# A Study on Common Fixed Point of JointA;B Generalized Cyclic ϕ−abc Weak Nonexpansive Mappings and Generalized Cyclic abc;r Contractions in Quasi Metric Spaces

Authors: Sahar Mohamed Ali Abou Bakr
Journal: Abstract and Applied Analysis
 (2020)

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## Abstract

This manuscript proves the existence of a single common fixed point of two jointA,B generalized cyclic ϕ−abc weak nonexpansive mappings, where A and B are compact sets. The result in particular demonstrated a single fixed point of generalized cyclic ϕ−abc weak nonexpansive mappings, without the assumption of a single contracting point. Additionally, it introduces new types of generalized cyclic abc;r contraction mappings and describes the existence of a single fixed point of them in b-metric spaces. Finally, the presented results establish a simpler convergence theorem for a sequence of generalized cyclic abc;r contraction mappings, extend, and generalize some of the previous reported results.


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## Body

## 1. Introduction and Preliminaries

In 1922, Banach [1] introduced the concept of contraction mapping and formulated his famous Banach contraction principle, namely, the contraction mapping defined on complete metric space has a unique fixed point.Definition 1.
A self mappingS:X⟶X on a metric space X,d is said to be contraction iff there is a real number 0<r<1 such that
(1)dSx,Sy≤rdx,y∀x,y∈X.Since 1922 generalizations of the Banach contraction principle have historical achievements.In 1962, Edelstein [2] introduced the concept of contractive mapping and proved the existence of single (fp) of contractive mapping defined on compact metric space.Definition 2.
A self mappingS:X⟶X on a metric space X,d is said to be contractive iff S fulfills the condition:
(2)dSx,Sy<dx,y∀x≠y,x,y∈X.In 1972, Chatterjea [3] introduced the Chatterjea type mapping and showed that every Chatterjea mapping on a complete metric space has a single (fp).Definition 3.
LetX,d be a metric space. If T is a self mapping on X, then T is said to be Chatterjea contraction type iff there is real number 0≤b<1/2 such that
(3)dTx,Ty≤bdx,Ty+dy,Tx∀x,y∈X.In 1979, Browder [4] introduced the concept of a nonexpansive mapping and proved the existence of (fp) for nonexpansive mapping defined on bounded closed convex subset of uniformly convex Banach space.Definition 4.
A self mappingS:X⟶X on a metric space X,d is said to be nonexpansive iff S fulfills the condition:
(4)dSx,Sy≤dx,y∀x,y∈X.In 1997, Alber and Guerre-Delabriere [5] highlighted the weak ϕ contraction type of mapping and proved the existence of (fp) for this mapping in a Hilbert space setting.In 2001, Rhoades [6] showed analogous results in the case of ϕ-weak contraction mapping on metric spaces.Definition 5.
A self mappingS:X⟶X on a metric space X,d is said to be ϕ-weak contraction iff it fulfills the condition:
(5)dSx,Sy≤dx,y−ϕdx,y∀x,y∈X,where ϕ is a continuous nondecreasing function ϕ:0,∞⟶0,∞withϕt>0,forallt∈0,∞andϕ0=0.In 2003, Kirk et al. [7] generalized the above (fp) theorems for mappings satisfying cyclical contractive conditions in various settings.Definition 6.
LetX,d be a metric space, j be a positive integer, Aii=1j be a nonempty closed subsets of X, and X=∪i=1jAi. Then, a self mapping T on X is said to satisfy cyclical contractive condition iff there is a constant r∈0,1 such that:
(6)dTx,Ty≤rdx,yforeveryx∈Ai,y∈Ai+1,i=1,2,⋯,j,whereAj+1=A1.In 2007, Elshobaky [8] generalized the Banach contraction principle forabc type contraction mapping.Definition 7.
LetX,d be a metric space, T be a self mapping on X, a,b,c∈0,1, and T satisfy the condition:.(7)dTx,Ty≤adx,y+bdx,Tx+cdy,Ty∀x,y∈X.(1)
Ifa+c+b<1, then T is said to be generalized abc contraction(2)
Ifa+c+b=1, then T is said to be generalized abc nonexpansiveIn 2008, Dutta and Choudhury [8] proved the existence of (fp) for ψ−ϕ- weak contraction mappings.Definition 8.
LetX,d be a metric space. If T is a self mapping on X, then T is said to be ψ−ϕ weak contraction on X iff there are two nondecreasing mappings ψ,ϕ:0,+∞⟶0,+∞, ψ is continuous, ψt=0ifft=0andϕt=0ifft=0 such that the following is true:
(8)ψdTx,Ty≤ψdx,y−ϕdx,y∀x,y∈X.In 2009, Choudhury [9] introduced the weakly Chatterjea contractive type mappings and proved that every weakly Chatterjea contractive mapping on a complete metric space has single (fp).Definition 9.
IfT:X⟶X is a self mapping on a metric space X, then T is said to be weakly C-contractive (or weak Chatterjea type contraction) iff
(9)dTx,Ty≤12dx,Ty+dy,Tx−ϕdx,Ty,dx,Ty∀x,y∈X,where ϕ:0,∞×0,∞⟶0,∞[ is a continuous function satisfying ϕx,y=0 iff x=y=0.In 2009, Bakr [10] proved the existence of (fp) for abc generalized nonexpansive mappings with one contracting point.Definition 10.
LetX be a normed space, C be bounded closed subset of X, 0≤t,a,b,c<1, t,b,c≠1, and a+b+c=1. Then, the point x0∈C is said to be abct contracting point with respect to C iff there is λ<1−ta+b+c/1−t satisfying:
(10)b∥x0−x∥+c∥x0−y∥≤λ∥x−y∥∀x≠y∈C.This author showed that ifX is a normed space in which every bounded sequence has a weakly convergent subsequence and the parallelogram law hold in X, C is a subset of X which containing ab0r-contracting point, and T is a ab0-generalized-nonexpansive mappings from C into itself, then for every ab0r-contracting point of C, x0∈C, the following sequence converges strongly to the best approximation element of x0 in FT.
(11)xn=1−tnx0+tnTxn.In 2011, Karapinar [11] gave another generalization for cyclic weak ϕ-contraction.Definition 11.
LetX,d be a metric space, j be a positive integer, Aii=1j be nonempty closed subsets of X, and X=∪i=1jAi. Then a self mapping T on X is said to be cyclic weak contraction iff there is a continuous nondecreasing mapping ϕ:0,+∞⟶0,+∞ with ϕt=0ifft=0 such that:
(1)
X=∪i=1jAi is cyclic representation of X with respect to T(12)dTx,Ty≤dx,y−ϕdx,yforallx∈Ai,y∈Ai+1,i=1,2,⋯,j,where Aj+1=A1.At the same year, Karapinar and Sadarangani [12] gave (fp) for generalized cyclic ψ−ϕ-weak contraction mappings in complete metric spaces.Definition 12.
LetX,d be a metric space, j be a positive integer, Aii=1j be nonempty closed subsets of X, X=∪i=1jAi, ψ,ϕ:0,+∞⟶0,+∞ are two nondecreasing mappings, ψ is continuous, ψt=0ifft=0, and ϕt=0ifft=0. IfT is a self mapping on X, then, T is said to be generalized cyclic ψ−ϕ-weak contraction mapping iff the following are true:
(1)
X=∪i=1jAi is cyclic representation of X with respect to T(2)
ψdTx,Ty≤ψdx,y−ϕdx,yforallx∈Ai,y∈Ai+1,i=1,2,⋯,j, where Aj+1=A1In 2012, Karapinar and Sadarangani [13] enhanced the corrigendum (fp) theory for cyclic ϕ-weak contraction mapping. Additionally, Karapinar et al. [14] gave some (fp) for generalized ψ−ϕ-weak contraction mapping.At the same year, (2012), Karapinar and Kumar [15] introduced the notion of cyclic weakly Chatterjea type contraction in metric spaces and proved corresponding (fp) theorems.Definition 13.
LetX,d be a metric space, j be a positive integer, Aii=1j be nonempty subsets of X, X=∪i=1jAi, and T:X⟶X be a self mapping on X. Then, T is said to be cyclic weakly Chatterjea type contraction iff:
(1)
X=∪i=1jAi is cyclic representation of X with respect to T(2)
there are a real numberr∈0,1/2 and a continuous function ϕ:0,∞×0,∞⟶0,∞,ϕx,y=0 iff x=y=0 such that
(13)dTx,Ty≤rdx,Ty+dy,Tx−ϕdx,Ty,dy,Tx∀x∈Ai,y∈Ai+1,i=1,2,⋯,j,whereAj+1=A1.In 2013, Bakr [16] discussed the concepts of a,b,c-ntype and a,b,c-ctype of mappings and proved the existence of single (fp) for both a,b,c-ntype and a,b,c-ctype on closed convex weakly Cauchy subset of normed space.Definition 14.
LetC be a subset of a normed space X,∙ and T be a self mapping on C satisfying:.(14)Tx−Ty≤ax−y+by−Ty+cmax∥x−Tx∥,∥x−Ty∥∀x,y∈Canda,b,c∈0,1.(1)
If0≤c<1/2 and a+b+c<1, then T is said to be a,b,c-contraction type mapping(2)
If0<a<1,0<b,0≤c<1/2 and a+b+c=1, then T is said to be a,b,c-nonexpansive type mappingIn this paper, we are introducing the following:.LetS:X⟶X be a self mapping on a metric space X,d and a,b,c be three real numbers. Then, we define the real valued function ΓSabc:X×X⟶ℝ+ by.
(15)ΓSabcx,y=adx,y+bdx,Sx+cdy,Sy∀x,y∈X.Definition 15.
LetS,T:X⟶X be two self mappings on a metric space X,d and a,b,c be three real numbers. Then, we define the real valued function ΓS,Tabc:X×X⟶ℝ+by(16)ΓS,Tabcx,y=adx,y+bdx,Sx+cdy,Ty∀x,y∈X.Definition 16.
LetX,d be a metric space; X=A⋃B and S be a self mapping on X with
(1)
SA⊂B and SB⊂A and(2)
there are real constantsa,b,c∈0,1suchthat(17)dSx,Sy≤ΓSabcx,y∀x∈A,y∈B.(i)
Ifa+c+b<1, then S is said to be cyclic generalized abc contraction w.r.t A,B.(ii)
Ifa+c+b=1, then S is said to be cyclic generalized abc nonexpansive w.r.t A,B.Include the definition of lower semicontinuity of the function.Definition 17.
LetX be a linear space. Then, a function f from Xinto−∞,+∞ is said to be lower semicontinuous if for any real number α, the set x∈X:fx≤α is closed convex subset of X.Definition 18.
LetS:X⟶X be a self mapping on a metric space X,d such that
(18)dSx,Sy≤ΓSabcx,y−ϕΓSabcx,y∀x∈A,y∈B,where ϕ is lower semicontinuous nondecreasing function ϕ:0,∞⟶0,∞withϕt>0,forallt∈0,∞andϕ0=0.
(1)
Ifa+c+b<1, then S is said to be generalized cyclic ϕ−abc weak contraction w.r.t A,B.(2)
Ifa+c+b=1, then S is said to be generalized cyclic ϕ−abc weak nonexpansive w.r.t A,B.Additionally, we introduce the following fascinating concept of joint mappings:.Definition 19.
LetX,d be a metric space with X=A⋃B, S,T:X⟶X be two self mappings, and a,b,c∈0,1bethreerealnumberssatisfying:
(1)
The cyclic condition1:SA⊂B and TB⊂A(2)
The contractivity condition2:(19)dSx,Ty≤ΓS,Tabcx,y−ϕΓS,Tabcx,y∀x∈A,y∈B,where ϕ is lower semicontinuous nondecreasing function ϕ:0,∞⟶0,∞withϕt>0,forallt∈0,∞andϕ0=0.
(i)
Ifa+c+b<1, then S and T are said to be joint A,B generalized cyclic ϕ−abc weak contraction type(ii)
Ifa+c+b=1, then S and T are said to be joint A,B generalized cyclic ϕ−abc weak nonexpansive typeIn 1993, Czerwik [17] verified the principle in a wider class of metric spaces, namely, quasi metric space, these spaces sometimes are called b-metric spaces, for some other types of quasi metric spaces one can see [18].Definition 20.
Letr be a real number, r≥1, Q be a mapping Q:X×X⟶ℝ+. Then, X,Q is called b-metric space iff
(1)
Qx,y=0 iff x=y(2)
Qx,y=Qy,x.(3)
Qx,y≤rQx,z+Qz,yforallx,y,z∈XIn 2017, Bakr [19] adjusted (fp) theorem by using both b-metric spaces and generalized abc;r contraction type of mappings.Definition 21.
Letr be a real number, r≥1 and X,Q be ab-metric space. Then, the mapping S is said to be abc;r-ctype mapping iff it satisfies:
(20)QSu,Sv≤aQu,v+bQv,Sv+cmaxQu,Su,Qu,Sv∀u,v∈X,where a,b,c∈0,1 are some real numbers with a+c<1 and ra+b+c<1.Now, the generalized cyclicabc;r contraction type mapping in b-metric spaces is defined as follows:.Definition 22.
Letr be a real number, r≥1 and X,Q be a b-metric space. Then, a self mapping S on X,Q is said to be generalized cyclic abc;r contraction w.r.tA,B iff S satisfies the following:
(1)
SA⊂B and SB⊂A and(2)
there are real constantsa,b,c∈0,1with(21)rmaxa+cr1−b+cr,a+b1−c<1and
(22)QSx,Sy≤aQx,y+bQx,Sx+cmaxQy,Sy,Q,y,Sx∀x∈A,y∈B.The purposes of this manuscript are designed as follows:Theorem23 together with Proposition 24 proves the existence of a single common (fp) for two joint A,B generalized cyclic ϕ−abc weak nonexpansive type mappings that have compact sets A,B. The result in particular presented a single (fp) for generalized cyclic ϕ−abc weak nonexpansive mapping, without the assumption of single contracting point, this result showed in Corollary 25.Additionally, we introduce a new type of generalized cyclicabc;r contraction mapping. Theorem 29 described the existence of a single (fp) of such mappings in b-metric spaces.Finally, Theorem31 established a simpler convergence theorem for a sequence of generalized cyclic abc;r contraction mappings. These results are extensions and generalizations of previous results reported in [10, 11, 17, 19, 20].Since every contraction mapping is nonexpansive and both are cyclic w.r.tX,X. We noticed the following:
(i)
The class of all generalizedabc nonexpansive mappings is wider than that of generalized abc contraction type(ii)
The class of all generalized cyclicabc contraction mappings is wider than that of generalized abc contraction type(iii)
The class of all generalized cyclicabc nonexpansive mappings is wider than that of generalized abc nonexpansive type(iv)
The class of all generalized cyclicabc nonexpansive mappings is wider than that of generalized cyclic abc contraction type(v)
The class of all generalized cyclicϕ−abc weak nonexpansive mappings is wider than that of generalized cyclic ϕ−abc weak contraction type(vi)
The class of all generalizedϕ−abc weak contraction is wider than that of ϕ weak contraction type(vii)
The class of all jointA,B generalized cyclic ϕ−abc weak nonexpansive types is wider than that of joint A,B generalized cyclic ϕ−abc weak contraction types(viii)
IfS,T are continuous self mappings on X,d, then the restriction of the mapping ΓS,Tabc:A×B⟶ℝ+ is continuous
(23)ΓS,Tabcx,y=adx,y+bdx,Sx+cdy,Ty,x∈A,y∈B.(ix)
IfA,B are two compact subsets of the metric space X,d and S,T are continuous self mappings on X, then the restriction of the mapping ΓS,Tabc:A×B⟶ℝ+ attains its infimum as well as its supremum at some points in A×B


## 2. Main Results

### 2.1. Common Fixed Point of Continuous Joint Generalized Cyclic Weak Nonexpansive Mappings

We have the following generalizations:Theorem 23.
LetX,d be a metric space and A,B be two compact subsets of which X=A∪B. IfS,T:X⟶X are two continuous joint A,B generalized cyclic ϕ−abc weak nonexpansive type of mappings on X, then there is single point u∈X such that Su=Tu=u∈A⋂B.Proof.
Letu0 be an arbitrarily chosen element in X. Then u0 is either in A or in B. If u0 is in B, then define u1=Tu0∈A, u2=Su1∈B, u3=Tu2∈A, and then by induction define:
(24)u2n+2=Su2n+1∈Bandu2n+1=Tu2n∈A∀n≥0.
First, supposen is an odd natural number. Then
(25)dun+1,un=dSun,Tun−1≤ΓS,Tabcun,un−1−ϕΓS,Tabcun,un−1.
Hence(26)dun+1,un≤ΓS,Tabcun,un−1=adun,un−1+bdSun,un+cdTun−1,un−1=adun,un−1+bdun+1,un+cdun,un−1=a+cdun,un−1+bdun+1,un.
Thus,(27)dun+1,un≤a+c1−bdun,un−1=dun,un−1.
Therefore,(28)ΓS,Tabcun,un−1≤a+cdun,un−1+bdun+1,un≤a+cdun,un−1+bdun,un−1=a+b+cdun,un−1=dun,un−1.
Hence(29)dun+1,un≤ΓS,Tabcun,un−1≤dun,un−1.
Continuing this process gives(30)dun+1,un≤ΓS,Tabcun,un−1≤dun,un−1≤ΓS,Tabcun−1,un−2.
Second, by a similar method whenn is an even natural number, we obtain
(31)dun+1,un≤ΓS,Tabcun,un−1≤dun,un−1≤ΓS,Tabcun−1,un−2.
Hence, the two sequencesdun+1,unn∈ℕ and ΓS,Tabcun+1,unn∈ℕ are monotonic nonincreasing and bounded below by 0 with the same infimum, if this infimum is r, then
(32)limn→∞dun+1,un=limn→∞ΓS,Tabcun,un−1=r.
Using the properties ofϕ,
(33)ϕr≤liminfn→∞ϕΓS,Tabcun,un−1.
Taking least upper limits on two sides of the inequalities (25) as n⟶∞ gives
(34)r≤r−liminfn→∞ϕΓS,Tabcun,un−1≤r−ϕr,that is ϕr≤0. Thus ϕr=0, this gives r=0. Hence,
(35)limn→∞dun+1,un=limn→∞ΓS,Tabcun,un−1=0.
This implies that(36)infΓS,Tabcx,y:x∈A,y∈B=0.
SinceΓS,Tabc attains its infimum on A×B, there are x0∈A and y0∈B such that
(37)ΓS,Tabcx0,y0=0.
This gives(38)adx0,y0+bdx0,Sx0+cdy0,T0y=ΓS,Tabcx0,y0=0,that is
(39)adx0,y0+bdx0,Sx0+cdTy0,y0=0,since all are nonnegative real numbers, clearly
(40)dx0,y0=dx0,Sx0=dy0,Ty0=0,and we have
(41)x0=y0=Sx0=Ty0.
Notice that the converse is also true, ifx0=y0=Sx0=Tx0=Sy0=Ty0, then ΓS,Tabcx0,y0=0 is clear. On the other side, this showed that x0=y0∈A⋂B is a common (fp) of S and T. If there exists another point u∈A⋂B such that Su=u=Tu, then we get
(42)du,y0=dSu,Ty0≤ΓS,Tabcu,y0−ϕΓS,Tabcu,y0≤adu,y0+bdSu,u+cdTy0,y0=adu,y0<du,y0.
This shows thatdu,y0=ΓS,Tabcu,y0=0, that is u=y0.Proposition 24.
The sequence defined iteratively by the induction (24) and by previous theorem notations is convergent to the unique common (fp) ofSandT.
(43)limn→∞un=u.Proof.
Letu be the common (fp) of S and T and limn→∞un=v with v≠u, then
(44)du2n+2,u=du2n+2,Tu=dSu2n+1,Tu≤ΓS,Tabcu2n+1,u≤adu2n+1,u+bdSu2n+1,u2n+1+cdTu,u≤adu2n+1,u+bdu2n+2,u2n+1+cdTu,u≤adu2n+1,u+bdu2n+2,u2n+1.
On limiting approach asn⟶∞, equation (35) proves that:
(45)dv,u≤adv,u.
Hence,1−adv,u≤0, since a≠1, we get dv,u=0, that is, v=u.Corollary 25.
IfX,dis metric space,X=A⋃B, A, andBare compact, andS:X⟶Xis continuous generalized cyclicϕ−abcweak nonexpansive mapping w.r.tA,B, then there is single pointu∈Xsuch thatSu=u. Moreover, for everyu0∈X, we havelimn→∞Snu0=u.


#### 2.1.1. Example

LetX=−1,1, A=−1,0, and B=0,1. Define S:X⟶X as:
(46)Sv=−v3,ifv∈A,−v2,ifv∈B.It is clear thatS is cyclic with respect to the representation A⋃B of X. Endow X with the metric du,v=u−v. The mapping S is having the unique (fp) u=0∈A⋂B and it is generalized cyclic ϕ−1/41/122/3 weak nonexpansive w.r.t A,B, where ϕ is defined by ϕt=1/13t. In fact, if u∈A and v∈B, then we have
(47)ΓS1411223u,v=14du,v+112du,Su+23dv,Sv=14v−u+112∣u−−u3∣+23∣v−−v2∣=14v−u+112∣u+u3∣+23∣v+v2∣=14v−u−112u+u3+23v+v2=14v−u−1124u3+233v2=14v−u−u9+v=54v−13u36=13645v−13u.Using this, we have(48)dSu,Sv=∣−u3−−v2∣=∣v2−u3∣=v2−u3=163v−2u=13618v−12u=13645v−13u−13627v−u≤13645v−13u−1364513v−u=13645v−13u−11313645v−13u=ΓS1411223u,v−113ΓS1411223u,v=ΓS1411223u,v−ϕΓS1411223u,v.


### 2.2. Generalized Results for Generalized abc;r Contraction.

Next, we generalize results for generalizedabc;r contraction [19] to those of generalized cyclic abc;r contraction in b-metric spaces. We have the following important lemmas:Lemma 26.
LetSbe generalized cyclicabc;rcontraction w.r.tA,Bof theb-metric spaceX. Then for everyy∈Xand any natural numbern∈ℕ, we have:(1)
The inequalities(49)QSn+1y,Sny≤a+b1−cQSny,Sn−1y,ifnisodd,a+cr1−b+crQSny,Sn−1y,ifniseven,are true(2)
There are two natural numbersl and m with n=l+m and
(50)QSn+1y,Sny≤a+b1−cla+cr1−b+crmQSy,y.Proof.
Lety∈X=A⋃B, y∈A (similar conclusion if y∈B). Then Sy∈B, using the contractivity condition of S gives
(51)QS2y,Sy=QSy,SSy≤aQy,Sy+bQy,Sy+cmaxQSy,SSy+QSy,Sy=a+bQy,Sy+cQSy,SSy,hence,
(52)QS2y,Sy≤a+b1−cQSy,y.
ForSy∈B, we have SSy∈A, thus
(53)QS3y,S2y=QSS2y,SSy≤aQS2y,Sy+bQS2y,SS2y+cmaxQSy,SSy,QSy,SS2y=aQS2y,Sy+bQS2y,S3y+cmaxQSy,SSy,rQSy,SSy+QSSy,SS2y=aQS2y,Sy+bQS2y,S3y+crQSy,SSy+QSSy,SS2y=a+crQS2y,Sy+b+crQS2y,S3y.
Becauser≥1, hence
(54)QS3y,S2y≤a+cr1−b+crQS2y,Sy.
Using the inequalities (52) and (54) gives
(55)QS3y,S2y≤a+cr1−b+cra+b1−cQSy,y.
ForS2y∈A, we have S3y∈B, and then
(56)QS4y,S3y=QSS2y,SS3y≤aQS2y,S3y+bQS2y,SS2y++cmaxQS3y,SS3y,QS3y,SS2y≤a+bQS2y,S3y+cQS3y,SS3y.
Consequently,(57)QS4y,S3y≤a+b1−cQS3y,S2y.
Using the inequalities (55) and (57) gives
(58)QS4y,S3y≤a+cr1−b+cra+b1−c2QSy,y.
ForS3y∈B, we have S4y∈A, and then
(59)QS5y,S4y=QSS4y,SS3y≤a+cr1−b+crQS4y,S3y.
Using the inequalities (58) and (59) gives
(60)QS5y,S4y≤a+cr1−b+cr2a+b1−c2QSy,y.
Continuing in this process, we see that(61)QS6y,S5y≤a+cr1−b+cr2a+b1−c3QSy,yQS7y,S6y≤a+cr1−b+cr3a+b1−c3QSy,y.
And so on, combining the above inequalities prove formulas (49) and (50).Lemma 27.
LetSbe generalized cyclicabc;rcontraction w.r.t any pairA,BofX. Then for everyy∈Xand any natural numbern∈ℕ, there is a real numberr0<1such that the inequalities(62)QSn+1y,Sny≤r0QSny,Sn−1y(63)QSn+1y,Sny≤r0nQSy,yare true.Proof.
Using Lemma26 with r0=maxa+cr/1−b+cr,a+b/1−c completes the proof.Lemma 28.
LetSbe generalized cyclicabc;rcontraction w.r.t any pairA,BofX. Then for everyy∈X, the sequenceSnyn∈ℕis Cauchy.Proof.
Letm, n be two natural numbers with m>n and y be an arbitrarily element in X. If y∈A or y∈B, then we will have the same conclusion
(64)QSny,Smy≤rQSny,Sn+1y+QSn+1y,Smy≤rQSny,Sn+1y+rQSn+1y,Sn+2y+QSn+2y,Smy≤rQSny,Sn+1y+r2QSn+1y,Sn+2y+r2rQSn+2y,Sn+3y+QSn+3y,Smy≤rQSny,Sn+1y+r2QSn+1y,Sn+2y+r3QSn+2y,Sn+3y+⋯+rm−nQSm−1y,Smy.
Using the inequalities (63) gives
(65)QSny,Smy≤rr0nQSy,y+r2r0n+1QSy,y+r3r0n+2QSy,y+⋯+rm−nr0n+m−1−nQSy,y=rr0nQSy,y+rr0QSy,y+rr02QSy,y+⋯+rr0m−1−nQSy,y=rr0nQSy,y1−rr0m−1−n1−rr0≤r1−rr0QSy,yr0n.
Clearly,r0<1, taking the limit as n⟶∞ completes the proof.
We have the following (fp) results:Theorem 29.
LetX,Qbe completeb-metric space,Sbe generalized cyclicabc;rcontraction w.r.tA,B. Then,Shas single (fp)u∈A⋂B. Moreover, for anyy∈X, the sequenceSnyn∈ℕis converging tou.Proof.
Lety be an arbitrarily chosen element of X, in either of the two cases y∈A or y∈B we have the same result, using inequalities (63) of Lemma 27 ensures that
(66)limn→∞QSn+1y,Sny=0,and Snyn∈ℕ is Cauchy by Lemma 28. Since X is complete b-metric space, Snyn∈ℕ is convergent to some element u∈X;
(67)limn→∞QSny,u=0.
Each subsequence is convergent to the same limit point, since each ofA and B contains infinite number of terms of Snyn∈ℕ. Hence, in A and in B, we construct a subsequence of Sniyi∈ℕ that converges to u. Since A and B are closed, we conclude that u is lying in the intersection of A and B. On the other side, if Sniy∈B, u∈A, (same conclusion if we consider Sniy∈A, u∈B), use the cyclic condition
(68)QSu,Sni+1y=QSu,SSniy≤aQu,Sniy+bQu,Su+cmaxQSniy,Sni+1y,QSniy,Su.
Taking the limit asi⟶∞, the inequalities (66) and (67) give
(69)limi→∞QSu,Sni+1y≤a0+bQu,Su+cmax0,limi→∞QSniy,Su.
Hence,(70)1−climi→∞QSu,Sniy≤bQu,Su,limi→∞QSu,Sniy≤b1−cQu,Su.
Since(71)QSu,u≤rQSu,Sniy+QSniy,u,we have
(72)QSu,u≤rlimi→∞QSu,Sniy+limi→∞QSniy,u,that is
(73)QSu,u≤rb1−cQu,Su+0.
Accordingly, we have(74)QSu,u≤rb1−cQu,Su<Qu,Su.
This shows thatQSu,u=0. Consequently Su=u, that is, u is a (fp) of S. Finally, to prove that such a (fp) is unique, let u and z be two distinct (fps) of S. Noting that both u and z are lying in the intersection of A and B (if z∈A, then z=Sz∈B and vise versa), the following strict inequality gives an obvious contradiction:
(75)Qu,z=QSu,Sz≤aQu,z+bQu,Su+cmaxQz,Sz+Qz,Su=aQu,z+bQu,u+cmaxQz,z+Qz,u=aQu,z+cQz,u=a+cQu,z<Qu,z.Remark 30.
Using the same steps ofproof, Theorem (29) can be easily extended to the case when S has Aii=1j (U) cyclic representations.Now, we are going to give a generalization of Theorem (31) [19], the prove is more simpler and in case of cyclic type mappings.Theorem 31.
IfSnn∈N and S are generalized cyclicabc;r contractions on the complete b-metric space X,Qw.r.tA,B, wn∗ is the unique fixed of each Sn, Snwn∗=wn∗∀n∈ℕ, andSnw∗n∈ℕ converges to w∗,
(76)limn→∞QSnw∗,Sw∗=0,where w∗ is the unique fixed of S; Sw∗=w∗, then wn∗n∈ℕ converges to w∗;
(77)limn→∞Qwn∗,w∗=0.Proof.
We have(78)Qwn∗,w∗=QSnwn∗,Sw∗≤rQwn∗,Snw∗+QSnw∗,Sw∗.
Taking in account thatwn∗n∈N⋃w∗⊆A⋂B, we have
(79)Qwn∗,Snw∗=QSnwn∗,Snw∗≤aQwn∗,w∗+bQw∗,Snw∗+cmaxQwn∗,Snwn∗,Qwn∗,Snw∗≤aQwn∗,w∗+bQw∗,Snw∗+cQwn∗,Snw∗Qwn∗,Snw∗≤11−caQwn∗,w∗+bQw∗,Snw∗.
Substituting from (79) in (78) gives
(80)Qwn∗,w∗=QSnwn∗,Sw∗≤r11−caQwn∗,w∗+bQw∗,Snw∗+QSnw∗,Sw∗≤ra1−cQwn∗,w∗+b1−cQw∗,Snw∗+QSnw∗,Sw∗≤ra1−cQwn∗,w∗+b1−c+1rQw∗,Snw∗.
Hence,(81)1−c−ra1−cQwn∗,w∗≤b−c+11−crQw∗,Snw∗.
Consequently, we have(82)Qwn∗,w∗≤b−c+11−c−rarQw∗,Snw∗.
Taking the limit asn⟶∞ and using assumption (76) prove (77).


## 2.1. Common Fixed Point of Continuous Joint Generalized Cyclic Weak Nonexpansive Mappings

We have the following generalizations:Theorem 23.
LetX,d be a metric space and A,B be two compact subsets of which X=A∪B. IfS,T:X⟶X are two continuous joint A,B generalized cyclic ϕ−abc weak nonexpansive type of mappings on X, then there is single point u∈X such that Su=Tu=u∈A⋂B.Proof.
Letu0 be an arbitrarily chosen element in X. Then u0 is either in A or in B. If u0 is in B, then define u1=Tu0∈A, u2=Su1∈B, u3=Tu2∈A, and then by induction define:
(24)u2n+2=Su2n+1∈Bandu2n+1=Tu2n∈A∀n≥0.
First, supposen is an odd natural number. Then
(25)dun+1,un=dSun,Tun−1≤ΓS,Tabcun,un−1−ϕΓS,Tabcun,un−1.
Hence(26)dun+1,un≤ΓS,Tabcun,un−1=adun,un−1+bdSun,un+cdTun−1,un−1=adun,un−1+bdun+1,un+cdun,un−1=a+cdun,un−1+bdun+1,un.
Thus,(27)dun+1,un≤a+c1−bdun,un−1=dun,un−1.
Therefore,(28)ΓS,Tabcun,un−1≤a+cdun,un−1+bdun+1,un≤a+cdun,un−1+bdun,un−1=a+b+cdun,un−1=dun,un−1.
Hence(29)dun+1,un≤ΓS,Tabcun,un−1≤dun,un−1.
Continuing this process gives(30)dun+1,un≤ΓS,Tabcun,un−1≤dun,un−1≤ΓS,Tabcun−1,un−2.
Second, by a similar method whenn is an even natural number, we obtain
(31)dun+1,un≤ΓS,Tabcun,un−1≤dun,un−1≤ΓS,Tabcun−1,un−2.
Hence, the two sequencesdun+1,unn∈ℕ and ΓS,Tabcun+1,unn∈ℕ are monotonic nonincreasing and bounded below by 0 with the same infimum, if this infimum is r, then
(32)limn→∞dun+1,un=limn→∞ΓS,Tabcun,un−1=r.
Using the properties ofϕ,
(33)ϕr≤liminfn→∞ϕΓS,Tabcun,un−1.
Taking least upper limits on two sides of the inequalities (25) as n⟶∞ gives
(34)r≤r−liminfn→∞ϕΓS,Tabcun,un−1≤r−ϕr,that is ϕr≤0. Thus ϕr=0, this gives r=0. Hence,
(35)limn→∞dun+1,un=limn→∞ΓS,Tabcun,un−1=0.
This implies that(36)infΓS,Tabcx,y:x∈A,y∈B=0.
SinceΓS,Tabc attains its infimum on A×B, there are x0∈A and y0∈B such that
(37)ΓS,Tabcx0,y0=0.
This gives(38)adx0,y0+bdx0,Sx0+cdy0,T0y=ΓS,Tabcx0,y0=0,that is
(39)adx0,y0+bdx0,Sx0+cdTy0,y0=0,since all are nonnegative real numbers, clearly
(40)dx0,y0=dx0,Sx0=dy0,Ty0=0,and we have
(41)x0=y0=Sx0=Ty0.
Notice that the converse is also true, ifx0=y0=Sx0=Tx0=Sy0=Ty0, then ΓS,Tabcx0,y0=0 is clear. On the other side, this showed that x0=y0∈A⋂B is a common (fp) of S and T. If there exists another point u∈A⋂B such that Su=u=Tu, then we get
(42)du,y0=dSu,Ty0≤ΓS,Tabcu,y0−ϕΓS,Tabcu,y0≤adu,y0+bdSu,u+cdTy0,y0=adu,y0<du,y0.
This shows thatdu,y0=ΓS,Tabcu,y0=0, that is u=y0.Proposition 24.
The sequence defined iteratively by the induction (24) and by previous theorem notations is convergent to the unique common (fp) ofSandT.
(43)limn→∞un=u.Proof.
Letu be the common (fp) of S and T and limn→∞un=v with v≠u, then
(44)du2n+2,u=du2n+2,Tu=dSu2n+1,Tu≤ΓS,Tabcu2n+1,u≤adu2n+1,u+bdSu2n+1,u2n+1+cdTu,u≤adu2n+1,u+bdu2n+2,u2n+1+cdTu,u≤adu2n+1,u+bdu2n+2,u2n+1.
On limiting approach asn⟶∞, equation (35) proves that:
(45)dv,u≤adv,u.
Hence,1−adv,u≤0, since a≠1, we get dv,u=0, that is, v=u.Corollary 25.
IfX,dis metric space,X=A⋃B, A, andBare compact, andS:X⟶Xis continuous generalized cyclicϕ−abcweak nonexpansive mapping w.r.tA,B, then there is single pointu∈Xsuch thatSu=u. Moreover, for everyu0∈X, we havelimn→∞Snu0=u.


### 2.1.1. Example

LetX=−1,1, A=−1,0, and B=0,1. Define S:X⟶X as:
(46)Sv=−v3,ifv∈A,−v2,ifv∈B.It is clear thatS is cyclic with respect to the representation A⋃B of X. Endow X with the metric du,v=u−v. The mapping S is having the unique (fp) u=0∈A⋂B and it is generalized cyclic ϕ−1/41/122/3 weak nonexpansive w.r.t A,B, where ϕ is defined by ϕt=1/13t. In fact, if u∈A and v∈B, then we have
(47)ΓS1411223u,v=14du,v+112du,Su+23dv,Sv=14v−u+112∣u−−u3∣+23∣v−−v2∣=14v−u+112∣u+u3∣+23∣v+v2∣=14v−u−112u+u3+23v+v2=14v−u−1124u3+233v2=14v−u−u9+v=54v−13u36=13645v−13u.Using this, we have(48)dSu,Sv=∣−u3−−v2∣=∣v2−u3∣=v2−u3=163v−2u=13618v−12u=13645v−13u−13627v−u≤13645v−13u−1364513v−u=13645v−13u−11313645v−13u=ΓS1411223u,v−113ΓS1411223u,v=ΓS1411223u,v−ϕΓS1411223u,v.


## 2.1.1. Example

LetX=−1,1, A=−1,0, and B=0,1. Define S:X⟶X as:
(46)Sv=−v3,ifv∈A,−v2,ifv∈B.It is clear thatS is cyclic with respect to the representation A⋃B of X. Endow X with the metric du,v=u−v. The mapping S is having the unique (fp) u=0∈A⋂B and it is generalized cyclic ϕ−1/41/122/3 weak nonexpansive w.r.t A,B, where ϕ is defined by ϕt=1/13t. In fact, if u∈A and v∈B, then we have
(47)ΓS1411223u,v=14du,v+112du,Su+23dv,Sv=14v−u+112∣u−−u3∣+23∣v−−v2∣=14v−u+112∣u+u3∣+23∣v+v2∣=14v−u−112u+u3+23v+v2=14v−u−1124u3+233v2=14v−u−u9+v=54v−13u36=13645v−13u.Using this, we have(48)dSu,Sv=∣−u3−−v2∣=∣v2−u3∣=v2−u3=163v−2u=13618v−12u=13645v−13u−13627v−u≤13645v−13u−1364513v−u=13645v−13u−11313645v−13u=ΓS1411223u,v−113ΓS1411223u,v=ΓS1411223u,v−ϕΓS1411223u,v.


## 2.2. Generalized Results for Generalized abc;r Contraction.

Next, we generalize results for generalizedabc;r contraction [19] to those of generalized cyclic abc;r contraction in b-metric spaces. We have the following important lemmas:Lemma 26.
LetSbe generalized cyclicabc;rcontraction w.r.tA,Bof theb-metric spaceX. Then for everyy∈Xand any natural numbern∈ℕ, we have:(1)
The inequalities(49)QSn+1y,Sny≤a+b1−cQSny,Sn−1y,ifnisodd,a+cr1−b+crQSny,Sn−1y,ifniseven,are true(2)
There are two natural numbersl and m with n=l+m and
(50)QSn+1y,Sny≤a+b1−cla+cr1−b+crmQSy,y.Proof.
Lety∈X=A⋃B, y∈A (similar conclusion if y∈B). Then Sy∈B, using the contractivity condition of S gives
(51)QS2y,Sy=QSy,SSy≤aQy,Sy+bQy,Sy+cmaxQSy,SSy+QSy,Sy=a+bQy,Sy+cQSy,SSy,hence,
(52)QS2y,Sy≤a+b1−cQSy,y.
ForSy∈B, we have SSy∈A, thus
(53)QS3y,S2y=QSS2y,SSy≤aQS2y,Sy+bQS2y,SS2y+cmaxQSy,SSy,QSy,SS2y=aQS2y,Sy+bQS2y,S3y+cmaxQSy,SSy,rQSy,SSy+QSSy,SS2y=aQS2y,Sy+bQS2y,S3y+crQSy,SSy+QSSy,SS2y=a+crQS2y,Sy+b+crQS2y,S3y.
Becauser≥1, hence
(54)QS3y,S2y≤a+cr1−b+crQS2y,Sy.
Using the inequalities (52) and (54) gives
(55)QS3y,S2y≤a+cr1−b+cra+b1−cQSy,y.
ForS2y∈A, we have S3y∈B, and then
(56)QS4y,S3y=QSS2y,SS3y≤aQS2y,S3y+bQS2y,SS2y++cmaxQS3y,SS3y,QS3y,SS2y≤a+bQS2y,S3y+cQS3y,SS3y.
Consequently,(57)QS4y,S3y≤a+b1−cQS3y,S2y.
Using the inequalities (55) and (57) gives
(58)QS4y,S3y≤a+cr1−b+cra+b1−c2QSy,y.
ForS3y∈B, we have S4y∈A, and then
(59)QS5y,S4y=QSS4y,SS3y≤a+cr1−b+crQS4y,S3y.
Using the inequalities (58) and (59) gives
(60)QS5y,S4y≤a+cr1−b+cr2a+b1−c2QSy,y.
Continuing in this process, we see that(61)QS6y,S5y≤a+cr1−b+cr2a+b1−c3QSy,yQS7y,S6y≤a+cr1−b+cr3a+b1−c3QSy,y.
And so on, combining the above inequalities prove formulas (49) and (50).Lemma 27.
LetSbe generalized cyclicabc;rcontraction w.r.t any pairA,BofX. Then for everyy∈Xand any natural numbern∈ℕ, there is a real numberr0<1such that the inequalities(62)QSn+1y,Sny≤r0QSny,Sn−1y(63)QSn+1y,Sny≤r0nQSy,yare true.Proof.
Using Lemma26 with r0=maxa+cr/1−b+cr,a+b/1−c completes the proof.Lemma 28.
LetSbe generalized cyclicabc;rcontraction w.r.t any pairA,BofX. Then for everyy∈X, the sequenceSnyn∈ℕis Cauchy.Proof.
Letm, n be two natural numbers with m>n and y be an arbitrarily element in X. If y∈A or y∈B, then we will have the same conclusion
(64)QSny,Smy≤rQSny,Sn+1y+QSn+1y,Smy≤rQSny,Sn+1y+rQSn+1y,Sn+2y+QSn+2y,Smy≤rQSny,Sn+1y+r2QSn+1y,Sn+2y+r2rQSn+2y,Sn+3y+QSn+3y,Smy≤rQSny,Sn+1y+r2QSn+1y,Sn+2y+r3QSn+2y,Sn+3y+⋯+rm−nQSm−1y,Smy.
Using the inequalities (63) gives
(65)QSny,Smy≤rr0nQSy,y+r2r0n+1QSy,y+r3r0n+2QSy,y+⋯+rm−nr0n+m−1−nQSy,y=rr0nQSy,y+rr0QSy,y+rr02QSy,y+⋯+rr0m−1−nQSy,y=rr0nQSy,y1−rr0m−1−n1−rr0≤r1−rr0QSy,yr0n.
Clearly,r0<1, taking the limit as n⟶∞ completes the proof.
We have the following (fp) results:Theorem 29.
LetX,Qbe completeb-metric space,Sbe generalized cyclicabc;rcontraction w.r.tA,B. Then,Shas single (fp)u∈A⋂B. Moreover, for anyy∈X, the sequenceSnyn∈ℕis converging tou.Proof.
Lety be an arbitrarily chosen element of X, in either of the two cases y∈A or y∈B we have the same result, using inequalities (63) of Lemma 27 ensures that
(66)limn→∞QSn+1y,Sny=0,and Snyn∈ℕ is Cauchy by Lemma 28. Since X is complete b-metric space, Snyn∈ℕ is convergent to some element u∈X;
(67)limn→∞QSny,u=0.
Each subsequence is convergent to the same limit point, since each ofA and B contains infinite number of terms of Snyn∈ℕ. Hence, in A and in B, we construct a subsequence of Sniyi∈ℕ that converges to u. Since A and B are closed, we conclude that u is lying in the intersection of A and B. On the other side, if Sniy∈B, u∈A, (same conclusion if we consider Sniy∈A, u∈B), use the cyclic condition
(68)QSu,Sni+1y=QSu,SSniy≤aQu,Sniy+bQu,Su+cmaxQSniy,Sni+1y,QSniy,Su.
Taking the limit asi⟶∞, the inequalities (66) and (67) give
(69)limi→∞QSu,Sni+1y≤a0+bQu,Su+cmax0,limi→∞QSniy,Su.
Hence,(70)1−climi→∞QSu,Sniy≤bQu,Su,limi→∞QSu,Sniy≤b1−cQu,Su.
Since(71)QSu,u≤rQSu,Sniy+QSniy,u,we have
(72)QSu,u≤rlimi→∞QSu,Sniy+limi→∞QSniy,u,that is
(73)QSu,u≤rb1−cQu,Su+0.
Accordingly, we have(74)QSu,u≤rb1−cQu,Su<Qu,Su.
This shows thatQSu,u=0. Consequently Su=u, that is, u is a (fp) of S. Finally, to prove that such a (fp) is unique, let u and z be two distinct (fps) of S. Noting that both u and z are lying in the intersection of A and B (if z∈A, then z=Sz∈B and vise versa), the following strict inequality gives an obvious contradiction:
(75)Qu,z=QSu,Sz≤aQu,z+bQu,Su+cmaxQz,Sz+Qz,Su=aQu,z+bQu,u+cmaxQz,z+Qz,u=aQu,z+cQz,u=a+cQu,z<Qu,z.Remark 30.
Using the same steps ofproof, Theorem (29) can be easily extended to the case when S has Aii=1j (U) cyclic representations.Now, we are going to give a generalization of Theorem (31) [19], the prove is more simpler and in case of cyclic type mappings.Theorem 31.
IfSnn∈N and S are generalized cyclicabc;r contractions on the complete b-metric space X,Qw.r.tA,B, wn∗ is the unique fixed of each Sn, Snwn∗=wn∗∀n∈ℕ, andSnw∗n∈ℕ converges to w∗,
(76)limn→∞QSnw∗,Sw∗=0,where w∗ is the unique fixed of S; Sw∗=w∗, then wn∗n∈ℕ converges to w∗;
(77)limn→∞Qwn∗,w∗=0.Proof.
We have(78)Qwn∗,w∗=QSnwn∗,Sw∗≤rQwn∗,Snw∗+QSnw∗,Sw∗.
Taking in account thatwn∗n∈N⋃w∗⊆A⋂B, we have
(79)Qwn∗,Snw∗=QSnwn∗,Snw∗≤aQwn∗,w∗+bQw∗,Snw∗+cmaxQwn∗,Snwn∗,Qwn∗,Snw∗≤aQwn∗,w∗+bQw∗,Snw∗+cQwn∗,Snw∗Qwn∗,Snw∗≤11−caQwn∗,w∗+bQw∗,Snw∗.
Substituting from (79) in (78) gives
(80)Qwn∗,w∗=QSnwn∗,Sw∗≤r11−caQwn∗,w∗+bQw∗,Snw∗+QSnw∗,Sw∗≤ra1−cQwn∗,w∗+b1−cQw∗,Snw∗+QSnw∗,Sw∗≤ra1−cQwn∗,w∗+b1−c+1rQw∗,Snw∗.
Hence,(81)1−c−ra1−cQwn∗,w∗≤b−c+11−crQw∗,Snw∗.
Consequently, we have(82)Qwn∗,w∗≤b−c+11−c−rarQw∗,Snw∗.
Taking the limit asn⟶∞ and using assumption (76) prove (77).


## 3. Conclusion

This manuscript indicates the existence of a single common (fp) for two jointA,B generalized cyclic ϕ−abc weak nonexpansive type mappings, where A and B are compact sets. The result presented a single (fp) for generalized cyclic ϕ−abc weak nonexpansive mapping, without the assumption of existence a single contracting point. Additionally, it introduces a new type of generalized cyclic abc;r contraction mapping and describes the existence of a single (fp) of such mappings in b-metric spaces. Finally, the presented results establish a simpler convergence theorem for a sequence of generalized cyclic abc;r contraction mappings. These results are extensions and generalizations of previous results reported in [10, 11, 17, 19, 20].



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