The object of this paper is to emphasize the role of ‘common limit range property’ and utilize the same with variants of
R
weakly commuting mappings for the existence of common fixed point under strict contractive conditions in metric spaces. We also furnish some interesting examples to validate our main result. Our results improve a host of previously known results including the ones contained in Pant [Contractive conditions and common fixed points, Acta Math. Acad. Paedagog. Nyhàzi. (N.S.) 24(2) (2008), 257–266 MR2461637 (2009h:54061)]. In the process, we also derive a fixed point result satisfying
ϕ
contractive condition.
1. INTRODUCTION AND PRELIMINARIES
The celebrated Banach fixed point theorem also known as Banach Contraction Principle appeared in it’s explicit form in the thesis of Banach
[4]
. Owing to its simplicity and usefulness, it became a very powerful tool in solving existence problems in pure and applied sciences which include biology, medicine, physics, computer science etc. This theorem asserts that every contraction mappings defined on a complete metric space has a unique fixed point and that fixed point can be explicitly obtained as limit of repeated iteration of the mapping at any point of the underlying space. Evidently, every contraction mapping is a continuous but not conversely. Some recent development in fixed point theory can be easily seen in
[2
,
3
,
6
,
9
,
10
,
13
,
21]
.
In 1976, Jungck
[7]
proved a fixed point theorem for a pair of commuting mappings in complete metric space. The first ever attempt to improve commutativity conditions in common fixed point theorems is due to Sessa
[23]
wherein he introduced the notion of weakly commuting mappings. Later on, Jungck
[8]
improved the notion of weak commutativity due to Sessa
[23]
by introducing the concept of compatible mappings. In the study of common fixed points of compatible mappings we often require assumptions on the completeness of the underlying space and continuity of the involved mappings. In an interesting note, Kannan
[11]
showed that there exists mappings that have a discontinuity in the domain but which have fixed points. Moreover, the involved mappings in every case were continuous at the fixed point. The study of common fixed points theorems for noncompatible mappings is firstly initiated by Pant
[15]
with the introduction of the notion of
R
weakly com muting mappings in metric spaces. Many mathematicians have contributed towards the vigorous development of fixed point theory (e.g.
[5
,
16
,
17
,
18]
). Further, Pathak et al. introduced the notion of
R
weakly commuting mappings of types (
A_{g}
) and (
A_{f}
) and generalized the result of Pant
[15]
.
A result on the existence and uniqueness of common fixed point in metric spaces, generally involves conditions on commutativity, continuity and contraction along with a suitable condition on the containment of range of one mapping into the range of other. Hence, one is always required to improve one or more of these conditions to prove a new fixed point theorem. In 2002, Amari and Moutawakil
[1]
introduced the notion of property (E.A) which generalized the concept of noncompatible mappings. The fixed point results proved under property (E.A) always require the closedness of the underlying subspaces for the existence of common fixed point. In 2011, Sintunavarat and Kumam
[27]
coined the idea of “common limit range property” (also see
[25
,
28
,
29
,
30
,
31]
).
In 2008, Pant
[19]
utilized the notion of property (E.A) with Rweakly commuting mappings of type (
A_{g}
) for the existence of common fixed point under strict contractive condition which improved and extended the results of Singh and Kumar
[24]
. Since then, Kumar
[12]
proved a common fixed point theorem for a pair of weakly compatible mappings along with property (E.A) in metric spaces which improves and generalizes the result of Jungck
[7]
without any continuity requirement besides relaxing the containment of the range of one mapping into the range of other mapping. In
[12]
, he also introduced the notion of
R
weakly commuting of type (
P
) in metric space and obtained some fixed point theorems for variants of
R
weakly commuting mappings with property (E.A).
In this paper, utilizing the notion of common limit range property due to Sintunavarat and Kumam
[27]
, we prove some common fixed point theorems for a pair of mappings under variants of
R
weakly commuting mappings. In process, many known results (especially the ones contained in Pant
[19]
) are enriched and improved. Some related results are also derived besides furnishing illustrative examples.
Throughout this paper, let
Y
be an arbitrary nonempty set and (
X
,
d
) a metric space.
Definition 1.1.
Let
f
,
g
:
X
→
X
be two self mappings of a metric space (
X
,
d
). Then the pair (
f
,
g
) is said to be

(1)commutingiffgx=gfx, for allx∈X.

(2)weakly commuting[23]ifd(fgx,gfx) ≤d(fx,gx), for allx∈X.

(3)Rweakly commuting[15]if there exists some real numberR> 0 such thatd(fgx,gfx) ≤Rd(fx,gx) for allx∈X.

(4)pointwise Rweakly commuting[15]if givenx∈Xthere exists some real numberR> 0 such thatd(fgx,gfx) ≤Rd(fx,gx).

(5)Rweakly commuting of type(Ag)[20]if there exists some real numberR> 0 such thatd(ffx,gfx) ≤Rd(fx,gx) for allx∈X.

(6)Rweakly commuting of type(Af)[20]if there exists some real number R > 0 such thatd(fgx,ggx) ≤Rd(fx,gx) for allx∈X.

(7)Rweakly commuting of type(P)[12]if there exists some real number R > 0 such thatd(ffx,ggx) ≤Rd(fx,gx) for allx∈X.

(8)compatible[8]iffor each sequence {xn} inXsuch that.

(9)noncompatible[15]if there exists a sequence {xn} inXsuch thatbutis either nonzero or nonexistent.
For more details on systematic comparisons and illustrations of earlier described notions, we refer to Singh and Tomar
[26]
, Murthy
[14]
and Kumar
[12]
.
Definition 1.2.
Let
f
and
g
be mappings on
Y
with values in
X
. Then
f
and
g
are said to satisfy the

(1)property (E.A)[22]if there exists a sequence {xn} inYsuch that

for somet∈X.

(2)the common limit range property[27]with respect to mappingg, denoted by (CLRg) in short, if there exists a sequence {xn} inYsuch that

for someu∈Y.
Remark 1.3
. If we take
Y
=
X
then we get the definition of property (E.A) for two self mappings of
X
studied by Aamri and Moutawakil
[1]
. In this case,
t
is called a tangent point by Sastry and Murthy
[22]
.
Example 1.4.
Consider
Y
=
X
= [0, 14] and d be the usual metric on
X
. Define the self mappings
f
and
g
on
X
as
Then
and
g
(
X
) = [0, 3). Now consider a sequence
in
X
. Then clearly,
Here it is pointed out that 3 ∉
g
(
X
) which shows that the pair (
f
,
g
) does not satisfy the (CLRg) property while it enjoys the property (E.A).
Example 1.5.
In the setting of Example 1.4, replace the self mapping
g
by the following besides retaining the rest:
Then
and
g
(
X
) = [0, 3]. Consider a sequence similar as Example 1.4, one can see that,
Hence the pair (
f
,
g
) enjoys the (CLRg) property.
Remark 1.6.
Thus, one can infer that a pair (
f
,
g
) satisfying the property (E.A) along with closedness of the underlying subspace
g
(
X
) always enjoys the (CLRg) property with respect to the mapping
g
.
2. MAIN RESULTS
In 2008, Pant
[19]
proved the following result for a pair of
R
weakly commuting mappings employing property (E.A).
Theorem 2.1
([19, Theorem 1]).
Let f and g be self mappings of a complete metric space
(
X
,
d
) such that
where
denotes the closure of range of the mapping f,
whenever the right hand side is positive and
1 ≤
k
< 2.
If f and g be Rweakly commuting of type of type
(
A_{g}
)
satisfying the property (E.A), then f and g have a unique common fixed point.
Now we prove a more general result by using the notion of common limit range property with variants of
R
weakly commuting mappings.
Theorem 2.2.
Let
(
X
,
d
)
be a metric space and
f
,
g
:
Y
→
X
.
Suppose that the following hypotheses hold:

(1)the pair(f,g)enjoys the(CLRg)property,

(2)for allx≠y∈Xand0 ≤k< 2,

whenever the right hand side of the above inequality is positive. Then the pair(f,g)has a coincidence point.
Moreover, if
Y
=
X
and
0 ≤
k
< 1,
then f and g have a unique common fixed point provided the pair
(
f
,
g
)
is Rweakly commuting or Rweakly commuting of type
(
A_{g}
) or
Rweakly commuting of type
(
A_{f}
) or
Rweakly commuting of type
(
P
).
Proof
. Since the pair (
f
,
g
) satisfies the (CLRg) property, there exists a sequence {
x_{n}
} in
Y
such that
for some
u
∈
Y
. First we show that
fu
=
gu
. If not, then using inequality (2.3) with
x
=
x_{n}
,
y
=
u
, we get
which on making
n
→ ∞, reduces to
which is a contradiction. Hence
fu
=
gu
which shows that
f
and
g
have a coincidence point.
Now consider
Y
=
X
and 0 ≤
k
< 1.
Case I: If the pair (
f
,
g
) is
R
weakly commuting, then we have
d
(
fgu
,
gfu
) ≤
Rd
(
fu
,
gu
) = 0,
that is,
fgu
=
gfu
. Therefore, we obtain
ffu
=
fgu
=
gfu
=
ggu
.
Case II: Suppose that the pair (
f
,
g
) is
R
weakly commuting of type (
A_{g}
), we obtain
d
(
ffu
,
gfu
) ≤
Rd
(
fu
,
gu
) = 0,
that is,
ffu
=
gfu
and so we get
fgu
=
ffu
=
gfu
=
ggu
.
Case III: Assume that the pair (
f
,
g
) is
R
weakly commuting of type (
A_{f}
), we have
d
(
fgu
,
ggu
) ≤
Rd
(
fu
,
gu
) = 0,
which implies that
fgu
=
ggu
. Hence
ffu
=
fgu
=
ggu
=
gfu
.
Case IV: Finally, if we consider the pair (
f
,
g
) is
R
weakly commuting of type (
P
), then we have
d
(
ffu
,
ggu
) ≤
Rd
(
fu
,
gu
) = 0,
that is,
ffu
=
ggu
. Therefore, we get
fgu
=
ffu
=
ggu
=
gfu
.
Now we assert that
fu
is a common fixed point of the mappings
f
and
g
. Suppose that
fu
≠
ffu
, using inequality (2.3) with
x
=
u
,
y
=
fu
, we get
which is a contradiction, we have
fu
=
ffu
. Therefore
fu
=
ffu
=
gfu
which shows that
fu
is a common fixed point of the pair (
f
,
g
).
Uniqueness of common fixed point is an easy consequence of inequality (2.3). This concludes the proof.
Theorem 2.3.
Let
(
X
,
d
)
be a metric space and
f
,
g
:
Y
→
X
satisfying inequality
(2.3).
Suppose that the pair
(
f
,
g
)
enjoys the property (E.A) whereas
g
(
X
)
is a closed subspace of
X
.
Then the pair
(
f
,
g
)
has a coincidence point
.
Moreover, if
Y
=
X
and 0 ≤
k
< 1,
then f and g have a unique common fixed point provided the pair
(
f
, g)
is either Rweakly commuting or Rweakly commuting of type
(
A_{g}
)
or Rweakly commuting of type
(
A_{f}
) or
Rweakly commuting of type
(
P
).
Proof.
If the pair (
f
,
g
) satisfies the property (E.A), then there exists a sequence {
x_{n}
} in
Y
such that
for some
t
∈
X
. Since it is assumed that
g
(
X
) is a closed subspace of
X
, there exists a point
u
∈
X
such that
gu
=
t
. Hence in view of Remark 1.6, the pair (
f
,
g
) also enjoys the (CLRg) property. By Theorem 2.2, we can obtain the mappings
f
and
g
have a unique common fixed point.
Since the pair of noncompatible mappings implies to the pair satisfying property (E.A), we get the following corollary.
Corollary 2.4.
Let
(
X
,
d
)
be a metric space and
f
,
g
:
Y
→
X
satisfying inequality
(2.3).
Suppose that the pair
(
f
,
g
)
is noncompatible whereas
g
(
X
)
is a closed subspace of
X
.
Then the pair
(
f
,
g
)
has a coincidence point.
Moreover, if
Y
=
X
and
0 ≤
k
< 1,
then f and g have a unique common fixed point provided the pair
(
f
,
g
)
is either Rweakly commuting or Rweakly commuting of type
(
A_{g}
)
or Rweakly commuting of type
(
A_{f}
)
or Rweakly commuting of type
(
P
).
Our next theorems involve a function
ϕ
ℝ
^{+}
→ ℝ
^{+}
which satisfies the following conditions:

(1)ϕis upper semicontinuous on ℝ+

(2) 0 <ϕ(s)
Theorem 2.5.
Let
(
X
,
d
)
be a metric space and
f
,
g
:
Y
→
X
.
Suppose that the following hypotheses hold
:

(1) the pair(f,g)enjoys the(CLRg)property,

(2) for all x, y∈Y,
Moreover, if
Y
=
X
,
then f and g have a unique common fixed point provided the pair
(
f
,
g
)
is either Rweakly commuting or Rweakly commuting of type
(
A_{g}
)
or Rweakly commuting of type
(
A_{f}
)
or Rweakly commuting of type
(
P
).
Proof.
Suppose that the pair (
f
,
g
) enjoys the (CLRg) property, there exists a sequence {
x_{n}
} in
Y
such that
for some
u
∈
Y
. We assert that
fu
=
gu
. Let on the contrary
fu
≠
gu
, then using inequality (2.4) with
x
=
x_{n}
,
y
=
u
, we get
Taking limit as
n
→ ∞, we have
which is a contradiction, we obtain
fu
=
gu
which shows that
f
and
g
have a coincidence point. The rest of the proof can be completed on the lines of the proof of Theorem 2.2, hence the details are omitted. ☐
Now, we introduce some example to support the useability of our results.
Example 2.6.
Let
X
= [0,+∞). Define
d
:
X
×
X
→
X
by
Also, define
f
,
g
:
X
→
X
by
and
gx
= 2
x
. Then

(1) The pair (f,g) enjoys the (CLRg) property.

(2) The pair (f,g) isRweakly commuting.

(3)d(gx,gy) > 0 for allx,y∈Xwithx≠y.

(4) for allx,y∈Xwithx≠y, we have
Proof.
The proof of (1), (2) and (3) are clear. To prove (4), given
x
,
y
∈
X
with
x
≠
y
. Without loss of generality, we may assume that
x
>
y
. Thus
Thus
f
and
g
satisfy all the hypotheses of Theorem 2.2. So
f
and
g
have a unique common fixed point. Here, 0 is the unique common fixed point of
f
and
g
. ☐
Example 2.7.
Let
X
= [0, 1]. Define
d
:
X
×
X
→
X
by
Also, define
f
,
g
:
X
→
X
by
and
gx
=
x
. Also, define
ϕ
: [0, +∞) → [0, +∞) by
Then

(1) The pair (f,g) enjoys the (CLRg) property.

(2) The pair (f,g) isRweakly commuting.

(3)d(gx,gy) > 0 for allx,y∈Xwithx≠y.

(4) for allx,y∈Xwithx≠y, we have
d
(
fx
,
fy
) ≤
ϕ
(
max
{
d
(
gx
,
gy
),
d
(
fx
,
gx
),
d
(
fy
,
gy
),
d
(
fy
,
gx
),
d
(
fx
,
gy
)} .
Proof.
The proof of (1), (2) and (3) are clear. To prove (4), given
x
,
y
∈
X
with
x
≠
y
. Without loss of generality, we may assume that
x
>
y
. Thus
Thus
f
and
g
satisfy all the hypotheses of Theorem 2.5. So
f
and
g
have a unique common fixed point. Here, 0 is the unique common fixed point of
f
and
g
. ☐
Remark 2.8.
It may be noticed that the earlier proved results namely; Theorem 2.2, Theorem 2.3 and Theorem 2.5 (also Corollary 2.4) remain valid in symmetric space (
X
,
d
) whenever
d
is continuous.
Acknowledgements
The authors are thankful to anonymous referees for their remarkable comments, suggestion and ideas that helps to improve this paper.
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