COMMON COUPLED FIXED POINT THEOREM UNDER GENERALIZED MIZOGUCHI-TAKAHASHI CONTRACTION FOR HYBRID PAIR OF MAPPINGS GENERALIZED MIZOGUCHI-TAKAHASHI CONTRACTION

Journal of the Korean Society of Mathematical Education Series B The Pure and Applied Mathematics.
2015.
Aug,
22(3):
199-214

- Received : November 09, 2014
- Accepted : July 20, 2015
- Published : August 31, 2015

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We establish a common coupled fixed point theorem for hybrid pair of mappings under generalized Mizoguchi-Takahashi contraction on a noncomplete metric space, which is not partially ordered. It is to be noted that to find coupled oincidence point, we do not employ the condition of continuity of any mapping involved therein. An example is also given to validate our results. We improve, extend and generalize several known results.
X
,
d
) be a metric space. We denote by 2
^{X}
the class of all nonempty subsets of
X
, by
CL
(
X
) the class of all nonempty closed subsets of
X
, by
CB
(
X
) the class of all nonempty closed bounded subsets of
X
and by
K
(
X
) the class of all nonempty compact subsets of
X
. A functional
H
:
CL
(
X
) ×
CL
(
X
) → ℝ
_{+}
∪ {+∞} is said to be the Pompeiu-Hausdorff generalized metric induced by
d
is given by
for all
A
,
B
∈
CB
(
X
), where
D
(
x
,
A
) = inf
_{a∈A}
d
(
x
,
a
) denote the distance from
x
to
A
⊂
X
. For simplicity, if
x
∈
X
, we denote
g
(
x
) by
gx
.
Markin
[23]
initiated to study the existence of fixed points for multivalued contractions and nonexpansive mappings using the Hausdorff metric which was further studied by many authors under different contractive conditions. The theory of multivalued mappings has application in control theory, convex optimization, differential inclusions and economics.
Bhaskar and Lakshmikantham
[6]
established some coupled fixed point theorems and applied these results to study the existence and uniqueness of solution for periodic boundary value problems, which were later extended by Lakshmikantham and Ciric
[19]
. For more details, see
[5
,
7
,
8
,
9
,
10
,
16
,
17
,
21
,
22
,
27
,
30]
.
Samet et al.
[28]
claimed that most of the coupled fixed point theorems for single valued mappings on ordered metric spaces are consequences of well-known fixed point theorems.
The concepts related to coupled fixed point theory for multivalued mappings were extended by Abbas et al.
[2]
and obtained coupled coincidence point and common coupled fixed point theorems involving hybrid pair of mappings satisfying generalized contractive conditions in complete metric spaces. Very few researcher gave attention to coupled fixed point problems for hybrid pair of mappings including
[1
,
2
,
11
,
12
,
13
,
14
,
15
,
20
,
29]
.
In
[2]
, Abbas et al. introduced the following for multivalued mappings:
Definition 1.1.
Let
X
be a nonempty set,
F
:
X
×
X
→ 2
^{X}
(a collection of all nonempty subsets of
X
) and
g
be a self-mapping on
X
. An element (
x
,
y
) ∈
X
×
X
is called
We denote the set of coupled coincidence points of mappings
F
and
g
by
C
(
F
,
g
). Note that if (
x
,
y
) ∈
C
(
F
,
g
), then (
y
,
x
) is also in
C
(
F
,
g
).
Definition 1.2.
Let
F
:
X
×
X
→ 2
^{X}
be a multivalued mapping and
g
be a self-mapping on
X
. The hybrid pair {
F
,
g
} is called
w−compatible
if
gF
(
x
,
y
) ⊆
F
(
gx
,
gy
) whenever (
x
,
y
) ∈
C
(
F
,
g
).
Definition 1.3.
Let
F
:
X
×
X
→ 2
^{X}
be a multivalued mapping and
g
be a self-mapping on
X
. The mapping
g
is called
F−weakly commuting
at some point (
x
,
y
) ∈
X
×
X
if
g
^{2}
x
∈
F
(
gx
,
gy
) and
g
^{2}
y
∈
F
(
gy
,
gx
).
Lemma 1.4
(
[26]
)
.
Let
(
X
,
d
)
be a metric space
.
Then, for each a
∈
X and B
∈
K(X), there is b
_{0}
∈
B such that D
(
a
,
B
) =
d
(
a
,
b
_{0}
),
where D
(
a
,
B
) = inf
_{b∈B}
d
(
a
,
b
).
Nadler
[25]
extended the famous Banach Contraction Principle
[4]
from single-valued mapping to multivalued mapping. Mizoguchi and Takahashi
[24]
proved the following generalization of Nadler’s fixed point theorem for weak contraction:
Theorem 1.5.
Let
(
X
,
d
)
be a complete metric space and
T
:
X
→
CB
(
X
)
be a multivalued mapping
.
Assume that
H (T x , T y ) ≤ Ψ (d (x , y ))d (x , y ),
for all x
,
y
∈
X
,
where Ψ is a function from
[0, ∞)
into
[0, 1)
satisfying
for all t
≥ 0.
Then T has a fixed point
.
Suzuki
[31]
gave its very simple proof. Amini-Harandi and O’Regan
[3]
obtained a generalization of Mizoguchi and Takahashi’s fixed point theorem.
In
[8]
, Ciric et al. proved coupled fixed point theorems for mixed monotone mappings satisfying a generalized Mizoguchi-Takahashi’s condition in the setting of ordered metric spaces. Main results of Ciric et al.
[8]
extended and generalized the results of Bhaskar and Lakshmikantham
[6]
, Du
[17]
and Harjani et al.
[18]
.
In this paper, we prove a common coupled fixed point theorem for hybrid pair of mappings under generalized Mizoguchi-Takahashi contraction on a noncomplete metric space, which is not partially ordered. It is to be noted that to find coupled coincidence point, we do not employ the condition of continuity of any mapping involved therein. We improve, extend and generalize the results of Amini-Harandi and O’Regan
[3]
, Bhaskar and Lakshmikantham
[6]
, Ciric et al.
[8]
, Du
[17]
, Harjani et al.
[18]
and Mizoguchi and Takahashi
[24]
. The effectiveness of our generalization is demonstrated with the help of an example.
φ
: [0, +∞) → [0, +∞) satisfying
Let Ψ denote the set of all functions
Ψ
: [0, +∞) → [0, 1) which satisfies lim
_{r→t+}
Ψ
(
r
) < 1 for all
t
≥ 0. For example, if
φ
(
t
) = ln(
t
+ 1) and
. Obviously, then
φ
∈ Φ and
Ψ
∈ Ψ, because
φ
is non-decreasing, positive in (0, +∞),
φ
(0) = 0 and
. Also,
Theorem 2.1.
Let
(
X
,
d
)
be a metric space
,
F
:
X
×
X
→
K
(
X
)
and g
:
X
→
X
be two mappings
.
Assume that there exist some
φ
∈ Φ
and
Ψ
∈ Ψ
such that
for all x
,
y
,
u
,
v
∈
X
.
Furthermore assume that F
(
X
×
X
) ⊆
g
(
X
)
and g
(
X
)
is a complete subset of X
.
Then F and g have a coupled coincidence point. Moreover
,
F and g have a common coupled fixed point
,
if one of the following conditions holds
:
Proof
. Let
x
_{0}
,
y
_{0}
∈
X
be arbitrary. Then
F
(
x
_{0}
,
y
_{0}
) and
F
(
y
_{0}
,
x
_{0}
) are well defined. Choose
gx
_{1}
∈
F
(
x
_{0}
,
y
_{0}
) and
gy
_{1}
∈
F
(
y
_{0}
,
x
_{0}
), because
F
(
X
×
X
) ⊆
g
(
X
). Since
F
:
X
×
X
→
K
(
X
), therefore by Lemma 1.4, there exist
z
_{1}
∈
F
(
x
_{1}
,
y
_{1}
) and
z
_{2}
∈
F
(
y
_{1}
,
x
_{1}
) such that
Since
F
(
X
×
X
) ⊆
g
(
X
), there exist
x
_{2}
,
y
_{2}
∈ X such that
z
_{1}
=
gx
_{2}
and
z
_{2}
=
gy
_{2}
.
Thus
Continuing this process, we obtain sequences {
x
_{n}
} and {
y
_{n}
} in
X
such that for all
n
∈ ℕ, we have
gx
_{n+1}
∈
F
(
x
_{n}
,
y
_{n}
) and
gy
_{n+1}
∈
F
(
y
_{n}
,
x
_{n}
) such that
which, by (
iφ
) and (2.1), implies
which, by the fact that
Ψ
< 1, implies
Similarly
Combining (2.2) and (2.3), we get
Since
φ
is non-decreasing, it follows that
Now (2.4) shows that {
φ
(max{
d
(
gx_{n}
,
gx
_{n+1}
),
d
(
gy_{n}
,
gy
_{n+1}
)})} is a non-increasing sequence. Therefore, there exists some
δ
≥ 0 such that
Since
Ψ
∈ Ψ, we have lim
_{r→δ+}
Ψ
(
r
) < 1 and
Ψ
(
δ
) < 1. Then there exists
α
∈ [0, 1) and
ε
> 0 such that
Ψ
(
r
) ≤
α
for all
r
∈ [
δ
,
δ
+
ε
). From (2.5), we can take
n
_{0}
≥ 0 such that
δ
≤
φ
(max{
d
(
gx_{n}
,
gx
_{n+1}
),
d
(
gy_{n}
,
gy
_{n+1}
)}) ≤
δ
+
ε
for all
n
≥
n
_{0}
. Then from (2.1), for all
n
≥
n
_{0}
, we have
Thus, for all
n
≥
n
_{0}
, we have
Similarly, for all
n
≥
n
_{0}
, we have
Combining (2.6) and (2.7), for all
n
≥
n
_{0}
, we get
Since
φ
is non-decreasing, it follows that, for all
n
≥
n
_{0}
,
Letting
n
→ ∞ in (2.8) and using (2.5), we obtain that
δ
≤
αδ
. Since
α
∈ [0, 1), therefore
δ
= 0. Thus
Since {
φ
(max{
d
(
gx_{n}
,
gx
_{n+1}
),
d
(
gy_{n}
,
gy
_{n+1}
)})} is a non-increasing sequence and
φ
is non-decreasing, then {max{
d
(
gx_{n}
,
gx
_{n+1}
),
d
(
gy_{n}
,
gy
_{n+1}
)}} is also a non-increasing sequence of positive numbers. This implies that there exists
θ
≥ 0 such that
Since
φ
is non-decreasing, we have
Letting
n
→ ∞ in this inequality, by using (2.9), we get 0 ≥
φ
(
θ
), which, by (
ii_{φ}
), implies that
θ
= 0. Thus, by (2.10), we get
Suppose that max{
d
(
gx_{n}
,
gx
_{n+1}
),
d
(
gy_{n}
,
gy
_{n+1}
)} = 0, for some
n
≥ 0. Then, we have
d
(
gx_{n}
,
gx
_{n+1}
) = 0 and
d
(
gy_{n}
,
gy
_{n+1}
) = 0 which implies that
gx_{n}
=
gx
_{n+1}
∈
F
(
x_{n}
,
y_{n}
) and
gy_{n}
=
gy
_{n+1}
∈
F
(
y_{n}
,
x_{n}
), that is, (
x_{n}
,
y_{n}
) is a coupled coincidence point of
F
and
g
. Now, suppose that max{
d
(
gx_{n}
,
gx
_{n+1}
),
d
(
gy_{n}
,
gy
_{n+1}
)} ≠ 0, for all
n
≥ 0.
Denote
From (2.8), we have
Then, we have
On the other hand, by (
iii_{φ}
), we have
Thus, by (2.12) and (2.13), we have ∑max{
d
(
gx_{n}
,
gx
_{n+1}
),
d
(
gy_{n}
,
gy
_{n+1}
)} < ∞. It means that
and
are Cauchy sequences in
g
(
X
). Since
g
(
X
) is complete, therefore there exist
x
,
y
∈
X
such that
Now, since
gx
_{n+1}
∈
F
(
x_{n}
,
y_{n}
) and
gy
_{n+1}
∈
F
(
y_{n}
,
x_{n}
), therefore by using condition (2.1) and (
i_{φ}
), we get
Since
φ
is non-decreasing, we have
Letting
n
→ ∞ in (2.15), by using (2.14), we obtain
which implies that
that is, (
x
,
y
) is a coupled coincidence point of
F
and
g
. Hence
C
(
F
,
g
) is nonempty.
Suppose now that (
a
) holds. Assume that for some (
x
,
y
) ∈
C
(
F
,
g
),
where
u
,
v
∈
X
. Since
g
is continuous at
u
and
v
. We have, by (2.16), that
u
and
v
are fixed points of
g
, that is,
As
F
and
g
are
w
−compatible, so
that is,
Now, by using (2.1) and (2.18), we obtain
Since
φ
is non-decreasing, we have
On taking limit as
n
→ ∞ in the above inequality, by using (2.16) and (2.17), we get
which implies that
Now, from (2.17) and (2.19), we have
that is, (
u
,
v
) is a common coupled fixed point of
F
and
g
.
Suppose now that (
b
) holds. Assume that for some (
x
,
y
) ∈
C
(
F
,
g
),
g
is
F
−weakly commuting, that is
g
^{2}
x
∈
F
(
gx
,
gy
),
g
^{2}
y
∈
F
(
gy
,
gx
) and
g
^{2}
x
=
gx
,
g
^{2}
y
=
gy
. Thus
gx
=
g
^{2}
x
∈
F
(
gx
,
gy
) and
gy
=
g
^{2}
y
∈
F
(
gy
,
gx
), that is, (
gx
,
gy
) is a common coupled fixed point of
F
and
g
.
Suppose now that (
c
) holds. Assume that for some (
x
,
y
) ∈
C
(
F
,
g
) and for some
u
,
v
∈
X
,
Since
g
is continuous at
x
and
y
, then
x
and
y
are fixed points of
g
, that is,
Since (
x
,
y
) ∈
C
(
F
,
g
), so we obtain
that is, (
x
,
y
) is a common coupled fixed point of
F
and
g
.
Finally, suppose that (
d
) holds. Let
g
(
C
(
F
,
g
)) = {(
x
,
x
)}. Then {
x
} = {
gx
} =
F
(
x
,
x
). Hence (
x
,
x
) is a common coupled fixed point of
F
and
g
. ⃞
Example.
Suppose that
X
= [0, 1], equipped with the metric
d
:
X
×
X
→ [0, +∞) defined as
d
(
x
,
y
) = max{
x
,
y
} and
d
(
x
,
x
) = 0 for all
x
,
y
∈
X
. Let
F
:
X
×
X
→
K
(
X
) be defined as
and
g
:
X
→
X
be defined as
Define
φ
: [0, +∞) → [0, +∞) by
and
Ψ
: [0, +∞) → [0, 1) defined by
Now, for all
x
,
y
,
u
,
v
∈
X
with
x
,
y
,
u
,
v
∈ [0, 1), we have
Case (
a
). If
x
=
u
, then
which implies that
Case (
b
). If
x
≠
u
with
x
<
u
, then
which implies that
Similarly, we obtain the same result for
u
<
x
. Thus the contractive condition (2.1) is satisfied for all
x
,
y
,
u
,
v
∈
X
with
x
,
y
,
u
,
v
∈ [0, 1). Again, for all
x
,
y
,
u
,
v
∈
X
with
x
,
y
∈ [0, 1) and
u
,
v
= 1, we have
which implies that
Thus the contractive condition (2.1) is satisfied for all
x
,
y
,
u
,
v
∈
X
with
x
,
y
∈ [0, 1) and
u
,
v
= 1. Similarly, we can see that the contractive condition (2.1) is satisfied for all
x
,
y
,
u
,
v
∈
X
with
x
,
y
,
u
,
v
= 1. Hence, the hybrid pair {
F
,
g
} satisfies the contractive condition (2.1), for all
x
,
y
,
u
,
v
∈
X
. In addition, all the other conditions of Theorem 2.1 are satisfied and
z
= (0, 0) is a common coupled fixed point of hybrid pair {
F
,
g
}. The function
F
:
X
×
X
→
K
(
X
) involved in this example is not continuous at the point (1, 1) ∈
X
×
X
.
Remark 2.2.
We improve, extend and generalize the results of Ciric et al.
[8]
in the sense that
If we put
g
=
I
(the identity mapping) in Theorem 2.1, we get the following result:
Corollary 2.3.
Let
(
X
,
d
)
be a complete metric space
,
F
:
X
×
X
→
K
(
X
)
be a mapping
.
Assume that there exist some
φ
∈ Φ
and
Ψ
∈ Ψ
such that
for all
x
,
y
,
u
,
v
∈
X
.
Then F has a coupled fixed point
.
If we put
for all
t
≥ 0 in Theorem 2.1, then we get the following result:
Corollary 2.4.
Let
(
X
,
d
)
be a metric space
,
F
:
X
×
X
→
K
(
X
)
and g
:
X
→
X
be two mappings
.
Assume that there exist some
φ
∈ Φ
and
such that
for all
x
,
y
,
u
,
v
∈
X
.
Furthermore assume that
F
(
X
×
X
) ⊆
g
(
X
)
and
g
(
X
)
is a complete subset of X
.
Then F and g have a coupled coincidence point
.
Moreover
,
F and g have a common coupled fixed point
,
if one of the following conditions holds
:
If we put
g
=
I
(the identity mapping) in the Corollary 2.4, we get the following result:
Corollary 2.5.
Let
(
X
,
d
)
be a complete metric space
,
F
:
X
×
X
→
K
(
X
)
be a mapping
.
Assume that there exist some
φ
∈ Φ
and
such that
for all x
,
y
,
u
,
v
∈
X
.
Then F has a coupled fixed point
.
If we put
φ
(
t
) = 2
t
for all
t
≥ 0 in Theorem 2.1, then we get the following result:
Corollary 2.6.
Let
(
X
,
d
)
be a metric space
,
F
:
X
×
X
→
K
(
X
)
and g
:
X
→
X
be two mappings
.
Assume that there exists some
Ψ
∈ Ψ
such that
for all
x
,
y
,
u
,
v
∈
X
.
Furthermore assume that
F
(
X
×
X
) ⊆
g
(
X
)
and
g
(
X
)
is a complete subset of X
.
Then F and g have a coupled coincidence point. Moreover
,
F and g have a common coupled fixed point, if one of the following conditions holds
:
If we put
g
=
I
(the identity mapping) in the Corollary 2.6, we get the following result:
Corollary 2.7.
Let
(
X
,
d
)
be a complete metric space
,
F
:
X
×
X
→
K
(
X
)
be a mapping
.
Assume that there exists some
Ψ
∈ Ψ
such that
H
(
F
(
x
,
y
),
F
(
u
,
v
)) ≤
Ψ
(2 max {
d
(
x
,
u
),
d
(
y
,
v
)}) max {
d
(
x
,
u
),
d
(
y
,
v
)} ,
for all
x
,
y
,
u
,
v
∈
X
.
Then F has a coupled fixed point
.
If we put
Ψ
(
t
) =
k
where 0 <
k
< 1, for all
t
≥ 0 in Corollary 2.6, then we get the following result:
Corollary 2.8.
Let
(
X
,
d
)
be a metric space
.
Assume
F
:
X
×
X
→
K
(
X
)
and
g
:
X
→
X
be two mappings satisfying
for all
x
,
y
,
u
,
v
∈
X
,
where
0 <
k
< 1.
Furthermore assume that
F
(
X
×
X
) ⊆
g
(
X
)
and
g
(
X
)
is a complete subset of
X
.
Then F and g have a coupled coincidence point
.
Moreover
,
F and g have a common coupled fixed point, if one of the following conditions holds
:
If we put
g
=
I
(the identity mapping) in the Corollary 2.8, we get the following result:
Corollary 2.9.
Let
(
X
,
d
)
be a complete metric space
.
Assume
F
:
X
×
X
→
K
(
X
)
be a mapping satisfying
for all
x
,
y
,
u
,
v
∈
X
,
where
0 <
k
< 1.
Then F has a coupled fixed point
.

coupled fixed point
;
coupled coincidence point
;
generalized Mizoguchi-Takahashi contraction
;
w−compatibility
;
F−weakly commutativity.

1. INTRODUCTION

Let (
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- (1) acoupled fixed pointofFifx∈F(x,y) andy∈F(y,x).
- (2) acoupled coincidence pointof hybrid pair {F,g} ifgx∈F(x,y) andgy∈F(y,x).
- (3) acommon coupled fixed pointof hybrid pair {F,g} ifx=gx∈F(x,y) andy=gy∈F(y,x).

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2. MAIN RESULTS

Let Ф denote the set of all functions
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- (a)F and g are w−compatible. limn→∞gnx=u andlimn→∞gny=v for some(x,y) ∈C(F,g)and for some u,v∈X and g is continuous at u and v.
- (b)g is F−weakly commuting for some(x,y) ∈C(F,g)and gx and gy are fixed points of g,that is,g2x=gx and g2y=gy.
- (c)g is continuous at x and y. limn→∞gnu=x andlimn→∞gnv=y for some(x,y) ∈C(F,g)and for some u,v∈X.
- (d)g(C(F,g))is a singleton subset of C(F,g).

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- (i) We prove our result for hybrid pair of mappings.
- (ii) We prove our result in the framework of non-complete metric space (X,d) and the product setX×Xis not empowered with any order.
- (iii) We prove our result without the assumption of continuity and mixed gmonotone property for mappingF:X×X→K(X).
- (iv) The functionsφ: [0, +∞) → [0, +∞) andΨ: [0, +∞) → [0, 1) involved in our theorem and example are discontinuous.

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- (a)F and g are w−compatible. limn→∞gnx=u andlimn→∞gny=v for some(x,y) ∈C(F,g)and for someu,v∈X and g is continuous at u and v.
- (b)g is F−weakly commuting for some(x,y) ∈C(F,g)and gx and gy are fixed points of g, that is,g2x=gxandg2y=gy.
- (c)g is continuous at x and y. limn→∞gnu=x andlimn→∞gnv=y for some(x,y) ∈C(F,g)and for someu,v∈X.
- (d)g(C(F,g))is a singleton subset ofC(F,g).

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- (a)F and g are w−compatible. limn→∞gnx=u andlimn→∞gny=v for some(x,y) ∈C(F,g)and for someu,v∈X and g is continuous at u and v.
- (b)g is F−weakly commuting for some(x,y) ∈C(F,g)and gx and gy are fixed points of g,that is,g2x=gx andg2y=gy.
- (c)g is continuous at x and y. limn→∞gnu=x andlimn→∞gnv=y for some(x,y) ∈C(F,g)and for someu,v∈X.
- (d)g(C(F,g))is a singleton subset ofC(F,g).

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- (a)F and g are w−compatible. limn→∞gnx=u andlimn→∞gny=v for some(x,y) ∈C(F,g)and for someu,v∈Xand g is continuous at u and v.
- (b)g is F−weakly commuting for some(x,y) ∈C(F,g)and gx and gy are fixed points of g,that is,g2x=gx andg2y=gy.
- (c)g is continuous at x and y. limn→∞gnu=x andlimn→∞gnv=y for some(x,y) ∈C(F,g)and for someu,v∈X.
- (d)g(C(F,g))is a singleton subset ofC(F,g).

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Citing 'COMMON COUPLED FIXED POINT THEOREM UNDER GENERALIZED MIZOGUCHI-TAKAHASHI CONTRACTION FOR HYBRID PAIR OF MAPPINGS GENERALIZED MIZOGUCHI-TAKAHASHI CONTRACTION
'

@article{ SHGHCX_2015_v22n3_199}
,title={COMMON COUPLED FIXED POINT THEOREM UNDER GENERALIZED MIZOGUCHI-TAKAHASHI CONTRACTION FOR HYBRID PAIR OF MAPPINGS GENERALIZED MIZOGUCHI-TAKAHASHI CONTRACTION}
,volume={3}
, url={http://dx.doi.org/10.7468/jksmeb.2015.22.3.199}, DOI={10.7468/jksmeb.2015.22.3.199}
, number= {3}
, journal={Journal of the Korean Society of Mathematical Education Series B The Pure and Applied Mathematics}
, publisher={Korean Society of Mathematical Education}
, author={DESHPANDE, BHAVANA
and
HANDA, AMRISH}
, year={2015}
, month={Aug}