@
Appl. Gen. Topol. 23, no. 1 (2022), 169-178

doi:10.4995/agt.2022.15214

© AGT, UPV, 2022

Some generalizations for mixed multivalued

mappings

Mustafa Aslantaş a , Hakan Sahin b and Ugur Sadullah a

a Department of Mathematics, Faculty of Science, Çankırı Karatekin University,

Çankırı, Turkey (maslantas@karatekin.edu.tr, ugur s 037@hotmail.com)
b Department of Mathematics, Faculty of Science and Arts, Amasya University,

Amasya, Turkey (hakan.sahin@amasya.edu.tr)

Communicated by I. Altun

Abstract

In this paper, we first introduce a new concept of KW -type m-
contraction mapping. Then, we obtain some fixed point results for these
mappings on M-metric spaces. Thus, we extend many well-known re-
sults for both single valued mappings and multivalued mappings such
as the main results of Klim and Wardowski [13] and Altun et al. [4].
Also, we provide an interesting example to show the effectiveness of our
result.

2020 MSC: 54H25; 47H10.

Keywords: fixed point; mixed multivalued mapping; M-metric space;
Pompeiu-Hausdorff metric.

1. Introduction

In 1922, Banach [7] proved an important theorem which is known as Banach
contraction principle. This principle is an important tool in the fixed point
theory and has been accepted as starting of the fixed point theory in metric
spaces. Due to its applicability, many authors have studied to generalize this
principle by considering different kinds of contractions or abstract spaces [2,
10, 11, 12, 19]. Taking into account multivalued mappings, Nadler [17] proved

Received 05 March 2021 – Accepted 14 January 2022

http://dx.doi.org/10.4995/agt.2022.15214
https://orcid.org/0000-0003-4338-3518
https://orcid.org/0000-0002-4671-7950
https://orcid.org/0000-0001-8463-2862


M. Aslantaş, H. Sahin and U. Sadullah

one of the interesting and famous generalizations of this result in metric spaces
as follows:

Theorem 1.1. Let T : X → CB(X) be a multivalued mapping on a complete
metric space (X,d) where CB(X) is the family of all nonempty bounded and
closed subsets of X. Suppose that there exists k in [0, 1) satisfying

Hd(Tx,Ty) ≤ kd(x,y)
for all x,y ∈ X where Hd : CB(X) × CB(X) → R is a Pompei-Hausdorff
metric defined as

Hd(A,B) = max

{
sup
x∈A

d(x,B), sup
y∈B

d(A,y)

}
for all A,B ∈ CB(X). Then, T has a fixed point in X.

Then, a lot of fixed point theorems for multivalued mappings have been
obtained [15, 20]. In this sense, Nadler’s result has been extended by Feng and
Liu [9] by taking into account C(X), which is the family of all nonempty closed
subsets of a metric space (X,d) valued mappings instead of CB(X) as follows:

Theorem 1.2. Let T : X → C(X) be a multivalued mapping on a com-
plete metric space (X,d). Suppose that for all x ∈ X there exists y ∈ Ixλ =
{z ∈ Tx : λd(x,z) ≤ d(x,Tx)} such that

d(y,Ty) ≤ γd(x,y).
If the function g(x) = d(x,Tx) is lower semicontinuous (briefly l.s.c.) on X
and 0 < γ < λ < 1, then T has a fixed point in X.

Later, Klim and Wardowski [13] generalized Theorem 1.2 by taking into
account a nonlinear contraction:

Theorem 1.3. Let T : X → C(X) be a multivalued mapping on a complete
metric space (X,d). If there exist λ in (0, 1) and ϕ : [0,∞) → [0,λ) such that
(1.1) lim

s→u+
sup ϕ(s) < λ

for all u ∈ [0,∞) and there is y ∈ Ixλ for all x ∈ X satisfying
d(y,Ty) ≤ ϕ (d(x,y)) d(x,y),

then T has a fixed point provided that g(x) = d(x,Tx) is lower-semicontinuous
function on X.

On the other hand, introducing the concept of partial metric, Matthews [14]
obtained another generalization of the Banach contraction principle. Now, we
give the definition of the partial metric space.

Definition 1.4 ([14]). Let X be a nonempty set and p : X ×X → [0,∞) be
a function satisfying following conditions for all x,y,z ∈ X.

p1) p(x,x) = p(x,y) = p(y,y) if and only if x = y
p2) p(x,x) ≤ p(x,y)

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Some generalizations for mixed multivalued mappings

p3) p(x,y) = p(y,x)
p4) p(x,z) ≤ p(x,y) + p(y,z) −p(y,y)

Then, p is said to be a partial metric. Also, the pair (X,p) is called partial
metric space.

It is clear that every metric space is a partial metric space, but the converse
may not be true. For some examples of partial metric space, we refer to [1,
5, 8, 21]. Let (X,p) be a partial metric space. Recently, Asadi et al. [6]
introduced a nice concept of M-metric which includes the notion of the partial
metric. Then, they proved a version of the Banach contraction principle on
these spaces. After that, many authors have proved many fixed point results
for multivalued and single valued mappings [3, 16, 18] on M-metric spaces.
Now, we recall some notations and properties of an M-metric space.

Definition 1.5. Let X be a nonempty set, and m : X × X → [0,∞) be a
function. Then, m is said to be an M-metric if the following conditions hold
for all x,y,z ∈ X:

m1) m(x,y) = m(x,x) = m(y,y) if and only if x = y,
m2) mxy = min{m(x,x),m(y,y)}≤ m(x,y),
m3) m(x,y) = m(y,x),
m4) m(x,y) −mxy ≤ (m(x,z) −mxz) + (m(z,y) −mzy) .

Also, (X,m) is called an M-metric space. It is obvius that every standard
metric and partial metric space is an M-metric space but the converse may
not be true. Indeed, let X = [0,∞) and m : X × X → [0,∞) be a function
defined by m(x,y) = x+y

2
. Hence, (X,m) is an M-metric space, but neither a

partial metric space nor a standard metric. Let (X,m) be an M-metric space.
Then, the M-metric m generates a T0 topology τm on X which has as a base
the family open balls

{Bm(x,r) : x ∈ X, r > 0}

where

Bm(x,r) = {y ∈ X : m(x,y) < mxy + r}

for all x ∈ X and r > 0. Let {xn} be a sequence in X and x ∈ X. It can be
seen that the sequence {xn} M-converges to x with respect to τm if and only
if

lim
n→∞

(m(xn,x) −mxnx) = 0.

If limn,m→∞m(xn,xm) exists and is finite, then {xn} is said to be an M-
Cauchy sequence. If every M-Cauchy sequence {xn} converges to a point x in
M such that

lim
n,k→∞

m(xn,xk) = m(x,x),

then (X,m) is said to be M-complete.
The following proposition is important for our main results.

© AGT, UPV, 2022 Appl. Gen. Topol. 23, no. 1 171



M. Aslantaş, H. Sahin and U. Sadullah

Proposition 1.6. Let (X,m) be an M-metric space, A ⊆ X and x ∈ X.

m(x,A) = 0 =⇒ x ∈ A
m

where A
m

is the closure of A with respect to τm.

The converse of Proposition 1.6 may not be true. Indeed, let X = [2,∞)
and m : X × X → R be a function defined by m(x,y) = min{x,y}. Then,
(X,m) is an M-metric space. Let A = (3, 5) and x = 2. It can be seen that

x ∈ A
m

, but m(x,A) = 2 > 0.
Note that, since every metric space is a T1-space, every singleton is a closed

set. Therefore, Theorem 1.2 is an extension of some fixed point result for
single-valued mappings. However, τm may not be a T1-space, and thus each
singleton does not have to be closed. Therefore, the fixed point results obtained
for multivalued mappings on an M-metric space may not be valid for single-
valued mappings unlike in the settings of metric spaces. To overcome this
problem, we will use the notion of mixed multivalued mapping introduced by
Romaguera [22].

In the current paper, we first introduce a new concept of KW-type m-
contraction for the mixed multivalued mapping. Then, we obtain some fixed
point results on M-metric spaces for these mappings. Hence, we extend some
well known results in the literature such as Theorem 1.3. Also, we provide a
noteworth example to show the effectiveness of our results.

2. Main Results

We start this section with the definition of KW-type m-contraction for
mixed multivalued mapping.

Definition 2.1. Let T : X → X ∪ Cm(X) be a mixed multivalued mapping
on an M-metric space (X,m) where Cm(X) is the family of all closed subsets
of X w.r.t. τm. Then, T is called KW-type m-contraction mapping if there
exists λ,α ∈ (0, 1) and ϕ : [0,∞) → [0,λ) satisfying lims→u+ sup ϕ(s) < λ for
all u ∈ [0,∞) and for all x ∈ X with m(x,Tx) > 0 there is y ∈ Txλ (m) = {z ∈
Tx : λm(x,z) ≤ m(x,Tx)} such that

m(y,Ty) ≤ ϕ(m(x,y))m(x,y)

and

αm(y,y) ≤ m(x,y).

Theorem 2.2. Let T : X → X ∪ Cm(X) be a KW -type m-contraction on
an M-complete M-metric space (X,m). If the function g : X → R defined by
g(x) = m(x,Tx) is l.s.c., then T has a fixed point in X.

Proof. Let x0 ∈ X be an arbitrary point. If there exists n0 ∈ N such that
m(xn0,Txn0 ) = 0, then xn0 ∈ Txn0 = Txn0 (or xn0 = Txn0 ), and so xn0 is a
fixed point of T. Assume that m(xn,Txn) > 0 for all n ≥ 1. Now, we consider
the following cases:

© AGT, UPV, 2022 Appl. Gen. Topol. 23, no. 1 172



Some generalizations for mixed multivalued mappings

Case 1. Let |Tx0| = 1. Since T is a KW-type m-contraction mapping, there
exists x1 = Tx0 such that

m(x1,Tx1) ≤ ϕ(m(x0,x1))m(x0,x1)
and

αm(x1,x1) ≤ m(x0,x1).
Now, if |Tx1| = 1, since T is a KW-type m-contraction mapping, there exists
x2 = Tx1 such that

m(x2,Tx2) ≤ ϕ(m(x1,x2))m(x1,x2)
and

αm(x2,x2) ≤ m(x1,x2),
If |Tx1| > 1, since T is a KW-type m-contraction mapping, there exists x2 ∈
Tx1λ (m) such that

m(x2,Tx2) ≤ ϕ(m(x1,x2))m(x1,x2)
and

αm(x2,x2) ≤ m(x1,x2).
Case 2. Let |Tx0| > 1. Since T is a KW-type m-contraction mapping, there
exists x1 ∈ Tx0λ (m) such that

m(x1,Tx1) ≤ ϕ(m(x0,x1))m(x0,x1)
and

αm(x1,x1) ≤ m(x0,x1).
If |Tx1| = 1. Since T is a KW-type m-contraction mapping, there exists

x2 = Tx1 such that

m(x2,Tx2) ≤ ϕ(m(x1,x2))m(x1,x2)
and

αm(x2,x2) ≤ m(x1,x2).
Now, if |Tx1| > 1. Since T is a KW-type m-contraction mapping, there

exists x2 ∈ Tx1λ (m) such that
m(x2,Tx2) ≤ ϕ(m(x1,x2))m(x1,x2)

and

αm(x2,x2) ≤ m(x1,x2).
Repeating this process, we can construct a sequence {xn} such that for xn+1 ∈
Txnλ (m),

m(xn+1,Txn+1) ≤ ϕ(m(xn,xn+1))m(xn,xn+1)
and

(2.1) αm(xn+1,xn+1) ≤ m(xn,xn+1)
for all n ≥ 1. Since xn+1 ∈ Txnλ (m) for all n ≥ 1, we have
(2.2) λm(xn,xn+1) ≤ m(xn,Txn)

© AGT, UPV, 2022 Appl. Gen. Topol. 23, no. 1 173



M. Aslantaş, H. Sahin and U. Sadullah

for all n ≥ 1. Hence, we get

m(xn,xn+1) ≤
m(xn,Txn)

λ

≤
ϕ(m(xn−1,xn))m(xn−1,xn)

λ
< m(xn−1,xn)(2.3)

and

m(xn,Txn) −m(xn+1,Txn+1) ≥ λm(xn,xn+1)
−ϕ(m(xn,xn+1))m(xn,xn+1)

= (λ−ϕ(m(xn,xn+1)))m(xn,xn+1)
> 0(2.4)

for all n ≥ 1. From inequalities (2.3) and (2.4), (m(xn,xn+1)) and (m(xn,Txn))
are decreasing sequences in R, and so they are convergent. Because of the fact
that

lim
n→∞

m(xn,xn+1) = r ≥ 0

and

lim
s→u+

sup ϕ(s) < λ,

we can find q ∈ [0,λ) satisfying

lim
n→∞

sup ϕ(m(xn,xn+1)) = q.

Therefore, for any λ0 ∈ (q,λ), there exists n0 ∈ N such that

ϕ(m(xn,xn+1)) < λ0

for all n ≥ n0. From (2.4), we have

m(xn,Txn) −m(xn+1,Txn+1) ≥ (λ−ϕ(m(xn,xn+1)))m(xn,xn+1)
≥ (λ−λ0)m(xn,xn+1)(2.5)

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Some generalizations for mixed multivalued mappings

for all n ≥ n0. Hence, for all n ≥ n0, we have

m(xn,Txn) ≤ ϕ (m(xn−1,xn)) m(xn−1,xn)

≤
ϕ(m(xn−1,xn))

λ
m(xn−1,Txn−1)

≤
ϕ(m(xn−1,xn))ϕ(m(xn−2,xn−1))

λ
m(xn−2,xn−1)

≤
ϕ(m(xn−1,xn))ϕ(m(xn−2,xn−1))

λ2
m(xn−2,Txn−2)

...

≤
ϕ(m(xn−1,xn)) · · ·ϕ(m(x0,x1))

λn
m(x0,Tx0)

=
ϕ(m(xn−1,xn)) · · ·ϕ(m(xn0,xn0+1))

λn−n0

×
ϕ(m(xn0−1,xn0 )) · · ·ϕ(m(x0,x1))

λn0
m(x0,Tx0)

<

(
λ0
λ

)n−n0 ϕ(m(xn0−1,xn0 )) · · ·ϕ(m(x0,x1))
λn0

m(x0,Tx0).

Then, since limn→∞
(
λ0
λ

)n−n0
= 0, we have

(2.6) lim
n→∞

m(xn,Txn) = 0.

Hence, for all k > n ≥ n0, from (2.5), we get

m(xn,xk) −mxnxk ≤
(
m(xn,xn+1) −mxnxn+1

)
+
(
m(xn+1,xn+2) −mxn+1xn+2

)
+ · · · +

(
m(xk−1,xk) −mxk−1xk

)
≤ m(xn,xn+1) + · · · + m(xk−1,xk)

=

k−1∑
j=n

m(xj,xj+1)

≤
1

λ−λ0

k−1∑
j=n

(m(xj,Txj) −m(xj+1,Txj+1))

=
1

λ−λ0
(m(xn,Txn) −m(xk,Txk))

From (2.6), we have

lim
n,k→∞

(m(xn,xk) −mxnxk ) = 0

Also, from (2.1), (2.2) and (2.6) we have limn→∞m(xn,xn) = 0, and so

lim
n,k→∞

m(xn,xk) = 0.

© AGT, UPV, 2022 Appl. Gen. Topol. 23, no. 1 175



M. Aslantaş, H. Sahin and U. Sadullah

Then, {xn} is an M-Cauchy sequence in (X,m). Since (X,m) is an M-complete
M-metric space, there exists x∗ ∈ M such that

lim
n→∞

(m(xn,x
∗) −mxnx∗ ) = 0

and
lim

n,k→∞
m(xn,xk) = m(x

∗,x∗)

Now, we shall show that x∗ is a fixed point of T. Since g(x) = m(x,Tx) is a
l.w.s.c. function and limn→∞m(xn,Txn) = 0, we have

0 ≤ m(x∗,Tx∗) = g(x∗) ≤ lim
n→∞

inf g(xn) = lim
n→∞

inf m(xn,Txn) = 0

Hence, m(x∗,Tx∗) = 0, and so we have x∗ ∈ Tx∗
m

= Tx∗. Therefore, x∗ is a
fixed point of T. �

The following example is important to show the effectiveness of our result.

Example 2.3. Let X = [0, 4] and m : X ×X → [0,∞) be a function defined
by

m(x,y) =
x + y

2
Then, (X,m) is an M-complete M-metric space. Define mappings T : X →
X ∪Cm(X) and ϕ : [0,∞) →

[
0, 3

4

)
by

Tx =

[
0,
x2

16

]
,

and

ϕ(u) =

{
3
4
u , u < 1

1
2

, u ≥ 1 ,

respectively. Then, we have

g(x) = m(x,Tx) =
x

2
for all x ∈ X. It can be seen that g is l.s.c. with respect to τm. Now, we shall
show that KW -type m-contraction mapping. Let x be an arbitrary point in X
with m(x,Tx) > 0. Also, we have

Tx3
4

=

{
y ∈ Tx :

3

4
m(x,y) ≤ m(x,Tx)

}
=

{
y ∈ Tx : y ≤

x

3

}
Choose α = 1

2
. Then, for y ∈ Tx3

4

, we have

m(y,Ty) =
y

2

≤
x

6

≤
3

8
(x + y)2

= ϕ (m(x,y)) m(x,y),

© AGT, UPV, 2022 Appl. Gen. Topol. 23, no. 1 176



Some generalizations for mixed multivalued mappings

and
1

2
m(y,y) ≤ m(x,y).

Hence, all hypotheses of Theorem 2.2 hold, and so T has a fixed point in X.

If we take ϕ(u) = γ ∈ (0,λ) for all u ∈ [0,∞) in Definition 2.1, we obtain
the following fixed point result which is a generalization result of [4].

Corollary 2.4. Let T : X → X ∪Cm(X) be a multivalued mapping on a M-
complete M-metric space (X,m). Assume that the following conditions hold:

(i) the function g : X → R defined by g(x) = m(x,Tx) is l.s.c.
(ii) there exist λ,γ,α ∈ (0, 1) with γ < λ such that for any x ∈ X with

m(x,Tx) > 0,

m(y,Ty) ≤ γm(x,y)
and

αm(y,y) ≤ m(x,y)
for some y ∈ Txλ (m).

Then, T has a fixed point in X.

3. Conclusion

In this paper, we extend the result given by Klim and Wardowski [13] to
M-metric spaces. Also, we generalize the main result of Altun et al. [4].
Since an M-metric space may not be a T1-space, we first introduce a new
concept of KW-type m-contraction mapping to obtain a real generalization of
the results obtained for the single valued mappings. Then we obtain some fixed
point results for these mappings in M-metric spaces. Moreover, we provide an
interesting example to show the effectiveness of our results.

Acknowledgements. The authors are thankful to the referees for making
valuable suggestions leading to the better presentations of the paper.

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