# Common Fixed Point Theorems of Greguš Type(

Authors: Penumarthy Parvateesam Murthy; Uma Devi Patel
Journal: Journal of Operators
 (2015)

---

## Abstract

The main purpose of this paper is to establish a common fixed point theorem for set valued mappings in 2-metric spaces by generalizing a theorem of Abd EL-Monsef et al. (2009) and Murthy and Tas (2009) by using(
ϕ
,
ψ
)-weak contraction in view of Greguš type condition for set valued mappings using R-weakly commuting maps.


---

## Body

## 1. Introduction and Preliminaries

The weak contraction condition in Hilbert Space was introduced by Alber and Guerre-Delabriere [1]. Later, Rhoades [2] has noticed that the results of Alber and Guerre-Delabriere [1] in Hilbert Spaces are also true in a complete metric space.Rhoades [2] established a fixed point theorem in a complete metric space by using the following contraction condition.LetT
:
X
→
X which satisfies the following condition:(1)
d
T
x
,
T
y
≤
d
x
,
y
-
φ
d
x
,
y
,where x
,
y
∈
X and φ
:
[
0
,
∞
)
→
[
0
,
∞
) is a continuous ad nondecreasing function such that φ
(
t
)
=
0 if and only if t
=
0.Remark 1.
In this above result, ifφ
(
t
)
=
(
1
-
k
)
t, where k
∈
(
0,1
), then we obtain condition (1) of Banach.In the recent years, Dutta and Choudhury [3], Zhang and Song [4], and Đorić [5] have given the results in (
ϕ
,
ψ
)-weak contractive mapping.The concept of 2-metric space is a natural generalization of the metric space. The concept of 2-metric spaces has been investigated initially in a series of papers (see Gahler [6–8]) and has been developed extensively by Gahler and many others. Gahler defined a 2-metric space as follows.Definition 2 (see [6]).
A 2-metric space on a setX with at least three points is nonnegative real-valued mapping d
:
X
×
X
×
X
→
R
+ satisfying the following conditions:(1)
For two distinct pointsx
,
y
∈
X, there exists a point z
∈
X such that d
(
x
,
y
,
z
)
≠
0.
(2)
d
(
x
,
y
,
z
)
=
0 if at least two of x, y, and z are equal.
(3)
d
(
x
,
y
,
z
)
=
d
(
x
,
z
,
y
)
=
d
(
y
,
x
,
z
).
(4)
d
(
x
,
y
,
z
)
≤
d
(
x
,
y
,
u
)
+
d
(
x
,
u
,
z
)
+
d
(
u
,
y
,
z
) for all x, y, z, and u in X.
The functiond is called a 2-metric for the space X and the pair (
X
,
d
) is then called a 2-metric space.Geometrically, the value of a 2-metricd
(
x
,
y
,
z
) represents the area of a triangle with vertices x, y, and c.After this, a number of fixed point theorems have been proved for 2-metric spaces by introducing compatible mappings, commuting and weakly commuting mappings. There were some generalizations of metric such as a 2-metric, aD-metric, a G-metric, a cone metric, and a complex-valued metric. Note that a 2-metric is not a continuous function of its variables, whereas an ordinary metric is. This led Dhage to introduce the notion of a D-metric in [9]. But, in 2003, Mustafa and Sims [10] demonstrated that most of the claims concerning the fundamental topological properties of D-metric spaces are incorrect. After that, in 2006, Mustafa and Sims [11] introduced the notion of G-metric spaces. Only a 2-metric space has not been known to be topologically equivalent to an ordinary metric. Then, there was no easy relationship between results obtained in 2-metric spaces and metric spaces. In particular, the fixed point theorems on 2-metric spaces and metric spaces may be unrelated easily. For more fixed point theorems on 2-metric spaces, the researchers may refer to [12–15].Throughout this paper,(
X
,
d
) is for a 2-metric space and B
(
X
) is the class of all nonempty bounded subsets of X.Definition 3 (see [15]).
A sequence{
x
n
} in (
X
,
d
) is said to be convergent to a point x
∈
X, denoted by lim
n
→
∞
⁡
x
n
=
x, if lim
n
→
∞
⁡
d
(
x
n
,
x
,
c
)
=
0 for all c
∈
X. The point x is called the limit of the sequence {
x
n
} in X.Definition 4 (see [15]).
A sequence{
x
n
} in (
X
,
d
) is said to be Cauchy sequence if lim
m
,
n
→
∞
⁡
d
(
x
m
,
x
n
,
c
)
=
0, for all c
∈
X.Definition 5 (see [15]).
The space(
X
,
d
) is said to be complete if every Cauchy sequence in X converges to a point of X.LetA, B, and C be nonempty sets in B
(
X
). Let δ
(
A
,
B
,
C
) and D
(
A
,
B
,
C
) be the functions defined by (2)
δ
A
,
B
,
C
=
sup
⁡
d
a
,
b
,
c
:
a
∈
A
,
b
∈
B
,
c
∈
C
,
D
A
,
B
,
C
=
inf
⁡
d
a
,
b
,
c
:
a
∈
A
,
b
∈
B
,
c
∈
C
.If A is a singleton set, then δ
(
A
,
B
,
C
)
=
δ
(
a
,
B
,
C
)
. In case B and C are also singleton sets, then (3)
δ
A
,
B
,
C
=
D
A
,
B
,
C
=
d
a
,
b
,
cfor every A
=
{
a
}, B
=
{
b
}, and C
=
{
c
}
. From the definition of δ, we can say that (4)
δ
A
,
B
,
C
=
δ
A
,
C
,
B
=
δ
C
,
A
,
B
=
δ
B
,
C
,
A
=
δ
C
,
B
,
A
=
δ
B
,
A
,
C
≥
0
.Also, (5)
δ
A
,
B
,
C
≤
δ
A
,
B
,
E
+
δ
A
,
E
,
C
+
δ
E
,
B
,
C
,for all A
,
B
,
C
,
E
∈
B
(
X
). Let us note that δ
(
A
,
B
,
C
)
=
0 if at least two of A, B, and C are equal singleton sets.Definition 6 (see [15]).
A sequence{
A
n
}
n
=
1
∞ of subset of a 2-metric space (
X
,
d
) is said to be convergent to a subset A of X if,(1)
givena
∈
A, there is a sequence {
a
n
} in X such that a
n
∈
A
n for n
=
0,1
,
2
,
… and lim
n
→
∞
⁡
d
(
a
n
,
a
,
c
)
=
0;
(2)
givenϵ
>
0, there exists a positive integer n
0 such that A
n
⊂
A
ϵ for n
>
n
0, where A
ϵ is the union of all open spheres with centers in A and radius ϵ
.Definition 7 (see [16]).
LetG
:
X
→
X and F
:
X
→
B
(
X
). Then, the pair {
G
,
F
} is said to be weakly commuting if G
F
(
X
)
∈
B
(
X
) and (6)
δ
F
G
x
,
G
F
x
,
C
≤
max
⁡
δ
G
x
,
F
x
,
C
,
δ
G
F
x
,
G
F
x
,
Cfor every x
∈
X, and C
∈
B
(
X
).Definition 8 (see [16]).
LetG
:
X
→
X and F
:
X
→
B
(
X
). Then, the pair {
G
,
F
} is said to be R-weakly commuting if (7)
δ
F
G
x
,
G
F
x
,
C
≤
R
·
max
⁡
δ
G
x
,
F
x
,
C
,
δ
G
F
x
,
G
F
x
,
Cfor every x
∈
X, and C
∈
B
(
X
) and R
>
0.Remark 9 (see [16]).
IfF is a single valued function, then Definitions 7 and 8 reduce to the following:(8)
δ
F
G
x
,
G
F
x
,
C
=
d
F
G
x
,
G
F
x
,
C
≤
d
G
x
,
F
x
,
C
=
δ
G
x
,
F
x
,
C
,
(9)
δ
F
G
x
,
G
F
x
,
C
=
d
F
G
x
,
G
F
x
,
C
≤
R
·
d
G
x
,
F
x
,
C
=
R
·
δ
G
x
,
F
x
,
C
,respectively.Common fixed points of Greguš type [17] have been proved by Diviccaro et al. [18], Fisher and Sessa [19], Mukherjee and Verma [20], Murthy et al. [21], and Singh et al. [14] under weaker conditions. Later, Murthy and Tas [16] generalized and extended the results of Singh et al. [14] and proved a theorem for set valued mapping in 2-metric space.In this paper, we generalize the results of Abd EL-Monsef et al. [15] and Murthy and Tas [16] by using (
ϕ
,
ψ
)-weak contraction with Greguš type condition in 2-metric spaces for set valued mapping for R-weakly commuting maps.


## 2. Main Results

LetS and T be mapping of 2-metric space (
X
,
d
) into itself and A
,
B
:
X
→
B
(
X
) are two set valued mappings satisfying the following condition:(10)
⋃
A
X
⊂
T
X
,
⋃
B
X
⊂
S
X
,for every x
,
y
∈
X, C
∈
B
(
X
), and p
>
0,(11)
ψ
δ
p
A
x
,
B
y
,
C
≤
ψ
M
x
,
y
,
C
-
ϕ
M
x
,
y
,
C
,where(12)
M
x
,
y
,
C
=
a
δ
p
S
x
,
T
y
,
C
+
1
-
a
max
⁡
δ
p
A
x
,
S
x
,
C
,
δ
p
B
y
,
T
y
,
C
,
b
D
p
S
x
,
B
y
,
C
+
c
D
p
T
y
,
A
x
,
C
,
a
∈
(
0,1
), 0
≤
b
+
c
≤
1
/
2, c
≥
0, and(1)
ψ
:
[
0
,
∞
)
→
[
0
,
∞
) is continuous monotone nondecreasing function with ψ
(
t
)
=
0 if and only if t
=
0;
(2)
ϕ
:
[
0
,
∞
)
→
[
0
,
∞
) is lower semicontinuous, monotone decreasing function with ϕ
(
t
)
=
0 if and only if t
=
0 and ϕ
(
t
)
>
0 for all t
∈
(
0
,
∞
)
.Letx
0 be an arbitrary point in X. Since ⋃
A
(
X
)
⊂
T
(
X
), then ∃ a point x
1
∈
X such that T
x
1
∈
A
x
0
=
y
0. Now again, since ⋃
B
(
X
)
⊂
S
(
X
), for the point x
1
∈
X
. we can find a point x
2
∈
X such that S
x
1
∈
B
x
0
=
y
0 and so on. Inductively, we can construct a sequence {
x
n
} in X such that (13)
T
x
n
+
1
∈
A
x
n
=
y
n
,
when
n
is  even,
S
x
n
+
1
∈
B
x
n
=
y
n
,
when
n
is  odd.Now we need to prove the following lemma for our main theorem.Lemma 10.
Let(
X
,
d
) be 2-metric space. Let S and T be self-maps of X and A
,
B
:
X
→
B
(
X
) satisfying conditions (10) and (11). Then, for every n
∈
N, one has (14)
lim
n
→
∞
⁡
δ
p
y
n
,
y
n
+
1
,
y
n
+
2
=
0
.Proof.
Since(15)
δ
p
y
2
n
+
2
,
y
2
n
+
1
,
y
2
n
=
δ
p
A
x
2
n
+
2
,
B
x
2
n
+
1
,
y
2
n
,we have(16)
ψ
δ
p
y
2
n
+
2
,
y
2
n
+
1
,
y
2
n
≤
ψ
a
δ
p
S
x
2
n
+
2
,
T
x
2
n
+
1
,
y
2
n
+
1
-
a
max
⁡
δ
p
S
x
2
n
+
2
,
A
x
2
n
+
2
,
y
2
n
,
δ
p
B
x
2
n
+
1
,
T
x
2
n
+
2
,
y
2
n
,
b
D
p
S
x
2
n
+
2
,
B
x
2
n
+
1
,
y
2
n
+
c
D
p
T
x
2
n
+
1
,
A
x
2
n
+
2
,
y
2
n
-
ϕ
a
δ
p
S
x
2
n
+
2
,
T
x
2
n
+
1
,
y
2
n
+
1
-
a
max
⁡
δ
p
A
x
2
n
+
2
,
S
x
2
n
+
2
,
y
2
n
,
δ
p
B
x
2
n
+
1
,
T
x
2
n
+
2
,
y
2
n
,
b
D
p
S
x
2
n
+
2
,
B
x
2
n
+
1
,
y
2
n
+
c
D
p
T
x
2
n
+
1
,
A
x
2
n
+
2
,
y
2
n
≤
ψ
a
δ
p
S
x
2
n
+
2
,
T
x
2
n
+
1
,
y
2
n
+
1
-
a
max
⁡
δ
p
S
x
2
n
+
2
,
A
x
2
n
+
2
,
y
2
n
,
δ
p
B
x
2
n
+
1
,
T
x
2
n
+
2
,
y
2
n
,
b
δ
p
S
x
2
n
+
2
,
B
x
2
n
+
1
,
y
2
n
+
c
δ
p
T
x
2
n
+
1
,
A
x
2
n
+
2
,
y
2
n
-
ϕ
a
δ
p
S
x
2
n
+
2
,
T
x
2
n
+
1
,
y
2
n
+
1
-
a
max
⁡
δ
p
A
x
2
n
+
2
,
S
x
2
n
+
2
,
y
2
n
,
δ
p
B
x
2
n
+
1
,
T
x
2
n
+
2
,
y
2
n
,
b
δ
p
S
x
2
n
+
2
,
B
x
2
n
+
1
,
y
2
n
+
c
δ
p
T
x
2
n
+
1
,
A
x
2
n
+
2
,
y
2
n
≤
ψ
a
δ
p
y
2
n
+
1
,
y
2
n
,
y
2
n
+
1
-
a
max
⁡
δ
p
y
2
n
+
1
,
y
2
n
+
2
,
y
2
n
,
δ
p
y
2
n
+
1
,
y
2
n
+
1
,
y
2
n
,
b
δ
p
y
2
n
+
1
,
y
2
n
+
1
,
y
2
n
+
c
δ
p
y
2
n
,
y
2
n
+
2
,
y
2
n
-
ϕ
a
δ
p
y
2
n
+
1
,
y
2
n
,
y
2
n
+
1
-
a
max
⁡
δ
p
y
2
n
+
1
,
y
2
n
+
2
,
y
2
n
,
δ
p
y
2
n
+
1
,
y
2
n
+
1
,
y
2
n
,
b
δ
p
y
2
n
+
1
,
y
2
n
+
1
,
y
2
n
+
c
δ
p
y
2
n
,
y
2
n
+
2
,
y
2
n
≤
ψ
1
-
a
δ
p
y
2
n
+
1
,
y
2
n
+
2
,
y
2
n
-
ϕ
1
-
a
δ
p
y
2
n
+
1
,
y
2
n
+
2
,
y
2
n
.Since ψ is nondecreasing function, we can write(17)
1
-
a
δ
p
y
2
n
+
1
,
y
2
n
+
2
,
y
2
n
≤
δ
p
y
2
n
+
1
,
y
2
n
+
2
,
y
2
n
⟹
ψ
1
-
a
δ
p
y
2
n
+
1
,
y
2
n
+
2
,
y
2
n
≤
ψ
δ
p
y
2
n
+
1
,
y
2
n
+
2
,
y
2
n
.Hence, we can write (18)
ψ
δ
p
y
2
n
+
2
,
y
2
n
+
1
,
y
2
n
≤
ψ
δ
p
y
2
n
+
1
,
y
2
n
+
2
,
y
2
n
-
ϕ
1
-
a
δ
p
y
2
n
+
1
,
y
2
n
+
2
,
y
2
n
,a contradiction as ϕ
(
t
)
>
0 for each t
∈
(
0
,
∞
).
Consider(19)
δ
p
y
2
n
+
3
,
y
2
n
+
2
,
y
2
n
+
1
=
δ
p
A
x
2
n
+
3
,
B
x
2
n
+
2
,
y
2
n
+
1
.We have(20)
ψ
δ
p
y
2
n
+
3
,
y
2
n
+
2
,
y
2
n
+
1
≤
ψ
a
δ
p
S
x
2
n
+
3
,
T
x
2
n
+
2
,
y
2
n
+
1
+
1
-
a
max
⁡
δ
p
A
x
2
n
+
3
,
S
x
2
n
+
3
,
y
2
n
+
1
,
δ
p
B
x
2
n
+
2
,
T
x
2
n
+
3
,
y
2
n
+
1
,
b
D
p
S
x
2
n
+
3
,
B
x
2
n
+
2
,
y
2
n
+
1
+
c
D
p
T
x
2
n
+
2
,
A
x
2
n
+
3
,
y
2
n
+
1
-
ϕ
a
δ
p
S
x
2
n
+
3
,
T
x
2
n
+
2
,
y
2
n
+
1
+
1
-
a
max
⁡
δ
p
A
x
2
n
+
3
,
S
x
2
n
+
3
,
y
2
n
+
1
,
δ
p
B
x
2
n
+
2
,
T
x
2
n
+
3
,
y
2
n
+
1
,
b
D
p
S
x
2
n
+
3
,
B
x
2
n
+
2
,
y
2
n
+
1
+
c
D
p
T
x
2
n
+
2
,
A
x
2
n
+
3
,
y
2
n
+
1
≤
ψ
a
δ
p
S
x
2
n
+
3
,
T
x
2
n
+
2
,
y
2
n
+
1
+
1
-
a
max
⁡
δ
p
A
x
2
n
+
3
,
S
x
2
n
+
3
,
y
2
n
+
1
,
δ
p
B
x
2
n
+
2
,
T
x
2
n
+
3
,
y
2
n
+
1
,
b
δ
p
S
x
2
n
+
3
,
B
x
2
n
+
2
,
y
2
n
+
1
+
c
δ
p
T
x
2
n
+
2
,
A
x
2
n
+
3
,
y
2
n
+
1
-
ϕ
a
δ
p
S
x
2
n
+
3
,
T
x
2
n
+
2
,
y
2
n
+
1
+
1
-
a
max
⁡
δ
p
A
x
2
n
+
3
,
S
x
2
n
+
3
,
y
2
n
+
1
,
δ
p
B
x
2
n
+
2
,
T
x
2
n
+
3
,
y
2
n
+
1
,
b
δ
p
S
x
2
n
+
3
,
B
x
2
n
+
2
,
y
2
n
+
1
+
c
δ
p
T
x
2
n
+
2
,
A
x
2
n
+
3
,
y
2
n
+
1
≤
ψ
a
δ
p
y
2
n
+
2
,
y
2
n
+
1
,
y
2
n
+
1
+
1
-
a
max
⁡
δ
p
y
2
n
+
2
,
y
2
n
+
3
,
y
2
n
+
1
,
δ
p
y
2
n
+
2
,
y
2
n
+
2
,
y
2
n
+
1
,
b
δ
p
y
2
n
+
2
,
y
2
n
+
2
,
y
2
n
+
1
+
c
δ
p
y
2
n
+
1
,
y
2
n
+
3
,
y
2
n
+
1
-
ϕ
a
δ
p
y
2
n
+
2
,
y
2
n
+
1
,
y
2
n
+
1
+
1
-
a
max
⁡
δ
p
y
2
n
+
2
,
y
2
n
+
3
,
y
2
n
+
1
,
δ
p
y
2
n
+
2
,
y
2
n
+
2
,
y
2
n
+
1
,
b
δ
p
y
2
n
+
2
,
y
2
n
+
2
,
y
2
n
+
1
+
c
δ
p
y
2
n
+
1
,
y
2
n
+
3
,
y
2
n
+
1
≤
ψ
1
-
a
δ
p
y
2
n
+
2
,
y
2
n
+
3
,
y
2
n
+
1
-
ϕ
1
-
a
δ
p
y
2
n
+
2
,
y
2
n
+
3
,
y
2
n
+
1
.Since ψ is nondecreasing function, (21)
1
-
a
δ
p
y
2
n
+
1
,
y
2
n
+
3
,
y
2
n
+
1
≤
δ
p
y
2
n
+
2
,
y
2
n
+
3
,
y
2
n
+
1
⟹
ψ
1
-
a
δ
p
y
2
n
+
2
,
y
2
n
+
3
,
y
2
n
+
1
≤
ψ
δ
p
y
2
n
+
2
,
y
2
n
+
3
,
y
2
n
+
1
.From the above conditions, we have(22)
ψ
δ
p
y
2
n
+
3
,
y
2
n
+
2
,
y
2
n
+
1
≤
ψ
δ
p
y
2
n
+
2
,
y
2
n
+
3
,
y
2
n
+
1
-
ϕ
1
-
a
δ
p
y
2
n
+
2
,
y
2
n
+
3
,
y
2
n
+
1
,a contradiction. Hence, we have (23)
lim
n
→
∞
⁡
δ
p
y
n
,
y
n
+
1
,
y
n
+
2
=
0
.Now we are ready to prove a common fixed point theorem by using the concept of R-weakly commuting maps theorem as follows.Theorem 11.
LetS and T be mapping of 2-metric space (
X
,
d
) into itself and A
,
B
:
X
→
B
(
X
) are two set valued mappings satisfying conditions (10), (11), and (12) and the following:(a)
S
(
X
) or T
(
X
) is a complete subspace of X.
(b)
The pair{
A
,
S
} and {
B
,
T
} are R-weakly commuting.
ThenA
,
B
,
S, and T have unique common fixed point in X.Proof.
Letx
0 be an arbitrary point in X. Since ⋃
A
(
X
)
⊂
T
(
X
), then there exists a point x
1
∈
X such that T
x
1
∈
A
x
0
=
y
0. Now again, since ⋃
B
(
X
)
⊂
S
(
X
), for the point x
1
∈
X
. we can find a point x
2
∈
X such that S
x
1
∈
B
x
0
=
y
0 and so on. Inductively, we can construct a sequence {
x
n
} in X such that(24)
T
x
n
+
1
∈
A
x
n
=
y
n
,
when
n
is  even
,
S
x
n
+
1
∈
B
x
n
=
y
n
,
when
n
is  odd
.Firstly, we have to prove that(25)
lim
n
→
∞
⁡
δ
p
y
n
,
y
n
+
1
,
C
=
0
.For this, assume V
n
=
δ
(
y
n
,
y
n
+
1
,
C
) for n
=
0,1
,
2
,
… by using (11):(26)
ψ
V
2
n
p
=
ψ
δ
p
y
2
n
,
y
2
n
+
1
,
C
=
ψ
δ
p
A
x
2
n
,
B
x
2
n
+
1
,
C
≤
ψ
M
x
2
n
,
x
2
n
+
1
,
C
-
ϕ
M
x
2
n
,
x
2
n
+
1
,
C
,where(27)
M
x
2
n
,
x
2
n
+
1
,
C
=
a
δ
p
S
x
2
n
,
T
x
2
n
+
1
,
C
+
1
-
a
max
⁡
δ
p
A
x
2
n
,
S
x
2
n
,
C
,
δ
p
B
x
2
n
+
1
,
T
x
2
n
+
1
,
C
,
b
D
p
S
x
2
n
,
B
x
2
n
+
1
,
C
+
c
D
p
T
x
2
n
+
1
,
A
x
2
n
,
C
=
a
δ
p
y
2
n
-
1
,
y
2
n
,
C
+
1
-
a
max
⁡
δ
p
y
2
n
,
y
2
n
-
1
,
C
,
δ
p
y
2
n
+
1
,
y
2
n
,
C
,
b
D
p
y
2
n
-
1
,
y
2
n
+
1
,
C
+
c
D
p
y
2
n
,
y
2
n
,
C
≤
a
δ
p
y
2
n
-
1
,
y
2
n
,
C
+
1
-
a
max
⁡
δ
p
y
2
n
,
y
2
n
-
1
,
C
,
δ
p
y
2
n
+
1
,
y
2
n
,
C
,
b
δ
p
y
2
n
-
1
,
y
2
n
+
1
,
C
+
c
δ
p
y
2
n
,
y
2
n
,
C
≤
a
δ
p
y
2
n
-
1
,
y
2
n
,
C
+
1
-
a
max
⁡
δ
p
y
2
n
,
y
2
n
-
1
,
C
,
δ
p
y
2
n
+
1
,
y
2
n
,
C
,
b
δ
p
y
2
n
-
1
,
y
2
n
+
1
,
y
2
n
+
δ
p
y
2
n
-
1
,
y
2
n
,
C
+
δ
p
y
2
n
,
y
2
n
+
1
,
C
=
a
V
2
n
-
1
p
+
1
-
a
max
⁡
V
2
n
-
1
p
,
V
2
n
p
,
b
V
2
n
-
1
p
+
V
2
n
p
.By (26), we get(28)
ψ
V
2
n
p
≤
ψ
a
V
2
n
-
1
p
+
1
-
a
max
⁡
V
2
n
-
1
p
,
V
2
n
p
,
b
V
2
n
-
1
p
+
V
2
n
p
-
ϕ
a
V
2
n
-
1
p
+
1
-
a
max
⁡
V
2
n
-
1
p
,
V
2
n
p
,
b
V
2
n
-
1
p
+
V
2
n
p
.If we take V
2
n
-
1
<
V
2
n, in the above equation by using the property of ϕ and ψ function, we can write (29)
ψ
V
2
n
p
≤
ψ
a
V
2
n
-
1
p
+
1
-
a
V
2
n
p
-
ϕ
a
V
2
n
-
1
p
+
1
-
a
V
2
n
p
.The above implies that (30)
ψ
V
2
n
p
≤
ψ
V
2
n
p
-
ϕ
V
2
n
p
,a contradiction. So we obtain(31)
V
2
n
-
1
≥
V
2
n
.By using (28) and (31) and employing the properties of ϕ and ψ function, we may write (32)
ψ
V
2
n
p
≤
ψ
a
V
2
n
-
1
p
+
1
-
a
max
⁡
V
2
n
-
1
p
,
2
b
V
2
n
-
1
p
-
ϕ
a
V
2
n
-
1
p
+
1
-
a
max
⁡
V
2
n
-
1
p
,
2
b
V
2
n
-
1
p
≤
ψ
a
V
2
n
-
1
p
+
1
-
a
V
2
n
-
1
p
-
ϕ
a
V
2
n
-
1
p
+
1
-
a
V
2
n
p
,
(33)
ψ
V
2
n
p
≤
ψ
V
2
n
-
1
p
-
ϕ
V
2
n
-
1
p
.Again, (31) implies that(34)
V
2
n
≤
V
2
n
-
1
⟹
δ
p
y
2
n
,
y
2
n
+
1
,
C
≤
δ
p
y
2
n
-
1
,
y
2
n
,
C
.Therefore, V
2
n
=
δ
p
(
y
2
n
-
1
,
y
2
n
,
C
) is a monotone decreasing sequence of nonnegative real number. There exists a nonnegative real number r
>
0 such that(35)
lim
n
→
∞
⁡
δ
p
y
2
n
-
1
,
y
2
n
,
C
=
r
.Letting limit n
→
∞ in (33), we get(36)
ψ
r
≤
ψ
r
-
ϕ
r
,a contradiction with the property of ϕ and ψ function. This implies that r
=
0. Thus, we have(37)
lim
n
→
∞
⁡
δ
p
y
2
n
-
1
,
y
2
n
,
C
=
0
.Now, repeating the above process by putting x
=
x
2
n
+
1 and y
=
x
2
n
+
2, we obtain(38)
lim
n
→
∞
⁡
δ
p
y
2
n
,
y
2
n
+
1
,
C
=
0
.Hence, for all n
≥
0, we can write(39)
lim
n
→
∞
⁡
δ
p
y
n
,
y
n
+
1
,
C
=
0
.Next, we will show that {
y
n
} is a Cauchy sequence. If, otherwise, there exists ϵ
>
0 and sequence of natural numbers {
m
(
k
)
} and {
n
(
k
)
} such that, for every natural number k, (40)
n
k
>
m
k
>
k
,
(41)
δ
p
y
m
k
,
y
n
k
,
C
≥
ϵ
,corresponding to m
(
k
), we can choose n
(
k
) to be the smallest integer such that (41) is satisfied. Then, we have(42)
δ
p
y
m
k
,
y
n
k
-
1
,
C
≤
ϵ
.Putting x
=
x
n
(
k
) and y
=
y
m
(
k
) in (11), we get(43)
ψ
δ
p
y
n
k
,
y
m
k
,
C
=
ψ
δ
p
A
x
n
k
,
B
x
m
k
,
C
≤
ψ
M
x
n
k
,
x
m
k
,
C
-
ϕ
M
x
n
k
,
x
m
k
,
C
,where(44)
M
x
n
k
,
x
m
k
,
C
=
a
δ
p
S
x
n
k
,
T
x
m
k
,
C
+
1
-
a
max
⁡
δ
p
A
x
n
k
,
S
x
n
k
,
C
,
δ
p
B
x
m
k
,
T
x
m
k
,
C
,
b
D
p
S
x
n
k
,
B
x
m
k
,
C
+
c
D
p
T
x
m
k
,
A
x
n
k
,
C
≤
a
δ
p
S
x
n
k
,
T
x
m
k
,
C
+
1
-
a
max
⁡
δ
p
A
x
n
k
,
S
x
n
k
,
C
,
δ
p
B
x
m
k
,
T
x
m
k
,
C
,
b
δ
p
S
x
n
k
,
B
x
m
k
,
C
+
c
δ
p
T
x
m
k
,
A
x
n
k
,
C
≤
a
δ
p
y
n
k
-
1
,
y
m
k
-
1
,
C
+
1
-
a
max
⁡
δ
p
y
n
k
,
y
n
k
-
1
,
C
,
δ
p
y
m
k
,
y
m
k
-
1
,
C
,
b
δ
p
y
n
k
-
1
,
y
m
k
,
C
+
c
δ
p
y
m
k
-
1
,
y
n
k
,
C
.Letting limit n
→
∞ in the above, (45)
M
x
n
k
,
x
m
k
,
C
=
a
δ
p
y
n
k
-
1
,
y
m
k
-
1
,
C
+
1
-
a
max
⁡
0,0
,
b
ϵ
+
c
δ
p
y
m
k
-
1
,
y
n
k
,
C
.Now(46)
ψ
δ
p
y
n
k
,
y
m
k
,
C
≤
ψ
a
δ
p
y
n
k
-
1
,
y
m
k
-
1
,
C
+
1
-
a
b
ϵ
+
c
δ
p
y
m
k
-
1
,
y
n
k
,
C
-
ϕ
a
δ
p
y
n
k
-
1
,
y
m
k
-
1
,
C
+
1
-
a
b
ϵ
+
c
δ
p
y
m
k
-
1
,
y
n
k
,
C
.We have to show that (47)
δ
p
y
m
k
,
y
n
k
,
C
⟶
ϵ
,
δ
p
y
m
k
-
1
,
y
n
k
-
1
,
C
⟶
ϵ
,
δ
p
y
m
k
-
1
,
y
n
k
,
C
⟶
ϵ
.Now, using properties of 2-metric space, we get (48)
δ
p
y
m
k
,
y
n
k
,
C
≤
δ
p
y
m
k
,
y
n
k
,
y
n
k
-
1
+
δ
p
y
m
k
,
y
n
k
-
1
,
C
+
δ
p
y
n
k
-
1
,
y
n
k
,
C
.Letting l
i
m
n
→
∞, we get(49)
δ
p
y
m
k
,
y
n
k
,
C
⟶
ϵ
.By using properties of 2-metric space, we can write (50)
δ
p
y
n
k
,
C
,
y
m
k
-
1
-
δ
p
y
n
k
,
C
,
y
m
k
≤
δ
p
y
m
k
-
1
,
y
m
k
,
y
n
k
+
δ
p
y
m
k
-
1
,
y
m
k
,
C
.Letting limit n
→
∞, we have (51)
lim
n
→
∞
⁡
δ
p
y
n
k
,
C
,
y
m
k
-
1
-
δ
p
y
n
k
,
C
,
y
m
k
=
0
or
δ
p
y
n
k
,
y
m
k
-
1
,
C
⟶
ϵ
.Again using properties of 2-metric space, we can write (52)
δ
p
y
m
k
,
y
n
k
-
1
,
C
≤
δ
p
y
m
k
,
y
n
k
-
1
,
y
n
k
+
δ
p
y
m
k
,
y
n
k
,
C
+
δ
p
y
n
k
,
y
n
k
-
1
,
C
.Letting limit n
→
∞, we have(53)
δ
p
y
m
k
,
y
n
k
-
1
,
C
⟶
ϵ
.Using (46), (49), (51), and (53), we have (54)
ψ
ϵ
≤
ψ
a
ϵ
+
1
-
a
b
ϵ
+
c
ϵ
-
ϕ
a
ϵ
+
1
-
a
b
ϵ
+
c
ϵ
.Since ψ is nondecreasing function, (
a
+
(
1
-
a
)
(
b
+
c
)
)
ϵ
≤
ϵ
⇒
ψ
(
(
a
+
(
1
-
a
)
(
b
+
c
)
)
ϵ
)
≤
ψ
(
ϵ
).
Therefore,(55)
ψ
ϵ
≤
ψ
ϵ
-
ϕ
a
ϵ
+
1
-
a
b
ϵ
+
c
ϵ
,a contradiction with ϕ function; hence, {
y
n
} is a Cauchy sequence.
AssumeT
(
X
) is a complete subspace X. Since the sequence {
x
n
} is Cauchy, then its subsequence T
x
2
n
+
1 is Cauchy and converges to a point z in T
(
X
). Since T
(
X
) is complete subspace of X, for some u
∈
X, (56)
T
x
2
n
+
1
⟶
z
=
T
u
.According to the construction of sequence, we can have (57)
δ
S
x
2
n
+
2
,
T
x
2
n
+
1
,
C
≤
δ
y
2
n
+
1
,
y
2
n
,
C
.Letting limit n
→
∞, (58)
lim
n
→
∞
⁡
δ
S
x
2
n
+
2
,
T
x
2
n
+
1
,
C
≤
lim
n
→
∞
⁡
δ
y
2
n
+
1
,
y
2
n
,
C
=
0
.The above implies that (59)
δ
S
x
2
n
+
2
,
T
x
2
n
+
1
,
C
=
0
.Therefore, we get(60)
lim
n
→
∂
⁡
S
x
2
n
+
2
=
lim
n
→
∂
⁡
T
x
2
n
+
1
=
z
.Similarly, (61)
δ
A
x
2
n
+
2
,
B
x
2
n
+
1
,
C
≤
δ
y
2
n
+
2
,
y
2
n
+
1
,
C
.Letting limit n
→
∞, (62)
lim
n
→
∞
⁡
δ
A
x
2
n
+
2
,
B
x
2
n
+
1
,
C
≤
lim
n
→
∞
⁡
δ
y
2
n
+
2
,
y
2
n
+
1
,
C
=
0
.Therefore, we get(63)
lim
n
→
∂
⁡
A
x
2
n
+
2
=
lim
n
→
∂
⁡
B
x
2
n
+
1
=
z
.Now we will show that u is a coincidence point of B and T. For n
=
0,1
,
2,3
,
… and putting x
=
x
2
n and y
=
u in (11) and (12), we have(64)
ψ
δ
p
A
x
2
n
,
B
u
,
C
≤
ψ
M
x
2
n
,
u
,
C
-
ϕ
M
x
2
n
,
u
,
C
,where(65)
M
x
2
n
,
u
,
C
=
a
δ
p
S
x
2
n
,
T
u
,
C
+
1
-
a
max
⁡
δ
p
A
x
2
n
,
S
x
2
n
,
C
,
δ
p
B
u
,
T
u
,
C
,
b
D
p
S
x
2
n
,
B
u
,
C
+
c
D
p
T
u
,
A
x
2
n
,
C
,
M
x
2
n
,
u
,
C
=
a
δ
p
S
x
2
n
,
T
u
,
C
+
1
-
a
max
⁡
δ
p
A
x
2
n
,
S
x
2
n
,
C
,
δ
p
B
u
,
T
u
,
C
,
b
δ
p
S
x
2
n
,
B
u
,
C
+
c
δ
p
T
u
,
A
x
2
n
,
C
.Letting limit n
→
∞ in the above equation, we have (66)
lim
n
→
∞
⁡
M
x
2
n
,
u
,
C
≤
1
-
a
max
⁡
δ
p
B
u
,
z
,
C
,
b
δ
p
z
,
B
u
,
C
.Letting limit n
→
∞ in (64), we get(67)
ψ
δ
p
z
,
B
u
,
C
≤
ψ
1
-
a
max
⁡
δ
p
B
u
,
z
,
C
,
b
δ
p
z
,
B
u
,
C
-
ϕ
1
-
a
max
⁡
δ
p
B
u
,
z
,
C
,
b
δ
p
z
,
B
u
,
C
.This implies that (68)
ψ
δ
p
z
,
B
u
,
C
≤
ψ
1
-
a
δ
p
B
u
,
z
,
C
-
ϕ
1
-
a
δ
p
B
u
,
z
,
C
.Since ψ is monotone nondecreasing function, we can write (69)
ψ
δ
p
z
,
B
u
,
C
≤
ψ
δ
p
B
u
,
z
,
C
-
ϕ
1
-
a
δ
p
B
u
,
z
,
C
,a contradiction. Hence, u is the coincidence point of B and T; that is, {
z
}
=
B
u
=
{
T
u
}
. Since ⋃
B
(
X
)
⊂
S
(
X
), for some v
∈
X, we have {
S
v
}
=
B
u
=
{
T
u
}. If A
v
≠
B
u, then, from (11) and (12), putting x
=
v and y
=
u, we get (70)
ψ
δ
p
A
v
,
B
u
,
C
≤
ψ
δ
p
v
,
u
,
C
-
ϕ
1
-
a
δ
p
v
,
u
,
C
,where (71)
M
v
,
u
,
C
=
a
δ
p
S
v
,
T
u
,
C
+
1
-
a
max
⁡
δ
p
A
v
,
S
v
,
C
,
δ
p
B
u
,
T
u
,
C
,
b
D
p
S
v
,
B
u
,
C
+
c
D
p
T
u
,
A
v
,
C
,
M
v
,
u
,
C
=
1
-
a
max
⁡
δ
p
A
v
,
S
v
,
C
,
c
δ
p
T
u
,
A
v
,
C
.Since 0
≤
b
+
c
≤
1
/
2, 0
<
a
<
1, and b
,
c
≥
0, we have (72)
ψ
δ
p
A
v
,
B
u
,
C
≤
ψ
1
-
a
δ
p
A
v
,
S
v
,
C
-
ϕ
1
-
a
δ
p
A
v
,
S
v
,
C
.Since ψ is monotone nondecreasing function, we can write (73)
ψ
δ
p
A
v
,
B
u
,
C
≤
ψ
δ
p
A
v
,
S
v
,
C
-
ϕ
1
-
a
δ
p
A
v
,
S
v
,
C
,a contradiction. Hence, A
v
=
B
u
=
{
S
v
}
=
{
T
u
}
=
z.
Since(
A
,
S
) are R-weakly commuting maps, then (74)
δ
A
S
v
,
S
A
u
,
C
≤
R
·
max
⁡
δ
A
v
,
S
v
,
C
,
δ
S
A
v
,
S
A
v
,
C
,which implies that (75)
A
S
v
=
S
A
v
⟹
A
z
=
S
z
.Again, using (11), putting x
=
z and y
=
u, we have(76)
ψ
δ
p
A
z
,
z
,
C
=
ψ
δ
p
A
z
,
B
u
,
C
≤
ψ
a
δ
p
S
z
,
T
u
,
C
+
1
-
a
max
⁡
δ
p
A
z
,
S
z
,
C
,
δ
p
B
u
,
T
u
,
C
,
b
D
p
S
z
,
B
u
,
C
+
c
D
p
T
u
,
A
z
,
C
-
ϕ
a
δ
p
S
z
,
T
u
,
C
+
1
-
a
max
⁡
δ
p
A
z
,
S
z
,
C
,
δ
p
B
u
,
T
u
,
C
,
b
D
p
S
z
,
B
u
,
C
+
c
D
p
T
u
,
A
z
,
C
≤
ψ
a
δ
p
A
z
,
z
,
C
+
1
-
a
max
⁡
0,0
,
b
δ
p
A
z
,
z
,
C
+
c
δ
p
A
z
,
z
,
C
-
ϕ
a
δ
p
A
z
,
z
,
C
+
1
-
a
max
⁡
0,0
,
b
δ
p
A
z
,
z
,
C
+
c
δ
p
A
z
,
z
,
C
,
ψ
δ
p
A
z
,
z
,
C
≤
ψ
a
+
1
-
a
b
+
c
δ
p
A
z
,
z
,
C
-
ϕ
a
+
1
-
a
b
+
c
δ
p
A
z
,
z
,
C
.Since ψ is nondecreasing function, we can write(77)
a
+
1
-
a
b
+
c
δ
p
A
z
,
z
,
C
≤
δ
p
A
z
,
z
,
C
⟹
ψ
a
+
1
-
a
b
+
c
δ
p
A
z
,
z
,
C
≤
ψ
δ
p
A
z
,
z
,
C
.This implies that (78)
ψ
δ
p
A
z
,
z
,
C
≤
ψ
δ
p
A
z
,
z
,
C
-
ϕ
a
+
1
-
a
b
+
c
δ
p
A
z
,
z
,
C
,a contradiction. Hence, we get A
z
=
{
S
z
}
=
{
z
} and z is a common fixed point of A and S.
Similarly, we can show that{
z
} is common fixed point of B and T by assuming {
B
,
T
} is a pair of R-weakly commuting maps. Hence, A
z
=
B
z
=
{
S
z
}
=
{
T
z
}
=
{
z
}.
For the uniqueness of common fixed pointz, let z
∗ be another fixed point of A, B, S, and T. By using (11), we have(79)
ψ
δ
p
A
z
,
B
z
∗
,
C
=
ψ
δ
p
z
,
z
∗
,
C
≤
ψ
a
δ
p
S
z
,
T
z
∗
,
C
+
1
-
a
max
⁡
δ
p
A
z
,
S
z
,
C
,
δ
p
B
z
∗
,
T
z
∗
,
C
,
b
D
p
S
z
,
B
z
∗
,
C
+
c
D
p
T
z
∗
,
A
z
,
C
-
ϕ
a
δ
p
S
z
,
T
z
∗
,
C
+
1
-
a
max
⁡
δ
p
A
z
,
S
z
,
C
,
δ
p
B
z
∗
,
T
z
∗
,
C
,
b
D
p
S
z
,
B
z
∗
,
C
+
c
D
p
T
z
∗
,
A
z
,
C
≤
ψ
a
δ
p
S
z
,
T
z
∗
,
C
+
1
-
a
max
⁡
δ
p
A
z
,
S
z
,
C
,
δ
p
B
z
∗
,
T
z
∗
,
C
,
b
δ
p
S
z
,
B
z
∗
,
C
+
c
δ
p
T
z
∗
,
A
z
,
C
-
ϕ
a
δ
p
S
z
,
T
z
∗
,
C
+
1
-
a
max
⁡
δ
p
A
z
,
S
z
,
C
,
δ
p
B
z
∗
,
T
z
∗
,
C
,
b
δ
p
S
z
,
B
z
∗
,
C
+
c
δ
p
T
z
∗
,
A
z
,
C
,
ψ
δ
p
z
,
z
∗
,
C
≤
ψ
a
δ
p
z
,
z
∗
,
C
+
1
-
a
max
⁡
0,0
,
b
δ
p
z
,
z
∗
,
C
+
c
δ
p
z
∗
,
z
,
C
-
ϕ
a
δ
p
z
,
z
∗
,
C
+
1
-
a
max
⁡
0,0
,
b
δ
p
z
,
z
∗
,
C
+
c
δ
p
z
∗
,
z
,
C
.Since ψ is nondecreasing function, we can write(80)
a
+
1
-
a
b
+
c
δ
p
z
,
z
∗
,
C
≤
δ
p
z
,
z
∗
,
C
⟹
ψ
a
+
1
-
a
b
+
c
δ
p
z
,
z
∗
,
C
≤
ψ
δ
p
z
,
z
∗
,
C
.This implies that (81)
ψ
δ
p
z
,
z
∗
,
C
≤
ψ
δ
p
z
,
z
∗
,
C
-
ϕ
a
+
1
-
a
b
+
c
δ
p
z
,
z
∗
,
C
,a contradiction. Hence, we have z
=
z
∗; that is, that z is unique common fixed point of A
,
B
,
S, and T in X.As an immediate consequence of the above theorem, we have the following corollaries.Corollary 12.
LetS and T be mapping of 2-metric space (
X
,
d
) into itself and A
,
B
:
X
→
B
(
X
) are two set valued mappings satisfying conditions (a) and (b) of Theorem 11 and the following conditions:(82)
⋃
A
X
⊂
T
X
,
⋃
B
X
⊂
S
X
,for every x
,
y
∈
X, C
∈
B
(
X
), and p
>
0,(83)
δ
p
A
x
,
B
y
,
C
≤
M
x
,
y
,
C
-
ϕ
M
x
,
y
,
z
,where(84)
M
x
,
y
,
z
=
a
δ
p
S
x
,
T
y
,
C
+
1
-
a
max
⁡
δ
p
A
x
,
S
x
,
C
,
δ
p
B
y
,
T
y
,
C
,
b
D
p
S
x
,
B
y
,
C
+
c
D
p
T
y
,
A
x
,
C
,
a
∈
(
0,1
), 0
≤
b
+
c
≤
1
/
2, c
≥
0, and ϕ
:
[
0
,
∞
)
→
[
0
,
∞
) is lower semicontinuous, monotone decreasing function with ϕ
(
t
)
=
0 if and only if t
=
0 and ϕ
(
t
)
>
0 for all t
∈
(
0
,
∞
)
.
ThenA
,
B
,
S, and T have unique common fixed point in X.Proof.
The proof follows from Theorem11 by taking ψ
(
t
)
=
t.Corollary 13.
LetS and T be mapping of 2-metric space (
X
,
d
) into itself and A
,
B
:
X
→
B
(
X
) are two set valued mappings satisfying conditions (a) and (b) of Theorem 11 and the following conditions:(85)
⋃
A
X
⊂
T
X
,
⋃
B
X
⊂
S
X
,for every x
,
y
∈
X, C
∈
B
(
X
), and p
>
0,(86)
δ
p
A
x
,
B
y
,
C
≤
k
M
x
,
y
,
z
,where(87)
M
x
,
y
,
z
=
a
δ
p
S
x
,
T
y
,
C
+
1
-
a
max
⁡
δ
p
A
x
,
S
x
,
C
,
δ
p
B
y
,
T
y
,
C
,
b
D
p
S
x
,
B
y
,
C
+
c
D
p
T
y
,
A
x
,
C
,
a
∈
(
0,1
), 0
≤
b
+
c
≤
1
/
2, c
≥
0, and k
∈
(
0,1
)
. Then, A
,
B
,
S, and T have unique common fixed point in X.Proof.
The proof follows from Corollary12 by taking ϕ
(
t
)
=
(
1
-
k
)
t, where k
∈
(
0,1
).



---

*Source: 195731-2015-10-20.xml*