abstract

The purpose of this note is to come up with some new directions in fuzzy fixed point theory. To this effect, notions of aC∗-algebra-valued fuzzy λ-contraction and related concepts in a convex C∗-algebra-valued metric space (C∗-AVMS) are set-up. In line with the view of a Hausdorff distance function, an idea of a distance between two approximate quantities is proposed. Consequently, two fixed point results of a C∗-algebra-valued fuzzy mapping (C∗-AVFM) for the new type of contractions are established using Mann and Ishikawa iterative schemes. For some future investigations of our results, two open problems are noted concerning sufficient criteria guaranteeing the existence of fixed points of a C∗-algebra-valued fuzzy λ-contraction and whether or not the Picard iteration for a C∗-algebra-valued fuzzy λ-contraction converges.


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## Body

## 1. Introduction and Preliminaries

We begin this section with specific notions ofC∗-algebras as follows.Definition 1 (see [1]).
LetA be a unital algebra with the unit IA. An involution on A is a conjugate linear map j↦j∗ such that j∗∗=j and jℓ∗=ℓ∗j∗, for all j,ℓ∈A. The pair A,∗ is called a ∗-algebra. A Banach ∗-algebra is a ∗-algebra A together with a submultiplicative norm such that ∥j∗∥=∥j∥, for all j∈A; where a norm . on an algebra A is said to be submultiplicative if jℓ≤jℓ, for all j,ℓ∈A. A C∗-algebra is a Banach ∗-algebra such that ∥j∗j∥=∥j∥2, for all j∈A.Throughout this paper,A represents a unital C∗-algebra with a unit IA. Also, we take Aa=j∈A:j=j∗ and denote the zero element in A by 0A. An element j∈A is called positive, written j±0A, if j∈Aa and σj⊆ℝ+=0,∞, where σj=λ∈ℂ:λI−jisnotinvertible is the spectrum of j. Availing positive elements, we set up a partial ordering ° on Aa as follows: j°ℓ if and only if ℓ−j±0A. Hereafter, by A+, we mean the set j∈A:j±0A and ∣j∣=j∗j1/2 (cf. [2]).Remark 2.
WhenA is a unital C∗-algebra, then for any j∈A+, we have j°IA if and only if ∥j∥≤1 (cf. [1]).With the aid of positive elements inA, Ma et al. [2] launched the concept of C∗-AVMS in the following manner.Definition 3 (see [2]).
Let℧ be a nonempty set. Suppose that the mapping σ:℧2⟶A satisfies the following conditions:
(c1)0A°σj,ℓ and σj,ℓ=0A if and only if j=ℓ
(c2)σj,ℓ=σℓ,j, for all j,ℓ∈℧
(c3)σj,ℓ°σj,z+σz,ℓ, for all j,ℓ,z∈℧
Then,σ is called a C∗-algebra-valued metric, and ℧,A,σ is known as a C∗-AVMS.It is clear that aC∗-AVMS generalizes the idea of a metric space, by replacing the set of real numbers with A+. For some recent fixed point results in C∗-AVMS, one can consult [3, 4] and the references therein.Definition 4 (see [2]).
Given aC∗-AVMS ℧,A,σ. Suppose that the sequence jnn∈ℕ⊂℧ and u∈℧. If for any ℘>0, there exists ζ∈ℕ such that for all n>ζ, ∥σjn,u∥≤℘, then jnn∈ℕ is said to be convergent to u with respect to A. In this case, we write limn⟶∞jn=u.
If for any℘>0, there is ζ∈ℕ such that n,m>ζ, ∥σjn,jm∥≤℘, then the sequence jnn∈ℕ is said to be Cauchy with respect to A. We say that ℧,A,σ is a complete C∗-AVMS, if every Cauchy sequence in ℧ is convergent with respect to A.Definition 5 (see [2]).
Given aC∗-AVMS ℧,A,σ, a mapping Y:℧⟶℧ is called a C∗-algebra-valued contractive mapping on ℧, if there exists λ∈A with ∥λ∥<1 such that for all j,ℓ∈℧,
(1)σYj,Yℓ°λ∗σj,ℓλ.The following lemma is useful in discussing our main results.Lemma 6 [1].
LetA be a C∗-algebra. Then:
(i)
For anyτ∈A, if p,q∈Aa with p°q, then r∗pr°r∗qr(ii)
For anyp,q∈Aa, if 0A°p°q, then 0≤∥p∥≤∥q∥In 1970, Takahashi [5] initiated the concept of convexity in metric spaces in the following fashion.Definition 7 (see [5]).
Let℧,σ be a metric space and D=0,1. A mapping Ψ:℧×℧×D⟶℧ is called a convex structure on ℧, if for all j,ℓ,a∈℧ and t∈D,
(2)σa,Ψj,ℓ,t≤tσa,j+1−tσa,ℓ.Using the notion of convexity, Takahashi [5] complemented some FP results originally obtained in Banach spaces. Following [5], several investigators have come up with FP notions in convex metric spaces; for such results, we can refer [6–9] and the references therein. Using the idea of a convex metric space and with the aid of positive elements in a C∗-algebra, Ghanifard et al. [10] brought up the next definition.Definition 8 (see [10]).
Let℧,A,σ be a C∗-AVMS. A mapping Ψ:℧×℧×D⟶℧ is called a convex structure on ℧ if for all j,ℓ,a∈℧, it satisfies the following:
(3)σa,Ψj,ℓ,t°tσa,j+1−tσa,ℓ.
AC∗-AVMS equipped with a convex structure is said to be a convex C∗-AVMS, denoted by ℧,A,Ψ,σ. A subset Θ of ℧ is called convex if for all j,ℓ∈Θ and t∈D, Ψj,ℓ,t∈Θ.As an attempt at reducing uncertainties in dealing with practical problems for which conventional mathematics cannot cope effectively, the evolvement of fuzzy mathematics started with the introduction of the concepts of fuzzy sets by Zadeh [11] in 1965. Fuzzy set theory is now well-known as one of the mathematical tools for handling information with nonstatistical uncertainty. As a result, the theory of fuzzy sets has gained greater applications in diverse domains such as management sciences, engineering, environmental sciences, medical sciences, and in other emerging fields. In the meantime, the basic notions of fuzzy sets have been modified and improved in various settings; for example, see [12–15]. Along the lane, Heilpern [16] employed the concept of fuzzy sets to come up with the notion of fuzzy mappings and established a FP result for fuzzy contraction mappings which is a fuzzy version of FP theorems established by Nadler [17] and Banach [18].Let℧ be a universal set. A fuzzy set in ℧ is a map with domain ℧ and range set D. Let I℧ be the collection of all fuzzy sets in ℧. If ∇ is a fuzzy set in ℧, then the function value ∇j is called the grade of membership of j in ∇. The α-level set of a fuzzy set ∇ is denoted by ∇α and is defined as follows:
(4)∇α=j∈℧:∇j>0¯,ifα=0,j∈℧:∇j≥α,ifα∈D∖0,where by P¯, we mean the closure of the crisp set P.Definition 9 (see [16]).
Let℧ be an arbitrary set and Y be a metric space. A mapping Y:℧⟶I℧ is called a fuzzy mapping. A fuzzy mapping Y is a fuzzy subset of ℧×Y with membership function Yjℓ. The function value Yjℓ is called the grade of membership of ℓ in Yj.Definition 10 (see [16]).
Let℧ be a nonempty set and Y:℧⟶I℧ be a fuzzy mapping. A point u∈℧ is said to be a fuzzy FP of Y if there exists an α∈D∖0 such that u∈Yuαu.Hereafter,FixY=u∈℧:u∈Yuαforsomeα∈D.Motivated by the ideas of fuzzy mappings andC∗-AVMSs due to Heilpern [16] and Ma et al. [2], respectively, the aim of this research is to initiate the study of fuzzy FP results in convex C∗-AVMSs. To this effect, some new concepts of C∗-algebra-valued fuzzy contractions in convex C∗-AVMSs are proposed, and related fuzzy FP theorems are established. The notions put forward herein are not only novel, but complement and unify a few corresponding results in the existing literature.


## 2. Main Results

In this section, we introduce notions ofC∗-algebra-valued fuzzy contractions and some corresponding fixed point results. First, a few requisite auxiliary concepts are initiated as follows.Definition 11.
A fuzzy setΩ in a C∗-AVMS ℧,A,σ is said to be convex if for all j,ℓ∈℧ and t∈D∖0,1, minΩj,Ωℓ°Ωtj+1−tℓ. A fuzzy set Ω in ℧ is called an approximate quantity if its α-level set is a compact convex subset of ℧ for each α∈D and supj∈℧AΩj=1.Throughout, the collection of all approximate quantities in℧,A,σ is denoted by WA℧. We define a distance function between two approximate quantities in WA℧ as follows.Definition 12.
LetF,G∈WA℧ and α∈D. Then, we define:
(5)Dα∗F,G=ℵFα,Gα,σ∞F,G=supαDα∗F,G,where the Hausdorff distance function ℵ:WA×WA⟶A is set-up as follows:
(6)ℵFα,Gα=maxsupj∈Fασj,Gα,supℓ∈FασFα,ℓIA,σa,ϖ=inf∥σa,ρ∥:ρ∈ϖIA.Consistent with Heilpern [16], we call the function Dα∗ an α,∗-distance and σ∞ a distance between two approximate quantities in WA℧.We say that a subsetΨ of a C∗-AVMS ℧,A,σ is bounded if supj,ℓ∈A∥σj,ℓ∥<∞. The collection of all closed and bounded subsets of ℧,A,σ is represented by BA℧.Note thatσ∞ is a C∗-algebra-valued metric on BA℧ (induced by the Hausdorff metric ℵ), and the completeness of ℧,A,σ∞ implies the completeness of the corresponding C∗-AVMS KΩ℧,A,σ∞. Moreover, ℧,A,σ∞↦BA℧,A,σ∞↦KΩ℧,A,σ∞ are isometric embedding via the relation j↦j (crisp set) and ⊑↦χ⊑, respectively, where χ⊑ is the characteristic function of ⊑, and
(7)KΩ℧=Ω∈I℧:Ωα∈BA℧,foreachα∈D.Similarly,(8)KΩ1,Ω2℧=Ω1,Ω2∈I℧:Ω1α,Ω2α∈BA℧,foreachα∈D.We now define the idea of aC∗-AVFM in the following manner.Definition 13.
Let℧ be an arbitrary set and Y,A,σ be a C∗-AVMS. A mapping Y:℧⟶WAY is called a C∗-AVFM.In line with the idea of fuzzyλ-contraction due to Heilpern [16], we introduce the next concept.Definition 14.
Let℧,A,σ be a C∗-AVMS. A C∗-AVFM Y:℧⟶WA℧ is called a C∗-algebra-valued fuzzy λ-contraction, if there exists λ∈A with ∥λ∥<1 such that for all j,ℓ∈℧,
(9)σ∞Yj,Yℓ°λ∗σj,ℓλ.Example 15.
Let℧=ℝ and A=M2ℝ (the collection all 2×2 matrices with real entries) with the norm Φ=maxp′,q′ςp′q′, where ςp′q′ are the entries of the matrix Φ∈M2ℝ and the involution given by Φ∗=ΦT. Define σ:℧×℧⟶A by σj,ℓ=diagj−ℓ,j−ℓ. Obviously, ℧,A,σ is a C∗-AVMS. We define a partial ordering on A as follows:
(10)p1p2p3p4=q1q2q3q4⇔pi≼λ∗qiλ,for i=1,2,3,4 and for some λ∈A. Let β∈0,1 and for each j∈℧, define a fuzzy mapping Yj:℧⟶0,1 as follows:
(11)Yjt=β,ifj7≤t≤j5β6,otherwise.
If we take the mappingα:℧⟶0,1 as αj=β for all j∈℧, then
(12)Yjαj=t∈℧:Yjt≥β=j7,j5.
Obviously,Yjαj∈BA℧, for each j∈℧. We see that for all j,ℓ∈℧,
(13)σ∞Yj,Yℓ=supβDβYj,Yℓ=ℵYjβ,Yℓβ=maxsupa∈Yjβσa,Yℓβ,supb∈YℓσYjβ,bIA=maxsupinfa∈Yjβ,ℓ∈Yℓβa−b,supinfb∈Yℓβ,a∈Yjβa−bIAmaxj7−ℓ7,j5−ℓ5IA=j5−ℓ5IA=diagj5−ℓ5.
It follows thatY is a C∗-algebra-valued fuzzy λ-contraction with λ=diag1/4,1/4. Clearly, λ<1.Definition 16.
Let℧ be a nonempty set. A point u∈℧ is called a stationary point of a fuzzy mapping Y:℧⟶I℧, if there exists an α∈D∖0 such that Yuα=u. We say that u is a common stationary point of any two fuzzy mappings Y,Λ:℧⟶I℧ if Yuα=u=Λuα, for some α∈D∖0.Our main result is provided hereunder.Theorem 17.
Let℧,A,Ψ,σ be a complete convex C∗-AVMS. Suppose that the mapping Y:℧⟶KY℧ is a C∗-algebra-valued fuzzy λ-contraction such that FixY≠∅ and every a∈FixY is a stationary point of Y. Let jn be the Mann iteration scheme given by
(14)jn+1=Ψℓn,jn,ηn,where ℓn∈Yjnαjn and ηn∈D. Then, jn converges to a fuzzy FP of Y, provided limn⟶∞σjn,FixY=0A.Proof.
Leta∈FixY and αj∈D∖0 for each j∈℧. Then, we have
(15)σjn+1,a=σΨℓn,jn,ηn,a≼ηnσℓn,a+1−ηnσjn,a.
From (15),
(16)σjn+1,a≤ηnσℓn,a+1−ηnσjn,a≤ηnsupℓn∈Yjnαjn,a∈Yaαaσℓn,a+1−ηnσjn,a=ηnσYjnαjn,Yaαa+1−ηnσjn,a≤ηnℵYjnαjn,Yaαa+1−ηnσjn,a≤ηnσ∞Yjn,Ya+1−ηnσjn,a≤ηnλ∗σjn,aλ+1−ηnσjn,a=ηnλ2σjn,a+1−ηnσjn,a<ηnσjn,a+1−ηnσjn,a=σjn,a.
We observe that the strict inequality in (16) is valid whenever jn≠a, for each n∈ℕ. Indeed, if we take jκ=a for a finite κ∈ℕ, then jn=a for each n≥κ, from which it yields that jnn∈ℕ converges to a for finite number of iterations, and hence, we obtain the conclusion of our result.
Now, we prove that the sequencejnn∈ℕ is Cauchy with respect to A. Since limn⟶∞σjn,FixY=0A, for each ℘>0, there exists m℘∈ℕ such that for all n≥m℘,
(17)σjn,FixY≤℘7.
By (17), there exists r1∈FixY such that for all n≥m℘,
(18)σjn,r1≤℘2.
Using the triangle inequality in℧,A,σ,
(19)σjn+k,jn≼σjn+k,r1+σr1,jn.
Therefore, taking (18) into consideration, we get
(20)∥σjn+k,jn∥≤∥σjn+k,r1∥+∥σr1,jn∥<∥σjn,r1∥+∥σr1,jn∥≤℘2+℘2=℘,for jn≠r1. This proves that jnn∈ℕ is a Cauchy sequence with respect to A. The completeness of ℧A implies that there exists a∗∈℧A such that limn⟶∞jn=a∗. Next, we establish that a∗ is a fuzzy FP of Y. For this, take ℘′>0. Since jn⟶a∗ as n⟶∞, there exists m℘′∈ℕ such that for all n≥m℘′,
(21)∥σjn,a∗∥≤℘′4.
Moreover,limn⟶∞σjn,FixY=0A yields that there exists m′℘≥m℘′ such that for all m℘≥m′℘,
(22)∥σjn,FixY∥≤℘′10.
Hence, there existsr2∈FixY such that for all m℘≥m′℘,
(23)∥σjn,r2∥≤℘′12.
By triangle inequality in℧A, there exists write
(24)σYa∗αa∗,a∗≼σYa∗αa∗,r2+σr2,Yj3αj3+σYj3αj3,r2+σr2,jn3+σjn3,a∗.
Consequently,(25)∥σYa∗αa∗,a∗∥≤∥σYa∗αa∗,r2∥+∥σr2,Yjn3αjn3∥+∥σYjn3αjn3,r2∥+∥σr2,jn3∥+∥σjn3,a∗∥≤∥ℵYa∗αa∗,Yr2αr2∥+2∥ℵYr2αr2,Yjn3αjn3∥+∥σr2,jn3∥+∥σjn3,a∗∥≤∥σ∞Ya∗,Yr2∥+2∥σ∞Yr2,Yjn3∥+∥σr2,jn3∥+∥σjn3,a∗∥≤∥λ∗∥∥σa∗,r2∥∥λ∥+2∥λ∗∥∥σr2,jn3∥∥λ∥+∥σjn,a∗∥+∥σr2,jn3∥+∥σjn3,a∗∥=∥λ∥2∥σa∗,r2∥+2∥λ∥2∥σr2,jn3∥+∥σjn3,a∗∥<∥σa∗,r2∥+2∥σr2,jn3∥+∥σjn3,a∗∥≤∥σa∗,jn3∥+∥σjn3,r2∥+∥σjn3,r2∥+2∥σjn3,r2∥+2∥σjn3,r2∥+∥σjn3,r2∥+∥σjn3,a∗∥=2∥σjn3,a∗∥+6∥σjn3,r2∥≤2℘′4+6℘′12=℘′,whenever jn3≠r2. It follows that σYa∗αa∗,a∗=0A, and thus, a∗∈Ya∗αa∗, for some αa∗∈D∖0.In what follows, we present a fuzzy coincidence theorem for twoC∗-algebra fuzzy mappings using Ishikawa iterative scheme.Theorem 18.
Let℧,A,σ be a complete convex C∗-AVMS and Λ,Y:℧⟶KY,Λ℧ be any two C∗-AVFMs satisfying:
(26)σ∞Yj,Λℓ≤λ∗σj,ℓλ,for all j,ℓ∈℧ with λ∈A such that ∥λ∥<1. Assume further that CY,Λ≔FixY∩FixΛ≠∅, and every a∈CY,Λ is a common stationary point of Y and Λ. Then, the sequence of Ishikawa iterative scheme set-up by
(27)jn+1=Ψzn,jn,ηn,ℓn=Ψz′,jn,ξn,where zn∈Λjnαjn,zn′∈Yjnαjn, and ηn,ξn∈D, converges to an element of CY,Λ, provided limn⟶∞σjn,CY,Λ=0A.Proof.
Leta∈CY,Λ. Assume that jn≠a for all n∈ℕ. Then
(28)σℓn,a=σΨzn′,jn,ξna≼ξnσzn′,a+1−ξnσjn,a,from which it follows that
(29)∥σℓn,a∥≤ξn∥σzn′,a∥+1−ξn∥σjn,a∥≤ξn∥ℵYjnαjn,Λaαa∥+1−ξn∥σjn,a∥≤ξn∥σ∞Yjn,Λa∥+1−ξn∥σjn,a∥≤∥λ∗σℓn,aλ∥+1−ξn∥σjn,a∥≤ξn∥λ∥2∥σjn,a∥+1−ξn∥σjn,a∥<ξn∥σjn,a∥+1−ξn∥σjn,a∥=∥σjn,a∥.
Similarly,(30)σjn+1,a=σΨzn,jn,ηn,a≼ηnσzn,a+1−ηnσjn,a,which gives
(31)∥σjn+1,a∥≤ηn∥σzn,a∥+1−ηn∥σjn,a∥≤ηn∥ℵΛjnαjn,Yaαa∥+1−ηn∥σjn,a∥≤ηn∥λ∥2∥σjn,a∥+1−ηn∥σjn,a∥<ηn∥σjn,a∥+1−ηn∥σjn,a∥=∥σjn,a∥.
Hence, in line with the proof of Theorem17, we can prove that jnn∈ℕ is a Cauchy sequence with respect to A, and the completeness of ℧,A,Ψ,σ implies that jnn∈ℕ converges to some u∗∈℧. Thus, consistent with Theorem 17, we obtain that u∗∈FixY∩FixΛ.As some consequences of Theorem17 and Theorem 18, the following two results, using C∗-algebra-valued Hausdorff distance function, can be deduced easily.Corollary 19.
Let℧,A,Ψ,σ be a complete convex C∗-AVMS and Y:℧⟶KY℧ be a C∗-AVFM. Suppose that FixY≠∅ and every a∈FixY is a stationary point of Y. Let jnn∈ℕ be the Mann iterative scheme set-up by (14). If there exist α∈D∖0 and λ∈A with ∥λ∥<1 such that for all j,ℓ∈℧,
(32)Dα∗Yj,Yℓ≤λ∗σj,ℓλ,then jnn∈ℕ converges to a fuzzy FP of Y, provided limn⟶∞σjn,FixY=0A.Proof.
Since for allj,ℓ∈℧ and α∈D∖0,
(33)∥Dα∗Yj,Yℓ∥≤∥σ∞Yj,Yℓ∥.
Theorem17 can be followed to complete the proof.On the same steps in deriving Corollary19, we can also deduce the following result.Corollary 20.
Let℧,A,Ψ,σ be a complete convex C∗-AVMS and Y,Λ:℧⟶KY,Λ℧ be any two C∗-AVFMs. Assume further that CY,Λ=FixY∩FixΛ≠∅ and every a∈FixY is a common stationary point of Y and Λ. Let jnn∈ℕ be the Ishikawa iterative scheme set-up by (23). If there exist α∈D∖0 and λ∈A with ∥λ∥<1 such that for all j,ℓ∈℧,
(34)Dα∗Yj,Λℓ≤λ∗σj,ℓλ,then jnn∈ℕ converges to a common fuzzy FP of Y and Λ, provided
(35)limn⟶∞σjn,CY,Λ=0A.


## 3. Open Problems

For some future examinations of our main results, the following two problems are highlighted:(P1) It is well-known that the importance of contractive mapping is to guarantee the existence and uniqueness of a fixed point of certain self-mappings in complete spaces. On this note, following Theorem17 and Theorem 18, sufficient criteria guaranteeing the existence of fixed points of C∗-algebra-valued fuzzy λ-contractions is still a gap that needed to be filled.(P2) In this article, Mann and Ishikawa iterations are used to develop the ideas ofC∗-algebra-valued fuzzy λ-contractions and associated fixed point theorems. Hence, it is natural to ask whether Picard iteration for C∗-algebra-valued fuzzy λ-contraction mapping converges or not.


## 4. Conclusions

Based on the ideas of fuzzy mappings andC∗-AVMSs in the sense of Heilpern [16] and Ma et al. [2], respectively, analogue notions of C∗-algebra-valued fuzzy contractions in convex C∗-AVMSs and associated FP theorems are established. The obtained fuzzy FP results are analysed using Mann and Ishikawa iterative schemes. It is pertinent to note that the ideas of this paper being discussed in fuzzy setting are very fundamental. Hence, it can be improved upon when presented in the framework of some generalized fuzzy mappings such as L-fuzzy, intuitionistic fuzzy, and soft set-valued mappings. The underlying space can also be fine-tuned in some other pseudo or quasi metric spaces. For some future considerations of our results, two open problems are posed regarding sufficient conditions under which C∗-algebra-valued fuzzy λ-contraction has a fixed point and whether or not the Picard iteration for C∗-algebra-valued fuzzy λ-contraction converges.



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