International Journal of Mathematical Analysis
Vol. 11, 2017, no. 4, 151 - 161
HIKARI Ltd, www.m-hikari.com
https://doi.org/10.12988/ijma.2017.713
A Generalization of Mizoguchi-Takahashi’s
Fixed Point Theorem and its Applications to
Fixed Point Theory
Wei-Shih Du and Yao-Lin Hung
Department of Mathematics
National Kaohsiung Normal University
Kaohsiung 82444, Taiwan
c 2017 Wei-Shih Du and Yao-Lin Hung. This article is distributed under the
Copyright Creative Commons Attribution License, which permits unrestricted use, distribution, and
reproduction in any medium, provided the original work is properly cited.
Abstract
In this paper, we establish a new generalization of Mizoguchi-Takahashi’s
fixed point theorem which is original and quite different from the well
known generalizations in the literature. As applications of our generalization, many new fixed point theorems are given.
Mathematics Subject Classification: 54H25, 47H10, 54C30
Keywords: MT -function (or R-function), Mizoguchi-Takahashi’s fixed
point theorem, Nadler’s fixed point theorem, Banach contraction principle
1. Introduction and preliminaries
We begin with some basic definitions and notation that will be needed
throughout this paper. Let (X, d) be a metric space. For each x ∈ X and any
subset A of X, let
d(x, A) = inf d(x, y).
y∈A
Denote by N (X) the family of all nonempty subsets of X and CB(X) the class
of all nonempty closed and bounded subsets of X. A function H : CB(X) ×
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Wei-Shih Du and Yao-Lin Hung
CB(X) → [0, ∞) defined by
H(A, B) = max sup d(x, A), sup d(x, B)
x∈B
x∈A
is said to be the Hausdorff metric on CB(X) induced by the metric d on X.
Let T : X → N (X) be a multivalued mapping. A point v in X is said to be
a fixed point of T if v ∈ T v. The set of fixed points of T is denoted by F(T ).
The symbols N and R are used to denote the sets of positive integers and real
numbers, respectively.
It is well-known that the Banach contraction principle [1] plays an important role in various fields of nonlinear analysis and applied mathematical
analysis. The Banach contraction principle has been generalized and improved
in many various different directions; see, e.g., [2-20] and references therein. The
famous multivalued version of the Banach contraction principle was proved by
Nadler [16] in 1969. Since then a number of extensions and generalizations
of Nadler’s fixed point theorem have been investigated by several authors; for
more detail, one can refer to [2-9, 11-15, 17-20] and references therein.
Definition 1.1 [4-10]. A function ϕ : [0, ∞) → [0, 1) is said to be a MT f unction if lim sup ϕ(s) < 1 for all t ∈ [0, ∞).
s→t+
Clearly, if ϕ : [0, ∞) → [0, 1) is a nondecreasing function or a nonincreasing
function, then ϕ is a MT -function. So the set of MT -functions is a rich class.
In 2012, Du [8] established the following characterizations of MT -functions.
Concerning the other interesting characterizations of MT -functions, we refer
to [10, Theorem 2.1].
Theorem 1.1 (Du [8, Theorem 2.1]). Let ϕ : [0, ∞) → [0, 1) be a function. Then the following statements are equivalent.
(a) ϕ is an MT -function.
(1)
∈ [0, 1) and εt
(2)
∈ [0, 1) and εt
(3)
∈ [0, 1) and εt
(b) For each t ∈ [0, ∞), there exist rt
(1)
(1)
ϕ(s) ≤ rt for all s ∈ (t, t + εt ).
(c) For each t ∈ [0, ∞), there exist rt
(2)
(2)
ϕ(s) ≤ rt for all s ∈ [t, t + εt ].
(d) For each t ∈ [0, ∞), there exist rt
(3)
(3)
ϕ(s) ≤ rt for all s ∈ (t, t + εt ].
(1)
> 0 such that
(2)
> 0 such that
(3)
> 0 such that
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A generalization of Mizoguchi-Takahashi’s fixed point theorem
(4)
(e) For each t ∈ [0, ∞), there exist rt
(4)
(4)
ϕ(s) ≤ rt for all s ∈ [t, t + εt ).
(4)
∈ [0, 1) and εt
> 0 such that
(f) For any nonincreasing sequence {xn }n∈N in [0, ∞), we have 0 ≤ sup ϕ(xn ) <
n∈N
1.
(g) ϕ is a function of contractive factor; that is, for any strictly decreasing
sequence {xn }n∈N in [0, ∞), we have 0 ≤ sup ϕ(xn ) < 1.
n∈N
In 1989, Mizoguchi and Takahashi [15] proved a famous generalization of
Nadler’s fixed point theorem which gave a partial answer of Problem 9 in Reich
[17].
Theorem 1.2 (Mizoguchi and Takahashi [15]). Let (X, d) be a complete
metric space, ϕ : [0, ∞) → [0, 1) be a MT -function and T : X → CB(X) be a
multivalued mapping. Assume that
H(T x, T y) ≤ ϕ(d(x, y))d(x, y),
for all x, y ∈ X. Then F(T ) 6= ∅.
In fact, the primitive proof of Mizoguchi-Takahashi’s fixed point theorem
in [15] is difficult. Later, Suzuki [19] gave a very simple proof of MizoguchiTakahashi’s fixed point theorem. There are several interesting generalizations
of this eminent theorem; see [2-4, 6-9, 11-14, 18] and references therein.
In this work, we first establish a new generalization of Mizoguchi-Takahashi’s
fixed point theorem. It should be mentioned that our new generalization of
Mizoguchi-Takahashi’s fixed point theorem is original and quite different from
the well known generalizations in the literature. We give an example which
can illustrate this new result but Mizoguchi-Takahashi’s fixed point theorem
is not applicable. As interesting applications of our new generalization, many
new fixed point theorems are given.
2. A new generalization of Mizoguchi-Takahashi’s fixed
point theorem
In this section, we establish a new fixed point theorem for hybrid MizoguchiTakahashi type mappings which generalizes and improves Mizoguchi-Takahashi’s
fixed point theorem. Please notice that this new generalization is original and
quite different from the well known generalizations of Mizoguchi-Takahashi’s
fixed point theorem in the literature.
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Wei-Shih Du and Yao-Lin Hung
Theorem 2.1. Let (X, d) be a complete metric space and T : X → CB(X)
be a multivalued mapping on X. Suppose that there exists an MT -function ϕ
: [0, ∞) → [0, 1) such that
min{H(T x, T y), d(x, T x)} ≤ ϕ(d(x, y))d(x, y)
for all x, y ∈ X with x 6= y.
(P )
Then T admits a fixed point in X.
Proof. Let µ : [0, ∞) → [0, 1) be defined by
µ(t) =
1 + ϕ(t)
2
for all t ∈ [0, ∞).
Clearly, 0 ≤ ϕ(t) < µ(t) < 1 for all t ∈ [0, ∞). Let u ∈ X. Take x0 = u ∈ X
and choose x1 ∈ T x0 . If x1 = x0 , then x0 ∈ T x0 and we are done. Otherwise,
if x1 6= x0 , then d(x1 , x0 ) > 0 and we obtain from (P ) that
min{H(T x1 , T x0 ), d(x1 , T x1 )} ≤ ϕ(d(x1 , x0 ))d(x1 , x0 ) < µ(d(x1 , x0 ))d(x1 , x0 ).
(2.1)
Since
d(x1 , T x1 ) ≤ sup d(a, T x1 ) ≤ H(T x0 , T x1 ),
a∈T x0
we know
min{H(T x1 , T x0 ), d(x1 , T x1 )} = d(x1 , T x1 ).
(2.2)
So, by (2.1) and (2.2), we have
d(x1 , T x1 ) < µ(d(x1 , x0 ))d(x1 , x0 ).
(2.3)
By (2.3), there exists x2 ∈ T x1 such that
d(x1 , x2 ) < µ(d(x1 , x0 ))d(x1 , x0 ).
If x2 = x1 , then x1 ∈ T x1 which means that x1 is a fixed point of T and the
desired conclusion is proved. Assume x2 6= x1 . By (P ) again, we obtain
d(x2 , T x2 ) = min{H(T x2 , T x1 ), d(x2 , T x2 )}
< µ(d(x2 , x1 ))d(x2 , x1 ).
So there exists x3 ∈ T x2 such that
d(x2 , x3 ) < µ(d(x2 , x1 ))d(x2 , x1 ).
By induction, we can obtain a sequence {xn }n∈N∪{0} satisfying the following:
for each n ∈ N,
(i) xn ∈ T xn−1 with xn 6= xn−1 ;
A generalization of Mizoguchi-Takahashi’s fixed point theorem
155
(ii) d(xn+1 , xn ) < µ(d(xn , xn−1 ))d(xn , xn−1 ).
Since µ(t) < 1 for all t ∈ [0, ∞), by (ii), we know that the sequence {d(xn , xn−1 )}n∈N
is strictly decreasing in [0, ∞). Since ϕ is an MT -function, by applying (g) of
Theorem 1.1, we have 0 ≤ supn∈N ϕ(d(xn , xn−1 )) < 1 and hence deduces
1
1 + sup ϕ(d(xn , xn−1 )) < 1.
0 < sup µ(d(xn , xn−1 )) =
2
n∈N
n∈N
Let γ := sup µ(d(xn , xn−1 )). So γ ∈ [0, 1). For any n ∈ N, by (ii) again, we
n∈N
have
d(xn+1 , xn ) < µ(d(xn , xn−1 ))d(xn , xn−1 ) ≤ γd(xn , xn−1 ).
(2.4)
So it follows from (2.4) that
d(xn+1 , xn ) < γd(xn , xn−1 ) < · · · < γ n d(x1 , x0 ) for each n ∈ N.
Let ξn =
that
γn
d(x1 , x0 ),
1−γ
(2.5)
n ∈ N. For m, n ∈ N with m > n, we get from (2.5)
d(xm , xn ) ≤
m−1
X
d(xj+1 , xj ) < ξn .
j=n
Since 0 < γ < 1, limn→∞ ξn = 0 and hence the last inequality implies
lim sup{d(xm , xn ) : m > n} = 0.
n→∞
This proves that {xn }n∈N∪{0} is a Cauchy sequence in X. By the completeness
of X, there exists v ∈ X such that xn → v as n → ∞. In order to finish the
proof it is sufficient to show v ∈ F(T ). Clearly, the condition (P ) still holds
for x = y ∈ X. So, we have
min{H(T xn , T v), d(v, T v)} ≤ ϕ(d(xn , v))d(xn , v) for all n ∈ N.
(2.6)
Let
A = {n ∈ N : min{H(T xn , T v), d(v, T v)} = H(T xn , T v)}.
We consider the following two possible cases:
Case 1. Assume that ](A) = ∞, where ](A) denotes the cardinal number of
A. Thus there exists {nj } ⊂ A such that
min{H(T xnj , T v), d(v, T v)} = H(T xnj , T v) for all j ∈ N.
Taking into account (2.6) and (2.7), we obtain
H(T xnj , T v) ≤ ϕ(d(xnj , v))d(xnj , v)
(2.7)
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Wei-Shih Du and Yao-Lin Hung
which implies
d(xnj +1 , T v) ≤ ϕ(d(xnj , v))d(xnj , v) < µ(d(xnj , v))d(xnj , v)
for each j ∈ N.
(2.8)
For any j ∈ N, from (2.8), there exists wnj ∈ T v such that
d(xnj +1 , wnj ) < µ(d(xnj , v))d(xnj , v) < d(xnj , v).
Since xnj → v as j → ∞, the last inequality deduces
lim d(xnj +1 , wnj ) = 0.
j→∞
Since
d(wnj , v) ≤ d(wnj , xnj +1 ) + d(xnj +1 , v)
for each j ∈ N,
taking the limit as j tends to infinity yields
lim d(wnj , v) = 0.
j→∞
Since {wnj } ⊂ T v, T v is closed and wnj → v as j → ∞, we get v ∈ F(T ).
Case 2. Suppose that ](A) < ∞. Then there exists a sequence {nk } of
natural numbers such that
min{H(T xnk , T v), d(v, T v)} = d(v, T v)
for all k ∈ N.
(2.9)
Taking into account (2.6) and (2.9), we have
d(v, T v) ≤ ϕ(d(xnk , v))d(xnk , v) < d(xnk , v)
for all k ∈ N.
Since xnk → v as k → ∞, the last inequality implies d(v, T v) = 0. By the
closedness of T v, we obtain v ∈ T v. The proof is completed.
Here, we give an example which can illustrate Theorem 2.1 but MizoguchiTakahashi’s fixed point theorem is not applicable. It is worth to mention that
the following example also shows that the considered mapping maybe has more
than one fixed point under the condition (P ).
Example 2.1. Let `∞ be the Banach space consisting of all bounded real
sequences with supremum norm d∞ and let {en } be the canonical basis of
`∞ . Let {τn } be a sequence of positive real numbers satisfying τ1 = τ2 and
τn+1 < τn for n ≥ 2 (for example, let τ1 = 21 and τn = n1 for n ∈ N with n ≥ 2).
Thus {τn } is convergent. Put vn = τn en for n ∈ N and let X = {vn }n∈N be a
bounded and complete subset of `∞ . Then (X, d∞ ) be a complete metric space
and d∞ (vn , vm ) = τn if m > n.
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A generalization of Mizoguchi-Takahashi’s fixed point theorem
Let T : X → CB(X) be defined by
{v1 , v2 }
, if n ∈ {1, 2},
T vn :=
X \ {v1 , v2 , · · · , vn , vn+1 }
, if n ≥ 3.
and define ϕ : [0, ∞) → [0, 1) by
τn+2
, if t = τn for some n ∈ N,
τn
ϕ(t) :=
0
, otherwise.
Since lim sup ϕ(s) = 0 < 1 for all t ∈ [0, ∞), ϕ is an MT -function. Clearly,
s→t+
F(T ) = {v1 , v2 }. We want to verify
min{H∞ (T x, T y), d∞ (x, T x)} ≤ ϕ(d∞ (x, y))d∞ (x, y) for all x, y ∈ X with x 6= y,
(2.10)
where H∞ is the Hausdorff metric induced by d∞ . In order to prove the
inequality (2.10), we consider the following four possible cases:
Case 1. min{H∞ (T v1 , T v2 ), d∞ (v1 , T v1 )} = 0 < τ3 = ϕ(d∞ (v1 , v2 ))d∞ (v1 , v2 ).
Case 2. For any m ≥ 3, we have
min{H∞ (T v1 , T vm ), d∞ (v1 , T v1 )} = 0 < τ3 = ϕ(d∞ (v1 , vm ))d∞ (v1 , vm ).
Case 3. For any m ≥ 3, we obtain
min{H∞ (T v2 , T vm ), d∞ (v2 , T v2 )} = 0 < τ4 = ϕ(d∞ (v2 , vm ))d∞ (v2 , vm ).
Case 4. For any n ≥ 3 and m > n, we get
min{H∞ (T vn , T vm ), d∞ (vn , T vn )} = τn+2 = ϕ(d∞ (vn , vm ))d∞ (vn , vm ).
Hence, by Cases 1, 2, 3 and 4, we prove that T satisfies (2.10). Therefore, all
the assumptions of Theorem 2.1 are satisfied. It is therefore possible to apply
Theorem 2.1 to get F(T ) 6= ∅. Notice that
H∞ (T v1 , T vm ) = τ1 > τ3 = ϕ(d∞ (v1 , vm ))d∞ (v1 , vm )
for all m ≥ 3,
so Mizoguchi-Takahashi’s fixed point theorem is not applicable here.
Remark 2.1. Mizoguchi-Takahashi’s fixed point theorem [15] is a special case
of Theorem 2.1.
The following conclusions are immediate consequences of Theorem 2.1.
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Wei-Shih Du and Yao-Lin Hung
Corollary 2.1. Let (X, d) be a complete metric space and T : X → X be
a self-mapping on X. Suppose that there exists an MT -function ϕ : [0, ∞) →
[0, 1) such that
min{d(T x, T y), d(x, T x)} ≤ ϕ(d(x, y))d(x, y)
for all x, y ∈ X with x 6= y.
Then T admits a fixed point in X.
Corollary 2.2. Let (X, d) be a complete metric space and T : X → CB(X)
be a multivalued mapping on X. Suppose that there exists an MT -function ϕ
: [0, ∞) → [0, 1) such that
d(x, T x) ≤ ϕ(d(x, y))d(x, y)
for all x, y ∈ X with x 6= y.
Then T admits a fixed point in X.
Corollary 2.3. Let (X, d) be a complete metric space and T : X → X be
a self-mapping on X. Suppose that there exists an MT -function ϕ : [0, ∞) →
[0, 1) such that
d(x, T x) ≤ ϕ(d(x, y))d(x, y)
for all x, y ∈ X with x 6= y.
Then T admits a fixed point in X.
3. Applications to fixed point theory
In this section, we establish some new fixed point theorems immediately
by applying Theorem 2.1.
Theorem 3.1. Let (X, d) be a complete metric space and T : X → CB(X)
be a multivalued mapping on X. Suppose that there exists an MT -function ϕ
: [0, ∞) → [0, 1) such that
H(T x, T y) + d(x, T x) ≤ 2ϕ(d(x, y))d(x, y)
for all x, y ∈ X with x 6= y.
(P 1)
Then T admits a fixed point in X.
Theorem 3.2. Let (X, d) be a complete metric space and T : X → CB(X)
be a multivalued mapping on X. Suppose that there exists an MT -function ϕ
: [0, ∞) → [0, 1) such that
p
H(T x, T y)d(x, T x) ≤ ϕ(d(x, y))d(x, y) for all x, y ∈ X with x 6= y.
(P 2)
Then T admits a fixed point in X.
A generalization of Mizoguchi-Takahashi’s fixed point theorem
159
Remark 3.1. Notice that condition (P 1) also implies condition (P 2). So
Theorem 3.1 can be proved by applying Theorem 3.2.
In fact, we can establish a wide generalization of Theorem 3.1 as follows.
Theorem 3.3. Let (X, d) be a complete metric space and T : X → CB(X)
be a multivalued mapping on X. Suppose that there exists an MT -function ϕ
: [0, ∞) → [0, 1) such that
αH(T x, T y) + βd(x, T x)
≤ ϕ(d(x, y))d(x, y)
α+β
for all x, y ∈ X with x 6= y,
(P 3)
where α and β are nonnegative real numbers with α + β > 0. Then T admits
a fixed point in X.
Remark 3.2.
(a) If we take α = 1 and β = 0 in Theorem 3.3, then we obtain MizoguchiTakahashi’s fixed point theorem.
(b) If we take α = 0 and β = 1 in Theorem 3.3, then we obtain Corollary
2.2.
(c) If we take α = β = 1 in Theorem 3.3, then we obtain Theorem 3.1.
(d) The conclusions of Theorems 2.1, 3.1, 3.2 and 3.3 still hold if
• ” min{H(T x, T y), d(x, T x)}” is replaced with ” min{H(T x, T y), d(y, T y)}”
in condition (P ) of Theorem 2.1;
• ”H(T x, T y) + d(x, T x)” is replaced with ”H(T x, T y) + d(y, T y)” in
condition (P 1) of Theorem 3.1;
p
p
• ” H(T x, T y)d(x, T x)” is replaced with ” H(T x, T y)d(y, T y)” in
condition (P 2) of Theorem 3.2;
y)+βd(x,T x)
y)+βd(y,T y)
• ” αH(T x,Tα+β
” is replaced with ” αH(T x,Tα+β
” in condition
(P 3) of Theorem 3.3.
(e) Obviously, we can establish single-valued versions of Theorems 2.1, 3.1,
3.2 and 3.3 if T is assumed to be a self-mapping on X.
Acknowledgements. The first author was supported by grant no. MOST
105-2115-M-017-002 of the Ministry of Science and Technology of the Republic
of China.
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Wei-Shih Du and Yao-Lin Hung
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Received: January 24, 2017; Published: February 10, 2017