# Fixed Point of Generalized Weak Contraction inb-Metric Spaces

Authors: Maryam Iqbal; Afshan Batool; Ozgur Ege; Manuel de la Sen
Journal: Journal of Function Spaces
 (2021)

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## Abstract

In this manuscript, a class of generalizedψ,α,β-weak contraction is introduced and some fixed point theorems in the framework of b-metric space are proved. The result presented in this paper generalizes some of the earlier results in the existing literature. Further, some examples and an application are provided to illustrate our main result.


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## Body

## 1. Introduction and Preliminaries

Fixed point theory plays a vital role in the development of nonlinear functional analysis. It has been used in various branches of engineering and sciences. Banach contraction principle is one of the most important results in fixed point theory introduced by great Polish mathematician Stefan Banach [1]. The concept of b-metric space or metric-type space was first introduced by Czerwik [2]. He provided a property which is weaker than the triangular inequality. The basic idea of b-metric was commenced by Bourbaki [3] and Bakhtin [4]. Later on, Khamsi and Hussain [5] reintroduced such spaces under the name of metric-type spaces for some results of fixed and common fixed points in the setting of b-metric spaces. Since then, several authors proved fixed point results of single valued and multivalued operators in b-metric space and its different type generalizations, we refer [6–22]. Every one of these applications captivated us to present the idea of b-metric space.Definition 1.
(see [23]). Let X be a nonempty set. A function d:X×X⟶0,∞ is said to be a b-metric if it satisfies the following conditions:(1)10≤dx,yanddx,y=0ifandonlyifx=y2dx,y=dy,x3dx,z≤sdx,y+dy,zforsomes≥1,for all x,y,z∈X. The pair X,d is called a b-metric with coefficient s.Here, we observe that every metric space is ab-metric with s=1. Conditions (1) and (2) of Definition 1 are similar to metric space but it is important how to use (3) effectively. An example is given to expound the concept of a third condition.Example 1.
LetX=ℝ. We define a mapping d:X×X⟶ℝ+ such that(2)dx,y=x−y2,∀x,y∈X.The first two conditions of Definition1 are clearly shown. The solution of third condition is as follows:(3)dx,z=x−z2=x2+z2−2xz≤x2+z2−2xz+x2+z2+4y2−4xy−2.Since(4)x2+z2+4y2>2x2y+z,x2+z2+4y2−2x2y+z>0,we have(5)dx,z≤x2+z2−2xz+x2+z2+4y2−4xy−2=2x2+4y2+2z2−4xy−4xz=2x2+2y2+z2−2xy−2xz=2x2+y2−2xy+y2+z2−2xz=2x−y2+y−z2=2dx,y+dy,z.Then, we obtain(6)dx,z≤2dx,y+dy,z.So the value of coefficient iss=2.In this section, the concept of generalizedψ,α,β-weak contraction for metric space is provided with some basic notions and results. In 1997, Alber and Guerre-Delabriere [24] suggested a generalization of Banach contraction mapping by introducing the concept of ϕ-weak contraction in Hilbert space. In 2008, Dutta and Choudhury [25] gave a generalization of weakly contractive mapping by defining ψ,ϕ-weak contraction in complete metric spaces.Definition 2.
(see [26]). Let Ψ denote the class of function ψ:0,∞⟶0,∞ which satisfies the following conditions:(1)
ψ is continuous and nondecreasing(7)ψt=0⟺t=0.Definition 3.
(see [26]). A self-map P is said to be a weakly contractive map if there exists a function ϕ:0,+∞⟶0,+∞ such that ϕ is continuous, nondecreasing, and ϕt=0⟺t=0 and satisfying(8)dPx,Py≤dx,y−ϕdx,y,∀x,y∈X.Theorem 4.
(see [26]). LetX,d be a complete metric space and P be a weakly contractive self-map on X. Then, P has a unique fixed point in X.Definition 5.
(see [25]). A self-map P is said to be ψ,ϕ-weak contraction, if for each x,y∈X,(9)ψdPx,Py≤ψdx,y−ϕdx,y,where ψ,ϕ:0,∞⟶0,∞ are both continuous and monotone nondecreasing functions with ψt=0=ϕt⟺t=0.Theorem 6.
(see [25]). Let X,d be a complete metric space and a self-map P be a ψ,ϕ-weak contraction. Then, P has a unique fixed point.Definition 7.
(see [27]) Two self-maps P and Q are said to be generalized ϕ-weakly contractive map if there exists a function ϕ:0,+∞⟶0,+∞ such that ϕ is continuous, nondecreasing, and ϕt=0⟺t=0 and satisfying(10)dPx,Qy≤Mx,y−ϕMx,y,where(11)Mx,y=maxdx,y,dx,Px,dy,Qy,dy,Px+dx,Qy2,∀x,y∈X.Theorem 8.
(see [27]). Let X,d be a complete metric space and T and R are generalized ϕ-weakly contractive self-maps on X. Then, P and Q have a unique common fixed point in X.Definition 9.
(see [28]). Two self-maps P and Q are said to be generalized ϕ,ψ-weakly contractive maps if they satisfy(12)ψPx,Qy≤Mx,y−ϕMx,y,
∀x,y∈X, where ϕ:0,+∞⟶0,+∞ such that ψ is continuous, nondecreasing, and ψt=0⟺t=0, ϕ:0,+∞⟶0,+∞ such that ϕ is a lower semicontinuous function, ϕt=0⟺t=0 and(13)Mx,y=maxdx,y,dx,Px,dy,Qy,dy,Px+dx,Qy2,∀x,y∈X.Theorem 10.
(see [28]). Let X,d be a complete metric space and P and Q are generalized ϕ,ψ-weakly contractive self-maps on X. Then, P and Q have a unique common fixed point in X.Definition 11.
(see [29]). Let X,≼ be a partially ordered set. Two mappings P,Q:X⟶X are said to be weakly increasing if Px≼QTx and Qx≼PQx for all x∈X.Remark 12.
(see [29]). Note that two weakly increasing mappings need not be nondecreasing.Definition 13.
(see [30]). Let X,d be a metric space and P,Q:X⟶X are given two self-mappings on X. The pair P,Q is said to be compatible if limn⟶∞dPQxn,QPxn=0, whenever xn is a sequence in X such that(14)limn⟶∞Pxn=limn⟶∞Qxn=t,forsomet∈X.Definition 14.
(see [1]). Let X,≼ be a partially ordered set and let X,d be a b-metric space with coefficient s≥1. Three maps T, R, and S are said to be a generalized ψ,α,β-weak contraction if for each x,y∈X,(15)ψdPx,Qy≤αdSx,SyβdSx,Sy,where ψ∈Ψ and β:0,∞⟶0,∞ is a continuous function with condition(16)0<βt<ψt,∀t>0.Theorem 15.
(see [1]). Let X,≼ be a partially ordered set and assume that there exists a metric function d in X such that X,d is a complete metric space. Let P,Q,S:X⟶X are generalized ψ,α,β-weak contraction mappings satisfying the following properties:(17)PX⊆SX,(1)
P, Q, and S are continuous(2)
The pairsP,S and Q,S are compatible(3)
P and Q weakly increasing with respect to S(4)
Sx and Sy are comparable(5)
∀x,y∈X×X, there exists u∈X such that Px≼Pu and Py≼Pu
Then,P, Q, and S have a unique common fixed point z∈X.The next section includes the concept of generalizedψ,α,β-weak contraction for b-metric space and theorem related to it.


## 2. Generalizedψ,α,β-Weak Contractions

Definition 16.
LetX,≼ be a partially ordered set and let X,d be a b-metric space with coefficient s≥1. Three maps P, Q, and S are said to be a generalized ψ,α,β-weak contraction if for each x,y∈X and b∈0,1.(18)bsψdPx,Qy≤αdSx,SyβdSx,Sy,∀x≥y,where α∈F,ψ∈Ψ and β:0,∞⟶0,∞ is a continuous function with the condition(19)0<βt<ψt,∀t>0.Theorem 17.
LetX,≼ be a partially ordered set and assume that there exist a b-metric function d in X such that X,d is complete b-metric space. Let T,R,S:X⟶X are generalized ψ,α,β-weak contraction mappings satisfying the following properties:(20)PX⊆SX,(1)
P, Q, and S are continuous(2)
The pairP,S and Q,S are compatible(3)
P and Q weakly increasing with respect to S(4)
Sx and Sy are comparable(5)
∀x,y∈X×X, there exist u∈X such that Px≼Pu and Py≼Pu
Then,P, Q, and S have a unique common fixed point z∈X.Proof.
The proof is done by using the concept of Banach contraction principle in which a Cauchy sequence is taken in completeb-metric space. Every Cauchy sequence is convergent in a complete metric space, and converging point of that sequence is proved to be a fixed point of contraction. ☐Let us assume thatx0∈X be an arbitrary point in X. By property (1), there exist x1,x2∈X such that Px0=Sx1 and Qx1=Sx2. Continuing this process, sequences xn and yn can be constructed in X, defined as(21)Sx2n+1=Px2n=y2n,Sx2n+1=Qx2n+1=y2n+1,∀n∈ℕ.By using property (4), we obtain(22)Sx1=Px0≼Qx1=Sx2.Similarly,(23)Sx2=Px1≼Qx2=Sx3.Continuing this process, we get(24)Sx1≼Sx2≼Sx3≼Sx4≼⋯≼Sx2n+1≼Sx2n+2≼⋯.Thus,(25)y0≼y1≼y2≼⋯≼y2n≼y2n+1≼⋯.According to our first supposition, if there existsn∈ℕ such that y2n−1=y2n, then from (18),(26)bsψdy2n,y2n+1=bsψdPx2n,Qx2n+1≤bsαdSx2n,Sx2n+1βdSx2n,Sx2n+1=bsαdy2n−1,y2nβdy2n−1,y2n=0,which implies that y2n+1=y2n. Consequently, ym=y2n−1 for any m≥2n. Hence, for every m≥2n, we have Sxm=Sx2n which implies that Sxn is a b-Cauchy sequence.According to our second supposition,yn≠yn+1 for any integer n. Let zn=dyn,yn+1. Now, we have to show that zn⟶0 as n⟶∞. Since Sx2n and Sx2n+1 are comparable, then from (18), we have(27)bsψdy2n+2,y2n+1=bsψdSx2n+3,Sx2n+2=bsψdPx2n+2,Qx2n+1≤bsαdSx2n+2,Sx2n+1βdSx2n+2,Sx2n+1=bsαdy2n+1,y2nβdy2n+1,y2n.By property (2) ofψ and the fact that α∈F, we get(28)dy2n+2,y2n+1≤dy2n+1,y2n.Similarly, we have(29)dy2n+1,y2n≤dy2n,y2n−1.By combining (28) and (29), we obtain(30)dy2n+2,y2n+1≤dy2n+1,y2n≤dy2n,y2n−1.This shows that the sequencezn is monotonically decreasing. So there exists r≥0 such that(31)limn⟶∞zn=dyn,yn+1=r.Supposer≥0. Then,(32)bsψdy2n+2,y2n+1≤bsαdy2n+1,y2nβdy2n+1,y2n.Taking the limit asn⟶∞, we get ψr≤αr≤βr. Since α∈F, by using (19), we have ψr<αr<βr but this is a contradiction. Then, r=0. Hence,(33)limn⟶∞zn=dyn,yn+1=0.Next, we have to show thatSxn is a b-Cauchy sequence. We prove this by contradiction. Now, we suppose that Sx2n is not a b-Cauchy sequence. Then, for any ε>0, there exist two subsequences of positive integers mk and nk such that nk>mk for all positive integers k,(34)dSx2mk,Sx2nk>ε,(35)dSx2mk,Sx2nk−2≤ε.From (34) and (35) and by using triangle inequality, we get(36)ε≤dSx2mk,Sx2nk−2≤dSx2mk,Sx2nk−2+dSx2nk−2,Sx2nk−1+dSx2nk−1,Sx2nk.Letk⟶∞ in the above inequality and by using (33), we obtain(37)limk⟶∞dSx2mk,Sx2nk=ε.Again by using triangle inequality, we have(38)dSx2nk,Sx2mk−1≤dSx2nk,Sx2mk+dSx2mk,Sx2mk−1.By taking limit ask⟶∞ in above inequality and using (33)–(35), we get(39)limk⟶∞dSx2nk,Sx2mk−1=ε.Moreover, we obtain(40)dSx2nk,Sx2mk≤dSx2nk,Sx2nk+1+dSx2nk+1,Sx2mk=dSx2nk,Sx2nk+1+dPx2nk,Qx2mk−1.Using inequalities (33)–(37) and letting limk⟶∞, we have(41)ε≤limk⟶∞dPx2nk,Qx2mk−1.However,ψ∈Ψ, therefore(42)ψε≤limk⟶∞ψdPx2nk,Qx2mk−1.From (18), we have(43)bsψdPx2nk,Qx2mk−1≤bsαdSx2nk,Sx2mk−1βdSx2nk,Sx2mk−1.Taking limit ask⟶∞ in the above inequality and using that fact that α∈F, we have(44)limk⟶∞ψdPx2nk,Qx2mk−1<βε.From (43) and (44), and using (19), we obtain(45)ψε≤limk⟶∞ψdPx2nk,Qx2mk−1<βε<ψε.This is a contradiction. Therefore,x2n is a b-Cauchy sequence, and hence, xn is a b-Cauchy sequence for all n∈ℕ. Hence, there exists u∈X such that(46)limn⟶∞Sxn=u.Next, claim thatu is a coincidence point of P, Q, and S. From (46) and the continuity of S, we get(47)limn⟶∞SSxn=Su.From triangle inequality, we have(48)dSu,Pu≤dSu,SSx2n+1+dSTx2n,PSx2n+dPSx2n,Tu,From (46) and (21), we have(49)Sx2n⟶u,Px2n⟶u.Since pairP,S are compatible, then(50)dSPx2n,PSx2n⟶0.Using the continuity ofP and (58), we get(51)dPSx2n,Pu⟶0,letting k⟶∞ in (48) and using (47)–(50) together with (51), we find(52)dSu,Pu≤0,which means that Su=Pu. Similarly, from triangle inequality, we have(53)dSu,Qu≤dSu,SSx2n+2+dSQx2n+1,QSx2n+1+dQSx2n+1,Qu.In a similar way, we obtaindSu,Qu≤0 which means that Su=Qu. Thus, we find that Su=Qu=Pu, that is, u is a coincidence point of P, Q, and S.Now, we use the property (6) to show thatu is a common fixed point of P, Q, and S. For this, we prove that P, Q, and S have a common fixed point. To prove this, we show that if p and q are coincidence points of P, Q, and S, i.e.,(54)Sp=Pp=Qp,Sq=Pq=Qq.Then,(55)Sp=Sq.From our assumption mentioned in property (6), there existsu0∈X such that(56)Tp≼Tu0,Tq≼Tu0.Now, we can define a sequenceSun as follows:(57)Su2n+1=Pu2n,Su2n+2=Qu2n+1,∀n∈ℕ.Again, we have(58)Pp=Sp≼Sun,Pq=Sq≼Su2n,∀n∈ℕ.Now, puttingx=un and y=p in (18), we get(59)bsψdSu2n+1,Sp=bsψdPu2n,Qp≤bsαdSu2n,SpβdSu2n,Sp.Sinceα∈F, we have the next inequality:(60)bsψdSu2n+1,Sp≤bsβdSu2n,Sp.Similarly, again writingy=u2n and x=p in (18), we find(61)bsψdSu2n+2,Sp≤bsβdSu2n+1,Sp.By combining (60) and (61), we obtain for all n∈ℕ,(62)bsψdSu2n+1,Sp≤bsβdSu2n,Sp.Consequently, by using property ofψt and βt, we get(63)dSu2n+1,Sp≤dSu2n,Sp.Therefore, there existsr≥0 such that(64)limn⟶∞dSun,Sp=r.Letr>0, then from (18), we obtain(65)bsψdSu2n+1,Sp≤bsαdSu2n,SpβdSu2n,Sp,on taking limit as n⟶∞ and using (19), we get(66)ψr<βr<ψr.This is a contradiction. Thus,r=0 and from (64), we obtain(67)limn⟶∞dSun,Sp=0.In the same pattern, we can show that(68)limn⟶∞dSun,Sq=0.Now, using the fact that the limit is unique and by using (55)–(68), we can write(69)limn⟶∞dSPu2n,PSu2n=0,limn⟶∞dSQu2n+1,QSu2n+1=0,limn⟶∞Pu2n=Sp=Sq,limn⟶∞Qu2n+1=Sp=Sq.Since the pairP,S and Q,S are compatible, we have(70)limn⟶∞dSPu2n,PSu2n=0,limn⟶∞dSQu2n+1,QSu2n+1=0.Let us takez=Sp. Consider(71)dSz,Pz≤dSz,SPu2n+dSTu2n,PSu2n+dPSu2n,Pz.Lettingn⟶∞ and using the continuity of P, we get the above inequality as(72)dSz,Pz≤0.That is,Sz=Pz and z is coincidence point of P and S. Similarly, dSz,Qz≤0. That is, Rz=Sz and z is the coincidence point of Q and S. Hence, from (55), we have(73)z=Sp=Sz=Pz=Qz.This proves thatz is a common fixed point of P, Q, and S. Now, we show that z is unique common fixed point. We will show this by contradiction. Assume that z is not unique, therefore, there exists another fixed point υ as(74)υ=Sp=Sυ=Pυ=Qυ.By using (55), we have Sυ=Sz. Hence, we get(75)υ=Sυ=Sz=z,but this is a contradiction to our assumption, and hence, a common fixed point is unique.An example is given to support the main result because there is limited examples in the literature.Example 2.
LetX=ℝ is the set of real numbers and a b-metric d:X×X⟶0,∞ is defined as(76)dx,y=x−y2,∀x,y∈X,with coefficient s=2 and b=0.01. Consider three self-maps P,Q,S:ℝ⟶ℝ defined as(77)Px=2x,Qx=5−2x,Sx=x3.
Define mapsψ,β:0,∞⟶0,∞ and α:0,∞⟶0,1 as(78)ψt=t2,.β=t3,αt=73.
After substituting values in (18),(79)bsψdPx,Qy≤αdSx,SyβdSx,Sy,we have(80)0.012ψ2x−5+2y2≤αdx3,y3βdx3,y3,0.0122x−5+2y22≤73x−y227.
Now, forx,y=0,1, we get(81)0.00045≤0.08,
Forx,y=1,2, we have(82)0.00005≤0.08,
Then, clearly, three mapsP, Q, and S are generalized ψ,α,β-weak contraction for all values of x,y∈ℝ.


## 3. An Application to Fredholm Integral Equations

In this section, applying Theorem17, we give an existence theorem for common solutions of Fredholm integral equations where the upper limits of equations are taken to be the coefficient of b-metric space s≥1. Here, we consider the following integral equations:(83)∫0sP1t,l,vldl+ϰt,(84)∫0sP2t,l,vldl+ϰt,for all t∈0,s where s≥1. Let us consider the space X=CII=0,s is a set of continuous functions defined on 0,s. Obviously, the space CI with the metric dp,q=maxt∈Ixt−yt for all x,y∈X is a complete metric space.Here,b-metric space is defined on the partially ordered set. So, X can be prepared with partial order ≼ given by(85)forallx,y∈X,x≼y:⟺xt≼yt,forallt∈I.Theorem 18.
Suppose the following hypotheses hold:(1)
P1,P2:I×I×R⟶R and λ:R⟶R are continuous(2)
The following inequalities hold:(86)P1t,l,ul≤P2t,l,∫0sP1l,r,urdr+ϰl,P2t,l,ul≤P1t,l,∫0sP2l,r,urdr+ϰl(3)
There exists a continuous functionξ:I×I⟶R+ such that(87)bsP1t,l,x−P2t,l,y≤ξt,llogx−y2+sx−y,for all t,l∈I and x,y∈X such that x≼y.(88)maxt∈Iξt,i∫0sξt,i2dl≤1s
Then, the integral equations (83) and (84) have the unique solution v∗∈CI.Proof.
Let us defineP,Q:CI⟶CI by(89)Pxt=∫0sT1t,l,vldl+ϰt,Qxt=∫0sT2t,l,vldl+ϰt,where t∈I and p∈CI. Here, P and Q are considered to be weakly increasing function according to the requirement of our result.Now, for allp,q∈CI such that p≼q, we have(90)bsPxt−Qyt=∫0sP1t,l,vl−P2t,l,vldl≤∫0sξt,llogxl−yl2+sxl−yldl,where b∈0,1.Using the Cauchy-Schwarz inequality in the R.H.S., we get(91)∫0sξt,llogxl−yl2+sxl−yldl≤∫0sξ2t,ldl∫0slogxl−yl2+sxl−yldl.By using hypotheses (4), we have(92)∫0sξt,llogxl−yl2+sxl−yldl≤1s∫0slogxl−yl2+sxl−yldl≤1slogdx,y2+sdx,ys≤logdx,y2+sdx,y.This implies the following:(93)bsPxt−Qyt=bsdPx,Py≤logdx,y2+sdx,y,bsdPx,Py2≤logdx,y2+sdx,y≤logdx,y2+sdx,ylogdx,y2+s,Suppose we choose the values ofα,β,ψ as logt2+s/t, logt2+s, and t2, respectively. Therefore, from inequality (93),(94)bsψdPx,Qy≤αdSx,SyβdSx,Sy,∀x≥ySince all the hypotheses of Theorem17 are satisfied so there exists a unique common fixed point v∗∈CI of P and Q, that is, the solution of equations (83) and (84).



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