# Common Fixed Point Theorems via InverseCk− Class Functions in Metric Spaces

Authors: Mi Zhou; Xiao-Lan Liu; Arslan Hojat Ansari; Mukesh Kumar Jain; Jia Deng
Journal: Journal of Mathematics
 (2021)

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## Abstract

In this paper, we firstly introduce a new notion of inverseCk−class functions which extends the notion of inverse C−class functions introduced by Saleem et al., 2018. Secondly, some common fixed point theorems are stated under some compatible conditions such as weak semicompatible of type A, weak semicompatibility, and conditional semicompatibility in metric spaces. Moreover, we introduce a new kind of compatibility called Sτ−compatibility which is weaker than E.A. property and also present a common fixed point theorem in metric spaces via inverse Ck−class functions. Some examples are provided to support our results.


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## Body

## 1. Introduction and Preliminaries

As a follow-up work of A.H. Ansari’s research on fixed point (or common fixed point) theory via auxiliary C-class functions, very recently, Saleem et al. [1] introduced the new concept of inverse C-class functions and obtained some corresponding fixed point theorems under certain weak compatibility assumption via inverse C-class functions. In 1976, Jungck [2] defined the concept of commutative maps and initiated the study of the existence of a common fixed point of such maps in metric spaces. After which, Sessa [3] introduced the weak version of commuting maps called weak commuting maps. Next, Jungck [4, 5] provided some generalizations of weak commuting maps by providing the notions of compatible maps and compatible maps of type A. Minor relaxations of compatible of type A are introduced by Pathak and Khan [6], which are well known as g−compatible and f−compatible (see [6], for more details).Singh et al. [7] proposed the notion of compatibility of type E by making a minor modification of compatibility of type A. By splitting the concept of compatibility of type E, Singh et al. [7] also gave some relaxations of compatibility type of E which are known as g−compatibility of type E and f−compatibility of type E.Definition 1.
(see [7]). Two self-maps f and g of a metric space X,d are said to be f−compatible of type E, if limn⟶+∞ffxn=limn⟶+∞fgxn=gt, whenever xn is a sequence in X such that limn⟶+∞fxn=limn⟶+∞gxn=t, for some t∈X. Similarly, two self-maps f and g of a metric space X,d are said to be g−compatible of type E, if limn⟶+∞ggxn=limn⟶+∞gfxn=ft, whenever xn is a sequence in X such that limn⟶+∞fxn=limn⟶+∞gxn=t, for some t∈X.
It is easy to see that compatibility of typeE implies both g−compatibility of type E and f−compatibility of type E; however, g−compatibility or f−compatibility of type E does not imply compatibility of type E.
In 1994, Pant [8] introduced the following definition.Definition 2.
(see [8]). Two self-maps f and g defined on a metric space X,d are said to be R−weakly commuting, if there exists a real number R>0 such that dfgx,gfx≤Rdfx,gx, for all x∈X.
Note thatR=1; then, f and g are weakly commuting.
In 1997, Pathak et al. [9] introduced the notions of R−weak commuting of type Ag and R−weak commuting of type Af as follows.Definition 3.
(see [9]). Two self-maps f and g of a metric space X,d are said to be R−weakly commuting of type Ag, if there exists a real number R>0 such that dgfx,ffx≤Rdfx,gx, for all x∈X.Definition 4.
(see [9]). Two self-maps f and g of a metric space X,d are said to be R−weakly commuting of type Af, if there exists a real number R>0 such that dfgx,ggx≤Rdfx,gx, for all x∈X.
It is noted that compatible mapsf and g are also R−weakly commuting of type Ag and R−weakly commuting of type Af. Moreover, we can find suitable examples which show that R−weakly commuting mappings and R−weakly commuting of type Af (or Ag) are independent concepts (see examples of [9, 10]).
In 2008, Gopal et al. [10] introduced the notions of g−absorbing and f−absorbing stated as follows.Definition 5.
(see [10]). Let f and gf≠g be two self-maps of a metric space X,d; then, f is said to be g-absorbing, if there exists a real number R>0 such that dgx,gfx≤Rdfx,gx for all x∈X. Similarly, let f and gf≠g be two self-maps of a metric space X,d; then, g is said to be f-absorbing, if there exists a real number R>0 such that dfx,fgx≤Rdfx,gx for all x∈X.
Jungck and Rhoades [11], in 1998, introduced the concept of weak compatibility which is weaker than the concept of compatibility.
Another generalization of compatible maps called semicompatible maps was firstly introduced by Cho et al. [12] under the setting of d−topological spaces in which a pair S,T of self-maps are called to be semicompatible if condition iSy=Ty implies that STy=TSy; ii for sequence xn in X and x∈X, whenever Sxn⟶x, Txn⟶x, and then STxn⟶Tx, as n⟶+∞, hold. However, Singh and Jain [13] redefined this concept by using condition ii only stated as follows.Definition 6.
(see [13]). A pair f,g of self-maps of a metric space X,d is said to be semicompatible, if limn⟶+∞fgxn=gt holds whenever xn is a sequence in X such that limn⟶+∞fxn=limn⟶+∞gxn=t for some t∈X.
It follows that iff,g is semicompatible and fx=gx, then fgx=gfx. It is also noted that if the pair f,g is semicompatible, then it is weak compatible; however, the converse is not true. Further, the semicompatibility of the pair f,g does not imply the semicompatibility of the pair g,f (see Example 3.2 in [13]).
Now, we make a minor modification of semicompatibility to introduce the notion of semicompatible of typeA as follows.Definition 7.
A pairf,g of self-maps of a metric space X,d is said to be semicompatible of type A, if limn⟶+∞fgxn=gt and limn⟶+∞gfxn=ft hold whenever xn is a sequence in X such that limn⟶+∞fxn=limn⟶+∞gxn=t for some t∈X.
It is obvious that semicompatibility of typeA implies semicompatibility; however, the converse is not true.
Recently, Saluja et al. [14, 15] introduced the weak semicompatible maps and conditional semicompatible maps and obtained corresponding fixed point theorems (see [14–16], for more details).Definition 8.
(see [14]). A pair f,g of self-maps of a metric space X,d is said to be weakly semicompatible, if limn⟶+∞fgxn=gt or limn⟶+∞gfxn=ft, whenever xn is a sequence in X such that limn⟶+∞fxn=limn⟶+∞gxn=t for some t∈X.Definition 9.
(see [15]). A pair f,g of self-maps of a metric space X,d is said to be conditionally semicompatible; if whenever the set of sequences xn satisfying limn⟶+∞fxn=limn⟶+∞gxn is nonempty, then there exists at least a sequence yn satisfying limn⟶+∞fyn=limn⟶+∞gyn=t such that limn⟶+∞fgyn=gt and limn⟶+∞gfyn=ft.
It is obvious that semicompatibility of typeA implies weak semicompatibility. From the definition itself, it is clear that if a pair f,g of self-maps is semicompatible of type A, then it is necessarily conditionally semicompatible; however, the conditionally semicompatible maps are not necessarily semicompatible of type A.Example 1.
LetX=1,+∞ and d be the usual metric on X. Define f,g:X↦X as follows:(1)fx=1,ifx=1,x+72,if1<x<3,3,ifx≥3,gx=1,ifx=1,x+3,if1<x<3,x,ifx≥3.
Let us consider the sequencexn=1+1/n; we have(2)limn⟶+∞fxn=limn⟶+∞f1+1n=4,limn⟶+∞gxn=limn⟶+∞g1+1n=4,limn⟶+∞fgxn=limn⟶+∞f4+1n=3≠g4,limn⟶+∞gfxn=limn⟶+∞g4+12n=4≠f4.
However, if we takeyn=3+1/n, we have that(3)limn⟶+∞fyn=limn⟶+∞f3+1n=3,limn⟶+∞gyn=limn⟶+∞g3+1n=3,limn⟶+∞fgxn=limn⟶+∞f3+1n=3=g3,limn⟶+∞gfxn=limn⟶+∞g3=3=f3.
Thus, the pairf,g is conditional semicompatible.
Finally, we introduce a new kind of compatibility of a pairf,g of self-maps called Sτ−compatible firstly proposed by Jain et al. [17] as follows.Definition 10.
Letτ be a self-map defined on X satisfying limn⟶+∞τxn=t for some sequence xn∈X and t∈X. Then, a pair f,g of self-maps defined on X is called Sτ−compatible if limn⟶+∞fτxn=limn⟶+∞gτxn=τt.Example 2.
LetX=ℝ, fx=2x, gx=4−2x, and τx=1+x. Take xn=1/n. Since limn⟶+∞τxn=1 with limn⟶+∞fτxn=2=τ1 and limn⟶+∞gτxn=2=τ1, then pair f,g is Sτ−compatible. However, limn⟶+∞τxn≠limn⟶+∞fxn and limn⟶+∞τxn≠limn⟶+∞gxn.
It is obvious thatSτ−compatibility of a pair f,g self-maps implies E.A. property of a pair f,g of self-maps by taking self-map τ as an identity map.
LetX=ℝ, fx=x, gx=x2, and τ=Ix(identity function on X). Take xn=1/n. Here, limn⟶+∞τxn=limn⟶+∞gτxn=limn⟶+∞fτxn=limn⟶+∞xn=0. Hence, pair self-maps f,g satisfy E.A. property.
Now, we introduce one more example ofSτ−compatibility including four maps as follows.Example 3.
IfX=1,+∞ with the usual metric. Define f,g,B,T:X↦X by(4)fx=x+1,ifx∈1,2,3,ifx∈2,+∞,gx=2,ifx∈1,2,x+42,ifx∈2,+∞,Bx=2x,ifx∈1,2,x+1,ifx∈2,+∞,Tx=3x−1,ifx∈1,2,2x−1,ifx∈2,+∞.
Choosexn=1+εn, where εn⟶0 when n⟶+∞, then limn⟶+∞fxn=2, limn⟶+∞gxn=2, limn⟶+∞Bxn=2, and limn⟶+∞Txn=2.
Sincelimn⟶+∞fBxn=3=B2 and limn⟶+∞gBxn=3=B2, then the pair f,g is SB−compatible. Next, since limn⟶+∞fTxn=3=T2 and limn⟶+∞gTxn=3=T2, then the pair f,g is ST−compatible. Further, since limn⟶+∞Bgxn=3=g2 and limn⟶+∞Tgxn=3=g2, then the pair B,T is Sg−compatible. Finally, since limn⟶+∞Bfxn=3=f2 and limn⟶+∞Tfxn=3=f2, then the pair B,T is Sf−compatible.
Ansari, in 2014, firstly [18], introduced the concept of C−class functions and proved some fixed point theorems via C−class functions (see [19, 20] for more details).Definition 11.
(see [18]). A mapping F:0,+∞2↦ℝ is called a C−class function if it is continuous and the following axioms hold:(1)
Fs,t≤s for all s,t∈0,+∞(2)
Fs,t=s implies that s=0 or t=0
Denote the family ofC−class functions by C.Example 4.
(see [18]). The following functions F:0,+∞2↦ℝ are elements of C, for all s,t∈0,+∞:(1)
Fs,t=s−t, Fs,t=s implies t=0(2)
Fs,t=ms, for some m∈0,1, Fs,t=s implies s=0(3)
Fs,t=s/1+tr, for some r∈0,+∞, Fs,t=s implies s=0 or t=0(4)
Fs,t=logat+as/1+t, for some a>1, Fs,t=s implies s=0 or t=0(5)
Fs,t=loga1+as/2, for a>e, Fs,1=s implies s=0(6)
Fs,t=s+l1/1+tr−l, l>1, for r∈0,+∞, Fs,t=s implies t=0(7)
Fs,t=slogt+aa, for a>1, Fs,t=s implies s=0 or t=0(8)
Fs,t=s−1+s/2+st/1+t, Fs,t=s implies t=0(9)
Fs,t=sβs, where β:0,+∞↦0,1 is continuous, Fs,t=s implies s=0(10)
Fs,t=s−t/k+t, Fs,t=s implies t=0(11)
Fs,t=s−φt, Fs,t=s implies s=0, where φ:0,+∞⟼0,+∞ is a continuous function such that φt=0 if and only if t=0(12)
Fs,t=shs,t, Fs,t=s implies s=0, where h:0,+∞×0,+∞↦0,+∞ is a continuous function such that ht,s<1 for all t,s>0(13)
Fs,t=s−2+t/1+tt, Fs,t=s implies t=0(14)
Fs,t=ln1+snn, Fs,t=s implies s=0(15)
Fs,t=ϕs, Fs,t=s implies s=0, where ϕ:0,+∞⟼0,+∞ is a continuous function such that ϕ0=0 and ϕt<t, for t>0
Afterward, by the motivation ofC−class functions, Saleem et al. [1, 21] introduced a new notion of inverse C−class functions as follows.Definition 12.
(see [1]). A mapping F:0,+∞2↦ℝ is called an inverse C−class function if it is continuous and the following axioms hold:(1)
Fs,t≥s for all s,t∈0,+∞(2)
Fs,t=s implies that s=0 or t=0
Denote the family of inverseC−class functions by Cinv.Example 5.
(see [1]). The following functions F:0,+∞2↦ℝ are elements of Cinv, for all s,t∈0,+∞:(1)
Fs,t=s+t,Fs,t=s implies t=0(2)
Fs,t=ms, for some m∈1,+∞,Fs,t=s implies s=0(3)
Fs,t=s1+tr, for some r∈0,+∞, Fs,t=s implies s=0 or t=0(4)
Fs,t=logat+as1+t, for some a>1, Fs,t=s implies t=0(5)
Fs,t=ϕs, Fs,t=s implies s=0, where ϕ:0,+∞↦0,+∞ is an upper semicontinuous function such that ϕ0=0 and ϕt>t, for t>0
Motivated by the above definition, we now define inverseCk−class functions as follows.Definition 13.
A mappingF:0,+∞2↦ℝ is called an inverse Ck−class function if it is continuous and the following axioms hold:(1)
Fs,t≥ks for all s,t∈0,+∞ and some k≥1(2)
Fs,t=ks implies that s=0 or t=0
Denote the family of inverseCk−class functions by Cinv−k. Every inverse C−class function and inverse Ck−class function are equivalent when k=1; however, an inverse Ck−class function may not be an inverse C−class function.Example 6.
A mappingF:0,+∞↦ℝ is defined by Fs,t=2s+t for all s,t∈0,+∞. Then, clearly, F is an inverse Ck−class function for k=2, but it is not an inverse C−class function.Example 7.
The following functionsF:0,+∞2↦ℝ are elements of Cinv−k, for all s,t∈0,+∞:(1)
Fs,t=ks+lt,Fs,t=ks implies t=0 for some k≥1 and l>0(2)
Fs,t=kms,Fs,t=ks implies s=0 for m∈1,+∞ and some k≥1(3)
Fs,t=ks1+ktr,Fs,t=ks implies s=0 or t=0 for r∈0,+∞ and some k≥1(4)
Fs,t=logakt+aks1+kt,Fs,t=ks implies t=0 for some a>1 and some k≥1(5)
Fs,t=kϕs, Fs,t=ks implies s=0, where ϕ:0,+∞↦0,+∞ is an upper semicontinuous function such that ϕ0=0 and ϕt>t, for t>0 and some k≥1Definition 14.
LetΨ denote the class of functions ψ:0,+∞↦0,+∞ which satisfy the following conditions:(a)
ψ is continuous and increasing with ψt≥t(b)
ψt=0⇔t=0Definition 15 (see [22]).
A functionϕ:0,+∞↦0,+∞ is said to be ultra-altering distance function if aϕ is nondecreasing and continuous; bϕt>0, for all t>0 and ϕt=0⟹t=0. Denote the class of ultra-altering distance functions by Φ.Lemma 1.
Every sequencexn in metric space X,d will be Cauchy if there exists λ∈0,1 such that dxn+1,xn≤λdxn,xn−1, for all n∈ℕ.
The aim of this presented paper is to provide some common fixed point theorems under several compatible conditions mentioned above via inverseCk−class functions, which extend, generalize, and improve the existing results in the literature. Some examples are provided to illustrate the validity of our results.


## 2. Main Results

Theorem 1.
LetX,d be a complete metric space and a let pair g,f of self-maps be semicompatible, satisfying the following assumptions:A1fX⊆gXA2kψdfx,fy≥FψMx,y,ϕMx,y+LNx,y
Here,Mx,y=a1dfx,gx+a2dfy,gy+a3dfx,gy+a4dfy,gx+a5dgx,gy and Nx,y=mindfx,gx,dfy,gx, for all x,y∈X. Moreover, ai>0,i=1,2,…,5, with 1≥a1−a3, a1+a2+a5>1, a3+a4>a1+a2, L∈ℝ, F∈Cinv−k, for some k≥1 and ψ∈Ψ, ϕ∈Φ. If the pair g,f is f−compatible of type E, then f and g have a unique common fixed point t in X.Proof.
Letx0 be any point in X. Since fX⊆gX, there exists x1∈X such that(5)fx1=gx0=y0.
Continuing this way, we can construct a sequencexn in X satisfying(6)fxn+1=gxn=yn.
By the assumption ofA2, we have(7)kψdfxn,fxn+1≥FψMxn,xn+1,ϕMxn,xn+1+LNxn,xn+1≥kψMxn,xn+1+LNxn,xn+1=kψa1dfxn,gxn+a2dfxn+1,gxn+1+a3dfxn,gxn+1+a4dfxn+1,gxn+a5dgxn,gxn+1+Lmindfxn,gxn,dfxn+1,gxn=kψa1dyn−1,yn+a2dyn,yn+1+a3dyn−1,yn+1+a4dyn,yn+a5dyn,yn+1+Lmindyn−1,yn,dyn,yn=kψa1dyn−1,yn+a2dyn,yn+1+a3dyn−1,yn+1+a5dyn,yn+1.
By the monotonicity ofψ, we have(8)dyn−1,yn≥a1dyn−1,yn+a2dyn,yn+1+a3dyn−1,yn+1+a5dyn,yn+1.
Again, from the triangle inequality, that is,dyn,yn+1≤dyn,yn−1+dyn−1,yn+1, we have(9)dyn−1,yn≥a1dyn−1,yn+a2dyn,yn+1+a3dyn,yn+1−dyn,yn−1+a5dyn,yn+1=a1−a3dyn−1,yn+a2+a3+a5dyn,yn+1,which further yields that(10)dyn,yn+1≤1−a1+a3a2+a3+a5dyn−1,yn.
Since1≥a1−a3, a1+a2+a5>1, then 0≤1−a1+a3/a2+a3+a5<1; from Lemma 1, it follows that sequence yn is a Cauchy sequence. Since X is complete, there exists a point t∈X such that(11)limn⟶+∞yn=limn⟶+∞fxn=limn⟶+∞gxn=t.
Now, we will show thatt is a common fixed point of f and g.
Since the pairg,f is semicompatible, we have(12)limn⟶+∞gfxn=ft.
Sincef and g are f−compatible of type E, it follows that(13)limn⟶+∞ffxn=limn⟶+∞fgxn=gt.
Now, by the definition ofψ and assumption A2, we obtain(14)kψdffxn,ft≥FψMfxn,t,ϕMfxn,t+LNfxn,t=Fψa1dffxn,gfxn+a2dft,gt+a3dffxn,gt+a4dft,gfxn+a5dgfxn,gt,ϕa1dffxn,gfxn+a2dft,gt+a3dffxn,gt+a4dft,gfxn+a5dgfxn,gt+Lmindffxn,gfxn,dft,gfxn.
Taking the limit asn⟶+∞ in above inequality, it follows that(15)kψdgt,ft≥Fψa1dgt,ft+a2dft,gt+a3dgt,gt+a4dft,ft+a5dft,gt,ϕa1dgt,ft+a2dft,gt+a3dgt,gt+a4dft,ft+a5dft,gt+Lmindgt,ft,dft,ft=Fψa1dft,gt+a2dft,gt+a5dft,gt,ϕa1dft,gt+a2dft,gt+a5dft,gt≥kψa1+a2+a5dgt,ft≥kψdgt,ft,which implies that(16)dft,gt=0orft=gt.
Again, it follows from the definition ofψ and assumption A2 that(17)kψdfxn,ft≥Fψa1dfxn,gxn+a2dft,gt+a3dfxn,gt+a4dft,gxn+a5dgxn,gt,ϕa1dfxn,gxn+a2dft,gt+a3dfxn,gt+a4dft,gxn+a5dgxn,gt+Lmindfxn,gxn,dft,gxn.
Taking the limit asn⟶+∞ in above inequality, we conclude that(18)kψdt,ft≥Fψa3dt,gt+a4dft,t+a5dt,gt,ϕa3dt,gt+a4dft,t+a5dt,gt=Fψa3dt,ft+a4dft,t+a5dt,ft,ϕa3dt,ft+a4dft,t+a5dt,ft≥kψa3+a4+a5dt,ft.
Sincea1+a2<a3+a4 and a1+a2+a5>1, it follows that(19)kψdt,ft≥Fψa3+a4+a5dt,ft,ϕa3+a4+a5dt,ft≥kψa3+a4+a5dt,ft≥kψa1+a2+a5dt,ft≥kψdt,ft,which implies that(20)Fψa3+a4+a5dt,ft,ϕa3+a4+a5dt,ft=kψdt,ft.
From the definition ofCinv−k−class functions, we have(21)ψa3+a4+a5dt,ft=0orϕa3+a4+a5dt,ft=0.
By the definitions ofψ and ϕ, we have that dft,t=0, which proves that ft=gt=t, that is, t is a common fixed point of f and g.
Next, we will prove the uniqueness of common fixed point off and g in X.
Suppose thatu is another common fixed point of f and g, that is fu=gu=u.
From above argument, it may be concluded that(22)kψdfu,ft≥FψMu,t,ϕMu,t+LNu,t=Fψa1dfu,gu+a2dft,gt+a3dfu,gt+a4dft,gu+a5dgu,gt,ϕa1dfu,gu+a2dft,gt+a3dfu,gt+a4dft,gu+a5dgu,gt+Lmindfu,gu,dft,gu.=Fψa3du,t+a4dt,u+a5du,t,ϕa3du,t+a4dt,u+a5du,t≥kψa3+a4+a5du,t.
Sincea1+a2<a3+a4 and a1+a2+a5>1, it follows that(23)kψdu,t≥Fψa3+a4+a5du,t,ϕa3+a4+a5du,t≥kψa3+a4+a5du,t≥kψa1+a2+a5du,t≥kψdu,t,which implies that(24)Fψa3+a4+a5du,t,ϕa3+a4+a5du,t=kψdu,t.
From the definition ofCinv−k−class functions, we have(25)ψa3+a4+a5du,t=0orϕa3+a4+a5du,t=0.
By the definitions ofψ and ϕ, we have that du,t=0, which proves that u=t.Remark 1.
(i)
The conclusion of Theorem1 still holds under the assumptions of semicompatibility of the pair f,g and g−compatibility of type E.(ii)
If the inequality in assumptionA2 is replaced by(26)kψdfx,gx≥FψMx,y,ϕMx,y,the conclusion still holds.
Here is an illustrated example to support the validity of Theorem1 as follows.Example 8.
LetX=0,2 be a usual metric space. We define f and g on X as follows:(27)fx=32,∀x∈X,gx=12,x∈0,1,32,x∈1,2.
To verify that the pairg,f is semicompatible as well as f−compatible of type E, we take any sequence xn∈1,2; then, limn⟶+∞fxn=limn⟶+∞gxn=3/2 and limn⟶+∞fgxn=3/2=g3/2,limn⟶+∞ffxn=3/2=g3/2,limn⟶+∞gfxn=3/2=g3/2. Define F∈Cinv−k by Fs,t=2s, with k=1 and ψt=t=ϕt, for all t∈0,+∞. Then, for all x,y∈0,1, we have(28)kψdfx,fy=0,FψMx,y,ϕMx,y+LNx,y=2ψMx,y+LNx,y=2a1dfx,gx+a2dfy,gy+a3dfx,gy+a4dfy,gx+a5dgx,gy+Lmindfx,gx,dfy,gx=2a132−12+a232−12+a332−12+a432−12+Lmin32−12,32−12=2a1+a2+a3+a4+L.
It is verified that assumptionA2 holds true provided by a1=2,a2=a3=1,a4=3, and a5>0 such that 1≥a1−a3, a1+a2+a5>1, a3+a4>a1+a2, and L≤−14.
Also, it is obvious that assumptionA2 holds true, for all x,y∈1,2.
Hence, functionsf and g satisfy all conditions of Theorem 1 with 3/2 are common fixed point.
Takingf as the identity map in Theorem 1, we have the following fixed point theorem.Corollary 1.
Letg be a self-map defined on a complete metric space X,d satisfying the following assumption:(29)kψdx,y≥FψMx,y,ϕMx,y+LNx,y,where Mx,y=a1dx,gx+a2dy,gy+a3dx,gy+a4dy,gx+a5dgx,gy and Nx,y=mindx,gx,dy,gy, for all x,y∈X. Moreover, ai>0,i=1,2,…,5 with 1≥a1−a3, a1+a2+a5>1, a1+a2<a3+a4, L∈ℝ, F∈Cinv−k for some k≥1 and ψ∈Ψ, ϕ∈Φ. Then, g has a unique fixed point in X.
Since semicompatibility of typeA implies semicompatibility, we can obtain the following theorem.Theorem 2.
LetX,d be a complete metric space and let a pair g,f of self-maps be semicompatible of type A satisfying the following assumptions:A1fX⊆gXA2kψdfx,fy≥FψMx,y,ϕMx,y+LNx,ywhere Mx,y=a1dfx,gx+a2dfy,gy+a3dfx,gy+a4dfy,gx+a5dgx,gy and Nx,y=mindfx,gx,dfy,gx, for all x,y∈X. Moreover, ai>0,i=1,2,…,5 with 1≥a1−a3, a1+a2+a5>1, a1+a2<a3+a4, L∈ℝ, F∈Cinv−k, for some k≥1 and ψ∈Ψ, ϕ∈Φ. If the pair f,g is either f−compatible of type E or g−compatible of type E, then f and g have a unique common fixed point t in X.
Now, we will present a common fixed point theorem under weakly semicompatible condition viaCinv−k-class functions as follows.Theorem 3.
LetX,d be a complete metric space and let a pair g,f of self-maps be weakly semicompatible and R−weakly commuting type of Af satisfying the following assumptions:A1fX⊆gXA2′kψdfx,fy≥FψMx,y,ϕMx,y+LNx,y
Here,Mx,y=a1dfx,gx+a2dfy,gy+a3dfx,gy+a4dfy,gx+a5dgx,gy and Nx,y=mindfx,gx,dfy,gx, for all x,y∈X. Moreover, ai>0,i=1,2,…,5, L∈ℝ, F∈Cinv−k for some k≥1 with 1≥a1−a3, a1+a2+a5>1, a1+a2<a3+a4, ka1+a2+a3+a4+L>1, and ψ∈Ψ, ϕ∈Φ. If the pair f,g is either f−compatible of type E or g−compatible of type E, then f and g have a unique common fixed point t in X.Proof.
Letx0 be any point in X. Since fX⊆gX, there exists x1∈X such that(30)fx1=gx0=y0.
Continuing this way, we can have a sequencexn in X satisfying(31)fxn+1=gxn=yn.
From the discussion of Theorem1, we have that there exists a point t∈X such that(32)limn⟶+∞yn=limn⟶+∞fxn=limn⟶+∞gxn=t.
Now consider the following possibilities:Case 1. (f and g are f−compatible of type E).Sincef and g are weakly semicompatible, it follows that(33)limn⟶+∞fgxn=gtorlimn⟶+∞gfxn=ft.Firstly, we take(34)limn⟶+∞gfxn=ft.Sincef and g are f−compatible of type E, it yields that(35)limn⟶+∞ffxn=limn⟶+∞fgxn=gt.Hence, the conclusion can directly follow from the proof of Theorem1.Secondly, we take(36)limn⟶+∞fgxn=gt.Sincef and g are f−compatible of type E, it yields that(37)limn⟶+∞ffxn=limn⟶+∞fgxn=gt.Again sincef and g are R−weakly commuting of type Af, which implies(38)dfgxn,ggxn≤Rdfxn,gxn.Taking limit asn⟶+∞ in above inequality, we have(39)limn⟶+∞ggxn=gt.From the monotonicity ofψ and assumption A2′, we have(40)kψdfgxn,fxn≥FψMgxn,xn,ϕMgxn,xn+LNgxn,xn=Fψa1dfgxn,ggxn+a2dfxn,gxn+a3dfgxn,gxn+a4dfxn,ggxn+a5dggxn,gxn,ϕa1dfgxn,ggxn+a2dfxn,gxn+a3dfgxn,gxn+a4dfxn,ggxn+a5dggxn,gxn+Lmindfgxn,ggxn,dfxn,ggxn.Taking limit asn⟶+∞ in above inequality, we have(41)kψdgt,t≥Fψa1dgt,gt+a2dt,t+a3dgt,t+a4dt,gt+a5dgt,t,ϕa1dgt,gt+a2dt,t+a3dgt,t+a4dt,gt+a5dgt,t+Lmindgt,gt,dt,gt=Fψa3+a4+a5dt,gt,ϕa3+a4+a5dt,gt≥kψa3+a4+a5dgt,t.Sincea1+a2<a3+a4 and a1+a2+a5>1, it follows that(42)kψdgt,t≥Fψa3+a4+a5dgt,t,ϕa3+a4+a5dgt,t≥kψa3+a4+a5dgt,t≥kψa1+a2+a5dgt,t≥kψdgt,t,which implies that(43)Fa3+a4+a5dgt,t,ϕa3+a4+a5dgt,t=kψdgt,t.From the definition ofCinv−k−class functions, we have(44)ψa3+a4+a5dgt,t=0orϕa3+a4+a5dgt,t=0.By the definitions ofψ and ϕ, we have that dgt,t=0, which proves that gt=t.Again, from the definition ofψ and assumption A2′, we have(45)0=kψdft,ft≥Fa1dft,gt+a2dft,gt+a3dft,gt+a4dft,gt+a5dgt,gt,ϕa1dft,gt+a2dft,gt+a3dft,gt+a4dft,gt+a5dgt,gt+Lmindft,gt,dft,gt=Fψa1dft,gt+a2dft,gt+a3dft,gt+a4dft,gt,ϕa1dft,gt+a2dft,gt+a3dft,gt+a4dft,gt+Ldft,t≥kψa1+a2+a3+a4dft,t+Ldft,t≥ka1+a2+a3+a4+Ldft,t.Together withai>0,i=1,2,…,5, ka1+a2+a3+a4+L>1, from the above inequality, it follows that dft,t=0, which proves that ft=t=gt. Hence, t is a common fixed point of f and g.Case 2 (f and g are g−compatible of type E).
Sincef and g are weakly compatible, it follows that(46)limn⟶+∞fgxn=gtorlimn⟶+∞gfxn=ft.
Firstly, we take(47)limn⟶+∞fgxn=gt.
Sincef and g are g−compatible of type E, it yields that(48)limn⟶+∞ggxn=limn⟶+∞gfxn=ft.
Therefore, the conclusion directly follows from Theorem1 and (i) of Remark 1.
Secondly, we take(49)limn⟶+∞gfxn=ft.
Sincef and g are g−compatible of type E, it yields that(50)limn⟶+∞ggxn=limn⟶+∞gfxn=ft.
Alsof and g are R−weakly commuting of type Af, which implies(51)dfgxn,ggxn≤Rdfxn,gxn.
Taking limit asn⟶+∞ in the above inequality, we have(52)limn⟶+∞fgxn=ft.
From the monotonicity ofψ and assumption A2′, we have(53)kψdfxn,fgxn≥FψMxn,gxn,ϕMxn,gxn+LNxn,gxn=Fψa1dfxn,gxn+a2dfgxn,ggxn+a3dfxn,ggxn+a4dfgxn,gxn+a5dgxn,ggxnϕa1dfxn,gxn+a2dfgxn,ggxn+a3dfxn,ggxn+a4dfgxn,gxn+a5gxn,ggxn+Lmindfxn,gxn,dfgxn,gxn.
Taking limit asn⟶+∞ in the above inequality, we have(54)kψdt,ft≥Fψa1dt,t+a2dft,ft+a3dt,ft+a4dft,t+a5dt,ft,ϕa1dt,t+a2dft,ft+a3dt,ft++a4dft,t+a5dt,ft+Lmindt,t,dft,t=Fψa3+a4+a5dt,ft,ϕa3+a4+a5dt,ft≥kψa3+a4+a5dt,ft.
Sincea1+a2<a3+a4 and a1+a2+a5>1, it follows that(55)kψdt,ft≥Fψa3+a4+a5dt,ft,ϕa3+a4+a5dt,ft≥kψa3+a4+a5dt,ft≥kψa1+a2+a5dt,ft≥kψdt,ft,which implies that(56)Fψa3+a4+a5dt,ft,ϕa3+a4+a5dt,ft=kψdt,ft.
From the definition ofCinv−k−class functions, we have(57)ψa3+a4+a5dt,ft=0orϕa3+a4+a5dt,ft=0.
By the definitions ofψ and ϕ, we have that dt,ft=0, which proves that ft=t.
Again, from the monotonicity ofψ and assumption A2′, we have(58)kψdfgxn,ft≥FψMgxn,t,ϕMgxn,t+LNgxn,t=Fψa1dfgxn,ggxn+a2dft,gt+a3dfgxn,gt+a4dft,ggxn+a5dggxn,gt,ϕa1dfgxn,ggxn+a2dft,gt+a3dfgxn,gt+a4dft,ggxn+a5dggxn,gt+Lmindfgxn,ggxn,dft,ggxn.
Taking limit asn⟶+∞ in above inequality and by the definition of ψ, we have(59)0=kψdft,ft≥Fψa1dft,ft+a2dft,gt+a3dft,gt+a4dft,ft+a5dft,gt,ϕa1dft,ft+a2dft,gt+a3dft,gt+a4dft,ft+a5dft,gt+Lmindft,ft,dft,ft=Fψa2+a3+a5dft,gt,ϕa2+a3+a5dft,gt≥kψa2+a3+a5dft,gt.
Sinceai>0,i=1,2,…,5, it yields that dft,gt=0 or ft=gt, which further implies ft=gt=t; hence, t is a common fixed point of f and g.
Finally, the uniqueness of the common fixed point off and g can be directly obtained from the second half proof of Theorem 1.Remark 2.
Iff and g are assumed to be R−weakly commuting of type Ag, the conclusion of Theorem 3 still holds true.
Now, we provide an example to verify the validity of Theorem3 as follows.Example 9.
LetX=1,2 be a usual metric space and x,y∈X. We define f and g on X as follows: fx=1 for all x∈X and gx=x for all x∈X. It is easy to check that fX⊆gX.
To show that the pair of mappingsf,g is weak semicompatible and f−compatible of type E, we take any sequence xn∈1,2, then limn⟶+∞fxn=limn⟶+∞gxn=1. Hence, limn⟶+∞fgxn=1=g1 and limn⟶+∞ffxn=1=g1. It also can be observed that pair f,g satisfy R−weak commuting of type Af for R=1. To satisfy condition A2′, we choose Fs,t=2s with k=1 and ψt=t/20, for all t≥0. Then, for all x,y∈1,2, we have(60)kψdfx,fy=0,FψMx,y,ϕMx,y+LNx,y=220a1dfx,gx+a2dfy,gy+a3dfx,gy+a4dfy,gx+a5dgx,gy+Lmindfx,gx,dfy,gx=110a11−x+a21−y+a31−y+a41−x+a5x−y+Lmin1−x,1−x=110a1x−1+a2y−1+a3y−1+a4x−1+a5x−y+Lx−1.
Here are three possible cases as follows:Case (1): ifx>y, taking a1=2,a2=2,a3=3,a4=3,a5=5, and −9<L≤−1 satisfying a1−a3≤1,a1+a2<a3+a4,a1+a2+a5>1, and ka1+a2+a3+a4+L>1, then we have(61)0≥FψMx,y,ϕMx,y+LNx,y=110a1x−1+a2y−1+a3y−1+a4x−1+a5x−y+Lx−1=x−1+Lx−1,which proves that conditionA2′ holds true.Case (2): ifx<y, taking a1=2,a2=2,a3=3,a4=3,a5=5, and −9<L≤−1 satisfying a1−a3≤1,a1+a2<a3+a4,a1+a2+a5>1, and ka1+a2+a3+a4+L>1, then we have(62)0≥FψMx,y,ϕMx,y+LNx,y=110a1x−1+a2y−1+a3y−1+a4x−1+a5y−x+Lx−1=y−1+Lx−1,which shows that conditionA2′ holds true.Case (3): ifx=y, taking a1=2,a2=2,a3=3,a4=3,a5=5, and −9<L≤−1 satisfying a1−a3≤1,a1+a2<a3+a4,a1+a2+a5>1, and ka1+a2+a3+a4+L>1, then we have(63)0≥FψMx,y,ϕMx,y+LNx,y=110a1+a2+a3+a4x−1+Lx−1=x−1+Lx−1,which shows that condition A2′ holds true.
Hence, the pairf,g satisfies all conditions of Theorems 3, with 1 being the unique common fixed point.
Next, we will present some common fixed point theorems under conditional semicompatibility as follows.Theorem 4.
LetX,d be a complete metric space and a pair g,f of self-maps be conditional semicompatible satisfying the following assumptions:A1fX⊆gXA2′kψdfx,fy≥FψMx,y,ϕMx,y+LNx,y
Here,Mx,y=a1dfx,gx+a2dfy,gy+a3dfx,gy+a4dfy,gx+a5dgx,gy and Nx,y=mindfx,gx,dfy,gx, for all x,y∈X. Moreover, ai>0,i=1,2,…,5, L∈ℝ, F∈Cinv−k for some k≥1 with 1≥a1−a3, a1+a2+a5>1, a1+a2<a3+a4, ka1+a2+a3+a4+L>1, and ψ∈Ψ, ϕ∈Φ. If f is g−absorbing or g is f−absorbing, then f and g have a unique common fixed point t in X.Proof.
Similar to the first part of the proof in Theorems1 or 3, we can construct a sequence xn in X such that(64)limn⟶+∞fxn=limn⟶+∞gxn=t,t∈X.
Again sincef and g are conditional semicompatible and limn⟶+∞fxn=limn⟶+∞gxn=t (nonempty), then there exists a sequence yn satisfying limn⟶+∞fyn=limn⟶+∞gyn=u such that limn⟶+∞fgyn=gu and limn⟶+∞gfyn=fu.Case 1:f is g−absorbing.Sincef is g−absorbing, this yields that(65)dgyn,gfyn≤Rdfyn,gyn.Taking limit asn⟶+∞ in the above inequality, it follows that(66)limn⟶+∞gyn=limn⟶+∞gfyn=u,fu=u.SincefX⊆gX, then there exists v in X such that fu=gv. Next, we will show that fv=u.By the definition ofψ and assumption A2′, we have(67)0=kψdfv,fv≥FψMv,v,ϕMv,v+LNv,v=Fψa1dfv,gv+a2dfv,gv+a3dfv,gv+a4dfv,gv+a5dgv,gv,ϕa1dfv,gv+a2dfv,gv+a3dfv,gv+a4dfv,gv+a5dgv,gv+Lmindfv,gv,dfv,gv=Fψa1dfv,gv+a2dfv,gv+a3dfv,gv+a4dfv,gv,ϕa1dfv,gv+a2dfv,gv+a3dfv,gv+a4dfv,gv+Ldfv,gv≥kψa1+a2+a3+a4dfv,u+Ldfv,u≥ka1+a2+a3+a4+Ldfv,u.Together withai>0,i=1,2,…,5, ka1+a2+a3+a4+L>1, from above inequality, it follows that dfv,u=0 or fv=u=gv.Again, sincef is g−absorbing, it yields(68)dgv,gfv≤Rdfv,gv,which implies thatgv=gfv or ggv=gv. Now, we will show fgv=gv. By the definition of ψ, s>1 and assumption A2′, we have(69)0=kψdfgv,fgv≥FψMgv,gv,ϕMgv,gv+LNgv,gv=Fψa1dfgv,ggv+a2dfgv,ggv+a3dfgv,ggv+a4dfgv,ggv+a5dggv,ggv,ϕa1dfgv,ggv+a2dfgv,ggv+a3dfgv,ggv+a4dfgv,ggv+a5dggv,ggv+Lmindfgv,ggv,dfgv,ggv=Fψa1dfgv,gv+a2dfgv,gv+a3dfgv,gv+a4dfgv,gv,ϕa1dfv,gv+a2dfgv,gv+a3dfgv,gv+a4dfgv,gv+Ldfgv,ggv≥kψa1+a2+a3+a4dfgv,gv+Ldfgv,gv≥ka1+a2+a3+a4+Ldfgv,gv.Together withai>0,i=1,2,…,5, ka1+a2+a3+a4+L>1, from the above inequality, it follows that dfgv,gv=0 or fgv=gv.Therefore,gv is a common fixed point of f and g.Case 2:g is f−absorbing.
Sinceg is f−absorbing, this yields(70)dfyn,fgyn≤Rdfyn,gyn.
Taking limit asn⟶+∞ in the above inequality, it follows that(71)limn⟶+∞gyn=limn⟶+∞fgyn=u,gu=u.
Next, we will showfu=gu.
By the definition ofψ and assumption A2′, we have(72)0=kψdfu,fu≥FψMu,u,ϕMu,u+LNu,u=Fψa1dfu,gu+a2dfu,gu+a3dfu,gu+a4dfu,gu+a5dgu,gu,ϕa1dfu,gu+a2dfu,gu+a3dfu,gu+a4dfu,gu+a5dgu,gu+Lmindfu,gu,dfu,gu=Fa1dfu,gu+a2dfu,gu+a3dfu,gu+a4dfu,gu,ϕa1dfu,gu+a2dfu,gu+a3dfu,gu+a4dfu,gu+Ldfu,gu≥kψa1dfu,gu+a2dfu,gu+a3dfu,gu+a4dfu,gu+Ldfu,gu≥ka1+a2+a3+a4+Ldfu,gu.
Together withai>0,i=1,2,…,5, ka1+a2+a3+a4+L>1, from the above inequality, it follows that dfu,gu=0 or fu=gu. Therefore, u is a common fixed point of f and g.
The uniqueness of common fixed point off and g can be directly obtained by following the same argument as Theorem 1.Remark 3.
The results of Theorem 4 still hold by replacing conditionA2′ with condition A2″ stated as follows.A2″: kθdfx,fy≥FθMx,y,ϕMx,y.
Here,Mx,y=a1dfx,gx+a2dfy,gy+a3dfx,gy+a4dfy,gx+a5dgx,gy, for all x,y∈X. Moreover, ai>0,i=1,2,…,5 with 1≥a1−a3, a1+a2+a5>1, a1+a2<a3+a4, F∈Cinv−k for some k≥1 and θ∈Θ, ϕ∈Φ.
Moreover,θ is said to be an altering distance function (see [19]), which satisfies iθ:0,+∞↦0,+∞ is continuous and increasing; iiθt=0⇔t=0. Denote the class of altering distance functions by Θ.
Now, we provide an example to verify the validity of Theorem4 as follows.Example 10.
LetX=2,+∞, x,y∈X with x>y and let d be the usual metric on X. Define f,g:X⟶X as follows:(73)fx=x+82,2≤x<5,5,x≥5,gx=x+3,2≤x<5,5,x≥5.
It is clear thatfX⊆gX.
Takingxn=2+εn, where εn⟶0 as n⟶+∞, we have limn⟶+∞fxn=5 and limn⟶+∞gxn=5. Therefore, limn⟶+∞fxn=limn⟶+∞gxn=5(nonempty). Then, we have a sequence yn=5+εn, where εn⟶0 as n⟶+∞, for which limn⟶+∞fyn=limn⟶+∞f5+εn=5 and limn⟶+∞gyn=limn⟶+∞g5+εn=5. Moreover, limn⟶+∞fgyn=limn⟶+∞fg5+εn=limn⟶+∞f5=5=g5 and limn⟶+∞gfyn=limn⟶+∞gf5+εn=limn⟶+∞g5=5=f5. Therefore, the pair f,g is conditional semicompatible.
For2≤x<5,gfx=gx+8/2=x+8/2, then dgx,gfx=x+3−x+8/2=x−2/2 and dgx,fx=x+3−x+8/2=x−2/2. Therefore, f and g satisfy dgx,gfx≤Rdfx,gx with R=1. Also, for x≥5, then 0=dgx,gfx=Rdgx,fx for all real number R. Therefore, f and g satisfy dgx,gfx≤Rdfx,gx, for all x∈X and R=1; that is, f is g−absorbing with R=1.
To satisfy conditionA2′, we choose Fs,t=s+t with k=1 and ψt=ϕt=t/40, for all t≥0. Then, for all x,y∈2,+∞ we have(74)kψdfx,fy=x−22,2≤y<x<5,0,5≤y<x<+∞,y−22,2≤y<5≤x<+∞.
Here are three possible cases:Case (1): if2≤y<x<5, we have(75)FψMx,y,ϕMx,y+LNx,y=120a1dfx,gx+a2dfy,gy+a3dfx,gy+a4dfy,gx+a5dgx,gy+Lmindfx,gx,dfy,gx=120a1x−22+a2y−22+a3x−2y+22+a4y−2x+22+a5y−22Lminx−22,y−2x+22≤120a1x−22+a2y−22+a3x−y2+y−22+a4x−y2+x−22+a5y−22Lx−22=a1+a420+Lx−22+a2+a3+a520y−22+a3+a420x−y2≤a1+a3+2a420+Lx−22+a2+2a3+a4+a520y−22≤a1+a2+3a3+3a4+a520+Lx−22.It is obvious thatkψdfx,fy≥FψMx,y,ϕMx,y+LNx,y holds true by taking a1=1,a2=a3=a4=2,a5=5, and −6<L≤−1 satisfying a1−a3≤1, a1+a2<a3+a4, a1+a2+a5>1, and ka1+a2+a3+a4+L>1.Case (2): if5≤y<x<+∞, we have(76)FψMx,y,ϕMx,y+LNx,y=0.It is obvious thatkψdfx,fy≥FψMx,y,ϕMx,y+LNx,y holds true by taking ai>0,i=1,2,…,5 and L∈ℝ.Case (3): if2≤y<5≤x<+∞, we have(77)FψMx,y,ϕMx,y+LNx,y=120a1dfx,gx+a2dfy,gy+a3dfx,gy+a4dfy,gx+a5dgx,gy+Lmindfx,gx,dfy,gx=120a2y−22+a3y−2+a4y−22+a5y−22=120a22+a3+a42+a52y−2.
It is obvious thatkψdfx,fy≥FψMx,y,ϕMx,y+LNx,y holds true by taking a1=1,a2=a3=a4=2,a5=5, and L∈ℝ satisfying a1−a3≤1, a1+a2<a3+a4, a1+a2+a5>1, and ka1+a2+a3+a4+L>1.
To sum up, conditionA2′ holds true by taking a1=1,a2=a3=a4=2,a5=5, and −6<L≤−1 satisfying a1−a3≤1, a1+a2<a3+a4, a1+a2+a5>1, and ka1+a2+a3+a4+L>1.
Therefore,f and g satisfy all the conditions of Theorems 4 with 5 being the unique common fixed point.Theorem 5.
LetX,d be a complete metric space and let a pair g,f of self-maps be conditional semicompatible satisfying the following assumptions:A1fX⊆gXA2′kψdfx,fy≥FψMx,y,ϕMx,y+LNx,y
Here,Mx,y=a1dfx,gx+a2dfy,gy+a3dfx,gy+a4dfy,gx+a5dgx,gy and Nx,y=mindfx,gx,dfy,gx, for all x,y∈X. Moreover, ai>0,i=1,2,…,5, L∈ℝ, F∈Cinv−k for some k≥1 with 1≥a1−a3, a1+a2+a5>1, a1+a2<a3+a4, ka1+a2+a3+a4+L>1, and ψ∈Ψ, ϕ∈Φ. If pair f,g is R−weak commuting either of type Af or of type Ag, then f and g have a unique common fixed point t in X.Proof.
From the part of the proof in Theorems1 or 3, we can construct a sequence xn in X such that limn⟶+∞fxn=limn⟶+∞gxn=t.
Again sincef and g are conditional semicompatible and limn⟶+∞fxn=limn⟶+∞gxn=t(nonempty), then there exists a sequence yn satisfying limn⟶+∞fyn=limn⟶+∞gyn=u such that limn⟶+∞fgyn=gu and limn⟶+∞gfyn=fu.Case 1: pairf,g is R−weak commuting of type Af.Since pairf,g is R−weak commuting of type Af, it yields(78)dfgyn,ggyn≤Rdfyn,gyn.Taking limit asn⟶+∞ in the above inequality, it follows that(79)limn⟶+∞fgyn=limn⟶+∞ggyn=gu.Now, we will show thatfu=gu. By the definition of ψ and assumption A2′, we have(80)0=kψdfu,fu≥FψMu,u,ϕMu,u+LNu,u=Fψa1dfu,gu+a2dfu,gu+a3dfu,gu+a4dfu,gu+a5dgu,gu,ϕa1dfu,gu+a2dfu,gu+a3dfu,gu+a4dfu,gu+a5dgu,gu+Lmindfu,gu,dfu,du=Fψa1dfu,gu+a2dfu,gu+a3dfu,gu+a4dfu,gu,ϕa1dfu,gu+a2dfu,gu+a3dfu,gu+a4dfu,gu+Ldfu,gu≥kψa1dfu,gu+a2dfu,gu+a3dfu,gu+a4dfu,gu+Ldfu,gu≥ka1+a2+a3+a4+Ldfu,gu.Together withai>0,i=1,2,…,5, ka1+a2+a3+a4+L>1, from the above inequality, it follows that dfu,gu=0 or fu=gu.Again, since pairf,g is R−weak commuting of type Af, then we have(81)dfgu,ggu≤Rdfu,gu,which impliesfgu=ggu.Now, we will showfgu=gu.By the definition ofψ and assumption A2′, we have(82)kψdfgu,fu≥FψMgu,u,ϕMgu,u+LNgu,u=Fψa1dfgu,ggu+a2dfu,gu+a3dfgu,gu+a4dfu,ggu+a5dggu,gu,ϕa1dfgu,ggu+a2dfu,gu+a3dfgu,gu+a4dfu,ggu+a5dggu,gu+Lmindfgu,ggu,dfu,ggu=Fϕa3dfgu,gu+a4dfgu,gu+a5dfgu,gu,ϕa3dfgu,gu+a4dfgu,gu+a5dfgu,gu≥kψa3+a4+a5dfgu,gu.Sincea1+a2+a5>1, a1+a2<a3+a4, and fu=gu, it follows that(83)kψdfgu,gu=Fψa3+a4+a5dfgu,gu,ϕa3+a4+a5dfgu,gu≥kψa3+a4+a5dfgu,gu≥kψa1+a2+a5dfgu,gu≥kψdfgu,gu,which implies that(84)Fψa3+a4+a5dfgu,gu,ϕa3+a4+a5dfgu,gu=kψdfgu,gu.From the definition ofCinv−k−class functions, we have(85)ψa3+a4+a5dfgu,gu=0orϕa3+a4+a5dfgu,gu=0.By the definitions ofψ and ϕ, we have that dfgu,gu=0, which proves that fgu=gu=ggu.Therefore,gu is a common fixed point of f and g.Case 2: pairf,g is R−weak commuting of type Ag.
Since pairf,g is R−weak commuting of type Af, it yields(86)dgfyn,ffyn≤Rdfyn,gyn.
Taking limit asn⟶+∞ in the above inequality, it follows that(87)limn⟶+∞gfyn=limn⟶+∞ffyn=fu.
Now, we will show thatfu=u.
By the definition ofψ and assumption A2′, we have(88)kψdffyn,fyn≥FψMfyn,yn,ϕMfyn,yn+LNfyn,yn=Fψa1dffyn,gfyn+a2dfyn,gyn+a3dffyn,gyn+a4dfyn,gfyn+a5dgfyn,gyn,ϕa1dffyn,gfyn+a2dfyn,gyn+a3dffyn,gyn+a4dfyn,gfyn+a5dgfyn,gyn+Lmindffyn,gfyn,dfyn,gyn.
Taking limit asn⟶+∞ in the above inequality, we have(89)kψdfu,u≥Fa1dfu,fu+a2du,u+a3dfu,u+a4du,fu+a5dfu,u,ϕa1dfu,fu+a2du,u+a3dfu,u+a4du,fu+a5dfu,u+Lmindfu,fu,du,u=Fψa3dfu,u+a4du,fu+a5dfu,u,ϕa3dfu,u+a4du,fu+a5dfu,u.
Sincea1+a2+a5>1, a1+a2<a3+a4, it follows that(90)kψdfu,u≥Fψa3dfu,u+a4du,fu+a5dfu,u,ϕa3dfu,u+a4du,fu+a5dfu,u≥kψa3+a4+a5dfu,u≥kψa1+a2+a5dfu,u≥kψdfu,u,which implies that(91)Fψa3+a4+a5dfu,u,ϕa3+a4+a5dfu,u=kψdfu,u.
From the definition ofCinv−k−class functions, we have(92)ψa3+a4+a5dfu,u=0orϕa3+a4+a5dfu,u=0.
By the definitions ofψ and ϕ, we have that dfu,u=0, which proves that fu=u.
SincefX⊆gX, there exists v∈X such that fu=gv. Now, we will show that fv=u.
By the definition ofψ, fu=gv=u, and assumption A2′, we have(93)0=kψdfv,fv≥FψMv,v,ϕMv,v+LNv,v=Fψa1dfv,gv+a2dfv,gv+a3dfv,gv+a4dfv,gv+a5dgv,gv,ϕa1dfv,gv+a2dfv,gv+a3dfv,gv+a4dfv,gv+a5dgv,gv+Lmindfv,gv,dfv,gv=Fψa1dfv,gv+a2dfv,gv+a3dfv,gv+a4dfv,gv,ϕa1dfv,gv+a2dfv,gv+a3dfv,gv+a4dfv,gv+Ldfv,gv≥kψa1dfv,gv+a2dfv,gv+a3dfv,gv+a4dfv,gv+Ldfv,gv≥ka1+a2+a3+a4+Ldfv,gv.
Together withai>0,i=1,2,…,5, ka1+a2+a3+a4+L>1, from above inequality, it follows that dfv,gv=dfv,u=0 or fv=gv=u.
Since pairf,g is R−weak commuting of type Af, it yields(94)dgfv,ffv≤Rdfv,gv,
which yieldsffv=gfv.
Next, we will showgfv=fv.
By the definition ofψ and assumption A2′, we have(95)0=kψdffv,fv≥FψMfv,v,ϕMfv,v+LNfv,v=Fψa1dffv,gfv+a2dfv,gv+a3dffv,gv+a4dfv,gfv+a5dgfv,gv,ϕa1dffv,gfv+a2dfv,gv+a3dffv,gv+a4dfv,gfv+a5dgfv,gv+Lmindffv,gfv,dfv,gv=Fψa3dffv,gv+a4dfv,gfv+a5dgfv,gv,ϕa3dffv,gv+a4dfv,gfv+a5dgfv,gv≥kψa3+a4+a5dffv,gv.
Sincea1+a2+a5>1, a1+a2<a3+a4, ffv=gfv, and fv=gv, it follows that(96)kψdgfv,fv=Fψa3+a4+a5dgfv,fv,ϕa3+a4+a5dgfv,fv≥kψa3+a4+a5dgfv,fv≥kψa1+a2+a5dgfv,fv≥kψdgfv,gv,which implies that(97)Fψa3+a4+a5dgfv,fv,ϕa3+a4+a5dgfv,fv=kψdgfv,fv.
From the definition ofCinv−k−class functions, we have(98)ψa3+a4+a5dgfv,fv=0orϕa3+a4+a5dgfv,fv=0.
By the definitions ofψ and ϕ, we have that dgfv,fv=0, which proves that gfv=fv=ffv. Therefore, fv is a common fixed point of f and g.
The uniqueness of common fixed point off and g is also obtained by following the same proof of Theorem 2.Remark 4.
The conclusions of Theorem5 still hold true by replacing condition A2′ with condition A2″ stated as before.
Now, we will provide an example to verify the validity of Theorem5 as follows.Example 11.
LetX=0,1 with the usual metric and x,y∈X. We define f and g on X as follows:(99)fx=45,x∈0,12,34,x∈12,1,gx=45,x∈0,12,12,x=12,34,x∈12,1.
It is clear thatfX⊆gX.
To show that the pairf,g of self-maps is conditional semicompatible, we take any sequence xn∈0,1/2, then limn⟶+∞fxn=limn⟶+∞gxn=4/5 (nonempty). Now, we choose another sequence yn∈1/2,1, then limn⟶+∞fgyn=3/4=g3/2 and limn⟶+∞gfyn=3/4=g3/2. It can be observed that the pair f,g satisfies R−weak commuting of type Af in intervals 0,1/2 and 1/2,1 for all real numbers. Also, the pair f,g satisfies R−weak commuting of type Af at x=1/2 with R=1. To verify assumption A2′, we define Fs,t=2s∈Cinv−k with k=2, ψt=t/10, for all t≥0. For x=y=1/2, we have(100)kψdfx,fy=0,FψMx,y,ϕMx,y=15a1dfx,gx+a2dfy,gy+a3dfx,gy+a4dfy,gx+a5dgx,gy+Lmindfx,gx,dfy,gx=15a145−12+a245−12+a345−12+a445−12+Lmin45−12,45−12=15310a1+a2+a3+a4+310L=31015a1+a2+a3+a4+L.
It is obvious that0≥3/101/5a1+a2+a3+a4+L holds true by taking a1=a2=a3=1, a4=2, a5>0, and −9<L≤−1 such that a1−a3≤1, a1+a2<a3+a4, a1+a2+a5>1, and ka1+a2+a3+a4+L>1. It is also clear that the above inequality holds true for all x,y∈0,1/2 and x,y∈1/2,1.
Therefore,f and g satisfy all the conditions of Theorem 5 with 3/4 being the unique common fixed point.
Finally, we will present a common fixed point theorem underSτ−compatibility as follows.Theorem 6.
LetX,d be a complete metric space and let f,g,B,T be self-maps satisfying the following assumptions:A1′fX⊆TX, gX⊆BXA2″kθdfx,By≥FθM′x,y,ϕM′x,y
Here,M′x,y=a1dfx,gx+a2dBy,Ty+a3dfx,Ty+a4dgx,By+a5dgx,Ty, for all x,y∈X. Moreover, ai>0,i=1,2,…,5 with 1≥a1−a3, a1+a2+a5>1, a1+a2<a3+a4, F∈Cinv−k for some k≥1 and θ∈Θ, ϕ∈Φ. If the pair f,g is SB−compatible and B,T is Sg−compatible, then f,g,B,T have a unique common fixed point t in X.Proof.
Letx0∈X. Since fX⊆TX, then there exists x1∈X such that fx1=Tx0=y0. For x1∈X and gX⊆BX, there exists x2∈X such that Bx2=gx1=y1. Continuing this way, we can construct the sequence xn in X such that fxn=Txn−1=yn−1 and Bxn+1=gxn=yn.
By the assumption ofA2″ and properties of θ and F, we have(101)kθdfxn,Bxn+1≥FθM′xn,xn+1,ϕM′xn,xn+1≥kθM′xn,xn+1=kθa1dfxn,gxn+a2dBxn+1,Txn+1+a3dfxn,Txn+1+a4dgxn,Bxn+1+a5dgxn,Txn+1=kθa1dyn−1,yn+a2dyn,yn+1+a3dyn−1,yn+1+a4dyn,yn+a5dyn,yn+1=kθa1dyn−1,yn+a2dyn,yn+1+a3dyn−1,yn+1+a5dyn,yn+1.
By the similar procedure demonstrated in Theorem1, we can get dyn,yn+1≤1−a1+a3/a2+a3+a5dyn−1,yn.
From Lemma1, it follows that sequence yn is a Cauchy sequence. Since X is complete, there exists a point t∈X such that(102)limn⟶+∞yn=limn⟶+∞fxn=limn⟶+∞gxn=limn⟶+∞Bxn=limn⟶+∞Txn=t.
Next, we will show thatt is a common fixed point of f,g,B,T.
Since the pairf,g is SB−compatible and the pair B,T is Sg−compatible and by equation (102), we have the following outcomes:(103)limn⟶+∞fBxn=limn⟶+∞gBxn=Bt,(104)limn⟶+∞Bgxn=limn⟶+∞Tgxn=gt.
By the assumption ofA2″ and properties of θ and F, we have(105)kθdfBxn,Bxn≥FθM′Bxn,xn,ϕM′Bxn,xn≥kθM′Bxn,xn=kθa1dfBxn,gBxn+a2dBxn,Txn+a3dfBxn,Txn+a4dgBxn,Bxn+a5dgBxn,Txn.
Lettingn⟶+∞ in the above inequality, with equation (103), we have(106)kθdBt,t≥Fθa3dBt,t+a4dBt,t+a5dBt,t,ϕa3dBt,t+a4dBt,t+a5dBt,t≥kθa3dBt,t+a4dBt,t+a5dBt,t=kθa3+a4+a5dBt,t.
Sincea1+a2+a5>1, a1+a2<a3+a4, we have(107)kθdBt,t≥Fθa3dBt,t+a4dBt,t+a5dBt,t,ϕa3dBt,t+a4dBt,t+a5dBt,t≥kθa3+a4+a5dBt,t≥kθa1+a2+a5dBt,t≥kθdBt,t,which implies that(108)Fθa3+a4+a5dBt,t,ϕa3+a4+a5dBt,t=kθdBt,t.
From the definition ofCinv−k−class functions, we have(109)θa3+a4+a5dBt,t=0orϕa3+a4+a5dBt,t=0.
By the definitions ofθ and ϕ, we have that dBt,t=0, which proves that Bt=t.
We also have(110)kθdfxn,Bgxn≥FθM′xn,gxn,ϕM′xn,gxn≥kθM′xn,gxn=kθa1dfxn,gxn+a2dBgxn,Tgxn+a3dfxn,Tgxn+a4dgxn,Bgxn+a5dgxn,Tgxn.
Lettingn⟶+∞ in the above inequality, with equation (104), we have(111)kθdt,gt≥Fθa3dt,gt+a4dt,gt+a5dt,gt,ϕa3dt,gt+a4dt,gt+a5dt,gt≥kθa3dt,gt+a4dt,gt+a5dt,gt=kθa3+a4+a5dt,gt.
Sincea1+a2+a5>1, a1+a2<a3+a4, we have(112)kθdt,gt≥Fθa3dt,gt+a4dt,gt+a5dt,gt,ϕa3dt,gt+a4dt,gt+a5dt,gt≥kθa3+a4+a5dt,gt≥kθa1+a2+a5dt,gt≥kθdt,gt,which implies that(113)Fθa3+a4+a5dt,gt,ϕa3+a4+a5dBt,t=kθdt,gt.
From the definition ofCinv−k−class functions, we have(114)θa3+a4+a5dt,gt=0orϕa3+a4+a5dt,gt=0.
By the definitions ofθ and ϕ, we have that dt,gt=0, which proves that t=gt.
Now, we will proveTt=t. If not, by the assumption of A2″ and properties of θ and F, we have(115)kθdfxn,Bt≥FθM′xn,t,ϕM′xn,t≥kθM′xn,t=kθa1dfxn,gxn+a2dBt,Tt+a3dfxn,Tt+a4dgxn,Bt+a5dgxn,Tt.
Lettingn⟶+∞ in the above inequality, we have(116)0=kθdt,t≥Fθa1dt,t+a2dt,Tta3dt,Tt+a4dt,t+a5dt,Tt,ϕa1dt,t+a2dt,Tta3dt,Tt+a4dt,t+a5dt,Tt≥kθa1dt,t+a2dt,Tta3dt,Tt+a4dt,t+a5dt,Tt=kθa2+a2+a5dt,Tt.which implies a2+a2+a5dt,Tt=0; that is, t=Tt.
In the same manner, we can also gett=ft.
Hence,t is a common fixed point of f,g,B,T.
The uniqueness oft can also be obtained by the similar way stated in Theorem 1.
Now, we present the following example to support the validity of Theorem6.Example 12.
LetX=0,10 with the usual metric. We define f,g,B,T on X as follows:(117)fx=gx=x2,for∀x∈X,Bx=Tx=x,for∀x∈X.
It is clear thatfX⊆TX and gX⊆BX.
To show that pairf,g is SB−compatible and pair B,T is Sg−compatible, we take xn=1/n; then, limn⟶+∞fxn=0, limn⟶+∞gxn=0, limn⟶+∞Bxn=0, limn⟶+∞Txn=0 with limn⟶+∞fBxn=limn⟶+∞gBxn=B0, and limn⟶+∞Bgxn=limn⟶+∞Tgxn=g0.
Hence, the pairf,g is SB−compatible and pair B,T is Sg−compatible.
To verify assumptionA2″, we define Fs,t=ks∈Cinv−k with k≥1, where 0<β=a3+a4+a5<1 and ai>0,i=1,2,…,5 satisfying 1≥a1−a3, a1+a2+a5>1 and a1+a2<a3+a4. Define θt=t/2β2, for all t≥0. Then, for all x,y∈X, we have(118)kθdfx,By=kdfx,By2β2=k2β2x2−y,FθM′x,y,ϕM′x,y=k2β2a1dfx,gx+a2dBy,Ty+a3dfx,Ty+a4dgx,By+a5dgx,Ty=k2βx2−y.
It is easy to verify thatkθdfx,By≥FθM′x,y,ϕM′x,y for all x,y∈X satisfying ai>0,i=1,2,…,5 with 0<β=a3+a4+a5≤1, 1≥a1−a3, a1+a2+a5>1, and a1+a2<a3+a4; that is, assumption A2″ holds true.
Then,f,g,B, and T  satisfy all the conditions of Theorem 6; moreover, 0 is the unique common fixed point of f,g,B, and T.


## 3. Conclusion

In this paper, concepts of semicompatibility of type (A),Sτ–compatibility are introduced, in which Sτ–compatibility is weaker than (E.A) property. We also give a brief discussion on the relation between new notions and other existing types of compatibility. Motivated by the notion of inverse C-class functions, a distinct concept of inverse Ck class functions is introduced which extends the notion of inverse C-class functions introduced by Saleem et al. [1, 21]. Moreover, some common fixed point theorems are stated under some compatible conditions such as semicompatibility, semicompatibility of type (A), weak semicompatibility, conditional semicompatibility, and Sτ–compatibility in metric spaces via inverse Ck class functions which are a valuable supplement to the common fixed point theory.



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