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Fixed point theorems for monotone multivalued mappings in partially ordered metric spaces
Fixed Point Theory and Applications volume 2014, Article number: 110 (2014)
Abstract
In this paper, we introduce two new types of monotone multivalued mappings in partially ordered metric spaces and prove some fixed point theorems of those two types of mappings under some contraction conditions. Our main results extend many known results in the literature. Moreover, we also give an example which satisfies our main theorem but Nadler’s theorem cannot be applied.
MSC:47H04, 47H10.
1 Introduction
For a metric space $(X,d)$, we let $\mathit{CB}(X)$ and $Comp(X)$ to be the set of all nonempty closed bounded subsets of X and the set of all nonempty compact subsets of X, respectively.
Fixed point theorems of multivalued mappings play extremely important roles in economics and engineering [1, 2], especially in game theory [3, 4]. They can be used for proving the existence of Nash equilibria of a noncooperative game. The first wellknown theorem for multivalued contraction mappings was given by Nadler [5] in 1967.
Theorem 1.1 ([5])
Let $(X,d)$ be a complete metric space and let T be a mapping from X into $\mathit{CB}(X)$. Assume that there exists $k\in [0,1)$ such that
Then there exists $z\in X$ such that $z\in Tz$.
The Nadler’s fixed point theorem for multivalued contractive mappings has been extended in many directions (see [2, 4–17]). Reich [18] proved the following fixed point theorem for multivalued φ contraction mappings.
Theorem 1.2 ([18])
Let $(X,d)$ be a complete metric space and let T be a mapping from X into $Comp(X)$. Assume that there exists a function $\phi :(0,\mathrm{\infty})\to [0,1)$ such that
and
Then there exists $z\in X$ such that $z\in Tz$.
The multivalued mapping T considered by Reich in Theorem 2 has compact value, that is, Tx is a nonempty compact subset of X for all $x\in X$. In 1989, Mizoguchi and Takahashi [16] relaxed the compactness of value of T to closed and bounded subsets of X. They proved the following theorem, which is a generalization of Nadler’s theorem.
Theorem 1.3 ([16])
Let $(X,d)$ be a complete metric space and let $T:X\to \mathit{CB}(X)$. Assume that there exists a function $\phi :(0,\mathrm{\infty})\to [0,1)$ such that
and
Then there exists $z\in X$ such that $z\in Tz$.
Recently, many fixed point theorems have been extended to partially ordered space (see [2–4, 8–16, 19–22]). Some fixed point theorems in partially ordered metric spaces can be applied to study a problem of ordinary differential equations. In 2004, Ran and Reurings [23] proved Banach’s fixed point theorem in partially ordered metric spaces.
Theorem 1.4 ([23])
Let X be a partially ordered set such that every pair $x,y\in X$ has a lower bound and upper bound. Furthermore, let d be a metric on T such that $(X,d)$ is a complete metric space. If T is a continuous, monotone (i.e., either orderpreserving or orderreversing) map from X into X such that

(1)
there exists $c\in (0,1)$ such that
$$d(T(x),T(y))\le cd(x,y)\phantom{\rule{1em}{0ex}}\mathit{\text{for all}}x\ge y,$$ 
(2)
there exists ${x}_{0}\in T{x}_{0}$ such that ${x}_{0}\le T({x}_{0})$ or ${x}_{0}\ge T({x}_{0})$,
then T has a unique fixed point $\overline{x}$. Moreover, for every $x\in X$,
In 2005, Nieto and RodriguezLopez [24] extended the above result for a mapping T without continuity.
Theorem 1.5 ([24])
Let $(X,\le )$ be a partially ordered set and suppose that there exists a metric d in X such that $(X,d)$ is a complete metric space. Assume that X satisfies
Let $f:X\to X$ be a monotone nondecreasing mapping such that there exists $k\in [0,1)$ with
If there exists ${x}_{0}\in X$ with ${x}_{0}\le f({x}_{0})$, then f has a fixed point.
Motivated by these works, we are interested to define two types of monotone multivalued mappings in partially ordered metric spaces and prove some fixed point theorems of these mappings under some contraction conditions considered by Mizoguchi and Takahashi [16].
2 Preliminaries
Let $(X,d)$ be a metric space and $\mathit{CB}(X)$ be the set of all nonempty closed bounded subsets of X. For $x\in X$ and $A,B\in \mathit{CB}(X)$, define
Denote H the Hausdorff metric induced by d, that is,
The following two lemmas which can be found in [5] or [16] are useful for our main results.
Lemma 2.1 ([5])
Let $(X,d)$ be a metric space. If $A,B\in \mathit{CB}(X)$ and $a\in A$, then, for each $\u03f5>0$, there exists $b\in B$ such that
Lemma 2.2 ([16])
Let $\{{A}_{n}\}$ be a sequence of sets in $\mathit{CB}(X)$, and suppose ${lim}_{n\to \mathrm{\infty}}H({A}_{n},{A}_{0})=0$ where ${A}_{0}\in \mathit{CB}(X)$. Then if ${x}_{n}\in {A}_{n}$, $n=1,2,3,\dots $ , and if ${lim}_{n\to \mathrm{\infty}}{x}_{n}={x}_{0}$, it follows that ${x}_{0}\in {A}_{0}$.
Definition 2.3 Let $T:X\to {2}^{X}$ be a mapping. A point $x\in X$ is said to be a fixed point of T if $x\in Tx$.
Definition 2.4 A preorder is a binary relation ≤ in a set X which satisfies the followings conditions:

(1)
$x\le x$ (reflexivity);

(2)
if $x\le y$ and $y\le z$, then $x\le z$ (transitivity)
for all $x,y\in X$.
A set with a preorder ≤ is called a preordered set.
For $x,y\in X$, we write
A partial ordering in a set X is a preordered ≤ in X with the additional property:
A set together with a definite partial ordering is called a partially ordered set. Let $(X,\le )$ be a partially ordered set. In 2010, Beg and Butt [8] defined relations between two sets. For $A,B\subset X$, the relations ${\le}^{(\mathrm{I})}$ and ${\le}^{(\mathrm{II})}$ between A and B are defined as follows:

(1)
$A{\le}^{(\mathrm{I})}B$ if $a\le b$ for all $a\in A$ and $b\in B$.

(2)
$A{\le}^{(\mathrm{II})}B$ if for each $a\in A$ there exists $b\in B$ such that $a\le b$.
Note that if A is a nonempty subset of X with $A{\le}^{(\mathrm{I})}A$, then A is singleton.
Next, we define two types of monotone mappings by using the relations ${\le}^{(\mathrm{I})}$ and ${\le}^{(\mathrm{II})}$.
Definition 2.5 Let $(X,d)$ be a metric space endowed with a partial order ≤ and $T:X\to \mathit{CB}(X)$. Then T is said to be

(i)
monotone nondecreasing of type (I) if
$$x,y\in X,\phantom{\rule{1em}{0ex}}x<y\phantom{\rule{1em}{0ex}}\Rightarrow \phantom{\rule{1em}{0ex}}Tx{\le}^{(\mathrm{I})}Ty;$$ 
(ii)
monotone nondecreasing of type (II) if
$$x,y\in X,\phantom{\rule{1em}{0ex}}x\le y\phantom{\rule{1em}{0ex}}\Rightarrow \phantom{\rule{1em}{0ex}}Tx{\le}^{(\mathrm{II})}Ty.$$
The concept of monotonicity of type (II) was first introduced by Fujimoto [21]. Instead of monotonicity of type (II), he used the notion of isotone or orderedincreasing upward. He proved the following fixed point theorem known as the FujimotoTarski fixed point theorem.
Theorem 2.6 ([21])
Let $(X,\le )$ be a complete lattice and $F:X\to {2}^{X}\{\mathrm{\varnothing}\}$ a multivalued mapping. If F satisfies the following two conditions:

(1)
F is isotone, that is, (F is orderedincreasing upward) if $x\le y$, then for any $z\in F(x)$ there is a $w\in F(y)$ such that $z\le w$.

(2)
$SF(x)=\{z\in X:z\le u\mathit{\text{for some}}u\in F(x)\}$ is an inductively ordered set for each $x\in X$.
Then F has a fixed point.
He also applied this result to study solvability of vectorcomplementarity problems. In [4], Li applied this theorem to investigate the existence of generalized and extended Nash equilibria of nonmonetized, noncooperative games on chaincomplete lattices.
Very recently, Li [4] extended Theorem 2.6 to the following theorems.
Theorem 2.7 Let $(P,\le )$ be a chaincomplete poset and let $P:P\to {2}^{P}\setminus \{\mathrm{\varnothing}\}$ be a setvalued mapping. If F satisfies the following three conditions:
(A1) F is orderincreasing upward;
(A2) $(SF(x),\le )$ is an inductively ordered set, for each $x\in P$;
(A3) There is a y in P with $y\le u$ for some $u\in F(y)$.
Then F has a fixed point.
He also used this theorem to study the existence of generalized and extended Nash equilibrium problems.
It is easy to see that monotone nondecreasing of type (I) is of type (II) but the converse is not true as seen in the following example.
Example 2.8 Let $X=[0,\mathrm{\infty})$ with the usual relation ≤ defined on it. Let $T:X\to {2}^{X}$ be defined by
It is easy to see that T is monotone nondecreasing of type (II) but not type (I).
3 Main results
We first prove the existence theorem for monotone multivalued mappings of type (II) involving the delta distance. Throughout this paper, we use $\mathcal{J}$ to denote the class function $\phi :[0,\mathrm{\infty})\to [0,1)$ such that ${lim}_{r\to {t}^{+}}sup\phi (r)<1$ for each $t\in [0,\mathrm{\infty})$.
Theorem 3.1 Let $(X,d)$ be a complete metric space endowed with a partial ordered ≤ and $T:X\to \mathit{CB}(X)$ be multivalued mapping. Suppose that:

(1)
T is monotone of type (II).

(2)
There exists ${x}_{0}\in X$ such that ${x}_{0}{\le}^{(\mathrm{II})}T{x}_{0}$.

(3)
For each sequence $\{{x}_{n}\}$ such that ${x}_{n}\le {x}_{n+1}$ for all $n\in \mathbb{N}$ and ${x}_{n}$ converges to x, for some $x\in X$, then ${x}_{n}\le x$ for all $n\in \mathbb{N}$.

(4)
There exists a function $\phi \in \mathcal{J}$ such that
$$\delta (Tx,Ty)\le \phi (d(x,y))(d(x,y))\phantom{\rule{1em}{0ex}}\mathit{\text{for all}}x,y\in X\mathit{\text{with}}x\le y.$$
Then there exists $x\in X$ such that $x\in Tx$.
Proof By assumption (2), there exists ${x}_{1}\in T{x}_{0}$ such that ${x}_{0}\le {x}_{1}$. Moreover, by monotonicity of T, we have $T{x}_{0}\le T{x}_{1}$ i.e. there exists ${x}_{2}\in T{x}_{1}$ such that ${x}_{1}\le {x}_{2}$. By (4), we have
By induction, we obtain a sequence $\{{x}_{n}\}$ in X with the property that ${x}_{n+1}\in T{x}_{n}$, ${x}_{n}\le {x}_{n+1}$ and
We see that $\{d({x}_{n},{x}_{n+1})\}$ is strictly decreasing and bounded below so
Suppose $r>0$. Then
Therefore
which is a contradiction. We conclude that $r=0$ i.e., ${lim}_{n\to \mathrm{\infty}}d({x}_{n},{x}_{n+1})=0$. We will show that $\{{x}_{n}\}$ is a Cauchy sequence. To show this, suppose not. Then there exist $\u03f5>0$ and integers ${m}_{k},{n}_{k}\in \mathbb{N}$ such that ${m}_{k}>{n}_{k}\ge k$ and
Also choosing ${m}_{k}$ as small as possible, it may be assumed that
Hence, for each $k\in \mathbb{N}$, we have
Since ${lim}_{k\to \mathrm{\infty}}d({x}_{{m}_{k}1},{x}_{{m}_{k}})=0$, we have ${lim}_{k\to \mathrm{\infty}}d({x}_{{m}_{k}}.{x}_{{n}_{k}})=\u03f5$. By transitivity of ≤, we have ${x}_{{n}_{k}}\le {x}_{{m}_{k}}$. By using the triangle inequality and assumption (4), we have
Letting $k\to \mathrm{\infty}$ and using the property of φ, we obtain
which is a contradiction. Hence $\{{x}_{n}\}$ is a Cauchy sequence in X. Since X is complete, there exists $x\in X$ such that ${lim}_{n\to \mathrm{\infty}}{x}_{n}=x$. Moreover, we have ${x}_{n}\le x$ by (3) and
Taking $n\to \mathrm{\infty}$, we obtain $d(x,Tx)=0$. That is, $x\in Tx$. □
In the next theorem, we prove a fixed point theorem of monotone multivalued mapping of type (I) under some contraction conditions involving the Hausdorff distance.
Theorem 3.2 Let $(X,d)$ be a complete metric space endowed with a partial ordered ≤ and $T:X\to \mathit{CB}(X)$ be multivalued mapping. Suppose that:

(1)
T is monotone nondecreasing of type (I).

(2)
There exists ${x}_{0}\in X$ such that ${x}_{0}{\le}^{(\mathrm{II})}T{x}_{0}$.

(3)
For each sequence $\{{x}_{n}\}$ such that ${x}_{n}\le {x}_{n+1}$ for all $n\in \mathbb{N}$ and ${x}_{n}$ converges to x, for some $x\in X$, then ${x}_{n}\le x$ for all $n\in \mathbb{N}$.

(4)
There exists a function $\phi \in \mathcal{J}$ such that
$$H(Tx,Ty)\le \phi (d(x,y))(d(x,y))\phantom{\rule{1em}{0ex}}\mathit{\text{for all}}x,y\in X\mathit{\text{with}}x\le y.$$
Then there exists $x\in X$ such that $x\in Tx$.
Proof Suppose that T has no fixed point, i.e.,
By assumption (4), for any $t>0$, there exist $M(t)>0$ and $\delta (t)>0$ such that
Since ${x}_{0}{\le}^{(\mathrm{II})}T{x}_{0}$, there exists ${x}_{1}\in T{x}_{0}$ such that ${x}_{0}\le {x}_{1}$. By monotonicity of T, we have $T{x}_{0}{\le}^{(\mathrm{I})}T{x}_{1}$. Put ${t}_{1}=d({x}_{1},T{x}_{1})$. Note that $d({x}_{1},T{x}_{1})\le d(x,y)$ for all $y\in T{x}_{1}$. We consider the following cases:
Case (I): $d({x}_{1},T{x}_{1})<d({x}_{1},y)$ for all $y\in T{x}_{1}$. Choose $d({t}_{1})$ such that
and then put
Then there exists ${x}_{2}\in T{x}_{1}$ such that
By the hypothesis that T has no fixed point, we have ${x}_{1}\ne {x}_{2}$ so ${x}_{1}<{x}_{2}$ and by (4), we obtain
This implies
By (3.3) and (3.4) we obtain
This implies by (3.1) that $\phi (d({x}_{1},{x}_{2}))\le M({t}_{1})<1$. Since
we have $\frac{1}{1+\u03f5({x}_{1})}>M({t}_{1})$. Hence
It follows from (3.5) that $d({x}_{2},T{x}_{2})<d({x}_{1},T{x}_{1})$.
Case (II): $d({x}_{1},T{x}_{1})=d({x}_{1},{x}_{2})$ for some ${x}_{2}\in T{x}_{1}$. Since $T{x}_{0}{\le}^{(\mathrm{I})}T{x}_{1}$, we have ${x}_{1}\le {x}_{2}$. By (4), we have
Therefore $d({x}_{2},T{x}_{2})<d({x}_{1},T{x}_{1})$. Next, let ${t}_{2}=d({x}_{2},T{x}_{2})$. Then $d({x}_{2},T{x}_{2})\le d({x}_{2},y)$ for all $y\in T{x}_{2}$. Again we consider the following two cases:
Case (A): $d({x}_{2},T{x}_{2})<d({x}_{2},y)$ for all $y\in T{x}_{2}$. For $\delta ({t}_{2})$ and $M({t}_{2})$, choose $d({t}_{2})$ with
and set
By using the argument as above, we obtain ${x}_{3}\in T{x}_{2}$ such that
and
Hence $d({x}_{3},T{x}_{3}<d({x}_{2},T{x}_{2}))$. From $\u03f5({x}_{2})\le \frac{{t}_{1}}{{t}_{2}}1$, it follows that
Case (B): $d({x}_{2},T{x}_{2})=d({x}_{2},{x}_{3})$ for some ${x}_{3}\in T{x}_{2}$. Using the same method as above, we can show that
and
Hence, $d({x}_{3},T{x}_{3})<d({x}_{2},T{x}_{2})$ and $d({x}_{2},{x}_{3})<d({x}_{1},{x}_{2})$. By continuing in this way, we can construct a sequence $\{{x}_{n}\}$ in X with ${x}_{n+1}\in T{x}_{n}$, ${x}_{n}\le {x}_{n+1}$ for all $n\in \mathbb{N}$ such that $\{d({x}_{n},{x}_{n+1})\}$ and $\{d({x}_{n},T{x}_{n})\}$ are decreasing sequences of positive numbers and
where $\gamma ({x}_{n})$ is a real number with $0\le \gamma ({x}_{n})\le \frac{1}{n}$ ($n=1,2,\dots $). Since $\{d({x}_{n},{x}_{n+1})\}$ is decreasing, there exists $t\in [0,\mathrm{\infty})$ such that ${lim}_{n\to \mathrm{\infty}}d({x}_{n},{x}_{n+1})=t$. By the property of φ, we obtain
Putting ${a}_{n}=\frac{1}{1+\gamma ({x}_{n})}\phi (d({x}_{n},{x}_{n+1}))$, we have
This implies by (3.7) that there exists $b>0$ such that
for large enough n. Since $\{d({x}_{n},T{x}_{n})\}$ is decreasing, it is convergent. It follows that
as $m,n\to \mathrm{\infty}$. Hence $\{{x}_{n}\}$ is Cauchy. So $\{{x}_{n}\}$ converges to some $x\in X$. By (3), we obtain ${x}_{n}\le x$ for all $n\in \mathbb{N}$. By (4), we get
Hence ${lim}_{n\to \mathrm{\infty}}H(Tx,T{x}_{n})=0$. Since ${x}_{n+1}\in T{x}_{n}$ and $\{{x}_{n+1}\}$ converges to x, it follows from Lemma 2.2 that $x\in Tx$. This contradicts that T has no fixed point. This complete the proof. □
Corollary 3.3 Let $(X,d)$ be a complete metric space endowed with a partial ordered ≤ and $T:X\to \mathit{CB}(X)$ be multivalued mapping. Suppose that:

(1)
T is monotone of type (II).

(2)
There exists ${x}_{0}\in X$ such that ${x}_{0}{\le}^{(\mathrm{II})}T{x}_{0}$.

(3)
For each sequence $\{{x}_{n}\}$ such that ${x}_{n}\le {x}_{n+1}$ for all $n\in \mathbb{N}$ and ${x}_{n}$ converges to x, for some $x\in X$, then ${x}_{n}\le x$ for all $n\in \mathbb{N}$.

(4)
There exists $k\in (0,1)$ such that
$$\delta (Tx,Ty)\le k(d(x,y))\phantom{\rule{1em}{0ex}}\mathit{\text{for each}}x\le y.$$
Then there exists $x\in X$ such that $x\in Tx$.
Proof We denote $\phi (r)=k$ for all $r\in (0,\mathrm{\infty})$, so, for any $t\in [0,\mathrm{\infty})$,
Moreover, for each $x,y\in X$ with $x\le y$, we obtain
By Theorem 3.2, we obtain the desired result. □
Remark 3.4 The reader interested in common fixed point results of multivalued mappings taking closed values in ordered generalized metric spaces is referred to [6].
Example 3.5 Let $X=\{\frac{1}{2},\frac{1}{4},\dots ,\frac{1}{{2}^{n}},\dots \}\cup \{0,1\}$, $d(x,y)=xy$, for $x,y\in X$; then X is a complete metric space. Define a relation ≤ by
It is easy to check that this relation is a partially ordered. Note that $0\nleqq 1$.
Define the mapping $T:X\to \mathit{CB}(X)$ by
Clearly, T is increasing of type (I). Moreover, if $\frac{1}{{2}^{n}}\le \frac{1}{{2}^{m}}$,
and
Moreover,
Thus T is not a contraction mapping. So Nadler’s theorem and Mizoguchi and Takahashi’s theorem [16] cannot guarantee the existence of its fixed point. However, T satisfies all conditions of Theorem 3.2. So it has a fixed point and we note that 0 and 1 are its fixed points.
4 Some applications
In this section, we will recall some concepts of generalized and extended Nash equilibria of nonmonetized, noncooperative game which are defined in [3, 4] and we point out that our main result can be applied to prove some existence theorems for generalized and extended Nash equilibria problems.
Definition 4.1 Let n be a positive integer greater than 1. An nperson nonmonetized noncooperative game consists of the following elements:

1.
the set of n players, which is denoted by $N=\{1,2,\dots ,n\}$;

2.
the collection of n strategy sets $\{{S}_{1},{S}_{2},\dots ,{S}_{n}\}$, for the n player, respectively, such that $({S}_{i},{\u2ab0}_{i})$ is a chaincomplete poset, for every player $i=1,2,\dots ,n$, with notation $S={S}_{1}\times {S}_{2}\times \cdots \times {S}_{n}$;

3.
the outcome space $(U,{\u2ab0}^{U})$ that is a poset;

4.
the n payoff function ${P}_{1},{P}_{2},\dots ,{P}_{n}$, where ${P}_{i}$ is the payoff function for player i that is a mapping from ${S}_{1}\times {S}_{2}\times \cdots \times {S}_{n}$ to the poset $(U,{\u2ab0}^{U})$, for $i=1,2,\dots ,n$. We denote $P=\{{P}_{1},{P}_{2},\dots ,{P}_{n}\}$.
This game is denoted by $\mathrm{\Gamma}=(N,S,P,U)$.
The rule to play in an nperson nonmonetized, noncooperative game $\mathrm{\Gamma}=(N,S,P,U)$ is that when all the n players $1,2,\dots ,n$ simultaneously and independently choose their own strategies ${x}_{1},{x}_{2},\dots ,{x}_{n}$ where ${x}_{i}\in {S}_{i}$, for $i=1,2,\dots ,n$, then player i will receive his or her utility (payoff) ${P}_{i}({x}_{1},{x}_{2},\dots ,{x}_{n})\in U$. For any $x=({x}_{1},{x}_{2},\dots ,{x}_{n})\in S$, and for every given $i=1,2,\dots ,n$, as usual, we denote
Then ${x}_{i}\in {S}_{i}$. We use the following notations: $x=({x}_{i},{x}_{i})$ and ${P}_{i}({S}_{i},{x}_{i})=\{{P}_{i}({t}_{i},{x}_{i}):{t}_{i}\in {S}_{i}\}$.
Next, we recall the concept of generalized and extended Nash equilibrium of nonmonetized, noncooperative games on preordered sets.
Definition 4.2 In an nperson nonmonetized, noncooperative game $\mathrm{\Gamma}=(N,S,P,U)$, a selection of strategies $\stackrel{\u02c6}{x}=({\stackrel{\u02c6}{x}}_{1},{\stackrel{\u02c6}{x}}_{2},\dots ,{\stackrel{\u02c6}{x}}_{n})\in ({S}_{1}\times {S}_{2}\times \cdots \times {S}_{n})$ is called

1.
a generalized Nash equilibrium of this game, if, for every $i=1,2,\dots ,n$, the following order inequality holds:
$${P}_{i}({x}_{i},{\stackrel{\u02c6}{x}}_{i}){\u2aaf}^{U}{P}_{i}({\stackrel{\u02c6}{x}}_{i},{\stackrel{\u02c6}{x}}_{i})\phantom{\rule{1em}{0ex}}\text{for all}{x}_{i}\in {S}_{i};$$ 
2.
an extended Nash equilibrium of this game, if, for every $i=1,2,\dots ,n$, the following order inequality holds:
$${P}_{i}({x}_{i},{\stackrel{\u02c6}{x}}_{i}){\nsucc}^{U}{P}_{i}({\stackrel{\u02c6}{x}}_{i},{\stackrel{\u02c6}{x}}_{i})\phantom{\rule{1em}{0ex}}\text{for all}{x}_{i}\in {S}_{i}.$$
We note that the conditions (A1) and (A2) of Theorem 2.7 are those (1) and (3) of Theorem 3.1, respectively, while we have no assumption (A2) in Theorem 2.7. However, we need to assume conditions (3) and (4) because the studied space of our main result is more general than that of Theorem 2.7. We observe that in our study, we need some contractive condition on a mapping $T:X\to \mathit{CB}(X)$.
By using the same as the application of Theorem 2.7, under some modifications of the problem setting, Theorem 3.1 can be applied to prove the existence of generalized and extended Nash equilibria of nonmonetized, noncooperative game on a partially ordered metric space.
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Acknowledgements
This work was supported by Chiang Mai University, Chiang Mai, Thailand.
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Tiammee, J., Suantai, S. Fixed point theorems for monotone multivalued mappings in partially ordered metric spaces. Fixed Point Theory Appl 2014, 110 (2014). https://doi.org/10.1186/168718122014110
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Keywords
 fixed point theorems
 multivalued mappings
 partially ordered set
 monotone mappings