# Two New Types of Fixed Point Theorems in Complete Metric Spaces

Authors: Farshid Khojasteh; Mujahid Abbas; Simona Costache
Journal: Abstract and Applied Analysis
 (2014)

---

## Abstract

We introduce two new types of fixed point theorems in the collection of multivalued and single-valued mappings in complete metric spaces.


---

## Body

## 1. Introduction

LetT be a mapping on a complete (or compact) metric space (
X
,
d
). We do not assume richer structure such as convex metric spaces and Banach spaces. There are thousands of theorems which assure the existence of a fixed point of T. We can categorize these theorems into the following four types.(T1)
Leader type [1]: T has a unique fixed point and {
T
n
x
} converges to the fixed point for all x
∈
X. Such a mapping is called aPicard operator in [2].
(T2)
Unnamed type:T has a unique fixed point and {
T
n
x
} does not necessarily converge to the fixed point.
(T3)
Subrahmanyam type [3]: T may have more than one fixed point and {
T
n
x
} converges to a fixed point for all x
∈
X. Such a mapping is called aweakly Picard operator in [3, 4].
(T4)
Caristi type [5, 6]: T may have more than one fixed point and {
T
n
x
} does not necessarily converge to a fixed point.We know that most of the theorems such as Banach’s [7], Ćirić’s [8], Kannan’s [9], Kirk’s [10], Matkowski’s [11], Meir and Keeler’s [12], and Suzuki’s [13, 14] belong to (
T
1
). Also, very recently, Suzuki [15] characterized (
T
1
). Subrahmanyam’s theorem [3] belongs to (
T
3
), and Caristi’s theorem [5, 6] and its generalizations [15–17] belong to (
T
4
). On the other hand, as far as the authors do know, there are no theorems belonging to (
T
2
); see Kirk’s survey [18]. Also, recently many interesting fixed point theorems are proved in the framework of ordered metric spaces; see [18–35] and others.In this paper, motivated by the above, we introduce two new types of fixed point theorems in the collection of multivalued and single-valued mappings and will prove them, which belong to(
T
3
).Let(
X
,
d
) be a metric space, and let P
cl
⁡
,
bd
(
X
) denote the class of all nonempty, closed, and bounded subsets of X. Let T
:
X
→
P
cl
⁡
,
bd
(
X
) be a multivalued mapping on X. A point x
∈
X is called a fixed point of T if x
∈
T
x. Set Fix
⁡
(
T
)
=
{
x
∈
X
:
x
∈
T
x
}.A famous theorem on multivalued mappings is due to Nadler [36], which extended the Banach contraction principle to multivalued mappings. Many authors have studied the existence and uniqueness of strict fixed points for multivalued mappings in metric spaces; see, for example, [37–44] and references therein.LetH be the Hausdorff metric on P
cl
⁡
,
bd
(
X
) induced by d; that is,
(1)
H
(
A
,
B
)
∶
=
max
⁡
{
sup
⁡
x
∈
B
d
(
x
,
A
)
,
sup
⁡
x
∈
A
d
(
x
,
B
)
}
,
s
s
s
s
s
s
s
s
s
s
s
s
s
s
s
s
s
s
s
s
s
s
s
s
s
s
s
s
s
s
s
A
,
B
∈
P
cl
⁡
,
bd
(
X
)
.
Denote δ
(
x
,
A
)
=
sup
⁡
{
d
(
x
,
y
)
:
y
∈
A
} and D
(
x
,
A
)
=
inf
⁡
{
d
(
x
,
y
)
:
y
∈
A
}, where A
∈
P
cl
⁡
,
bd
(
X
).


## 2. Main Results

The following is the first our main results.Theorem 1.
Let(
X
,
d
) be a complete metric space and let T be a mapping from X into itself. Suppose that T satisfies the following condition:
(2)
d
(
T
x
,
T
y
)
≤
(
d
(
x
,
T
y
)
+
d
(
y
,
T
x
)
d
(
x
,
T
x
)
+
d
(
y
,
T
y
)
+
1
)
d
(
x
,
y
)
,
for all x
,
y
∈
X. Then(a)
T has at least one fixed point x
˙
∈
X;
(b)
{
T
n
x
} converges to a fixed point, for all x
∈
X;
(c)
ifx
˙
,
y
˙ are two distinct fixed points of T, then d
(
x
˙
,
y
˙
)
≥
1
/
2.Proof.
Letx
0
∈
X be arbitrary and choose a sequence {
x
n
} such that x
n
+
1
=
T
x
n. We have
(3)
d
(
x
n
+
1
,
x
n
)
=
d
(
T
x
n
,
T
x
n
-
1
)
≤
(
d
(
x
n
,
x
n
)
+
d
(
x
n
-
1
,
x
n
+
1
)
d
(
x
n
,
x
n
+
1
)
+
d
(
x
n
-
1
,
x
n
)
+
1
)
d
(
x
n
,
x
n
-
1
)
=
(
d
(
x
n
-
1
,
x
n
+
1
)
d
(
x
n
,
x
n
+
1
)
+
d
(
x
n
-
1
,
x
n
)
+
1
)
d
(
x
n
,
x
n
-
1
)
≤

(
d
(
x
n
-
1
,
x
n
)
+
d
(
x
n
,
x
n
+
1
)
d
(
x
n
,
x
n
+
1
)
+
d
(
x
n
-
1
,
x
n
)
+
1
)
d
(
x
n
,
x
n
-
1
)
.
Given
(4)
β
n
=
d
(
x
n
-
1
,
x
n
)
+
d
(
x
n
,
x
n
+
1
)
d
(
x
n
,
x
n
+
1
)
+
d
(
x
n
-
1
,
x
n
)
+
1
,
we have
(5)
d
(
x
n
+
1
,
x
n
)
≤
β
n
d
(
x
n
,
x
n
-
1
)
≤
(
β
n
β
n
-
1
)
d
(
x
n
-
1
,
x
n
-
2
)
⋮
≤
(
β
n
β
n
-
1
⋯
β
1
)
d
(
x
1
,
x
0
)
.
Observe that(
β
n
) is nonincreasing, with positive terms. So β
1
⋯
β
n
≤
β
1
n and β
1
n
→
0. It follows that
(6)
lim
⁡
n
→
∞
(
β
1
β
2
⋯
β
n
)
=
0
.
Thus, it is verified that
(7)
lim
⁡
n
→
∞
d
(
x
n
+
1
,
x
n
)
=
0
.
Now for allm
,
n
∈
N we have
(8)
d
(
x
m
,
x
n
)
≤
d
(
x
n
,
x
n
+
1
)
+
d
(
x
n
+
1
,
x
n
+
2
)
+
⋯
+
d
(
x
m
-
1
,
x
m
)
≤
[
(
β
n
β
n
-
1
⋯
β
1
)
+
(
β
n
+
1
β
n
⋯
β
1
)
+
⋯
+
(
β
m
-
1
β
m
-
2
⋯
β
1
)
]
d
(
x
1
,
x
0
)
=
∑
k
=
n
m
-
1
‍
(
β
k
β
k
-
1
⋯
β
1
)
d
(
x
1
,
x
0
)
.
Suppose that a
k
=
(
β
k
β
k
-
1
⋯
β
1
). Since
(9)
lim
⁡
k
→
∞
a
k
+
1
a
k
=
0
∑
k
=
1
∞
‍
a
k
<
∞. It means that
(10)
∑
k
=
n
m
-
1
‍
(
β
k
β
k
-
1
⋯
β
1
)
⟶
0
,
as m
,
n
→
∞. In other words, {
x
n
} is a Cauchy sequence and so converges to x
˙
∈
X.
We claim thatx
˙ is a fixed point.
Note that(11)
d
(
x
n
+
1
,
T
x
˙
)
≤
(
d
(
x
n
,
T
x
˙
)
+
d
(
x
˙
,
T
x
n
)
d
(
x
˙
,
T
x
˙
)
+
d
(
x
n
,
T
x
n
)
+
1
)
d
(
x
n
,
x
˙
)
.
On taking limit on both sides of (11), we have d
(
x
˙
,
T
x
˙
)
=
0. Thus, T
x
˙
=
x
˙.
If there exist two distinct fixed pointsx
˙
,
y
˙
∈
X, then
(12)
d
(
x
˙
,
y
˙
)
=
d
(
T
x
˙
,
T
y
˙
)
≤
[
d
(
x
˙
,
T
y
˙
)
+
d
(
T
x
˙
,
y
˙
)
]
d
(
x
˙
,
y
˙
)
=
2
[
d
(
x
˙
,
y
˙
)
]
2
.
Therefore,d
(
x
˙
,
y
˙
)
≥
1
/
2 and we find the desired results.In the following, two examples of such type of mappings, which satisfy (2), are given.Example 2.
LetX
=
{
0
,
1
/
2
,
1
} and let d
:
X
×
X
→
[
0
,
∞
) be defined by
(13)
d
(
0
,
1
2
)
=
2
,
d
(
1
,
1
2
)
=
5
2
,
d
(
0,1
)
=
3
,
d
(
0,0
)
=
d
(
1
2
,
1
2
)
=
d
(
1,1
)
=
0
,
d
(
a
,
b
)
=
d
(
b
,
a
)
,
∀
a
,
b
∈
X
.
(
X
,
d
) is a complete metric space. Let T
:
X
→
X be defined by
(14)
T
(
0
)
=
0
,
T
(
1
2
)
=
1
2
,
T
(
1
)
=
0
d
(
T
0
,
T
1
)
=
d
(
0,0
)
=
0
,
d
(
T
0
,
T
(
1
2
)
)
=
d
(
0
,
1
2
)
=
2
,
d
(
T
1
,
T
(
1
2
)
)
=
d
(
0
,
1
2
)
=
2
,
and we have
(15)
d
(
T
0
,
T
(
1
2
)
)
=
d
(
0
,
1
2
)
=
2
≤
(
d
(
0
,
T
(
1
/
2
)
)
+
d
(
1
/
2
,
T
(
0
)
)
d
(
0
,
T
0
)
+
d
(
1
/
2
,
T
(
1
/
2
)
)
+
1
)
×
d
(
0
,
1
2
)
=
8
and also
(16)
d
(
T
1
,
T
(
1
2
)
)
=
d
(
0
,
1
2
)
=
2
≤
(
d
(
1
,
T
(
1
/
2
)
)
+
d
(
1
/
2
,
T
(
1
)
)
d
(
1
,
T
1
)
+
d
(
1
/
2
,
T
(
1
/
2
)
)
+
1
)
×
d
(
1
,
1
2
)
=
(
5
/
2
+
2
4
)
×
5
2
=
45
16
.
Therefore, T satisfies all the conditions of Theorem 1. Also, T has two distinct fixed points {
0
,
1
/
2
} and d
(
0
,
1
/
2
)
=
2
≥
1
/
2.Example 3.
LetX
=
[
0,2
-
3
] be endowed with Euclidean metric and let T
:
X
→
X be defined by
(17)
T
x
=
{
0
0
≤
x
<
2
-
3
2
-
3
x
=
2
-
3
.
Then we claim that T satisfies all the conditions of Theorem 1.
Ifx
=
2
-
3 and 0
≤
y
<
2
-
3, we have
(18)
|
T
x
-
T
y
|
(
|
x
-
T
x
|
+
|
y
-
T
y
|
+
1
)
=
(
2
-
3
)
(
|
y
|
+
1
)
=
(
2
-
3
)
(
y
+
1
)
≤
(
2
-
3
-
y
)
2
-
(
2
-
3
)
(
2
-
3
-
y
)
=
(
|
x
-
T
y
|
+
|
y
-
T
x
|
)
|
x
-
y
|
.
Thus,
(19)
|
T
x
-
T
y
|
≤
(
|
x
-
T
y
|
+
|
y
-
T
x
|
|
x
-
T
x
|
+
|
y
-
T
y
|
+
1
)
|
x
-
y
|
.
Similar argument holds for the other conditions.Remark 4.
Note that in (2) the ratio
(20)
d
(
x
,
T
y
)
+
d
(
y
,
T
x
)
d
(
x
,
T
x
)
+
d
(
y
,
T
y
)
+
1
might be greater or less than 1 and has not introduced an upper bound. Note that if, for every x
,
y
∈
X, d
(
x
,
y
)
<
1
/
2, then we have
(21)
d
(
x
,
T
y
)
+
d
(
y
,
T
x
)
≤
2
d
(
x
,
y
)
+
d
(
x
,
T
x
)
+
d
(
y
,
T
y
)
<
d
(
x
,
T
x
)
+
d
(
y
,
T
y
)
+
1
.
It means that
(22)
d
(
x
,
T
y
)
+
d
(
y
,
T
x
)
d
(
x
,
T
x
)
+
d
(
y
,
T
y
)
+
1
<
1
,
and thus Theorem 1 is a special case of Banach contraction principle. Therefore, when (
X
,
d
) is a complete metric space such that, for all x
,
y
∈
X, d
(
x
,
y
)
≥
1
/
2, Theorem 1 is valuable because (20) might be greater than 1. Example 2 shows this note precisely.The following is the second in our main results.Theorem 5.
Let(
X
,
d
) be a complete metric space and let T be a multivalued mapping from X into P
c
l
,
b
d
(
X
). Let T satisfy the following:
(23)
H
(
T
x
,
T
y
)
≤
(
D
(
x
,
T
y
)
+
D
(
y
,
T
x
)
δ
(
x
,
T
x
)
+
δ
(
y
,
T
y
)
+
1
)
d
(
x
,
y
)
,
for all x
,
y
∈
X. Then T has a fixed point x
˙
∈
X.Proof.
Letx
0
∈
X and x
1
∈
T
x
0. For each 0
<
h
1
<
1 one can choose x
2
∈
T
x
1 such that
(24)
d
(
x
1
,
x
2
)
<
H
(
T
x
0
,
T
x
1
)
+
(
1
-
1
h
1
)
H
(
T
x
0
,
T
x
1
)
=
1
h
1
H
(
T
x
0
,
T
x
1
)
.
For each 0
<
h
n
<
1 we can choose x
n
+
1
∈
T
x
n such that
(25)
d
(
x
n
,
x
n
+
1
)
<
H
(
T
x
n
-
1
,
T
x
n
)
+
(
1
-
1
h
n
)
H
(
T
x
0
,
T
x
1
)
=
1
h
n
H
(
T
x
0
,
T
x
1
)
.
Specifically if
(26)
h
n
=
d
(
x
n
-
1
+
x
n
+
1
)
d
(
x
n
-
1
+
x
n
)
+
d
(
x
n
+
x
n
+
1
)
+
1
=
β
n
,
then
(27)
d
(
x
n
,
x
n
+
1
)
≤
β
n
d
(
x
n
-
1
,
x
n
)
≤
β
n
d
(
x
n
-
1
,
x
n
)
.
Therefore,
(28)
d
(
x
n
+
1
,
x
n
)
≤
β
n
d
(
x
n
,
x
n
-
1
)
≤
(
β
n
β
n
-
1
)
d
(
x
n
-
1
,
x
n
-
2
)
⋮
≤
(
β
n
β
n
-
1
⋯
β
1
)
d
(
x
1
,
x
0
)
.
It can easily be seen that
(29)
lim
⁡
n
→
∞
(
β
1
β
2
⋯
β
n
)
=
0
.
Thus, it is easily verified that
(30)
lim
⁡
n
→
∞
d
(
x
n
+
1
,
x
n
)
=
0
.
Now for all m
,
n
∈
N we have
(31)
d
(
x
m
,
x
n
)
≤
d
(
x
n
,
x
n
+
1
)
+
d
(
x
n
,
x
n
+
1
)
+
⋯
+
d
(
x
m
-
1
,
x
m
)
≤
[
(
β
n
β
n
-
1
⋯
β
1
)
+
(
β
n
+
1
β
n
⋯
β
1
)
+
⋯
+
(
β
m
-
1
β
m
-
2
⋯
β
1
)
]
d
(
x
1
,
x
0
)
=
∑
k
=
n
m
-
1
‍
(
β
k
β
k
-
1
⋯
β
1
)
d
(
x
1
,
x
0
)
.
Suppose that a
k
=
(
β
k
β
k
-
1
⋯
β
1
). Since
(32)
lim
⁡
k
→
∞
a
k
+
1
a
k
=
0
,
∑
k
=
1
∞
‍
a
k
<
∞. It means that
(33)
∑
k
=
n
m
-
1
‍
(
β
k
β
k
-
1
⋯
β
1
)
⟶
0
,
as m
,
n
→
∞. In other words, {
x
n
} is a Cauchy sequence and so converges to x
˙
∈
X. We claim that x
˙ is a fixed point. Consider
(34)
D
(
x
˙
,
T
x
˙
)
≤
d
(
x
˙
,
x
n
+
1
)
+
D
(
x
n
+
1
,
T
x
˙
)
≤
H
(
T
x
n
,
T
x
˙
)
+
d
(
x
˙
,
x
n
+
1
)
≤
(
D
(
x
˙
,
T
x
n
)
+
D
(
x
n
,
T
x
˙
)
δ
(
x
˙
,
T
x
˙
)
+
δ
(
x
n
,
T
x
n
)
+
1
)
×
d
(
x
n
,
x
˙
)
+
d
(
x
˙
,
x
n
+
1
)
≤
[
D
(
x
˙
,
x
n
+
1
)
+
D
(
x
n
,
T
x
˙
)
]
×
d
(
x
n
,
x
˙
)
+
d
(
x
˙
,
x
n
+
1
)
.
On taking limit on both sides of (31) we have D
(
x
˙
,
T
x
˙
)
=
0. It means that x
˙
∈
T
x
˙.Remark 6.
Note that Theorem5 is a generalization of Theorem 1 because by taking F
x
=
{
T
x
} and applying Theorem 5 for F we obtain Theorem 1.



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*Source: 325840-2014-06-26.xml*