It is shown that occasionally 𝒥ℋ operators as well as occasionally weakly biased
mappings reduce to weakly compatible mappings in the presence of a unique point
of coincidence (and a unique common fixed point) of the given maps.

1. Introduction and Preliminaries

The study of finding a common fixed point of pair of commuting mappings was initiated
by Jungck [1]. Later, on this condition was
weakened in various ways. Sessa [2]
introduced the notion of weakly commuting maps. Jungck [3] gave the notion of compatible mappings in order to generalize
the concept of weak commutativity. One of the conditions that was used most often
was the weak compatibility, introduced by Jungck in [4] fixed point results for various classes of mappings on a metric
space, utilizing these concepts. Jungck and Pathak [5] defined the concept of a weakly biased maps in order to generalize
the concept of weak compatibility. In the paper [6], published in 2008, Al-Thagafi and Shahzad introduced an even weaker
condition which they called occasionally weak compatibility (see also [7]). Many authors (see, e.g., [8–13]) used this condition to obtain common fixed point results, sometimes
trying to generalize results that were known to use (formally stronger) condition of
weak compatibility. Recently, Hussain et al. [14] have introduced two new and different classes of noncommuting
self-maps: 𝒥ℋ-operators and occasionally weakly biased mappings.
These classes contain the occasionally weakly compatible and weakly biased self-maps
as proper subclasses. For these new classes, authors have proved common fixed point
results on the space (X,d) which is more general than metric space. We will
show in this short note that in the presence of a unique point of coincidence (and a
unique common fixed point) of the given mappings, occasionally
𝒥ℋ-operators as well as occasionally weakly biased
mappings reduce to weakly compatible mappings (and so occasionally weakly compatible
mappings). For more details on the subject, we refer the reader to [15–18].

Let X be a nonempty set and let f and g be two self-mappings on X. The set of fixed points of
f (resp., g) is denoted by F(f) (resp., F(g)). A point x∈X is called a coincidence point
(CP) of the pair (f,g) if fx=gx(=w). The point w is then called a point of coincidence
(POC) for (f,g). The set of coincidence points of
(f,g) will be denoted as C(f,g). Let PC(f,g) represent the set of points of coincidence of the
pair (f,g). A point x∈X is a common fixed point of
f and g if fx=gx=x. The self-maps f and g on X are called

commuting if fgx=gfx for all x∈X;

weakly compatible (WC) if they commute at their coincidence points, that
is, if fgx=gfx whenever fx=gx [4];

occasionally weakly compatible (OWC) if fgx=gfx for some x∈X with fx=gx [6].

Let d be symmetric on X. Then f and g are called

𝒫-operators if there is a point
x∈X such that x∈C(f,g) and d(x,fx)≤δ(C(f,g)), where δ(A)=sup{max{d(x,y),d(y,x)}:x,y∈A} [19];

𝒥ℋ-operators if there is a point
w=fx=gx in PC(f,g) such that d(w,x)≤δ(PC(f,g)) [14];

weakly g-biased, if d(gfx,gx)≤d(fgx,fx) whenever fx=gx [5];

occasionally weakly g-biased, if there exists some
x∈X such that fx=gx and d(gfx,gx)≤d(fgx,fx) [14].

Let d:X×X→[0,+∞) be a mapping such that d(x,y)=0 if and only if x=y. Then f and g are called

𝒥ℋ-operators if there is a point
w=fx=gx in PC(f,g) such that d(w,x)≤δ(PC(f,g)) and d(x,w)≤δ(PC(f,g)), where

δ(A)=sup{max{d(x,y),d(y,x)}:x,y∈A}.2. Results

We begin with the following results.

Lemma 2.1 (see [<xref ref-type="bibr" rid="B2">20</xref>]).

If a WC pair (f,g) of self-maps on X has a unique POC, then it has a unique common
fixed point.

The following lemma is according to Jungck and Rhoades [12].

Lemma 2.2 (see [<xref ref-type="bibr" rid="B10">12</xref>]).

If an OWC pair (f,g) of self-maps on X has a unique POC, then it has a unique common
fixed point.

Proof.

Since (f,g) is an OWC, there exists
x∈C(f,g) such that fx=gx=:w and fw=gfx=gfx=gw. Hence, fw=gw is also a POC for (f,g), and since it must be unique, we have that
w=fw=gw,thatis,w is a common fixed point for
(f,g). If z is any common fixed point for
(f,g) (i.e., fz=gz=z), then, again by the uniqueness of POC, it must
be z=w.

The following result is due to Ðorić et al. [21]. It shows that the results of Jungck and Rhoades are not
generalizations of results obtained from Lemma 2.1.

Proposition 2.3 (see [<xref ref-type="bibr" rid="B5">21</xref>]).

Let a pair of mappings (f,g) have a unique POC. Then it is WC if and only if
it is OWC.

Proof.

In this case, we have only to prove that OWC implies WC. Let
w1=fx=gx be the given POC, and let
(f,g) be OWC. Let y∈C(f,g),y≠x. We have to prove that
fgy=gfy. Now w2=fy=gy is a POC for the pair (f,g). By the assumption, w2=w1,thatis,fy=gy=fx=gx. Since, by Lemma 2.2, w1 is a unique common fixed point of the pair
(f,g), it follows that w1=fw1=fgy and w1=gw1=gfy, hence fgy=gfy. The pair (f,g) is WC.

Proposition 2.4.

Let d:X×X→[0,+∞) be a mapping such that
d(x,y)=0 if and only if x=y. Let a pair of mappings
(f,g) have a unique POC. If it is a pair of
𝒥ℋ-operators, then it is WC.

Proof.

Let (f,g) be a pair of 𝒥ℋ-operators. Then there is a point
w=fx=gx in PC(f,g) such that d(x,w)≤δ(PC(f,g)) and d(w,x)≤δ(PC(f,g)). Clearly PC(f,g) is a singleton. If not, then
w1=fy=gy is a POC for the pair (f,g). By the assumption, w=w1. As a result, we have δ(PC(f,g))=0, which implies d(x,w)=d(w,x)=0, that is, x=fx=gx. Consequently, we have
gfx=gx=fx=fgx and thus (f,g) is OWC. By Proposition 2.3, (f,g) is WC.

It is worth mentioning that if iX is the identity mapping, then the pair
(f,iX) is always WC, but it is a pair of
𝒥ℋ-operators if and only if f has a fixed point.

Proposition 2.5.

Let d:X×X→[0,+∞) be a mapping such that
d(x,y)=0 if and only if x=y. Suppose (f,g) is a pair of 𝒥ℋ-operators satisfying d(fx,fy)≤ad(gx,gy)+bmax{d(fx,gx),d(fy,gy)}+cmax{d(gx,gy),d(gx,fx),d(gy,fy)} for each x,y∈X, where a,b,care real numbers such that
0<a+c<1.Then (f,g) is WC.

Proof.

By hypothesis, there exists some x∈X such that w=fx=gx. It remains to show that
(f,g) has a unique POC. Suppose there exists another
point w1=fy=gy with w≠w1. Then, we have d(w,w1)=d(fx,fy)≤(a+c)d(fx,fy)=(a+c)d(w,w1), which is a contradiction since
a+c<1. Thus (f,g) has a unique POC. By Proposition 2.4, (f,g) is WC.

Proposition 2.6.

Let d be symmetric on X. Let a pair of mappings
(f,g) have a unique POC which belongs to
F(f). If it is a pair of occasionally weakly
g-biased mappings, then it is WC.

Proof.

Let (f,g) be a pair of occasionally weakly
g-biased mappings. Then there exists some
x∈X such that fx=gx and d(gfx,gx)≤d(fgx,fx). Since w=fx=gx belong to F(f), then fw=w,thatis,fgx=fx=gx. Thus d(gfx,gx)≤d(fgx,fx)=0 and thus gfx=gx=fx=fgx. Hence (f,g) is OWC. By Proposition 2.3, (f,g) is WC.

Let ϕ:[0,+∞)→[0,+∞) be a nondecreasing function satisfying the
condition ϕ(t)<t for each t>0.

Proposition 2.7.

Let f,g be self-maps of symmetric space
X and let the pair (f,g) be occasionally weakly
g-biased. If for the control function
ϕ, we have d(fx,fy)≤ϕ(max{d(gx,gy),d(gx,fy),d(gy,fx),d(gy,fy)}), for each x,y∈X, then (f,g) is WC.

Proof.

It remains to show that (f,g) has a unique POC which belongs to
F(f). Since (f,g) are occasionally weakly
g-biased mappings, there exists some
x∈X such that w=fx=gx and d(gfx,gx)≤d(fgx,fx). If w1=fy=gy and w≠w1, then d(fx,fy)≤ϕ(max{d(gx,gy),d(gx,fy),d(gy,fx),d(gy,fy)})=ϕ(d(fx,fy))<d(fx,fy), which is a contradiction. Also if
ffx≠fx, we have d(ffx,fx)≤ϕ(max{d(gfx,gx),d(gfx,fx),d(gx,ffx),d(gx,fx)})≤ϕ(max{d(fgx,fx),d(fgx,fx),d(gx,ffx),d(gx,fx)})=ϕ(d(ffx,fx))<d(ffx,fx), which is a contradiction. By Proposition 2.6, (f,g) is WC.

Remark 2.8.

According to Propositions 2.4,
2.5, 2.6, and 2.7, it follows that results from
[14]: (Theorems 2.8, 2.9, 2.10, 2.11,
2.12, 3.7, 3.9 and Corollary 3.8) are not generalizations (extensions) of some
common fixed point theorems due to Bhatt et al. [9], Jungck and Rhoades [12,
13], and Imdad and Soliman [11]. Moreover, all mappings in these results
are WC.

Proposition 2.9.

Let d be symmetric on X, and let a pair of mappings
(f,g) have a unique (CP), that is, C(f,g) is a singleton. If (f,g) is 𝒫-operator pair, then it is WC.

Proof.

According to (4), there is a point x∈X such that x∈C(f,g)={x} and d(x,fx)≤δ(C(f,g))=δ({x})=0. Hence, x=fx=gx is a unique POC of pair
(f,g) and since gfx=gx=x=fx=fgx,(f,g) is OWC. By Proposition 2.3 it is WC.

Remark 2.10.

By Proposition 2.9, it follows
that Theorem 2.1 from [19] is not a
generalization result of [6], the main
result of Jungck [1], and other
results.

Proposition 2.11.

Let d be symmetric on X, and let a pair of mappings
(f,g) have a unique POC. Then it is weakly
g-biased if and only if it is occasionally weakly
g-biased.

Proof.

In this case, we have only to prove that (7) implies (6). Let
w=fx=gx be the given POC. Let y∈C(f,g),y≠x. We have to prove that
d(gfy,gy)≤d(fgy,fy). Now w1=fy=gy is a POC for the pair (f,g). By the assumption, w=w1, that is, fy=gy=fx=gx. Further, we have gfy=gfx and gy=gx, which implies that d(gfy,gy)=d(gfx,gx)≤d(fgx,fx)=d(fgy,fy), that is, the pair (f,g) satisfies (6).

The following example shows that the assumption about the uniqueness of POC in
Propositions 2.3, 2.4, 2.6, and 2.11 cannot be removed.

Example 2.12.

Let X=[1,+∞),d(x,y)=|x-y|,fx=3x-2,gx=x2 (see [21]). It is obvious that C(f,g)={1,2}, the pair (f,g) is occasionally weakly
g-biased, but it is not weakly
g-biased. Also, (f,g) is occasionally weakly compatible, but it is
not weakly compatible. However, the pair (f,g) has not the unique POC.

Acknowledgments

The authors are very grateful to the anonymous referees for their valuable comments
and suggestions. The second author is thankful to the Ministry of Science and
Technological Development of Serbia.

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