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@ Appl. Gen. Topol. 18, no. 2 (2017), 317-330doi:10.4995/agt.2017.7067
c© AGT, UPV, 2017

Multiple fixed point theorems for contractive

and Meir-Keeler type mappings defined on

partially ordered spaces with a distance

Mitrofan M. Choban
a
and Vasile Berinde

b

a
Department of Physics, Mathematics and Information Technologies, Tiraspol State University,

Gh. Iablocikin 5., MD2069 Chişinău, Republic of Moldova (mmchoban@gmail.com)
b
Department of Mathematics and Computer Science, Technical University of Cluj-Napoca, North

University Center at Baia Mare, Victoriei 76, 430072 Baia Mare, Romania (vberinde@cunbm.utcluj.ro)

Communicated by I. Altun

Abstract

We introduce and study a general concept of multiple fixed point for
mappings defined on partially ordered distance spaces in the presence of
a contraction type condition and appropriate monotonicity properties.
This notion and the obtained results complement the corresponding
ones from [M. Choban, V. Berinde, A general concept of multiple fixed
point for mappings defined on spaces with a distance, Carpathian J.
Math. 33 (2017), no. 3, 275–286] and also simplifies some concepts of
multiple fixed point considered by various authors in the last decade or
so.

2010 MSC: 47H10; 47H09.

Keywords: distance space; partial order; symmetric space; quasi-metric
space; H-distance; contraction condition; multiple fixed point.

1. Introduction

In a previous paper [32], the authors have introduced and studied a general
concept of multidimensional fixed point for mappings defined on a distance
space and satisfying a certain contraction condition.

Received 02 January 2017 – Accepted 12 July 2017

http://dx.doi.org/10.4995/agt.2017.7067


M. M. Choban and V. Berinde

Several interesting new results that generalise, extend and unify correspond-
ing related results from literature for the case of non ordered distance spaces
were obtained. However, the great majority of the multidimensional fixed point
theorems existing in literature were established in the setting of a partially or-
dered metric space or of a partially ordered generalised metric space. Therefore,
the main aim of this paper is to study the concept of multidimensional fixed
point introduced in [32] for the case of mappings defined on partially ordered
distance space, thus extending and complementing most of the results estab-
lished in [32].

We start by presenting a brief survey on the notion of multidimensional fixed
point, which naturally emerged from the rich literature produced in the last four
decades devoted to coupled fixed points. The concept itself of coupled fixed
point has been first introduced and studied by Opoitsev, in a series of papers
he published in the period 1975-1986, see [61]-[64]. Opoitsev has been inspired
by some concrete problems arising in the dynamics of collective behaviour in
mathematical economics and considered the coupled fixed point problem for
mixed monotone nonlinear operators which also satisfy a nonexpansive type
condition.

In 1987, Guo and Lakshmikantham [41], apparently not being aware of
Opoitsev’s previous results [61]-[64], have studied coupled fixed points in con-
nection with coupled quasi-solutions of an initial value problem for ordinary dif-
ferential equations. Amongst the subsequent developments we quote the follow-
ing works: [40]; [30], containing coupled fixed point results of 1

2
-α-condensing

and mixed monotone operators, where α denotes the Kuratowski’s measure of
non compactness, thus extending some previous results from [41] and [79]; [29],
which discusses some existence results and iterative approximation of coupled
fixed points for mixed monotone condensing set-valued operators; [28] where
the authors obtained coupled fixed point results of 1

2
-α-contractive and gener-

alized condensing mixed monotone operators.
More recently, Gnana Bhaskar and Lakshmikantham in [37] established cou-

pled fixed point results for mixed monotone operators in partially ordered met-
ric spaces in the presence of a Bancah contraction type condition. Essentialy,
the results by Bhaskar and Lakshmikantham in [37] combined, in the context
of bivariable mixed monotone mappings, the main fixed point results previ-
ously obtained by Nieto and Rodriguez-Lopez [58] and [59], for the case of one
variable increasing and decreasing nonlinear operator, respectively. The last
two papers are, in turn, a continuation of the hybrid fixed point theorem es-
tablished in the seminal paper of Ran and Reurings [66], which has the merit
to combine a metrical fixed point theorem (the contraction mapping principle)
and an order theoretic fixed point result (Tarski’s fixed point theorem).

Various applications of the theoretical results in coupled fixed point theory
were also considered, for the case of: a) Uryson integral equations [63]; b) a
system of Volterra integral equations [30], [28]; c) a class of functional equations
arising in dynamic programming [29]; d) initial value problems for first order
differential equations with discontinuous right hand side [41]; e) (two point)

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Multiple fixed point theorems for contractive and Meir-Keeler type mappings

periodic boundary value problems [17], [37], [33], [82]; f) integral equations
and systems of integral equations [3], [6], [9], [24], [39], [42], [78], [80], [85];
g) nonlinear elliptic problems and delayed hematopoesis models [84]; h) non-
linear Hammerstein integral equations [76]; i) nonlinear matrix and nonlinear
quadratic equations [4], [24]; j) initial value problems for ODE [8], [75] etc.

For a very recent account on the developments of coupled fixed point theory,
we also refer to [22].

On the other hand, in 2010, Samet and Vetro [74] apart of some coupled
fixed point results they have established, considered a concept of fixed point of
m-order as a natural extension of the notion of coupled fixed point. Then, in
2011, mainly inspired by [37], Berinde and Borcut [18] introduced the concept
of triple fixed point and proved existence as well as existence and uniqueness
triple fixed point theorems for three-variable mixed monotone mappings, while,
in 2012, Karapinar and Berinde [47], have studied quadruple fixed points of
nonlinear contractions in partially ordered metric spaces.

After these starting papers, a substantial number of articles were dedicated
to the study of triple fixed points, quadruple fixed points, as well as to multiple
fixed points (also called fixed point of m-order, or ”a multidimensional fixed
point”, or ”an m-tuplet fixed point”, or ”an m-tuple fixed point”), see [1], [2],
[7], [48], [49], [50], [53], [60], [67]-[72], [81], [83], [86], which form a very selective
list contributions.

Starting from this background, the main aim of the present paper is to study
the concept of multidimensional fixed point introduced in [32] but for mappings
defined on partially ordered distance space, in the presence of a contraction
type condition and appropriate monotonicity properties, thus extending and
complementing the results established in [32].

This approach is based on the idea to reduce the study of multidimensional
fixed points and coincidence points to the study of usual one-dimensional fixed
points for an associate operator. Note that, the first author who reduced the
problem of finding a coupled fixed point of mixed monotone operators to the
problem of finding a fixed point of an increasing one variable operator was
Opoitsev, see for example [63].

2. Preliminaries

By a space we understand a topological T0-space. We use the terminology
from [36, 38, 73, 31].

Let X be a non-empty set and d : X × X → R be a mapping such that:

(im) d(x, y) ≥ 0, for all x, y ∈ X;
(iim) d(x, y) + d(y, x) = 0 if and only if x = y.

Then d is called a distance on X, while (X, d) is called a distance space.
Let d be a distance on X and B(x, d, r) = {y ∈ X : d(x, y) < r} be the

ball with the center x and radius r > 0. The set U ⊂ X is called d-open if for
any x ∈ U there exists r > 0 such that B(x, d, r) ⊂ U. The family T (d) of
all d-open subsets is the topology on X generated by d. A distance space is

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M. M. Choban and V. Berinde

a sequential space, i.e., a space for which a set B ⊆ X is closed if and only if
together with any sequence it contains all its limits [36].

Let (X, d) be a distance space, {xn}n∈N be a sequence in X and x ∈ X. We
say that the sequence {xn}n∈N is:

1) convergent to x if and only if limn→∞ d(x, xn) = 0. We denote this by
xn → x or x = limn→∞ xn (really, we may denote x ∈ limn→∞ xn);

2) convergent if it converges to some point x in X;
3) Cauchy or fundamental if limn,m→∞ d(xn, xm) = 0.

A distance space (X, d) is called complete if every Cauchy sequence in X con-
verges to some point x in X.

Let X be a non-empty set and d be a distance on X. Then:

• (X, d) is called a symmetric space and d is called a symmetric on X if

(iiim) d(x, y) = d(y, x), for all x, y ∈ X;

• (X, d) is called a quasimetric space and d is called a quasimetric on X
if

(ivm) d(x, z) ≤ d(x, y) + d(y, z), for all x, y, z ∈ X;

• (X, d) is called a metric space and d is called a metric if d is a symmetric
and a quasimetric, simultaneously.

Let X be a non-empty set and d(x, y) be a distance on X with the following
property:

(N) for each point x ∈ X and any ε > 0 there exists δ = δ(x, ε) > 0 such
that from d(x, y) ≤ δ and d(y, z) ≤ δ it follows d(x, z) ≤ ε.

Then (X, d) is called an N-distance space and d is called an N-distance on X.
If d is a symmetric, then we say that d is an N-symmetric.

Spaces with N-distances were studied by Niemyzki [56] and by Nedev [55].
If d satisfies the condition

(F) for any ε > 0 there exists δ = δ(ε) > 0 such that from d(x, y) ≤ δ and
d(y, z) ≤ δ it follows d(x, z) ≤ ε,

then d is called an F-distance or a Fréchet distance and (X, d) is called an
F-distance space. Any F-distance d is an N-distance, too. If d is a symmetric
and an F-distance on a space X, then we say that d is an F-symmetric.

Remark 2.1. If (X, d) is an F-symmetric space, then any convergent sequence
is a Cauchy sequence. For N-symmetric spaces and for quasimetric spaces this
assertion is not more true.

If s > 0 and d(x, y) ≤ s[d(x, z) + d(z, y)] for all points x, y, z ∈ X, then we
say that d is an s-distance. Any s-distance is an F-distance.

A distance space (X, d) is called an H-distance space if, for any two distinct
points x, y ∈ X, there exists δ = δ(x, y) > 0 such that B(x, d, δ) ∩ B(y, d, δ) =
∅.

We say that (X, d) is a C-distance space or a Cauchy distance space if any
convergent Cauchy sequence has a unique limit point.

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Multiple fixed point theorems for contractive and Meir-Keeler type mappings

Remark 2.2. A distance space (X, d) is an H-distance space if and only if any
convergent sequence in X has a unique limit point. Hence, any H-distance
space is a C-distance space.

3. Ordering on Cartesian product of distance spaces

Let (X, d) be a distance space, m ∈ N = {1, 2, ...}. On Xm consider the
distances

dm((x1, ..., xm), (y1, ..., ym)) = sup{d(xi, yi) : i ≤ m}

and

d̄m((x1, ..., xm), (y1, ..., ym)) =

m∑

i=1

d(xi, yi).

Obviously, (Xm, dm) and (Xm, d̄m) are distance spaces, too.

Proposition 3.1 ([32]). Let (X, d) be a distance space. Then:

1. If d is a symmetric, then (Xm, dm) and (Xm, d̄m) are symmetric spaces,
too.

2. If d is a quasimetric, then (Xm, dm) and (Xm, d̄m) are quasimetric
spaces, too.

3. If d is a metric, then (Xm, dm) and (Xm, d̄m) are metric spaces, too.
4. If d is an F-distance space, then (Xm, dm) and (Xm, d̄m) are F-distance

spaces, too.

5. If d is an N-distance space, then (Xm, dm) and (Xm, d̄m) are N-
distance spaces, too.

6. If d is an H-distance space, then (Xm, dm) and (Xm, d̄m) are H-
distance spaces, too.

7. If (X, d) is a C-distance space, then (Xm, dm) and (Xm, d̄m) are C-
distance spaces, too.

8. If (X, d) is a complete distance space, then (Xm, dm) and (Xm, d̄m)
are complete distance spaces, too.

9. If d is an s-distance space, then (Xm, dm) and (Xm, d̄m) are s-distance
spaces, too.

10. The spaces (Xm, dm) and (Xm, d̄m) share the same convergent se-
quences and the same Cauchy sequences. Moreover, the distances dm

and d̄m are uniformly equivalent, i.e., for each ε > 0, there exists δ =
δ(ε) > 0 such that:

- from dm(x, y) ≤ δ it follows d̄m(x, y) ≤ ε;
- from d̄m(x, y) ≤ δ it follows dm(x, y) ≤ ε.

Let � be a (partial) order on a distance space (X, d). A sequence {xn}n∈N
is called:

• non-decreasing if xn � xn+1 for each n ∈ N;
• non-increasing if xn � xn+1 for each n ∈ N;
• monotone if it is either non-decreasing or non-increasing.

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M. M. Choban and V. Berinde

If (X, d, �) is an ordered distance space and g : X → X is a mapping,
then (X, d, �) is said to have the sequential g-monotone property [37, 69] if it
verifies:

i) if {xn}n∈N is a non-decreasing sequence and lim
n→∞

d(x, xn) = 0, then

g(xn) � g(x) for all n ∈ N;
ii) if {xn}n∈N is a non-increasing sequence and lim

n→∞
d(x, xn) = 0, then

g(xn) � g(x) for all n ∈ N.

An ordered distance space (X, d, �) is called monotonically complete if any
monotone Cauchy sequence in X converges to some point in X.

Fix m ∈ N and a subset L ⊆ {1, 2, ..., m}. Like in [67, 69, 70], we introduce
on X the ordering �L: (x1, ..., xm) �L (y1, ..., ym) iff xi � yi for i ∈ L and
yj � xj for j /∈ L.

By construction, (Xm, dm, �L) and (X
m, d̄m, �L) are ordered distance spaces.

If L ⊆ {1, 2, ..., m} and M = {1, 2, ..., m} \ L, then x �L y if and only if
y �M x for x, y ∈ X

m. Hence �M is the dual (inverse) order of the order �L.

Proposition 3.2. Let (X, d, �) be a monotonically complete distance space.
Then (Xm, dm, �L) and (X

m, d̄m, �L) are ordered monotonically complete dis-
tance spaces, too.

Proof. It is obvious. �

4. Multiple fixed point principles for monotone type operators

Fix m ∈ N. Denote by λ = (λ1, ..., λm) a collection of mappings {λi :
{1, 2, ..., m} −→ {1, 2, ..., m} : 1 ≤ i ≤ m}.

Let (X, d) be a distance space and F : Xm −→ X be an operator. The
operator F and the mappings λ generate the operator λF : Xm −→ Xm,
where

λF(x1, ...., xm) = (y1, ..., ym) and yi = F(xλi(1), ..., xλi(m)),

for each point (x1, ..., xm) ∈ X
m and any index i ∈ {1, 2, ..., m}.

A point a = (a1, ..., am) ∈ X
m is called a λ-multiple fixed point of the

operator F if a = λF(a), i.e., ai = F(aλi(1), ..., aλi(m)) for any i ∈ {1, 2, ..., m}.
Let (X, d, �) be a (partially) ordered distance space, m ∈ N, L ⊆ {1, 2, ..., m}

and F : Xm −→ X be an operator. In this context, we consider the following
sets of assumptions that include a symmetric type contractive condition, similar
to the symmetric contraction introduced and used by the second author in [14].

Conditions Ω1:

1. (X, �) is a lattice;
2. If x, y, z ∈ X and x � y � z, then d(x, y) + d(y, x) ≤ d(x, z) + d(z, x);
3. If x, y ∈ Xm, x 6= y and x �L y, then λF(x) �L λF(y) and d

m(λF(x), λF(y))
+ dm(λF(y), λF(x)) < dm(x, y) + dm(y, x).

Conditions Ω2:

1. (X, �) is a lattice;

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Multiple fixed point theorems for contractive and Meir-Keeler type mappings

2. If x, y, z ∈ X and x � y � z, then d(x, y) + d(y, x) ≤ d(x, z) + d(z, x);
3. If x, y ∈ Xm, x 6= y and x �L y, then λF(y) �L λF(x) and d

m(λF(x), λF(y))
+ dm(λF(y), λF(x)) < dm(x, y) + dm(y, x).

Conditions Ω3:

1. (X, �) is a lattice;
2. If x, y, z ∈ X and x � y � z, then d(x, y) + d(y, x) ≤ d(x, z) + d(z, x);
3. For any i ∈ {1, 2, . . . , m} the mapping λi is a surjection or, more gen-

erally, | ∪ {λ−1
i

(j) : 1 ≤ j ≤ m}| = m, for each i ∈ {1, 2, . . . , m};
4. If x, y ∈ Xm, x 6= y and x �L y, then λF(x) �L λF(y) and d̄

m(λF(x), λF(y))
+ d̄m(λF(y), λF(x)) < d̄m(x, y) + d̄m(y, x).

Conditions Ω4:

1. (X, �) is a lattice;
2. If x, y, z ∈ X and x � y � z, then d(x, y) + d(y, x) ≤ d(x, z) + d(z, x);
3. For any i ∈ {1, 2, . . . , m} the mapping λi is a surjection or, more gen-

erally, | ∪ {λ−1i (j) : 1 ≤ j ≤ m}| = m, for each i ∈ {1, 2, . . . , m};
4. If x, y ∈ Xm, x 6= y and x �L y, then λF(y) �L λF(x) and d̄

m(λF(x), λF(y))
+ d̄m(λF(y), λF(x)) < d̄m(x, y) + d̄m(y, x).

Now we can state concisely the following general and comprehensive multidi-
mensional fixed point result.

Theorem 4.1. Let a ∈ Xm be a multidimensional fixed point of the operator
of F : Xm −→ X. If any of the Conditions Ωi, i ∈ {1, 2, 3, 4}, is satisfied, then
the operator F has a unique multidimensional fixed point.

Proof. Obviously, (Xm, �L) is a lattice, too. Let ρ = d
m, for i ∈ {1, 2}, and

ρ = d̄m, for i ∈ {3, 4}. Then for x, y, z ∈ Xm and x �L y �L z we have
ρ(x, y)+ ρ(y, x) ≤ ρ(x, z)+ ρ(z, x). Assume that b ∈ Xm is a multidimensional
fixed point of the operator F with b 6= a.

Case 1. The points a and b are comparable.
Assume that a �L b. Then ρ(a, b) + ρ(b, a) = ρ(λF(a), λF(b)) + ρ(λF(b), λF(a))

< ρ(a, b) + ρ(b, a), a contradiction.
Case 2. The points a and b are not comparable.
From any of the conditions Ωi it follows that (X, �) is a lattice. Hence

(Xm, �L) is a lattice too, as the Cartesian product of lattices.
Fix c = max{a, b} ∈ Xm. We put d = λF(λF(c)). By construction, a �L d

and b �L d. Hence, c �L d and ρ(a, c) ≤ ρ(a, d), ρ(b, c) ≤ ρ(b, d). Therefore
ρ(a, c)+ρ(c, a) ≤ ρ(a, d)+ρ(d, a). By virtue of the conditions Ωi, we then have
ρ(a, d) + ρ(d, a) < ρ(a, c) + ρ(c, a), a contradiction. �

Now, according to [52], consider the following two classes of Meir-Keeler
type assumptions.

Conditions MK1:

1. For any two points x, y ∈ X there exist an upper bound and a lower
bound;

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M. M. Choban and V. Berinde

2 (Meir-Keeler monotone contraction condition). There exists a function
δ : (0, +∞) −→ (0, +∞) such that from r > 0, x, y ∈ X, d(x, y) <
r + δ(r) and x � y it follows that d(x, y) < r;

3. If x, y ∈ Xm, x �L y, then λF(x) �L λF(y).

Conditions MK2:

1. For any two points x, y ∈ X there exist an upper bound and a lower
bound;

2 (Meir-Keeler monotone contraction condition). There exists a function
δ : (0, +∞) −→ (0, +∞) such that from r > 0, x, y ∈ X, d(x, y) <
r + δ(r) and x � y it follows that d(x, y) < r;

3. If x, y ∈ Xm, x �L y, then λF(y) �L λF(x).

Theorem 4.2. Let (X, d) be an H-distance space and let a ∈ Xm be a mul-
tidimensional fixed point of the operator of F : Xm → X. Then in any of
the Conditions MKi, i ∈ {1, 2}, the operator F has a unique multidimensional
fixed point.

Proof. Obviously, in (Xm, �L), for any two points x, y ∈ X
m, there exist an

upper bound and a lower bound. Let ρ = dm. In this case for any two points
x, y ∈ Xm, from the condition ρ(x, y) < r + δ(r) and x � y, it follows that
ρ(x, y) < r.

Assume that b ∈ Xm is a multidimensional fixed point of the operator of F
and that b 6= a.

Case 1. The points a and b are comparable.
Assume that a �L b and ρ(a, b) = r > 0. Since ρ(a, b) < r + δ(r), we have r

= ρ(a, b) = ρ(λF(a), λF(b)) < r, a contradiction.
Case 2. The points a and b are not comparable. We put r = inf{max{ρ(a, c),

ρ(b, c)} : c ∈ Xm, a �L c, b �L c}. We claim that r = 0. Assume that r > 0.
Then δ(r) > 0 and there exists c such that max{ρ(a, c), ρ(b, c)} < r + δ(r),
a �L c, b �L c. We put e = λF(λF(c)). Then a �L e and b �L e. Since
λF(λF(a)) = a and λF(λF(b)) = b, we have max{ρ(a, e), ρ(b, e)} < r, a con-
tradiction. Thus r = 0. For each n ∈ N there exists a point cn ∈ X

m such
that a �L cn, b �L cn and max{ρ(a, cn), ρ(b, cn)} < 2

−n. We can construct
a sequence {cn : n ∈ N} for which a = limn→∞ cn and b = limn→∞ cn, a
contradiction. �

In the particular case m = 2, the following theorems were proved in [21].
Their proofs in the general case are similar and we omit them.

Theorem 4.3. Let (X, d, �) be an ordered metric space, m ∈ N, F : Xm → X
be an operator. Suppose that:

a) there exists a function δ : (0, +∞) −→ (0, +∞) such that from r > 0,
x, y ∈ Xm, dm(x, y) < r+δ(r) and x � y it follows that dm(λF(x), λF(y)) <
r;

b) for any two points x, y ∈ X there exists an upper bound and a lower
bound.

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Multiple fixed point theorems for contractive and Meir-Keeler type mappings

Suppose also that one of the following sets of conditions is satisfied:

1. (X, d, �) is monotonically complete; from x, y ∈ Xm and x �L y it
follows that λF(x) �L λF(y); there exists a ∈ X

m such that a �L
λF(a).

2. (X, d, �) is complete; from x, y ∈ Xm and x �L y it follows that
λF(y) �L λF(x); there exists a ∈ X

m such that a �L λF(a) or
F(a) �L a.

Then there exists a unique multidimensional fixed point of the operator of F.

Theorem 4.4. Let (X, d, �) be an ordered metric space, m ∈ N, F : Xm → X
be an operator. Suppose that:

a) there exists a function δ : (0, +∞) −→ (0, +∞) such that from r > 0,
x, y ∈ Xm, d̄m(x, y) < r+δ(r) and x � y it follows that d̄m(λF(x), λF(y)) <
r;

b) for any two points x, y ∈ X, there exist an upper bound and a lower
bound; for any i ∈ {1, 2, . . . , m}, the mapping λi is a surjection or,
more generally, |∪{λ−1i (j) : 1 ≤ j ≤ m}| = m, for each i ∈ {1, 2, . . . , m}.

Suppose also that one of the following sets of conditions is satisfied:

1. (X, d, �) is monotonically complete; from x, y ∈ Xm and x �L y it
follows that λF(x) �L λF(y); there exists a ∈ X

m such that a �L
λF(a).

2. (X, d, �) is complete; from x, y ∈ Xm and x �L y it follows that
λF(y) �L λF(x); there exists a ∈ X

m such that a �L λF(a) or
F(a) �L a.

Then there exists a unique multidimensional fixed point of the operator of F.

5. Some particular cases and a generic application of multiple
fixed points

If we take concrete values of m ∈ N and consider various particular functions
λ = {λi : {1, ..., m} → {1, ..., m} : 1 ≤ i ≤ m} then, most of the concepts of
coupled, triple, quadruple,..., multiple fixed points existing in literature are
obtained as particular cases of the concept of multiple fixed point considered
in [32] and the present paper.

For example, if m = 2, λ1(1) = 1, λ1(2) = 2; λ2(1) = 2, λ2(2) = 1,
we obtain the concept of coupled fixed point studied in [37] and in various
subsequent papers. If m = 3, λ1(1) = 1, λ1(2) = 2, λ1(3) = 3; λ2(1) = 2,
λ2(2) = 1, λ2(3) = 2; λ3(1) = 3, λ3(2) = 2, λ3(3) = 1, then the concept of
multiple fixed point studied in the present paper reduces to that of triple fixed
point, first introduced in [18] and intensively studied in many other research
works emerging from it.

We note that, as pointed out in [77], the notion of tripled fixed point due to
Berinde and Borcut [18] is different from the one defined by Samet and Vetro
[74] for N = 3, since in the case of ordered metric spaces, in order to keep the

c© AGT, UPV, 2017 Appl. Gen. Topol. 18, no. 2 325



M. M. Choban and V. Berinde

mixed monotone property working, it is necessary to take λ2(3) = 2 and not
λ2(3) = 3.

It is also important to mention here that some cases of multidimensional
coincidence point results (that extend multiple fixed point theorems) are not
compatible with the mixed monotone property (see [7]).

For other concepts of multiple fixed points considered in literature the con-
dition ”λi is a surjection, for each i ≤ m” is no more valid, see for example
[18] and the research papers emerging from it, while the second condition,
| ∪ {λ−1i (j) : 1 ≤ j ≤ m}| = m, for each i ≤ m, is satisfied.

Finally, we point out the fact that our approach in [32] and in this paper is
based on the idea to obtain general multiple fixed point theorems by reducing
this problem to a unidimensional fixed point problem and by simultaneously
working in a more general and very reliable setting, i.e., that of distance space.

Many other related and relevant results could be obtained in the same way,
by reducing the multidimensional fixed point problem to many other indepen-
dent unidimensional fixed point principles, like the ones established in [5], [11],
[12], [13], [15], [16], [19], [21], [23] etc.

We end the paper by indicating an interesting generic application of multiple
fixed points in game theory.

Fix an orderable distance space (X, d, �) and a positive integer number
m ≥ 2. We put Nm = {1, 2, ..., m}. For L ⊆ Nm, we introduce on X

m the
ordering �L: (x1, ..., xm) �L (y1, ..., ym) iff xi � yi for i ∈ L and yj � xj for
j /∈ L.

Denote by Λ = (λ1, ..., λm) a collection of mappings {λi : Nm −→ Nm : i ≤
m}.

Let F : Xm −→ X be an operator. The operator F and the mappings Λ
generate the operator ΛF : Xm −→ Xm, where λF(x1, ...., xm) = (y1, ..., ym)
and yi = F(xλi(1), ..., xλi(m)) for each point (x1, ..., xm) ∈ X

m and any index
i ≤ m.

Assume now that Nm is the set of players and i ∈ Nm is the symbol of the
ith player. In this case we say that:

- X is the space of the positions (decisions) of the players;
- d(x, y) is the measure of the non-convenience of the position x relatively
to the position y;

- ordering ≤ is the relation of domination of positions;
- a point x = (x1, x2, ..., xm) ∈ X

m is a selection of positions, where xi
is the position of the player i;

- the operator F is the operator of correction of the positions.

Every selection of positions x = (x1, x2, ..., xm) ∈ X
m determines the selection

of positions y = (y1, y2, ..., ym) = ΛF(x) ∈ X
m.

For any player i the number d(xi, yi) is the measure of the non-convenience
of the position xi relatively to the position yi for the i

th player.
One considers that the selection of the positions x = (x1, x2, ..., xm) ∈ X

m

is optimal if d(xi, yi) is minimal for each i.

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Multiple fixed point theorems for contractive and Meir-Keeler type mappings

In particular, if ΛF(x) = x, then the selection of positions x is optimal.
One can find distinct concrete examples of the above general model in [54],

[51], [61], [62].

Acknowledgements. This second author acknowledges the support provided

by the Deanship of Scientific Research at King Fahd University of Petroleum

and Minerals for funding this work through the projects IN151014 and IN141047.

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