# On Fuzzy Fixed Points for Fuzzy Maps with Generalized Weak Property

Authors: Marwan Amin Kutbi; Jamshaid Ahmad; Akbar Azam; Nawab Hussain
Journal: Journal of Applied Mathematics
 (2014)

---

## Abstract

Let(
X
,
d
) be a complex valued metric space and let S
,
T be mappings from X to a set of all fuzzy subsets of X. We present sufficient conditions for the existence of a common α-fuzzy fixed point of S and T. Our results improve and extend certain recent results in literature. Moreover, we discuss an illustrative example to highlight the realized improvements.


---

## Body

## 1. Introduction

In 1981, Heilpern [1] used the concept of fuzzy set to introduce a class of fuzzy mappings, which is a generalization of the set-valued mapping, and proved a fixed point theorem for fuzzy contraction mappings in a metric linear space. It is worth noting that the result announced by Heilpern [1] forms a fuzzy extension of the Banach contraction principle. Subsequently, several other authors have studied existence of fixed points of fuzzy mappings or in fuzzy metric spaces; for example, see the work of Azam et al. [2, 3], Bose et al. [4], Chang et al. [5], Cho and Petrot [6], Hussain et al. [7], Qiu and Shu [8], Rashwan and Ahmed [9], and Zhang [10].Recently, Azam et al. [11] introduced the concept of complex valued metric space and obtained sufficient conditions for the existence of common fixed points of a pair of mappings satisfying contractive type condition involving rational expressions. For more details on complex valued metric space we refer the reader to [12–17].In [18], Azam obtained some common fuzzy fixed points for fuzzy mappings under a rational contractive condition on a metric space in connection with the Hausdorff metric on the family of fuzzy sets.The aim of this paper is to obtain a commonα-fuzzy fixed point of a pair of fuzzy mappings S and T on a complete complex valued metric space under a generalized rational contractive condition for α-level sets. Our results generalize the results proved by Azam et al. [11, 18].


## 2. Preliminaries

LetC be the set of complex numbersandz
1
,
z
2
∈
C. Define a partial order ≾ on C as follows:
(1)
z
1
≾
z
2
iff
Re
(
z
1
)
⩽
Re
(
z
2
)
,
vv
Im
⁡
(
z
1
)
⩽
Im
⁡
(
z
2
)
.It follows that(2)
z
1
≾
z
2
,
if one of the following conditions is satisfied:(i)
Re
(
z
1
)
=
Re
(
z
2
)
,
Im
⁡
(
z
1
)
<
Im
⁡
(
z
2
),
(ii)
Re
(
z
1
)
<
Re
(
z
2
)
,
Im
⁡
(
z
1
)
=
Im
⁡
(
z
2
),
(iii)
Re
(
z
1
)
<
Re
(
z
2
)
,
Im
⁡
(
z
1
)
<
Im
⁡
(
z
2
),
(iv)
Re
(
z
1
)
=
Re
(
z
2
)
,
Im
⁡
(
z
1
)
=
Im
⁡
(
z
2
).In particular, we will write z
1
⋨
z
2 ifz
1
≠
z
2 and one of (i), (ii), and (iii) is satisfied and we will write z
1
≺
z
2 if only (iii) is satisfied. Note that
(3)
0
≾
z
1
⋨
z
2
⟹
|
z
1
|
<
|
z
2
|
,
z
1
⪯
z
2
,
z
2
≺
z
3
⟹
z
1
≺
z
3
.Definition 1.
LetX be a nonempty set. Suppose that the mapping
(4)
d
:
X
×
X
⟶
C
,
satisfies(1)
0
≾
d
(
x
,
y
)
,for all x
,
y
∈
X and d
(
x
,
y
)
=
0 if and only if x
=
y;
(2)
d
(
x
,
y
)
=
d
(
y
,
x
) for all x
,
y
∈
X;
(3)
d
(
x
,
y
)
≾
d
(
x
,
z
)
+
d
(
z
,
y
), for all x
,
y
,
z
∈
X.Then d is called a complex valued metric on X, and (
X
,
d
)is called a complex valued metric space. A point x
∈
X is called interior point of a set A
⊆
X whenever there exists 0
≺
r
∈
C such that
(5)
B
(
x
,
r
)
=
{
y
∈
X
:
d
(
x
,
y
)
≺
r
}
⊆
A
.A pointx
∈
X is called a limit point of A whenever, for every 0
≺
r
∈
C,
(6)
B
(
x
,
r
)
∩
(
A
∖
{
x
}
)
≠
ϕ
.A is called open whenever each element of A is an interior point of A. Moreover, a subset B
⊆
X is called closed whenever each limit point of B belongs to B. The family
(7)
F
=
{
B
(
x
,
r
)
:
x
∈
X
,
0
≺
r
}
is a subbasis for a Hausdorff topology τ on X.Letx
nbe a sequence in Xand x
∈
X. If for every c
∈
C with 0
≺
c there is n
0
∈
N such that, for all n
>
n
0
,
d
(
x
n
,
x
)
≺
c, then {
x
n
} is said to be convergent, {
x
n
}converges to x, and xis the limit point of{
x
n
}. We denote this by lim
⁡
n
→
∞
x
n
=
x, or x
n
→
x
,as n
→
∞. If for every c
∈
C with 0
≺
cthere is n
0
∈
N such that, for all n
>
n
0
,
d
(
x
n
,
x
n
+
m
)
≺
c, where m
∈
N, then {
x
n
}is called a Cauchy sequence in (
X
,
d
). If every Cauchy sequence is convergent in (
X
,
d
), then (
X
,
d
) is called a complete complex valued metric space. We require the following lemmas.Lemma 2 (see [11]).
Let(
X
,
d
)be a complex valued metric space and let {
x
n
} be a sequence in X. Then {
x
n
} converges to x if and only if |
d
(
x
n
,
x
)
|
→
0 as n
→
∞.Lemma 3 (see [11]).
Let(
X
,
d
)be a complex valued metric space and let {
x
n
} be a sequence in X. Then {
x
n
} is a Cauchy sequence if and only if |
d
(
x
n
,
x
n
+
m
)
|
→
0 as n
→
∞
,where m
∈
N.A fuzzy set inX is a function with domain X and values in [
0,1
]; I
X is the collection of all fuzzy sets in X. If A is a fuzzy set and x
∈
X, then the function values A
(
x
) are called the grade of membership of x in A. The α-level set of A is denoted by [
A
]
α and is defined as follows:
(8)
[
A
]
α
=
{
x
:
A
(
x
)
≥
α
}
if
α
∈
(
0,1
]
,
[
A
]
0
=
{
x
:
A
(
x
)
>
0
}
.
¯HereB
¯ denotes the closure of the set B. Let F
(
X
) be the collection of all fuzzy sets in a metric space X. For A
,
B
∈
F
(
X
)
,
A
⊂
B means A
(
x
)
≤
B
(
x
) for each x
∈
X. We denote the fuzzy set χ
{
x
} by {
x
} unless and until it is stated, where χ
{
A
} is the characteristic function of the crisp set A. A fuzzy setA in a metric linear space V is said to be an approximate quantity if and only if [
A
]
α is compact and convex in V for each α
∈
[
0,1
] andsup
⁡
x
∈
V
A
(
x
)
=
1. The collection of all approximate quantitiesinVis denoted by W
(
V
).Definition 4.
LetX be a nonempty set and let (
Y
,
d
) be a complex valued metric space. A mapping T is called fuzzy mapping if T is a mapping from X into (
Y
). A fuzzy mapping T is a fuzzy subset on X
×
Ywith membership function T
(
x
)
(
y
). The function T
(
x
)
(
y
)is the grade of membership of y in T
(
x
).Definition 5.
Let(
X
,
d
) be a complex valued metric space and let S
,
T be fuzzy mappings from X into (
X
). A point z
∈
X is called a fuzzy fixed point of  T if z
∈
[
T
z
]
α, for some α
∈
[
0,1
]. The point z
∈
X is called a common fuzzy fixed point of S and T if z
∈
[
S
z
]
α
∩
[
T
z
]
α for some α
∈
[
0,1
]. When α
=
1
,
z is called a common fixed point of fuzzy mappings.


## 3. Main Result

Let(
X
,
d
) be a complex valued metric space. We denote the family of all nonempty, closed and bounded subsets of a complex valued metric space X by C
B
(
X
).From now on, we denotes
(
z
1
)
=
{
z
2
∈
C
:
z
1
⪯
z
2
} for z
1
∈
C and s
(
a
,
B
)
=
∪
b
∈
B
s
(
d
(
a
,
b
)
)
=
∪
b
∈
B
{
z
∈
C
:
d
(
a
,
b
)
⪯
z
} for a
∈
X and B
∈
C
B
(
X
).ForA
,
B
∈
C
B
(
X
), we denote
(9)
s
(
A
,
B
)
=
(
⋂
a
∈
A
s
(
a
,
B
)
)
∩
(
⋂
b
∈
B
s
(
b
,
A
)
)
.Lemma 6.
Let(
X
,
d
) be a complex valued metric space.(i)
Letp
,
q
∈
C. If p
⪯
q, then s
(
q
)
⊂
s
(
p
).
(ii)
Letx
∈
X and A
∈
N
(
X
). If θ
∈
s
(
x
,
A
)
, then x
∈
A.
(iii)
Letq
∈
C and let A
,
B
∈
C
B
(
X
) and a
∈
A. If q
∈
s
(
A
,
B
), then q
∈
s
(
a
,
B
) for all a
∈
A or q
∈
s
(
A
,
b
) for all b
∈
B.Remark 7.
If(
X
,
d
) is a metric space, for A
,
B
∈
C
B
(
X
)
,
H
(
A
,
B
)
=
inf
⁡
s
(
A
,
B
) is the Hausdorff distance induced by the metric d.Let(
X
,
d
) be a complex valued metric space and C
(
X
) be a collection of nonempty closed subsets of X. Let T
:
X
→
C
B
(
X
)be a multivalued map. For x
∈
X and A
∈
C
B
(
X
), define
(10)
W
x
(
A
)
=
{
d
(
x
,
a
)
:
a
∈
A
}
.
Thus for x
,
y
∈
X
(11)
W
x
(
T
y
)
=
{
d
(
x
,
u
)
:
u
∈
T
y
}
.Definition 8.
Let(
X
,
d
) be a complex valued metric space. A subset A of X is called bounded from below if there exists some z
∈
X, such that z
⪯
a for all a
∈
A.Definition 9.
Let(
X
,
d
) be a complex valued metric space. A multivalued mapping F
:
X
→
2
C is called bounded from below if for each x
∈
X there exists z
x
∈
C such that
(12)
z
x
⪯
u
,
for all u
∈
F
x.Definition 10.
Let(
X
,
d
) be a complex valued metric space. The fuzzy mapping T
:
X
→
F
(
X
) is said to have lower bound property (l.b property) on (
X
,
d
), if, for any x
∈
X associated with some α
∈
(
0,1
], the multivalued mapping F
x
:
X
→
2
C defined by
(13)
F
x
(
y
)
=
W
x
(
[
T
y
]
α
)
is bounded from below. That is, for x
,
y
∈
X there exists an element l
x
(
[
T
y
]
α
)
∈
C such that
(14)
l
x
(
[
T
y
]
α
)
⪯
u
,
for all u
∈
W
x
(
[
T
y
]
α
), where l
x
(
[
T
y
]
α
) is called lower bound of T associated with (
x
,
y
).Definition 11.
Let(
X
,
d
) be a complex valued metric space. The fuzzy mapping T
:
X
→
F
(
X
) is said to have greatest lower bound property (g.l.b property) on (
X
,
d
), if for any x
∈
X and any α
∈
(
0,1
], greatest lower bound of W
x
(
[
T
y
]
α
) exists in C for all y
∈
X. One denotes d
(
x
,
[
T
y
]
α
) by the g.l.b of W
x
(
[
T
y
]
α
). That is,
(15)
d
(
x
,
[
T
y
]
α
)
=
inf
⁡
{
d
(
x
,
u
)
:
u
∈
[
T
y
]
α
}
.


### 3.1. Banach Type Fuzzy Fixed Point Result

Theorem 12.
Let(
X
,
d
) be a complete complex valued metric space and let S
,
T be fuzzy mappings from X into F
(
X
). Assume that there exists some α
∈
(
0,1
], such that, for each x
∈
X
,
[
S
x
]
α and [
T
x
]
α are nonempty closed bounded subsets of X; greatest lower bound of W
x
(
[
T
y
]
α
)
,
W
x
(
[
S
y
]
α
)exists in C for all y
∈
X and
(16)
ζ
d
(
x
,
y
)
+
κ
d
(
x
,
[
S
x
]
α
)
d
(
y
,
[
T
y
]
α
)
+
ς
d
(
y
,
[
S
x
]
α
)
d
(
x
,
[
T
y
]
α
)
1
+
d
(
x
,
y
)
∈
s
(
[
S
x
]
α
,
[
T
y
]
α
)
,
for all x
,
y
∈
X, where ζ
,
κ
,
ς are nonnegative real numbers with ζ
+
κ
+
ς
<
1. Then there exists some u
∈
[
S
u
]
α
∩
[
T
u
]
α.Proof.
Letx
0 be an arbitrary point in X. By assumption, we can find x
1
∈
[
S
x
0
]
α. So, we have
(17)
ζ
d
(
x
0
,
x
1
)
+
(
(
1
+
d
(
x
0
,
x
1
)
)
-
1
(
κ
d
(
x
0
,
[
S
x
0
]
α
)
d
(
x
1
,
[
T
x
1
]
α
)
v
v
v

+

ς
d
(
x
1
,
[
S
x
0
]
α
)
d
(
x
0
,
[
T
x
1
]
α
)
)
×
(
1
+
d
(
x
0
,
x
1
)
)
-
1
)
∈
s
(
[
S
x
0
]
α
,
[
T
x
1
]
α
)
.
By Lemma 6(iii), we have
(18)
ζ
d
(
x
0
,
x
1
)
+
(
(
1
+
d
(
x
0
,
x
1
)
)
-
1
(
κ
d
(
x
0
,
[
S
x
0
]
α
)
d
(
x
1
,
[
T
x
1
]
α
)
vvv
+
ς
d
(
x
1
,
[
S
x
0
]
α
)
d
(
x
0
,
[
T
x
1
]
α
)
)
v
×
(
1
+
d
(
x
0
,
x
1
)
)
-
1
)
∈
s
(
x
1
,
[
T
x
1
]
α
)
.
By definition there exists some x
2
∈
[
T
x
1
]
α, such that
(19)
ζ
d
(
x
0
,
x
1
)
+
(
(
1
+
d
(
x
0
,
x
1
)
)
-
1
(
κ
d
(
x
0
,
[
S
x
0
]
α
)
d
(
x
1
,
[
T
x
1
]
α
)
v
v

+

ς
d
(
x
1
,
[
S
x
0
]
α
)
d
(
x
0
,
[
T
x
1
]
α
)
)
v
×
(
1
+
d
(
x
0
,
x
1
)
)
-
1
)
∈
s
(
d
(
x
1
,
x
2
)
)
.
That is,
(20)
d
(
x
1
,
x
2
)
⪯
ζ
d
(
x
0
,
x
1
)
+
(
(
1
+
d
(
x
0
,
x
1
)
)
-
1
(
κ
d
(
x
0
,
[
S
x
0
]
α
)
d
(
x
1
,
[
T
x
1
]
α
)
v
v

+

ς
d
(
x
1
,
[
S
x
0
]
α
)
d
(
x
0
,
[
T
x
1
]
α
)
)
×
(
1
+
d
(
x
0
,
x
1
)
)
-
1
)
.
By the meaning of W
x
(
[
T
y
]
α
) and W
x
(
[
S
y
]
α
) for x
,
y
∈
X, we get
(21)
d
(
x
1
,
x
2
)
⪯
ζ
d
(
x
0
,
x
1
)
+
κ
d
(
x
0
,
x
1
)
d
(
x
1
,
x
2
)
+
ς
d
(
x
1
,
x
1
)
d
(
x
0
,
x
2
)
1
+
d
(
x
0
,
x
1
)
=
ζ
d
(
x
0
,
x
1
)
+
κ
d
(
x
0
,
x
1
)
d
(
x
1
,
x
2
)
1
+
d
(
x
0
,
x
1
)
,
which implies that
(22)
|
d
(
x
1
,
x
2
)
|
≤
ζ
|
d
(
x
0
,
x
1
)
|
+
κ
|
d
(
x
0
,
x
1
)
|
|
d
(
x
1
,
x
2
)
|
|
1
+
d
(
x
0
,
x
1
)
|
=
ζ
|
d
(
x
0
,
x
1
)
|
+
κ
|
d
(
x
1
,
x
2
)
|
|
d
(
x
0
,
x
1
)
1
+
d
(
x
0
,
x
1
)
|
,
|
d
(
x
1
,
x
2
)
|
≤
ζ
|
d
(
x
0
,
x
1
)
|
+
κ
|
d
(
x
1
,
x
2
)
|
,
(
1
-
κ
)
|
d
(
x
1
,
x
2
)
|
≤
ζ
|
d
(
x
0
,
x
1
)
|
,
where
(23)
h
=
ζ
1
-
κ
<
1
.
Inductively, we can construct a sequence {
x
n
} in X such that, for n
=
0,1
,
2
,
…,
(24)
|
d
(
x
n
,
x
n
+
1
)
|
≤
h
n
|
d
(
x
0
,
x
1
)
|
,
with h
=
ζ
/
(
1
-
κ
)
<
1, for x
2
n
+
1
∈
[
S
x
2
n
]
α and x
2
n
+
2
∈
[
T
x
2
n
+
1
]
α.
Now form
>
n, we get
(25)
|
d
(
x
n
,
x
m
)
|
≤
|
d
(
x
n
,
x
n
+
1
)
|
+
|
d
(
x
n
+
1
,
x
n
+
2
)
|
+
⋯
+
|
d
(
x
m
-
1
,
x
m
)
|
≤
[
h
n
+
h
n
+
1
+
⋯
+
h
m
-
1
]
|
d
(
x
0
,
x
1
)
|
≤
[
h
n
1
-
h
]
|
d
(
x
0
,
x
1
)
|
,
and so
(26)
|
d
(
x
n
,
x
m
)
|
≤
h
n
1
-
h
|
d
(
x
0
,
x
1
)
|
⟶
0
,
as
m
,
n
⟶
∞
.
This implies that {
x
n
} is a Cauchy sequence in X. Since X is complete, so there exists u
∈
X such that x
n
→
u as n
→
∞
. We now show that u
∈
[
T
u
]
α and u
∈
[
S
u
]
α. From (16), we have
(27)
ζ
d
(
x
2
k
,
u
)
+
(
×
(
1
+
d
(
x
2
k
,
u
)
)
-
1
(
κ
d
(
x
2
k
,
[
S
x
2
k
]
α
)
d
(
u
,
[
T
u
]
α
)
vvx
+
ς
d
(
u
,
[
S
x
2
k
]
α
)
d
(
x
2
k
,
[
T
u
]
α
)
)
v
×
(
1
+
d
(
x
2
k
,
u
)
)
-
1
)
∈
s
(
[
S
x
2
k
]
α
,
[
T
u
]
α
)
.
By Lemma 6(iii), we have
(28)
ζ
d
(
x
2
k
,
u
)
+
(
(
1
+
d
(
x
2
k
,
u
)
)
-
1
(
κ
d
(
x
2
k
,
[
S
x
2
k
]
α
)
d
(
u
,
[
T
u
]
α
)
v
v
v
+
ς
d
(
u
,
[
S
x
2
k
]
α
)
d
(
x
2
k
,
[
T
u
]
α
)
)
v
×
(
1
+
d
(
x
2
k
,
u
)
)
-
1
)
∈
s
(
x
2
k
+
1
,
[
T
u
]
α
)
.
By definition there exists some u
k
∈
[
T
u
]
α such that
(29)
ζ
d
(
x
2
k
,
u
)
+
(
(
1
+
d
(
x
2
k
,
u
)
)
-
1
(
κ
d
(
x
2
k
,
[
S
x
2
k
]
α
)
d
(
u
,
[
T
u
]
α
)
v
v
v
+
ς
d
(
u
,
[
S
x
2
k
]
α
)
d
(
x
2
k
,
[
T
u
]
α
)
)
v
×
(
1
+
d
(
x
2
k
,
u
)
)
-
1
)
∈
s
(
d
(
x
2
k
+
1
,
u
k
)
)
.
That is,
(30)
d
(
x
2
k
+
1
,
u
k
)
⪯
ζ
d
(
x
2
k
,
u
)

+

(
(
1
+
d
(
x
2
k
,
u
)
)
-
1
(
κ
d
(
x
2
k
,
[
S
x
2
k
]
α
)
d
(
u
,
[
T
u
]
α
)
.
v
+
ς
d
(
u
,
[
S
x
2
k
]
α
)
d
(
x
2
k
,
[
T
u
]
α
)
)
v
×
(
1
+
d
(
x
2
k
,
u
)
)
-
1
)
.
By the meaning of W
x
(
[
T
y
]
α
) and W
x
(
[
S
y
]
α
) for x
,
y
∈
X, we get
(31)
d
(
x
2
k
+
1
,
u
k
)
⪯
ζ
d
(
x
2
k
,
u
)
+
κ
d
(
x
2
k
,
x
2
k
+
1
)
d
(
u
,
u
k
)
+
ς
d
(
u
,
x
2
k
+
1
)
d
(
x
2
k
,
u
k
)
1
+
d
(
x
2
k
u
)
.
Since by triangle inequality, we get
(32)
d
(
u
,
u
k
)
⪯
d
(
u
,
x
2
k
+
1
)
+
d
(
x
2
k
+
1
,
u
k
)
.
So using (31) in (32), we get
(33)
d
(
u
,
u
k
)
⪯
d
(
u
,
x
2
k
+
1
)
+
ζ
d
(
u
,
x
2
k
+
1
)
+
κ
d
(
x
2
k
,
x
2
k
+
1
)
d
(
u
,
u
k
)
+
ς
d
(
u
,
x
2
k
+
1
)
d
(
x
2
k
,
u
k
)
1
+
d
(
x
2
k
,
u
)
,
|
d
(
u
,
u
k
)
|
≤
|
d
(
u
,
x
2
k
+
1
)
|
+
ζ
|
d
(
u
,
x
2
k
+
1
)
|
+
κ
|
d
(
x
2
k
,
x
2
k
+
1
)
|
|
d
(
u
,
u
k
)
|

+

ς
|
d
(
u
,
x
2
k
+
1
)
|
|
d
(
x
2
k
,
u
k
)
|
|
1
+
d
(
x
2
k
,
u
)
|
.
Taking the limit as k
→
∞, we get |
d
(
u
,
u
k
)
|
→
0 as k
→
∞
. By Lemma 2 [11], we have u
k
→
u as k
→
∞
. Since [
T
u
]
α is closed, so u
∈
[
T
u
]
α. Similarly, it follows that u
∈
[
S
u
]
α. ThusS and T have a common fuzzy fixed point.By settingς
=
0 in Theorem 12, we get the following corollary.Corollary 13.
Let(
X
,
d
) be a complete complex valued metric space and let S
,
T be fuzzy mappings from X into F
(
X
). Assume that there exists some α
∈
(
0,1
], such that, for each x
∈
X,[
S
x
]
α and [
T
x
]
α are nonempty closed bounded subsets of X; greatest lower bound of W
x
(
[
T
y
]
α
)
,
W
x
(
[
S
y
]
α
)exists in C for all y
∈
Xand
(34)
ζ
d
(
x
,
y
)
+
κ
d
(
x
,
[
S
x
]
α
)
d
(
y
,
[
T
y
]
α
)
1
+
d
(
x
,
y
)
∈
s
(
[
S
x
]
α
,
[
T
y
]
α
)
,
for all x
,
y
∈
X, where ζ and κ are nonnegative real numbers with ζ
+
κ
<
1. Then there exists some u
∈
[
S
u
]
α
∩
[
T
u
]
α.By settingS
=
T in Theorem 12, we get the following corollary.Corollary 14.
Let(
X
,
d
) be a complete complex valued metric space and let T be fuzzy mapping from X into F
(
X
). Assume that there exists some α
∈
(
0,1
], such that, for each x
∈
X,[
T
x
]
α is nonempty closed bounded subset of X; greatest lower bound of W
x
(
[
T
y
]
α
)exists in C for all y
∈
Xand
(35)
ζ
d
(
x
,
y
)
+
κ
d
(
x
,
[
T
x
]
α
)
d
(
y
,
[
T
y
]
α
)
+
ς
d
(
y
,
[
T
x
]
α
)
d
(
x
,
[
T
y
]
α
)
1
+
d
(
x
,
y
)
∈
s
(
[
T
x
]
α
,
[
T
y
]
α
)
,
for all x
,
y
∈
X, where ζ
,
κ, and ς are nonnegative real numbers with ζ
+
κ
+
ς
<
1. Then there exists some u
∈
[
T
u
]
α.By Definition11, one can have the following corollaries easily from Theorem 12.Corollary 15.
Let(
X
,
d
) be a complete complex valued metric space and let S
,
T be fuzzy mappings from X into F
(
X
) with g.l.b property such that, for each x
,
y
∈
X and α
∈
(
0,1
] and [
S
x
]
α
,
[
T
y
]
α are nonempty closed bounded subsets of X and
(36)
ζ
d
(
x
,
y
)
+
κ
d
(
x
,
[
S
x
]
α
)
d
(
y
,
[
T
y
]
α
)
+
ς
d
(
y
,
[
S
x
]
α
)
d
(
x
,
[
T
y
]
α
)
1
+
d
(
x
,
y
)
∈
s
(
[
S
x
]
α
,
[
T
y
]
α
)
,
for all x
,
y
∈
X, and ζ
,
κ, and ς are nonnegative real numbers with ζ
+
κ
+
ς
<
1. Then there exists some u
∈
[
S
u
]
α
∩
[
T
u
]
α.Corollary 16.
Let(
X
,
d
) be a complete complex valued metric space and let S
,
T be fuzzy mappings from X into F
(
X
) with g.l.b property such that, for each x
,
y
∈
X and α
∈
(
0,1
] and [
S
x
]
α
,
[
T
y
]
α are nonempty closed bounded subsets of X and
(37)
ζ
d
(
x
,
y
)
+
κ
d
(
x
,
[
S
x
]
α
)
d
(
y
,
[
T
y
]
α
)
1
+
d
(
x
,
y
)
∈
s
(
[
S
x
]
α
,
[
T
y
]
α
)
,
for all x
,
y
∈
X, and ζ and κ are nonnegative real numbers with ζ
+
κ
<
1. Then there exists some u
∈
[
S
u
]
α
∩
[
T
u
]
α.Corollary 17.
Let(
X
,
d
) be a complete complex valued metric space and let T be fuzzy mapping from X into F
(
X
) with g.l.b property such that, for each x
,
y
∈
X and α
∈
(
0,1
]
,
[
T
y
]
α is nonempty closed bounded subset of X and
(38)
ζ
d
(
x
,
y
)
+
κ
d
(
x
,
[
T
x
]
α
)
d
(
y
,
[
T
y
]
α
)
+
ς
d
(
y
,
[
T
x
]
α
)
d
(
x
,
[
T
y
]
α
)
1
+
d
(
x
,
y
)
∈
s
(
[
T
x
]
α
,
[
T
y
]
α
)
,
for all x
,
y
∈
X, and ζ
,
κ, and ς are nonnegative real numbers with ζ
+
κ
+
ς
<
1. Then there exists some u
∈
[
T
u
]
α.Corollary 18 (see [19]).
Let(
X
,
d
) be a complete complex valued metric space and let F
,
G
:
X
→
C
B
(
X
) be multivalued mappings with g.l.b property such that
(39)
ζ
d
(
x
,
y
)
+
κ
d
(
x
,
F
x
)
d
(
y
,
G
y
)
+
ς
d
(
y
,
F
x
)
d
(
x
,
G
y
)
1
+
d
(
x
,
y
)
∈
s
(
F
x
,
G
y
)
,
for all x
,
y
∈
X, where ζ
,
κ, and ς are nonnegative real numbers with ζ
+
κ
+
ς
<
1. Then there exists some u
∈
F
u
∩
T
u.Proof.
Consider a pair of fuzzy mappingsS
,
T
:
X
→
F
(
X
) defined by(40)
S
(
x
)
(
t
)
=
{
α
,
t
∈
F
x
0
,
t
∉
F
x
,
T
(
x
)
(
t
)
=
{
α
,
t
∈
G
x
0
,
t
∉
G
x
,
where α
∈
(
0,1
]. Then
(41)
[
S
x
]
α
=
{
t
:
S
(
x
)
(
t
)
≥
α
}
=
F
x
,
[
T
x
]
α
=
G
x
.
Thus, Theorem 12 can be applied to obtain u
∈
X such that
(42)
u
∈
[
S
u
]
α
∩
[
T
u
]
α
=
F
u
∩
G
u
.


### 3.2. Kannan Type Fuzzy Fixed Point Result

Theorem 19.
Let(
X
,
d
) be a complete complex valued metric space and let S
,
T be fuzzy mappings from X into F
(
X
). Assume that there exists some α
∈
(
0,1
], such that, for each x
∈
X
,
[
S
x
]
α and [
T
x
]
α are nonempty closed bounded subsets of X; greatest lower bound of W
x
(
[
T
y
]
α
),W
x
(
[
S
y
]
α
)exists in C for all y
∈
Xand
(43)
β
d
(
x
,
[
S
x
]
α
)
+
γ
d
(
y
,
[
T
y
]
α
)
+
η
d
(
x
,
[
S
x
]
α
)
d
(
y
,
[
T
y
]
α
)
1
+
d
(
x
,
y
)
∈
s
(
[
S
x
]
α
,
[
T
y
]
α
)
,
for all x
,
y
∈
X and nonnegative real numbers β
,
γ, and η with β
+
γ
+
η
<
1. Then there exists some v
∈
[
S
v
]
α
∩
[
T
v
]
α.Proof.
Letx
0 be an arbitrary point in X. By assumption, we can find x
1
∈
[
S
x
0
]
α. So, we have
(44)
β
d
(
x
0
,
[
S
x
0
]
α
)
+
γ
d
(
x
1
,
[
T
x
1
]
α
)
+
η
d
(
x
0
,
[
S
x
0
]
α
)
d
(
x
1
,
[
T
x
1
]
α
)
1
+
d
(
x
0
,
x
1
)
∈
s
(
[
S
x
0
]
α
,
[
T
x
1
]
α
)
.
By Lemma 6(iii), we have
(45)
β
d
(
x
0
,
[
S
x
0
]
α
)
+
γ
d
(
x
1
,
[
T
x
1
]
α
)
+
η
d
(
x
0
,
[
S
x
0
]
α
)
d
(
x
1
,
[
T
x
1
]
α
)
1
+
d
(
x
0
,
x
1
)
∈
s
(
x
1
,
[
T
x
1
]
α
)
.
By definition there exists somex
2
∈
[
T
x
1
]
α, such that
(46)
β
d
(
x
0
,
[
S
x
0
]
α
)
+
γ
d
(
x
1
,
[
T
x
1
]
α
)
+
η
d
(
x
0
,
[
S
x
0
]
α
)
d
(
x
1
,
[
T
x
1
]
α
)
1
+
d
(
x
0
,
x
1
)
∈
s
(
d
(
x
1
,
x
2
)
)
.
That is,(47)
d
(
x
1
,
x
2
)
⪯
β
d
(
x
0
,
[
S
x
0
]
α
)
+
γ
d
(
x
1
,
[
T
x
1
]
α
)
+
η
d
(
x
0
,
[
S
x
0
]
α
)
d
(
x
1
,
[
T
x
1
]
α
)
1
+
d
(
x
0
,
x
1
)
.
By the meaning of W
x
(
[
T
y
]
α
) and W
x
(
[
S
y
]
α
) for x
,
y
∈
X, we get
(48)
d
(
x
1
,
x
2
)
⪯
β
d
(
x
0
,
x
1
)
+
γ
d
(
x
1
,
x
2
)
+
η
d
(
x
0
,
x
1
)
d
(
x
1
,
x
2
)
1
+
d
(
x
0
,
x
1
)
,
which implies that
(49)
|
d
(
x
1
,
x
2
)
|
≤
β
|
d
(
x
0
,
x
1
)
|
+
γ
|
d
(
x
1
,
x
2
)
|
+
η
|
d
(
x
0
,
x
1
)
|
|
d
(
x
1
,
x
2
)
|
|
1
+
d
(
x
0
,
x
1
)
|
.
Thus(50)
|
d
(
x
1
,
x
2
)
|
≤
l
|
d
(
x
0
,
x
1
)
|
,
wherel
=
β
/
(
1
-
γ
-
η
)
<
1. Inductively, we can construct a sequence {
x
n
} in X such that, for n
=
0,1
,
2
,
…,
(51)
|
d
(
x
n
,
x
n
+
1
)
|
≤
l
n
|
d
(
x
0
,
x
1
)
|
,
with l
=
β
/
(
1
-
γ
-
η
)
<
1, for x
2
n
+
1
∈
[
S
x
2
n
]
α and x
2
n
+
2
∈
[
T
x
2
n
+
1
]. Now for m
>
n, we get
(52)
|
d
(
x
n
,
x
m
)
|
≤
|
d
(
x
n
,
x
n
+
1
)
|
+
|
d
(
x
n
+
1
,
x
n
+
2
)
|
+
⋯
+
|
d
(
x
m
-
1
,
x
m
)
|
≤
[
l
n
+
l
n
+
1
+
⋯
+
l
m
-
1
]
|
d
(
x
0
,
x
1
)
|
≤
[
l
n
1
-
l
]
|
d
(
x
0
,
x
1
)
|
,
and so
(53)
|
d
(
x
n
,
x
m
)
|
≤
l
n
1
-
l
|
d
(
x
0
,
x
1
)
|
⟶
0
as
m
,
n
⟶
∞
.
This implies that {
x
n
} is a Cauchy sequence in X. Since X is complete, there exists υ
∈
X such that x
n
→
υ as n
→
∞. We now show that υ
∈
[
T
υ
]
α and υ
∈
[
S
υ
]
α. From (43), we get
(54)
β
d
(
x
2
n
,
[
S
x
2
n
]
α
)
+
γ
d
(
υ
,
[
T
υ
]
α
)
+
η
d
(
x
2
n
,
[
S
x
2
n
]
α
)
d
(
υ
,
[
T
υ
]
α
)
1
+
d
(
x
2
n
,
υ
)
∈
s
(
[
S
x
2
n
]
α
,
[
T
υ
]
α
)
.
By Lemma6 (iii), we have
(55)
β
d
(
x
2
n
,
[
S
x
2
n
]
α
)
+
γ
d
(
υ
,
[
T
υ
]
α
)
+
η
d
(
x
2
n
,
[
S
x
2
n
]
α
)
d
(
υ
,
[
T
υ
]
α
)
1
+
d
(
x
2
n
,
υ
)
∈
s
(
x
2
n
+
1
,
[
T
υ
]
α
)
.
By definition there exists someυ
n
∈
[
T
υ
]
α such that
(56)
β
d
(
x
2
n
,
[
S
x
2
n
]
α
)
+
γ
d
(
υ
,
[
T
υ
]
α
)
+
η
d
(
x
2
n
,
[
S
x
2
n
]
α
)
d
(
υ
,
[
T
υ
]
α
)
1
+
d
(
x
2
n
,
υ
)
∈
s
(
d
(
x
2
n
+
1
,
υ
n
)
)
.
That is,
(57)
d
(
x
2
n
+
1
,
υ
n
)
⪯
β
d
(
x
2
n
,
[
S
x
2
n
]
α
)
+
γ
d
(
υ
,
[
T
υ
]
α
)
vvvvvvvv
+
η
d
(
x
2
n
,
[
S
x
2
n
]
α
)
d
(
υ
,
[
T
υ
]
α
)
1
+
d
(
x
2
n
,
υ
)
.
By the meaning of W
x
(
[
T
y
]
α
) and W
x
(
[
S
y
]
α
) for x
,
y
∈
X, we get
(58)
d
(
x
2
n
+
1
,
υ
n
)
⪯
β
d
(
x
2
n
,
x
2
n
+
1
)
+
γ
d
(
υ
,
υ
n
)
+
η
d
(
x
2
n
,
x
2
n
+
1
)
d
(
υ
,
υ
n
)
1
+
d
(
x
2
n
,
υ
)
.
Now by using (58) and the triangular inequality, we get
(59)
d
(
υ
,
υ
n
)
⪯
d
(
υ
,
x
2
n
+
1
)
+
d
(
x
2
n
+
1
,
υ
n
)
⪯
d
(
υ
,
x
2
n
+
1
)
+
β
d
(
x
2
n
,
x
2
n
+
1
)
+
γ
d
(
υ
,
υ
n
)
+
η
d
(
x
2
n
,
x
2
n
+
1
)
d
(
υ
,
υ
n
)
1
+
d
(
x
2
n
,
υ
)
,
which implies that
(60)
(
1
-
γ
)
|
d
(
υ
,
υ
n
)
|
≤
|
d
(
υ
,
x
2
n
+
1
)
|
vv
+
β
|
d
(
x
2
n
,
x
2
n
+
1
)
|
vv
+
η
|
d
(
x
2
n
,
x
2
n
+
1
)
d
(
υ
,
υ
n
)
1
+
d
(
x
2
n
,
υ
)
|
,
|
d
(
υ
,
υ
n
)
|
≤
1
(
1
-
γ
)
|
d
(
υ
,
x
2
n
+
1
)
|
vv
+
β
(
1
-
γ
)
|
d
(
x
2
n
,
x
2
n
+
1
)
|
vv
+
η
(
1
-
γ
)
|
d
(
x
2
n
,
x
2
n
+
1
)
|
|
d
(
υ
,
υ
n
)
|
|
1
+
d
(
x
2
n
,
υ
)
|
.
By letting n
→
∞ in above inequality, we get
(61)
|
d
(
υ
,
υ
n
)
|
⟶
0
as
n
⟶
∞
.
By Lemma 2 [11], we have υ
n
→
υ as n
→
∞. Since[
T
υ
]
α is closed, so υ
∈
[
T
υ
]
α. Similarly, it follows that υ
∈
[
S
υ
]
α. Thus there exists some v
∈
[
S
v
]
α
∩
[
T
v
]
α.By settingη
=
0 and k
=
β
=
γ in Theorem 19, we get the following corollary.Corollary 20.
Let(
X
,
d
) be a complete complex valued metric space and let S
,
T be fuzzy mappings from X into F
(
X
). Assume that there exists some α
∈
(
0,1
], such that, for each x
∈
X
,
[
S
x
]
α and [
T
x
]
α are nonempty closed bounded subsets of X; greatest lower bound of W
x
(
[
T
y
]
α
),W
x
(
[
S
y
]
α
)exists in C for all y
∈
Xand
(62)
k
(
d
(
x
,
[
S
x
]
α
)
+
d
(
y
,
[
T
y
]
α
)
)
∈
s
(
[
S
x
]
α
,
[
T
y
]
α
)
for all x
,
y
∈
X and 0
≤
k
<
1
/
2. Then there exists some v
∈
[
S
v
]
α
∩
[
T
v
]
α.By settingS
=
T in Theorem 19, we get the following corollary.Corollary 21.
Let(
X
,
d
) be a complete complex valued metric space and let T be fuzzy mapping from X into F
(
X
). Assume that there exists some α
∈
(
0,1
], such that, for each x
∈
X,[
T
x
]
α is nonempty closed bounded subset of X; greatest lower bound of W
x
(
[
T
y
]
α
)exists in C for all y
∈
Xand
(63)
β
d
(
x
,
[
T
x
]
α
)
+
γ
d
(
y
,
[
T
y
]
α
)
+
η
d
(
x
,
[
T
x
]
α
)
d
(
y
,
[
T
y
]
α
)
1
+
d
(
x
,
y
)
∈
s
(
[
T
x
]
α
,
[
T
y
]
α
)
,
for all x
,
y
∈
X and nonnegative reals β
,
γ, and η with β
+
γ
+
η
<
1. Then there exists some v
∈
[
T
v
]
α.By Definition11, one can have the following corollaries easily from Theorem 19.Corollary 22.
Let(
X
,
d
) be a complete complex valued metric space and let S
,
T be fuzzy mappings from X into F
(
X
) with g.l.b property such that, for each x
,
y
∈
X and α
∈
(
0,1
]
,
[
S
x
]
α and [
T
y
]
α are nonempty closed bounded subsets of X and
(64)
β
d
(
x
,
[
S
x
]
α
)
+
γ
d
(
y
,
[
T
y
]
α
)
+
η
d
(
x
,
[
S
x
]
α
)
d
(
y
,
[
T
y
]
α
)
1
+
d
(
x
,
y
)
∈
s
(
[
S
x
]
α
,
[
T
y
]
α
)
,
for all x
,
y
∈
X and nonnegative real numbers β
,
γ, and η with β
+
γ
+
η
<
1. Then there exists some v
∈
[
S
v
]
α
∩
[
T
v
]
α.Remark 23.
By Definition11, one can have a host of corollaries of Kannan type contractive fuzzy mappings with g.l.b property easily from Theorem 19.Corollary 24 (see[20]).
Let(
X
,
d
) be a complete complex valued metric space and let F
,
G
:
X
→
C
B
(
X
) be multivalued mappings with g.l.b property such that
(65)
β
d
(
x
,
F
x
)
+
γ
d
(
y
,
G
y
)
+
η
d
(
x
,
F
x
)
d
(
y
,
G
y
)
1
+
d
(
x
,
y
)
∈
s
(
F
x
,
G
y
)
,
for all x
,
y
∈
X and nonnegative real numbers β
,
γ, and η with β
+
γ
+
η
<
1. Then there exists some v
∈
F
v
∩
G
v.Proof.
Consider a pair of fuzzy mappingsS
,
T
:
X
→
F
(
X
) defined by(66)
S
(
x
)
(
t
)
=
{
α
,
t
∈
F
x
0
,
t
∉
F
x
,
T
(
x
)
(
t
)
=
{
α
,
t
∈
G
x
0
,
t
∉
G
x
,
where α
∈
(
0,1
]. Then
(67)
[
S
x
]
α
=
{
t
:
S
(
x
)
(
t
)
≥
α
}
=
F
x
,
[
T
x
]
α
=
G
x
.
Thus, Theorem 19 can be applied to obtain v
∈
X such that
(68)
v
∈
[
S
v
]
α
∩
[
T
v
]
α
=
F
v
∩
G
v
.


### 3.3. Chatterjea Type Fuzzy Fixed Point Result

Theorem 25.
Let(
X
,
d
) be a complete complex valued metric space and let S
,
T be fuzzy mappings from X into F
(
X
). Assume that there exists some α
∈
(
0,1
], such that, for each x
∈
X
,
[
S
x
]
α, and [
T
x
]
α are nonempty closed bounded subsets of X; greatest lower bound of W
x
(
[
T
y
]
α
),W
x
(
[
S
y
]
α
)exists in C for all y
∈
Xand
(69)
a
d
(
x
,
[
T
y
]
α
)
+
b
d
(
y
,
[
S
x
]
α
)
+
c
d
(
x
,
[
T
y
]
α
)
d
(
y
,
[
S
x
]
α
)
1
+
d
(
x
,
y
)
∈
s
(
[
S
x
]
α
,
[
T
y
]
α
)
,
for all x
,
y
∈
X and nonnegative reals a
,
b, and c with a
+
b
+
c
<
1. Then there exists some w
∈
[
S
w
]
α
∩
[
T
w
]
α.Proof.
Letx
0 be an arbitrary point in X. By assumption, we can find x
1
∈
[
S
x
0
]
α. So, we have
(70)
a
d
(
x
0
,
[
T
x
1
]
α
)
+
b
d
(
x
1
,
[
S
x
0
]
α
)
+
c
d
(
x
0
,
[
T
x
1
]
α
)
d
(
x
1
,
[
S
x
0
]
α
)
1
+
d
(
x
0
,
x
1
)
∈
s
(
[
S
x
0
]
α
,
[
T
x
1
]
α
)
.
By Lemma 6(iii), we have
(71)
a
d
(
x
0
,
[
T
x
1
]
α
)
+
b
d
(
x
1
,
[
S
x
0
]
α
)
+
c
d
(
x
0
,
[
T
x
1
]
α
)
d
(
x
1
,
[
S
x
0
]
α
)
1
+
d
(
x
0
,
x
1
)
∈
s
(
x
1
,
[
T
x
1
]
α
)
.
By definition there exists some x
2
∈
[
T
x
1
]
α, such that
(72)
a
d
(
x
0
,
[
T
x
1
]
α
)
+
b
d
(
x
1
,
[
S
x
0
]
α
)
+
c
d
(
x
0
,
[
T
x
1
]
α
)
d
(
x
1
,
[
S
x
0
]
α
)
1
+
d
(
x
0
,
x
1
)
∈
s
(
d
(
x
1
,
x
2
)
)
.
That is,
(73)
d
(
x
1
,
x
2
)
⪯
a
d
(
x
0
,
[
T
x
1
]
α
)
+
b
d
(
x
1
,
[
S
x
0
]
α
)
+
c
d
(
x
0
,
[
T
x
1
]
α
)
d
(
x
1
,
[
S
x
0
]
α
)
1
+
d
(
x
0
,
x
1
)
.
By the meaning of W
x
(
[
T
y
]
α
) and W
x
(
[
S
y
]
α
) for x
,
y
∈
X, we get
(74)
d
(
x
1
,
x
2
)
⪯
a
d
(
x
0
,
x
2
)
+
b
d
(
x
1
,
x
1
)
+
c
d
(
x
0
,
x
2
)
d
(
x
1
,
x
1
)
1
+
d
(
x
0
,
x
1
)
,
which implies that
(75)
|
d
(
x
1
,
x
2
)
|
≤
a
1
-
a
|
d
(
x
0
,
x
1
)
|
.
Similarly from (69), we have
(76)
a
d
(
x
2
,
[
T
x
1
]
α
)
+
b
d
(
x
1
,
[
S
x
2
]
α
)
+
c
d
(
x
2
,
[
T
x
1
]
α
)
d
(
x
1
,
[
S
x
2
]
α
)
1
+
d
(
x
1
,
x
2
)
∈
s
(
[
T
x
1
]
α
,
[
S
x
2
]
α
)
.
By Lemma 6(iii), we have
(77)
a
d
(
x
2
,
[
T
x
1
]
α
)
+
b
d
(
x
1
,
[
S
x
2
]
α
)
+
c
d
(
x
2
,
[
T
x
1
]
α
)
d
(
x
1
,
[
S
x
2
]
α
)
1
+
d
(
x
1
,
x
2
)
∈
s
(
x
2
,
[
S
x
2
]
α
)
.
By definition there exists some x
3
∈
[
S
x
2
]
α, such that
(78)
a
d
(
x
2
,
[
T
x
1
]
α
)
+
b
d
(
x
1
,
[
S
x
2
]
α
)
+
c
d
(
x
2
,
[
T
x
1
]
α
)
d
(
x
1
,
[
S
x
2
]
α
)
1
+
d
(
x
1
,
x
2
)
∈
s
(
d
(
x
2
,
x
3
)
)
.
That is,
(79)
d
(
x
2
,
x
3
)
⪯
a
d
(
x
2
,
[
T
x
1
]
α
)
+
b
d
(
x
1
,
[
S
x
2
]
α
)
+
c
d
(
x
2
,
[
T
x
1
]
α
)
d
(
x
1
,
[
S
x
2
]
α
)
1
+
d
(
x
1
,
x
2
)
.
By the meaning of W
x
(
[
T
y
]
α
) and W
x
(
[
S
y
]
α
) for x
,
y
∈
X, we get
(80)
d
(
x
2
,
x
3
)
⪯
a
d
(
x
2
,
x
2
)
+
b
d
(
x
1
,
x
3
)
+
c
d
(
x
2
,
x
2
)
d
(
x
1
,
x
3
)
1
+
d
(
x
1
,
x
2
)
,
which implies that
(81)
|
d
(
x
2
,
x
3
)
|
≤
b
1
-
b
|
d
(
x
1
,
x
2
)
|
.
Inductively, we can construct a sequence {
x
n
} in X such that, for n
=
0,1
,
2
,
…,
(82)
|
d
(
x
n
,
x
n
+
1
)
|
≤
q
n
|
d
(
x
0
,
x
1
)
|
,
with q
=
max
⁡
{
a
/
(
1
-
a
)
,
b
/
(
1
-
b
)
}
<
1
,
x
2
n
+
1
∈
[
S
x
2
n
]
α, and x
2
n
+
2
∈
[
T
x
2
n
+
1
]
α. Now for m
>
n, we get
(83)
|
d
(
x
n
,
x
m
)
|
≤
|
d
(
x
n
,
x
n
+
1
)
|
+
|
d
(
x
n
+
1
,
x
n
+
2
)
|
+
⋯
+
|
d
(
x
m
-
1
,
x
m
)
|
≤
[
q
n
+
q
n
+
1
+
⋯
+
q
m
-
1
]
|
d
(
x
0
,
x
1
)
|
=
[
q
n
1
-
q
]
|
d
(
x
0
,
x
1
)
|
,
and so
(84)
|
d
(
x
n
,
x
m
)
|
≤
q
n
1
-
q
|
d
(
x
0
,
x
1
)
|
⟶
0
as
m
,
n
⟶
∞
.
This implies that {
x
n
} is a Cauchy sequence in X. Since X is complete, there exists w
∈
X such that x
n
→
w as n
→
∞. We now show that w
∈
[
T
w
]
α and w
∈
[
S
w
]
α. From (69), we get
(85)
a
d
(
x
2
n
,
[
T
w
]
α
)
+
b
d
(
w
,
[
S
x
2
n
]
α
)
+
c
d
(
x
2
n
,
[
T
w
]
α
)
d
(
w
,
[
S
x
2
n
]
α
)
1
+
d
(
x
2
n
,
w
)
∈
s
(
[
S
x
2
n
]
α
,
[
T
w
]
α
)
.
By Lemma6(iii), we have
(86)
a
d
(
x
2
n
,
[
T
w
]
α
)
+
b
d
(
w
,
[
S
x
2
n
]
α
)
+
c
d
(
x
2
n
,
[
T
w
]
α
)
d
(
w
,
[
S
x
2
n
]
α
)
1
+
d
(
x
2
n
,
w
)
∈
s
(
x
2
n
+
1
,
[
T
w
]
α
)
.
By definition there exists some w
n
∈
[
T
w
]
α, such that
(87)
a
d
(
x
2
n
,
[
T
w
]
α
)
+
b
d
(
w
,
[
S
x
2
n
]
α
)
+
c
d
(
x
2
n
,
[
T
w
]
α
)
d
(
w
,
[
S
x
2
n
]
α
)
1
+
d
(
x
2
n
,
w
)
∈
s
(
x
2
n
+
1
,
[
T
w
]
α
)
∈
s
(
d
(
x
2
n
+
1
,
w
n
)
)
.
That is,
(88)
d
(
x
2
n
+
1
,
w
n
)
⪯
a
d
(
x
2
n
,
[
T
w
]
α
)
+
b
d
(
w
,
[
S
x
2
n
]
α
)
+
c
d
(
x
2
n
,
[
T
w
]
α
)
d
(
w
,
[
S
x
2
n
]
α
)
1
+
d
(
x
2
n
,
w
)
.
By the meaning of W
x
(
[
T
y
]
α
) and W
x
(
[
S
y
]
α
) for x
,
y
∈
X, we get
(89)
d
(
x
2
n
+
1
,
v
n
)
⪯
a
d
(
x
2
n
,
w
n
)
+
b
d
(
w
,
x
2
n
+
1
)
+
c
d
(
x
2
n
,
w
n
)
d
(
w
,
x
2
n
+
1
)
1
+
d
(
x
2
n
,
w
)
.
Now by using the triangular inequality, we get
(90)
d
(
x
2
n
+
1
,
w
n
)
⪯
a
d
(
x
2
n
,
x
2
n
+
1
)
+
a
d
(
x
2
n
+
1
,
w
n
)
+
b
d
(
w
,
x
2
n
+
1
)
+
c
d
(
x
2
n
,
w
n
)
d
(
w
,
x
2
n
+
1
)
1
+
d
(
x
2
n
,
w
)
,
and it follows that
(91)
d
(
x
2
n
+
1
,
w
n
)
⪯
a
1
-
a
d
(
x
2
n
,
x
2
n
+
1
)
+
b
1
-
a
d
(
w
,
x
2
n
+
1
)
+
c
1
-
a
d
(
x
2
n
,
w
n
)
d
(
w
,
x
2
n
+
1
)
1
+
d
(
x
2
n
,
w
)
.
By using again triangular inequality, we get
(92)
d
(
w
,
w
n
)
⪯
d
(
w
,
x
2
n
+
1
)
+
d
(
x
2
n
+
1
,
w
n
)
⪯
d
(
w
,
x
2
n
+
1
)
+
a
1
-
a
d
(
x
2
n
,
x
2
n
+
1
)
+
b
1
-
a
d
(
w
,
x
2
n
+
1
)
+
c
1
-
a
d
(
x
2
n
,
w
n
)
d
(
w
,
x
2
n
+
1
)
1
+
d
(
x
2
n
,
w
)
,
and it follows that
(93)
|
d
(
w
,
w
n
)
|
≤
|
d
(
w
,
x
2
n
+
1
)
|
+
a
1
-
a
|
d
(
x
2
n
,
x
2
n
+
1
)
|
+
b
1
-
a
|
d
(
w
,
x
2
n
+
1
)
|
+
c
1
-
a
|
d
(
x
2
n
,
w
n
)
|
|
d
(
w
,
x
2
n
+
1
)
|
|
1
+
d
(
x
2
n
,
w
)
|
.
By letting n
→
∞ in above inequality, we get |
d
(
w
,
w
n
)
|
→
0 as n
→
∞. By Lemma 2 [11], we have w
n
→
w as n
→
∞. Since[
T
w
]
α is closed, so w
∈
[
T
w
]
α. Similarly, it follows that w
∈
[
S
w
]
α. Thus there exists some w
∈
[
S
w
]
α
∩
[
T
w
]
α.Corollary 26.
Let(
X
,
d
) be a complete complex valued metric space and let S
,
T be fuzzy mappings from X into F
(
X
). Assume that there exists some α
∈
(
0,1
], such that, for each x
∈
X
,
[
S
x
]
α and [
T
x
]
α are nonempty closed bounded subsets of X; greatest lower bound of W
x
(
[
T
y
]
α
),W
x
(
[
S
y
]
α
)exists in C for all y
∈
Xand
(94)
h
(
d
(
x
,
[
T
y
]
α
)
+
d
(
y
,
[
S
x
]
α
)
)
∈
s
(
[
S
x
]
α
,
[
T
y
]
α
)
,
for all x
,
y
∈
X and 0
≤
h
<
1
/
2. Then there exists some w
∈
[
S
w
]
α
∩
[
T
w
]
α.By takingS
=
T in Theorem 25, we get the following corollary.Corollary 27.
Let(
X
,
d
) be a complete complex valued metric space and let T be fuzzy mapping from X into F
(
X
). Assume that there exists some α
∈
(
0,1
] such that, for each x
∈
X,[
T
x
]
α is nonempty closed bounded subset of X; greatest lower bound of W
x
(
[
T
y
]
α
)exists in C for all y
∈
Xand
(95)
a
d
(
x
,
[
T
y
]
α
)
+
b
d
(
y
,
[
T
x
]
α
)
+
c
d
(
x
,
[
T
y
]
α
)
d
(
y
,
[
T
x
]
α
)
1
+
d
(
x
,
y
)
∈
s
(
[
T
x
]
α
,
[
T
y
]
α
)
,
for all x
,
y
∈
X and nonnegative reals a
,
b, and c with a
+
b
+
c
<
1. Then there exists some w
∈
[
T
w
]
α.Corollary 28.
Let(
X
,
d
) be a complete complex valued metric space and let S
,
T be fuzzy mappings from X into F
(
X
) with g.l.b property such that, for each x
,
y
∈
X and α
∈
(
0,1
]
,
[
S
x
]
α and [
T
y
]
α are nonempty closed bounded subsets of X and
(96)
a
d
(
x
,
[
T
y
]
α
)
+
b
d
(
y
,
[
S
x
]
α
)
+
c
d
(
x
,
[
T
y
]
α
)
d
(
y
,
[
S
x
]
α
)
1
+
d
(
x
,
y
)
∈
s
(
[
S
x
]
α
,
[
T
y
]
α
)
,
for all x
,
y
∈
X and nonnegative reals a
,
b, and c with a
+
b
+
c
<
1. Then there exists some w
∈
[
S
w
]
α
∩
[
T
w
]
α.Remark 29.
By Definition11, one can have a host of corollaries of Chatterjea type contractive fuzzy mappings with g.l.b property easily from Theorem 25.Corollary 30 (see [20]).
Let(
X
,
d
) be a complete complex valued metric space and let F
,
G
:
X
→
C
B
(
X
) be multivalued mappings with g.l.b property such that
(97)
a
d
(
x
,
G
y
)
+
b
d
(
y
,
F
x
)
+
c
d
(
x
,
G
y
)
d
(
y
,
F
x
)
1
+
d
(
x
,
y
)
∈
s
(
F
x
,
G
y
)
,
for all x
,
y
∈
X and nonnegative reals a
,
b, and c with a
+
b
+
c
<
1. Then there exists some w
∈
F
w
∩
G
w.Proof.
Consider a pair of fuzzy mappingsS
,
T
:
X
→
F
(
X
) defined by
(98)
S
(
x
)
(
t
)
=
{
α
,
t
∈
F
x
0
,
t
∉
F
x
,
T
(
x
)
(
t
)
=
{
α
,
t
∈
G
x
0
,
t
∉
G
x
,
where α
∈
(
0,1
]. Then
(99)
[
S
x
]
α
=
{
t
:
S
(
x
)
(
t
)
≥
α
}
=
F
x
,
[
T
x
]
α
=
G
x
.
Thus, Theorem 25 can be applied to obtain w
∈
X such that
(100)
w
∈
[
S
w
]
α
∩
[
T
w
]
α
=
F
w
∩
G
w
.Remark 31.
Consider the following.(i)
By settingζ
,
κ, and ς in Corollary 18, β
,
γ, and η in Corollary 24, and a
,
b, and c in Corollary 30 with different combinations, one can get corresponding results in [19, 20] as corollaries.
(ii)
By Remark7 and Corollaries 18, 24, and 30, one can easily get the results of [13, 18–21].Example 32.
LetX
=
[
0,1
]and let C be the set of complexnumbers; define d
:
X
×
X
→
C as follows:
(101)
d
(
x
,
y
)
=
|
x
-
y
|
e
i
θ
,
where
θ
=
Arg
(
z
)
,
v
v
v
v
v
z
=
x
+
i
y
,
and
|
·
|
is
modulus
function
.
Then (
X
,
d
)is a complete complex valued metric space. Define a pair of mappings S
,
T
:
X
→
(
X
)
,for α
∈
(
0,1
]as follows.
Forx
,
y
∈
X with x
≤
y, we have(102)
S
(
x
)
(
t
)
=
{
α
if
0
≤
t
≤
x
40
α
2
if
x
40
<
t
≤
x
30
α
3
if
x
30
<
t
≤
x
20
α
5
if
x
20
<
t
≤
1
,
(103)
T
(
x
)
(
t
)
=
{
α
if
0
≤
t
≤
x
20
α
3
if
x
20
<
t
≤
x
10
α
4
if
x
10
<
t
≤
x
5
α
7
if
x
5
<
t
≤
1
,
such that
(104)
[
T
x
]
α
=
[
0
,
x
20
]
,
[
S
x
]
α
=
[
0
,
x
40
]
,
and then
(105)
W
x
(
[
T
y
]
α
)
=
{
d
(
x
,
u
)
:
u
∈
[
0
,
y
20
]
}
,
W
y
(
[
S
x
]
α
)
=
{
d
(
y
,
v
)
:
v
∈
[
0
,
x
40
]
}
.
Denoted
(
x
,
[
T
x
]
α
)and d
(
x
,
[
S
x
]
α
)bythe greatest lower boundsof W
x
(
[
T
x
]
α
) and W
x
(
[
S
x
]
α
). Then
(106)
d
(
x
,
[
T
y
]
α
)
(
z
)
=
{
0
if
x
<
y
20
(
x
-
y
20
)
e
i
θ
if
x
>
y
20
,
d
(
y
,
[
S
x
]
α
)
(
z
)
=
{
(
y
-
x
40
)
e
i
θ
,
as
y
>
x
40
,
and also
(107)
d
(
y
,
[
T
y
]
α
)
(
z
)
=
(
19
y
20
)
e
i
θ
,
d
(
x
,
[
S
x
]
α
)
(
z
)
=
(
39
x
40
)
e
i
θ
.
Moreover, if w
y
x
∈
C such that
(108)
w
y
x
=
|
y
20
-
x
40
|
e
θ
,
then
(109)
s
(
[
T
y
]
α
,
[
S
x
]
α
)
=
{
w
∈
C
:
w
x
y
⪯
w
}
.
Forx
>
y
/
20, we have
(110)
d
(
y
,
[
T
y
]
α
)
+
d
(
x
,
[
S
x
]
α
)
3
(
z
)
=
1
3
(
19
y
20
+
39
x
40
)
e
i
θ
=
1
3
(
y
-
y
20
+
x
-
x
40
)
e
i
θ
=
1
3
(
y
-
x
40
+
x
-
y
20
)
e
i
θ
=
1
3
(
y
-
x
40
)
e
i
θ
+
(
x
-
y
20
)
e
i
θ
=
d
(
y
,
[
S
x
]
α
)
+
d
(
x
,
[
T
y
]
α
)
3
(
z
)
=
1
3
(
6
y
20
+
13
y
20
+
6
x
40
+
33
x
40
)
e
i
θ
=
1
3
(
6
y
20
+
6
x
40
)
e
i
θ
+
(
13
y
20
+
33
x
40
)
e
i
θ
≻
1
3
(
6
y
20
+
6
x
40
)
e
i
θ
=
(
2
y
20
+
2
x
40
)
e
i
θ
>
(
y
20
+
x
40
)
e
i
θ
≻
|
y
20
-
x
40
|
e
i
θ
=
w
x
y
,
also as
(111)
1
2
d
(
x
,
y
)
=
1
2
|
x
-
y
|
e
i
θ
⪰
|
y
20
-
x
40
|
e
i
θ
.
It follows that, with ζ
=
κ
=
1
/
3
,
ς
≠
0, such that ζ
+
κ
+
ς
<
1, we have
(112)
ζ
d
(
x
,
[
S
x
]
α
)
+
κ
d
(
y
,
[
T
y
]
α
)
+
ς
d
(
x
,
[
S
x
]
α
)
d
(
y
,
[
T
y
]
α
)
1
+
d
(
x
,
y
)
∈
s
(
[
S
x
]
α
,
[
T
y
]
α
)
,
(113)
ζ
d
(
x
,
[
T
y
]
α
)
+
κ
d
(
y
,
[
S
x
]
α
)
+
ς
d
(
x
,
[
T
y
]
α
)
d
(
y
,
[
S
x
]
α
)
1
+
d
(
x
,
y
)
∈
s
(
[
S
x
]
α
,
[
T
y
]
α
)
,
and for ζ
=
1
/
2with κ
≠
0 and ς
≠
0, such that ζ
+
κ
+
ς
<
1, we have
(114)
ζ
d
(
x
,
y
)
+
κ
d
(
x
,
[
S
x
]
α
)
d
(
y
,
[
T
y
]
α
)
+
ς
d
(
y
,
[
S
x
]
α
)
d
(
x
,
[
T
y
]
α
)
1
+
d
(
x
,
y
)
∈
s
(
[
S
x
]
α
,
[
T
y
]
α
)
.
Hence T and S satisfy all the conditions of our main Theorem 12 to obtain 0
∈
[
S
0
]
α
∩
[
T
0
]
α.


## 3.1. Banach Type Fuzzy Fixed Point Result

Theorem 12.
Let(
X
,
d
) be a complete complex valued metric space and let S
,
T be fuzzy mappings from X into F
(
X
). Assume that there exists some α
∈
(
0,1
], such that, for each x
∈
X
,
[
S
x
]
α and [
T
x
]
α are nonempty closed bounded subsets of X; greatest lower bound of W
x
(
[
T
y
]
α
)
,
W
x
(
[
S
y
]
α
)exists in C for all y
∈
X and
(16)
ζ
d
(
x
,
y
)
+
κ
d
(
x
,
[
S
x
]
α
)
d
(
y
,
[
T
y
]
α
)
+
ς
d
(
y
,
[
S
x
]
α
)
d
(
x
,
[
T
y
]
α
)
1
+
d
(
x
,
y
)
∈
s
(
[
S
x
]
α
,
[
T
y
]
α
)
,
for all x
,
y
∈
X, where ζ
,
κ
,
ς are nonnegative real numbers with ζ
+
κ
+
ς
<
1. Then there exists some u
∈
[
S
u
]
α
∩
[
T
u
]
α.Proof.
Letx
0 be an arbitrary point in X. By assumption, we can find x
1
∈
[
S
x
0
]
α. So, we have
(17)
ζ
d
(
x
0
,
x
1
)
+
(
(
1
+
d
(
x
0
,
x
1
)
)
-
1
(
κ
d
(
x
0
,
[
S
x
0
]
α
)
d
(
x
1
,
[
T
x
1
]
α
)
v
v
v

+

ς
d
(
x
1
,
[
S
x
0
]
α
)
d
(
x
0
,
[
T
x
1
]
α
)
)
×
(
1
+
d
(
x
0
,
x
1
)
)
-
1
)
∈
s
(
[
S
x
0
]
α
,
[
T
x
1
]
α
)
.
By Lemma 6(iii), we have
(18)
ζ
d
(
x
0
,
x
1
)
+
(
(
1
+
d
(
x
0
,
x
1
)
)
-
1
(
κ
d
(
x
0
,
[
S
x
0
]
α
)
d
(
x
1
,
[
T
x
1
]
α
)
vvv
+
ς
d
(
x
1
,
[
S
x
0
]
α
)
d
(
x
0
,
[
T
x
1
]
α
)
)
v
×
(
1
+
d
(
x
0
,
x
1
)
)
-
1
)
∈
s
(
x
1
,
[
T
x
1
]
α
)
.
By definition there exists some x
2
∈
[
T
x
1
]
α, such that
(19)
ζ
d
(
x
0
,
x
1
)
+
(
(
1
+
d
(
x
0
,
x
1
)
)
-
1
(
κ
d
(
x
0
,
[
S
x
0
]
α
)
d
(
x
1
,
[
T
x
1
]
α
)
v
v

+

ς
d
(
x
1
,
[
S
x
0
]
α
)
d
(
x
0
,
[
T
x
1
]
α
)
)
v
×
(
1
+
d
(
x
0
,
x
1
)
)
-
1
)
∈
s
(
d
(
x
1
,
x
2
)
)
.
That is,
(20)
d
(
x
1
,
x
2
)
⪯
ζ
d
(
x
0
,
x
1
)
+
(
(
1
+
d
(
x
0
,
x
1
)
)
-
1
(
κ
d
(
x
0
,
[
S
x
0
]
α
)
d
(
x
1
,
[
T
x
1
]
α
)
v
v

+

ς
d
(
x
1
,
[
S
x
0
]
α
)
d
(
x
0
,
[
T
x
1
]
α
)
)
×
(
1
+
d
(
x
0
,
x
1
)
)
-
1
)
.
By the meaning of W
x
(
[
T
y
]
α
) and W
x
(
[
S
y
]
α
) for x
,
y
∈
X, we get
(21)
d
(
x
1
,
x
2
)
⪯
ζ
d
(
x
0
,
x
1
)
+
κ
d
(
x
0
,
x
1
)
d
(
x
1
,
x
2
)
+
ς
d
(
x
1
,
x
1
)
d
(
x
0
,
x
2
)
1
+
d
(
x
0
,
x
1
)
=
ζ
d
(
x
0
,
x
1
)
+
κ
d
(
x
0
,
x
1
)
d
(
x
1
,
x
2
)
1
+
d
(
x
0
,
x
1
)
,
which implies that
(22)
|
d
(
x
1
,
x
2
)
|
≤
ζ
|
d
(
x
0
,
x
1
)
|
+
κ
|
d
(
x
0
,
x
1
)
|
|
d
(
x
1
,
x
2
)
|
|
1
+
d
(
x
0
,
x
1
)
|
=
ζ
|
d
(
x
0
,
x
1
)
|
+
κ
|
d
(
x
1
,
x
2
)
|
|
d
(
x
0
,
x
1
)
1
+
d
(
x
0
,
x
1
)
|
,
|
d
(
x
1
,
x
2
)
|
≤
ζ
|
d
(
x
0
,
x
1
)
|
+
κ
|
d
(
x
1
,
x
2
)
|
,
(
1
-
κ
)
|
d
(
x
1
,
x
2
)
|
≤
ζ
|
d
(
x
0
,
x
1
)
|
,
where
(23)
h
=
ζ
1
-
κ
<
1
.
Inductively, we can construct a sequence {
x
n
} in X such that, for n
=
0,1
,
2
,
…,
(24)
|
d
(
x
n
,
x
n
+
1
)
|
≤
h
n
|
d
(
x
0
,
x
1
)
|
,
with h
=
ζ
/
(
1
-
κ
)
<
1, for x
2
n
+
1
∈
[
S
x
2
n
]
α and x
2
n
+
2
∈
[
T
x
2
n
+
1
]
α.
Now form
>
n, we get
(25)
|
d
(
x
n
,
x
m
)
|
≤
|
d
(
x
n
,
x
n
+
1
)
|
+
|
d
(
x
n
+
1
,
x
n
+
2
)
|
+
⋯
+
|
d
(
x
m
-
1
,
x
m
)
|
≤
[
h
n
+
h
n
+
1
+
⋯
+
h
m
-
1
]
|
d
(
x
0
,
x
1
)
|
≤
[
h
n
1
-
h
]
|
d
(
x
0
,
x
1
)
|
,
and so
(26)
|
d
(
x
n
,
x
m
)
|
≤
h
n
1
-
h
|
d
(
x
0
,
x
1
)
|
⟶
0
,
as
m
,
n
⟶
∞
.
This implies that {
x
n
} is a Cauchy sequence in X. Since X is complete, so there exists u
∈
X such that x
n
→
u as n
→
∞
. We now show that u
∈
[
T
u
]
α and u
∈
[
S
u
]
α. From (16), we have
(27)
ζ
d
(
x
2
k
,
u
)
+
(
×
(
1
+
d
(
x
2
k
,
u
)
)
-
1
(
κ
d
(
x
2
k
,
[
S
x
2
k
]
α
)
d
(
u
,
[
T
u
]
α
)
vvx
+
ς
d
(
u
,
[
S
x
2
k
]
α
)
d
(
x
2
k
,
[
T
u
]
α
)
)
v
×
(
1
+
d
(
x
2
k
,
u
)
)
-
1
)
∈
s
(
[
S
x
2
k
]
α
,
[
T
u
]
α
)
.
By Lemma 6(iii), we have
(28)
ζ
d
(
x
2
k
,
u
)
+
(
(
1
+
d
(
x
2
k
,
u
)
)
-
1
(
κ
d
(
x
2
k
,
[
S
x
2
k
]
α
)
d
(
u
,
[
T
u
]
α
)
v
v
v
+
ς
d
(
u
,
[
S
x
2
k
]
α
)
d
(
x
2
k
,
[
T
u
]
α
)
)
v
×
(
1
+
d
(
x
2
k
,
u
)
)
-
1
)
∈
s
(
x
2
k
+
1
,
[
T
u
]
α
)
.
By definition there exists some u
k
∈
[
T
u
]
α such that
(29)
ζ
d
(
x
2
k
,
u
)
+
(
(
1
+
d
(
x
2
k
,
u
)
)
-
1
(
κ
d
(
x
2
k
,
[
S
x
2
k
]
α
)
d
(
u
,
[
T
u
]
α
)
v
v
v
+
ς
d
(
u
,
[
S
x
2
k
]
α
)
d
(
x
2
k
,
[
T
u
]
α
)
)
v
×
(
1
+
d
(
x
2
k
,
u
)
)
-
1
)
∈
s
(
d
(
x
2
k
+
1
,
u
k
)
)
.
That is,
(30)
d
(
x
2
k
+
1
,
u
k
)
⪯
ζ
d
(
x
2
k
,
u
)

+

(
(
1
+
d
(
x
2
k
,
u
)
)
-
1
(
κ
d
(
x
2
k
,
[
S
x
2
k
]
α
)
d
(
u
,
[
T
u
]
α
)
.
v
+
ς
d
(
u
,
[
S
x
2
k
]
α
)
d
(
x
2
k
,
[
T
u
]
α
)
)
v
×
(
1
+
d
(
x
2
k
,
u
)
)
-
1
)
.
By the meaning of W
x
(
[
T
y
]
α
) and W
x
(
[
S
y
]
α
) for x
,
y
∈
X, we get
(31)
d
(
x
2
k
+
1
,
u
k
)
⪯
ζ
d
(
x
2
k
,
u
)
+
κ
d
(
x
2
k
,
x
2
k
+
1
)
d
(
u
,
u
k
)
+
ς
d
(
u
,
x
2
k
+
1
)
d
(
x
2
k
,
u
k
)
1
+
d
(
x
2
k
u
)
.
Since by triangle inequality, we get
(32)
d
(
u
,
u
k
)
⪯
d
(
u
,
x
2
k
+
1
)
+
d
(
x
2
k
+
1
,
u
k
)
.
So using (31) in (32), we get
(33)
d
(
u
,
u
k
)
⪯
d
(
u
,
x
2
k
+
1
)
+
ζ
d
(
u
,
x
2
k
+
1
)
+
κ
d
(
x
2
k
,
x
2
k
+
1
)
d
(
u
,
u
k
)
+
ς
d
(
u
,
x
2
k
+
1
)
d
(
x
2
k
,
u
k
)
1
+
d
(
x
2
k
,
u
)
,
|
d
(
u
,
u
k
)
|
≤
|
d
(
u
,
x
2
k
+
1
)
|
+
ζ
|
d
(
u
,
x
2
k
+
1
)
|
+
κ
|
d
(
x
2
k
,
x
2
k
+
1
)
|
|
d
(
u
,
u
k
)
|

+

ς
|
d
(
u
,
x
2
k
+
1
)
|
|
d
(
x
2
k
,
u
k
)
|
|
1
+
d
(
x
2
k
,
u
)
|
.
Taking the limit as k
→
∞, we get |
d
(
u
,
u
k
)
|
→
0 as k
→
∞
. By Lemma 2 [11], we have u
k
→
u as k
→
∞
. Since [
T
u
]
α is closed, so u
∈
[
T
u
]
α. Similarly, it follows that u
∈
[
S
u
]
α. ThusS and T have a common fuzzy fixed point.By settingς
=
0 in Theorem 12, we get the following corollary.Corollary 13.
Let(
X
,
d
) be a complete complex valued metric space and let S
,
T be fuzzy mappings from X into F
(
X
). Assume that there exists some α
∈
(
0,1
], such that, for each x
∈
X,[
S
x
]
α and [
T
x
]
α are nonempty closed bounded subsets of X; greatest lower bound of W
x
(
[
T
y
]
α
)
,
W
x
(
[
S
y
]
α
)exists in C for all y
∈
Xand
(34)
ζ
d
(
x
,
y
)
+
κ
d
(
x
,
[
S
x
]
α
)
d
(
y
,
[
T
y
]
α
)
1
+
d
(
x
,
y
)
∈
s
(
[
S
x
]
α
,
[
T
y
]
α
)
,
for all x
,
y
∈
X, where ζ and κ are nonnegative real numbers with ζ
+
κ
<
1. Then there exists some u
∈
[
S
u
]
α
∩
[
T
u
]
α.By settingS
=
T in Theorem 12, we get the following corollary.Corollary 14.
Let(
X
,
d
) be a complete complex valued metric space and let T be fuzzy mapping from X into F
(
X
). Assume that there exists some α
∈
(
0,1
], such that, for each x
∈
X,[
T
x
]
α is nonempty closed bounded subset of X; greatest lower bound of W
x
(
[
T
y
]
α
)exists in C for all y
∈
Xand
(35)
ζ
d
(
x
,
y
)
+
κ
d
(
x
,
[
T
x
]
α
)
d
(
y
,
[
T
y
]
α
)
+
ς
d
(
y
,
[
T
x
]
α
)
d
(
x
,
[
T
y
]
α
)
1
+
d
(
x
,
y
)
∈
s
(
[
T
x
]
α
,
[
T
y
]
α
)
,
for all x
,
y
∈
X, where ζ
,
κ, and ς are nonnegative real numbers with ζ
+
κ
+
ς
<
1. Then there exists some u
∈
[
T
u
]
α.By Definition11, one can have the following corollaries easily from Theorem 12.Corollary 15.
Let(
X
,
d
) be a complete complex valued metric space and let S
,
T be fuzzy mappings from X into F
(
X
) with g.l.b property such that, for each x
,
y
∈
X and α
∈
(
0,1
] and [
S
x
]
α
,
[
T
y
]
α are nonempty closed bounded subsets of X and
(36)
ζ
d
(
x
,
y
)
+
κ
d
(
x
,
[
S
x
]
α
)
d
(
y
,
[
T
y
]
α
)
+
ς
d
(
y
,
[
S
x
]
α
)
d
(
x
,
[
T
y
]
α
)
1
+
d
(
x
,
y
)
∈
s
(
[
S
x
]
α
,
[
T
y
]
α
)
,
for all x
,
y
∈
X, and ζ
,
κ, and ς are nonnegative real numbers with ζ
+
κ
+
ς
<
1. Then there exists some u
∈
[
S
u
]
α
∩
[
T
u
]
α.Corollary 16.
Let(
X
,
d
) be a complete complex valued metric space and let S
,
T be fuzzy mappings from X into F
(
X
) with g.l.b property such that, for each x
,
y
∈
X and α
∈
(
0,1
] and [
S
x
]
α
,
[
T
y
]
α are nonempty closed bounded subsets of X and
(37)
ζ
d
(
x
,
y
)
+
κ
d
(
x
,
[
S
x
]
α
)
d
(
y
,
[
T
y
]
α
)
1
+
d
(
x
,
y
)
∈
s
(
[
S
x
]
α
,
[
T
y
]
α
)
,
for all x
,
y
∈
X, and ζ and κ are nonnegative real numbers with ζ
+
κ
<
1. Then there exists some u
∈
[
S
u
]
α
∩
[
T
u
]
α.Corollary 17.
Let(
X
,
d
) be a complete complex valued metric space and let T be fuzzy mapping from X into F
(
X
) with g.l.b property such that, for each x
,
y
∈
X and α
∈
(
0,1
]
,
[
T
y
]
α is nonempty closed bounded subset of X and
(38)
ζ
d
(
x
,
y
)
+
κ
d
(
x
,
[
T
x
]
α
)
d
(
y
,
[
T
y
]
α
)
+
ς
d
(
y
,
[
T
x
]
α
)
d
(
x
,
[
T
y
]
α
)
1
+
d
(
x
,
y
)
∈
s
(
[
T
x
]
α
,
[
T
y
]
α
)
,
for all x
,
y
∈
X, and ζ
,
κ, and ς are nonnegative real numbers with ζ
+
κ
+
ς
<
1. Then there exists some u
∈
[
T
u
]
α.Corollary 18 (see [19]).
Let(
X
,
d
) be a complete complex valued metric space and let F
,
G
:
X
→
C
B
(
X
) be multivalued mappings with g.l.b property such that
(39)
ζ
d
(
x
,
y
)
+
κ
d
(
x
,
F
x
)
d
(
y
,
G
y
)
+
ς
d
(
y
,
F
x
)
d
(
x
,
G
y
)
1
+
d
(
x
,
y
)
∈
s
(
F
x
,
G
y
)
,
for all x
,
y
∈
X, where ζ
,
κ, and ς are nonnegative real numbers with ζ
+
κ
+
ς
<
1. Then there exists some u
∈
F
u
∩
T
u.Proof.
Consider a pair of fuzzy mappingsS
,
T
:
X
→
F
(
X
) defined by(40)
S
(
x
)
(
t
)
=
{
α
,
t
∈
F
x
0
,
t
∉
F
x
,
T
(
x
)
(
t
)
=
{
α
,
t
∈
G
x
0
,
t
∉
G
x
,
where α
∈
(
0,1
]. Then
(41)
[
S
x
]
α
=
{
t
:
S
(
x
)
(
t
)
≥
α
}
=
F
x
,
[
T
x
]
α
=
G
x
.
Thus, Theorem 12 can be applied to obtain u
∈
X such that
(42)
u
∈
[
S
u
]
α
∩
[
T
u
]
α
=
F
u
∩
G
u
.


## 3.2. Kannan Type Fuzzy Fixed Point Result

Theorem 19.
Let(
X
,
d
) be a complete complex valued metric space and let S
,
T be fuzzy mappings from X into F
(
X
). Assume that there exists some α
∈
(
0,1
], such that, for each x
∈
X
,
[
S
x
]
α and [
T
x
]
α are nonempty closed bounded subsets of X; greatest lower bound of W
x
(
[
T
y
]
α
),W
x
(
[
S
y
]
α
)exists in C for all y
∈
Xand
(43)
β
d
(
x
,
[
S
x
]
α
)
+
γ
d
(
y
,
[
T
y
]
α
)
+
η
d
(
x
,
[
S
x
]
α
)
d
(
y
,
[
T
y
]
α
)
1
+
d
(
x
,
y
)
∈
s
(
[
S
x
]
α
,
[
T
y
]
α
)
,
for all x
,
y
∈
X and nonnegative real numbers β
,
γ, and η with β
+
γ
+
η
<
1. Then there exists some v
∈
[
S
v
]
α
∩
[
T
v
]
α.Proof.
Letx
0 be an arbitrary point in X. By assumption, we can find x
1
∈
[
S
x
0
]
α. So, we have
(44)
β
d
(
x
0
,
[
S
x
0
]
α
)
+
γ
d
(
x
1
,
[
T
x
1
]
α
)
+
η
d
(
x
0
,
[
S
x
0
]
α
)
d
(
x
1
,
[
T
x
1
]
α
)
1
+
d
(
x
0
,
x
1
)
∈
s
(
[
S
x
0
]
α
,
[
T
x
1
]
α
)
.
By Lemma 6(iii), we have
(45)
β
d
(
x
0
,
[
S
x
0
]
α
)
+
γ
d
(
x
1
,
[
T
x
1
]
α
)
+
η
d
(
x
0
,
[
S
x
0
]
α
)
d
(
x
1
,
[
T
x
1
]
α
)
1
+
d
(
x
0
,
x
1
)
∈
s
(
x
1
,
[
T
x
1
]
α
)
.
By definition there exists somex
2
∈
[
T
x
1
]
α, such that
(46)
β
d
(
x
0
,
[
S
x
0
]
α
)
+
γ
d
(
x
1
,
[
T
x
1
]
α
)
+
η
d
(
x
0
,
[
S
x
0
]
α
)
d
(
x
1
,
[
T
x
1
]
α
)
1
+
d
(
x
0
,
x
1
)
∈
s
(
d
(
x
1
,
x
2
)
)
.
That is,(47)
d
(
x
1
,
x
2
)
⪯
β
d
(
x
0
,
[
S
x
0
]
α
)
+
γ
d
(
x
1
,
[
T
x
1
]
α
)
+
η
d
(
x
0
,
[
S
x
0
]
α
)
d
(
x
1
,
[
T
x
1
]
α
)
1
+
d
(
x
0
,
x
1
)
.
By the meaning of W
x
(
[
T
y
]
α
) and W
x
(
[
S
y
]
α
) for x
,
y
∈
X, we get
(48)
d
(
x
1
,
x
2
)
⪯
β
d
(
x
0
,
x
1
)
+
γ
d
(
x
1
,
x
2
)
+
η
d
(
x
0
,
x
1
)
d
(
x
1
,
x
2
)
1
+
d
(
x
0
,
x
1
)
,
which implies that
(49)
|
d
(
x
1
,
x
2
)
|
≤
β
|
d
(
x
0
,
x
1
)
|
+
γ
|
d
(
x
1
,
x
2
)
|
+
η
|
d
(
x
0
,
x
1
)
|
|
d
(
x
1
,
x
2
)
|
|
1
+
d
(
x
0
,
x
1
)
|
.
Thus(50)
|
d
(
x
1
,
x
2
)
|
≤
l
|
d
(
x
0
,
x
1
)
|
,
wherel
=
β
/
(
1
-
γ
-
η
)
<
1. Inductively, we can construct a sequence {
x
n
} in X such that, for n
=
0,1
,
2
,
…,
(51)
|
d
(
x
n
,
x
n
+
1
)
|
≤
l
n
|
d
(
x
0
,
x
1
)
|
,
with l
=
β
/
(
1
-
γ
-
η
)
<
1, for x
2
n
+
1
∈
[
S
x
2
n
]
α and x
2
n
+
2
∈
[
T
x
2
n
+
1
]. Now for m
>
n, we get
(52)
|
d
(
x
n
,
x
m
)
|
≤
|
d
(
x
n
,
x
n
+
1
)
|
+
|
d
(
x
n
+
1
,
x
n
+
2
)
|
+
⋯
+
|
d
(
x
m
-
1
,
x
m
)
|
≤
[
l
n
+
l
n
+
1
+
⋯
+
l
m
-
1
]
|
d
(
x
0
,
x
1
)
|
≤
[
l
n
1
-
l
]
|
d
(
x
0
,
x
1
)
|
,
and so
(53)
|
d
(
x
n
,
x
m
)
|
≤
l
n
1
-
l
|
d
(
x
0
,
x
1
)
|
⟶
0
as
m
,
n
⟶
∞
.
This implies that {
x
n
} is a Cauchy sequence in X. Since X is complete, there exists υ
∈
X such that x
n
→
υ as n
→
∞. We now show that υ
∈
[
T
υ
]
α and υ
∈
[
S
υ
]
α. From (43), we get
(54)
β
d
(
x
2
n
,
[
S
x
2
n
]
α
)
+
γ
d
(
υ
,
[
T
υ
]
α
)
+
η
d
(
x
2
n
,
[
S
x
2
n
]
α
)
d
(
υ
,
[
T
υ
]
α
)
1
+
d
(
x
2
n
,
υ
)
∈
s
(
[
S
x
2
n
]
α
,
[
T
υ
]
α
)
.
By Lemma6 (iii), we have
(55)
β
d
(
x
2
n
,
[
S
x
2
n
]
α
)
+
γ
d
(
υ
,
[
T
υ
]
α
)
+
η
d
(
x
2
n
,
[
S
x
2
n
]
α
)
d
(
υ
,
[
T
υ
]
α
)
1
+
d
(
x
2
n
,
υ
)
∈
s
(
x
2
n
+
1
,
[
T
υ
]
α
)
.
By definition there exists someυ
n
∈
[
T
υ
]
α such that
(56)
β
d
(
x
2
n
,
[
S
x
2
n
]
α
)
+
γ
d
(
υ
,
[
T
υ
]
α
)
+
η
d
(
x
2
n
,
[
S
x
2
n
]
α
)
d
(
υ
,
[
T
υ
]
α
)
1
+
d
(
x
2
n
,
υ
)
∈
s
(
d
(
x
2
n
+
1
,
υ
n
)
)
.
That is,
(57)
d
(
x
2
n
+
1
,
υ
n
)
⪯
β
d
(
x
2
n
,
[
S
x
2
n
]
α
)
+
γ
d
(
υ
,
[
T
υ
]
α
)
vvvvvvvv
+
η
d
(
x
2
n
,
[
S
x
2
n
]
α
)
d
(
υ
,
[
T
υ
]
α
)
1
+
d
(
x
2
n
,
υ
)
.
By the meaning of W
x
(
[
T
y
]
α
) and W
x
(
[
S
y
]
α
) for x
,
y
∈
X, we get
(58)
d
(
x
2
n
+
1
,
υ
n
)
⪯
β
d
(
x
2
n
,
x
2
n
+
1
)
+
γ
d
(
υ
,
υ
n
)
+
η
d
(
x
2
n
,
x
2
n
+
1
)
d
(
υ
,
υ
n
)
1
+
d
(
x
2
n
,
υ
)
.
Now by using (58) and the triangular inequality, we get
(59)
d
(
υ
,
υ
n
)
⪯
d
(
υ
,
x
2
n
+
1
)
+
d
(
x
2
n
+
1
,
υ
n
)
⪯
d
(
υ
,
x
2
n
+
1
)
+
β
d
(
x
2
n
,
x
2
n
+
1
)
+
γ
d
(
υ
,
υ
n
)
+
η
d
(
x
2
n
,
x
2
n
+
1
)
d
(
υ
,
υ
n
)
1
+
d
(
x
2
n
,
υ
)
,
which implies that
(60)
(
1
-
γ
)
|
d
(
υ
,
υ
n
)
|
≤
|
d
(
υ
,
x
2
n
+
1
)
|
vv
+
β
|
d
(
x
2
n
,
x
2
n
+
1
)
|
vv
+
η
|
d
(
x
2
n
,
x
2
n
+
1
)
d
(
υ
,
υ
n
)
1
+
d
(
x
2
n
,
υ
)
|
,
|
d
(
υ
,
υ
n
)
|
≤
1
(
1
-
γ
)
|
d
(
υ
,
x
2
n
+
1
)
|
vv
+
β
(
1
-
γ
)
|
d
(
x
2
n
,
x
2
n
+
1
)
|
vv
+
η
(
1
-
γ
)
|
d
(
x
2
n
,
x
2
n
+
1
)
|
|
d
(
υ
,
υ
n
)
|
|
1
+
d
(
x
2
n
,
υ
)
|
.
By letting n
→
∞ in above inequality, we get
(61)
|
d
(
υ
,
υ
n
)
|
⟶
0
as
n
⟶
∞
.
By Lemma 2 [11], we have υ
n
→
υ as n
→
∞. Since[
T
υ
]
α is closed, so υ
∈
[
T
υ
]
α. Similarly, it follows that υ
∈
[
S
υ
]
α. Thus there exists some v
∈
[
S
v
]
α
∩
[
T
v
]
α.By settingη
=
0 and k
=
β
=
γ in Theorem 19, we get the following corollary.Corollary 20.
Let(
X
,
d
) be a complete complex valued metric space and let S
,
T be fuzzy mappings from X into F
(
X
). Assume that there exists some α
∈
(
0,1
], such that, for each x
∈
X
,
[
S
x
]
α and [
T
x
]
α are nonempty closed bounded subsets of X; greatest lower bound of W
x
(
[
T
y
]
α
),W
x
(
[
S
y
]
α
)exists in C for all y
∈
Xand
(62)
k
(
d
(
x
,
[
S
x
]
α
)
+
d
(
y
,
[
T
y
]
α
)
)
∈
s
(
[
S
x
]
α
,
[
T
y
]
α
)
for all x
,
y
∈
X and 0
≤
k
<
1
/
2. Then there exists some v
∈
[
S
v
]
α
∩
[
T
v
]
α.By settingS
=
T in Theorem 19, we get the following corollary.Corollary 21.
Let(
X
,
d
) be a complete complex valued metric space and let T be fuzzy mapping from X into F
(
X
). Assume that there exists some α
∈
(
0,1
], such that, for each x
∈
X,[
T
x
]
α is nonempty closed bounded subset of X; greatest lower bound of W
x
(
[
T
y
]
α
)exists in C for all y
∈
Xand
(63)
β
d
(
x
,
[
T
x
]
α
)
+
γ
d
(
y
,
[
T
y
]
α
)
+
η
d
(
x
,
[
T
x
]
α
)
d
(
y
,
[
T
y
]
α
)
1
+
d
(
x
,
y
)
∈
s
(
[
T
x
]
α
,
[
T
y
]
α
)
,
for all x
,
y
∈
X and nonnegative reals β
,
γ, and η with β
+
γ
+
η
<
1. Then there exists some v
∈
[
T
v
]
α.By Definition11, one can have the following corollaries easily from Theorem 19.Corollary 22.
Let(
X
,
d
) be a complete complex valued metric space and let S
,
T be fuzzy mappings from X into F
(
X
) with g.l.b property such that, for each x
,
y
∈
X and α
∈
(
0,1
]
,
[
S
x
]
α and [
T
y
]
α are nonempty closed bounded subsets of X and
(64)
β
d
(
x
,
[
S
x
]
α
)
+
γ
d
(
y
,
[
T
y
]
α
)
+
η
d
(
x
,
[
S
x
]
α
)
d
(
y
,
[
T
y
]
α
)
1
+
d
(
x
,
y
)
∈
s
(
[
S
x
]
α
,
[
T
y
]
α
)
,
for all x
,
y
∈
X and nonnegative real numbers β
,
γ, and η with β
+
γ
+
η
<
1. Then there exists some v
∈
[
S
v
]
α
∩
[
T
v
]
α.Remark 23.
By Definition11, one can have a host of corollaries of Kannan type contractive fuzzy mappings with g.l.b property easily from Theorem 19.Corollary 24 (see[20]).
Let(
X
,
d
) be a complete complex valued metric space and let F
,
G
:
X
→
C
B
(
X
) be multivalued mappings with g.l.b property such that
(65)
β
d
(
x
,
F
x
)
+
γ
d
(
y
,
G
y
)
+
η
d
(
x
,
F
x
)
d
(
y
,
G
y
)
1
+
d
(
x
,
y
)
∈
s
(
F
x
,
G
y
)
,
for all x
,
y
∈
X and nonnegative real numbers β
,
γ, and η with β
+
γ
+
η
<
1. Then there exists some v
∈
F
v
∩
G
v.Proof.
Consider a pair of fuzzy mappingsS
,
T
:
X
→
F
(
X
) defined by(66)
S
(
x
)
(
t
)
=
{
α
,
t
∈
F
x
0
,
t
∉
F
x
,
T
(
x
)
(
t
)
=
{
α
,
t
∈
G
x
0
,
t
∉
G
x
,
where α
∈
(
0,1
]. Then
(67)
[
S
x
]
α
=
{
t
:
S
(
x
)
(
t
)
≥
α
}
=
F
x
,
[
T
x
]
α
=
G
x
.
Thus, Theorem 19 can be applied to obtain v
∈
X such that
(68)
v
∈
[
S
v
]
α
∩
[
T
v
]
α
=
F
v
∩
G
v
.


## 3.3. Chatterjea Type Fuzzy Fixed Point Result

Theorem 25.
Let(
X
,
d
) be a complete complex valued metric space and let S
,
T be fuzzy mappings from X into F
(
X
). Assume that there exists some α
∈
(
0,1
], such that, for each x
∈
X
,
[
S
x
]
α, and [
T
x
]
α are nonempty closed bounded subsets of X; greatest lower bound of W
x
(
[
T
y
]
α
),W
x
(
[
S
y
]
α
)exists in C for all y
∈
Xand
(69)
a
d
(
x
,
[
T
y
]
α
)
+
b
d
(
y
,
[
S
x
]
α
)
+
c
d
(
x
,
[
T
y
]
α
)
d
(
y
,
[
S
x
]
α
)
1
+
d
(
x
,
y
)
∈
s
(
[
S
x
]
α
,
[
T
y
]
α
)
,
for all x
,
y
∈
X and nonnegative reals a
,
b, and c with a
+
b
+
c
<
1. Then there exists some w
∈
[
S
w
]
α
∩
[
T
w
]
α.Proof.
Letx
0 be an arbitrary point in X. By assumption, we can find x
1
∈
[
S
x
0
]
α. So, we have
(70)
a
d
(
x
0
,
[
T
x
1
]
α
)
+
b
d
(
x
1
,
[
S
x
0
]
α
)
+
c
d
(
x
0
,
[
T
x
1
]
α
)
d
(
x
1
,
[
S
x
0
]
α
)
1
+
d
(
x
0
,
x
1
)
∈
s
(
[
S
x
0
]
α
,
[
T
x
1
]
α
)
.
By Lemma 6(iii), we have
(71)
a
d
(
x
0
,
[
T
x
1
]
α
)
+
b
d
(
x
1
,
[
S
x
0
]
α
)
+
c
d
(
x
0
,
[
T
x
1
]
α
)
d
(
x
1
,
[
S
x
0
]
α
)
1
+
d
(
x
0
,
x
1
)
∈
s
(
x
1
,
[
T
x
1
]
α
)
.
By definition there exists some x
2
∈
[
T
x
1
]
α, such that
(72)
a
d
(
x
0
,
[
T
x
1
]
α
)
+
b
d
(
x
1
,
[
S
x
0
]
α
)
+
c
d
(
x
0
,
[
T
x
1
]
α
)
d
(
x
1
,
[
S
x
0
]
α
)
1
+
d
(
x
0
,
x
1
)
∈
s
(
d
(
x
1
,
x
2
)
)
.
That is,
(73)
d
(
x
1
,
x
2
)
⪯
a
d
(
x
0
,
[
T
x
1
]
α
)
+
b
d
(
x
1
,
[
S
x
0
]
α
)
+
c
d
(
x
0
,
[
T
x
1
]
α
)
d
(
x
1
,
[
S
x
0
]
α
)
1
+
d
(
x
0
,
x
1
)
.
By the meaning of W
x
(
[
T
y
]
α
) and W
x
(
[
S
y
]
α
) for x
,
y
∈
X, we get
(74)
d
(
x
1
,
x
2
)
⪯
a
d
(
x
0
,
x
2
)
+
b
d
(
x
1
,
x
1
)
+
c
d
(
x
0
,
x
2
)
d
(
x
1
,
x
1
)
1
+
d
(
x
0
,
x
1
)
,
which implies that
(75)
|
d
(
x
1
,
x
2
)
|
≤
a
1
-
a
|
d
(
x
0
,
x
1
)
|
.
Similarly from (69), we have
(76)
a
d
(
x
2
,
[
T
x
1
]
α
)
+
b
d
(
x
1
,
[
S
x
2
]
α
)
+
c
d
(
x
2
,
[
T
x
1
]
α
)
d
(
x
1
,
[
S
x
2
]
α
)
1
+
d
(
x
1
,
x
2
)
∈
s
(
[
T
x
1
]
α
,
[
S
x
2
]
α
)
.
By Lemma 6(iii), we have
(77)
a
d
(
x
2
,
[
T
x
1
]
α
)
+
b
d
(
x
1
,
[
S
x
2
]
α
)
+
c
d
(
x
2
,
[
T
x
1
]
α
)
d
(
x
1
,
[
S
x
2
]
α
)
1
+
d
(
x
1
,
x
2
)
∈
s
(
x
2
,
[
S
x
2
]
α
)
.
By definition there exists some x
3
∈
[
S
x
2
]
α, such that
(78)
a
d
(
x
2
,
[
T
x
1
]
α
)
+
b
d
(
x
1
,
[
S
x
2
]
α
)
+
c
d
(
x
2
,
[
T
x
1
]
α
)
d
(
x
1
,
[
S
x
2
]
α
)
1
+
d
(
x
1
,
x
2
)
∈
s
(
d
(
x
2
,
x
3
)
)
.
That is,
(79)
d
(
x
2
,
x
3
)
⪯
a
d
(
x
2
,
[
T
x
1
]
α
)
+
b
d
(
x
1
,
[
S
x
2
]
α
)
+
c
d
(
x
2
,
[
T
x
1
]
α
)
d
(
x
1
,
[
S
x
2
]
α
)
1
+
d
(
x
1
,
x
2
)
.
By the meaning of W
x
(
[
T
y
]
α
) and W
x
(
[
S
y
]
α
) for x
,
y
∈
X, we get
(80)
d
(
x
2
,
x
3
)
⪯
a
d
(
x
2
,
x
2
)
+
b
d
(
x
1
,
x
3
)
+
c
d
(
x
2
,
x
2
)
d
(
x
1
,
x
3
)
1
+
d
(
x
1
,
x
2
)
,
which implies that
(81)
|
d
(
x
2
,
x
3
)
|
≤
b
1
-
b
|
d
(
x
1
,
x
2
)
|
.
Inductively, we can construct a sequence {
x
n
} in X such that, for n
=
0,1
,
2
,
…,
(82)
|
d
(
x
n
,
x
n
+
1
)
|
≤
q
n
|
d
(
x
0
,
x
1
)
|
,
with q
=
max
⁡
{
a
/
(
1
-
a
)
,
b
/
(
1
-
b
)
}
<
1
,
x
2
n
+
1
∈
[
S
x
2
n
]
α, and x
2
n
+
2
∈
[
T
x
2
n
+
1
]
α. Now for m
>
n, we get
(83)
|
d
(
x
n
,
x
m
)
|
≤
|
d
(
x
n
,
x
n
+
1
)
|
+
|
d
(
x
n
+
1
,
x
n
+
2
)
|
+
⋯
+
|
d
(
x
m
-
1
,
x
m
)
|
≤
[
q
n
+
q
n
+
1
+
⋯
+
q
m
-
1
]
|
d
(
x
0
,
x
1
)
|
=
[
q
n
1
-
q
]
|
d
(
x
0
,
x
1
)
|
,
and so
(84)
|
d
(
x
n
,
x
m
)
|
≤
q
n
1
-
q
|
d
(
x
0
,
x
1
)
|
⟶
0
as
m
,
n
⟶
∞
.
This implies that {
x
n
} is a Cauchy sequence in X. Since X is complete, there exists w
∈
X such that x
n
→
w as n
→
∞. We now show that w
∈
[
T
w
]
α and w
∈
[
S
w
]
α. From (69), we get
(85)
a
d
(
x
2
n
,
[
T
w
]
α
)
+
b
d
(
w
,
[
S
x
2
n
]
α
)
+
c
d
(
x
2
n
,
[
T
w
]
α
)
d
(
w
,
[
S
x
2
n
]
α
)
1
+
d
(
x
2
n
,
w
)
∈
s
(
[
S
x
2
n
]
α
,
[
T
w
]
α
)
.
By Lemma6(iii), we have
(86)
a
d
(
x
2
n
,
[
T
w
]
α
)
+
b
d
(
w
,
[
S
x
2
n
]
α
)
+
c
d
(
x
2
n
,
[
T
w
]
α
)
d
(
w
,
[
S
x
2
n
]
α
)
1
+
d
(
x
2
n
,
w
)
∈
s
(
x
2
n
+
1
,
[
T
w
]
α
)
.
By definition there exists some w
n
∈
[
T
w
]
α, such that
(87)
a
d
(
x
2
n
,
[
T
w
]
α
)
+
b
d
(
w
,
[
S
x
2
n
]
α
)
+
c
d
(
x
2
n
,
[
T
w
]
α
)
d
(
w
,
[
S
x
2
n
]
α
)
1
+
d
(
x
2
n
,
w
)
∈
s
(
x
2
n
+
1
,
[
T
w
]
α
)
∈
s
(
d
(
x
2
n
+
1
,
w
n
)
)
.
That is,
(88)
d
(
x
2
n
+
1
,
w
n
)
⪯
a
d
(
x
2
n
,
[
T
w
]
α
)
+
b
d
(
w
,
[
S
x
2
n
]
α
)
+
c
d
(
x
2
n
,
[
T
w
]
α
)
d
(
w
,
[
S
x
2
n
]
α
)
1
+
d
(
x
2
n
,
w
)
.
By the meaning of W
x
(
[
T
y
]
α
) and W
x
(
[
S
y
]
α
) for x
,
y
∈
X, we get
(89)
d
(
x
2
n
+
1
,
v
n
)
⪯
a
d
(
x
2
n
,
w
n
)
+
b
d
(
w
,
x
2
n
+
1
)
+
c
d
(
x
2
n
,
w
n
)
d
(
w
,
x
2
n
+
1
)
1
+
d
(
x
2
n
,
w
)
.
Now by using the triangular inequality, we get
(90)
d
(
x
2
n
+
1
,
w
n
)
⪯
a
d
(
x
2
n
,
x
2
n
+
1
)
+
a
d
(
x
2
n
+
1
,
w
n
)
+
b
d
(
w
,
x
2
n
+
1
)
+
c
d
(
x
2
n
,
w
n
)
d
(
w
,
x
2
n
+
1
)
1
+
d
(
x
2
n
,
w
)
,
and it follows that
(91)
d
(
x
2
n
+
1
,
w
n
)
⪯
a
1
-
a
d
(
x
2
n
,
x
2
n
+
1
)
+
b
1
-
a
d
(
w
,
x
2
n
+
1
)
+
c
1
-
a
d
(
x
2
n
,
w
n
)
d
(
w
,
x
2
n
+
1
)
1
+
d
(
x
2
n
,
w
)
.
By using again triangular inequality, we get
(92)
d
(
w
,
w
n
)
⪯
d
(
w
,
x
2
n
+
1
)
+
d
(
x
2
n
+
1
,
w
n
)
⪯
d
(
w
,
x
2
n
+
1
)
+
a
1
-
a
d
(
x
2
n
,
x
2
n
+
1
)
+
b
1
-
a
d
(
w
,
x
2
n
+
1
)
+
c
1
-
a
d
(
x
2
n
,
w
n
)
d
(
w
,
x
2
n
+
1
)
1
+
d
(
x
2
n
,
w
)
,
and it follows that
(93)
|
d
(
w
,
w
n
)
|
≤
|
d
(
w
,
x
2
n
+
1
)
|
+
a
1
-
a
|
d
(
x
2
n
,
x
2
n
+
1
)
|
+
b
1
-
a
|
d
(
w
,
x
2
n
+
1
)
|
+
c
1
-
a
|
d
(
x
2
n
,
w
n
)
|
|
d
(
w
,
x
2
n
+
1
)
|
|
1
+
d
(
x
2
n
,
w
)
|
.
By letting n
→
∞ in above inequality, we get |
d
(
w
,
w
n
)
|
→
0 as n
→
∞. By Lemma 2 [11], we have w
n
→
w as n
→
∞. Since[
T
w
]
α is closed, so w
∈
[
T
w
]
α. Similarly, it follows that w
∈
[
S
w
]
α. Thus there exists some w
∈
[
S
w
]
α
∩
[
T
w
]
α.Corollary 26.
Let(
X
,
d
) be a complete complex valued metric space and let S
,
T be fuzzy mappings from X into F
(
X
). Assume that there exists some α
∈
(
0,1
], such that, for each x
∈
X
,
[
S
x
]
α and [
T
x
]
α are nonempty closed bounded subsets of X; greatest lower bound of W
x
(
[
T
y
]
α
),W
x
(
[
S
y
]
α
)exists in C for all y
∈
Xand
(94)
h
(
d
(
x
,
[
T
y
]
α
)
+
d
(
y
,
[
S
x
]
α
)
)
∈
s
(
[
S
x
]
α
,
[
T
y
]
α
)
,
for all x
,
y
∈
X and 0
≤
h
<
1
/
2. Then there exists some w
∈
[
S
w
]
α
∩
[
T
w
]
α.By takingS
=
T in Theorem 25, we get the following corollary.Corollary 27.
Let(
X
,
d
) be a complete complex valued metric space and let T be fuzzy mapping from X into F
(
X
). Assume that there exists some α
∈
(
0,1
] such that, for each x
∈
X,[
T
x
]
α is nonempty closed bounded subset of X; greatest lower bound of W
x
(
[
T
y
]
α
)exists in C for all y
∈
Xand
(95)
a
d
(
x
,
[
T
y
]
α
)
+
b
d
(
y
,
[
T
x
]
α
)
+
c
d
(
x
,
[
T
y
]
α
)
d
(
y
,
[
T
x
]
α
)
1
+
d
(
x
,
y
)
∈
s
(
[
T
x
]
α
,
[
T
y
]
α
)
,
for all x
,
y
∈
X and nonnegative reals a
,
b, and c with a
+
b
+
c
<
1. Then there exists some w
∈
[
T
w
]
α.Corollary 28.
Let(
X
,
d
) be a complete complex valued metric space and let S
,
T be fuzzy mappings from X into F
(
X
) with g.l.b property such that, for each x
,
y
∈
X and α
∈
(
0,1
]
,
[
S
x
]
α and [
T
y
]
α are nonempty closed bounded subsets of X and
(96)
a
d
(
x
,
[
T
y
]
α
)
+
b
d
(
y
,
[
S
x
]
α
)
+
c
d
(
x
,
[
T
y
]
α
)
d
(
y
,
[
S
x
]
α
)
1
+
d
(
x
,
y
)
∈
s
(
[
S
x
]
α
,
[
T
y
]
α
)
,
for all x
,
y
∈
X and nonnegative reals a
,
b, and c with a
+
b
+
c
<
1. Then there exists some w
∈
[
S
w
]
α
∩
[
T
w
]
α.Remark 29.
By Definition11, one can have a host of corollaries of Chatterjea type contractive fuzzy mappings with g.l.b property easily from Theorem 25.Corollary 30 (see [20]).
Let(
X
,
d
) be a complete complex valued metric space and let F
,
G
:
X
→
C
B
(
X
) be multivalued mappings with g.l.b property such that
(97)
a
d
(
x
,
G
y
)
+
b
d
(
y
,
F
x
)
+
c
d
(
x
,
G
y
)
d
(
y
,
F
x
)
1
+
d
(
x
,
y
)
∈
s
(
F
x
,
G
y
)
,
for all x
,
y
∈
X and nonnegative reals a
,
b, and c with a
+
b
+
c
<
1. Then there exists some w
∈
F
w
∩
G
w.Proof.
Consider a pair of fuzzy mappingsS
,
T
:
X
→
F
(
X
) defined by
(98)
S
(
x
)
(
t
)
=
{
α
,
t
∈
F
x
0
,
t
∉
F
x
,
T
(
x
)
(
t
)
=
{
α
,
t
∈
G
x
0
,
t
∉
G
x
,
where α
∈
(
0,1
]. Then
(99)
[
S
x
]
α
=
{
t
:
S
(
x
)
(
t
)
≥
α
}
=
F
x
,
[
T
x
]
α
=
G
x
.
Thus, Theorem 25 can be applied to obtain w
∈
X such that
(100)
w
∈
[
S
w
]
α
∩
[
T
w
]
α
=
F
w
∩
G
w
.Remark 31.
Consider the following.(i)
By settingζ
,
κ, and ς in Corollary 18, β
,
γ, and η in Corollary 24, and a
,
b, and c in Corollary 30 with different combinations, one can get corresponding results in [19, 20] as corollaries.
(ii)
By Remark7 and Corollaries 18, 24, and 30, one can easily get the results of [13, 18–21].Example 32.
LetX
=
[
0,1
]and let C be the set of complexnumbers; define d
:
X
×
X
→
C as follows:
(101)
d
(
x
,
y
)
=
|
x
-
y
|
e
i
θ
,
where
θ
=
Arg
(
z
)
,
v
v
v
v
v
z
=
x
+
i
y
,
and
|
·
|
is
modulus
function
.
Then (
X
,
d
)is a complete complex valued metric space. Define a pair of mappings S
,
T
:
X
→
(
X
)
,for α
∈
(
0,1
]as follows.
Forx
,
y
∈
X with x
≤
y, we have(102)
S
(
x
)
(
t
)
=
{
α
if
0
≤
t
≤
x
40
α
2
if
x
40
<
t
≤
x
30
α
3
if
x
30
<
t
≤
x
20
α
5
if
x
20
<
t
≤
1
,
(103)
T
(
x
)
(
t
)
=
{
α
if
0
≤
t
≤
x
20
α
3
if
x
20
<
t
≤
x
10
α
4
if
x
10
<
t
≤
x
5
α
7
if
x
5
<
t
≤
1
,
such that
(104)
[
T
x
]
α
=
[
0
,
x
20
]
,
[
S
x
]
α
=
[
0
,
x
40
]
,
and then
(105)
W
x
(
[
T
y
]
α
)
=
{
d
(
x
,
u
)
:
u
∈
[
0
,
y
20
]
}
,
W
y
(
[
S
x
]
α
)
=
{
d
(
y
,
v
)
:
v
∈
[
0
,
x
40
]
}
.
Denoted
(
x
,
[
T
x
]
α
)and d
(
x
,
[
S
x
]
α
)bythe greatest lower boundsof W
x
(
[
T
x
]
α
) and W
x
(
[
S
x
]
α
). Then
(106)
d
(
x
,
[
T
y
]
α
)
(
z
)
=
{
0
if
x
<
y
20
(
x
-
y
20
)
e
i
θ
if
x
>
y
20
,
d
(
y
,
[
S
x
]
α
)
(
z
)
=
{
(
y
-
x
40
)
e
i
θ
,
as
y
>
x
40
,
and also
(107)
d
(
y
,
[
T
y
]
α
)
(
z
)
=
(
19
y
20
)
e
i
θ
,
d
(
x
,
[
S
x
]
α
)
(
z
)
=
(
39
x
40
)
e
i
θ
.
Moreover, if w
y
x
∈
C such that
(108)
w
y
x
=
|
y
20
-
x
40
|
e
θ
,
then
(109)
s
(
[
T
y
]
α
,
[
S
x
]
α
)
=
{
w
∈
C
:
w
x
y
⪯
w
}
.
Forx
>
y
/
20, we have
(110)
d
(
y
,
[
T
y
]
α
)
+
d
(
x
,
[
S
x
]
α
)
3
(
z
)
=
1
3
(
19
y
20
+
39
x
40
)
e
i
θ
=
1
3
(
y
-
y
20
+
x
-
x
40
)
e
i
θ
=
1
3
(
y
-
x
40
+
x
-
y
20
)
e
i
θ
=
1
3
(
y
-
x
40
)
e
i
θ
+
(
x
-
y
20
)
e
i
θ
=
d
(
y
,
[
S
x
]
α
)
+
d
(
x
,
[
T
y
]
α
)
3
(
z
)
=
1
3
(
6
y
20
+
13
y
20
+
6
x
40
+
33
x
40
)
e
i
θ
=
1
3
(
6
y
20
+
6
x
40
)
e
i
θ
+
(
13
y
20
+
33
x
40
)
e
i
θ
≻
1
3
(
6
y
20
+
6
x
40
)
e
i
θ
=
(
2
y
20
+
2
x
40
)
e
i
θ
>
(
y
20
+
x
40
)
e
i
θ
≻
|
y
20
-
x
40
|
e
i
θ
=
w
x
y
,
also as
(111)
1
2
d
(
x
,
y
)
=
1
2
|
x
-
y
|
e
i
θ
⪰
|
y
20
-
x
40
|
e
i
θ
.
It follows that, with ζ
=
κ
=
1
/
3
,
ς
≠
0, such that ζ
+
κ
+
ς
<
1, we have
(112)
ζ
d
(
x
,
[
S
x
]
α
)
+
κ
d
(
y
,
[
T
y
]
α
)
+
ς
d
(
x
,
[
S
x
]
α
)
d
(
y
,
[
T
y
]
α
)
1
+
d
(
x
,
y
)
∈
s
(
[
S
x
]
α
,
[
T
y
]
α
)
,
(113)
ζ
d
(
x
,
[
T
y
]
α
)
+
κ
d
(
y
,
[
S
x
]
α
)
+
ς
d
(
x
,
[
T
y
]
α
)
d
(
y
,
[
S
x
]
α
)
1
+
d
(
x
,
y
)
∈
s
(
[
S
x
]
α
,
[
T
y
]
α
)
,
and for ζ
=
1
/
2with κ
≠
0 and ς
≠
0, such that ζ
+
κ
+
ς
<
1, we have
(114)
ζ
d
(
x
,
y
)
+
κ
d
(
x
,
[
S
x
]
α
)
d
(
y
,
[
T
y
]
α
)
+
ς
d
(
y
,
[
S
x
]
α
)
d
(
x
,
[
T
y
]
α
)
1
+
d
(
x
,
y
)
∈
s
(
[
S
x
]
α
,
[
T
y
]
α
)
.
Hence T and S satisfy all the conditions of our main Theorem 12 to obtain 0
∈
[
S
0
]
α
∩
[
T
0
]
α.



---

*Source: 549504-2014-05-20.xml*