# Fixed Point Theorems for Generalized Mizoguchi-Takahashi Graphic Contractions

Authors: Nawab Hussain; Jamshaid Ahmad; Marwan Amin Kutbi
Journal: Journal of Function Spaces
 (2016)

---

## Abstract

Remarkable feature of contractions is associated with the concept Mizoguchi-Takahashi function. For the purpose of extension and modification of classical ideas related with Mizoguchi-Takahashi contraction, we define generalized Mizoguchi-TakahashiG-contractions and establish some generalized fixed point theorems regarding these contractions in this paper. Some applications to the construction of a fixed point of multivalued mappings in ε-chainable metric space are also discussed.


---

## Body

## 1. Introduction

In 1922, Banach [1] presented a revolutionary contraction principle (namely, Banach contraction principle) in which Picard iteration process was used for the evaluation of a fixed point. This principle guarantees the existence and uniqueness of fixed points of certain self-mappings on metric spaces and provides a constructive method to find those fixed points. The Banach contraction principle was also used to establish the existence of a unique solution for a nonlinear integral equation [2]. For instance, it has been used to show the existence of solutions of nonlinear Volterra integral equations and nonlinear integrodifferential equations in Banach spaces and to show the convergence of algorithms in computational mathematics. Because of its importance and usefulness for mathematical theory, it has become a very popular tool in solving existence problems in many directions. Several authors have obtained various extensions and generalizations of Banach’s theorem by defining a variety of contractive type conditions for self- and non-self-mappings on metric spaces.Following the Banach contraction principle, Nadler [3] introduced the concept of multivalued contractions using the Hausdorff metric and established that a multivalued contraction possesses a fixed point in a complete metric space (see [4–14]). Subsequently, in 1969, Mizoguchi and Takahashi [15] proved the following famous result as a generalization of Nadler’s fixed point theorem [3].Theorem 1 (see [15]).
Let(
X
,
d
) be a complete metric space and let S
:
X
→
C
B
(
X
) be a multivalued mapping. Assume that(1)
H
S
x
,
S
y
≤
φ
d
x
,
y
d
x
,
yfor all x
,
y
∈
X, where φ
:
[
0
,
∞
)
→
[
0,1
) is such that(2)
lim
v
→
u
+
⁡
sup
⁡
φ
v
<
1for all u
∈
[
0
,
∞
)
. Then, S has a fixed point.We denote byΩ the set of all functions φ
:
[
0
,
∞
)
→
[
0,1
) satisfying inequalities (1) and (2).Recently, Javahernia et al. [16] generalized the abovementioned function by introducing the notion of generalized Mizoguchi-Takahashi function in such a way.Definition 2 (see [16]).
A functionβ
:
R
×
R
→
R is called a generalized Mizoguchi-Takahashi function (shortly, generalized M
T-function) if the following conditions hold: (
a
1
)
0
<
β
(
u
,
v
)
<
1 for all u
,
v
>
0;
(
a
2
)
for any bounded sequence{
u
n
}
⊂
(
0
,
+
∞
) and any nonincreasing sequence {
v
n
}
⊂
(
0
,
+
∞
), one has (3)
lim
n
→
∞
⁡
sup
⁡
β
u
n
,
v
n
<
1
.Consistent with Javahernia et al. [16], we denote by Λ the set of all functions β
:
R
×
R
→
R satisfying conditions (
a
1
)-(
a
2
).On the other hand, Jachymski [17] proved some fixed point results in metric spaces endowed with a graph and generalized simultaneously Banach’s contraction principle from metric and partially ordered metric spaces in 2008. Consistent with Jachymski, let (
X
,
d
) be a metric space and let Δ denote the diagonal of the Cartesian product X
×
X. Consider a directed graph G such that the set V
(
G
) of its vertices coincides with X and the set E
(
G
) of its edges contains all loops; that is, E
(
G
)
⊇
Δ. We assume G has no parallel edges and so we can identify G with the pair (
V
(
G
)
,
E
(
G
)
). Moreover, we may treat G as a weighted graph (see [17]) by assigning to each edge the distance between its vertices. If x and y are vertices in a graph G, then a path in G from x to y of length N
(
N
∈
N
) is a sequence x
i
i
=
0
N of N
+
1 vertices such that x
0
=
x, x
N
=
y, and (
x
n
-
1
,
x
n
)
∈
E
(
G
) for each i
=
1
,
…
,
N
. A graph G isconnected if there is a path between any two vertices. G isweakly connected if G
~ is connected.In 2010, Beg et al. [18] obtained sufficient conditions for the existence of a fixed point of multivalued mapping in the complete metric space endowed with a graph. For more details in this direction, we refer the reader to [19–24].Recently, Sultana and Vetrivel [25] introduced the notion of Mizoguchi-Takahashi G-contraction in this way.Definition 3 (see [25]).
Let(
X
,
d
) be metric space endowed with a graph and let S
:
X
→
C
B
(
X
) be a multivalued mapping. Then, S is called a Mizoguchi-Takahashi G
-contraction if (1)
for allx
,
y
∈
X, with x
≠
y and (
x
,
y
)
∈
E
(
G
), (4)
H
S
x
,
S
y
≤
φ
d
x
,
y
d
x
,
y
,
whereφ
∈
Ω;
(2)
u
∈
S
(
x
) and v
∈
S
(
y
) are such that d
u
,
v
≤
d
x
,
y; then, (
u
,
v
)
∈
E
(
G
)
.They established the following fixed point theorem as main result regarding Mizoguchi-TakahashiG-contraction.Theorem 4 (see [25]).
Let(
X
,
d
) be a complete metric space endowed with a graph and S
:
X
→
C
B
(
X
) a Mizoguchi-Takahashi G-contraction. Assume that there exist N
∈
N and x
0
∈
X such that(1)
x
0
G
N
∩
S
(
x
0
)
≠
0;
(2)
for any sequencex
n
n
∈
N
∈
X, if x
n
→
x and x
n
∈
x
n
-
1
G
N
∩
S
(
x
n
-
1
) for all n
∈
N, then, there is a subsequence x
n
m
m
∈
N
∈
X such that (
x
n
m
,
x
)
∈
E
(
G
) for each m
∈
N
.Then, S has a fixed point and the sequence x
n
n
∈
N
∈
X converges to the fixed point of S.The purpose of this paper is to define generalized Mizoguchi-TakahashiG-contractions and prove some generalized fixed point theorems for these contractions.The following lemma of Sultana and Vetrivel [25] for single-valued mappings is crucial for some of our proofs.Lemma 5.
LetS
:
X
→
X be a single-valued mapping satisfying (5)
d
S
x
,
S
y
≤
φ
d
x
,
y
d
x
,
y for all (
x
,
y
)
∈
E
(
G
) with x
≠
y
. Let p
,
q
∈
X be two points such that there is a path x
i
i
=
0
N from p to q. Then, there exist λ
∈
[
0,1
) and n
0
∈
N such that (6)
d
S
n
p
,
S
n
q
≤
C
λ
n for all n
≥
n
0, where C
≥
0
.


## 2. Main Results

We start this section with the definition of a generalized Mizoguchi-TakahashiG-contraction.Definition 6.
Let(
X
,
d
) be metric space endowed with a graph and S
:
X
→
C
B
(
X
). Then, S is said to be generalized Mizoguchi-Takahashi G-contraction if (1)
for allx
,
y
∈
X, with x
≠
y and (
x
,
y
)
∈
E
(
G
),(7)
H
S
x
,
S
y
≤
β
H
S
x
,
S
y
,
d
x
,
y
d
x
,
y
,
whereβ
∈
Λ;
(2)
u
∈
S
(
x
) and v
∈
S
(
y
) are such that d
(
u
,
v
)
≤
d
(
x
,
y
); then, (
u
,
v
)
∈
E
(
G
)
.The following remark illustrates our main definition.Remark 7.
Takingβ
(
u
,
v
)
=
φ
(
v
) in Definition 6, we get Definition 3 which is the main definition of [25].Now, we state our main result of this section.Theorem 8.
Let(
X
,
d
) be a complete metric space endowed with a graph and S
:
X
→
C
B
(
X
) a generalized Mizoguchi-Takahashi G-contraction. Assume that there exist N
∈
N and x
0
∈
X such that(1)
x
0
G
N
∩
S
(
x
0
)
≠
0;
(2)
for any sequencex
n
n
∈
N
∈
X, if x
n
→
x and x
n
∈
x
n
-
1
G
N
∩
S
(
x
n
-
1
) for all n
∈
N, then there is a subsequence x
n
m
m
∈
N
∈
X such that (
x
n
m
,
x
)
∈
E
(
G
) for each m
∈
N
.Then, S has a fixed point and the sequence x
n
n
∈
N
∈
X converges to a fixed point of S.Proof.
Letx
1
∈
x
0
G
N
∩
S
(
x
0
)
. Then, there is a path y
j
j
=
0
N in G from x
0 to x
1; that is, y
0
=
x
0, y
N
=
x
1, and (
y
j
-
1
,
y
j
)
∈
E
(
G
) for each j
=
1,2
,
…
,
N
. Select a positive integer m
1
1 such that (8)
β
m
1
1
H
S
y
0
,
S
y
1
,
d
y
0
,
y
1
+
β
H
S
y
0
,
S
y
1
,
d
y
0
,
y
1
d
y
0
,
y
1
<
d
y
0
,
y
1
.Since y
0
,
y
1
∈
E
(
G
), it follows from (7) that(9)
H
S
y
0
,
S
y
1
≤
β
H
S
y
0
,
S
y
1
,
d
y
0
,
y
1
d
y
0
,
y
1and, also, since x
1
∈
S
(
x
0
=
y
0
), by the definition of Hausdorff metric, there exists y
1
1
∈
S
(
y
1
) such that (10)
d
x
1
,
y
1
1
≤
H
S
y
0
,
S
y
1
+
β
m
1
1
H
S
y
0
,
S
y
1
,
d
y
0
,
y
1
. From (9), we have (11)
d
x
1
,
y
1
1
≤
β
H
S
y
0
,
S
y
1
,
d
y
0
,
y
1
d
y
0
,
y
1
+
β
m
1
1
H
S
y
0
,
S
y
1
,
d
y
0
,
y
1
.It follows from (8) and (11) that d
(
x
1
,
y
1
1
)
<
d
(
y
0
,
y
1
)
. Thus, (
x
1
,
y
1
1
)
∈
E
(
G
)
. Select a positive integer m
1
2 such that (12)
β
m
1
2
H
S
y
1
,
S
y
2
,
d
y
1
,
y
2
+
β
H
S
y
1
,
S
y
2
,
d
y
1
,
y
2
d
y
1
,
y
2
<
d
y
1
,
y
2
.Since (
x
1
,
y
1
1
)
∈
E
(
G
), it follows from (7) that(13)
H
S
y
1
,
S
y
2
≤
β
H
S
y
1
,
S
y
2
,
d
y
1
,
y
2
d
y
1
,
y
2
.Since y
1
1
∈
S
(
y
1
) , by the definition of Hausdorff metric, there exists y
1
2
∈
S
(
y
2
) such that(14)
d
y
1
1
,
y
1
2
≤
H
S
y
1
,
S
y
2
+
β
m
1
2
H
S
y
1
,
S
y
2
,
d
y
1
,
y
2
.From (13), we have (15)
d
y
1
1
,
y
1
2
≤
β
H
S
y
1
,
S
y
2
,
d
y
1
,
y
2
d
y
1
,
y
2
+
β
m
1
2
H
S
y
1
,
S
y
2
,
d
y
1
,
y
2
.It follows from (12) and (15) that d
(
y
1
1
,
y
1
2
)
<
d
(
y
1
,
y
2
)
. Thus, (
y
1
1
,
y
1
2
)
∈
E
(
G
)
. Continuing the abovementioned procedure, we have (16)
y
1
j
-
1
∈
S
y
j
-
1
,
y
1
j
-
2
,
y
1
j
-
1
∈
E
G and, for a chosen positive integer m
1
j for each j
=
3,4
,
…
,
N, (17)
β
H
S
y
j
-
1
,
S
y
j
,
d
y
j
-
1
,
y
j
d
y
j
-
1
,
y
j
+
β
m
1
j
H
S
y
j
-
1
,
S
y
j
,
d
y
j
-
1
,
y
j
<
d
y
j
-
1
,
y
j
.Since (
y
1
j
-
2
,
y
1
j
-
1
)
∈
E
(
G
), it follows from (7) that(18)
H
S
y
j
-
1
,
S
y
j
≤
β
H
S
y
j
-
1
,
S
y
j
,
d
y
j
-
1
,
y
j
d
y
j
-
1
,
y
j
.Since y
1
j
-
1
∈
S
(
y
j
-
1
) , by the definition of Hausdorff metric, there exists y
1
j
∈
S
(
y
j
) such that (19)
d
y
1
j
-
1
,
y
1
j
≤
H
S
y
j
-
1
,
S
y
j
+
β
m
1
j
H
S
y
j
-
1
,
S
y
j
,
d
y
j
-
1
,
y
j
≤
β
H
S
y
j
-
1
,
S
y
j
,
d
y
j
-
1
,
y
j
d
y
j
-
1
,
y
j
+
β
m
1
j
H
S
y
j
-
1
,
S
y
j
,
d
y
j
-
1
,
y
j
<
d
y
j
-
1
,
y
j
; that is, (
y
1
j
-
1
,
y
1
j
)
∈
E
(
G
) for each j
=
3,4
,
…
,
N
. Set x
2
=
y
1
N
∈
S
(
y
N
=
x
1
)
. Then, there is a path y
1
j
j
=
0
N in G from x
1 to x
2; that is, y
1
0
=
x
1
,
y
1
N
=
x
2, and (
y
1
j
-
1
,
y
1
j
)
∈
E
(
G
) for each j
=
1,2
,
…
,
N
. Hence, x
2
∈
x
1
G
N
∩
S
(
x
1
)
. Select a positive integer m
2
1
>
m
1
1 such that(20)
β
m
2
1
H
S
y
1
0
,
S
y
1
1
,
d
y
1
0
,
y
1
1
+
β
H
S
y
1
0
,
S
y
1
1
,
d
y
1
0
,
y
1
1
d
y
1
0
,
y
1
1
<
d
y
1
0
,
y
1
1
.Since (
y
1
0
,
y
1
1
)
∈
E
(
G
), it follows from (7) that(21)
H
S
y
1
0
,
S
y
1
1
≤
β
H
S
y
1
0
,
S
y
1
1
,
d
y
1
0
,
y
1
1
d
y
1
0
,
y
1
1and, also, x
2
∈
S
(
x
1
=
y
1
0
), which gives, by the definition of Hausdorff metric, the notion that there exists y
2
1
∈
S
(
y
1
1
) such that (22)
d
x
2
,
y
2
1
≤
H
S
y
1
0
,
S
y
1
1
+
β
m
2
1
H
S
y
1
0
,
S
y
1
1
,
d
y
1
0
,
y
1
1
≤
β
H
S
y
1
0
,
S
y
1
1
,
d
y
1
0
,
y
1
1
d
y
1
0
,
y
1
1
+
β
m
2
1
H
S
y
1
0
,
S
y
1
1
,
d
y
1
0
,
y
1
1
.It follows from (20) and (22) that d
(
x
2
,
y
2
1
)
<
d
(
y
1
0
,
y
1
1
) and so (
x
2
,
y
2
1
)
∈
E
(
G
)
. Select a positive integer m
2
2
>
m
1
2 such that (23)
β
m
2
2
H
S
y
1
1
,
S
y
1
2
,
d
y
1
1
,
y
1
2
+
β
H
S
y
1
1
,
S
y
1
2
,
d
y
1
1
,
y
1
2
d
y
1
1
,
y
1
2
<
d
y
1
1
,
y
1
2
.Since x
2
,
y
2
1
∈
E
(
G
) and y
2
1
∈
S
(
y
1
1
) , it follows from (7) that(24)
H
S
y
1
1
,
S
y
1
2
≤
β
H
S
y
1
1
,
S
y
1
2
,
d
y
1
1
,
y
1
2
d
y
1
1
,
y
1
2
.Since y
2
1
∈
S
(
y
1
1
), by the definition of Hausdorff metric, there exists y
2
2
∈
S
(
y
1
2
) such that (25)
d
y
2
1
,
y
2
2
≤
H
S
y
1
1
,
S
y
1
2
+
β
m
2
2
H
S
y
1
1
,
S
y
1
2
,
d
y
1
1
,
y
1
2
≤
β
H
S
y
1
1
,
S
y
1
2
,
d
y
1
1
,
y
1
2
d
y
1
1
,
y
1
2
+
β
m
2
2
H
S
y
1
1
,
S
y
1
2
,
d
y
1
1
,
y
1
2
.It follows from (23) and (25) that d
(
y
2
1
,
y
2
2
)
<
d
(
y
1
1
,
y
1
2
) and so (
y
2
1
,
y
2
2
)
∈
E
(
G
)
. Continuing the abovementioned procedure, we have (26)
y
2
j
-
1
∈
S
y
1
j
-
1
,
y
2
j
-
2
,
y
2
j
-
1
∈
E
G and, for a chosen positive integer m
2
j
>
m
1
j for each j
=
3,4
,
…
,
N, (27)
β
H
S
y
1
j
-
1
,
S
y
1
j
,
d
y
1
j
-
1
,
y
1
j
d
y
1
j
-
1
,
y
1
j
+
β
m
1
j
H
S
y
1
j
-
1
,
S
y
1
j
,
d
y
1
j
-
1
,
y
1
j
<
d
y
1
j
-
1
,
y
1
j
.As (
y
2
j
-
2
,
y
2
j
-
1
)
∈
E
(
G
), from (7), we have(28)
H
S
y
1
j
-
1
,
S
y
1
j
≤
β
H
S
y
1
j
-
1
,
S
y
1
j
,
d
y
1
j
-
1
,
y
1
j
d
y
1
j
-
1
,
y
1
j
.Since y
2
j
-
1
∈
S
y
1
j
-
1, by the definition of Hausdorff metric, there exists y
2
j
∈
S
y
1
j such that (29)
d
y
2
j
-
1
,
y
2
j
≤
H
S
y
1
j
-
1
,
S
y
1
j
+
β
m
1
j
H
S
y
1
j
-
1
,
S
y
1
j
,
d
y
1
j
-
1
,
y
1
j
≤
β
H
S
y
1
j
-
1
,
S
y
1
j
,
d
y
1
j
-
1
,
y
1
j
d
y
1
j
-
1
,
y
1
j
+
β
m
1
j
H
S
y
1
j
-
1
,
S
y
1
j
,
d
y
1
j
-
1
,
y
1
j
<
d
y
1
j
-
1
,
y
1
j
. Hence, (
y
2
j
-
1
,
y
2
j
)
∈
E
(
G
) for each j
=
3,4
,
…
,
N
. Set x
3
=
y
2
N
∈
S
(
y
1
N
=
x
2
)
. Then, there is a path y
2
j
j
=
0
N in G from x
2 to x
3; that is, y
2
0
=
x
2
,
y
2
N
=
x
3, and (
y
2
j
-
1
,
y
2
j
)
∈
E
(
G
) for each j
=
1,2
,
…
,
N
. Hence, x
3
∈
x
2
G
N
∩
S
(
x
2
)
. Continuing in this way for each n
∈
N, we construct x
n
+
1
∈
x
n
G
N
∩
S
(
x
n
) by generating a path y
n
j
j
=
0
N in G from x
n to x
n
+
1; that is, y
n
0
=
x
n
,
y
n
N
=
x
n
+
1, and y
n
j
∈
S
(
y
n
-
1
j
) for each j
=
1,2
,
…
,
N, and, for a chosen positive integer m
n
j with m
n
j
>
m
n
-
1
j for 1
≤
j
≤
N, we have (30)
β
H
S
y
n
-
1
j
-
1
,
S
y
n
-
1
j
,
d
y
n
-
1
j
-
1
,
y
n
-
1
j
d
y
n
-
1
j
-
1
,
y
n
-
1
j
+
β
m
n
j
H
S
y
n
-
1
j
-
1
,
S
y
n
-
1
j
,
d
y
n
-
1
j
-
1
,
y
n
-
1
j
<
d
y
n
-
1
j
-
1
,
y
n
-
1
j
.Since (
y
n
j
-
2
,
y
n
j
-
1
)
∈
E
(
G
) and y
n
j
-
1
∈
S
(
y
n
-
1
j
-
1
) , it follows from (7) that(31)
H
S
y
n
-
1
j
-
1
,
S
y
n
-
1
j
≤
β
H
S
y
n
-
1
j
-
1
,
S
y
n
-
1
j
,
d
y
n
-
1
j
-
1
,
y
n
-
1
j
d
y
n
-
1
j
-
1
,
y
n
-
1
j
.Since y
n
j
-
1
∈
S
(
y
n
j
-
1
), by the definition of Hausdorff metric, there exists y
n
j
∈
S
(
y
n
-
1
j
) such that (32)
d
y
n
j
-
1
,
y
n
j
≤
H
S
y
n
-
1
j
-
1
,
S
y
n
-
1
j
+
β
m
n
j
H
S
y
n
-
1
j
-
1
,
S
y
n
-
1
j
,
d
y
n
-
1
j
-
1
,
y
n
-
1
j
≤
β
H
S
y
n
-
1
j
-
1
,
S
y
n
-
1
j
,
d
y
n
-
1
j
-
1
,
y
n
-
1
j
d
y
n
-
1
j
-
1
,
y
n
-
1
j
+
β
m
n
j
H
S
y
n
-
1
j
-
1
,
S
y
n
-
1
j
,
d
y
n
-
1
j
-
1
,
y
n
-
1
j
<
d
y
n
-
1
j
-
1
,
y
n
-
1
j
.Hence, for 1
≤
j
≤
N, we have (
y
n
j
-
1
,
y
n
j
)
∈
E
(
G
)
. For 1
≤
j
≤
N, we denote d
0
j
=
d
(
y
j
-
1
,
y
j
) and d
n
j
=
d
(
y
n
j
-
1
,
y
n
j
)
. From the abovementioned inequality, we have the notion that d
n
j
n
∈
N is a monotone nonincreasing sequence of nonnegative real numbers for each j
≥
1. Let d
n
j
→
r
j
≥
0
. By (
a
1
) and inequality (31), it follows that {
H
(
S
(
y
n
-
1
j
-
1
)
,
S
(
y
n
-
1
j
)
)
} is a bounded sequence. By (
a
2
), we have(33)
lim
n
→
∞
⁡
sup
⁡
β
H
S
y
n
-
1
j
-
1
,
S
y
n
-
1
j
,
d
y
n
j
-
1
,
y
n
j
<
1
.Let π
n
j
=
H
(
S
(
y
n
-
1
j
-
1
)
,
S
(
y
n
-
1
j
)
) for each 1
≤
j
≤
N
. Then, there exist positive integers n
0
j and real numbers γ
j
∈
[
0,1
) such that β
(
π
n
j
,
d
n
j
)
<
γ
j for each n
≥
n
0
j, where (34)
lim
n
→
∞
⁡
sup
⁡
β
π
n
j
,
d
n
j
<
γ
j
<
1
. Hence, for each 1
≤
j
≤
N, we have(35)
β
π
n
j
,
d
n
j
<
γfor all n
≥
n
0, where γ
=
max
1
≤
j
≤
N
⁡
γ
j and n
0
=
max
1
≤
j
≤
N
⁡
n
0
j
. For 1
≤
j
≤
N and n
≥
n
0
+
1, we have (36)
d
n
j
≤
β
π
n
-
1
j
,
d
n
-
1
j
d
n
j
+
β
m
n
j
π
n
-
1
j
,
d
n
-
1
j
≤
⋯
≤
∏
s
=
0
n
-
1
β
π
s
j
,
d
s
j
d
0
j
+
∑
k
=
1
n
-
1
∏
s
=
k
+
1
n
β
π
s
-
1
j
,
d
s
-
1
j
β
m
k
j
π
k
-
1
j
,
d
k
-
1
j
+
β
m
n
j
π
n
-
1
j
,
d
n
-
1
j
≤
γ
n
-
n
0
∏
s
=
0
n
0
β
π
s
j
,
d
s
j
d
0
j
+
∑
k
=
1
n
-
1
∏
s
=
max
⁡
n
0
+
1
,
k
+
1
n
β
π
s
-
1
j
,
d
s
-
1
j
β
m
k
j
π
k
-
1
j
,
d
k
-
1
j
+
γ
n
.Let us denote the second term of the last inequality by B to get (37)
B
=
∑
k
=
1
n
-
1
∏
s
=
max
⁡
n
0
+
1
,
k
+
1
n
β
π
s
-
1
j
,
d
s
-
1
j
β
m
k
j
π
k
-
1
j
,
d
k
-
1
j
≤
γ
n
-
n
0
∑
k
=
1
n
0
β
m
k
j
π
k
-
1
j
,
d
k
-
1
j
+
∑
k
=
n
0
+
1
n
-
1
γ
n
-
k
β
m
k
j
π
k
-
1
j
,
d
k
-
1
j
≤
Q
1
j
γ
n
+
∑
k
=
n
0
+
1
n
-
1
γ
n
-
k
+
m
k
j
≤
Q
1
j
γ
n
+
γ
n
γ
m
n
0
j
-
n
0
-
1
1
-
γ
≤
Q
2
j
γ
n
, where Q
1
j and Q
2
j are nonnegative real numbers. Hence, we have(38)
d
n
j
≤
γ
n
-
n
0
∏
s
=
0
n
0
β
π
s
j
,
d
s
j
d
0
j
+
Q
2
j
γ
n
+
γ
n
≤
Q
j
γ
n
,where Q
j are nonnegative real numbers. Thus, for each n
≥
n
0
+
1, (39)
d
x
n
,
x
n
+
1
=
d
y
n
0
,
y
n
N
≤
∑
j
=
1
N
d
n
j
≤
∑
j
=
1
N
Q
j
γ
n
. Hence, for each n
≥
n
0
+
1 and m
∈
N, we have (40)
d
x
n
,
x
n
+
m
≤
d
x
n
,
x
n
+
1
+
⋯
+
d
x
n
+
m
-
1
,
x
n
+
m
≤
∑
j
=
1
N
Q
j
γ
n
+
⋯
+
γ
n
+
m
-
1
≤
∑
j
=
1
N
Q
j
γ
n
1
-
γ
. Since γ
n
→
0 as n
→
+
∞, {
x
n
} is a Cauchy sequence and so x
n
→
x
∗
∈
X. If (41)
E
=
n
∈
N
:
x
n
=
x
∗ and #
(
E
)
=
∞, where #
(
E
) is the cardinal number of E, then x
n
+
1
∈
S
(
x
n
)
=
S
(
x
∗
) and x
∗
∈
S
(
x
∗
)
. If #
(
E
)
<
∞, then one can deduce that there exists l
∈
N such that x
n
≠
x
∗ for all n
∈
N with n
≥
l
. This means H
(
S
(
x
n
)
,
S
(
x
∗
)
)
>
0 and d
(
x
n
,
x
∗
)
>
0 for each n
≥
l
. Since x
n
→
x
∗
∈
X, for a sequence {
x
n
} with x
n
∈
x
n
G
N
∩
S
(
x
n
) for each n
∈
N, there is a subsequence x
n
m
m
∈
N
∈
X such that x
n
m
,
x
∈
E
(
G
) for each m
∈
N
. Thus, we have (42)
d
x
∗
,
S
x
∗
≤
d
x
∗
,
x
n
m
+
1
+
d
x
n
m
+
1
,
S
x
∗
≤
d
x
∗
,
x
n
m
+
1
+
H
S
x
n
m
,
S
x
∗
≤
d
x
∗
,
x
n
m
+
1
+
β
H
S
x
n
m
,
S
x
∗
,
d
x
n
m
,
x
∗
d
x
n
m
,
x
∗
. Taking the limit as m
→
∞, in the last inequality, we have d
(
x
∗
,
S
(
x
∗
)
)
=
0
. Since S
(
x
∗
) is closed, it follows that x
∗
∈
S
(
x
∗
)
. This completes the proof.Remark 9.
Takingβ
(
u
,
v
)
=
φ
(
v
) in Theorem 8, we get main result of Sultana and Vetrivel’s work which is Theorem 4.Remark 10.
We have Theorem1 as a corollary to our Theorem 8 by taking β
(
u
,
v
)
=
φ
(
v
)
,
E
(
G
)
=
X
×
X, and N
=
1
.Remark 11.
Takingβ
(
u
,
v
)
=
μ in Theorem 8, we can get the main result of [18].Javahernia et al. [16] introduced the concept of weak lower semicontinuity in the following way.Definition 12.
A functionϕ
:
[
0
,
∞
)
→
[
0
,
∞
) is said to be weak lower semicontinuous function if, for each bounded sequence {
u
n
}
⊂
(
0
,
+
∞
), one has (43)
lim
n
→
∞
⁡
inf
⁡
ϕ
u
n
>
0
.Consistent with Javahernia et al. [16], we denote by Ϝ the set of all functions ϕ
:
[
0
,
∞
)
→
[
0
,
∞
) satisfying the abovementioned condition.Theorem 13.
Let(
X
,
d
) be a complete metric space endowed with a graph and let S
:
X
→
C
B
(
X
) be a multivalued mapping such that(1)
for allx
,
y
∈
X, with x
≠
y and (
x
,
y
)
∈
E
(
G
), (44)
H
S
x
,
S
y
≤
d
x
,
y
-
ϕ
d
x
,
y
,
whereϕ
:
[
0
,
∞
)
→
[
0
,
∞
) is such that ϕ
(
0
)
=
0, ϕ
(
v
)
<
v, and ϕ
∈
Ϝ;
(2)
ifu
∈
S
(
x
) and v
∈
S
(
y
) are such that d
(
u
,
v
)
≤
d
(
x
,
y
), then (
u
,
v
)
∈
E
(
G
)
.Assume that there exist N
∈
N and x
0
∈
X such that(a)
x
0
G
N
∩
S
x
0
≠
0;
(b)
for any sequencex
n
n
∈
N
∈
X, if x
n
→
x and x
n
∈
x
n
-
1
G
N
∩
S
(
x
n
-
1
) for each n
∈
N, then there is a subsequence x
n
m
m
∈
N
∈
X such that (
x
n
m
,
x
)
∈
E
(
G
) for each m
∈
N
.Then, S has a fixed point and the sequence x
n
n
∈
N
∈
X converges to the fixed point of S.Proof.
Defineβ
(
u
,
v
)
=
1
-
ϕ
(
u
)
/
u for all u
,
v
>
0
. Since, for each bounded sequence {
u
n
}
⊂
(
0
,
+
∞
), we have l
i
m
n
→
∞
inf
⁡
ϕ
(
u
n
)
>
0, we have l
i
m
n
→
∞
inf
⁡
ϕ
(
u
n
)
/
u
n
>
0
. Thus, we have (45)
lim
n
→
∞
⁡
sup
⁡
1
-
ϕ
u
n
u
n
=
1
-
lim
n
→
∞
⁡
inf
⁡
ϕ
u
n
u
n
<
0
. This shows that β
∈
Λ
. Also, we have (46)
H
S
x
,
S
y
≤
β
H
S
x
,
S
y
,
d
x
,
y
d
x
,
y
. Thus, by Theorem 8, w is fixed point of S
. This completes the proof.


## 3. Applications

As an application of Theorem8, we now establish the following theorem for mappings satisfying inequality (7) uniformly locally on an ε-chainable metric space.Theorem 14.
Let(
X
,
d
) be a complete ε-chainable metric space and let S
:
X
→
C
B
(
X
) be a multivalued mapping such that, for all x
,
y
∈
X with x
≠
y and d
(
x
,
y
)
<
ε, there exists β
∈
Λ satisfying (47)
H
S
x
,
S
y
≤
β
H
S
x
,
S
y
,
d
x
,
y
d
x
,
y
. Then, S has a fixed point.Proof.
We consider the graphG with V
(
G
)
=
X and E
(
G
)
=
(
x
,
y
)
∈
X
×
X
:
d
(
x
,
y
)
<
ε
. Evidently, Δ
⊆
E
(
G
) and G has no parallel edges. Then, one can easily check that S is a generalized Mizoguchi-Takahashi G-contraction for the abovementioned chosen graph G. Let x
0
∈
X and x
1
∈
S
(
x
0
)
. Since (
X
,
d
) is ε-chainable, there is a path y
i
i
=
0
N from x
0 to x
1; that is, y
0
=
x
0
,
y
N
=
x
1, and (
y
i
-
1
,
y
i
)
∈
E
(
G
) for each i
=
1,2
,
…
,
N
. Thus, there exists N
∈
N such that x
0
G
N
∩
S
(
x
0
)
≠
∅
. If {
x
n
} is a sequence in X such that {
x
n
} converges to some x
∗
∈
X, then there exists a positive integer n
0 such that d
(
x
n
,
x
∗
)
<
ε for each n
≥
n
0
. Therefore, we can obtain a subsequence {
x
n
m
} of {
x
n
} such that d
(
x
n
m
,
x
∗
)
∈
E
(
G
) for all m
∈
N
. Thus, all the conditions of Theorem 8 hold and S has a fixed point. This completes the proof.Corollary 15 (see [25]).
Let(
X
,
d
) be a complete ε-chainable metric space and let S
:
X
→
C
B
(
X
) be a multivalued mapping such that, for all x
,
y
∈
X with x
≠
y and d
(
x
,
y
)
<
ε, there exists φ
∈
Ω satisfying (48)
H
S
x
,
S
y
≤
φ
d
x
,
y
d
x
,
y
. Then, S has a fixed point.Proof.
Takeβ
(
u
,
v
)
=
φ
(
v
) in Theorem 14.



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*Source: 6514920-2016-07-31.xml*