# On Some Common Fixed Point Results for Weakly Contraction Mappings with Application

Authors: Yan Hao; Hongyan Guan
Journal: Journal of Function Spaces
 (2021)

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## Abstract

In this paper, we introduce a new class of generalized weakly contractive mappings and prove common fixed point results by using different algorithms involving this new class of mappings in the framework ofb-metric spaces, which generalize the results of Cho. We also provide two examples to show the applicability and validity of our results. As an application of our result, we obtain a solution to an integral equation. Our results extend and improve several comparable results in the existing literature.


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## Body

## 1. Introduction

The Banach fixed point theorem [1] popularly known as the Banach contraction mapping principle is a rewarding result in fixed point theory. It has widespread applications in both pure and applied mathematics and has been extended in many different directions. One of the most popular and interesting topics among them is the study of new classes of spaces and their fundamental properties.In 1993, Czerwik [2] introduced firstly the concept of b-metric space and proved some fixed point theorems of contractive mappings in b-metric space. After that, some authors have researched on the fixed point theorems of various new types of contractive conditions in b-metric space. Aydi et al. in [3] proved common fixed point results for single-valued and multivalued mappings satisfying a weak ϕ-contraction in b-metric spaces. Starting from the results of Berinde [4], Pacurar [5] proved the existence and uniqueness of the fixed point of ϕ-contractions and Zada et al. [6] established fixed point results satisfying contractive conditions of rational type. In 2019, Hussain et al. studied the existence and uniqueness of periodic common fixed point for pairs of mappings via rational type contraction in [7]. After that, in [8], the authors obtained fixed point theorems for cyclic α,β−ψ,ϕs-rational type contractions and discussed the existence of a unique solution to nonlinear fractional differential equations. Also using rational type contractive conditions, Hussain et al. [9] got the existence and uniqueness of a common n-tupled fixed point for a pair of mappings. Using a contraction condition defined by means of a comparison function, [10] established results regarding the common fixed points of two mappings. In 2014, Abbas et al. obtained the results on common fixed point of four mappings in b-metric space in [11]. Iqbal et al. [12] introduced a generalized multivalued α,L-almost contraction and proved the existence and uniqueness of the fixed point for a specific mapping in the b-metric space.Inspired by Czerwik’s results, 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. Younis et al. [14] studied the existence of fixed points of a new class of generalized F-contraction in partial b-metric space. In [15], some fixed point results for weakly contractive mappings in ordered partial metric space were obtained. Recently, Samet et al. [16] introduced the concept of α-admissible and α−ψ-contractive mappings and presented fixed point theorems for them. In [17, 18], Zoto et al. studied generalized αsp contractive mappings and α−ψ,ϕ-contractions in b-metric-like space. In 2020, Isik et al. [19] firstly introduced the structure of extended quasi b-metric-like spaces as a generalization of both quasi metric-like spaces and quasi b-metric-like spaces. Also, they presented the notion of JSR-contractive mappings in the setup of extended quasi b-metric-like spaces and investigated the existence of fixed point for such mappings. Abu-Donia et al. [20] 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. In 2015, Ege [21] introduced complex valued rectangular b-metric space and proved an analogue of the Banach contraction principle in this space. Recently, Younis et al. [22] provided much simpler and shorter proofs of some new results in rectangular metric spaces, and Mitrovic et al. [23] gave a proof of the results of Miculescu and Mihail [24] and Suzuki [25] in extended b-metric spaces. In graphical b-metric spaces, Younis et al. presented fixed point results for Kannan-type and Reich-type mappings in [26, 27]. Lately, Gholidahneh et al. [28] introduced the notion of a modular p-metric space (an extended modular b-metric space) and established some fixed point results for α-v^-Meir-Keeler contractions in this new space.In 1997, Alber and Guerre-Delabriere [29] generalized the Banach fixed point theorem by introducing the concept of weak contraction mappings in Hilbert spaces. Weak contraction principle states that every weak contraction mapping on a complete Hilbert space has a unique fixed point. Rhoades [30] extended weak contraction principle in Hilbert spaces to metric spaces. Since then, many authors (for example, [31–42]) obtained generalizations and extensions of the weak contraction principle. Recently, in [43], 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 particular, Choudhury et al. [36] obtained a generalization of the weak contraction principle in metric spaces by using altering distance functions as follows:Theorem 1 (see [36]).
Suppose that a mappingg:X⟶X, where X is a metric space with metric d, satisfies the following condition:(1)ψdgx,gy≤ψmaxdx,y,dx,gx,dy,gy,12dx,gy+dy,gx−φmaxdx,y,dy,gy,for all x,y∈X, where φ:0,+∞⟶0,+∞ is a continuous function and ψ:0,+∞⟶0,+∞ is an altering function, that is, ψ is a nondecreasing and continuous function, and ψt=0 if and only if t=0. Then, g has a unique fixed point.LetX be a metric space with metric d, let T:X⟶X, and let φ:X⟶0,+∞ be a lower semicontinuous function. Then, T is called a generalized weakly contractive mapping if it satisfies the following condition:(2)ψdTx,Ty+φTx+φTy≤ψmx,y,d,T,φ−ϕlx,y,d,T,φ,where ψ∈Ψ,ϕ∈Φ, and(3)mx,y,d,T,φ=maxdx,y+φx+φy,dx,Tx+φx+φTx,dy,Ty+φy+φTy,12dx,Ty+φx+φTy+dy,Tx+φy+φTx,(4)lx,y,d,T,φ=maxdx,y+φx+φy,dy,Ty+φy+φTy.Cho [44] extended the results of Choudhury et al. [36] to generalized weakly contractive mappings in the setting of metric spaces and obtained the following result:Theorem 2 (see [44]).
LetX be complete. If T is a generalized weakly contractive mapping, then there exists a unique z∈X such that z=Tz and φz=0.Motivated and inspired by Theorem 2.1 in [44], in this paper, our purpose is to introduce a new class of generalized weakly contractive mappings and obtain a few of common fixed point results by using different algorithms involving generalized weakly contractive conditions in the framework of b-metric space, which generalize the results of Cho. 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

In this section, in order to get our main results, we will introduce some definitions and lemmas first.Definition 3 (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.In general,X,d is called a b-metric space with parameter s≥1.Remark 4.
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 [45]).Example 5 (see [46]).
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 6 (see [11]).
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 addition, ab-metric space is called complete if and only if each Cauchy sequence in this space is b-convergent.Definition 7 (see [47]).
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 8 (see [47]).
Letf and g be two self-mappings defined on a nonempty set X. Then, f and g is said to be weakly compatible if they commute at every coincidence point, that is, fx=gx⇒fgx=gfx for every x∈Cf,g.The following lemma plays an important role to obtain our main results:Lemma 9 (see [46]).
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(5)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(6)1sdx,z≤liminfn⟶+∞dxn,z≤limsupn⟶+∞dxn,z≤sdx,z.


## 3. Main Results

In this section, we will establish common fixed point theorems for generalized weakly contractive mappings in completeb-metric space. Furthermore, we also provide two examples to support our results.A functionf:X⟶0,+∞, where X,d is a b-metric space, is called lower semicontinuous if, for all x∈X and xn are b-convergent to x, we have(7)fx≤liminfn⟶+∞fxn.We shall consider that the contractive conditions in this section are constructed via auxiliary functions defined with the familiesΨ,Φ, respectively:(8)Ψ=ψ:0,+∞→0,+∞isanondecreasingandcontinuousfunction,(9)Φ=ϕ:0,+∞⟶0,+∞isanondecreasingandlowersemicontinuousfunctionandϕt=0ifandonlyift=0.Theorem 10.
LetX,d be a complete b-metric space with parameter s≥1, and let f,g:X⟶X be given self-mappings satisfying g as injective and fX⊂gX where gX is closed. Suppose φ:X⟶0,+∞ is a lower semicontinuous function and p≥2 is a constant. If there are functions ψ∈Ψ and ϕ∈Φ such that(10)ψspdfx,fy+φfx+φfy≤ψmx,y,d,f,g,φ−ϕlx,y,d,f,g,φ,where(11)mx,y,d,f,g,φ=maxdgx,gy+φgx+φgy,12dfx,gx+φfx+φgx+dfy,gy+φfy+φgy,12sdfx,gy+φfx+φgy+dfy,gx+φfy+φgx,lx,y,d,f,g,φ=maxdgx,gy+φgx+φgy,dfy,gy+φfy+φgy,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 with 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 for all n∈ℕ. Applying (10) with x=xn and y=xn+1, we obtain(12)ψdyn,yn+1+φyn+φyn+1≤ψspdyn,yn+1+φyn+φyn+1=ψspdfxn,fxn+1+φfxn+φfxn+1≤ψmxn,xn+1,d,f,g,φ−ϕlxn,xn+1,d,f,g,φ, where(13)mxn,xn+1,d,f,g,φ=maxdgxn,gxn+1+φgxn+φgxn+1,12dfxn,gxn+φfxn+φgxn+dfxn+1,gxn+1+φfxn+1+φgxn+1,12sdfxn,gxn+1+φfxn+φgxn+1+dfxn+1,gxn+φfxn+1+φgxn≤maxdyn−1,yn+φyn−1+φyn,12dyn,yn−1+φyn+φyn−1+dyn+1,yn+φyn+1+φyn,12sdyn,yn+φyn+φyn+dyn+1,yn−1+φyn+1+φyn−1≤maxdyn−1,yn+φyn−1+φyn,dyn+1,yn+φyn+1+φyn,(14)lxn,xn+1,d,f,g,φ=maxdgxn,gxn+1+φgxn+φgxn+1,dfxn+1,gxn+1+φfxn+1+φgxn+1=maxdyn−1,yn+φyn−1+φyn,dyn+1,yn+φyn+1+φyn.
Ifdyn,yn+1+φyn+φyn+1>dyn,yn−1+φyn+φyn−1, for some n∈ℕ, in view of (12), (13), and (14), we have(15)ψdyn,yn+1+φyn+φyn+1≤ψmxn,xn+1,d,f,g,φ−ϕlxn,xn+1,d,f,g,φ≤ψdyn+1,yn+φyn+1+φyn−ϕdyn+1,yn+φyn+1+φyn,which implies ϕdyn,yn+1+φyn+φyn+1=0. Hence, yn=yn+1, a contradiction.
Thus, we have(16)dyn,yn+1+φyn+φyn+1≤dyn,yn−1+φyn+φyn−1,(17)mxn,xn+1,d,f,g,φ≤dyn,yn−1+φyn+φyn−1,(18)lxn,xn+1,d,f,g,φ=dyn,yn−1+φyn+φyn−1.
It follows from (16) that dyn,yn+1+φyn+φyn+1 is a nonincreasing sequence, and so there exists r≥0 such that(19)limn⟶+∞dyn,yn+1+φyn+φyn+1=r.
By virtue of (12), (17), and (18), one can obtain(20)ψdyn,yn+1+φyn+φyn+1≤ψmxn,xn+1,d,f,g,φ−ϕlxn,xn+1,d,f,g,φ≤ψdyn,yn−1+φyn+φyn−1−ϕdyn,yn−1+φyn+φyn−1.
Now assume thatr>0. Taking the upper limit as n⟶∞ in (20), we have(21)limsupn⟶+∞ψdyn,yn+1+φyn+φyn+1≤limsupn⟶+∞ψmxn,xn+1,d,f,g,φ−limsupn⟶+∞ϕlxn,xn+1,d,f,g,φ≤limsupn⟶+∞ψdyn,yn−1+φyn+φyn−1−liminfn⟶+∞ϕdyn,yn−1+φyn+φyn−1,which implies that ψr≤ψr−ϕr, a contradiction. This yields that(22)limn⟶+∞dyn,yn+1+φyn+φyn+1=r=0.
It follows thatlimn⟶+∞dyn,yn+1=0 and limn⟶+∞φyn=0.
Now we shall prove thatyn is a Cauchy sequence in X. Suppose on the contrary that yn is not Cauchy. It follows that 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.
By the triangle inequality inb-metric space and (23) and (24), we have(25)ε≤dymk,ynk≤sdymk,ynk−1+sdynk−1,ynk<sε+sdynk−1,ynk.
Taking the upper limit ask⟶+∞ in the above inequality, 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 (23), (24), and (27), we obtain(30)εs≤limsupk⟶+∞dymk,ynk−1≤ε.
Using (23), (28), and (29), we get(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.
Using the same method, one can obtain that(34)ε≤liminfk⟶+∞dymk,ynk≤sε,εs≤liminfk⟶+∞dymk,ynk−1≤ε,εs≤liminfk⟶+∞dymk−1,ynk≤s2ε,εs2≤liminfk⟶+∞dymk−1,ynk−1≤sε.
In view of the definition ofmx,y,d,f,g,φ, we deduce(35)mxmk,xnk,d,f,g,φ=maxdgxmk,gxnk+φgxmk+φgxnk,12dfxmk,gxmk+φfxmk+φgxmk+dfxnk,gxnk+φfxnk+φgxnk,12sdfxmk,gxnk+φfxmk+φgxnk+dfxnk,gxmk+φfxnk+φgxmk=maxdymk−1,ynk−1+φymk−1+φynk−1,12dymk,ymk−1+φymk+φymk−1+dynk,ynk−1+φynk+φynk−1,12sdymk,ynk−1+φymk+φynk−1+dynk,ymk−1+φynk+φymk−1.Taking the upper limit as k⟶+∞ in (35), we obtain(36)limsupk⟶+∞mxmk,xnk,d,f,g,φ≤maxsε,0,ε+s2ε2s=sε.
Also, we have(37)lxmk,xnk,d,f,g,φ=maxdgxmk,gxnk+φgxmk+φgxnk,dfxnk,gxnk+φfxnk+φgxnk=maxdymk−1,ynk−1+φymk−1+φynk−1,dynk,ynk−1+φynk+φynk−1.
It follows that(38)sε≥liminfk⟶+∞lxmk,xnk,d,f,g,φ≥εs2.
Applying (10) with x=xmk and y=xnk, one can get(39)ψsε≤ψspε≤ψsplimsupn⟶+∞dymk,ynk+φymk+φynk≤ψlimsupn⟶+∞mxmk,xnk,d,f,g,φ−liminfn⟶+∞ϕlxmk,xnk,d,f,g,φ≤ψsε−ϕliminfn⟶+∞lxmk,xnk,d,f,g,φ,which implies that(40)liminfn⟶+∞lxmk,xnk,d,f,g,φ=0,a contradiction to (38). It follows that yn is a Cauchy sequence in X. The completeness of X ensures that there exists a u∈X such that(41)limn⟶+∞dyn,u=limn⟶+∞dfxn,u=limn⟶+∞dgxn+1,u=limn,m⟶+∞dyn,ym=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 (41) as(42)limn⟶+∞dyn,gz=limn⟶+∞dfxn,gz=limn⟶+∞dgxn+1,gz=0.
Following from the definition ofφ, we get (43)φgz=φu≤liminfn⟶+∞φyn=0.
That is,φgz=φu=0.
Iffz≠gz, taking x=xnk and y=z in contractive condition (10), we deduce that(44)ψdfxnk,fz+φfxnk+φfz≤ψspdfxnk,fz+φfxnk+φfz≤ψmxnk,z,d,f,g,φ−ϕlxnk,z,d,f,g,φ,where(45)mxnk,z,d,f,g,φ=maxdgxnk,gz+φgxnk+φgz,12dfxnk,gxnk+φfxnk+φgxnk+dfz,gz+φfz+φgz,12sdfxnk,gz+φfxnk+φgz+dfz,gxnk+φfz+φgxnk=maxdynk−1,gz+φynk−1+φgz,12dynk,ynk−1+φynk+φynk−1+dfz,gz+φfz+φgz,12sdynk,gz+φynk+φgz+dfz,ynk−1+φfz+φynk−1,lxnk,z,d,f,g,φ=maxdgxnk,gz+φgxnk+φgz,dfz,gz+φfz+φgz=maxdynk−1,gz+φynk−1+φgz,dfz,gz+φfz+φgz.
By simple calculation, we obtain(46)limsupk⟶+∞mxnk,z,d,f,g,φ≤dfz,gz+φfz,(47)liminfk⟶+∞lxnk,z,d,f,g,φ≥dfz,gz+φfz.
By taking the upper limit ask⟶+∞ in (44) and using (46) and (47), one can get(48)ψdfz,gz+φfz≤ψdfz,gz+φfz−ϕdfz,gz+φfz.
Hence,dfz,gz+φfz=0, which implies that fz=gz and φfz=0.
Now we claim thatz is the unique coincidence point of f and g. If not, there exist z,z′∈Cf,g and z≠z′; applying (10) with x=z and y=z′, we obtain that(49)ψdfz,fz′+φfz+φfz′≤ψspdfz,fz′+φfz+φfz′≤ψmz,z′,d,f,g,φ−ϕlz,z′,d,f,g,φ.
Here,(50)mz,z′,d,f,g,φ=maxdgz,gz′+φgz+φgz′,12dfz,gz+φfz+φgz+dfz′,gz′+φfz′+φgz′,12sdfz,gz′+φfz+φgz′+dfz′,gz+φfz′+φgz≤dgz,gz′+φgz′,lz,z′,d,f,g,φ=maxdgz,gz′+φgz+φgz′,dfz′,gz′+φfz′+φgz′≥dgz,gz′+φgz′.
It follows from (49) that(51)ψdgz,gz′+φgz′≤ψdgz,gz′+φgz′−ϕdgz,gz′+φgz′.
Hence, we get thatdgz,gz′+φgz′=0, which implies that gz=gz′ and φgz′=0. Since g is an injective mapping, then z=z′; that is, z is a unique coincidence point of f and g. Further, if f and g are weakly compatible, then it is easy to show that z is a unique common fixed point of f and g. This completes the proof.Example 11.
LetX=0,+∞ and dx,y=x−y2. Define mappings f,g,φ:X⟶X by(52)fx=x+x28,gx=7x+x28,φx=x2,x∈0,+∞.
Define mappingsψ,ϕ:0,+∞⟶0,+∞ with ψt=t,ϕt=35t/98.It is clear thatfX⊂gX. For all x,y∈X, we have(53)ψspdfx,fy+φfx+φfy=4·x+x28−y+y282+x+x282+y+y282≤4·164·2x+x22+y+y22=18x+x22+y+y22,ψmx,y,d,f,g,φ≥ψ12dfx,gx+φfx+φgx+dfy,gy+φfy+φgy=12x+x28−7x+x282+x+x282+7x+x282+y+y28−7y+y282+y+y282+7y+y282=4364x+x22+y+y22,ϕlx,y,d,f,g,φ=3598maxdgx,gy+φgx+φgy,dfy,gy+φfy+φgy=3598max7x+x28−7y+y282+7x+x282+7y+y282,y+y28−7y+y282+y+y282+7y+y282≤3598max4932x+x22+y+y22,4332y+y22≤3564x+x22+y+y22.According to above inequalities, we get that(54)ψspdfx,fy+φfx+φfy≤18x+x22+y+y22=4364x+x22+y+y22−3564x+x22+y+y22≤ψmx,y,d,f,g,φ−ϕlx,y,d,f,g,φ.It follows that all conditions of Theorem10 are satisfied with s=2,p=2. It is easy to obtain that 0 is the unique common fixed point of f and g.Note that, forx=0,y∈0,+∞, one can calculate that(55)sdfx,gy=49y+y2232=max0,49y+y2232,L113y+y2264>maxsdx,fx,sdy,gy,Ldx,gy+dfx,y>φmaxsdx,fx,sdy,gy,Ldx,gy+dfx,y,which implies that Theorem 1 of [10] cannot be applied to testify the existence of common fixed points of the mappings f and g in X.Ifφ=0 in Theorem 10, we can get the following result:Corollary 12.
LetX,d be a complete b-metric space with parameter s≥1, and let f,g:X⟶X be given self-mappings satisfying g as injective and fX⊂gX where gX is closed. Suppose p≥2 is a constant. If there are functions ψ∈Ψ and ϕ∈Φ such that(56)ψspdfx,fy≤ψm1x,y,d,f,g−ϕl1x,y,d,f,g,where(57)m1x,y,d,f,g=maxdgx,gy,12dfx,gx+dfy,gy,12sdfx,gy+dfy,gx,l1x,y,d,f,g=maxdgx,gy,dfy,gy,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.If we consider the corresponding problem in the setting of metric space, that is,s=1 in Theorem 10, one can obtain the following:Corollary 13.
LetX,d be a complete metric space, and let f,g:X⟶X be given self-mappings satisfying g as injective and fX⊂gX where gX is closed. Suppose φ:X⟶0,+∞ is a lower semicontinuous function. If there are functions ψ∈Ψ and ϕ∈Φ such that(58)ψdfx,fy+φfx+φfy≤ψm2x,y,d,f,g,φ−ϕlx,y,d,f,g,φ,where(59)m2x,y,d,f,g,φ=maxdgx,gy+φgx+φgy,12dfx,gx+φfx+φgx+dfy,gy+φfy+φgy,12dfx,gy+φfx+φgy+dfy,gx+φfy+φgx,and lx,y,d,f,g,φ is the same as Theorem 10, 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.Theorem 14.
LetX,d be a complete b-metric space with parameter s≥1 and let f,g:X⟶X be given self-mappings, and one of f and g is continuous. Suppose φ:X⟶0,+∞ is a lower semicontinuous function and p≥3,0<λ≤1/4 are two constants. If there are functions ψ∈Ψ and ϕ∈Φ such that(60)ψspdfx,gy+φfx+φgy≤ψnx,y,d,f,g,φ−ϕrx,y,d,g,φ,where(61)nx,y,d,f,g,φ=λmaxdx,y+φx+φy,dfx,x+φfx+φx+dy,gy+φy+φgy,1sdfx,y+φfx+φy+dx,gy+φx+φgy,rx,y,d,g,φ=maxdx,y+φx+φy,dy,gy+φy+φgy,then f and g have a unique common fixed point in X.Proof.
Letx0∈X be an arbitrary point. Define a sequence xn in X by x2i+1=fx2i,x2i+2=gx2i+1 for i=0,1,2,⋯. Firstly, we prove that f and g have at most one common fixed point. Suppose that v and w are two different common fixed points; then, fv=v≠w=gw. It follows that dfv,gw=dv,w>0. Applying (60) with x=v and y=w, we obtain(62)ψdv,w+φv+φw≤ψspdfv,gw+φfv+φgw≤ψnv,w,d,f,g,φ−ϕrv,w,d,g,φ,where(63)nv,w,d,f,g,φ=λmaxdv,w+φv+φw,dfv,v+φfv+φv+dw,gw+φw+φgw,1sdfv,w+φfv+φw+dv,gw+φv+φgw=λmaxdv,w+φv+φw,dv,v+φv+φv+dw,w+φw+φw,1sdv,w+φv+φw+dv,w+φv+φw≤dv,w+φv+φw,rv,w,d,g,φ=maxdv,w+φv+φw,dw,gw+φw+φgw=maxdv,w+φv+φw,dw,w+φw+φw≥dv,w+φv+φw.
It follows from (62) that(64)ψdv,w+φv+φw≤ψdv,w+φv+φw−ϕdv,w+φv+φw,which implies that dv,w+φv+φw=0. That is, v=w and φv=0. Hence, the pair f,g has at most one common fixed point.
We suppose thatdxn,xn+1>0 for all n∈ℕ. If not, for some k, x2k=x2k+1 and from (60), we obtain(65)ψdx2k+1,x2k+2+φx2k+1+φx2k+2≤ψspdfx2k,gx2k+1+φfx2k+φgx2k+1≤ψnx2k,x2k+1,d,f,g,φ−ϕrx2k,x2k+1,d,g,φ,where(66)nx2k,x2k+1,d,f,g,φ=λmaxdx2k,x2k+1+φx2k+φx2k+1,dfx2k,x2k+φfx2k+φx2k+dx2k+1,gx2k+1+φx2k+1+φgx2k+1,1sdfx2k,x2k+1+φfx2k+φx2k+1+dx2k,gx2k+1+φx2k+φgx2k+1=λmaxdx2k,x2k+1+φx2k+φx2k+1,dx2k+1,x2k+φx2k+1+φx2k+dx2k+1,x2k+2+φx2k+1+φx2k+2,1sdx2k+1,x2k+1+φx2k+1+φx2k+1+dx2k,x2k+2+φx2k+φx2k+2≤12maxdx2k+1,x2k+φx2k+1+φx2k,dx2k+1,x2k+2+φx2k+1+φx2k+2≤dx2k+1,x2k+2+φx2k+1+φx2k+2,rx2k,x2k+1,d,g,φ=maxdx2k,x2k+1+φx2k+φx2k+1,dx2k+1,gx2k+1+φx2k+1+φgx2k+1=maxdx2k,x2k+1+φx2k+φx2k+1,dx2k+1,x2k+2+φx2k+1+φx2k+2≥dx2k+1,x2k+2+φx2k+1+φx2k+2.
By virtue of (65) and the above inequalities, we have(67)ψdx2k+1,x2k+2+φx2k+1+φx2k+2≤ψnx2k,x2k+1,d,f,g,φ−ϕrx2k,x2k+1,d,g,φ≤ψdx2k+1,x2k+2+φx2k+1+φx2k+2−ϕdx2k+1,x2k+2+φx2k+1+φx2k+2,which implies that dx2k+1,x2k+2+φx2k+1+φx2k+2=0. That is, x2k+1=x2k+2. Thus, x2k is a common fixed point of f and g. If x2k+1=x2k+2, then using the same arguments as in the case x2k=x2k+1, it can be shown that x2k+1 is a common fixed point of f and g.
Now takedxn,xn+1>0 for each n∈ℕ. Letting x=x2n,y=x2n+1 in (60), as the same arguments, we obtain(68)ψdx2n+1,x2n+2+φx2n+1+φx2n+2≤ψnx2n,x2n+1,d,f,g,φ−ϕrx2n,x2n+1,d,g,φ,where(69)nx2n,x2n+1,d,f,g,φ=λmaxdx2n,x2n+1+φx2n+φx2n+1,dx2n+1,x2n+φx2n+1+φx2n+dx2n+1,x2n+2+φx2n+1+φx2n+2,1sdx2n+1,x2n+1+φx2n+1+φx2n+1+dx2n,x2n+2+φx2n+φx2n+2≤12maxdx2n+1,x2n+φx2n+1+φx2n,dx2n+1,x2n+2+φx2n+1+φx2n+2,(70)rx2n,x2n+1,d,g,φ=maxdx2n,x2n+1+φx2n+φx2n+1,dx2n+1,x2n+2+φx2n+1+φx2n+2.
If for somen, dx2n+1,x2n+2+φx2n+1+φx2n+2>dx2n,x2n+1+φx2n+φx2n+1, then it follows from (68), (69), and (70) that(71)ψdx2n+1,x2n+2+φx2n+1+φx2n+2≤ψdx2n+1,x2n+2+φx2n+1+φx2n+2−ϕdx2n+1,x2n+2+φx2n+1+φx2n+2,which yields that ϕdx2n+1,x2n+2+φx2n+1=0 or equivalently(72)dx2n+1,x2n+2+φx2n+1+φx2n+2=0.
That is,dx2n+1,x2n+2=0, a contradiction. Hence,(73)dx2n+1,x2n+2+φx2n+1+φx2n+2≤dx2n,x2n+1+φx2n+φx2n+1,for all n∈N. By similar arguments, we get(74)dx2n+2,x2n+3+φx2n+2+φx2n+3≤dx2n+1,x2n+2+φx2n+1+φx2n+2.
Therefore,dxn,xn+1+φxn+φxn+1 is a nonincreasing sequence, and there exists a r≥0 such that(75)limn⟶+∞dxn,xn+1+φxn+φxn+1=r.
Ifr>0, by virtue of (68), (69), (70), and (73), one can obtain that(76)ψdx2n+1,x2n+2+φx2n+1+φx2n+2≤ψdx2n,x2n+1+φx2n+φx2n+1−ϕdx2n,x2n+1+φx2n+φx2n+1.
Taking the upper limit asn⟶+∞ in (76), we have(77)limsupn⟶+∞ψdx2n+1,x2n+2+φx2n+1+φx2n+2≤limsupn⟶+∞ψdx2n,x2n+1+φx2n+φx2n+1−limsupn⟶+∞ϕdx2n,x2n+1+φx2n+φx2n+1≤limsupn⟶+∞ψdx2n,x2n+1+φx2n+φx2n+1−liminfn⟶+∞ϕdx2n,x2n+1+φx2n+φx2n+1,which implies that ψr≤ψr−ϕr, a contradiction. It follows that(78)limn⟶+∞dxn,xn+1+φxn+φxn+1=0,which yields that(79)limn⟶+∞dxn,xn+1=0,limn⟶+∞φxn=0.
Now we shall prove thatxn is a Cauchy sequence in X. To do this, it is sufficient to prove that x2n is Cauchy. Suppose on the contrary that x2n is not Cauchy. It follows that there exists ε>0 for which one can find sequences x2mk and x2nk of x2n satisfying nk as the smallest index for which 2mk>2nk>k,(80)ε≤dx2mk,x2nk,dx2mk−2,x2nk<ε.
Using the same technique in the proof of Theorem10, we can deduce that(81)ε≤liminfk⟶+∞dx2mk,x2nk≤limsupk⟶+∞dx2mk,x2nk≤sε,(82)εs≤liminfk⟶+∞dx2mk−1,x2nk≤limsupk⟶+∞dx2mk−1,x2nk≤s2ε,(83)εs2≤liminfk⟶+∞dx2mk−1,x2nk+1≤limsupk⟶+∞dx2mk−1,x2nk+1≤s3ε,(84)εs≤liminfk⟶+∞dx2mk,x2nk+1≤limsupk⟶+∞dx2mk,x2nk+1.
Lettingx=x2nk and y=x2mk−1 in (60), we obtain(85)ψdx2nk+1,x2mk+φx2nk+1+φx2mk≤ψspdfx2nk,gx2mk−1+φfx2nk+φgx2mk−1≤ψnx2nk,x2mk−1,d,f,g,φ−ϕrx2nk,x2mk−1,d,g,φ.
Here,(86)nx2nk,x2mk−1,d,f,g,φ=λmaxdx2nk,x2mk−1+φx2nk+φx2mk−1,dfx2nk,x2nk+φfx2nk+φx2nk+dx2mk−1,gx2mk−1+φx2mk−1+φgx2mk−1,1sdfx2nk,x2mk−1+φfx2nk+φx2mk−1+dx2nk,gx2mk−1+φx2nk+φgx2mk−1=λmaxdx2nk,x2mk−1+φx2nk+φx2mk−1,dx2nk+1,x2nk+φx2nk+1+φx2nk+dx2mk−1,x2mk+φx2mk−1+φx2mk,1sdx2nk+1,x2mk−1+φx2nk+1+φx2mk−1+dx2nk,x2mk+φx2nk+φx2mk,rx2nk,x2mk−1,d,g,φ=maxdx2nk,x2mk−1+φx2nk+φx2mk−1,dx2mk−1,gx2mk−1+φx2mk−1+φgx2mk−1=maxdx2nk,x2mk−1+φx2nk+φx2mk−1,dx2mk−1,x2mk+φx2mk−1+φx2mk.
It follows from (79) to (84) that(87)limsupk⟶+∞nx2nk,x2mk−1,d,f,g,φ≤λmaxs2ε,0,s3ε+sεs≤s2ε,(88)s2ε≥liminfk⟶+∞rx2nk,x2mk−1,d,g,φ≥εs.
By virtue of (85), (87), and (88), we have(89)ψs2ε=ψs3εs≤ψsplimsupk⟶+∞dx2nk+1,x2mk+φx2nk+1+φx2mk≤ψs2ε−liminfk⟶+∞ϕrx2nk,x2mk−1,d,g,φ,which implies that(90)liminfk⟶+∞rx2nk,x2mk−1,d,g,φ=0,a contradiction to (88). Hence, xn is a Cauchy sequence. The completeness of X ensures that there exists a x∗ in X such that(91)limn⟶+∞fx2n=limn⟶+∞gx2n+1=x∗.
By the definition ofφ, we deduce that(92)φx∗≤liminfn⟶+∞φxn=0.
Now we will show that if one of the mappingsf and g is continuous, then fx∗=gx∗=x∗. Without loss of generality, we can suppose that f is continuous. It follows from (91) that(93)x∗=limn⟶+∞fx2n=flimn⟶+∞x2n=fx∗.
That is,x∗ is a fixed point of f.
From the contractive conditions (60), we get(94)ψdx∗,gx∗+φx∗+φgx∗≤ψspdfx∗,gx∗+φfx∗+φgx∗≤ψnx∗,x∗,d,f,g,φ−ϕrx∗,x∗,d,g,φ,where(95)nx∗,x∗,d,f,g,φ=λmaxdx∗,x∗+φx∗+φx∗,dfx∗,x∗+φfx∗+φx∗+dx∗,gx∗+φx∗+φgx∗,1sdfx∗,x∗+φfx∗+φx∗+dx∗,gx∗+φx∗+φgx∗=λmax0,dx∗,gx∗+φgx∗,1sdx∗,gx∗+φgx∗≤dx∗,gx∗+φgx∗,rx∗,x∗,d,g,φ=maxdx∗,x∗+φx∗+φx∗,dx∗,gx∗+φx∗+φgx∗≥dx∗,gx∗+φgx∗.
It follows from (94) that(96)ψdx∗,gx∗+φx∗+φgx∗≤ψdx∗,gx∗+φgx∗−ϕdx∗,gx∗+φgx∗,
Hence,dx∗,gx∗+φgx∗=0, that is, x∗=gx∗ and φgx∗=0. This implies that x∗ is the unique common fixed point of f and g. This completes the proof.Example 15.
LetX=0,1 and dx,y=x−y2. Define mappings f,g,φ:X⟶X by(97)fx=0,x∈0,12x16,x∈12,1,gx=x16,φx=x24,x∈0,1.Define mappingsψ,ϕ:0,+∞⟶0,+∞ with ψt=2t,ϕt=3·256t/8·1413.Now we consider two cases:Case 1.
x∈0,1/2. For y∈X, we have(98)ψspdfx,gy+φfx+φgy=2·8·y162+14y162=564y2≤145x28+1157y22048,ψnx,y,d,f,g,φ≥2·14dfx,x+φfx+φx+dy,gy+φy+φgy=12x2+x24+y−y162+y24+14y162=58x2+11572048y2,ϕrx,y,d,g,φ=3·2568·1413maxdx,y+φx+φy,dy,gy+φy+φgy≤3·11574·6922maxx−y2+x24+y24,1157y21024≤3·11574·69222x2+2y2+x24+y24+1157y21024≤3·11574·692294x2+3461y21024≤3458x2+11572048y2.
It follows that(99)ψspdfx,gy+φfx+φgy≤145x28+1157y22048=5x28+1157y22048−345x28+1157y22048≤ψnx,y,d,f,g,φ−ϕrx,y,d,g,φ.Case 2.
x∈1/2,1. For y∈X, one can obtain (100)ψspdfx,gy+φfx+φgy=2·8·x16−y162+14x162+14y162=116x−y2+x2+y2128≤18x−y2+x2+y24,ψnx,y,d,f,g,φ≥2·14dx,y+φx+φy=12x−y2+x2+y24,ϕrx,y,d,g,φ=3·2568·1413maxdx,y+φx+φy,dy,gy+φy+φgy=3·2568·1413maxx−y2+x24+y24,1157y21024≤38x−y2+x2+y24.
It follows that(101)ψspdfx,gy+φfx+φgy≤18x−y2+x2+y24=12x−y2+x2+y24−38x−y2+x2+y24≤ψnx,y,d,f,g,φ−ϕrx,y,d,g,φ.Therefore, all conditions of Theorem10 are satisfied with λ=1/4, s=2,p=3. Theorem 10 ensures that f and g has a unique common fixed point. It is easy to get that 0 is the unique common fixed point of f and g.Note that, takingS=T=Ix in Theorem 2.1 of [11], Roshan et al. give the existence of common fixed point for mappings f,g such that(102)dfx,gy≤qs4maxdx,y,dfx,x,dgy,y,12dx,fy+dfx,y,where q∈0,1 is a constant. For x=0,y∈1/2,1, it is easy to calculate that for q<1/16,(103)dfx,gy=y2256>q16y2=q16maxy2,0,225y2256,y22·256+y22=qs4maxdx,y,dfx,x,dgy,y,12dx,fy+dfx,y,which implies that Theorem 2.1 of [11] cannot be applied to testify the existence of common fixed points of the mappings f and g in X.Ifφ=0 in Theorem 10, we can get the following result:Corollary 16.
LetX,d be a complete b-metric space with parameter s≥1 and let f,g:X⟶X be given self-mappings, and one of f and g is continuous. Suppose p≥3,0<λ≤1/4 are two constants. If there are functions ψ∈Ψ and ϕ∈Φ such that(104)ψspdfx,gy≤ψn1x,y,d,f,g,φ−ϕr1x,y,d,g,φ,where(105)n1x,y,d,f,g,φ=λmaxdx,y,dfx,x+dy,gy,1sdfx,y+dx,gy,r1x,y,d,g,φ=maxdx,y,dy,gy,then f and g have a unique common fixed point in X.If we consider the corresponding problem in the setting of metric space, that is,s=1 in Theorem 10, we get the following:Corollary 17.
LetX,d be a complete metric space and let f,g:X⟶X be given self-mappings, and one of f and g is continuous. Suppose φ:X⟶0,∞ is a lower semicontinuous function and 0<λ≤1/4 is a constant. If there are functions ψ∈Ψ and ϕ∈Φ such that(106)ψdfx,gy+φfx+φgy≤ψn2x,y,d,f,g,φ−ϕrx,y,d,g,φ,where(107)n2x,y,d,f,g,φ=λmaxdx,y+φx+φy,dfx,x+φfx+φx+dy,gy+φy+φgy,1sdfx,y+φfx+φy+dx,gy+φx+φgy,and rx,y,d,g,φ is the same as Theorem 10, then f and g have a unique common fixed point in X.Theorem 18.
LetX,d be a complete b-metric space with parameter s≥1, and let f:X⟶X be a given self-mapping and φ:X⟶0,+∞ be a lower semicontinuous function with φt=0 for t∈Fixf. Suppose p≥2 is a constant. If there are functions ψ∈Ψ and ϕ∈Φ such that(108)ψspdfx,fy+φfx+φfy≤ψhx,y,d,f,φ−ϕqx,y,d,f,φ,where(109)hx,y,d,f,φ=maxdx,y+φx+φy,dx,fx+φx+φfx,dy,fy+φy+φfy,12sdx,fy+φx+φfy+dy,fx+φy+φfx,qx,y,d,f,φ=maxdx,y+φx+φy,dy,fy+φy+φfy,then f has a unique fixed point z∈X.Proof.
The proof is similar to that of Theorem10, so we omit it.Remark 19.
(i)
Ifs=1 in Theorem 18, then we get Theorem 2.(ii)
Ifs=1 and φ=0 in Theorem 18, then we get Theorem 1.According to Theorem18, we can obtain the following result:Corollary 20.
LetX,d be a complete b-metric space with parameter s≥1, and let f:X⟶X be a given self-mapping and φ:X⟶0,+∞ be a lower semicontinuous function with φt=0 for t∈Fixf. Suppose p≥2 is a constant. If there are functions ψ∈Ψ and ϕ∈Φ such that(110)ψspdfx,fy+φfx+φfy≤ψhx,y,d,f,φ−ϕhx,y,d,f,φ,where hx,y,d,f,φ is the same as Theorem 18, then f has a unique fixed point z∈X.


## 4. Application

It is well known that an automobile suspension system is the realistic application for the spring mass system in engineering problems. Consider the motion of a spring of a car when it moves along a rough and pitted road, where the forcing term is the rough road and shock absorbers provide the damping. The external forces under which the system operates may be gravity, ground vibrations, earthquake, tension force, etc. Letm be the mass of the spring and F be the external force acting on it; then, the critical damped motion of this system subjected to the external force F is governed by the following initial value problem:(111)md2xdt2+ldxdt−mFt,xt=0,x0=0,x′0=0,where l>0 is the damping constant and F:0,T×ℝ+⟶ℝ is a continuous function.It is easy to show that the problem (111) is equivalent to the integral equation:(112)xt=∫0Tγt,rFr,xrdr,t∈0,T,where γt,r is Green’s function given by(113)γt,r=1−eμt−rμ,0≤r≤t≤T0,0≤t≤r≤T,where μ=l/m is a constant.In this section, by using Corollary20, we will show the existence of a solution to the integral equation:(114)xt=∫0TGt,r,xrdr.LetX=C0,T be the set of real continuous functions defined on 0,T. For p≥1, we define(115)dx,y=ρx,yp=supt∈0,Txt−ytpforallx,y∈X.It is easy to prove thatX,d is a complete b-metric space with s=2p−1.Consider the mappingf:X⟶X defined by(116)fxt=∫0TGt,r,xrdr.Theorem 21.
Consider equation (114) and suppose that(i)
G:0,T×0,T×ℝ⟶ℝ+ is continuous(ii)
there exists a continuous functionγ:0,T×0,T⟶ℝ+ such that(117)supt∈0,T∫0Tγt,rdr≤1,(iii)
there exists a constantL∈0,1 such that for t,r∈0,T×0,T,(118)Gt,r,xr−Gt,r,yr≤1−Lsppγt,rxr−yr.Then, the integral equation (114) has a unique solution x∈X.Proof.
Forx,y∈X, by virtue of assumptions (i)–(iii), we have(119)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,r∣xr−yr∣drp≤spsupt∈0,T∫0T1−Lsppγt,rdrpsupt∈0,Txt−ytp≤1−Lhxt,yt,d,f,φ,which implies that(120)spdfxt,fyt≤1−Lhxt,yt,d,f,φ.Therefore, lettingψt=t, ϕt=Lt, and φt=0, all the conditions of Corollary 20 are satisfied. As a result, the mapping f has a unique fixed point x∈X, which is a solution of the integral equation (114).Remark 22.
If we letGt,r,xr=γt,rFr,xr, ∣Fr,xr−Fr,yr∣≤1−L/spp∣xr−yr∣ and μ,T satisfy μ≥T, then all the conditions of Theorem 21 are satisfied, which implies that the problem (111) has a unique solution.


## 5. Conclusions

In this manuscript, we introduced a new class of generalized weakly contractive mappings and established common fixed point results involving this new class of mappings in the framework ofb-metric spaces. Further, we provided examples that elaborated the useability of our results. Meanwhile, we presented an application to the existence of solutions to an integral equation by means of one of our results.



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