# Some Fixed Point Results in Function Weighted Metric Spaces

Authors: Awais Asif; Nawab Hussain; Hamed Al-Sulami; Muahammad Arshad
Journal: Journal of Mathematics
 (2021)

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## Abstract

After the establishment of the Banach contraction principle, the notion of metric space has been expanded to more concise and applicable versions. One of them is the conception ofℱ-metric, presented by Jleli and Samet. Following the work of Jleli and Samet, in this article, we establish common fixed points results of Reich-type contraction in the setting of ℱ-metric spaces. Also, it is proved that a unique common fixed point can be obtained if the contractive condition is restricted only to a subset closed ball of the whole ℱ-metric space. Furthermore, some important corollaries are extracted from the main results that describe fixed point results for a single mapping. The corollaries also discuss the iteration of fixed point for Kannan-type contraction in the closed ball as well as in the whole ℱ-metric space. To show the usability of our results, we present two examples in the paper. At last, we render application of our results.


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## Body

## 1. Introduction and Preliminaries

In recent years, along withℱ-metric presented by Jleli et al. [1], many authors presented interesting generalizations of metric spaces [2–9]. Jleli and Samet introduced generalized metric spaces, known as ℱ-metric spaces, and proved their generality to metric spaces with the help of concrete examples. The idea of ℱ-metric spaces was compared with b-metric and s-relaxed metric spaces, and hence, the Banach contraction principle was established in the frame of ℱ-metric spaces.Banach contraction principle states that any contraction on a complete metric space has a unique ﬁxed point. This principle guarantees the existence and uniqueness of the solution of considerable problems arising in mathematics. Because of its importance for mathematical theory, the Banach contraction principle has been extended and generalized in many directions [10, 11]. The ﬁxed point theory of multivalued contraction mappings using the Hausdorﬀ metric was initiated by Nadler [12], who extended the Banach contraction principle to multivalued mappings. Since then, many authors have studied various ﬁxed point results for multivalued mappings. Nazam et al. [13] proved fixed point theorems for Kannan-type contractions on closed balls in complete partial metric spaces. The abovementioned results and its generalizations are recently investigated for fixed point in the setting of F-metric space (see [14–16]).In this article, we prove fixed point and common fixed points results of Reich-type contractions for single-valued mappings inℱ-metric spaces.This article is organized into three sections. Section2 contains a short history of the previous literature that becomes a motivation for this article. There are some basic definitions which help readers to understand our results easily. In Section 3, we established theorems of fixed points and common fixed points of single-valued Reich contractions in ℱ-metric spaces. An example is provided to explain our results. Section 4 deals with fixed point theorems of contractions with respect to closed balls in ℱ-metric spaces along with an example.


## 2. Basic Relevant Notions

Definition 1.
(see [1]). A self-mapping g on a nonempty set A is said to be Kannan contraction if there exists a number k, 0<k<1/2, such that, for each a,b∈A, we have(1)dga,gb≤kda,ga+db,gb.
Letf:0,∞⟶ℝ with following characteristics:(F1)
f is strictly increasing(F2)
For any sequencetn⊂0,∞, we have(2)limn⟶∞tn=0⇔limn⟶∞ftn=−∞.
The collection of all such functions satisfying (F1) and (F2) is denoted byℱ. The concept of ℱ-metric is generalized as follows:Definition 2.
(see [1]). SupposeAis a nonempty set and f,α∈ℱ×0,∞. Let the function d:A×A⟶0,∞ be such that(d1)
For alla,b∈A×A,da,b=0⇔a=b(d2)
For alla,b∈A×A,da,b=db,a(d3) {tn}i=1n⊂X
For everya,b∈A×A,for each N′∈ℕ,N′≥2 and for every with t1,tN′=a,b, we have(3)da,b>0⇒fda,b≤f∑i=1N′−1dti,ti+1+α.Then,d is known as anℱ-metric on A, and the pair A,d is called anℱ-metric space.Example 1.
(see [1]). Let A=ℕ (set of natural numbers) and d:A×A⟶0,∞be defined by(4)da,b=a−b2,ifa,b∈0,3×0,3,a−b,ifa,b∉0,3×0,3,for all a,b∈A×A. It can easily be seen that d is an ℱ-metric with fx=lnx.Example 2.
(see [1]). Let A=ℕ and d:A×A⟶0,∞ is defined as(5)da,b=0,ifa=b,ea−b,ifa≠b,for all a,b∈A×A. Then, d is ℱ-metric on A.Definition 3.
(see [1]). Supposeanis a sequence in A. Then,(i)
an is ℱ-convergent to a point a∈A if limn⟶∞dan,a=0(ii)
an is anℱ-Cauchy sequence if limn,m⟶∞dan,am=0(iii)
The spaceA,d is ℱ-complete if everyℱ-Cauchy sequencean⊂A is ℱ-convergent to a point a∈ADefinition 4.
(see [1]). Let A,dbe an ℱ-metric space. A subset Oof A is said to be ℱ-open if, for every a∈O, there is some r>0 such that Ba,r⊂O, where(6)Ba,r=b∈A:da,b<r.
We say that a subsetC of A is ℱ-closed if A\C is ℱ-open.Definition 5.
(see [1]). Let A,dbe an ℱ-metric space andB be a nonempty subset of A. Then, the following statements are equivalent:(i)
Bisℱ-closed.(ii)
For any sequencean⊂B, we have(7)limn⟶∞dan,a=0,a∈A⟹a∈B.Theorem 1.
(see [1]). Supposef,α∈ℱ×0,∞ and A,d is an ℱ-complete ℱ-metric space. Letg:A⟶A be a given mapping. Suppose that there exists k∈0,1 such that(8)dga,gb≤kda,b,a,b∈A×A.Then,g has a unique fixed point a∗∈A. Moreover, for any a0∈A, the sequencean⊂A defined by an+1  =  gan,n∈ℕ is F-convergent to a∗.Theorem 2.
(see [17]). Suppose A is a complete metric space with metric d, and let g:A⟶A be a function such that(9)dga,gb≤αda,b+βda,ga+γdb,gb,for all a,b∈A, where α,β,andγ are nonnegative integers and satisfy α+β+γ<1. Then, g has a unique fixed point.Lemma 1.
(see [18]). The Banach space BW,⋅along with the metric d defined by(10)dg,h=g−h=maxa∈Wga−ha,g,h∈BW.is an ℱ-metric space.


## 3. Fixed Points of Reich-Type Contractions inF−Metric Spaces

In this section, we construct fixed point and common fixed points results for single-valued Reich-type and Kannan-type contractions in the setting ofℱ-metric space.Theorem 3.
Supposef,α∈ℱ×0,∞ and X,d is an ℱ-complete ℱ-metric space. Let S,T:X⟶Xbe self-mappings such that(11)dSx,Ty≤adx,y+bdx,Sx+cdy,Ty,fora,b,c∈0,∞ such a+b+c<1, for all x,y∈X×X. Then, S and Thave at most one common fixed point in X.Proof.
Supposex0 is an arbitrary point and define a sequence xn by(12)Sx2j=x2j+1,Tx2j+1=x2j+2,j=0,1,2,…,
Using (11) and (12), we can write(13)dx2j+1,x2j+2=dSx2j,Tx2j+1≤adx2j,x2j+1+bdx2j,Sx2j+cdx2j+1,Tx2j+1=adx2j,x2j+1+bdx2j,x2j+1+cdx2j+1,x2j+2.
This implies(14)1−cdx2j+1,x2j+2≤a+bdx2j,x2j+1dx2j+1,x2j+2<a+b1−cdx2j,x2j+1=λdx2j,x2j+1,where a+b/1−c=λ
Similarly,(15)dx2j+2,x2j+3<a+b1−cdx2j+1,x2j+2=λdx2j+1,x2j+2.
Continuing this way, we get(16)dxn,xn+1<λdxn−1,xn,for allnɛℕ,which yields(17)dxn,xn+1<λdxn−1,xn<λ2dxn−2,xn−1<⋯<λndx0,x1.
Hence,(18)dxn,xn+1<λndx0,x1,n∈ℕ.
Using (18), we can write(19)∑k=nm−1dxk,xk+1=dxn,xn+1+dxn+1,xn+2+⋯+dxm−1,xm<λndx0,x1+λn+1+⋯+λm−1dx0,x1<λn1+λ+λ2+⋯+λm−n−1dx0,x1≤λn1−λdx0,x1,m>n.
Sincelimn⟶∞λn/1−λdx0,x1=0, for any δ>0, there exists some n′∈ℕ such that(20)0<λn1−λdx0,x1<δ,n≥n′.
Furthermore, supposef,α∈ℱ×0,∞ satisfies (d3) andɛ>0is fixed. By F2, there is someδ>0 such that(21)0<t<δ⟹ft<fɛ−α.
By (21), we write(22)fλn1−λdx0,x1<fɛ−α,m>n≥n′.
Using (20), we write(23)f∑k=nm−1dxk,xk+1≤fλn1−λdx0,x1<fɛ−α,m>n≥n′.
By (d3) and above equation, we obtain(24)dxn,xm>0,m>n>n′⟹fdxn,xm<fɛ.
This shows that(25)dxn,xm<ɛ,m>n≥n′.
Hence, we showed thatxn is anℱ-Cauchy sequence in X. Since X,d is ℱ-complete, there exists z∗∈X such that xn is ℱ-convergent to z∗, i.e.,(26)limn⟶∞dxn,z∗=0.
To prove thatz∗ is the fixed point of S, assume dSz∗,z∗>0. Then,(27)dSz∗,x2j+2=dSz∗,Tx2j+1≤adz∗,x2j+1+bdz∗,Sz∗+cdx2j+1,Tx2j+1=adz∗,x2j+1+bdz∗,Sz∗+cdx2j+1,x2j+2,which implies 1−bdSz∗,z∗<0, which is a contradiction. Hence, dSz∗,z∗=0, i.e., Sz∗=z∗. Similarly, suppose dz∗,Tz∗>0:(28)dTz∗,x2j+1=dTz∗,Sx2j≤adz∗,x2j+bdx2j,Sx2j+cdz∗,Tz∗=adz∗,x2j+bdz∗,Sz∗+cdx2j,x2j+1,i.e.,(29)1−cdz∗,Tz∗<0,which is contradiction to the assumption. Therefore, we get Tz∗=z∗. Hence, Tz∗=Sz∗=z∗.
Uniqueness. Assume thatz∗∗is also a common fixed point of S and T andz∗≠z∗∗.Then,(30)dz∗,z∗∗=dSz∗,Tz∗∗≤adz∗,z∗∗+bdz∗,Sz∗+cdz∗∗,Tz∗∗=adz∗,z∗∗+bdz∗,z∗+cdz∗∗,z∗∗.
We get (1-a)dz∗,z∗∗<0, which is a contradiction. Hence, z∗=z∗∗Example 3.
Suppose(31)Y=Yj≔6j+12,j∈ℕ,dx,y=0,ifx=yex−y,ifx≠y.
Letfx=lnx and S,T:X⟶X are defined by(32)TYj=Y1ifj=1,2,Yj−1ifj>2,SYj=Y1ifj=1,Y2ifj=2,Yj−2ifj>2.
It can be easily verified thatd is an ℱ-metric and f satisfies F1−F2. Fix b=c=0 and x,y∈X×X. Suppose i≠j, then(33)dSYj,TYi=dYj−2,Yi−1=eYj−2−Yi−1=e6j−12+1/2−6i−6+1/2=e3j−i−3<e−2⋅e3j−i=adYj,Yi=adYj,Yi+bdYj,SYj+cdYi,TYi,where a=e−2. The inequality (11) holds true. Moreover, it is clear that Y1 is the only common fixed point of S and T.
Takinga=0 in Theorem 1, we get the following result of Kannan contractions.
ReplacingS in Theorem 3, we get the following corollary.Corollary 1.
Supposef,α∈ℱ×0,∞ and X,d is an ℱ-complete ℱ-metric space. Let T:X⟶Xis a self-mapping such that(34)dTx,Ty≤adx,y+bdx,Tx+cdy,Ty,for a,b,c∈0,∞ such a+b+c<1,for all x,y∈X×X. Then, Thas at most one fixed point in X.
Takingb=c=0 in Corollary 1, we get the following result.Corollary 2.
Supposef,α∈ℱ×0,∞ and X,d is an ℱ-complete ℱ-metric space. Let T:X⟶Xis a self-mapping such that(35)dTx,Ty≤adx,y,for a∈0,∞and x,y∈X×X. Then, Thas at most one fixed point in X.
Besides the above important results, Theorem3 also led us to the following fixed point result of Kannan-type contraction.Corollary 3.
Supposef,α∈ℱ×0,∞ and X,d is an ℱ-complete ℱ-metric space. Let S,T:X⟶Xbe self-mappings. Suppose that, for k∈0,1such that(36)dSx,Ty≤k2dx,Sx+dy,Ty,for all x,y∈X×X, then S and Thave at most one common fixed point in X.Proof.
Supposex0 is an arbitrary point and define a sequence xn by Sx2j=x2j+1 and Tx2j+1=x2j+2 ; j=0,1,2,…,
Using the contraction and the iteration given above, we can write(37)dx2j+1,x2j+2=dSx2j,Tx2j+1=k2dx2j,Sx2j+dx2j+1,Tx2j+1=k2dx2j,x2j+1+dx2j+1,x2j+2.
This implies(38)1−k2dx2j+1,x2j+2≤k2dx2j,Sx2j.or(39)dx2j+1,x2j+2≤k2−kdx2j,Sx2jdx2j+1,x2j+2≤λdx2j,x2j+1,where k/2−k=λ. Similarly,(40)dx2j+2,x2j+3<k2−kdx2j+1,x2j+2=λdx2j+1,x2j+2.
Continuing the same way as in Theorem3, we get the common fixed point of S and T.
ReplacingS with T, we get the following result of single mapping.Corollary 4.
Supposef,α∈ℱ×0,∞ and X,d is an ℱ-complete ℱ-metric space. Let T:X⟶Xbe a self-mapping. Suppose that, fork∈0,1such that(41)dTx,Ty≤k2dx,Tx+dy,Ty,for all x,y∈X×X, then Thas at most one fixed point in X.


## 4. Fixed Points of Reich-Type Contractions onF-Closed Balls

This portion of the paper deals with the fixed points theorems of Reich-type contractions that hold true only on the closed balls rather than on the whole spaceX.Definition 6.
LetX,d be an ℱ-complete ℱ-metric space and S,T:X⟶Xbe self-mappings. Suppose that a+b+c<1 fora,b,c∈0,∞. Then, the mappings S and T are called Reich-type contractions on Bx0,r⊆X such that(42)dSx,Ty≤adx,y+bdx,Sx+cdy,Ty,∀x,y∈Bx0,r.Theorem 4.
Supposef,α∈ℱ×0,∞ and X,d is an ℱ-complete ℱ-metric space. Let S and Tbe Reich-type F-contractions on Bx0,r. Suppose that forx0∈X and r>0, the following conditions are satisfied:(a)
Bx0,r is ℱ-closed(b)
dx0,x1≤1−λr, for x1∈X and λ=a+b/1−c(c)
There exist0<ϵ<r such as f1−λk+1r≤fɛ−α, where k∈ℕ
Then,S and Thave at most one common fixed point in Bx0,r.Proof:.
Supposex0 is an arbitrary point and define a sequence xn by Tx2j=x2j+1 and Sx2j+1=x2j+2 ; j=0,1,2,…,.
We need to show thatxn is in Bx0,r for all n∈ℕ. We show it by mathematical induction. By (b), we write(43)dx0,x1<r.
Therefore,x1∈Bx0,r. We know by previous theorems that(44)dx1,x2≤λdx0,x1.
Now,(45)fdx0,x2≤fdx0,x1+dx1,x2+α≤fdx0,x1+λdx0,x1+α=f1+λdx0,x1+α=f1+λ1−λr+α=f1−λ2r+α≤fε<fr.
This implies that(46)dx0,x2<r,i.e., x2∈Bx0,r. Suppose x3,…,xk∈Bx0,r for some k∈ℕ. Now, if x2j+1≤xk, then by (42), we can write(47)dx2j,x2j+1=dSx2j−1,Tx2j≤adx2j−1,x2j+bdx2j−1,Sx2j−1+cdx2j,Tx2j=adx2j−1,x2j+bdx2j−1,x2j+cdx2j,x2j+1.
This implies(48)1−cdx2j,x2j+1≤a+bdx2j−1,x2j,or(49)dx2j,x2j+1<a+b1−cdx2j−1,x2j.
Leta+b/1−c=λ, we get(50)dx2j,x2j+1<λdx2j−1,x2j.
Similarly, ifx2j≤xk, then(51)dx2j−1,x2j<a+b1−cdx2j−2,x2j−1=λdx2j−2,x2j−1.
Therefore, from inequality (50) and (51), we write(52)dx2j,x2j+1<λdx2j−1,x2j<⋯<λ2jdx0,x1and(53)dx2j−1,x2j<λdx2j−2,x2j−1<⋯<λ2j−1dx0,x1.
From (52) and (53), we write(54)dxk,xk+1≤λkdx0,x1,for somek∈ℕ.
Now, using (54), we have(55)fdx0,xk+1≤f∑i=1k+1dxi−1,xi+α=fdx0,x1+⋯+dxk,xk+1+α≤f1+λ+λ2+⋯+λkdx0,x1+α=f1−λk+11−λdx0,x1+α.
Using (b), we write(56)fdx0,xk+1≤f1−λk+11−λ1−λr+α=f1−λk+1r+α.
Using (c), we deduce that(57)fdx0,xk+1≤fɛ<fr.
Hence, by (F1), we notice that(58)dx0,xk+1≤r.
This implies thatxk+1∈Bx0,r. Therefore, xn∈Bx0,r for all n∈ℕ. Now, we have by (42)(59)dx2i+1,x2i+2=dSx2i,Tx2i+1≤adx2i,x2i+1+bdx2i,Sx2i+cx2i+1,Tx2i+1=adx2i,x2i+1+bdx2i,x2i+1+cdx2i+1,x2i+2.
Following the same steps of proof of Theorem3 and using (a), we obtain that the sequence xn is ℱ-convergent to some z∗ in Bx0,r⋅z∗ can be proved as common fixed point of S and T in the same way as in Theorem 3.
TakingS=T in Theorem 4, we get the following result of single mappings.Corollary 5.
Supposef,α∈ℱ×0,∞,X,d is an ℱ-complete ℱ-metric space and T:X⟶Xis a self-mapping. Suppose that a+b+c<1,fora,b,c∈0,∞. Suppose that forx0∈X and r>0, the following conditions are satisfied:(a)
Bx0,r⊆X is ℱ-closed(b)
dTx,Ty≤adx,y+bdx,Tx+cdy,Ty, for all x,y∈Bx0,r(c)
dx0,x1≤1−λr, for x1∈X and λ=a+b/1−c(d)
There exists0<ɛ<r such as f1−λk+1r≤fɛ−α, where k∈ℕ
Then,Thas at most one fixed point in Bx0,r.Example 4.
LetX=0,∞ andfx=lnx. Define T:X⟶X by(60)Tx=x3,ifx∈0,1,x3ifx∈1,∞,and define dby(61)dx,y=x−y2,ifx,y∈0,1×0,1,x−y,ifx,y∉0,1×0,1.
It can be easily verified thatd is an ℱ-metric and function f satisfies F1−F2. Fix x0=r=1/4, then Bx0,r=0,1/2. Clearly, Bx0,r is ℱ-closed so condition (a) of Corollary 5 is satisfied. Now, if a=3/4,b=c=0, then λ=a and(62)dx0,x1=dx0,Tx0=14−1122=136<1−3414=1−λr.
This shows that condition (b) is fulfilled. Furthermore, supposek=1, then(63)f1−λk+1r=ln1−34214=ln764ln864−ln87=fε−α,i.e.,(64)f1−λk+1r=fε−α.
Hence, condition (d) is satisfied forε=8/64≤1/4=r and α=ln8/7 Similarly, for all values of k∈N, we can find some 0<ɛ<r and α such that condition (d) is fulfilled. Now, checking for condition (b), we have two cases:(i)
Ifx,y∈Bx0,r×Bx0,r, then(65)dTx,Ty=x3−y32=19x−y2<34x−y2=adx,y+0.dx,Tx+0.dy,Ty=adx,y+bdx,Tx+cdy,Ty,asb=c=0.Therefore, for allx,y∈Bx0,r×Bx0,r, condition (d) is also satisfied.(ii)
Ifx,y∉Bx0,r×Bx0,r, e.g., x=2 and y=3, then(66)dTx,Ty=23−33>342−3=adx,y=adx,y+0.dx,Tx+0.dy,Ty=adx,y+bdx,Tx+cdy,Ty.Hence, condition (b) holds only forBx0,r and not on X×X. Moreover, 0∈Bx0,r is the fixed point of T.Corollary 6.
Supposef,α∈ℱ×0,∞and X,d is an ℱ-complete ℱ-metric space. Let S,T:X⟶Xare self-mappings and k∈0,1, assume that, for x0∈X and r>0, the following conditions are satisfied:(a)
Bx0,r⊆X is ℱ-closed(b)
dSx,Ty≤k/2dx,Sx+dy,Ty, for all x,y∈Bx0,r(c)
dx0,x1≤1−λr, for x1∈X and λ=k/2−k(d)
there exist0<ɛ<r such as f1−λk+1r≤fɛ−α, where k∈N
Then,S and Ta11a12a21a22 have at most one common fixed point in Bx0,r.Corollary 7.
Supposef,α∈ℱ×0,∞and X,d is an ℱ-complete ℱ-metric space. Let S,T:X⟶Xare self-mappings and k∈0,1, assume that, for x0∈X and r>0, the following conditions are satisfied:(a)
Bx0,r⊆X is ℱ-closed(b)
dSx,Ty≤kdx,y, for all x,y∈Bx0,r(c)
dx0,x1≤1−λr, for x1∈X and λ=k/2−k(d)
there exist0<ϵ<r such as f1−λk+1r≤fɛ−α, where k∈N
Then,S and Thave at most one common fixed point in Bx0,r.
An example can be proved in a similar way as that to the previous examples.


## 5. Application

This section is concerned with the application of the main result proved in Section2, in finding a unique common solution of the functional equations that are used in dynamic programming.The two main components of dynamic programming are decision space (DS) and a state space (SS). The SS includes different states such as transitional states, initial, and action states, while the DS is composed of the steps that are taken for locating the possible solution point of the problem. Optimization and computer programming are based on this system. In particular, a problem of dynamic programming is converted to functional equations as(67)pu=maxv∈VFu,v+f1u,v,pηu,v,foru∈U,(68)xu=maxv∈VFu,v+f2u,v,pηu,v,foru∈U,where Y and Z are Banach spaces such as U⊆Y and V⊆Z and(69)η:U×V⟶UF:U×V⟶R,f1,f2:U×V×R⟶R.SupposeU and V are the DS and SS, respectively. We aim to locate a single common solution point for equations (67) and (68). We denote the set of all bounded real-valued mappings on U by WU. Let j be arbitrary member of WU and say j=maxu∈Uju. Then, the duplet WU,⋅ is a Banach space with ddefined by(70)dj,k=maxu∈Uju−ku.Let the following conditions holds true:(C1)
F,f1,f2 are bounded.(C2)Foru∈U and j∈WU, define S,T:WU⟶WU by(71)Sju=maxv∈VFu,v+f1u,v,jηu,v,foru∈U,Tju=maxv∈VFu,v+f2u,v,jηu,v,foru∈U.Observe that,S and T are well-defined whenever the functions F,f1 and f2 are bounded.(C3)
Foru,v∈U×V, j,k∈WU and l∈U, we write(72)f1u,v,jl−f1u,v,kl≤Mj,k,where(73)Mj,k=αdj,k+βdj,Sj+γdk,Tk,forα,β,γ∈0,∞ and α+2β+2γ<1Now, we develop the following theorem.Theorem 5.
Suppose conditions (C1)–(C3) hold true, then there exists a single bounded common solution of equations (67) and (68).Proof:.
From Lemma 1.10, we haveWU,d is an F-complete F−MS. d is defined by (70), and from (C1), we deduce that S and T are self-mappings on WU. Let ω be an arbitrary positive number and j1,j2∈WU. Take u∈U and v1,v2∈V such as(74)Sjx<Fu,vx+f1u,vx,jxηu,vx+ω,(75)Tjx<Fu,vx+f2u,vx,jxηu,vx+ω,and(76)Sj1≥Fu,v2+f1u,v2,j1ηu,v2,(77)Tj2≥Fu,v1+f1u,v,j2ηu,v1.
Then, using (74) and (77), we obtain(78)Sj1u−Tj2u<f1u,v1,j1ηu,v1−f1u,v1,j2ηu,v1+ω≤f1u,v1,j1ηu,v1−f1u,v1,j2ηu,v1+ω≤Mj1u,j2u+ω.
Also, from (75) and (76), we get(79)Tj2u−Sj1u<Mj1u,j2u+ω.
Merging the above two inequalities, we write(80)Sj1u−Tj2u<Mj1u,j2u+ω,for all ω>0. Thus,(81)dSj1u,Tj2u≤Mj1u,j2u,i.e.,(82)dSj1,Tj2≤Mj1,j2,for every∈U. All the requirements of Theorem 3 are fulfilled. Therefore, by using Theorem 3, S and T have a unique bounded and common solution for equations (67) and (68).


## 6. Conclusion

This article has furthered the idea ofF-metric space and fixed point and common fixed point results are elaborated in the setting of F-metric space. It is obtained that the fixed point and common fixed point of a contraction mapping can be availed even if the contractive condition is restricted to only a subset closed ball of the whole F-metric space. Examples have been provided for both locally and globally contractions and a comparison between them is made for better understanding. Some important corollaries have been developed from the proved results. At last, application of the main result in finding a unique solution of the functional equation is given. In future, we opt to explore similar results in the frame of fuzzy cone metric space. Fixed point of Reich-type contractions will be investigated in picture fuzzy metric space, fuzzy soft sets, and other applicable abstract spaces. The proposed research will be primarily based upon some existing literature on the topics ([19–22]).



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