# New Results of Fixed-Point Theorems in Complete Metric Spaces

Authors: Mustafa T. Yaseen; Ali Hasan Ali; Areej A. Al-moneef; Omer Bazighifan; Taher A. Nofal; F. Ghanim
Journal: Mathematical Problems in Engineering
 (2022)

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## Abstract

In this paper, fixed point theorems of multiple values of self-mappings have been investigated in Banach space. In addition, a new modification of the same mentioned study has been considered in a metric space with at most three valued self-mappings. Moreover, we have extended and improved some existent results in a complete metric space. Finally, we present a slight improvement on some well-known lemmas that are related to nonexpansive mappings and nonincreasing distances mappings.


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## Body

## 1. Introduction

In mathematics and related fields, the fixed point theory plays a significant role in several fields, see [1–4]. In this recent work, we continue studying some existent results by many researchers on fixed point theorems in Banach and metric spaces. In 2014, a general fixed point theorem for multivalued mappings that is not always a contraction was proposed by Abdul Latif[2]. In addition, he derived some modern studies on contraction mappings. In 2020, Singh and Anthony [5] studied and extended the concept of more than one fixed point into mappings on a cone Sb-metric space. Moreover, they provided some robust examples to validate their results.The structure of our proposed paper is organized as follows:In Section 2, we present our first generalization of results by Ray [6], where we provide some well-known results consisting of multivalued mappings in a complete metric space X,d. Moreover, we present the same study for the results provided by Taskovic [7].In section 3, we provide some results about the fixed point theorem of three mappingsf1,f2, and f3 in a complete metric space X,d. Furthermore, a brief discussion of few common fixed point theorems is going to be presented. Also, we will give a study of the three mappings f1,f2, and f3 that satisfy the following equation:(1)df1x,f2y2≤ϕdf3x,f1xdf3y,f2y,df3x,f2ydf3y,f1x,df3x,f1xdf3x,f2y,df3y,f1xdf3y,f2y.wherex,y∈X; f1f3=f3f1, f2f3=f3f2, f1x⊂f3x; f2x⊂f3x; and ϕ are continuous function that satisfy the following condition:(2)ϕs,s,a1s,a2s<sfor anys>0andai∈0,1,2such thata1+a2=2.More studies have been done and considered by many authors regarding theorems that satisfy different functional inequalities and in terms of fixed point theorems in different spaces as well as several applications such as solving nonlinear and differential equations, see [8], [3, 9, 10], and [11–15].In Section 4, we mainly extend the results that have been obtained by Kirk in [16] and in [1]. In fact, we work on modifying the given lemmas that are related to multivalued mappings in order to extend the results for the sake of generalization.


## 2. Fixed Point in Metric Space

Jleli and Bessem in [17] introduced a novel definition of generalized metric spaces, where they worked on extending several existing fixed-point findings along with the Banach contraction fundamental theorem [7]. In addition, Taskovic presented and proved a fixed-point theorem on a metric space X,d along with a mapping T, which is not necessarily continuous and satisfy a condition of the type,(3)adTx,Ty+bdx,Tx+cdy,Ty−mindx,Ty,y,Tx≤qdx,y,for all x,y∈X.In the following theorem, we present the first result for two mappingsf1 and f2 that generalizes the abovementioned result.Theorem 1.
Suppose thatf! and f2 are a couple self-functionals of a complete metric space X,d with the following condition:(4)adf!x,f2y+bdx,f!x+cdy,f2y−mindx,f2y,dy,f1x≤qdx,y,for all x,y∈X such that the numbers a,b,c≥0 and q>0 with a>q+1 and a+c>0. Then f1 and f2 will contain a unique common fixed point.Now, we introduce the below generalized theorem of Ray [6] for a couple of self-mapping on a metric space.Theorem 2.
Suppose thatf1 is a continuous functional in a complete metric space X,d and suppose that f2⊂f1 is another mapping in the same space where the two mappings commute with each other, and f2 satisfies the following condition:(5)df2x,f2y≤αdf1x,f1ydf1xf1y+βdf1x,f1ydf1xf2xdf1y,f2yγdf1x,f1ydf1xgy+df1y,f2x,for each f1x≠f1y, where α,β, and γ are monotonically decreasing functions from 0,∞ into 0,1 with αt+2βs+2γs<1 and s∈0,∞. Then, the two mappings f1 and f2 have only one common fixed point in X.The proof of the above theorem is not going to be presented as it goes in a similar way as that in [18]. Therefore, the proof is omitted.Remark 1.
Iff1 happens to be an identity mapping, we get Theorem 2 of Ray [6] without the diagram of f2 being closed. Further, if β=γ=0, we have the result of Rokotch [19].
Another generalization of Theorem2 of Ray goes as follows:Theorem 3.
Suppose thatf1 and f2 are two self-functionals in a complete metric space X,d, where the two mappings satisfy the following condition:(6)df1x,f2y≤αdx,ydx,y+βdx,ydx,f1x+dy,f2y+γdx,ydx,f2y+dy,f1x,for every x≠y∈X, such that α,β, and γ are monotonically decreasing functions from 0,∞ into 0,1 with αs+2βs+2γs<1, s∈0,∞ Then the two mappings f1 and f2 have only one common fixed point in X.


## 3. Consequences

Let us assume thatϕ represent an upper semicontinuous function from the space R+4 to the space R+, where this function is monotonically increasing in every variable and for each number s>0 the following condition is satisfied:(7)ϕs,s,a1s⋅a2s<s,where ai∈0,1,2 such that a1+a2=2.In the following, we present the main theorem of this work.Theorem 4.
Suppose thatf1,f2, and f3 are three self-functionals of a complete metric space X,d, which satisfy the following conditions:(8)f1f3=f3f1,f2f3=f3f2,f1x⊂f3xandf2x⊂f3xdf1x,f2y2≤ϕdf3x,f1ydf3y,f2y,df3x,f2ydf3y,f1x,df3x,f1xdf3x,f2y,df3y,f1xdf3y,f2y,for all x,y∈X. Further, let f3 be continuous, then f1,f2, and f3 have only one common fixed point in X. The proof of this theorem goes in a similar way to the one by Devi Prasad [9]. Therefore, we omit the proof.The following result is a corollary to the previous theorem by takingf3=I, where I represents the identity map.Corollary 1.
Letf1 and f2 be a couple of mappings from a complete metric space X,d into itself, and they satisfy the following condition:(9)df1x,f2y2≤ϕdx,f1xdy,f2y,dx,f2ydx,f2y,dy,f1xdx,f2y,dy,f1xdy,f2y,for all x,y∈X, where ϕ is defined in (7), then f1 and f2 have only one common fixed point.Now, we present the following three theorems, where the proof of each one follows from Theorem4.Theorem 5.
Suppose thatf1,f2, and f3 are three self-functionals of a complete metric space X,d sin such a way that,
f1f3=f3f1,f2f3=f3f2,f1x⊂f3rx,andf2x⊂f3rx, then there is a non-negative and nonzero integers p,q, and r satisfying the following condition:(10)df1px,f2qy2=ϕdf3rx,f1pxdf3ry,f2qy,df3px,f2qydf3py,f1px,df3rx,f1pxdf3rx,f2qy,df3ry,f1pxdf3ry,f2qy,for all x,y∈X, and if f3r is continuous, then f1,f2, and f3 have only one common fixed point.Theorem 6.
Suppose thatf1n,f2n, and f3n are sequences of self-functionals of a complete metric space X,d such that f1n, f2n, and f3n converge uniformly to self-mappings f1,f2, and f3 on X with f3 continuous. Suppose that for each n≥1, xn is a common fixed point of f1n and f3n, and yn is a common fixed point of f2n and f3n. Furthermore, let f1,f2, and f3 satisfy conditions (8) and (7). If x0 is the common fixed point of f1,f2, and f3, and supdxnx0<∞ and supdyn,x0<∞. Then, xn⟶x0 and yn⟶y0.Theorem 7.
Letf1n,f2n, and f3n be sequences of self-mappings of a complete metric space X,d such that f3n is continuous for each n and f1n,f2n, and f3n satisfy conditions (8) and (7) for each n≥1. If f1,f2, and f3 are uniform limits of f1n,f2n, and f3n respectively, then, f1,f2, and f3 also satisfy conditions (8) and (7). Also, let xn be a sequence of the only one common fixed point of f1n,f2n, and f3n converges to the only one common fixed point x0 of f1,f2, and f3, whenever supdxn,x0<∞.


## 4. Main Results

Suppose thatΩ,d is a metric space and let Q be a class of countably compact subsets of Ω, and suppose that this class is stable under intersections, normal, and contains all the closed balls in Ω. Also, suppose that Λi, i∈N0 are subsets of Ω. Moreover, we define ζ as ζΛ=rΛ/δΛifδΛ>01ifδΛ=0 such that δΛ=supdλ1,λ2:λ1,λ2∈ΛruΛ=supdλ1,λ2:λ2∈Λλ1∈ΛrΛ=infrλ1Λ:λ1∈ΛWe shall take this assumption in order to initiate our proposed theorems.Theorem 8.
IfT:Ω⟶Fx be a mapping such that,(11)Tx∩​Ω≠ϕ∀x∈Ω,and for all x∈Ω.(12)Tx∩​Q=Tx∩​Λ:Λ∈Q,is a compact class of which each nonempty member is a compact subset of X. For any G∈Q satisfying Tξ∩​G≠ϕ, for all ξ∈G,(13)HTx∩​G,Ty∩​G≤dx,y,∀x,y∈G.Then,T has a fixed point in Ω.Proof.
A slight modification on the main lemma by kirk [1], we show that for each ε>0, there exists a nonempty set Ωε∈Q such that,(14)Tx∩​Ωε≠ϕ,for all x∈Ωε and for which δΩε≤ζΩ+εδΩ. For this, we take Ωε=Ω if δΩ=0. Otherwise, construct Ωε as follows:
Now, letr=ζΩ+εδΩ. By the definition of ζ, the set C=z∈Ω:Ω⊂Bα,r is nonempty, where Bz,p represents the closed ball with the center α and the radius r. Let(15)F=Λ∈Q;C⊂Λ,Tx∩​Λ≠ϕ,∀x∈Λ,and let τΛii∈N0 be a decreasing chain in F. Let Λ0=∩​ΛI, then Λ0∈Q and C⊂Λ0. Furthermore, since τΛii∈N0 is decreasing, it follows that for each x∈Λ0, the family Tx∩​Λii∈N0 has a finite intersection property. So, by hypothesis (13),(16)Tx∩​∩​i∈N0Λi≠ϕ.
That isTx∩​Λ0≠ϕ, thus every decreasing chain in F has a lower bound. Therefore, by Zorn’s lemma, F has a minimal element L. Let A=C∪​TL, where(17)TL=∪​x∈ΛTx∩​L.
Then,TL⊂L. So, covA=∩​Λ∈Q;A⊂Λ⊂L. Also, for x∈covA,Tx∩​L≠ϕ and Tx∩​L⊂TL. So, Tx∩​A≠ϕ. Hence, Tx∩​covA≠ϕ. Thus, covA∈F.
SinceL is a minimal member of F and covA⊂L, it follows that covA=L. Let(18)Ωε=x∈L;L⊂Bx,r.
Then,Ωε≠ϕ. Since Ωε⊂C, and let x∈Ωε. Then, Tx∩​L≠ϕ. Take x′∈Tx∩​L. Let 0=L∩​Bx′,r. Next take η∈0, then η∈L and by (13),(19)HTx∩​L,Tη∩​L≤dx,η≤r∵x∈Ωε,η∈L.
SinceTx∩​L and Tη∩​L are nonempty compact sets and x′∈Tx∩​L, there exists η′∈Tη∩​L, such that(20)dx′,η′≤HTx∩​L,Tη∩​L≤r.
So,η′∈0, hence, Tη∩​0≠ϕ. Thus, 0∈F and 0⊂L. Since L is a minimal element in F, it follows that 0=L. So dx′,y′≤r∀y∈L, i.e., Bx′,r⊂L⋅ So, x′∈Ωε, which implies Tx∩​Ωε≠ϕ. Now, let m=Λ∈Q;Λ≠ϕ,Tx∩​Λ≠ϕ,∀x∈Λ and for each Λ∈m, let(21)δ0Λ=infδF;F∈m,F⊂Λ.
The rest of the proof is omitted as it goes in a similar way as the one by Kirk in [1].Remark 2.
The necessity of condition (14) of the theorem has been studied by Samanta in [20, 21].
In the following, we mention two theorems by Kirk in [1] and Samanta in [20] as we consider them corollaries of Theorem 4.Corollary 2 (see [1]).
Suppose thatΩ,d is a metric space, which is bounded and nonempty and let Q be a countably compact class of subsets in Ω, which is normal and stable under random intersections. In addition, suppose that we have closed balls of Ω that are contained in Q. Then, there exists a fixed point in each nonexpansive mapping T of Ω into itself.Corollary 3 (see [20]).
LetS be bounded and a closed convex subset in a reflexive Banach space X, and if Ψ:S⟶2x is a mapping such that,(22)Ψx∩​S≠ϕ,∀x∈S,for any closed convex subset G of S satisfying lξ∩​G≠ϕ,∀ξ∈G(23)Hψx∩​G,ψy∩​G≤x−ywheneverx,y≠x∈G.Then,l has a fixed point.


## 5. Conclusions

We have presented in this paper generalizations of various fixed-point theorems in a metric space and in a Banach space. A generalization with multiple values of self-mappings in those mentioned spaces have been intensively studied. We saw that mapping scheme of the fixed-point theorem is vast no doubt. Therefore, the existent result by Kirk was improved by extending the proof and working on the fixed-point theorem of multiself-mappings.



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