# Common Fixed Point Results for Generalizedg−αsp,ψ,φ Contractive Mappings with Applications

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

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## Abstract

In this paper, we introduce a new class ofg−αsp−admissible mappings and prove some common fixed point theorems involving this new class of mappings which satisfy generalized contractive conditions in the framework of b−metric spaces. We also provide two examples to show the applicability and validity of our results. Meanwhile, we present an application to the existence of solutions to an integral equation by means of one of our results.


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## Body

## 1. Introduction

The Banach contraction principle [1] is one of the essential pillars of the theory of metric fixed points. Many authors have obtained generalizations, extensions, and applications of their findings by investigating the Banach contraction principle in many directions. One of the most popular and interesting topics among them is the study of new classes of spaces and their fundamental properties.Czerwik [2] introduced the concept of b−metric space and proved some fixed point theorems of contractive mappings in b−metric space. Subsequently, some authors have studied on the fixed point theorems of a various new type of contractive conditions in b−metric space. Aydi et al. in [3] proved common fixed point results for mappings satisfying a weak ϕ−contraction in b−metric spaces. Following the results of Berinde [4], Pacurar [5] obtained the existence and uniqueness of fixed point of ϕ−contractions, and Zada et al. [6] got fixed point results satisfying contractive conditions of rational type. In 2019, Hussain et al. studied the existence and uniqueness of a periodic common fixed point for pairs of mappings via rational type contraction in [7]. After that, authors obtained fixed point theorems for L−cyclic α,βs−contractions and cyclic α,β−ψ,ϕs−rational type contractions and discussed the existence of a unique solution to nonlinear fractional differential equations in [8, 9], respectively. Also using rational type contractive conditions, Hussain et al. [10] got the existence and uniqueness of common n−tupled fixed point for a pair of mappings. Using a contractive condition defined by means of a comparison function, [11] established results regarding the common fixed points of two mappings. In 2014, Abbas et al. obtained the results on common fixed points of four mappings in b−metric space in [12].To generalize the concept ofb−metric spaces, Hussain and Shah in [13] introduced the notion of a cone b−metric space, which means that it is a generalization of b−metric spaces and cone metric spaces; they considered topological properties of cone b−metric spaces and obtained some results on KKM mappings in the setting of cone b−metric spaces. In [14], some fixed point results for weakly contractive mappings in ordered partial metric space were obtained. Recently, Samet et al. [15] introduced the concept of α−admissible and α−ψ−contractive mappings and presented fixed point theorems for them. In [16], Jamal et al. used ψ,ϕ−weak contraction to generalize coincidence point results which are established in the context of partially ordered b−metric spaces. In [17, 18], Zoto et al. studied generalized αsp contractive mappings and α−ψ,ϕ−contractions in b−metric−like space. Recently, in [16], Jamal et al. used ψ,ϕ−weak contraction to generalize coincidence point results which are established in the context of partially ordered b−metric spaces. Abu-Donia et al. [19] proved the uniqueness and existence of the fixed points for five mappings from a complete intuitionistic fuzzy 3−metric space into itself under weak compatible of type α and asymptotically regular. For recent development on fixed point theory, we refer to [20–26].Motivated and inspired by Theorems 27 and 29 in [17], Theorem 3.13 in [18], and Theorem 2.1 in [20], in this paper, our purpose is to introduce the concept of g−αsp−admissible mappings and obtain a few common fixed point results involving generalized contractive conditions in the framework of b−metric space. Furthermore, we provide examples that elaborated the useability of our results. Meanwhile, we present an application to the existence of solutions to an integral equation by means of one of our results.


## 2. Preliminaries

First of all, we introduce some definitions as follows:Definition 1 (see [2]).
LetX 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:
(i)
dx,y=0 if and only if x=y(ii)
dx,y=dy,x(iii)
dx,y≤sdx,z+dy,z
Generally, we callX,d a b−metric space with parameter s≥1.Remark 2.
We should note that ab−metric space with s=1 is a metric space. We can find several examples of b−metric spaces which are not metric spaces. (see [24]).Example 3 (see [20]).
LetX,ρ be a metric space, and dx,y=ρx,yp, where p>1 is a real number
Then,dx,y is a b−metric space with s=2p−1.Definition 4 (see [12]).
LetX,d be a b−metric space with parameter s≥1. Then, a sequence xn in X is said to be:
(i)
b−convergent if and only if there exists x∈X such that dxn,x⟶0 as n⟶+∞(ii)
a Cauchy sequence if and only ifdxn,xm⟶0 when n,m⟶+∞
In general, ab−metric space is called complete if and only if each Cauchy sequence in this space is b−convergent.Definition 5 (see [21]).
Letf 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 6 (see [21]).
Letf 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.We need the following lemma to obtain our main results:Lemma 7 (see [20]).
LetX,d be a b−metric space with parameter s≥1. Assume that xn and yn are b−convergent to x and y, respectively. Then, we have
(1)1s2dx,y≤liminfn⟶+∞dxn,yn≤limsupn⟶+∞dxn,yn≤s2dx,y.
In particular, ifx=y, then we have limn⟶+∞dxn,yn=0. Moreover, for each z∈X, we have
(2)1sdx,z≤liminfn⟶+∞dxn,z≤limsupn⟶+∞dxn,z≤sdx,z.


## 3. Main Results

In this section, we will show the existence and uniqueness of common fixed point for generalizedg−αsp,ψ,φ contractive mappings in complete b−metric space. Meanwhile, we give two examples to support our results.Definition 8.
LetX,d be a b−metric space with parameter s≥1, and let f,g:X⟶X and α:X×X⟶0,+∞ be given mappings and p≥1 be an arbitrary constant. The mapping f:X⟶X is said to be g−αsp−admissible if, for all x,y∈X,αgx,gy≥sp implies αfx,fy≥sp.Remark 9.
(i)
Note that, forg=I, the definition reduces to an αsp-admissible mapping in a b−metric space(ii)
Fors=1, the definition reduces to the definition of an α−admissible mapping in a metric space
LetX,d be a complete b−metric space with parameter s≥1 and α:X×X⟶0,+∞ be a function. Then,
Hsp If xn is a sequence in X such that gxn⟶gx as n⟶+∞, then there exists a subsequence gxnk of gxn with αgxnk,gx≥sp for all k∈N
Usp For all u,v∈Cf,g, we have the condition of αgu,gv≥sp or αgv,gu≥sp.We know that contraction-type mappings are extended in several directions. Since Samet introduced the concept ofα−admissible mappings and α−ψ−contractive mapping, some papers have been published to study a series of generalizations. Afterwards, these classes of mappings are used under generalized weakly contractive conditions.We shall consider the contractive conditions in this section are constructed via auxiliary functions defined with the familiesΨ,Φ, respectively:
(3)Ψ=ψ:0,+∞⟶0,+∞isanincreasingandcontinuousfunction,Φ=φ:0,+∞⟶0,+∞isanincreasingandcontinuousfunctionandφt=0ifft=0.Now, we introduce the notion of rationalg−αsp,ψ,φ contraction in the setting of b−metric spaces.Definition 10.
LetX,d be a b−metric space with parameter s≥1, and let f,g:X⟶X be two self-mappings. Assume that α:X×X⟶0,+∞ and p≥1 is a constant. A mapping f is called a generalized g−αsp,ψ,φ contractive mapping, if there exist ψ∈Ψ,φ∈Φ such that
(4)ψαgx,gydfx,fy≤ψNx,y−φMx,y,for all x,y∈X with αx,y≥sp, where
(5)Nx,y=maxdgx,gy,dfx,gx,dfy,gy,dgx,fy+dfx,gy4s,dfx,gxdfy,gy1+dfx,fy,dfy,gy1+dfx,gx1+dgx,gy,Mx,y=maxdfx,gy,dgx,gy,dfx,gx,dfy,gy,dfx,gx1+dgx,gy1+dfx,gy,dfx,gx1+dfx,gx1+dfx,gy,dfx,gx1+dfy,gy1+dfx,gy.Example 11.
LetX=0,+∞ and dx,y=x−y2. Define mappings f,g:X⟶X by
(6)fx=x+x28,x∈0,12x,x>1,andgx=7x+x28,x∈0,17x4,x>1.Define mappingsα:gX×gX⟶0,+∞ by
(7)αx,y=s2,x,y∈0,740,otherwise,and ψ,φ:0,+∞⟶0,+∞ with ψt=t/2,φt=64t/585.It is clear thatfX⊂gX. For x,y∈X such that αgx,gy≥s2, we can know that gx,gy∈0,7/4 and this implies that x,y∈0,1. By definitions, we obtain fx,fy∈0,7/4 and αfx,fy≥s2. That is, f is a g−αs2− admissible mapping. For all x,y∈0,1, we have
(8)ψαgx,gydfx,fy=12·4·dfx,fy=2·164x+x2-y+y22≤132maxx+x22,y+y22,ψNx,y≥ψmaxdfx,gx,dfy,gy=932maxx+x22y+y22,φMx,y=φmaxx+x28-7y+y282,7x+x28-7y+y282,x+x28-7x+x282,y+y28-7y+y282,x+x2/8-7x+x2/821+7x+x2/8-7y+y2/821+x+x2/8-7y+y2/82,x+x2/8-7x+x2/821+x+x2/8-7x+x2/821+x+x2/8-7y+y2/82,x+x2/8-7x+x2/821+y+y2/8-7y+y2/821+x+x2/8-7y+y2/82≤φmax4964maxx+x22,y+y22,4964maxx+x22,y+y22,916x+x22,916y+y22,916x+x221+4964maxx+x22,y+y22,916x+x221+916x+x22,916x+x221+916y+y22≤64585·916·6516maxx+x22,y+y22=14maxx+x22,y+y22.According to the above inequalities, we get that(9)ψαgx,gydfx,fy≤132maxx+x22,y+y22=932maxx+x22,y+y22−14maxx+x22,y+y22≤ψNx,y−φMx,y.It follows thatf is a generalized g−αsp,ψ,φ contractive mapping.Theorem 12.
LetX,d be a complete b−metric space with parameter s≥1 and let f,g:X⟶X be given self-mappings on X such that fX⊂gX. Also, gX is a closed subset of X, and α:X×X⟶0,+∞ is a given mapping. If the following conditions are satisfied:
(i)
f is a g−αsp− admissible mapping(ii)
f is a generalized g−αsp,ψ,φ contractive mapping(iii)
there isx0∈X with αgx0,fx0≥sp(iv)
propertiesHsp and Usp are satisfied(v)
α has a transitive property type sp, that is, for x,y,z∈X(10)αx,y≥spandαy,z≥sp⇒αx,z≥sp.
Then,f and g have a unique point of coincidence in X. Moreover, f and g have a unique common fixed point provided that f and g are weakly compatible.Proof.
According to condition (3), there exists anx0∈X such that αgx0,fx0≥sp. 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 it is easy to see that f and g have a point of coincidence. Without loss of generality, assume that yn≠yn+1 for all n∈N. By the condition (1), we get
(11)αgx0,gx1=αgx0,fx0≥sp,αgx1,gx2=αfx0,fx1≥sp,αgx2,gx3=αfx1,fx2≥sp.Therefore, by induction, we obtain αgxn,gxn+1=αyn−1,yn≥sp for all n∈N. It follows from (4) that
(12)ψdyn,yn+1≤ψspdyn,yn+1≤ψαgxn,gxn+1dfxn,fxn+1≤ψNxn,xn+1−φMxn,xn+1,where
(13)Nxn,xn+1=maxdyn−1,yn,dyn,yn−1,dyn+1,yn,dyn−1,yn+1+dyn,yn4s,dyn−1,yndyn+1,yn1+dyn,yn+1,dyn+1,yn1+dyn,yn−11+dyn−1,yn≤maxdyn−1,yn,dyn,yn+1,sdyn−1,yn+dyn,yn+14s=maxdyn−1,yn,dyn,yn+1,(14)Mxn,xn+1=maxdyn,yn,dyn−1,yn,dyn,yn−1,dyn+1,yn,dyn,yn−11+dyn−1,yn1+dyn,yn,dyn,yn−11+dyn,yn−11+dyn,yn,dyn,yn−11+dyn+1,yn1+dyn,yn.
If we assume that, for somen∈N,
(15)dyn,yn+1≥dyn,yn−1>0,then from inequalities (13) and (14), we have
(16)Nxn,xn+1≤dyn,yn+1,(17)Mxn,xn+1≥maxdyn,yn+1,dyn,yn−1=dyn,yn+1.
Using (12), (16), and (17), one can obtain that
(18)ψdyn,yn+1≤ψNxn,xn+1−φMxn,xn+1≤ψdyn,yn+1−φdyn,yn+1,which gives φdyn,yn+1≤0 and then yn=yn+1, a contradiction. It follows that dyn,yn+1<dyn,yn−1, that is, dyn,yn+1 is a nonincreasing sequence and so there exists r≥0 such that
(19)limn⟶+∞dyn,yn+1=r.
By virtue of (13) and (14) again, we have
(20)Nxn,xn+1≤dyn,yn−1,Mxn,xn+1≥dyn,yn−1.
It follows that(21)ψdyn,yn+1≤ψNxn,xn+1−φMxn,xn+1≤ψdyn,yn−1−φdyn,yn−1.
Now suppose thatr>0, then taking the limit as n⟶+∞ in above inequality, we have ψr≤ψspr≤ψr−φr, which gives a contradiction. Hence,
(22)limn⟶+∞dyn,yn+1=0.
Next, we aim to prove thatyn is a Cauchy sequence. Suppose on the contrary that, limn,m⟶+∞dyn,ym≠0, then there exists ε>0 for which one can find sequences ymk and ynk of yn satisfying nk is the smallest index for which nk>mk>k,
(23)ε≤dymk,ynk,(24)dymk,ynk−1<ε.
In view of the triangle inequality, we have(25)ε≤dymk,ynk≤sdymk,ynk−1+sdynk−1,ynk<sε+sdynk,ynk−1.
Taking the upper limit ask⟶+∞ in the above inequality and using (22), we have
(26)ε≤limsupk⟶+∞dymk,ynk≤sε.
Also,(27)dymk,ynk≤sdymk,ynk−1+sdynk−1,ynk,(28)dymk,ynk≤sdymk,ymk−1+sdymk−1,ynk,(29)dymk−1,ynk≤sdymk−1,ymk+sdymk,ynk.
From (24) and (27), we obtain
(30)εs≤limsupk⟶+∞dymk,ynk−1≤ε.
Using (28) and (29), we have
(31)εs≤limsupk⟶+∞dymk−1,ynk≤s2ε.
Similarly,(32)dymk−1,ynk−1≤sdymk−1,ymk+sdymk,ynk−1,dymk,ynk≤sdymk,ymk−1+s2dymk−1,ynk−1+s2dynk−1,ynk,so there is
(33)εs2≤limsupk⟶+∞dymk−1,ynk−1≤sε.
In view of the definition ofNx,y, one can deduce that
(34)Nxmk,xnk=maxdymk−1,ynk−1,dymk,ymk−1,dynk,ynk−1,dymk−1,ynk+dymk,ynk−14s,dymk−1,ymkdynk,ynk−11+dymk,ynk,dynk,ynk−11+dymk,ymk−11+dymk−1,ynk−1,which yields that
(35)limsupk⟶+∞Nxmk,xnk≤maxεs,0,0,εs2+ε4s,0,0=εs.
Similarly, we obtain(36)Mxmk,xnk=maxdymk,ynk−1,dymk−1,ynk−1,dymk,ymk−1,dynk,ynk−1,dymk,ymk−11+dymk−1,ynk−11+dymk,ynk−1,dymk,ymk−11+dymk,ymk−11+dymk,ynk−1,dymk,ymk−11+dynk,ynk−11+dymk,ynk−1.
So there is(37)liminfk⟶+∞Mxmk,xnk≥maxεs,εs2,0,0,0,0,0≥εs2,liminfk⟶+∞Mxmk,xnk≤maxε,sε,0,0,0,0,0=sε,that is,
(38)εs2≤liminfk⟶+∞Mxmk,xnk≤sε.
Using the transitive property typesp of α, we get
(39)αxmk,xnk≥sp.
Applying (4) with x=xnk and y=xmk,we get
(40)ψdymk,ynk≤ψspdfxmk,fxnk≤ψαgxmk,gxnkdfxmk,fxnk≤ψNxmk,xnk−φMxmk,xnk.
By (35) and (38), we have
(41)ψsε≤ψspε≤ψsplimsupk⟶+∞dfxmk,fxnk≤ψlimsupk⟶+∞Nxmk,xnk−φliminfk⟶+∞Mxmk,xnk≤ψsε−φliminfk⟶+∞Mxmk,xnk,which implies that
(42)liminfk⟶+∞Mxmk,xnk=0,a contradiction to (38). Therefore, yn is a Cauchy sequence in X. The completeness of X ensures that there exists a u∈X such that
(43)limn⟶+∞dyn,u=limn⟶+∞dfxn,u=limn⟶+∞dgxn+1,u=0.
SincegX is closed, we have u∈gX. It follows that one can choose a z∈X such that u=gz, and we can write (43) as
(44)limn⟶+∞dyn,gz=limn⟶+∞dfxn,gz=limn⟶+∞dgxn+1,gz=0.
The propertyHsp yields that there exists a subsequence ynk of yn so that αynk−1,gz≥sp for all k∈N. If fz≠gz, applying contractive condition (4) with x=xnk and y=z, we have
(45)ψdynk,fz=ψdfxnk,fz≤ψspdfxnk,fz≤ψαgxnk,gzdfxnk,fz≤ψNxnk,z−φMxnk,z,where
(46)Nxnk,z=maxdynk−1,gz,dynk,ynk−1,dfz,gz,dynk−1,fz+dynk,gz4s,dynk,ynk−1dfz,gz1+dynk,fz,dfz,gz1+dynk,ynk−11+dynk−1,gz,Mxnk,z=maxdynk,gz,dynk−1,gz,dynk,ynk−1,dfz,gz,dynk,ynk−11+dynk−1,gz1+dynk,gz,dynk,ynk−11+dynk,ynk−11+dynk,gz,dynk,ynk−11+dfz,gz1+dynk,gz.
It is easy to show that(47)limsupk⟶+∞Nxnk,z≤max0,0,dgz,fz,sdfz,gz4s,0,dgz,fz=dgz,fz,liminfk⟶+∞Mxnk,z=max0,0,0,dfz,gz,0,0,0=dfz,gz.
Taking the upper limit ask⟶+∞ in (45), we have
(48)ψdgz,fz≤ψsp−1dgz,fz=ψsp1sdgz,fz≤ψsplimsupk⟶+∞dfxnk,fz≤ψlimsupk⟶+∞Nxnk,z−φliminfk⟶+∞Mxnk,z≤ψdgz,fz−φdgz,fz,which implies that
(49)dfz,gz=0.
That is,fz=gz. Therefore, u=fz=gz is a point of coincidence for f and g. By using contractive condition (4) and the property Usp, one can conclude that the point of coincidence is unique. Assume on the contrary that, there exist z,z′∈Cf,g and z≠z′. According to the property of Usp, without loss of generality, we assume that
(50)αgz,gz′≥sp.
Applying (4) with x=z and y=z′, we obtain that
(51)dfz,fz′=0,that is, fz=fz′. By the weak compatibility of f and g, it is easy to show that z is a unique common fixed point. This completes the proof.Remark 13.
It is obvious that the mappings defined in Example11 satisfy the conditions of Theorem 12, so f and g have a unique common fixed point 0.In Theorem12, put ψt=t, one can get the following result.Corollary 14.
LetX,d be a complete b−metric space with parameter s≥1 and let f,g:X⟶X be given self-mappings on X with fX⊂gX. Also, gX is a closed subset of X, and α:X×X⟶0,+∞ is a given mapping. If the following conditions are satisfied:
(i)
f is a g−αsp− admissible mapping(ii)
there is functionφ∈Φ such that(52)αgx,gydfx,fy≤Nx,y−φMx,y,where Nx,y,Mx,y are same as Theorem 12,
(iii)
there existsx0∈X with αgx0,fx0≥sp(iv)
propertiesHsp and Usp are satisfied(v)
α has a transitive property type sp, that is, for x,y,z∈X(53)αx,y≥spandαy,z≥sp⇒αx,z≥sp.
Then,f and g have a unique point of coincidence in X. Moreover, f and g have a unique common fixed point provided that f and g are weakly compatible.Theorem 15.
LetX,d be a complete b−metric space with parameter s≥1, and let f,g:X⟶X be given self-mappings on X with fX⊂gX. Also, gX is a closed subset of X, and α:X×X⟶0,+∞ is a given mapping. Suppose that the following conditions are satisfied:
(i)
f is a g−αsp−admissible mapping(ii)
there are functionsψ∈Ψ and φ∈Φ such that for all x,y∈X(54)ψαgx,gydfx,fy≤ψLx,y−φMx,y,where Mx,y is same as Theorem 12 and
(55)Lx,y=maxdfx,gy,dfx,gx,dfy,gy,dgx,gy+dfx,gy2s.(iii)
there existsx0∈X with αgx0,fx0≥sp(iv)
propertiesHsp and Usp are satisfied(v)
α has a transitive property type sp, that is, for x,y,z∈X(56)αx,y≥spandαy,z≥sp⇒αx,z≥sp,
thenf and g have a unique point of coincidence 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 Theorem12, we also define the sequences xn and yn in X by yn=fxn=gxn+1 for n∈N and suppose that yn≠yn+1 for each n∈N, so one can get that
(57)αyn−1,yn=αgxn,gxn+1≥sp.
It follows from (54) that
(58)ψdyn,yn+1≤ψspdyn,yn+1≤ψαgxn,gxn+1dfxn,fxn+1≤ψLxn,xn+1−φMxn,xn+1,where
(59)Lxn,xn+1=maxdyn,yn,dyn,yn−1,dyn+1,yn,dyn−1,yn+dyn,yn2s≤maxdyn+1,yn,dyn,yn−1,(60)Mxn,xn+1=maxdyn,yn,dyn−1,yn,dyn,yn−1,dyn+1,yn,dyn,yn−11+dyn−1,yn1+dyn,yn,dyn,yn−11+dyn,yn−11+dyn,yn,dyn,yn−11+dyn+1,yn1+dyn,yn.
If we assume that, for somen∈N(61)dyn,yn+1≥dyn−1,yn>0,then according to inequalities (59) and (60), we obtain
(62)Lxn,xn+1≤dyn+1,yn,Mxn,xn+1≥dyn+1,yn.
In view of (58), we get
(63)ψdyn,yn+1≤ψLxn,xn+1−φMxn,xn+1≤ψdyn,yn+1−φdyn,yn+1,which implies 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.
By (59) and (60), we have
(64)Lxn,xn+1≤dyn,yn−1,Mxn,xn+1≥dyn,yn−1.
So,(65)ψdyn,yn+1≤ψLxn,xn+1−φMxn,xn+1≤ψdyn,yn−1−φdyn,yn−1.
Ifr>0, then letting n⟶+∞ in above inequality, we obtain that ψr=ψr−φr, which implies that r=0, i.e.,
(66)limn⟶+∞dyn,yn+1=0.
Now, we prove thatyn is a Cauchy sequence. If not, as the proof of Theorem 12, 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:
(67)ε≤limsupk⟶+∞dymk,ynk≤sε,(68)εs≤limsupk⟶+∞dymk,ynk−1≤ε,(69)εs≤limsupk⟶+∞dymk−1,ynk≤s2ε,(70)εs2≤limsupk⟶+∞dymk−1,ynk−1≤sε.
In view of the definitions ofLx,y and Mx,y, we have
(71)Lxmk,xnk=maxdymk,ynk−1,dymk−1,ymk,dynk,ynk−1,dymk−1,ynk−1+dymk,ynk−12s.Mxmk,xnk=maxdymk,ynk−1,dymk−1,ynk−1,dymk−1,ymk,dynk,ynk−1,dymk,ymk−11+dymk−1,ynk−11+dymk,ynk−1,dymk,ymk−11+dymk,ymk−11+dymk,ynk−1,dymk,ymk−11+dynk,ynk−11+dymk,ynk−1.
Lettingk⟶+∞ and using (67)–(70), one can obtain
(72)limsupk⟶+∞Lxmk,xnk≤maxε,0,0,sε+s2s=ε.
Similarly, we get that(73)liminfk⟶+∞Mxmk,xnk≤maxε,sε,0,0,0,0,0=sε,liminfk⟶+∞Mxmk,xnk≥maxεs,εs2,0,0,0,0,0=εs2.
That is,(74)εs2≤liminfk⟶+∞Mxmk,xnk≤sε.
Using the transitive property typesp of α, we have
(75)αxmk,xnk≥sp.
Takingx=xnk and y=xmk in (54), one can deduce that
(76)ψdymk,ynk≤ψspdfxmk,fxnk≤ψαgxmk,gxnkdfxmk,fxnk≤ψLxmk,xnk−φMxmk,xnk.
Therefore,(77)ψε≤ψspε≤ψsplimsupk⟶+∞dfxmk,fxnk≤ψlimsupk⟶+∞Lxmk,xnk−φliminfk⟶+∞Mxmk,xnk≤ψε−φliminfk⟶+∞Mxmk,xnk.
It follows thatliminfk⟶+∞Mxmk,xnk=0, and which gives a contradiction to (74). Hence,
(78)limn,m⟶+∞dyn,ym=0.
From the completeness ofX and the closure of gX, there exists u∈X such that
(79)limn⟶+∞dyn,u=limn⟶+∞dfxn,u=limn⟶+∞dgxn+1,u=0.
It follows that one can choose az∈X such that u=gz, and write the above equality as
(80)limn⟶+∞dyn,gz=limn⟶+∞dfxn,gz=limn⟶+∞dgxn+1,gz=0.
In view of the propertyHsp, one can get a subsequence ynk of yn with αynk−1,gz≥sp for all k∈N. If fz≠gz, taking x=xnk and y=z in contractive condition (54), we have
(81)ψdynk,fz=ψdfxnk,fz≤ψspdfxnk,fz≤ψαgxnk,gzdfxnk,fz≤ψLxnk,z−φMxnk,z,where
(82)Lxnk,z=maxdynk,gz,dynk,ynk−1,dfz,gz,dynk−1,gz+dynk,gz2s.Mxnk,z=maxdynk,gz,dynk−1,gz,dynk,ynk−1,dfz,gz,dynk,ynk−11+dynk−1,gz1+dynk,gz,dynk,ynk−11+dynk,ynk−11+dynk,gz,dynk,ynk−11+dfz,gz1+dynk,gz.
Consequently,(83)limsupk⟶+∞Lxnk,z≤max0,0,dfz,gz,0=dfz,gz.liminfk⟶+∞Mxnk,z=dfz,gz.
Taking the upper limit ask⟶+∞ in (81), we get
(84)ψdgz,fz≤ψsp−1dgz,fz=ψsp1sdgz,fz≤ψsplimsupk⟶+∞dfxnk,fz≤ψlimsupk⟶+∞Lxnk,z−φliminfk⟶+∞Mxnk,z≤ψdgz,fz−φdgz,fz.
It follows thatdfz,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 12, one can complete the proof.Example 16.
LetX=0,+∞ and dx,y=x−y2. Define mappings f,g:X⟶X by
(85)fx=x64,x∈0,1ex−e+12,x>1,andgx=x2,x∈0,1e2x−e2+12,x>1.
Define mappingsα:gX×gX⟶0,+∞ by
(86)αx,y=s2,x,y∈0,120,otherwise,and ψ,φ:0,+∞⟶0,+∞ with ψt=t,φt=3828t/4805.
It is clear thatfX⊂gX and gX is closed. For x,y∈X such that αgx,gy≥s2, we can know that gx,gy∈0,1/2 and which implies that x,y∈0,1. It follows that fx,fy∈0,1/2 and αfx,fy≥s2, that is, f is a g−αsp−admissible mapping.
Forx,y∈0,1, we have
(87)ψαgx,gydfx,fy=4·x64−y642≤4642maxx2,y2,ψLx,y≥ψmaxdfx,gx,dfy,gy=31642maxx2,y2,φMx,y=φmaxx64−y22,x2−y22,x64−x22,y64−y22,x/64−x/221+x/2−y/221+x/64−y/22,x/64−x/221+x/2−x/6421+x/64−y/22,x/64−x/221+y/2−y/6421+x/64−y/22≤φmax14maxx2,y2,14maxx2,y2,31642maxx2,y2,31642maxx2,y2,31642maxx2,y2·54,31642maxx2,y2·50574096,31642maxx2,y2·50574096=38284805·31642maxx2,y2·54.
Obviously, we conclude(88)ψαgx,gydfx,fy≤4642maxx2,y2=31642maxx2,y2−38284805·31642maxx2,y2·54≤ψLx,y−φMx,y.
It follows that all conditions of Theorem15 are satisfied. It is obvious that 0 is the unique common fixed point of f and g.Remark 17.
TakingS=T=Ix in Theorem 2.1 of [12], Roshan et al. give that the existence of common fixed point for mappings f,g such that
(89)dfx,gy≤qs4maxdx,y,dfx,x,dgy,y,12dx,fy+dfx,y,where q∈0,1 is a constant. Suppose all hypotheses in Example 16 are true. For x=0,y∈0,1/2, it is easy to calculate that
(90)dfx,gy=y24>y216≥q16maxy2,0,y24,y22·642+y22=qs4maxdx,y,dfx,x,dgy,y,12dx,fy+dfx,y,which implies that Theorem 2.1 of [12] cannot be applied to testify the existence of common fixed points of the mappings f and g in X.Ifψt=t and φt=t in Theorem 15, we get the following result immediately:Corollary 18.
LetX,d be a complete b−metric space with parameter s≥1 and let f,g:X⟶X be given self-mappings on X such that fX⊂gX. Also, gX is a closed subset of X, and α:X×X⟶0,+∞ is a given mapping. If the following conditions are satisfied:
(i)
f is a g−αsp−admissible mapping,(ii)
forx,y∈X(91)αgx,gydfx,fy≤Lx,y−Mx,y,where Lx,y,Mx,y are same as Theorem 15,
(iii)
there existsx0∈X with αgx0,fx0≥sp(iv)
propertiesHsp and Usp are satisfied(v)
α has a transitive property type sp, that is, for x,y,z∈X(92)αx,y≥spandαy,z≥sp⇒αx,z≥sp.
Then,f and g have a unique point of coincidence in X. Moreover, if f and g are weakly compatible, then f and g have a unique common fixed point.Letg=I, ψt=t, and φt=Lt (L>0 is a constant), we obtain thatTheorem 19.
LetX,d be a complete b−metric space with parameter s≥1 and f:X⟶X be a given self-mapping on X. Let α:X×X⟶0,+∞ be a given mapping. If the following conditions are satisfied:
(i)
f is a αsp−admissible mapping(ii)
forx,y∈X(93)αx,ydfx,fy≤1−LK∗x,y,where
(94)K∗x,y=maxdx,y,dfx,x,dfy,y,dfx,y,L∈0,1,(iii)
there existsx0∈X with αx0,fx0≥sp(iv)
propertiesHsp and Usp are satisfied when g=I(v)
α has a transitive property type sp, that is, for x,y,z∈X(95)αx,y≥spandαy,z≥sp⇒αx,z≥sp.
Then,f has a unique fixed point.Proof.
The proof of Theorem19 is similar to that of Theorem 15, we omit it.Remark 20.
Since ab−metric space is a metric space when s=1, so our results can be viewed as the generalization and the extension of comparable results.


## 4. Application

In this section, we will use Theorem19 to show that there is a solution to the integral equation:
(96)xt=∫0TGt,r,xrdr.LetX=C0,T be the set of real continuous functions defined on 0,T. The standard metric given by
(97)ρx,y=supt∈0,T∣xt−yt∣for allx,y∈X.Now forp≥1, we define
(98)dx,y=ρx,yp=supt∈0,Txt−ytpfor allx,y∈X.It is obvious thatX,d is a complete b−metric space with s=2p−1.Consider the mappingf:X⟶X defined by
(99)fxt=∫0TGt,r,xrdr,and let ξ:R×R⟶R be a given function.Theorem 21.
Consider equation (96) and suppose that
(i)
G:0,T×0,T×R⟶R+ is continuous(ii)
there existsx0∈X such that ξx0t,fx0t≥0 for all t∈0,T(iii)
for allt∈0,T and x,y∈X, ξxt,yt≥0 implies ξfxt,fyt≥0(iv)
propertiesHsp and Usp are satisfied when g=I(v)
there exists a continuous functionγ:0,T×0,T⟶R+ such that(100)supt∈0,T∫0Tγt,rdr≤1,(vi)
there exists a constantL∈0,1 such that for t,r∈0,T×0,T(101)Gt,r,xr−Gt,r,yr≤1−Lsppγt,rxr−yr.
Then, the integral equation (96) has a unique solution x∈X.Proof.
Defineα:X×X⟶0,+∞ by
(102)αx,y=sp,ifξxt,yt≥00,otherwise.
It is easy to prove thatf is αsp-admissible. For x,y∈X, by virtue of assumptions (1)–(6), we have
(103)spdfxt,fyt=spsupt∈0,Tfxt−fytp=spsupt∈0,T∫0TGt,r,xrdr−∫0TGt,r,yrdrp≤spsupt∈0,T∫0TGt,r,xr−Gt,r,yrdrp≤spsupt∈0,T∫0T1−Lsppγt,rxr−yrdrp≤spsupt∈0,T∫0T1−Lsppγt,rdrpsupt∈0,Txt−ytp≤1−LK∗xt,yt,which implies that
(104)αxt,ytdfxt,fyt≤1−LK∗xt,yt.
Therefore, all the conditions of Theorem19 hold. As a result, the mapping f has a unique fixed point x∈X, which is a solution of the integral equation (96).


## 5. Conclusions

In this manuscript, we introduced a new class ofg−αsp−admissible mappings and obtained common fixed point theorems for generalized g−αsp,ψ,φ contractive mappings in the framework of b−metric space. Further, we provided examples that elaborated the useability of our results. As an application of our result, we obtained a solution to an integral equation. The obtained results will be helpful for the variational iteration method, so we are going to study this topic in future.



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*Source: 5020027-2021-06-16.xml*