# A Common Fixed Point Theorem for Nonlinear Quasi-Contractions onb-Metric Spaces with Application in Integral Equations

Authors: Zoran D. Mitrović; Ivan D. Aranđelović; Vesna Mišić; Abdollah Dinmohammadi; Vahid Parvaneh
Journal: Advances in Mathematical Physics
 (2020)

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## Abstract

In this paper, we present a common fixed point result for a pair of mappings defined on a b-metric space, which satisfies quasi-contractive inequality with nonlinear comparison functions. An application in solving a class of integral equations will support our results.


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## Body

## 1. Introduction

In 1974, Ćirić presented the first fixed point result for quasi-contractive mappings. This Ćirić’s theorem is one of the most general results with linear comparison function in classical metrical fixed point theory (see [1, 2]). The existence and uniqueness of fixed point for mappings defined on metric spaces, which satisfies a quasi-contractive inequality with a nonlinear comparison function, were considered by Danes [3], Ivanov [4], Aranđelović et al. [5], and Bessenyei [6]. Alshehri et al. [7] proved a fixed point theorem for quasi-contractive mappings, defined by linear quasi-contractive conditions on b-metric spaces. Common fixed point generalizations of Ćirić result was obtained by Das and Naik [8], with linear comparison functions and by Di Bari and Vetro [9], with a nonlinear comparison function.The notion of symmetric spaces, which is the oldest and one of the most important generalizations of metric spaces, was introduced by Fréchet [10]. He used the name E-space for a symmetric space. In the last 50 years, many authors (see [11–16]) called them semimetric (in German halb-metrisher) spaces. Now, the term symmetric space is usual. After 1955, the term semimetric space is widely used to denote a symmetric space in which the closure operator is idempotent, which started the papers of Heath, Brown, Mc Auley, Jones, and Burke (see [17, 18]). Fixed point investigation was started by Cicchese [19] and Jachymski et al. [18] on semimetric spaces and by Hicks and Rhoades [20] on symmetric spaces.In [10], Fréchet also considered the class of E-spaces with regular écart which include the class of b-metric spaces. Important examples of b-metric spaces are quasi-normed spaces introduced by Bourgin [21] and Hyers [22] and spaces of homogeneous type which have many applications in the theory of analytic functions (see Coifman and Weiss [23]). First, fixed point results on b-metric spaces were presented by Bakhtin [24] and Czerwik [25].In this paper, we present a common fixed point result for a pair of mappings defined on ab-metric space, which satisfies a quasi-contractive inequality with a nonlinear comparison function.


## 2. Symmetric Spaces andb-Metric Spaces

The ordered pairΔ,μ, where Δ is a nonempty set and μ:Δ2→0,∞, is a symmetric space, if and only if it satisfies:
(W1)
μι,κ=0 if and only if ι=κ(W2)
μι,κ=μκ,ι for any ι,κ∈ΔThe difference between symmetric spaces and more convenient metric spaces is in the absence of triangle inequality, but many notions in symmetric spaces are defined similar to those in metric spaces. For instance, in symmetric spaceΔ,μ, the limit point of a sequence ιn is defined by
(1)limμιn,ι=0⇔limιn=ι.Also, we say that a sequenceιn⊆Δ is a Cauchy sequence, if for any given ε>0, there exists a positive integer n0 such that μιm,ιn<ε for every m,n≥n0. If each Cauchy sequence in symmetric space Δ,μ is convergent, then we say that Δ,μ is a complete symmetric space.By(2)diamA=supι,κ∈Aμι,κ,we indicate the diameter of the set A.LetΔ,μ be a symmetric space. We can introduce the topology τd by defining the family of all closed sets as follows: a set A⊆Δ is closed if and only if for each ι∈Δ, μι,A=0 implies ι∈A, where
(3)μι,A=infμι,a:a∈A.The convergence of a sequenceιn in the topology τd need not imply μιn,ι→0, but the converse is true.LetΔ,μ be a symmetric space, ιn,κn,zn⊆Δ and ι,κ∈Δ. We considered the following seven properties as partial replacements for the triangle inequality:(W3)limμιn,ι=0∧limμιn,κ=0⇒ι=κ(W4)limμιn,ι=0∧limμιn,κn=0⇒limμκn,ι=0(HE)limμιn,ι=0∧limμκn,ι=0⇒limμιn,κn=0(CC)limμιn,ι=0⇒limμιn,κ=μι,κ(W)limμιn,κn=0∧limμκn,zn=0⇒limμιn,zn=0(JMS)limμιn,κn=0∧limμκn,zn=0⇒lim¯μιn,zn≠∞;(SC)limn→∞μιn,ι=0 implies lim¯n→∞μιn,κ=μι,κThe property (W3) has been introduced by Fréchet [10]; (W4), (HE), and (W) by Pitcher and Chittenden [14]; (CC) by Sims [15]; (JMS) by Jachymski et al. [18]; and (SC) by Aranđelović and Kečkić [17]. Note that W⇒W4⇒W3, W⇒JMS, W⇒HE, CC⇒W3, and CC⇒SC (see [17, 26]).In [26], the authors give examples for the following relationships: W3⇒W4, W4⇒HE, W4⇒CC, W3⇒HE, W3⇒CC, CC⇒W4, HE⇒CC, HE⇒W3, HE⇒W4, and CC⇒HE. The fact that W⇒CC has been proved in [27].Definition 1.
LetΔ be a nonempty set, μ:Δ×Δ→0,∞. Δ,μ is said to be a b-metric space if there exists s∈0,∞ such that
(1)
μι,κ=0 if and only if ι=κ(2)
μι,κ=μκ,ι for any ι,κ∈Δ(3)
μι,z≤sμι,κ+μκ,zfor all ι,κ,z∈Δ.
Anys∈0,∞ which satisfies inequality (3) of Definition 1 for all ι,κ,z∈Δ, where Δ,μ is a b-metric space, is said to be the b constant of space Δ,μ. It is clear that if s=1, then Δ,μ is a metric space.Lemma 2.
LetΔ,μ be a b-metric space with b constant s. Then, s≥1.Proof.
Letι,κ∈Δ. Then, μι,κ≤sμι,κ+μκ,κ=sμι,κ, which implies that s≥1.
In [17], the following result was proved.Lemma 3 (see [17]).
LetΔ,μ be a b-metric space. Then, Δ,μ is a symmetric space which satisfies the properties (W3), (W4), (HE), (W), and (JMS).


## 3. Comparison Functions

Letχ:0,∞→0,∞ be a function such that χι=0 if and only if ι=0. Define:
(1)
χ∈Ξ0 if and only if χr<r for each r>0(2)
χ∈Ξ1 if and only if limt→r+¯χt=χr for each r>0(3)
χ∈Ξ2 if and only if limt→r¯χt≤χr for any r>0(4)
χ∈Ξ3 if and only if limt→r+¯χt<r for any r>0(5)
χ∈Ξ4 if and only if limt→r¯χt<r for all r>0(6)
χ∈Ξ5 if and only if limt→∞t−χt=∞(7)
χ∈Ξ6 if and only if I−χ:0,∞→0,∞ is a strictly increasing surjection(8)
χ∈Ξ7 if and only if ι:I−χι<r is bounded for every r>0(9)
χ∈Ξ8 if and only if χ is monotone nondecreasingIfχ∈∩i=07Ξi, then we say that χ is a comparison function.Ifχ∈Ξ1, then χ is continuous from the right on 0,∞.Ifχ∈Ξ2, then χ is upper semicontinuous on 0,∞.The class ofΞ0∩Ξ1∩Ξ6∩Ξ8 has been applied in the theory of nonlinear quasi-contractions by Danes [18],Ξ0∩Ξ1∩Ξ5∩Ξ8 by Ivanov [4], Ξ0∩Ξ3∩Ξ5∩Ξ8 by Aranđelović et al. [5] and Di Bari and Vetro [9], Ξ0∩Ξ4∩Ξ5 by Aranđelović et al. [5], and the class of Ξ0∩Ξ2∩Ξ7∩Ξ8 by Bessenyei [6].Note thatΞ4⊆Ξ3, Ξ1∩Ξ0⊆Ξ3∩Ξ0, and Ξ2∩Ξ0⊆Ξ4∩Ξ0. Some further inclusion between different classes of comparison functions will be presented in the next statements.Proposition 4.
Ifχ∈Ξ3∩Ξ8, then χ∈Ξ4.Proof.
For anyr>0, from limt→r+¯χt<r, we get that limt→r−¯χt<r, because χ is monotone nondecreasing. So, we obtain that limt→r¯χt<r, for every r>0. ⋄Proposition 5.
Ξ5=Ξ7.Proof.
Letχ∈Ξ5. If there exists r>0 such that ι:ι−χι<r is unbounded; then, for every M>0, there exists t>0 such that t−χt<r. So,
(4)lim¯t→∞χt≤r<∞,which is a contradiction with χ∈Ξ5.
Letχ∈Ξ7. Suppose that there exists an increasing sequence tn⊆0,∞ such that limtn=∞ and R>0 such that tn−χtn<R, for each n. Hence, tn⊆ι:ι−χι<R. So, ι:ι−χι<R is unbounded which implies that χ∈Ξ7.Proposition 6.
Ifχ∈Ξ6, then χ∈Ξ5.Proof.
Letχ∈Ξ6. Suppose that there exists a strictly increasing sequence tn⊆0,∞ such that limtn=∞ and R>0 such that tn−χtn<R, for each positive integer n. So, for any ι>0, we have ι−χι<R, because there exists tn such that ι<tn, which implies that χ is not a surjection. Hence, limt→∞t−χt=∞.⋄Two following two lemmas have been proved in [5].Lemma 7.
Letχ∈Ξ0∩Ξ4∩Ξ5. Then, there exists Ω∈Ξ0∩Ξ3∩Ξ5∩Ξ8 such that
(5)χι≤Ωι<ι,for each ι>0.Lemma 8.
Letχ1,⋯,χn∈Ξ0∩Ξ3∩Ξ5∩Ξ8. Then, there exists Ω∈Ξ0∩Ξ3∩Ξ5∩Ξ8 such that
(6)χkι≤Ωι<ι,for each 1≤k≤n and ι>0.


## 4. Main Results

First, recall some standard terminology and notations from the fixed point theory.LetΔ be a nonempty set, and let Y:Δ→Δ be an arbitrary mapping.LetΔ and Λ be nonempty sets, Y,Γ:Δ→Λ, and YΔ⊆ΓΔ. Choose a point ι1∈Δ such that Yι0=Γι1. Continuing this process, having ιn∈Δ, we obtain ιn+1∈Δ such that Yιn=Γιn+1. Yιn is called a Jungck sequence with an initial point ι0. Note that a Jungck sequence might not be determined by its initial point ι0.LetΔ be a nonempty set and Y,Γ:Δ→Δ. Y and Γ are called weakly compatible if they commute at their coincidence points.Lemma 9 (see [28]).
LetΔ be a nonempty set and let Y,Γ:Δ→Δ be weakly compatible self mappings. If Y and Γ have a unique point of coincidence κ=Yι=Γι, then κ is the unique common fixed point of Y and Γ.
Now, we present our main result. Before stating the result, we make a convention to abbreviateYι and Γι in order to avoid too much parenthesis.Theorem 10.
LetΛ,μ be a b-metric space with b constant s and let Y,Γ:Δ→Λ be two mappings. Suppose that the range of Γ contains the range of Y and that ΓΔ is a complete subspace of Λ. If there exist χ1,χ2,χ3,χ4,χ5:0,∞→0,∞ such that
(7)s·χ1,s·χ2,s·χ3,s·χ4,s·χ5∈Ξ0∩Ξ4∩Ξ5and(8)μYι,Yκ≤maxχ1μΓι,Γκ,χ2μΓι,Yι,χ3μΓκ,Yκ,χ4μΓι,Yκ,χ5μYι,Γκ,for any ι,κ∈Δ, then there exists z∈Λ which is the limit of every Jungck sequence defined by Y and Γ. Further, z is the unique point of coincidence of Y and Γ. Moreover, if Δ=Λ and Y, Γ are weakly compatible, then z is the unique common fixed point for Y and Γ.Proof.
We shall, first, reduce the statement to the caseχ1=⋯=χ5 and s·χi∈Ξ0∩Ξ3∩Ξ5∩Ξ8. Indeed, from Lemma 7, it follows that there exist functions χk∗:0,∞→0,∞ such that s·χk∗∈Ξ0∩Ξ3∩Ξ5∩Ξ8 and
(9)χkι≤χk∗ι<ι,for each ι>0 and for all 1≤k≤5, whereas from Lemma 8, it follows that there exists a real function χ:0,∞→0,∞ such that s·χ∈Ξ0∩Ξ3∩Ξ5∩Ξ8 and
(10)χk∗ι≤χι<ιs,1≤k≤5foreachι>0,which implies
(11)μYι,Yκ≤maxχμΓι,Γκ,χμΓι,Yι,χμΓκ,Yκ,χμΓι,Yκ,χμYι,Γκ.
Thus, we can assume thatχj=χ for all 1≤j≤5 and s·χ∈Ξ0∩Ξ3∩Ξ5∩Ξ8.
Letι0∈Δ be arbitrary and let ιn be an arbitrary sequence such that Yιn is a Jungck sequence with an initial point ι0.
Letd0=μΓι0,Yι0. We will prove that there exists a real number r0>0 such that:
(12)r0−s·χr0≤d0andr−s·χr>d0forr>r0.
Consider the setD=r∣t−s·χt>d0∀t>r which is nonempty, since r−χr→∞ as r→∞. Also, if q∈D and p>q imply p∈D, and hence, D is an unbounded interval. Set r0=infD. For each positive integer n, there is rn∉D such that r0−1/n<rn, and therefore, there is r0≥tn>rn>r0−1/n such that tn−s·χtn≤d0. Since χ is nondecreasing, we have s·χtn≤s·χr0 which implies that tn−s·χr0≤d0. Taking the limit as n→∞, we get r0−s·χr0≤d0.
For anyj≥0, define Onιj=Yιk∣k=j,j+1,j+2,⋯,j+n and Oιj=Yιk∣k=j,j+1,j+2,⋯. Also, let diamA denote the diameter of A.
Next, we prove that(13)δOnιk≤χδOn+1ιk−1,
for all positive integerk,n.
Sinceχ is nondecreasing, it commutes with max, and for all k≤i,j≤k+n, we have
(14)μYιi,Yιj≤χmaxμΓιi,Γιj,μΓιi,Yιi,μΓιj,Yιj,μΓιi,Yιj,μΓιj,Yιi=χmaxμYιi−1,Yιj−1,μYιi−1,Yιi,μYιj−1,Yιj,μYιi−1,Yιj,μYιj−1,Yιi≤χdiamOn+1ιk−1.
By induction, from (13), we obtain that
(15)δOnιk≤χldiamOn+lιk−l.
For1≤i,j≤n, we have Yιi,Yιj∈On−1ι1, and hence, by (13)
(16)μYιi,Yιj≤diamOn−1ι1≤χdiamOnι0<diamOnι0.
Therefore, there is1≤k≤n such that
(17)diamOnι0=μYι0,Yιk≤s·μYι0,Yι1+μYι1,Yιk≤s·d0+s·diamOn−1ι1≤s·d0+s·χdiamOnι0.
Hence, we get(18)diamOnι0−s·χdiamOnι0≤d0,which implies that diamOnι0≤r0, and hence
(19)diamOι0=supndiamOnι0≤r0.
Hence, all Jungck sequences defined byY and Γ are bounded.
Now, we shall prove that our Jungck sequence is a Cauchy sequence. Letm>n be positive integers. Then, Yιn,Yιm∈Om−n+1ιn. Using (15) (with l=n) and (19), we get
(20)μYιn,Yιm≤diamOm−n+1ιn≤χndiamOm+1ι0≤χnr0→0,as m,n→∞. Since YΔ⊆ΓΔ, and ΓΔ is complete, it follows that Yιn is convergent. Let κ∈Δ be its limit.
Clearly,κ∈ΓΔ. So, there is z∈Δ such that Γz=κ. Let us prove that Yz is also equal to κ. By (8), we have
(21)μYιn,Yz≤χmaxμΓιn,Γz,μΓιn,Yιn,μΓz,Yz,μΓιn,Yz,μΓz,Yιn=χmaxμYιn−1,κ,μYιn−1,Yιn,μκ,Yz,μYιn−1,Yz,μκ,Yιn.
Ifn→∞, then the left-hand side in the previous inequality tends to μκ,Yz, and the first, the second, and the fifth argument of max tend to μκ,κ=0, whereas the third and the fourth tend to μκ,Yz. Thus, we have
(22)μκ,Yz≤χμκ,Yz,which is impossible, unless μκ,Yz=0.
Finally, we prove that the point of coincidence is unique. Suppose that there is two points of coincidenceκ and κ′ obtained by z and z′, i.e., Yz=Γz=κ and Yz′=Γz′=κ′. Then, by (8) we have
(23)μκ,κ′=μYz,Yz′≤χmaxμΓz,Γz′,μΓz,Yz,μΓz′,Yz′,μΓz,Yz′,μΓz′,Yz=χmaxμκ,κ′,0,0,μκ,κ′,μκ′,κ=χμκ,κ′<μκ,κ′,
unlessμκ,κ′=0. Since every Jungck sequence converges to some point of coincidence, and the point of coincidence is unique, it follows that all Jungck sequences converge to the same limit.
LetΔ=Λ and let Y, Γ be weakly compatible. By Lemma 3, we get that κ=z which is the unique common fixed point of Y and Γ.
The previous theorem extended earlier results for nonlinear contractions on metric spaces obtained by Danes [3], Ivanov [4], Aranđelović et al. [5], and Bessenyei [6] and common fixed point results of Das and Naik [8] and Di Bari and Vetro [9]. It also generalizes the fixed point theorem of Aleksić et al. [7] which proved the fixed point theorems for quasi-contractive mappings on b-metric spaces, defined by linear quasi-contractive conditions.Example 1.
LetΔ=Λ=0,1,2,3 be equipped with the following b-metric μ:X×X→ℝ+ by μι,κ=ι−κ2.
It is easy to see thatΔ=Λ,μ is a complete b-metric space with s=2.
Define the self-mapsY and Γ by
(24)Y=01230010,Γ=01230031.
We see thatΓΔ⊇YΔ.
Defineχi:0,∞→0,∞ by χt=t−sinh−1t. One can easily check that Y satisfies condition (8). Indeed, we have some cases as follows:
(1)
ι,κ=0,2. Then,
(25)μYι,Yκ=Y0−Y22=1≤9−sinh−19≤χ1μΓι,Γκ≤maxχ1μΓι,Γκ,χ2μΓι,Yι,χ3μΓκ,Yκ,χ4μΓι,Yκ,χ5μYι,Γκ.(2)
ι,κ=1,2. Then,
(26)μYι,Yκ=Y1−Y22=1≤9−sinh−19=χ1μΓι,Γκ≤maxχ1μΓι,Γκ,χ2μΓι,Yι,χ3μΓκ,Yκ,χ4μΓι,Yκ,χ5μYι,Γκ.(3)
ι,κ=3,2. Then,
(27)μYι,Yκ=Y1−Y22=1≤4−sinh−14=χ1μΓι,Γκ≤maxχ1μΓι,Γκ,χ2μΓι,Yι,χ3μΓκ,Yκ,χ4μΓι,Yκ,χ5μYι,Γκ.
Thus, all the conditions of Theorem10 are satisfied, and hence, Y and Γ have a common fixed point. Indeed, 0 is the unique common fixed point of Y and Γ.


## 5. Application

The existence of the solution for the following integral equation is the main purpose in this section.(28)σι=fι,∫0ριgι,κ,σρκdκ,whereι∈0,∞.We will ensure such an existence by applying Theorem10.LetBC0,∞ be the space of all real, bounded and continuous functions on the interval 0,∞. We endow it with the b-metric
(29)dι,κ=supιt−κtp:t∈0,∞,where p≥1.Theorem 11.
Suppose that the following assumptions are satisfied:(i)
ρ,ϱ:0,∞→0,∞ are continuous functions so that
(30)Λp=supϱt:t∈0,∞<1,(ii)
The functionf:0,∞×R→R is continuous so that
(31)fι,σ1−fι,σ2≤σ1−σ2,(iii)
For allι∈0,∞ and σi∈ℝ(32)gι,κ,σ1ρκ−gι,κ,σ2ρκ≤σ1ρκ−σ2ρκ,where g:0,∞2×R→R is continuous.(iv)
M=maxfι,0:ι∈0,∞<∞ and G=supgι,κ,0:ι∈0,∞<∞
Then, the integral equation (28) admits at least one solution in the space BC0,∞.Proof.
Let us consider the operatorY:BC0,∞→BC0,∞ defined by
(33)Yσι=fι,∫0ριgι,κ,σρκdκ.
In view of the given assumptions, we infer that the functionYσ is continuous for arbitrarily σ∈BC0,∞. Now, we show that Yσ is bounded in BC0,∞. As
(34)Yσι=fι,∫0ριgι,κ,σρκdκ≤fι,∫0ριgι,κ,σρκdκ−fι,0+fι,0,we have
(35)fι,∫0ριgι,κ,σρκdκ−fι,0≤∫0ριgι,κ,σρκdκ≤Λσ+ΛG.
Thus,(36)fι,∫0ριgι,κ,σρκdκ−fι,0≤Λσ+ΛG.
From the above calculations, we have(37)Yσι≤Λσ+ΛG+M.
Due to the above inequality, the functionY is bounded.
Now, we show thatY satisfies all the conditions of Theorem 10. Let σ1,σ2 be some elements of BC0,∞. Then, we have
(38)Yσ1ι−Yσ2ιp≤fι,∫0ριgι,κ,σ1ρκdκ−fι,∫0ριgι,κ,σ2ρκdκp≤∫0ριgι,κ,σ1ρκdκ−∫0ριgι,κ,σ2ρκdκp≤∫0ρι1qdκ1/q∫0ριgι,κ,σ1ρκ−gι,κ,σ2ρκpdκ1/pp≤ριp/q∫0ριχ1σ1ρκdκ−σ2ρκpdκ1/pp≤ριp/q+1χ1dσ1,σ2≤Λpχ1dσ1,σ2≤Mσ1,σ2,
whereMσ1,σ2 is defined by
(39)Mσ1,σ2=maxχ1μι,κ,χ2μι,Yι,χ3μκ,Yκ,χ4μι,Yκ,χ5μYι,κ.
Thus, we obtain that(40)μYσ1,Yσ2≤Mσ1,σ2.
Using Theorem10, we obtain that the operator Y admits a fixed point. Thus, the functional integral equation (28) admits at least one solution in BC0,∞.



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