# CyclicG‐Ω-Weak Contraction-Weak Nonexpansive Mappings and Some Fixed Point Theorems in Metric Spaces

Authors: Sahar Mohamed Ali Abou Bakr
Journal: Abstract and Applied Analysis
 (2021)

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## Abstract

This paper introduces novel concepts of jointY,Z cyclic G‐ΩS,T,abef-weak contraction and joint Y,Z cyclic G‐ΩS,T,abef-weak nonexpansive mappings and then proves the existence of a unique common fixed point of such mappings in case of complete and compact metric spaces, respectively, in particular, it proves the existence of a unique fixed point for both cyclic G‐ΩS,abef-weak contraction and cyclic G‐ΩS,abef-weak nonexpansive mappings, and hence, it also proves the existence of a unique fixed point for both cyclic ΩS,abef-weak contraction and cyclic ΩS,abef-weak nonexpansive mappings. The results of this research paper extend and generalize some fixed point theorems previously proved via the attached references.


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## Body

## 1. Introduction, Notations, and Basic Definitions

In 1922, Banach [1] proved the highly important fixed point principle for contraction mappings; this principle has many applications in different branches of mathematics, including variational linear inequalities, optimization, differential equations, approximation theory, and in minimum norm problems.Since then, exclusive interest has been shown by researchers and many research papers have been written to generalize this Banach contraction principle. Step by step, we have the following historical generalizations:LetX,D be a metric space and T be a self mapping on X, T:X⟶X satisfying
(1)DTx,Ty≤aDx,y+bDy,Ty+cDx,Tx++eDTx,y+fDTy,x∀x,y∈X.In case of normed space, letX,. be a normed space, C be a subset of X, and T be a self-mapping on C satisfying
(2)Tx−Ty≤ax−y+by−Ty+cx−Tx+eTx−y+fTy−x∀x,y∈C, for some real numbers a,b,c,e,f∈0,1. Then, the class of all mappings defined by (1) or by (2) includes the following types of mappings.The contraction type: [0≤a<1, b=c=e=f=0]. In 1922, Banach [1] initiated the field of fixed point theory. Indeed, his famous Banach contraction principle proved that every contraction mapping on a complete metric space has a ufp.The nonexpansive type:a=1,b=c=e=f=0. In 1965, Kirk [2] proved interesting fixed point theorems (fp may not be unique) concerning nonexpansive type provided that C is bounded closed and convex subset of a Banach space X.The Kannan type:a=0,b=c=1/2,e=f=0. In 1971 and 1975, Kannan and Wong [3, 4], respectively, enriched the fixed point theory by some results concerning such type.The Chatterjea type:b=c∈0,1/2,a=e=f=0. In 1972, Chatterjea [5] proved that every Chatterjea type contraction on a complete metric space has a ufp.The Hardy-Rogers type: [a,b,c,e, and f∈0,1 with a+b+c+e+f<1]. In 1973, Hardy and Rogers [6] proved that Hardy-Rogers type on a complete metric space has a ufp. In 2007, Sahar [7] proved that (e=f=0) this type has a ufp in weakly Cauchy normed space, and recently, in 2020 and 2021, Sahar [8, 9] proved the existence of a ufp in complete theta-cone metric spaces.The general-Gregus type: [0<a<1, b,c≥0, and a+b+c+e+f=1]. In 1973, Goebel et al. [10] proved the existence of fixed points of this type, where X is assumed to be uniformly convex Banach space, T is continuous, and C is bounded closed and convex subset of X.The Gregus type: [0<a<1, b≥c≥0,e=f=0, and a+b+c=1]. In 1980, Gregus [11] proved the existence of a ufp of this type provided that C is a closed convex subset of a Banach space X. His theorem is now known as the Gregus fixed point theorem.The general-Gregus type: [0<a<1, b,c≥0, and a+b+c+e+f=1]. In 1977/1980/1981, Gregus, Park, and Rhoades [12, 13] independently proved the existence of a ufp for such a type provided that X is a compact metric space and T is continuous.In 2013, another extension of the Banach contraction principle and Gregus fixed point theorem was given by Sahar [14]; it states the following.Theorem 1 (see [14]).
LetC be a closed convex and weakly Cauchy subset of a normed space X and T:C⟶C be a self-mapping satisfying
(3)Tx−Ty≤ax−y+bx−Tx+cmaxy−Ty,∥Tx−y∥∀x,y∈C,for some real numbers a,b,c∈0,1 with 0<a<1, 0<b, 0≤c<1/2, and a+b+c≤1. Then, T has a ufp.We recall the following definition and classes of mappings.From now on, the classesΦ, Ψ, and Φ0 denote the following:
(4)Φ≔ϕ:ϕ:0,+∞⟶0,+∞,nondecreasing,ϕt=0⇔t=0,Ψ≔ψ:ψ∈Φ,ψcontinuous,Φ0≔ϕ:ϕ∈Φ,ϕlowersemicontinuous.Definition 2.
LetX be a normed space. Then, a function f from X into −∞,+∞ is said to be lower semicontinuous if and only if for any real number α, the level set x∈X:fx≤α is a closed convex subset of X.In 1997, Alber and Guerre-Delabriere [15] introduced the concept of weakly ϕ-contractive mappings and proved some fixed point theorems concerning them.Definition 3. (see [15]).
A self-mappingS:X⟶X is said to be weakly ϕ-contractive if there is ϕ∈Ψ such that
(5)DSx,Sy≤Dx,y−ϕDx,y∀x,y∈X.Remark 4.
Every contraction mapping is weaklyϕ-contractive with ϕt=1−at,t∈0,+∞, where a is a real number with 0<a<1. The converse is not true.In 2001, Rhoades [16] delivered a study on the existence of fixed points of weakly ϕ-contractive mappings and gave further generalized version of Banach’s fixed point theorem. He showed that every weakly ϕ-contractive self-mapping on a complete metric space has a ufp.In 2008, Dutta and Choudhury [17] generalized the concept of weakly ϕ-contractive mappings and proved the existence of fp for ψ‐ϕ-weak contraction mappings.Definition 5.
A self-mappingT:X⟶X is said to be a ψ‐ϕ-weak contraction iff there are two mappings ψ∈Ψ and ϕ∈Φ such that the following is true:
(6)ψDTx,Ty≤ψDx,y−ϕDx,y∀x,y∈X.
In 2011, Karapinar and Sadarangani [18, 19] gave some generalized fixed point theorems for cyclic ψ‐ϕ-weak contraction type of mappings.In 2012, Karapinar and Kumar [20] introduced the notion of cyclic weakly generalized Chatterjea mappings in metric spaces and proved corresponding fp theorems.Definition 6.
LetY and Z be nonempty subsets of X, X=Y∪Z and T:X⟶X be a self mapping. Then X=Y∪Z is said to be a cyclic representation of X with respect to TiffTY⊂Z and TZ⊂Y.Definition 7.
LetY and Z be nonempty subsets of X, and X=Y∪Z be a cyclic representation of X with respect to T. If there is a real number r∈0,1/2 and there is a continuous function ϕ:0,∞×0,∞⟶0,∞ satisfying ϕt,s=0ifft=s=0, then T is said to be cyclic weakly Chatterjea type contraction iffT satisfies
(7)DTx,Ty≤rDx,Ty+Dy,Tx−ϕDx,Ty,Dy,Tx∀x∈Y,y∈Z.
Recall in [21] that a real valued function ΓS,abc:X×X⟶ℝ+ is defined by
(8)ΓS,abcx,y=aDx,y+bDx,Sx+cDy,Sy∀x,y∈X.Definition 8.
LetX=Y∪Z be a cyclic representation of X with respect to S and a,b,c∈0,1 be real constants such that
(9)DSx,Sy≤ΓS,abcx,y∀x∈Y,y∈Z.(1)
Ifa+c+b<1, then S is said to be cyclic generalized abc-contraction w.r.t Y,Z(2)
Ifa+c+b=1, then S is said to be cyclic generalized abc-nonexpansive w.r.t Y,ZDefinition 9 (see [21]).
Letϕ be an element belonging to Φ0 such that
(10)DSx,Sy≤ΓS,abcx,y−ϕΓS,abcx,y∀x∈Y,y∈Z.(1)
Ifa+c+b<1, then S is said to be generalized cyclic ϕ−abc weak contraction w.r.t Y,Z.(2)
Ifa+c+b=1, then S is said to be generalized cyclic ϕ−abc weak non-expansive w.r.t Y,Z.In 2020, for these generalized types of contraction mappings, Sahar [21] gave the following result.Theorem 10.
IfX,D is a metric space, X=Y∪Z, Y, and Z are compact, and S:X⟶X is continuous generalized cyclic ϕ‐abc-weak nonexpansive mapping w.r.t Y,Z, then there is a single point u∈Y∩Z such that Su=u. Moreover, for every u0∈X, we have limn⟶∞Snu0=u.Fixed point theorems corresponding to these types of contraction mappings have many applications, not exclusively; see [22, 23] and also see the applications of the prior results given via the references mentioned in these research papers.Motivated from the useful applications of the fixed point theorems corresponding to such types of mappings in variety fields and areas of mathematics, we introduce the following novel concepts and definitions.Definition 11.
LetS,T:X⟶X be two self-mappings on a metric space X,D and a,b,e,f∈0,1 be real numbers.
Then,(1)
A functionΩS,abef:X×X⟶ℝ+ is supposed to be given by(11)ΩS,abefx,y≔aDx,y+bmaxDx,Sx,Dy,Sy+eDy,Sx+fDx,Sy∀x,y∈X(2)
A functionΩS,T,abef:X×X⟶ℝ+ is supposed to be given by(12)ΩS,T,abefx,y≔aDx,y+bmaxDx,Sx,Dy,Ty+eDy,Sx+fDx,Ty∀x,y∈XDefinition 12.
LetS:X⟶X be a self-map satisfying
(13)DSx,Sy≤ΩS,abefx,y∀x,y∈X.(1)
Ifa+b+e+f<1, then S is said to be Ω contraction type(2)
Ifa+b+e+f=1, then S is said to be Ω non-expansive typeRemark 13.
Every Hardy-Roger mapping isΩ contraction type and every general Gregus mapping is Ω nonexpansive type. In fact, we observe that
(14)DSx,Sy≤aDx,y+bDx,Sx+cDy,Sy+eDy,Sx+fDx,Sy≤aDx,y+b+cmaxDx,Sx,Dy,Sy+eDy,Sx+fDx,Sy=ΩS,ab+cefx,y∀x,y∈X.Definition 14.
LetX=Y∪Z be a cyclic representation of X with respect to S and a,b,e,f∈0,1 be real constants such that
(15)DSx,Sy≤ΩS,abefx,y∀x∈Y,y∈Z.(1)
Ifa+b+2f<1, then S is said to be cyclic Ω contraction(2)
Ifa+b+2f=1, then S is said to be cyclic Ω nonexpansiveRemark 15.
The class of all mappings defined by (15) includes the class of all mappings defined by (13).Definition 16.
LetX=Y∪Z be a cyclic representation of X with respect to S, a,b,e,f∈0,1 be real constants, and ϕ∈Φ0 be such that
(16)DSx,Sy≤ΩS,abefx,y−ϕΩS,abefx,y∀x∈Y,y∈Z.(1)
Ifa+b+2f<1, then S is said to be cyclic Ω‐ϕ-weak contraction w.r.t Y,Z(2)
Ifa+b+2f=1, then S is said to be cyclic Ω‐ϕ-weak nonexpansive w.r.t Y,ZDefinition 17.
LetX=Y∪Z be a cyclic representation of X with respect to S, ψandϕ∈Φ0, and a,b,e,f∈0,1 be real constants such that
(17)ψDSx,Sy≤ψΩS,abefx,y−ϕΩS,abefx,y∀x∈Y,y∈Z.(1)
Ifa+b+2f<1, then S is said to be Y,Z cyclic ΩS,abef-weak contraction(2)
Ifa+b+2f=1, then S is said to be Y,Z cyclic ΩS,abef-weak nonexpansiveRemark 18.
The class of all mappings defined by (17) includes the class of all mappings defined by (16).For more generalized types of contraction mappings we use the notion ofC class functions, see [24] and its included references.Definition 19.
AC class functions is the set of all real valued continuous functions G, G:0,∞×0,∞⟶ℝ satisfying the conditions
(1)
Gs,t≤s for all s,t∈0,∞(2)
IfGs,t=s, then either s=0 or t=0
Now, we introduce the following main definitions:Definition 20.
LetX=Y∪Z be a cyclic representation of X with respect to S, a,b,e,f∈0,1 be real constants, ψ,ϕ∈Φ0, and G∈C be such that
(18)ψDSx,Sy≤GψΩS,abefx,y,ϕΩS,abefx,y∀x∈Y,y∈Z.(1)
Ifa+b+2f<1, then S is said to be Y,Z cyclic G‐ΩS,abef-weak contraction(2)
Ifa+b+2f=1, then S is said to be Y,Z cyclic G‐ΩS,abef-weak nonexpansiveRemark 21.
The class of allG‐ΩS,abef-weak types of mappings includes all the previously mentioned classes. In fact; in particular replacing G by Gs,t=s−t for s,t∈0,∞ in Definition 20 ensures that the class of all mappings defined by (18) includes the class of all mappings defined by ((17)).We also introduce the following.(1)
Ifa+c+2f<1, then S and T are said to be joint Y,Z cyclic-G‐ΩS,T,abef-weak contraction types(2)
Ifa+c+2f=1, then S and T are said to be joint Y,Z cyclic-G‐ΩS,T,abef-weak nonexpansive typesDefinition 22.
LetX,D be a metric space with X=Y∪Z, S,T:X⟶X be two self-mappings, and a,b,e,f∈0,1 be real numbers satisfying
(1)
The cyclic condition:SY⊂Z and TZ⊂Y(2)
The following contractivity condition:(19)ψDSx,Ty≤GψΩS,T,abefx,y,ϕΩS,T,abefx,y∀x∈Y,y∈Z,where ψ∈Ψ, ϕ∈Φ0, and G∈C.This paper proves the existence of a unique common fixed point between any two joint cyclicG‐ΩS,T,abef-weak contraction mappings in complete metric spaces and proves the existence of a unique common fixed point of two joint cyclic G‐ΩS,T,abef-weak nonexpansive mappings in compact metric spaces. In particular, it proves the existence of a ufp of both cyclic G‐ΩS,abef-weak contraction and cyclic G‐ΩS,abef-weak nonexpansive mappings and hence proves the existence of a ufp of cyclic ΩS,abef-weak contraction and weak nonexpansive mappings. These results extend and generalize some fixed point theorems previously proved and given in the attached references.


## 2. Main Results

The following fixed point theorem is for contraction mappings:Theorem 23.
LetX,D be a complete metric space and Y,Z be two closed subsets of which X=Y∪Z. If S,T:X⟶X are continuous joint Y,Z cyclic G‐ΩS,T,abef-weak contraction mappings on X, e≤b, then there is only one point y∈Y∩Z such that Sy=y=Ty.Proof.
Letw0 be an arbitrarily chosen element in X. Then, w0 is lying either in Y or in Z, if w0 is in Z, then define w1=Tw0∈Y, w2=Sw1∈Z, w3=Tw2∈Z, and then by induction define
(20)w2n+2=Sw2n+1∈Z,(21)w2n+1=Tw2n∈Y∀n≥0.
First, suppose thatn is an odd natural number. Then,
(22)ψDwn+1,wn=ψDSwn,Twn−1≤GψΩS,T,abefwn,wn−1,ϕΩS,T,abefwn,wn−1≤ψΩS,T,abefwn,wn−1.
Sinceψ is nondecreasing, we see that
(23)Dwn+1,wn≤ΩS,T,abefwn,wn−1=aDwn,wn−1+bmaxDSwn−1,wn−1,DTwn,wn++eDTwn−1,wn+fDwn−1,Swn≤aDwn,wn−1+bmaxDSwn−1,wn−1,DTwn,wn+eDwn,wn+fDwn−1,wn+fDwn,Swn=a+fDwn,wn−1+bmaxDwn,wn−1,Dwn+1,wn+fDwn+1,wn.
We have the following:(i)
IfmaxDwn,wn−1,Dwn+1,wn=Dwn,wn−1, then(24)Dwn+1,wn≤a+b+f1−fDwn,wn−1
Back to the inequalities (23) with the help of inequalities (24), we have
(25)ΩS,T,abefwn,wn−1≤a+b+fDwn,wn−1+fDwn+1,wn≤a+b+fDwn,wn−1+fDwn,wn−1≤a+b+2fDwn,wn−1.(ii)
IfmaxDwn,wn−1,Dwn+1,wn=Dwn+1,wn, then(26)Dwn+1,wn≤a+f1−b+fDwn,wn−1
Back to the inequalities (23) with the help of inequalities (24), we have
(27)ΩS,T,abefwn,wn−1≤a+fDwn,wn−1+b+fDwn+1,wn≤a+fDwn,wn−1+b+fDwn,wn−1≤a+b+2fDwn,wn−1.
Letk=maxa+b+2f,a+b+f/1−f,a+f/1−b+f, clearly k<1, we see that
(28)Dwn+1,wn≤kDwn,wn−1∀n∈ℕ,ΩS,T,abefwn,wn−1≤kDwn,wn−1∀n∈ℕ.Continuing this process, we see that(29)Dwn+1,wn≤knDw1,w0∀n∈ℕ,(30)ΩS,T,abefwn,wn−1≤knDw1,w0∀n∈ℕ.Now, we use the inequalities (29) to show that the sequence wnn∈ℕ is Cauchy, let n,m∈ℕ, n≤m. Then,
(31)Dwm,wn≤Dwn,wn+1+Dwn+1,wn+2+⋯+Dwm−2,wm−1+Dwm−1,wm≤kn+kn+1+⋯+km−2+km−1Dw1,w0≤kn1+k1+⋯+km−n+2+km−n+1Dw1,w0≤kn11−kDw1,w0.Taking the limit of the two sides of inequalities (31) as n⟶∞ proves that limn,m⟶∞Dwn,wm=0. This proves that wnn∈ℕ is Cauchy sequence. Since X is complete, there is y∈X such that limn⟶∞Dwn,y=0 that is; limn⟶∞wn=y. We notice that limn⟶∞w2n=limn⟶∞w2n+1=y because they are subsequences of the same convergent sequence, thus y∈Y⋂Z because Y and Z are closed. We will prove that such a limit point is a common fixed point of S and T. In fact, we have
(32)ψDw2n+2,Ty=ψDSw2n+1,Ty≤GψΩS,T,abefw2n+1,y,ϕΩS,T,abefw2n+1,y≤ψΩS,T,abefw2n+1,y,thus,Taking the limit asn⟶∞ implies the following:
(33)Dy,Ty≤limn⟶∞aDw2n+1,y+bmaxDw2n+1,Sw2n+1,Dy,Ty+limn⟶∞eDw2n+1,Ty+fDy,Sw2n+1=bDy,Ty+eDy,Ty=b+eDy,Ty.That is,1−b+eDy,Ty≤0, thus Dy,Ty=0 because 1<b+e; hence, Ty=y and y is a fixed point of T. On the other side,
(34)0=Dy,Ty≤limn⟶∞ΩS,T,abefw2n+1,y=ΩS,T,abeflimn⟶∞w2n+1,y=ΩS,T,abefy,y=b+eDy,Ty=0,proves that ΩS,T,abefy,y=0. Therefore,
(35)aDy,y+bmaxDy,Sy,Dy,Ty+eDy,Sy+fDy,Ty=0,accordingly Sy=y and y is a fixed point of S. Now, we prove that such a common fixed point is unique. In fact, if there exists another point w∈Y∩Z such that Sw=w=Tw with w≠y, then we get
(36)ψDw,y=ψDSw,Ty≤GψΩS,T,abefw,y,ϕΩS,T,abefw,y≤ψΩS,T,abefw,y.Hence, the following gives a contradiction:(37)Dw,y≤ΩS,T,abefw,y≤aDw,y+bmaxDSw,w,Dy,Ty+eDw,Ty+fDy,Sw=aDw,y+e+fDw,y=a+e+fDw,y<a+b+2fDw,y=Dw,y.This shows thatDw,y=0, that is w=y.Corollary 24.
LetX,D be a complete metric space and Y,Z be two closed subsets of which X=Y∪Z. If S:X⟶X is continuous Y,Z cyclic G‐ΩS,abef-weak contraction mapping, e≤b, then there is only one point y∈Y∩Z such that Sy=y.Proof.
Using Theorem23 with S=T completes the proof.Corollary 25.
LetX,D be a complete metric space and Y,Z be two closed subsets of which X=Y∪Z. If S:X⟶X is continuous Y,Z cyclic ΩS,abef-weak contraction mapping, e≤b, then there is only one point y∈Y∩Z such that Sy=y. Moreover, limn⟶∞Snw0=y for any w0∈X.Proof.
Using Corollary (24) with Gt,s=t−s completes the proof.Furthermore, the following fixed point theorem is for nonexpansive mappings.Theorem 26.
LetX,D be a compact metric space and Y,Z be two closed subsets of which X=Y∪Z. If S,T:X⟶X are continuous joint Y,Z cyclic G‐ΩS,T,abef-weak nonexpansive mappings on X, then S and T have common fixed points y∈Y∩Z. Moreover, if e≤b, then they have a unique common fixed point.Proof.
Letw0 be an arbitrarily chosen element in X and consider the sequence defined by (20). Back to the inequalities (23) with the help of inequalities (24), we have
(38)ΩS,T,abefwn,wn−1≤a+b+2fDwn,wn−1=Dwn,wn−1.
Using the inequalities (27), we see that
(39)ΩS,T,abefwn,wn−1≤a+b+2fDwn,wn−1=Dwn,wn−1.
Consequently, we have(40)Dwn+1,wn≤ΩS,T,abefwn,wn−1≤Dwn,wn−1.
Continuing gives(41)Dwn+1,wn≤ΩS,T,abefwn,wn−1≤Dwn,wn−1≤ΩS,T,abefwn−1,wn−2.
Similarly, ifn is an even natural number, we get same inequalities as (41). These show that both of the two real valued sequences Dwn+1,wnn∈N and ΩS,T,abefwn+1,wnn∈N are monotonic nonincreasing sequences; both are bounded below by 0; then, the limit of each one exists and equals its infimum. On the other hand, the inequalities (41) prove that their infimums should coincide. Let r be such coinciding infimums (limits). Then,
(42)limn⟶∞Dwn+1,wn=limn⟶∞ΩS,T,abefwn,wn−1=r.
Using the lower semicontinuity property ofϕ gives
(43)ϕr≤liminfn⟶∞ϕΩS,T,abefwn,wn−1.
Taking the least upper limits on the two sides of the inequalities (22) as n⟶∞ gives
(44)ψr≤Gψr,liminfn⟶∞ϕΩS,T,abefwn,wn−1≤ψr.
Thus,(45)Gψr,liminfn⟶∞ϕΩS,T,abefwn,wn−1=ψr.
SinceG is C class function, then either liminfn⟶∞ϕΩS,T,abefwn,wn−1=0 or ψr=0. If ψr=0, then r=0. If liminfn⟶∞ϕΩS,T,abefwn,wn−1=0 while r>0, then the following is a contradiction
(46)0<ϕr≤liminfn⟶∞ϕΩS,T,abefwn,wn−1=0.
Hence,(47)limn⟶∞Dwn+1,wn=limn⟶∞ΩS,T,abefwn,wn−1=0.
Therefore,(48)infΩS,T,abefw,z:w∈Y,z∈Z=0.
There isx∈Y and y∈Z such that
(49)ΩS,T,abefx,y=0.
This gives(50)aDx,y+bmaxDx,Sx,Dy,Ty+eDx,Ty+fDy,Sx=0.
Consequently,(51)Dx,y=Dx,Sx=Dy,Ty=Dy,Sx=Dx,Ty=0,hence, x=y=Sx=Sy and x=y∈Y∩Z. To prove that such a fixed point is unique, we use analogous contradiction given by inequalities (37).We have the following algorithm and convergent proposition:Proposition 27.
The iterated sequence defined by (20) converges strongly to the unique common fixed point of S and T.
(52)limn⟶∞wn=y.Proof.
Lety be the unique common fixed point of S and T, in addition suppose that limn⟶∞wn=u with u≠y. Then, for each n∈N, we have
(53)ψDwn,y=ψDSwn−1,Ty≤GψΩS,T,abefwn−1,y,ϕΩS,T,abefwn−1,y≤ψΩS,T,abefwn−1,y.
Hence,(54)Dwn,y=DSwn−1,Ty≤ΩS,T,abefwn−1,y≤aDwn−1,y+bmaxDSwn−1,wn−1,DTy,y+eDwn−1,Ty+fDSwn−1,y≤aDwn−1,y+bDwn,wn−1+eDwn−1,y+fDwn,y≤a+eDwn−1,y+bDwn,wn−1+fDwn,y.
That is,(55)Dwn,y≤11−fa+eDwn−1,y+bDwn,wn−1.
Using equation (47) with the limiting approach as n⟶∞ prove that
(56)Du,y≤a+e1−fDu,y,hence, 1−a+e/1−fDu,y≤0, since a+e≠1−f, we get Du,y=0, that is, u=y.Corollary 28.
LetX,Dbe compact metric space andY,Zbe two closed subsets of whichX=Y⋃Z. IfS:X⟶Xis continuousY,ZcyclicG−ΩS,abef-weak nonexpansive mapping onX, e≤b, thenShas a ufpy∈Y⋂Z. Moreover, for anyw0∈X, we havelimn⟶∞Snw0=y.Proof.
Using Theorem26 with S=T completes the proof.Corollary 29.
LetX,Dbe compact metric space andY,Zbe two closed subsets of whichX=Y⋃Z. IfS:X⟶Xis continuousY,Zgeneralized cyclicΩS,abef-weak nonexpansive mapping onX, e≤b, thenShas a ufpy∈Y⋂Z. Moreover, for anyw0∈X, we havelimn⟶∞Snw0=y.Proof.
Using Theorem (24) with S=T and Gt,s=t−s completes the proof.We have the following examples.Example 1.
LetX=−1,1, Y=−1,0, andZ=0,1. EndowXwith the metricDx,y=∣x−y∣and defineS:X⟶Xas(57)Sx=−x3,ifx∈Y,−x2,ifx∈Z.
Then,S is cyclic G−ΩS,1000-weak nonexpansive type with the unique fixed point 0∈Y⋂‍Z, where Gs,t=s−t, ψt=tet, and ϕt=1/3te2/3t. Indeed, if x∈Y and y∈Z, then we have
(58)ψDSx,Sy=DSx,SyeDSx,Sy=∣Sx−Sy∣e∣Sx−Sy∣=∣−x3−−y2∣e−x/3−−y/2=∣y2−x3∣e∣y/2−x/3∣=y−x−y2−2x3ey−x−y/2−2x/3≤∣y−x∣ey−x−2/6y−x−13∣y−x∣ey−x−2/6y−x≤Dx,yeDx,y−13Dx,ye2/3Dx,y=GψΩS,1000x,y,ϕΩS,1000x,y.
On the other hand, we noticed thatDS−1,S1=∣1/2−−1/3∣=5/6, D−1,1=∣1−−1∣=2, and ϕD−1,1=1/3D−1,1e2/3D−1,1=1/32e4/3=2/3e4/3. Thus, DS−1,S1=5/6>2−2/3e4/3≅−0.53; hence,
(59)DS−1,S1>D−1,1−ϕD−1,1.
That is,S is not weakly ϕ-contractive.
This mapping is also cyclic-G‐ΩS,1000-weak nonexpansive type, where Gs,t=s−t, ψt=t2, and ϕt=5/9t2. Indeed, if x∈Y and y∈Z, then we have
(60)ψDSx,Sy=DSx,Sy2=Sx−Sy2=−x3−−y22=y2−x32=3y−2x62≤4y−2x62≤4y−22x62=4262y−x2=49y−x2=y−x2−59y−x2=GψΩS,1000x,y,ϕΩS,1000x,y.
On the other hand, we noticed thatDS−1,S1=5/6, D−1,1=∣1−−1∣=2, and ϕD−1,1=5/922=20/9. Thus, DS−1,S1=5/6>2−20/9=−2/9; hence,
(61)DS−1,S1>D−1,1−ϕD−1,1.
That is,S is not weakly ϕ-contractive.


## 3. Conclusion

This paper suggests two new theorems (Theorems23 and 26); these theorems prove the existence of a common ufp of novel introduced joint Y,Z cyclic G‐ΩS,T,abef-weak contraction and weak nonexpansive mappings in complete and compact metric spaces, respectively.Their corollaries, Corollary24 and Corollary 28 in particular, prove the existence of a ufp for both cyclic G‐ΩS,abef-weak contraction and cyclic G‐ΩS,abef-weak nonexpansive mappings in complete and compact metric spaces, respectively.In addition, Corollary25 and Corollary 29 prove the existence of a ufp for both cyclic ΩS,abef-weak contraction and cyclic ΩS,abef-weak nonexpansive mappings in complete and compact metric spaces, respectively. These results extend and generalize previously proved fixed point theorems given in the references.



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*Source: 6642564-2021-05-21.xml*