# New Fixed Point Results inF-Quasi-Metric Spaces and an Application

Authors: Ehsan Lotfali Ghasab; Hamid Majani; Erdal Karapinar; Ghasem Soleimani Rad
Journal: Advances in Mathematical Physics
 (2020)

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## Abstract

The main goal of the present paper is to obtain several fixed point theorems in the framework ofF-quasi-metric spaces, which is an extension of F-metric spaces. Also, a Hausdorff δ-distance in these spaces is introduced, and a coincidence point theorem regarding this distance is proved. We also present some examples for the validity of the given results and consider an application to the Volterra-type integral equation.


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## Body

## 1. Introduction and Preliminaries

In the last century, nonlinear functional analysis has experienced many advances. One of these improvements is the introduction of various spaces and is the proof of fixed point results in these spaces along with its applications in engineering science. One of these spaces is function weighted metric space introduced by Jleli and Samet [1]. This is a generalization of metric spaces.Definition 1 [1, 2].
A functionf:0,+∞⟶ℝ is named called a nondecreasing function if fs1≤fs2 for every s1,s2∈0,+∞. Also, f is said to be logarithmic-like when every positive sequence tn satisfies limn⟶∞tn=0 iff limn⟶∞ftn=−∞.In the sequel, we applyF for the set of all nondecreasing functions that are logarithmic-like.In 2019, some of researchers such as Alqahtani et al. [2], Aydi et al. [3], and Bera et al. [4] discussed on the structure of this space and on the fixed points of mappings satisfying in various contractive conditions.Definition 2 [1].
Consider a mappingδ:X×X⟶0,+∞, a constant B∈0,+∞ and a f∈F so that
(η1) δx1,x2=0⇔x1=x2 for all x1,x2∈X;
(η2) δx1,x2=δx1,x2 for all x1,x2∈X; and
(η3) δx1,x2>0 implies that fδx1,x2≤f∑i=1N−1δvi,vi+1+B for every N∈ℕ with N≥2, for every x1,x2∈X, and for all vii=1N⊂X with v1,vN=x1,x2.
Then, the functionδ is named as a function weighted metric or a F-metric on X, and the pair X,δ is called a F-metric space.Definition 3 [5].
Consider a mappingδq:X×X⟶0,+∞ satisfies the properties η1 and η3 from the definition of a F-metric. Then, δq is named a F-quasi-metric on X and X,δq is called a F-quasi-metric space.In [5], Karapinar et al. showed that δq generally induces other F-quasi-metrics such as δq−1,δq∗:X×X⟶0,+∞ defined by δq−1s,t=δqt,s and δq∗s,t=maxδqs,t,δq−1s,t. Regarding the discussion above, we conclude that any quasi-metric is a F-quasi-metric by choosing ft=lnt for the axiom η3 with B=0.Definition 4 [5].
LetX,δq be a F-quasi-metric space. For x∈X, the right (left) centered ball at x and of radius e>0 is the set Brs,e=t∈X:δqs,t<e (Bls,e=t∈X:δqt,s<e).Definition 5 [5].
Consider aF-quasi-metric space X,δq with a sequence xn therein. Then, xn is said to be aright-convergent sequence (left-convergent sequence) to x∈X if limn⟶∞δqx,xn=0 (limn⟶∞δqxn,x=0). Further, xn is said to be a biconvergent sequence (in summary, convergent sequence) when it is both right-convergent and left-convergent.Proposition 6 [5].
Consider aF-quasi-metric space X,δq with a sequence xn therein. Also, let limn⟶∞δqs,xn=limn⟶∞δqxn,t=0 for every s,t∈X. Then s=t.Definition 7 [5].
Consider aF-quasi-metric space X,δq with a sequence xn therein. Then, xn is a right-Cauchy sequence (a left-Cauchy sequence) if limn,m⟶∞δqxn,xm=0 (limn,m⟶∞δqxm,xn=0). With this interpretation, xn is bi-Cauchy sequence (in summary, Cauchy sequence) if it is both left-Cauchy sequence and right-Cauchy sequence.Now, aF-quasi-metric space X,δq is named right-complete (left-complete) if every right-Cauchy sequence (left-Cauchy sequence) in X is a right-convergent sequence (left-convergent sequence) in X. Further, X,δq is bicomplete (in short, complete) if it is both left-complete and right-complete.Example 8.
Defineδq:ℕ×ℕ⟶0,+∞ by
(1)δqs,t=0whens=tes+∣s−t∣otherwise,for every s,t∈ℕ. Evidently, ℕ,δq is a bicomplete F-quasi-metric space.On the other hand, Bhaskar and Lakshmikantham [6] defined the notion of coupled fixed point and presented several coupled fixed point propositions for a mixed monotone mapping in partially ordered matric spaces. Also, they studied the existence and uniqueness of a solution to a periodic boundary value problem. For more details on coupled, tripled, and n-tupled fixed point assertions, one can see [7] and references therein.Definition 9 [8, 9].
LetF:X×X⟶X and g:X⟶X be two optional mappings. An element u,v∈X×X is said to be a coupled coincidence point of F and g if Fu,v=gu and Fv,u=gv. Further, an element u,v∈X×X is named acommon fixed point of F and g if Fu,v=gu=u and Fv,u=gv=v.
Note that ifg is the identity mapping, then x,y is called a coupled fixed point of F [6].Definition 10 [9].
LetF:X×X⟶X and g:X⟶X be two optional mappings. Then F and g is said to be commutative if Fgu,gv=gFu,v for every u,v∈X.In this paper, we introduce several common fixed point and common coupled fixed point theorems in such spaces and prove them. In Section2, we prove a common fixed point theorem and a common coupled fixed point result in this space. In Section 3, we obtain a coincidence point result for single-valued and multivalued mappings regarding a Hausdorff δ-distance. Ultimately, as an application of these results, the existence of solution of the Volterra-type integral equation is investigated in Section 4.


## 2.F-Quasi-Metric Space and Fixed Point Theory

Theorem 11.
LetX,δq be a bicomplete F-quasi-metric space. Also, let g,T:X⟶X be two arbitrary mappings so that T and g are commutative, TX⊂gX, gX is closed, and
(2)δqTx,Ty≤kδqgx,gy,for every x,y∈X, where k∈0,1. Then T and g contain a unique common fixed point in X.Proof.
Due toTX⊂gX, we select a point x1∈X such thatTx0=gx1 for a given x0∈X. By continuing this process, we can construct a sequence yn in X by yn=Txn=gxn+1 for n=0,1,⋯. First, note that T and g possess a unique coincidence point. On the contrary, assume that u1,v1∈X are two different coincidence points of T and g. Then, δqu2,v2>0 with gu1=Tu1=u2 and gv1=Tv1=v2. Now, by (2), we get
(3)δqu2,v2=δqTu1,Tv1≤kδqgu1,gv1=kδqu2,v2<δqu2,v2,which is a contradiction.
Assumef,B∈F×0,+∞ so that (η3) is complied. For an arbitrary ε>0 and because of (η3), there exists γ>0 such that
(4)0<t<γ⟹ft<fε−B.
Now, letyn be a sequence in X. Without loss of totality, suppose that δqTx0,Tx1>0. Otherwise, x1 is a coincidence point of T and g. Now, using (2), we obtain
(5)δqTxn,Txn+1≤kδqgxn,gxn+1=kδqTxn−1,Txn≤k2δqgxn−1,gxn,which implies by induction that
(6)δqTxn,Txn+1≤knδqTx0,Tx1for every n in ℕ. Hence, for every m and n in ℕ so that m>n, we get
(7)∑i=nm−1δqTxi,Txi+1≤kn1−kδqTx0,Tx1.
Since(8)limn⟶∞kn1−kδqTx0,Tx1=0,there is some N∈ℕ so that 0<kn/1−kδqTx0,Tx1<γ for every n≥N. Hence, from (4) and (η1), we observe that
(9)f∑i=nm−1δqTxi,Txi+1≤fkn1−kδqTx0,Tx1<fε−B,for m>n≥N. Employing (η3) together with (9), we obtain
(10)δqTxn,Txm>0⟹fδqTxn,Txm≤f∑i=nm−1δqTxi,Txi+1+B<fε.
It follows thatδqTxn,Txm<ε. Therefore, yn=Txn is right-Cauchy. Similarly, by changing the order of the pairs xi+1,xi in the above process, we conclude that yn is also a left-Cauchy sequence. Hence, it is a Cauchy sequence. Now, since X,δq is a bicomplete space, there exists z∈X such that yn is convergent to z. Since Txn=gxn+1⊂gX and gX is closed, we have limn⟶∞δgxn,gz=0. As a next step, we show that z is a coincidence point of T and g. On the contrary, consider δqTz,gz>0. Then we have
(11)fδqTz,gz≤fδqTz,Txn+δqTxn,gz+B≤fkδqgz,gxn+δqgxn+1,gz+B.
Asn⟶∞ in the inequality above, we obtain
(12)limn⟶∞fkδqgz,gxn+δqgxn+1,gz+B=−∞,which is a contradiction. Hence, δqTz,gz=0; that is, z is a unique coincidence point of T and g. Therefore, g and T contian a unique point of coincidence w=gz=Tz. By commutativity of the mapping T and g, we have gw=ggz=gTz=Tgz=Tw. Hence, gw is another point of coincidence of g and T. Now, by the uniqueness of the point of coincidence of g and T, we have w=gw=Tw; that is, g and T contain a unique common fixed point. This completes the proof.In the sequel, denote for simplicityX×⋯×X by Xn, where X is a nonempty set and n∈ℕ.Lemma 12.
Consider aF-quasi-metric space X,δq. Then, the following assertions hold:
(1)
Xn,Δq is a F-quasi-metric space with
(13)Δqu1,⋯,un,v1,⋯,vn=maxδqu1,v1,δqu2,v2,⋯,δqun,vn.(2)
The mappingf:Xn⟶X and g:X⟶X contain ann-tuple common fixed point iff the mapping F:Xn⟶Xn and G:Xn⟶Xn defined by
(14)Fu1,u2,⋯,un=fu1,u2,⋯,un,fu2,⋯,un,u1,⋯,fun,u1,⋯,un−1Gu1,u2,⋯,un=gu1,gu2,⋯,gunpossess a common fixed point in Xn.(3)
X,δq is bicomplete iff Xn,Δq is bicomplete.Proof.
Clearly,Δq satisfies in (η1). We show that Δq satisfies in (η3). For every xi,j⊂X for 1≤i≤N and 1≤j≤n, consider x1j,xN−1j=uj,vj. Suppose that
(15)δquj,vj=maxδqu1,v1,δqu2,v2,⋯,δqun,vn.
Then, we have(16)fjδquj,vj≤fj∑i=1N−1δqxi,j,xi+1,j+Bj,where fj∈F and Bj∈0,+∞. Therefore, we obtain
(17)fjΔqu1,u2,⋯,un,v1,v2,⋯,vn=fjmaxδqu1,v1,δqu2,v2,⋯,δqun,vn=fjδquj,vj≤fj∑i=1N−1δqxi,j,xi+1,j+Bj≤fj∑i=1N−1Δqxi,1,xi,2,⋯,xi,n,xi+1,1,xi+1,2,⋯,xi+1,n+Bj.The proofs of (2) and (3) are straightforward and left to the reader.Remember that Lemma12 is a two-way relationship. Consequently, we can establish n-tuple fixed point propositions from fixed point assertions and conversely. Now, set n=2 in Lemma 12. Then, we have the following theorem.Theorem 13.
LetX,δq be a bicomplete F-quasi-metric space. Also, let g:X⟶X and T:X2⟶X be two arbitrary mappings so that T and g are commutative, TX2⊂gX, gX is closed, and
(18)δqTx,y,Tx∗,y∗≤k2δqgx,gx∗+δqgy,gy∗,for all x,y and x∗,y∗ in X2, where k∈0,1. Then, T and g contain a unique common coupled fixed point in X2.Proof.
Let us defineΔq:X2×X2⟶0,∞ by Δqu1,u2,v1,v2=maxδqu1,v1,δqu2,v2 for all u1,u2,v1,v2∈X. Further, we consider F:X2⟶X2 by Fx,y=Tx,y,Ty,x and G:X2⟶X2 by Gx,y=gx,gy for all x,y∈X. Using Lemma 12, X2,Δq is a bicomplete F-quasi-metric space. Also, x,y∈X2 is a common coupled fixed point of T and g iff it is a common fixed point of F and G. On the other hand, from (18), we have either
(19)ΔqFx,y,Fx∗,y∗=ΔqTx,y,Ty,x,Tx∗,y∗,Ty∗,x∗=maxδqTx,y,Tx∗,y∗,δqTy,x,Ty∗,x∗=δqTx,y,Tx∗,y∗≤k2δqgx,gx∗+δqgy,gy∗≤kmaxδqgx,gx∗,δqgy,gy∗=kΔqGx,y,Gx∗,y∗,or
(20)ΔqFx,y,Fx∗,y∗=ΔqTx,y,Ty,x,Tx∗,y∗,Ty∗,x∗=maxδqTx,y,Tx∗,y∗,δqTy,x,Ty∗,x∗=δqTy,x,Ty∗,x∗≤k2δqgy,gy∗+δqgx,gx∗≤kmaxδqgy,gy∗,δqgx,gx∗=kΔqGy,x,Gy∗,x∗.
Now, by Theorem11, F and G have a common fixed point and by Lemma 12, T and g have a common coupled fixed point.Example 14.
LetX=0,1. Define δq:X×X⟶0,∞ by
(21)δqs,t=0whens=t∣s∣+∣s−t∣otherwise,for every x,y∈X. Evidently, δq is a bicomplete F-quasi-metric with ft=lnt and B=0. Consider T:X2⟶X by Tx,y=x/2+y/2 and define g:X⟶X by gx=2x. Clearly T and g are commutative. Also, we have
(22)δqTx,y,Tx∗,y∗=x2+y2−x∗2+y∗2+x2+y2=142x−2x∗+2y−2y∗+2x+2y≤142x−2x∗+2y−2y∗+2x+2y=14δqgx,gx∗+14δqgy,gy∗.
Therefore, by lettingk=1/2, all of the hypotheses of Theorem 13 hold. Thus, T and g possess a common coupled fixed point in X2.


## 3. Fixed Point Theorem and Hausdorffδq-Distance

Let us start with the following definition:Consider aF-quasi-metric space X,δq, and denote thefamily of all nonempty bounded closed subsets of X by CBX. Then, H·,· is said to be a Hausdorff δq-distance on CBX, if
(23)HδqA,B=maxsupx∈Aδqx,B,supx∈BδqA,x,where δqx,B=infδqx,y,y∈B.Definition 15 [10].
LetX be a nonempty set, g:X⟶X be a single-valued mapping, and T:X⟶CBX be a multivalued mapping. Also, let w=gx∈Tx for some x∈X. Then w is said to be a point of coincidence of g and T, and x is said to be a coincidence point of g and T.Theorem 16.
LetX,δq be a bicomplete F-quasi-metric space. Also, let g:X⟶X be a single-valued mapping and T:X⟶CBXbe a multivalued mapping so that TX⊂gX, gX is closed, and g is continuous. Assume that there exists k∈0,1 such that
(24)HδqTx,Ty≤kδqgx,gy,for all x,y∈X. Then T and g have a coincidence point in X.Proof.
Due toTX⊂gX, we select a point x1∈X such that gx1∈Tx0 for a given x0∈X. By continuing this procedure, we can construct a sequence xn in X such that gxn+1∈Txn for n=0,1,⋯. Suppose that f,B∈F×0,+∞ so that (η3) holds. For an arbitrary ε>0 and due to (η3), there exists γ>0 such that
(25)0<t<γ⟹ft<fε−B.
Consider the sequencegxn⊂X. Now, without loss of generality, suppose that HδqTx0,Tx1>0. Otherwise, x1 is a coincidence point of T and g. Now, from (24), we have
(26)δqgxn+1,gxn+2≤HδqTxn,Txn+1≤kδqgxn,gxn+1≤kHδqTxn−1,Txn≤k2δqgxn−1,gxn.
Hence, we haveδqgxn,gxn+1≤knδqgx0,x1 for all n∈ℕ. Now, let m and n be two natural numbers with m>n. Then, we have
(27)∑i=nm−1δqgxi,gxi+1≤kn1−kδqgx0,gx1.
On the other hand, sincelimn⟶∞kn/1−kδqgx0,gx1=0, there exists N∈ℕ so that
(28)0<kn1−kδqgx0,gx1<γ,for n≥N. Hence, by (25) and (η1), we have
(29)f∑i=nm−1δqgxi,gxi+1≤fkn1−kδqgx0,gx1<fε−B,for all m>n≥N. Employing (η3) together with (29), we obtain
(30)δqgxn,gxm>0⟹fδqgxn,gxm≤f∑i=nm−1δqgxi,gxi+1+B<fε.
Now, by (η1), we have δqgxn,gxm<ε. This proves that gxn is right-Cauchy. Similarly, by changing the order of the pairs xi+1,xi in the above process, we conclude that gxn is also a left-Cauchy sequence. Therefore, it is a Cauchy sequence. Note that X,δq is bicomplete and gX is closed. Thus, there exists x∈X such that limn⟶∞gxn=gx. Now, we shall show that gx∈Tx. For this purpose, using (24), we have
(31)δqgxn+1,Tx≤HδqTxn,Tx≤kδqgxn,gx.
Thus,(32)limn⟶∞δqgxn+1,Tx=δqgx,Tx=0.
Hence,gx∈Tx. Consequently, T and g possess a point of coincidence.Example 17.
ConsiderX=0,1. Define T:X⟶CBX by Tx=0,1/16x and g:X⟶X by gx=x/2. Also, define δq:X×X⟶0,∞ by
(33)δqs,t=0ifs=ts+s−totherwise.
Clearly,δq is a bicomplete F-quasi-metric with ft=lnt and B=0. Also, evidently, TX⊂gX and gX is closed. First, let x=y=0. Then
(34)HδqTx,Ty=0≤kδqgx,gy.
Therefore, we may considerx and y are not zero. Without loss of totality, suppose that x≤y. Then
(35)HδqTx,Ty=Hδq0,116x,0,116y=maxsup0≤a≤1/16xδqa,0,116y,sup0≤b≤1/16yδq0,116x,b.
Due tox≤y, we have 0,1/16x⊂0,1/16y. For every a∈0,1/16x, we have δqa,0,1/16y=0. Also, for every b∈0,1/16y, we obtain
(36)δq0,116x,b=0b≤x16∣0∣+∣x8−2b∣b≥x16.
This yields that(37)sup0≤b≤1/16yδq0,116x,b=x8−y8.
We deduce that(38)HδqTx,Ty=x8−y8≤14x2+x2−y2=14δqgx,gy.
Obviously, all other hypotheses of Theorem16 hold. Hence, g and T possess a coincidence point in X.


## 4. An Application

As an application of our results, we consider the following Volterra-integral equation:(39)xt=∫0tKt,s,xsds+vt,where t∈I=0,1, K∈CI×I×ℝ,ℝ, and v∈CI,ℝ.Consider a Banach space of all real continuous functions defined onI (CI,ℝ) with norm x∞=maxt∈Ixt for every x∈CI,ℝ. Also, let CI×I×CI,ℝ,ℝ be the space of all continuous functions defined on I×I×CI,ℝ. On the other hand, the Banach space CI,ℝ can be dedicated with the Bielecki norm xB=supt∈Ixte−τt for all x∈CI,ℝ and τ>0, and the derived metric δBx,y=x−yB for all x,y∈CI,ℝ. Define δq:X×X⟶0,∞ by
(40)δqx,y=supt∈Ixt−yte−∥x∥Bt.Also, defineT:CI,ℝ⟶CI,ℝ by
(41)Txt=∫0tKt,s,xsds+vt,v∈CI,ℝ.Theorem 18.
Consider a bicompleteF-quasi-metric space CI,ℝ,δB with ft=lnt. Also, let T be an operator from CI,ℝ into CI,ℝ with Txt=∫0tKt,s,xsds+vt, and let gx=Ix. Assume that K∈CI×I×ℝ,ℝis an operator such that
(i)
K is continuous;(ii)
∫0tKt,s,· for all t,s∈I is increasing; and(iii)
for everyx and y in CI,ℝ, and t and s in I, we have
(42)Kt,s,xs−Kt,s,ys≤e−xBxs−ys.Then, the integral equation (39) possesses an answer in CI,ℝ.Proof.
By definition ofT, we have
(43)δqTx,Ty=supt∈I∫0tKt,s,xsds−∫0tKt,s,ysdse−∥x∥Bt≤supt∈I∫0tKt,s,xs−Kt,s,yse−∥x∥Btds≤supt∈I∫0te−∥x∥Bxs−yse−∥x∥Btds≤x−yBsupt∈I∫0te−∥x∥Bds≤e−∥x∥Bδqx,y.Now, we consider that the functionft=lnt for every t∈I, B=0, and k=e−∥x∥B. Therefore, all assertions of Theorem 11 hold. As a result, Theorem 11 confirms the existence of fixed point of T so that this fixed point is the answer of the integral equation.



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