TRIPLED FIXED POINT THEOREM FOR HYBRID PAIR OF MAPPINGS UNDER GENERALIZED NONLINEAR CONTRACTION

Journal of the Korean Society of Mathematical Education Series B The Pure and Applied Mathematics.
2014.
Jan,
21(1):
23-38

- Received : August 02, 2013
- Accepted : November 25, 2013
- Published : January 30, 2014

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In this paper, we introduce the concept of ω - compatibility and weakly commutativity for hybrid pair of mappings
F : X × X × X × → 2^{X}
and
g : X → X
and establish a common tripled fixed point theorem under generalized nonlinear contraction. An example is also given to validate our result. We improve, extend and generalize various known results.
X, d
) be a metric space and
CB
(
X
) be the set of all nonempty closed bounded subsets of
X
. Let
D
(
x, A
) denote the distance from
x
to
A
⊂ X and
H
denote the Hausdorff metric induced by
d
, that is,
and for all A, B ∈ CB (X ).
Markin
[23]
initiated the study of fixed points for multivalued contractions and non-expansive maps using the Hausdorff metric. Fixed points existence for various multivalued contractive mappings has been studied by several authors under different conditions. For details, we refer the reader to
[1
,
2
,
12
,
13
,
14
,
15
,
16
,
18
,
19
,
20
,
21
,
25
,
26
,
27]
and the reference therein. Multivalued maps theory has application in control theory, convex optimization, differential equations and economics.
Bhaskar and Lakshmikantham
[10]
, established some coupled fixed point theorems and apply these to study the existence and uniqueness of solution for periodic boundary value problems. Lakshmikantham and Ciric
[22]
proved coupled coinci-dence and common coupled fixed point theorems for nonlinear contractive mappings in partially ordered complete metric spaces and extended the results of Bhaskar and Lakshmikantham
[10]
.
Berinde and Borcut
[8]
introduced the concept of tripled fixed point for single valued mappings in partially ordered metric spaces. In
[8]
, Berinde and Borcut established the existence of tripled fixed point of single-valued mappings in partially ordered metric spaces. For more details on tripled fixed point theory, we also refer the reader to
[3
,
4
,
5
,
6
,
7
,
9
,
11]
: Samet and Vetro
[24]
introduced the notion of fixed point of N order in case of single-valued mappings. In particular for N=3 (tripled case), we have the following definition:
Definition 1.1
(
[24]
). Let
X
be a non-empty set and
F :
X
×
X
×
X
→
X
be a given mapping. An element (
x, y, z
) ∈
X
×
X
×
X
is called
a tripled fixed point
of the mapping
F
if
F (x, y, z ) = x, F (y, z, x ) = y and F (z, x, y ) = z .
In this paper, we prove a common tripled fixed point for hybrid pair of mappings under generalized nonlinear contraction. We improve, extend and generalize the results of Ding, Li and Radenovic
[17]
and Abbas, Ciric, Damjanovic and Khan
[2]
. The effectiveness of the present work is validated with the help of suitable example.
Definition 2.1.
Let
X
be a nonempty set,
F
:
X
×
X
×
X
→ 2
^{X}
(a collection of all nonempty subsets of
X
) and
g
be a self-map on
X
. An element (
x, y, z
)∈
X
×
X
×
X
is called
We denote the set of tripled coincidence points of mappings
F
and
g
by
C
(
F, g
). Note that if (
x, y, z
) ∈
C
(
F, g
), then (
y, z, x
) and (
z, x, y
) are also in
C
(
F, g
).
Deginition 2.2
. Let
F : X × X × X→ 2^{X}
be a multivalued mapping and
g
be a self-map on
X
. The hybrid pair {
F, g
} is called
w-compatible
if
g
(
F
(
x, y, z
)) ⊆
F
(
gx, gy, gz
) whenever (
x, y, z
) ∈
C
(
F, g
).
Deginition 2.3
. Let
F : X × X × X→ 2^{X}
be a multivalued mapping and
g
be a self-map on
X
. The mapping
g
is called
F-weakly commuting
at some point (x, y, z) ∈
X
^{3}
if
g
^{2}
x
∈
F
(
gx, gy, gz
), g
^{2}
y ∈
F
(
gy, gz, gx
) and
g
^{2}
z
∈
F
(
gz, gx, gy
).
Lemma 2.1
. Let (
X, d
)
be a metric space. Then, for each a
∈
X and B
∈
CB
(
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
).
Proof. Let
a
∈
X
and
B
∈
CB
(
X
). Since the function
d
is continuous. Thus, by the closedness of
B
, there exists
b
_{0}
∈
B
such that inf
_{b ∈B}
d
(
a, b
) =
d
(
a, b
_{0}
), that is,
D
(
a,B
) =
d
(
a, b
_{0}
).
It is clear that
φ(t
) <
t
for each
t
> 0: In fact, if
φ
(
t
_{0}
) ≥
t
_{0}
for some
t
_{0 }
> 0. then, since
φ
is non-decreasing,
φ
^{n}
(
t
_{0}
) ≥
t
_{0}
for all
n
∈
, which contradicts with lim
_{n→∞}
φ
^{n}
(
t
_{0}
) = 0. In addition, it is easy to see that
φ
(0) = 0. □
Theorem 2.1
. Let (
X
,
d
) be a metric space. Assume
F : X × X × X
→
CB
(
X
)
and g : X → X be two mappings satisfying
for all x, y, z, u, v, w ∈ X, where φ
∈ Φ.
Furthermore assume that F
(
X×X×X
) ⊆
g
(
X
)
and g
(
X
)
is a complete subset of X. Then F and g have a tripled coincidence point. Moreover, F and g have a common tripled fixed point, if one of the following conditions holds.
Proof. Let x_{0}, y_{0}, z_{0}
∈
X
be arbitrary. Then
F
(
x_{0}, y_{0}, z_{0}
),
F
(
y_{0}, z_{0}, x_{0}
) and
F
(
z_{0}, x_{0}, y_{0}
) are well defined. Choose
gx_{1}
∈
F
(
x_{0}, y_{0}, z_{0}
),
gy
_{1}
∈ F(
y_{0}, z_{0}, x_{0}
) and
gz
_{1}
∈
F
(
z_{0}, x_{0}, y_{0}
), because
F
(
X×X×X
) ⊆
g
(
X
). Since
F . X×X×X
→
CB
(
X
), therefore by Lemma 2.1, there exist
u
_{1}
∈
F
(
x
_{1}
,
y
_{1}
,
z
_{1}
),
u
_{2}
∈
F
(
y
_{1}
,
z
_{1}
,
x
_{1}
) and
u
_{3}
∈
F
(
z
_{1}
,
x
_{1}
,
y
_{1}
) such that
Since
F
(
X×X×X
) ⊆
g
(
X
), there exist
x
_{2}
,
y
_{2}
,
z
_{2}
∈
X
such that
u
_{1}
=
gx
^{2}
,
u
_{2}
=
gy
_{2}
and
u
_{3}
=
gz
_{2}
, Thus
Continuing this process, we obtain sequences {
x_{n}
}, {
y_{n}
} and {
z_{n}
} in
X
such that for all
n
∈
, we have
gx
_{n+1}
∈
F
(
x_{n}, y_{n}, z_{n}
),
gy
_{n+1}
∈
F
(
y_{n}, z_{n}, x_{n}
) and
gz
_{n+1}
∈
F
(
z_{n}, x_{n}, y_{n}
) such that
Thus,
Similarly
Combining (2,2), (2,3) and (2,4), we get
Thus
If we suppose that
then by (2, 5), (
i_{φ}
) and (
ii_{φ}
), we have
which is a contradiction. Thus, we must have
Hence by (2,5), we have for all
n
∈
,
Thus
where
δ = max {d (gx _{0}, gx _{1}), d (gy _{0}, gy _{1}), d (gz _{0}, gz _{1})} .
Without loss of generality, one can assume that max max {
d
(
gx
_{0}
,
gx
_{1}
),
d
(
gy
_{0}
,
gy
_{1}
),
d
(
gz
_{0}
,
gz
_{1}
)} ≠ In fact, if this is not true, then
gx
_{0}
=
gx
_{1}
∈
F
(
x
_{0}
,
y
_{0}
,
z
_{0}
),
gy
_{0}
=
gy
_{1}
∈
F
(
y
_{0}
,
z
_{0}
,
x
_{0}
) and
gz
_{0}
=
gz
_{1}
∈
F
(
z
_{0}
,
x
_{0}
,
y
_{0}
) that (
x
_{0}
,
y
_{0}
,
z
_{0}
) s a tripled coincidence point of
F
and
g
.
Thus, for
m, n
∈
with m >
n
, by triangle inequality and (2,6), we get
which implies, by (
ii_{φ}
), that {
gx_{n}
} is a Cauchy sequence in
g
(
X
). Similarly we obtain that {
gy_{n}
} and {
gz_{n}
} are Cauchy sequences in
g
(
X
). Since
g
(
X
) is complete, there exist
x, y, z
∈
X
such that
Now, since
gx
_{n+1}
∈
F
(
x_{n}, y_{n,} z_{n}
),
gy
_{n+1}
∈
F
(
y_{n}, z_{n,} x_{n}
) and
gz
_{n+1}
∈
F
(
z_{n}, x_{n,} y_{n}
), therefore by using condition (2,1), we get
where
Since lim
_{n→∞}
gx_{n}
=
gx
, lim
_{n→∞}
gy_{n}
=
gy
and lim
_{n→∞}
gz_{n}
=
gz
there exists
n
_{0}
∈
such that for all
n
>
n
_{0}
,
Combining this with (2,8), (2,9) and (2,10), we get for all
n
>
n
_{0}
,
Now, we claim that
If this is not true, then
Thus, by (2,11), (
i_{φ}
) and (
ii_{φ}
), we get for all
n
>
n
_{0}
,
Thus
Letting
n
→ ∞ in (2:13), by using (2,7), we obtain
which is a contradiction. So (2, 12) holds. Thus, it follows that
gx ∈ F (x, y, z ), gy ∈ F (y, z, x ) and gz ∈ F (z, x, y ),
that is, (
x, y, z
) is a tripled coincidence point of
F
and
g
. Hence
C
(
F, g
) is nonempty. Suppose now that (a) holds. Assume that for some (
x, y, z
) ∈
C
(
F, g
),
where
u, v, w
∈
X
. Since
g
is continuous at
u, v
and
w
. We have, by (2,14), that
u,v
and
w
are fixed points of
g
, that is,
As
F
and
g
are
w
-compatible, so for all
n
≥ 1,
Now, by using (2,1) and (2,16), we obtain
where
By (2,14) and (2,15), there exists
n
_{0}
∈
such that for all
n
>
n
_{0}
,
Combining this with (2,17), we get for all
n
>
n
_{0}
,
Now, we claim that
If this is not true, then
Thus, by (2,18), (
i_{φ}
) and (
ii_{φ}
), we get for all
n
>
n
_{0}
,
On taking limit as
n
→ ∞ in (2, 20), by using (2, 14) and (2, 15), we get
which is a contradiction. So (2, 19) holds. Thus, it follows that
Now, from (2, 15) and (2, 21), we have
that is, (
u, v, w
) is a common tripled fixed point of
F
and
g
.
Suppose now that (
b
) holds. Assume that for some (
x, y, z
) ∈
C
(
F, g
),
g
is F- weakly commuting, that is,
g
^{2}
x
∈
F
(
gx, gy, gz
),
g
^{2}
y
∈
F
(
gy, gz, gx
),
g
^{2}
z
∈
F
(
gz, gx, gy
), and
g
^{2}
x
=
gx
,
g
^{2}
y
=
gy
,
g
^{2}
z
=
gz
. Thus
gx
= g
^{2}
x ∈
F
(
gx, gy, gz
),
gy
= g
^{2}
y ∈
F
(
gy, gz, gx
) and
gz
= g
^{2}
z ∈
F
(
gz, gx, gy
), that is (
gx, gy, gz
) is a common tripled fixed point of
F
and
g
.
Suppose now that (
c
) holds. Assume that for some (
x, y, z
) ∈
C
(
F, g
) and for some
u, v, w
∈
X
, lim
_{n→∞}
g^{n}u
=
x
, lim
_{n→∞}
g^{n}v
=
y
and lim
_{n→∞}
g^{n}w
=
z
. Since
g
is continuous at
x, y
and
z
. We have that
x, y
and
z
are fixed point of
g
, that is,
gx
=
x
,
gy
=
y
and
gz
=
z
. Since (
x, y, z
) ∈ C(
F, g
), therefore, we obtain
x
=
gx
∈
F(x, y, z), y
=
gy
∈
F(y, z, x)
and
z
=
gz
∈
F(z, x, y)
,
Finally, suppose that (
d
) holds. Let
g
(
C
(
F, g
)) = {(
x, x, x
)}. Then {
x
} = {
gx
} = F(
x, x, x
): Hence (
x, x, x
) is tripled fixed point of
F
and
g
. □
Example 2.1.
Suppose that
X
= [0, 1], equipped with the metric
d
:
X ×X
→[0, +∞) defined by
d
(
x, y
) = max{
x, y
} and
d
(
x, x
) = 0 for all
x, y
∈
X
. Let
F
:
X×X×X
→
CB
(
X
) be defined as
and
g
:
X
→
X
be defined as
g(x)
=
x
^{2}
, for all
x
∈
X
.
Define
φ
: [0, ∞ ] → [0, ∞ ] by
Now, for all
x, y, z, u, v, w
∈
X
with
x, y, z, u, v, w
∈ [0, 1), we have
Case (a ) If x ^{2} + y ^{2} + z ^{2} = u ^{2} + v ^{2} + w ^{2}, then
Case (
b
) If
x
^{2}
+
y
^{2}
+
z
^{2}
≠
u
^{2}
+
v
^{2}
+
w
^{2}
with
x
^{2}
+
y
^{2}
+
z
^{2}
<
u
^{2}
+
v
^{2}
+
w
^{2}
, then
Similarly, we obtain the same result for
u
^{2}
+
v
^{2}
+
w
^{2}
<
x
^{2}
+
y
^{2}
+
z
^{2}
. Thus the contractive condition (2,1) is satisfied for all
x, y, z, u, v, w ∈ X
with
x, y, z, u, v, w
∈ [0, 1). Again for all
x, y, z, u, v, w
∈
X
with
x, y, z
, ∈ [0, 1) and
u, v, w
= 1, we have
Thus the contractive condition (2:1) is satisfied for all
x, y, z, u, v, w
∈
X
with
x, y z
∈ [0, 1) and
u, v, w
= 1, Similarly, we can see that the contractive condition (2,1) is satisfied for all
x, y, z, u, v, w
∈
X
with
x, y, z, u, v, w
= 1. Hence, the hybrid pair {F, g} satisfies the contractive condition (2,1), for all
x, y, z, u, v, w
∈
X
. In addition, all the other conditions of Theorem 2.1 are satisfied and z = (0, 0, 0) is a common tripled fixed point of hybrid pair {
F, g
} . The function
F : X×X×X
→
CB
(
X
) involved in this example is not continuous at the point (1, 1, 1) ∈
X×X×X
.
Remark 2.1
. We improve, extend and generalize the result of Ding, Li and Radenovic
[17]
in the following sense:
If we put
g
=
I
(
I
is the identity mapping) in Theorem 2.1, then we have the following result:
Corollary 2.2
. Let (
X, d
)
be a complete metric space, F : X × X× X
→
CB
(
X) be a mapping satisfying
for all x, y, z, u, v, w
∈
X
,
where φ
∈ Φ.
Then F has a tripled fixed point.
If we put
φ
(
t
) =
kt
where 0 <
k
< 1 in Theorem 2.1, then we have the following result:
Corollary 2.3
. Let (
X, d
)
be a metric space. Assume F : X × X× X
→
CB
(
X
) and
g
:
X → X be two mapping satisfying
for all x, y, z, u, v, w
∈
X where
0 <
k
< 1.
Furthermore assume that F
(
X ×X×X
) ⊆
g
(
X
) and
g
(
X
)
is a complete subset of X. Then F and g have a tripled coincidence point. Moreover, F and g have a common tripled fixed point, if one of the following conditions holds.
If we put
g
=
I
(
I
is the identity mapping) in Corollary 2.3, then we have the following result:
Corollary 2.4
.
Let
(
X, d
)
be a complete metric space, F : X × X × X → CB(X) be a mapping satisfying
for all x, y, z, u, v, w ∈ X . Then F has a tripled fixed point.

1. INTRODUCTION AND PRELIMINARIES

Let (
2. MAIN RESULTS

First we introduce the following:
- (1)a tripled fixed pointofFifx∈F(x, y, z),y∈F(y, z, x) and z ∈F(z, x, y).
- (2)a tripled coincidence pointof hybrid pair {F, g} ifg(x) ∈F(x, y, z),g(y) ∈F(y, z, x) andg(z) ∈F(z, x, y).
- (3)a common tripled fixed pointof hybrid pair {F, g} ifx=g(x) ∈F(x, y, z),y=g(y) ∈F(y, z, x) andz=g(z) ∈F(z, x, y).

- Let Φ denote the set of all functionsφ: [0, ∞) → [0, ∞) satisfying
- (iφ)φis non-decreasing,
- (iiφ) limn→ ∞φn(t) = 0 for allt> 0.

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- (a)F and g are w-compatible.limn→∞gnx= u, limn→∞gny=v andlimn→∞gnz=w for some(x, y, z) ∈C(F, g)and for some u, v, w∈Xand g is continuous at u, v and w.
- (b)g is F-weakly commuting for some(x, y, z) ∈C(F, g)and gx, gy and gz are fixed points of g, that is, g2x = gx, g2y = gy and g2z = gz.
- (c)g is continuous at x, y and z.limn→∞gnu = x, limn→∞gnv = y andlimn→∞gnw = z for some(x, y, z) ∈C(F, g) and for someu, v, w∈X:
- (d)g(C(g, F))is singleton subset of C(g, F):

- d(gx1,u1) ≤H(F(x0,y0,z0),F(x1,y1,z1)),
- d(gy1,u2) ≤H(F(y0,z0,x0),F(y1,z1,x1)),
- d(gz1,u3) ≤H(F(z0,x0,y0),F(z1,x1,y1)).

- d(gx1,gx2) ≤H(F(x0,y0,z0),F(x1,y1,z1)),
- d(gy1,gy2) ≤H(F(y0,z0,x0),F(y1,z1,x1)),
- d(gz1,gz2) ≤H(F(z0,x0,y0),F(z1,x1,y1)).

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- (i) We prove our result in the settings of multivalued mapping and for hybrid pair of mappings while Ding, Li and Radenovic[17]proved result for single valued mappings.
- (ii) We prove tripled coincidence and common tripled fixed point theorem while Ding, Li and Radenovic[17]proved coupled coincidence and common coupled fixed point theorems.
- (iii) To prove the result we consider non complete metric space and the space is also not partially ordered.
- (iv) The mappingF : X ×X ×X→CB(X) is discontinuous and not satisfying mixed g-monotone property.
- (v) The functionφ: [0, ∞) → [0, ∞) involved in our theorem and example is discontinuous.
- (vi) Our proof is simple and different from the other results in the existing literature.

- (a)F and g are w-compatible. limn→∞gnx=u, limn→∞gny=vandlimn→∞gnz=wfor some (x, y, z)∈C(F, g)and for some u, v, w ∈ X and g is con-tinuous at u, v and w.
- (b)g is F-weakly commuting for some (x, y, z) ∈ C(F, g) and gx, gy and gz are fixed points of g, that is, g2x = gx, g2y = gyandg2z = gz.
- (c)g is continuous at x, y and z. limn→∞gnu=x, limn→∞gnv=yandlimn→∞gnw=z for som (x, y, z ) ∈ C(F, g) and for some u, v, w ∈ X.
- (d)g(C(g, F)) is singleton subset of C(g, F).

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Citing 'TRIPLED FIXED POINT THEOREM FOR HYBRID PAIR OF MAPPINGS UNDER GENERALIZED NONLINEAR CONTRACTION
'

@article{ SHGHCX_2014_v21n1_23}
,title={TRIPLED FIXED POINT THEOREM FOR HYBRID PAIR OF MAPPINGS UNDER GENERALIZED NONLINEAR CONTRACTION}
,volume={1}
, url={http://dx.doi.org/10.7468/jksmeb.2014.21.1.23}, DOI={10.7468/jksmeb.2014.21.1.23}
, number= {1}
, 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
SHARMA, SUSHIL
and
HANDA, AMRISH}
, year={2014}
, month={Jan}