# Common Fixed-Point Theorems of Generalizedψ,φ−Weakly Contractive Mappings in b−Metric-Like Spaces and Application

Authors: Hongyan Guan; Jianju Li
Journal: Journal of Mathematics
 (2021)

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## Abstract

In this paper, we prove some common fixed-point theorems of generalizedψ,φ−weakly contractive mappings in b−metric-like spaces. We also give two examples to support our results. Meanwhile, we present an application to the existence of solutions for a system of integral equations by means of our results.


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## Body

## 1. Introduction

The Banach contraction mapping theorem [1] popularly known as Banach contraction mapping principle is a rewarding result in fixed-point theory. It has widespread applications in both pure and applied mathematics. This celebrated principle has been generalized by several authors. Recently, Saleem et al. [2] obtained fixed-point results of Suzuki-type generalized multivalued f,θ,L-almost contractions and coincidence and common fixed-point results of Suzuki-type generalized multivalued f,θ,L-almost contraction mapping in the setting of metric spaces. In 2019, Li et al. [3] defined a new contractive-type mapping called Zθ-contraction and proved some fixed-point and Suzuki-type fixed-point results in the context of complete metric spaces. Another extension idea is the promotion of spaces. In 1993, Czerwik [4] introduced the concept of the b−metric space, and he also obtained some fixed-point theorems of contractive mappings in b−metric space. Since then, some papers have generalized the fixed-point of different contractive conditions in b−metric space. For example, Aydi et al. [5] proved common fixed-point results for weak ϕ−contractions on b−metric space, and Abbas et al. [6] also proved common fixed-point theorems of four mappings in b−metric space. In [7], Hussain et al. established some best proximity point and coupled best proximity point results in the context of complete b-metric spaces based on the concepts of α-proximal admissible mappings and simulation function. Lately, Lael et al. [8] replied to an open problem related to a b-metric version of Banach’s fixed-point theorem and obtained some new fixed-point theorems for multivalued mappings in b-metric spaces.Inspired by Czerwik’s work, some researchers had studied the fixed-point theorems of various new types of contractive conditions in the generalized metric space andb−metric space. In 2011, Hussain and Shah in [9] introduced the notion of cone b−metric spaces, which means that it is a generalization of b−metric spaces and cone metric spaces. They also considered topological properties of cone b−metric spaces and results on KKM mappings in the setting of cone b−metric spaces. Fernandez et al. [10] introduced the concept of Nb-cone metric spaces over a Banach algebra as a generalization of N-cone metric spaces over a Banach algebra and b-metric spaces and studied some coupled common fixed-point theorems for generalized Lipschitz mappings in this framework. In 2019, Kanwal et al. [11] generalized Nadler’s theorem in weak partial b-metric space by using weak partial Hausdorff b-metric spaces. In 2020, Abbas et al. [12] introduced the concepts of ψ-contraction and monotone ψ-contraction correspondence in fuzzy b-metric spaces and obtained fixed-point results for these contractive mappings. Ansari et al. [13] introduced the concept of inverse C-class function in G-metric setting and established some fixed-point theorems. Recently, Saleem et al. [14] proved some new fixed-point theorems, coincidence point theorems, and common fixed-point theorems for multivalued F-contractions involving a binary relation that is not necessarily a partial order, in the context of generalized metric spaces (in the sense of Jleli and Samet).The concept ofφ−contractive mappings was introduced by Rhoades [15]. Afterwards, some researchers introduced a few φ− and ψ,φ−weakly contractive conditions and discussed the existence of fixed and common fixed-point for these mappings [16–18]. In particular, Aghajani et al. [19] presented several common fixed-point results of generalized weak contractive mappings in partially ordered b–metric spaces.Our main concern is to study common fixed-point results involving generalizedψ,φ–weakly contractive conditions in b−metric-like space. Furthermore, we apply the given results to obtain existence of solutions of integral equations.


## 2. Preliminaries

Throughout this paper, letℕ denote the set of all positive integers, ℝ+=0,+∞ and ℝ=−∞,+∞. Put(1)Ψ=ψ:0,+∞⟶0,+∞is an increasing and continuous function,Φ=φ:0,+∞⟶0,+∞is an increasing and continuous function and,φt=0if and only ift=0.In order to get our main results, we introduce some definitions and lemmas as follows.Definition 1.
(see [4]). Let X be a nonempty set and s≥1 be a given real number. A mapping d:X×X⟶0,+∞ is said to be a b−metric if and only if, for all x,y,z∈X, the following conditions are satisfied:(1)
dx,y=0 if and only if x=y(2)
dx,y=dy,x(3)
dx,y≤sdx,z+dy,zThe pairX,d is called a b−metric space with parameter s≥1.In general, the class ofb−metric space is effectively larger than that of metric space, since a b−metric is a metric with s=1. We can find several examples of b−metric spaces which are not metric spaces (see [20]).Definition 2.
(see [21]). Let X be a nonempty set and s≥1 be a given real number. A mapping d:X×X⟶0,+∞ is said to be a b−metric-like if and only if, for all x,y,z∈X, the following conditions are satisfied:(1)
dx,y=0 implies x=y(2)
dx,y=dy,x(3)
dx,y≤sdx,z+dy,zThe pairX,d is called a b−metric-like space with parameter s≥1.Remark 1.
We should note that in ab−metric-like space X,d, if x,y∈X and dx,y=0, then x=y. But the converse need not be true, and dx,x may be positive for x∈X.Example 1.
LetX=ℝ+, and let the mapping d:X×X⟶ℝ+ be defined by dx,y=x+y2 for all x,y∈X. Then, X,d is a b−metric-like space with parameter s≥2.Proof.
We can infer from the convexity of the functionfx=x2 (x>0) that a+b2≤2a2+b2 holds. Then, for x,y,z∈X, we have(2)dx,y=x+y2≤x+z+y+z2≤2x+z2+y+z2=2dx,z+2dz,y.
In this case,X,d is a b−metric-like space with parameter s≥2.Definition 3.
(see [21]). Let X,d be a b−metric-like space with parameter s≥1 and xn be a sequence in X.(1)
The sequencexn is said to be convergent to x if limn⟶+∞dxn,x=dx,x.(2)
The sequencexn is said to be a Cauchy sequence if and only if limn⟶+∞dxn,xm exists and is finite.(3)
X,d is said to be complete if for each Cauchy sequence xn in X, there exists an x∈X such that limn,m⟶+∞dxn,xm=limn⟶+∞dxn,x=dx,x.Definition 4.
(see [22]). Let f and g be two self-mappings on a nonempty set X. If w=fx=gx, for some x∈X, then x is said to be the coincidence point of f and g, where w is called the point of coincidence of f and g. Let Cf,g denote the set of all coincidence points of f and g.Definition 5.
(see [22]). Let f and g be two self-mappings defined on a nonempty set X. Then, f and g are said to be weakly compatible if they commute at every coincidence point, that is, fx=gx⇒fgx=gfx for every x∈Cf,g.Lemma 1.
(see [21]). Let X,d be a b−metric-like space with s≥1. We assume that xn and yn are convergent to x and y, respectively. Then, we have(3)1s2dx,y−1sdx,x−dy,y≤liminfn⟶+∞dxn,yn≤limsupn⟶+∞dxn,yn≤sdx,x+s2dy,y+s2dx,y.
In particular, ifdx,y=0, then we have limn⟶+∞dxn,yn=0. Moreover, for each z∈X, we have(4)1sdx,z−dx,x≤liminfn⟶+∞dxn,z≤limsupn⟶+∞dxn,z,≤sdx,z+sdx,x.
In particular, ifdx,x=0, then(5)1sdx,z≤liminfn⟶+∞dxn,z≤limsupn⟶+∞dxn,z≤sdx,z.Lemma 2.
(see [23]). Let X,d be a b−metric-like space with s≥1. Then,(1)
Ifdx,y=0, then dx,x=dy,y=0(2)
Ifxn is a sequence such that limn⟶+∞dxn,xn+1=0, then we have limn⟶+∞dxn,xn=limn⟶+∞dxn+1,xn+1=0(3)
Ifx≠y, then dx,y>0


## 3. Main Results

In this section, we will show the existence and uniqueness of common fixed-point for generalizedψ,φ−weakly contractive mappings in complete b−metric-like space. Meanwhile, we give two examples to support our results.Theorem 1.
LetX,d be a complete b−metric-like space with parameter s≥1 and let f,g:X⟶X be given self-mappings satisfying fX⊂gX where gX is a closed subset of X. If there are functions ψ∈Ψ and φ∈Φ such that(6)ψs2dfx,fy2≤ψN1x,y−φM1x,y,where(7)N1x,y=maxdfx,gx2,dgx,gy2,dfy,gy2,dfx,gxdfx,fy,dfx,gxdgx,gy,M1x,y=maxdfy,gy2,dfx,gy2,dgx,gy2,dfx,gx21+dgx,gy21+dfx,gy2,then f and g have a unique coincidence point in X. Moreover, f and g have a unique common fixed-point provided that f and g are weakly compatible.Proof.
Letx0∈X. As fX⊂gX, there exists x1∈X such that fx0=gx1. Now we define the sequences xn and yn in X by yn=fxn=gxn+1 for all n∈ℕ. If yn=yn+1 for some n∈ℕ, then we have yn=yn+1=fxn+1=gxn+1 and f and g have a coincidence point. Without loss of generality, we assume that yn≠yn+1 (by Lemma 2, we know that dyn,yn+1>0) for all n∈ℕ. Applying (6) with x=xn and y=xn+1, we obtain(8)ψs2dyn,yn+12=ψs2dfxn,fxn+12≤ψN1xn,xn+1−φM1xn,xn+1,where(9)N1xn,xn+1=maxdyn,yn−12,dyn−1,yn2,dyn+1,yn2,dyn,yn−1dyn,yn+1,dyn,yn−12,(10)M1xn,xn+1=maxdyn+1,yn2,dyn,yn2,dyn−1,yn2,dyn,yn−121+dyn−1,yn21+dyn,yn2.
Ifdyn,yn+1≥dyn,yn−1>0, for some n∈ℕ, in view of (9) and (10), we have(11)N1xn,xn+1=dyn,yn+12,M1xn,xn+1≥dyn,yn+12.
It follows from inequality (8) and the above inequalities that(12)ψdyn,yn+12≤ψs2dyn,yn+12≤ψN1xn,xn+1−φM1xn,xn+1≤ψdyn,yn+12−φdyn,yn+12,which implies φdyn,yn+12=0, that is, yn=yn+1, a contradiction. Hence, dyn,yn+1<dyn,yn−1 and dyn,yn+1 is a nonincreasing sequence, and so there exists r≥0 such that limn⟶+∞dyn,yn+1=r.
By virtue of (9) and (10), we have(13)N1xn,xn+1=dyn,yn−12,M1xn,xn+1=dyn,yn−12.
It follows that(14)ψdyn,yn+12≤ψN1xn,xn+1−φM1xn,xn+1≤ψdyn,yn−12−φdyn,yn−12.
Now suppose thatr>0. By taking the limit as n⟶+∞ in (12), we have ψr2≤ψr2−φr2, a contradiction. This yields that(15)limn⟶+∞dyn,yn+1=r=0.
Now we shall prove thatlimn,m⟶+∞dyn,ym=0. Suppose on the contrary that limn,m⟶+∞dyn,ym≠0. It follows that there exists ε>0 for which one can find sequences ymk and ynk of yn where nk is the smallest index for which nk>mk>k, ε≤dymk,ynk, and dymk,ynk−1<ε.
In view of the triangle inequality inb−metric-like space, we get(16)ε2≤dymk,ynk2≤sdymk,ynk−1+sdynk−1,ynk2=s2dymk,ynk−12+s2dynk−1,ynk2+2s2dymk,ynk−1dynk−1,ynk≤s2ε2+s2dynk−1,ynk2+2s2dymk,ynk−1dynk−1,ynk.
Using equality (15) and taking the upper limit as k⟶+∞ in the above inequality, we obtain(17)ε2≤limsupk⟶+∞dymk,ynk2≤s2ε2.
As the same arguments, we deduce the following results:(18)ε2≤dymk,ynk2≤sdymk,ynk−1+sdynk−1,ynk2=s2dymk,ynk−12+s2dynk−1,ynk2+2s2dymk,ynk−1dynk−1,ynk,(19)dymk,ynk2≤sdymk,ymk−1+sdymk−1,ynk2=s2dymk,ymk−12+s2dymk−1,ynk2+2s2dymk,ymk−1dymk−1,ynk,(20)dymk−1,ynk2≤sdymk−1,ymk+sdymk,ynk2=s2dymk−1,ymk2+s2dymk,ynk2+2s2dymk−1,ymkdymk,ynk.
In view of (18), we have(21)ε2s2≤limsupk⟶+∞dymk,ynk−12≤ε2.
Using (19) and (20), we obtain(22)ε2s2≤limsupk⟶+∞dymk−1,ynk2≤s4ε2.
Similarly, we deduce that(23)dymk−1,ynk−12≤sdymk−1,ymk+sdymk,ynk−12=s2dymk−1,ymk2+s2dymk,ynk−12+2s2dymk−1,ymkdymk,ynk−1,dymk,ynk2≤sdymk,ymk−1+sdymk−1,ynk2=s2dymk,ymk−12+s2dymk−1,ynk2+2s2dymk,ymk−1dymk−1,ynk≤s2dymk,ymk−12+s2sdymk−1,ynk−1+sdynk−1,ynk2+2s2dymk,ymk−1sdymk−1,ynk−1+sdynk−1,ynk.
It follows that(24)ε2s4≤limsupk⟶+∞dymk−1,ynk−12≤s2ε2.
Through the definition ofN1x,y, we have(25)N1xmk,xnk=maxdymk,ymk−12,dymk−1,ynk−12,dynk,ynk−12,dymk,ymk−1dymk,ynk,dymk,ymk−1dymk−1,ynk−1,which yields that(26)limsupk⟶+∞N1xmk,xnk≤max0,s2ε2,0,0,0=ε2s2.
Also,(27)M1xmk,xnk=maxdynk,ynk−12,dymk,ynk−12,dymk−1,ynk−12,dymk,ymk−121+dymk−1,ynk−121+dymk,ynk−12.
It is easy to show that(28)liminfk⟶+∞M1xmk,xnk≥max0,ε2s2,ε2s4,0≥ε2s4.
Applying (6) with x=xmk and y=xnk, we get(29)ψdymk,ynk2≤ψs2dymk,ynk2≤ψN1xmk,xnk−φM1xmk,xnk.
In light of (26), one can obtain(30)ψs2ε2≤ψs2limsupk⟶+∞dfxmk,fxnk2≤ψlimsupk⟶+∞N1xmk,xnk−φliminfk⟶+∞M1xmk,xnk≤ψs2ε2−φliminfk⟶+∞M1xmk,xnk,which implies that(31)liminfk⟶+∞M1xmk,xnk=0,a contradiction to (28). It follows that yn is a Cauchy sequence in X and limn,m⟶+∞dyn,ym=0. Since X is complete b−metric-like space, there exists u∈X such that(32)limn⟶+∞dyn,u=limn⟶+∞dfxn,u=limn⟶+∞dgxn+1,u=limn,m⟶+∞dyn,ym=du,u=0.
Furthermore, we haveu∈gX since gX is closed. It follows that one can choose a z∈X such that u=gz, and one can write (32) as(33)limn⟶+∞dyn,gz=limn⟶+∞dfxn,gz=limn⟶+∞dgxn+1,gz=0.
Iffz≠gz, taking x=xnk and y=z in contractive condition (6), we get(34)ψs2dynk,fz2=ψs2dfxnk,fz2≤ψN1xnk,z−φM1xnk,z,where(35)N1xnk,z=maxdynk,ynk−12,dynk−1,gz2,dfz,gz2,dynk,ynk−1dynk,fz,dynk,ynk−1dynk−1,gz,M1xnk,z=maxdfz,gz2,dynk,gz2,dynk−1,gz2,dynk,ynk−121+dynk−1,gz21+dynk,gz2,and then we obtain(36)limsupk⟶+∞N1xnk,z=max0,0,dgz,fz2,0,0=dgz,fz2,liminfk⟶+∞M1xnk,z=maxdgz,fz2,0,0,0=dgz,fz2.
Taking the upper limit ask⟶+∞ in (34),(37)ψdgz,fz2=ψs2⋅1s2dgz,fz2≤ψs2limsupk⟶+∞dfxnk,fz2≤ψlimsupk⟶+∞N1xnk,z−φliminfk⟶+∞M1xnk,z=ψdgz,fz2−φdgz,fz2,which implies that ϕdfz,gz2=0.
It follows thatdfz,gz=0. That is, fz=gz. Therefore, u=fz=gz is a point of coincidence for f and g. We also conclude that the point of coincidence is unique. Assume on the contrary that there exist z,z′∈Cf,g and z≠z′; applying (6) with x=z and y=z′, we obtain that(38)ψdfz,fz′2≤ψs2dfz,fz′2≤ψN1z,z′−φM1z,z′≤ψdfz,fz′2−φdfz,fz2.
Hence,fz=fz′. That is, the point of coincidence is unique. Considering the weak compatibility of f and g, it can be shown that z is a unique common fixed-point. This completes the proof.Theorem 2.
LetX,d be a complete b−metric-like space with parameter s≥1 and let f,g:X⟶X be given self-mappings satisfying fX⊂gX where gX is a closed subset of X. If there are functions ψ∈Ψ and φ∈Φ such that(39)ψs2dfx,fy2≤ψN2x,y−φM2x,y,where(40)N2x,y=maxdfx,fydgx,gy,dgx,gy2,dfy,gy2+dfx,gy21+4s2,M2x,y=maxdfy,gy2,dfx,gy2,dgx,gy2,dfx,gx21+dgx,gy21+dfx,gy2,dgx,gy21+dgx,gy21+dfx,gx2,then f and g have a unique coincidence point in X. Moreover, f and g have a unique common fixed-point provided that f and g are weakly compatible.Proof.
It is the same as the proof of Theorem1, and we also define the sequences xn and yn in X by yn=fxn=gxn+1 for all n∈ℕ. We also suppose that yn≠yn+1 for each n∈ℕ, and it follows from (39) that(41)ψs2dyn,yn+12=ψs2dfxn,fxn+12≤ψN2xn,xn+1−φM2xn,xn+1,where(42)N2xn,xn+1=maxdyn,yn+1dyn−1,yn,dyn−1,yn2,dyn+1,yn2+dyn,yn21+4s2,(43)M2xn,xn+1=maxdyn+1,yn2,dyn,yn2,dyn−1,yn2,dyn,yn−121+dyn−1,yn21+dyn,yn2,dyn−1,yn21+dyn−1,yn21+dyn,yn−12.
If we assume that for somen∈ℕ,(44)dyn,yn+1≥dyn−1,yn>0,then from inequalities (42) and (43), we get that(45)N2xn,xn+1≤dyn+1,yn2,M2xn,xn+1≥dyn+1,yn2.
In view of (41), we have the following inequality:(46)ψdyn,yn+12≤ψs2dyn,yn+12≤ψN2xn,xn+1−φM2xn,xn+1≤ψdyn,yn+12−φdyn,yn+12,which gives that dyn,yn+1=0, a contradiction to dyn,yn+1>0. It follows that dyn,yn+1<dyn,yn−1. Hence, dyn,yn+1 is a nonincreasing sequence. Consequently, the limit of the sequence is a nonnegative number, say r≥0. That is, limn⟶+∞dyn,yn+1=r.
According to (42) and (43), we have(47)N2xn,xn+1≤dyn,yn−12,M2xn,xn+1≥dyn,yn−12.
So,(48)ψdyn,yn+12≤ψN2xn,xn+1−φM2xn,xn+1≤ψdyn,yn−12−φdyn,yn−12.
Ifr>0, then letting n⟶+∞ in above inequality, we obtain that ψr2=ψr2−φr2 which implies that r=0, i.e., limn⟶+∞dyn,yn+1=0.
Now we prove thatlimn,m⟶+∞dyn,ym=0. If not, as the proof of Theorem 1, there exists ε>0 for which one can find sequences ymk and ynk of yn so that nk is the smallest index for which nk>mk>k, and the following inequalities hold:(49)ε≤limsupk⟶+∞dymk,ynk≤sε,εs≤limsupk⟶+∞dymk,ynk−1≤ε,εs≤limsupk⟶+∞dymk−1,ynk≤s2ε,εs2≤limsupk⟶+∞dymk−1,ynk−1≤sε.
We deduce the following equations according to the definitions ofN2x,y and M2x,y:(50)N2xmk,xnk=maxdymk,ynkdymk−1,ynk−1,dymk−1,ynk−12,dynk,ynk−12+dymk,ynk−121+4s2,M2xmk,xnk=maxdynk,ynk−12,dymk,ynk−12,dymk−1,ynk−12,dymk,ymk−121+dymk−1,ynk−121+dymk,ynk−12,dymk−1,ynk−121+dymk−1,ynk−121+dymk,ymk−12,
using (49), one can obtain that(51)limsupk⟶+∞N2xmk,xnk≤maxs2ε2,s2ε2,ε21+4s2=s2ε2,(52)liminfk⟶+∞M2xmk,xnk≥max0,ε2s2,ε2s4,0,ε2s41+ε2s4≥ε2s4.
Takingx=xmk and y=xnk in (39), we get(53)ψdymk,ynk2≤ψs2dymk,ynk2≤ψN2xmk,xnk−φM2xmk,xnk.
Therefore, we have(54)ψs2ε2≤ψs2limsupk⟶+∞dfxmk,fxnk2≤ψlimsupk⟶+∞N2xmk,xnk−φliminfk⟶+∞M2xmk,xnk≤ψs2ε2−φliminfk⟶+∞M2xmk,xnk,and we conclude that liminfk⟶+∞M2xmk,xnk=0 which gives a contradiction to (52). Hence, limn,m⟶+∞dyn,ym=0.
The completeness ofX ensures that there exists u∈X such that(55)limn⟶+∞dyn,u=limn⟶+∞dfxn,u=limn⟶+∞dgxn+1,u=limn,m⟶+∞dyn,ym=du,u=0.
In view of the hypothesisgX is closed, we obtain that u∈gX. It follows that one can choose z∈X such that u=gz, and we write the above equality as(56)limn⟶+∞dyn,gz=limn⟶+∞dfxn,gz=limn⟶+∞dgxn+1,gz=0.
Iffz≠gz, putting x=xnk and y=z into contractive condition (39), we have(57)ψs2dfxnk,fz2≤ψN2xnk,z−φM2xnk,z,where(58)N2xnk,z=maxdynk,fzdynk−1,gz,dynk−1,gz2,dfz,gz2+dynk,gz21+4s2,M2xnk,z=maxdfz,gz2,dynk,gz2,dynk−1,gz2,dynk,ynk−121+dynk−1,gz21+dynk,gz2,dynk−1,gz21+dynk−1,gz21+dynk,ynk−1.
Consequently, we get(59)limsupk⟶+∞N2xnk,z=max0,0,dfz,gz21+4s2≤dfz,gz2,liminfk⟶+∞M2xnk,z=maxdfz,gz2,0,0,0,0=dfz,gz2.
Taking the upper limit ask⟶+∞ in (57), we have(60)ψdgz,fz2=ψs2⋅1s2dgz,fz2≤ψs2limsupk⟶+∞dfxnk,fz2≤ψlimsupk⟶+∞N2xnk,z−φliminfk⟶+∞M2xnk,z≤ψdgz,fz2−φdgz,fz2,which implies that dfz,gz=0. That is, u=fz=gz is a point of coincidence for f and g. Using the same technique in the proof of Theorem 1, it can be proved that z is a unique common fixed-point. This completes the proof.The following examples support Theorems1 and 2.Example 2.
LetX=0,1 be endowed with the b−metric-like dx,y=x+y2 for all x,y∈X and s=2. Define mappings f,g:X⟶X by fx=x/64 and gx=x/2.
The control functionsψ,φ:0,+∞⟶0,+∞ are defined as ψt=5t/4,φt=48545t/87846 for all t∈0,+∞. It is clear that fX⊂gX and gX is closed. For all x,y∈X, we have (61)ψs2dfx,fy2=ψ4⋅x64+y644=54⋅4x64+y644=5644x+y4,ψN1x,y≥ψdgx,gy2=54⋅x2+y24=564x+y4,φM1x,y=maxy64+y24,x64+y24,x2+y24,x/64+x/241+x/2+y/241+x/64+y/24.
It follows that(62)φM1x,y≤φ2⋅33x64+33y644=5⋅643−52⋅334⋅2⋅33644x+y4=1310715644x+y4.
It is easy to see that(63)ψs2dfx,fy2=ψdgx,gy2−φ2⋅33644x+y4≤ψN1x,y−φM1x,y.
Therefore, the conditions of Theorem1 are satisfied. It is obvious that 0 is the unique common fixed-point of f and g.Example 3.
LetX=−1,1 and dx,y=x+y2 for all x,y∈X, so X,d is a b−metric-like space with parameter s=2. Let f,g:X⟶X be defined by the formulas(64)fx=0,x∈−1,1,18,x=1,gx=x22.
The control functionsψ,φ:0,+∞⟶0,+∞ are defined as ψt=bt and φt=b−1t, for all t∈0,+∞, where 1<b≤2048/2045.
Now we consider four cases:Case 1. x≠1,y≠1. It is clear that(65)ψs2dfx,fy2=ψ0=0,ψN2x,y=bN2x,y≥bdgx,gy2=bx22+y224,φM2x,y=b−1M2x,y≤b−1⋅2x22+y224,so we have(66)ψs2dfx,fy2=0≤bx22+y224−2b−1x22+y224≤ψN2x,y−φM2x,y.Case 2. x=1,y≠1. It follows that(67)ψs2dfx,fy2=4b184=b1024,ψN2x,y=bN2x,y≥bdfx,fydgx,gy=b6412+y222≥164⋅b4=b256,φM2x,y=b−1M2x,y≤2b−1,and we obtain that(68)ψs2dfx,fy2=b1024≤b256−2b−1≤ψN2x,y−φM2x,y.Case 3. x≠1,y=1. By concise calculation, we get(69)ψs2dfx,fy2=4b184=b1024,ψN2x,y=bN2x,y≥bdfx,fydgx,gy=b6412+x222≥164⋅b4=b256,φM2x,y=b−1M2x,y≤2b−1.It follows that(70)ψs2dfx,fy2=b1024≤b256−2b−1≤ψN2x,y−φM2x,y.Case 4. x=1,y=1. It is easy to see that(71)ψs2dfx,fy2=4b18+184=b64,ψN2x,y=bN2x,y≥bdfx,fydgx,gy=b18+182⋅12+122=b16φM2x,y=b−1M2x,y≤2b−1.Obviously, for allx,y∈−1,1, we obtain(72)ψs2dfx,fy2=b64≤b16−2b−1≤ψN2x,y−φM2x,y.It follows from Theorem2 that f and g have a unique common fixed-point in X. It is easy to show that 0 is the unique common fixed-point of f and g.According to Theorems1 and 2, one can get the following results.Corollary 1.
LetX,d be a complete b−metric-like space with constant s≥1 and let f,g:X⟶X be given self-mappings satisfying fX⊂gX where gX is a closed subset of X. If the following condition is satisfied:(73)s2dfx,fy2≤N1x,y−Ldgx,gy2,where L ∈ (0,1) is a constant and then f and g have a unique coincidence point in X. Moreover, f and g have a unique common fixed-point provided that f and g are weakly compatible.Corollary 2.
LetX,d be a complete b−metric-like space with constant s≥1 and let f,g:X⟶X be given self-mappings satisfying fX⊂gX where gX is a closed subset of X. If the following condition is satisfied:(74)s2dfx,fy2≤N2x,y−M2x,y,then f and g have a unique coincidence point in X. Moreover, if f and g are weakly compatible, then f and g have a unique common fixed-point.


## 4. Application

In this section, we will use Corollary1 to show that there is a solution to the following system of integral equations:(75)xt=∫0tKr,xrdr,xt=∫0txrdr.LetX=C0,T be the set of real continuous functions defined on 0,T for T>0. Define the b−metric-like mapping d: X×X⟶0,+∞ by(76)dx,y=maxt∈0,Txt+ytm/2,for all x,y∈X, where m>2. It is evident that X,d with s=2m/2−1 is a complete b−metric-like space. Consider the mappings f,g:X⟶X by(77)fxt=∫0tKr,xrdr,gxt=∫0txrdr.Theorem 3.
Consider the system of integral equations (75) and suppose that the following conditions hold:(i)
K:0,T×ℝ⟶ℝ+ is continuous(ii)
IfKr,xr=xr for all r∈0,T, then we have Kr,∫0txωdω=∫0tKω,xωdω(iii)
There exists a continuous functionγ:0,T⟶0,1 such that Kr,xr≤γrxr, for all r∈0,T,x,y∈X(iv)
There exists constantL such that for all r∈0,T,(78)supt∈0,Tγt≤1−Ls21/m.Then, the system of integral equations (75) has a unique solution in x∈X.Proof.
Obviously, by condition (ii),f,g are weakly compatible. Let x,y∈X; from conditions (i), (iii), and (iv), for all t∈0,T, we have(79)s2fxt+fytm=s2∫0tKr,xrdr+∫0tKr,yrdrm≤s2∫0tKr,xr+Kr,yrdrm≤s2∫0tγrxr+yrdrm≤s2supt∈0,Tγtm⋅dgxt,gyt2≤1−Ldgxt,gyt2≤N1xt,yt−Ldgxt,gyt2,which implies that(80)s2dfxt,fyt2≤N1xt,yt−Ldgxt,gyt2.
Consequently, lettingψx=x and φx=Lx, all of the conditions of Corollary 1 are satisfied. As a result, the mappings f,g have a unique common fixed-point in X, which is the solution of the system of integral equations (75).


## 5. Conclusions

In this paper, we introduced new generalizedψ,φ−weakly contractive mappings and obtained common fixed-point theorems in the framework of b−metric-like space. Further we provided examples that elaborated the useability of our results. As an application of our result, we obtained a solution of the system of integral equations.



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