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 Open Access
Fixed point theorems in $CAT(0)$ spaces with applications
 Haishu Lu^{1},
 Di Lan^{2}Email author,
 Qingwen Hu^{3} and
 George Yuan^{4}
https://doi.org/10.1186/1029242X2014320
© Lu et al.; licensee Springer. 2014
 Received: 20 March 2014
 Accepted: 30 July 2014
 Published: 21 August 2014
Abstract
In this paper, noncompact $CAT(0)$ versions of the FanBrowder fixed point theorem are established. As applications, we obtain new minimax inequalities, a saddle point theorem, a fixed point theorem for singlevalued mappings, best approximation theorems, and existence theorems of φequilibrium points for multiobjective noncooperative games in the setting of noncompact $CAT(0)$ spaces. These results generalize many wellknown theorems in the literature.
MSC:91A10, 47H04, 47H10, 54H25.
Keywords
 $CAT(0)$ space
 fixed point
 minimax inequality
 multiobjective game
 equilibrium point
1 Introduction
Fixed point theorems for setvalued mappings play a vital role in various fields of pure and applied mathematics. In 1968, Browder [1] proved that every setvalued mapping with convex values and open fibers from a compact Hausdorff topological vector space to a convex space has a continuous selection. By using this selection theorem and the Brouwer fixed point theorem, Browder [1] obtained the famous Browder fixed point theorem which is equivalent to the Fan section theorem established by Fan [2] in 1961. For this reason, the Browder fixed point theorem is also called in the literature the FanBrowder fixed point theorem. Since then, a body of generalizations and applications of the FanBrowder fixed point theorem have been extensively investigated by many authors; see, for example, [3–12] and the references therein. In particular, Park [13] discussed some updated unified forms of KKM theorems under the framework of abstract convex spaces, which include hyperconvex spaces as special cases.
We recall that a $CAT(0)$ space is a special metric space and it does not possess any linear structure. Many authors have made a lot of efforts to generalize the fixed point theory from Euclidean spaces to $CAT(0)$ spaces. Recently, a number of authors pay attention to establish fixed point theorems in $CAT(0)$ spaces. Kirk [14, 15] first studied the fixed point theory in $CAT(0)$ spaces. Since then, many authors have developed the fixed point theory for singlevalued and setvalued mappings in the setting of $CAT(0)$ spaces. Dhompongsa et al. [16] proved that a nonexpansive mapping from a nonempty bounded closed convex subset of a $CAT(0)$ space to the family of nonempty compact subsets of the $CAT(0)$ space has a fixed point under suitable conditions. Shahzad [17] obtained fixed point theorems for singlevalued and setvalued mappings in $CAT(0)$ spaces or ℝtrees. By using a Ky Fan type minimax inequality in $CAT(0)$ spaces, Shabanian and Vaezpour [18] proved fixed point theorems and best approximation theorems. More recently, Asadi [19] studied the existence problem of common fixed points for two mappings in $CAT(0)$ spaces. Other results, we refer the reader to the literature of Kirk [20], Shahzad and Markin [21], Shahzad [17], and many others.
We know that both $CAT(0)$ and hyperconvex spaces are two interesting classes of spaces. But a $CAT(0)$ space may not be a hyperconvex, indeed a $CAT(0)$ space is a hyperconvex space if and only if it is a complete ℝtree (see Kirk [22] and the references therein).
Inspired and motivated by the results mentioned above, in this paper, we first establish generalized $CAT(0)$ versions of the FanBrowder fixed point theorem. As applications, new minimax inequalities, a saddle point theorem, a fixed point theorem for singlevalued mappings, best approximation theorems, and existence theorems of φequilibrium points for multiobjective noncooperative games are obtained in the setting of noncompact $CAT(0)$ spaces.
2 Preliminaries
Let ℝ and ℕ denote the set of all real numbers and the set of natural numbers, respectively. Let X be a set. We will denote by ${2}^{X}$ the family of all subsets of X, by $\u3008X\u3009$ the family of nonempty finite subsets of X. Let A be a subset of a topological space X, we will denote the interior of A in X and the closure of A in X by ${int}_{X}A$ and ${cl}_{X}A$, respectively. Let X, Y be two nonempty sets and $T:X\to {2}^{Y}$ be a setvalued mapping. Then the setvalued mapping ${T}^{1}:Y\to {2}^{X}$ is defined by ${T}^{1}(y)=\{x\in X:y\in T(x)\}$ for every $y\in Y$.
Now we introduce some notation and concepts related to $CAT(0)$ spaces. For more details, the reader may consult [16–19, 21, 23–29] and the references therein.
Let $(E,d)$ be a metric space. A geodesic which joints the pair of points ${x}_{1},{x}_{2}\in E$ is a mapping $\gamma :[0,a]\subseteq \mathbb{R}\to E$ such that $\gamma (0)={x}_{1}$, $\gamma (a)={x}_{2}$, and $d(\gamma (t),\gamma ({t}^{\prime}))=t{t}^{\prime}$ for every $t,{t}^{\prime}\in [0,a]$. In particular, we have $a=d({x}_{1},{x}_{2})$. The image $\gamma ([0,a])$ of γ is said to be a geodesic segment joining ${x}_{1}$ and ${x}_{2}$. If the segment $\gamma ([0,a])$ is unique, then this geodesic segment is denoted by $[{x}_{1},{x}_{2}]$. The metric space $(E,d)$ is said to be a geodesic space if, for every $x,y\in E$, there is a geodesic jointing x and y, and $(E,d)$ is called to be uniquely geodesic if there is only one geodesic segment joining every pair of points $x,y\in E$.
Let D be a subset of a geodesic space $(E,d)$. Then D is said to be convex if every geodesic segment joining any two points in D is contained in D.
We point out that such a comparison triangle always exists (see [23]). A geodesic space is said to be a $CAT(0)$ space if the equality $d(x,y)\le {d}_{{\mathbb{R}}^{2}}(\overline{x},\overline{y})$ holds for every $x,y\in \Delta $ and every $\overline{x},\overline{y}\in \overline{\Delta}$. Every $CAT(0)$ space $(E,d)$ is uniquely geodesic (see [23]).
which is called the (CN) inequality of Bruhat and Tits [30].
where ${D}_{0}=D$ and for $n\ge 1$, the set ${D}_{n}$ consists of all points in E which lie on geodesics which start and end in ${D}_{n1}$.
Definition 2.2 ([29])
Let K be a nonempty subset of a topological space X. If every continuous mapping $\varphi :K\to K$ has a fixed point, then K is said to have the fixed point property.
Definition 2.3 ([18])
A $CAT(0)$ space $(E,d)$ is said to have the convex hull finite property if the closed convex hull of every nonempty finite subset of E has the fixed point property.
Lemma 2.1 ([29])
Let $(E,d)$ be a complete $CAT(0)$ space with the convex hull finite property and X be a nonempty subset of E. Suppose that $H:X\to {2}^{X}$ is a KKM mapping with closed values and $H(z)$ is compact for some $z\in X$. Then ${\bigcap}_{x\in X}H(x)\ne \mathrm{\varnothing}$.
Lemma 2.2 Let $(E,d)$ be a complete metric space. Then E is a geodesic space if and only if for every $x,y\in E$, there exists $m\in E$ such that $d(x,z)=d(z,y)=\frac{1}{2}d(x,y)$.
Proof The proof of sufficiency can be found in [[23], p.4]. Therefore, it suffices to prove the necessity. By the definition of a geodesic space, for every $x,y\in E$, there exists a mapping $\gamma :[0,a]\subseteq \mathbb{R}\to E$ such that $\gamma (0)=x$, $\gamma (a)=y$, and $d(\gamma (t),\gamma ({t}^{\prime}))=t{t}^{\prime}$ for every $t,{t}^{\prime}\in [0,a]$. Take ${t}_{0}=\frac{a}{2}\in [0,a]$ and $z=\gamma ({t}_{0})\in E$. Then we have $d(x,z)=d(\gamma (0),\gamma ({t}_{0}))=\frac{a}{2}$ and $d(z,y)=d(\gamma ({t}_{0}),\gamma (a))=\frac{a}{2}$. Since $d(x,y)=d(\gamma (0),\gamma (a))=a$, it follows that $d(x,z)=d(z,y)=\frac{1}{2}d(x,y)$. This completes the proof. □
Lemma 2.3 ([23])
A geodesic space is a $CAT(0)$ space if and only if it satisfies the (CN) inequality.
Lemma 2.4 ([31])
Every locally compact $CAT(0)$ space $(E,d)$ has the convex hull finite property.
Lemma 2.5 ([25])
Let $(E,d)$ be a $CAT(0)$ space and let $x,y\in E$. Then, for every $t\in [0,1]$, there exists a unique point $z\in [x,y]$ such that $d(x,z)=td(x,y)$ and $d(y,z)=(1t)d(x,y)$.
From now on, we will use the notation $(1t)x\oplus ty$ for the unique point z in Lemma 2.5.
Lemma 2.6 ([25])
Let $(E,d)$ be a $CAT(0)$ space and let $x,y\in E$ such that $x\ne y$. Then $[x,y]=\{(1t)x\oplus ty:t\in [0,1]\}$.
3 Fixed point theorems
In this section, we will develop four new versions of fixed point theorems in noncompact $CAT(0)$ spaces.
 (i)
for every $y\in E$, $F(y)\subseteq G(y)$ and $G(y)$ is convex;
 (ii)
for every $x\in E$, ${F}^{1}(x)$ is open in E;
 (iii)
for every $y\in K$, $F(y)\ne \mathrm{\varnothing}$;
 (iv)one of the following conditions holds:

(iv)_{1} for every $N\in \u3008E\u3009$, there exists a nonempty compact convex subset ${E}_{N}$ of E containing N such that${E}_{N}\setminus K\subseteq \bigcup _{x\in {E}_{N}}{int}_{{E}_{N}}({G}^{1}(x)\cap {E}_{N});$

(iv)_{2} there exists a point ${x}_{0}\in E$ such that ${cl}_{E}(E\setminus {G}^{1}({x}_{0}))\subseteq K$.

Then there exists $\stackrel{\u02c6}{y}\in E$ such that $\stackrel{\u02c6}{y}\in G(\stackrel{\u02c6}{y})$.
Proof We distinguish the following two cases (iv)_{1} and (iv)_{2} for the proof.
By (iii), for every $y\in K$, $F(y)\ne \mathrm{\varnothing}$ and so, $K\subseteq {\bigcup}_{x\in E}{F}^{1}(x)$, which is a contradiction. Therefore, there exists $\stackrel{\u02c6}{y}\in K$ such that $\stackrel{\u02c6}{y}\in G(\stackrel{\u02c6}{y})$. This completes the proof.
Taking ${y}_{0}\in K\cap ({\bigcap}_{x\in E}\tilde{F}(x))$, we have ${y}_{0}\in K$ and $x\notin F({y}_{0})$ for every $x\in E$. Hence, we have $F({y}_{0})=\mathrm{\varnothing}$, which contradicts (iii). Therefore, there exists $\stackrel{\u02c6}{y}\in K$ such that $\stackrel{\u02c6}{y}\in G(\stackrel{\u02c6}{y})$. This completes the proof. □
Remark 3.1 Theorem 3.1 can be regarded as a generalization of the FanBrowder fixed point theorem on Euclidean spaces to $CAT(0)$ spaces without any linear structure. Theorem 3.1 is different from Theorem 1 of Browder [1], Theorem 1 of Yannelis [3], and Theorem 2.4‴ of Tan and Yuan [32], which are established in the setting of topological vector spaces.
Remark 3.2 If only (iv)_{1} of Theorem 3.1 holds, then the E in Theorem 3.1 does not need to possess the convex hull finite property. In fact, from the first part of the proof of Theorem 3.1, we can see that for every $N\in \u3008E\u3009$, ${E}_{N}$ is a nonempty compact convex subset of E and thus, it is a compact $CAT(0)$ space with the induced metric. Hence, by Lemma 2.4, ${E}_{N}$ has the convex hull finite property. The key approach to the first part of the proof of Theorem 3.1 is to define two setvalued mappings on each ${E}_{N}$ and then apply the KKM lemma on ${E}_{N}$. Therefore, the E in Theorem 3.1 does not need to have the convex hull finite property.
Remark 3.3 If $F=G$, then (iv)_{1} and (iv)_{2} of Theorem 3.1 can be replaced by the following equivalent conditions, respectively:
${\text{(iv)}}_{1}^{\prime}$ for every $N\in \u3008E\u3009$, there exists a nonempty compact convex subset ${E}_{N}$ of E containing N such that ${E}_{N}\setminus K\subseteq {\bigcup}_{x\in {E}_{N}}{F}^{1}(x)$;
${\text{(iv)}}_{2}^{\prime}$ there exists a point ${x}_{0}\in E$ such that $E\setminus {F}^{1}({x}_{0})\subseteq K$.
 (i)
for every $y\in E$, $F(y)\subseteq G(y)$ and $G(y)$ is convex;
 (ii)
$K\subseteq {\bigcup}_{x\in E}{int}_{E}{F}^{1}(x)$;
 (iii)one of the following conditions holds:

(iii)_{1} for every $N\in \u3008E\u3009$, there exists a nonempty compact convex subset ${E}_{N}$ of E containing N such that${E}_{N}\setminus K\subseteq \bigcup _{x\in {E}_{N}}{int}_{{E}_{N}}({G}^{1}(x)\cap {E}_{N});$

(iii)_{2} there exists a point ${x}_{0}\in E$ such that ${cl}_{E}(E\setminus {G}^{1}({x}_{0}))\subseteq K$.

Then there exists $\stackrel{\u02c6}{y}\in E$ such that $\stackrel{\u02c6}{y}\in G(\stackrel{\u02c6}{y})$.
Proof Define $\tilde{F}:E\to {2}^{E}$ by $\tilde{F}(y)={({int}_{E}{F}^{1})}^{1}(y)$ for every $y\in E$. By (i), we have $\tilde{F}(y)\subseteq F(y)\subseteq G(y)$ for every $y\in E$. By the definition of $\tilde{F}$, we have ${\tilde{F}}^{1}(x)={int}_{E}{F}^{1}(x)$ for every $x\in E$, which is open in E. By (ii) and by the definition of $\tilde{F}$, we know that $\tilde{F}(y)\ne \mathrm{\varnothing}$ for every $y\in K$. Thus, all the hypotheses of Theorem 3.1 for $\tilde{F}$ and G are satisfied. Hence, by Theorem 3.1 for $\tilde{F}$ and G, the conclusion of Theorem 3.2 holds. □
Remark 3.4 We have shown that Theorem 3.1 implies Theorem 3.2. It is evident that Theorem 3.2 implies Theorem 3.1. Therefore, Theorem 3.1 is equivalent to Theorem 3.2.
By Theorem 3.1, we have the following maximal element theorem.
 (i)
for every $y\in E$, $F(y)\subseteq G(y)$ and $G(y)$ is convex;
 (ii)
for every $x\in E$, ${F}^{1}(x)$ is open in E;
 (iii)
for every $y\in E$, $y\notin G(y)$;
 (iv)one of the following conditions holds:

(iv)_{1} for every $N\in \u3008E\u3009$, there exists a nonempty compact convex subset ${E}_{N}$ of E containing N such that${E}_{N}\setminus K\subseteq \bigcup _{x\in {E}_{N}}{int}_{{E}_{N}}({G}^{1}(x)\cap {E}_{N});$

(iv)_{2} there exists a point ${x}_{0}\in E$ such that ${cl}_{E}(E\setminus {G}^{1}({x}_{0}))\subseteq K$.

Then there exists $\stackrel{\u02c6}{y}\in K$ such that $F(\stackrel{\u02c6}{y})=\mathrm{\varnothing}$.
Proof Suppose to the contrary that $F(y)\ne \mathrm{\varnothing}$ for every $y\in K$. Then, by Theorem 3.1, there exists $\stackrel{\u02c6}{y}\in E$ such that $\stackrel{\u02c6}{y}\in G(\stackrel{\u02c6}{y})$, which contradicts (iii) of Theorem 3.3. Therefore, the conclusion of Theorem 3.3 holds. This completes the proof. □
Remark 3.5 Theorem 3.3 is equivalent to Theorem 3.1. We have shown that Theorem 3.1 implies Theorem 3.3. So, it suffices to show that Theorem 3.3 implies Theorem 3.1. Suppose not. Then, for every $y\in E$, $y\notin G(y)$. By Theorem 3.3, there exists $\stackrel{\u02c6}{y}\in K$ such that $F(\stackrel{\u02c6}{y})=\mathrm{\varnothing}$, which contradicts (iii) of Theorem 3.1. Therefore, the conclusion of Theorem 3.1 holds.
Remark 3.6 Theorem 3.3 is established in the setting of noncompact $CAT(0)$ spaces which include Hadamard manifolds as special cases (see [23, 33] and the references therein). Therefore, Theorem 3.3 generalizes Theorem 3.1 of Yang and Pu [34] from Hadamard manifolds to noncompact $CAT(0)$ spaces. We point out that the proof of Theorem 3.3 is different from that of Theorem 3.1 of Yang and Pu [34].
Let I be a finite index set and ${\{({E}_{i},{d}_{i})\}}_{i\in I}$ be a family of metric spaces, where ${d}_{i}$ is the metric of ${E}_{i}$ for every $i\in I$. Let $(E,d)$ be the product space ${\prod}_{i\in I}({E}_{i},{d}_{i})$, where d is the metric of E. For every $i\in I$, every ${x}_{i}\in {E}_{i}$, and every $r>0$, let ${U}_{i}^{{d}_{i}}({x}_{i},r)\subseteq {E}_{i}$ denote the open ball centered at ${x}_{i}$ with radius r. For every $x\in E$ and every $r>0$, let ${U}^{d}(x,r)\subseteq E$ denote the open ball centered at x with radius r.
By Theorem 3.1, we have the following collectively fixed point theorem in noncompact $CAT(0)$ spaces.
 (i)
for every $i\in I$ and every $y\in E$, ${F}_{i}(y)\subseteq {G}_{i}(y)$ and ${G}_{i}(y)$ is convex;
 (ii)
for every $i\in I$ and every ${x}_{i}\in {E}_{i}$, ${F}_{i}^{1}({x}_{i})$ is open in E;
 (iii)
for every $i\in I$ and every $y\in K$, ${F}_{i}(y)\ne \mathrm{\varnothing}$;
 (iv)one of the following conditions holds:

(iv)_{1} for every $i\in I$ and every ${N}_{i}\in \u3008{E}_{i}\u3009$, there exists a nonempty compact convex subset ${E}_{{N}_{i}}$ of ${E}_{i}$ containing ${N}_{i}$ such that, for every $y={({y}_{i})}_{i\in I}\in {E}_{N}\setminus K$, there exist $r(y)>0$ and $\overline{x}(y)={({\overline{x}}_{i}(y))}_{i\in I}\in {E}_{N}$ such that$\prod _{i\in I}{U}_{i}^{{d}_{i}}({y}_{i},r(y))\cap {E}_{N}\subseteq \bigcap _{i\in I}{G}_{i}^{1}({\overline{x}}_{i}(y))\cap {E}_{N},$
where ${E}_{N}={\prod}_{i\in I}{E}_{{N}_{i}}$;

(iv)_{2} there exists a point ${x}_{0}={({x}_{0i})}_{i\in I}\in E$ such that ${cl}_{E}(E\setminus {\bigcap}_{i\in I}{G}_{i}^{1}({x}_{0i}))\subseteq K$.

Then there exists $\stackrel{\u02c6}{y}\in E$ such that ${\stackrel{\u02c6}{y}}_{i}\in {G}_{i}(\stackrel{\u02c6}{y})$ for every $i\in I$.
We prove Theorem 3.4 in the following four steps.
Step 1. Show that $(E,d)$ is a metric space.
Since ${d}_{i}({x}_{i},{y}_{i})\le {d}_{i}({x}_{i},{z}_{i})+{d}_{i}({z}_{i},{y}_{i})$ for every $i\in \{1,2,\dots ,n\}$, it follows from the above inequality that $d(x,y)\le d(x,z)+d(z,y)$.
Step 2. Show that $(E,d)$ is a complete locally compact space.
Therefore, the metric δ is equivalent to d and hence, ${\tau}_{\delta}={\tau}_{d}$. For every $r>0$ and every $x=({x}_{1},{x}_{2},\dots ,{x}_{n})\in E$, we have ${U}^{\delta}(x,r)={\prod}_{i=1}^{n}{U}_{i}^{{d}_{i}}({x}_{i},r)$. In fact, $y\in {U}^{\delta}(x,r)\iff max\{{d}_{i}({x}_{i},{y}_{i}):1\le i\le n\}<r\iff {d}_{i}({x}_{i},{y}_{i})<r$ for every $i\in \{1,2,\dots ,n\}$. So, $y\in {U}^{\delta}(x,r)\iff {y}_{i}\in {U}_{i}^{{d}_{i}}({x}_{i},r)$ for every $i\in \{1,2,\dots ,n\}\iff y\in {\prod}_{i=1}^{n}{U}_{i}^{{d}_{i}}({x}_{i},r)$. We can see that the collection $\{{U}^{\delta}(x,r):x\in E\text{and}r0\}$ forms a base for ${\tau}_{\delta}$ and the collection $\{{\prod}_{i=1}^{n}{U}_{i}^{{d}_{i}}({x}_{i},r):{x}_{i}\in {E}_{i}\text{and}r0\}$ forms a base for the product topology on E. Hence, $\{{U}^{\delta}(x,r):x\in E\text{and}r0\}$ also forms a base for the product topology on E. Therefore, the topology ${\tau}_{d}$ associated to the metric d is the product topology on E.
Thus, ${lim}_{k\to \mathrm{\infty}}{x}^{(k)}=x={({x}_{i})}_{i\in I}\in E$, which implies that E is a complete metric space.
Since ${\prod}_{i=1}^{n}{cl}_{{E}_{i}}{U}_{i}^{{d}_{i}}({x}_{i},{r}_{i})$ is compact and ${cl}_{E}{U}^{d}(x,r)$ is closed, it follows that ${cl}_{E}{U}^{d}(x,r)$ is compact. Hence, E is locally compact.
Step 3. Show that $(E,d)$ is a $CAT(0)$ space.
which implies that E satisfies the (CN) inequality. By Lemma 2.3 again, we know that E is a $CAT(0)$ space.
Step 4. Prove that there exists $\stackrel{\u02c6}{y}\in E$ such that ${\stackrel{\u02c6}{y}}_{i}\in {G}_{i}(\stackrel{\u02c6}{y})$ for every $i\in I$.
Therefore, by Theorem 3.1, there exists $\stackrel{\u02c6}{y}={({\stackrel{\u02c6}{y}}_{i})}_{i\in I}\in E$ such that $\stackrel{\u02c6}{y}\in G(\stackrel{\u02c6}{y})$; that is, ${\stackrel{\u02c6}{y}}_{i}\in {G}_{i}(\stackrel{\u02c6}{y})$ for every $i\in I$. This completes the proof. □
Remark 3.7 We can compare Theorem 3.4 with Theorem 3 of Prokopovych [35] in the following aspects: (1) every ${E}_{i}$ in Theorem 3.4 does not need to be compact and it does not possess any linear structure; (2) in Theorem 3.4, there are two setvalued mappings, but there is only one setvalued mapping in Theorem 3 of Prokopovych [35]; (3) (iii) of Theorem 3.4 is weaker than the corresponding condition of Theorem 3 of Prokopovych [35] because the domain of every ${F}_{i}$ does not need to be E.
4 Minimax inequalities with applications
In this section, by using Theorem 3.1, we will give minimax inequalities in noncompact $CAT(0)$ spaces. As applications of minimax inequalities, we obtain a saddle point theorem, a fixed point theorem for singlevalued mappings, and best approximation theorems in the setting of noncompact $CAT(0)$ spaces.
 (i)
for every $(x,y)\in E\times E$, $f(x,y)\le g(x,y)$;
 (ii)
for every $y\in E$, the set $\{x\in E:g(x,y)>0\}$ is convex;
 (iii)
for every $x\in E$, $y\mapsto f(x,y)$ is lower semicontinuous on E;
 (iv)
for every $y\in E$, $g(y,y)\le 0$;
 (v)one of the following conditions holds:

(v)_{1} for every $N\in \u3008E\u3009$, there exists a nonempty compact convex subset ${E}_{N}$ of E containing N such that${E}_{N}\setminus K\subseteq \bigcup _{x\in {E}_{N}}{int}_{{E}_{N}}(\{y\in E:g(x,y)>0\}\cap {E}_{N});$

(v)_{2} there exists a point ${x}_{0}\in E$ such that ${cl}_{E}(E\setminus \{y\in E:g({x}_{0},y)>0\})\subseteq K$.

Then there exists $\stackrel{\u02c6}{y}\in K$ such that $f(x,\stackrel{\u02c6}{y})\le 0$ for every $x\in E$.
 (a)for every $N\in \u3008E\u3009$, there exists a nonempty compact convex subset ${E}_{N}$ of E containing N such that${E}_{N}\setminus K\subseteq \bigcup _{x\in {E}_{N}}{int}_{{E}_{N}}({G}^{1}(x)\cap {E}_{N});$
 (b)
there exists a point ${x}_{0}\in E$ such that ${cl}_{E}(E\setminus {G}^{1}({x}_{0}))\subseteq K$.
By (iv), $y\notin G(y)$ for every $y\in E$, which implies that the conclusion of Theorem 3.1 does not hold. Hence, (iii) of Theorem 3.1 is not true. So, there exists $\stackrel{\u02c6}{y}\in K$ such that $F(\stackrel{\u02c6}{y})=\mathrm{\varnothing}$, which implies that $f(x,\stackrel{\u02c6}{y})\le 0$ for every $x\in E$. This completes the proof. □
Remark 4.1 If $f=g$, then (v)_{1} and (v)_{2} of Theorem 4.1 can be replaced by the following equivalent conditions, respectively:
${\text{(v)}}_{1}^{\prime}$ for every $N\in \u3008E\u3009$, there exists a nonempty compact convex subset ${E}_{N}$ of E containing N such that ${E}_{N}\setminus K\subseteq {\bigcup}_{x\in {E}_{N}}\{y\in E:f(x,y)>0\}$;
${\text{(v)}}_{2}^{\prime}$ there exists a point ${x}_{0}\in E$ such that $E\setminus \{y\in E:f({x}_{0},y)>0\}\subseteq K$.
Remark 4.2 (ii) of Theorem 4.1 can be replaced by the following condition:
which is a contraction. Hence, (ii) of Theorem 4.1 holds.
If (iv)_{2} of Theorem 3.1 holds, then, by (iv)_{2} of Theorem 3.1 and by the definition of g, there exists a point ${x}_{0}\in E$ such that ${cl}_{E}(E\setminus \{y\in E:g({x}_{0},y)>0\})\subseteq K$. Thus, all the hypotheses of Theorem 4.1 are satisfied. By Theorem 4.1, there exists $\stackrel{\u02c6}{y}\in K$ such that $f(x,\stackrel{\u02c6}{y})\le 0$ for every $x\in E$. Therefore, $x\notin F(\stackrel{\u02c6}{y})$ for every $x\in E$, which implies that $F(\stackrel{\u02c6}{y})=\mathrm{\varnothing}$. This contradicts (iii) of Theorem 3.1. Hence, the conclusion of Theorem 3.1 must hold.
Remark 4.4 Theorem 4.1 generalizes Theorem 5.3 of Yang and Pu [34] in the following aspects: (1) The underlying spaces of Theorem 4.1 and Theorem 5.3 of Yang and Pu [34] are $CAT(0)$ spaces and Hadamard manifolds, respectively. We can see that $CAT(0)$ spaces include Hadamard manifolds as special cases (see [23]); (2) the E in Theorem 4.1 does not need to be compact; (3) in Theorem 4.1, there are two functions, but there is only one function in Theorem 5.3 of Yang and Pu [34].
Remark 4.5 By Remarks 3.5 and 4.3, we know that Theorem 3.1, Theorem 3.3 and Theorem 4.1 are equivalent.
 (i)
for every $(x,y)\in C\times C$, $f(x,y)\le g(x,y)$;
 (ii)
for every $y\in C$, the set $\{x\in C:g(x,y)>{sup}_{y\in C}g(y,y)\}$ is convex;
 (iii)
for every $x\in C$, $y\mapsto f(x,y)$ is lower semicontinuous on C;
 (iv)one of the following conditions holds:

(iv)_{1} for every $N\in \u3008C\u3009$, there exists a nonempty compact convex subset ${C}_{N}$ of C containing N such that${C}_{N}\setminus K\subseteq \bigcup _{x\in {C}_{N}}{int}_{{C}_{N}}(\{y\in C:g(x,y)>\underset{y\in C}{sup}g(y,y)\}\cap {C}_{N});$

(iv)_{2} there exists a point ${x}_{0}\in C$ such that${cl}_{C}(C\setminus \{y\in C:g({x}_{0},y)>\underset{y\in C}{sup}g(y,y)\})\subseteq K.$

Then there exists $\stackrel{\u02c6}{y}\in K$ such that $f(x,\stackrel{\u02c6}{y})\le {sup}_{y\in C}g(y,y)$ for every $x\in C$.
We can easily check that ${f}^{\prime}$, ${g}^{\prime}$ satisfy all the hypotheses of Theorem 4.1. Therefore, by Theorem 4.1, we infer that there exists $\stackrel{\u02c6}{y}\in K$ such that ${f}^{\prime}(x,\stackrel{\u02c6}{y})\le 0$ for every $x\in C$; that is, $f(x,\stackrel{\u02c6}{y})\le {sup}_{y\in C}g(y,y)$ for every $x\in C$. This completes the proof. □
Remark 4.6 Corollary 4.1 generalizes Theorem 3.3 of Shabanian and Vaezpour [18] in the following aspects: (1) the set C in Corollary 4.1 does not need to be compact; (2) (ii) of Corollary 4.1 is weaker than the corresponding (2) of Theorem 3.3 of Shabanian and Vaezpour [18]; (3) in Corollary 4.1, there are two functions, but there is only one function in Theorem 3.3 of Shabanian and Vaezpour [18].
By Theorem 4.1, we get the following saddle point theorem in $CAT(0)$ spaces.
 (i)
for every $y\in E$, $f(y,y)=0$;
 (ii)
for every $y\in E$, the set $\{x\in E:f(x,y)>0\}$ is convex;
 (iii)
for every $x\in E$, the set $\{y\in E:f(x,y)<0\}$ is convex;
 (iv)one of the following conditions holds:

(iv)_{1} for every $N\in \u3008E\u3009$, there exist two nonempty compact convex subsets ${E}_{N}$, ${\tilde{E}}_{N}$ of E containing N such that${E}_{N}\setminus {K}_{1}\subseteq \bigcup _{x\in {E}_{N}}\{y\in E:f(x,y)>0\}$and${\tilde{E}}_{N}\setminus {K}_{2}\subseteq \bigcup _{y\in {\tilde{E}}_{N}}\{x\in E:f(x,y)<0\};$

(iv)_{2} there exist two points ${x}_{0},{y}_{0}\in E$ such that$E\setminus {K}_{1}\subseteq \{y\in E:f({x}_{0},y)>0\}\phantom{\rule{1em}{0ex}}\mathit{\text{and}}\phantom{\rule{1em}{0ex}}E\setminus {K}_{2}\subseteq \{x\in E:f(x,{y}_{0})<0\}.$

Then f has a saddle point $(\stackrel{\u02c6}{x},\stackrel{\u02c6}{y})\in {K}_{1}\times {K}_{2}$; that is, $f(x,\stackrel{\u02c6}{y})\le f(\stackrel{\u02c6}{x},\stackrel{\u02c6}{y})\le f(\stackrel{\u02c6}{x},y)$ for every $(x,y)\in E\times E$. In particular, ${inf}_{x\in E}{sup}_{y\in E}f(x,y)={sup}_{y\in E}{inf}_{x\in E}f(x,y)$.
This completes the proof. □
By Theorem 4.1, we have the following best approximation theorem in $CAT(0)$ spaces.
Theorem 4.3 Let $(E,d)$ be a complete $CAT(0)$ space, $C\subseteq E$ be a closed locally compact convex set, $H:C\to E$ be a continuous mapping. Suppose that there exists a nonempty compact subset K of C such that one of the following conditions holds:
Then there exists $\stackrel{\u02c6}{y}\in K$ such that $d(\stackrel{\u02c6}{y},H(\stackrel{\u02c6}{y}))={inf}_{x\in C}d(x,H(\stackrel{\u02c6}{y}))$.
 (a)
for every $N\in \u3008C\u3009$, there exists a nonempty compact convex subset ${C}_{N}$ of C containing N such that ${C}_{N}\setminus K\subseteq {\bigcup}_{x\in {C}_{N}}\{y\in C:f(x,y)>0\}$;
 (b)
there exists a point ${x}_{0}\in C$ such that $C\setminus \{y\in C:f({x}_{0},y)>0\}\subseteq K$.
which implies that $\gamma ({t}_{0})\in U(H(y),d(y,H(y)))$; that is, $d(y,H(y))>d(\gamma ({t}_{0}),H(y))$. This contradicts $d(y,H(y))\le d(\gamma ({t}_{0}),H(y))$. Therefore, for every $y\in C$, the set $\{x\in C:f(x,y)>0\}$ is convex. Thus, by Remark 4.1, all the requirements of Theorem 4.1 with $f=g$ are fulfilled. Hence, by Theorem 4.1 with $f=g$, there exists $\stackrel{\u02c6}{y}\in K$ such that $f(x,\stackrel{\u02c6}{y})\le 0$ for every $x\in C$; that is, $d(\stackrel{\u02c6}{y},H(\stackrel{\u02c6}{y}))\le d(x,H(\stackrel{\u02c6}{y}))$ for every $x\in C$, which implies that $d(\stackrel{\u02c6}{y},H(\stackrel{\u02c6}{y}))={inf}_{x\in C}d(x,H(\stackrel{\u02c6}{y}))$. This completes the proof. □
Remark 4.7 Theorem 4.3 generalizes Theorem 3.1 of Shabanian and Vaezpour [18] in the following aspects: (1) the C in Theorem 4.3 does not need to be compact; (2) the E in Theorem 4.3 does not need to have the convex hull finite property. We point out that the proof of Theorem 4.3 is different from that of Theorem 3.1 of Shabanian and Vaezpour [18].
As an application of Theorem 4.3, we have the following fixed point theorem for singlevalued mappings.
 (i)for every $c\in K$ with $c\ne H(c)$, there exists $t\in (0,1)$ such that$C\cap U(H(c),(1t)d(c,H(c)))\ne \mathrm{\varnothing},$
 (ii)one of the following conditions holds:

(ii)_{1} for every $N\in \u3008C\u3009$, there exists a nonempty compact convex subset ${C}_{N}$ of C containing N such that${C}_{N}\setminus K\subseteq \bigcup _{x\in {C}_{N}}\{y\in C:d(y,H(y))>d(x,H(y))\};$

(ii)_{2} there exists a point ${x}_{0}\in C$ such that$C\setminus \{y\in C:d(y,H(y))>d({x}_{0},H(y))\}\subseteq K.$

Then there exists $\stackrel{\u02c6}{y}\in K$ such that $\stackrel{\u02c6}{y}=H(\stackrel{\u02c6}{y})$.
which contradicts the fact that $d(\stackrel{\u02c6}{y},H(\stackrel{\u02c6}{y}))={inf}_{x\in C}d(x,H(\stackrel{\u02c6}{y}))$. Therefore, $\stackrel{\u02c6}{y}$ is a fixed point of H. This completes the proof. □
Remark 4.8 Theorem 4.4 generalizes Theorem 3.2 of Shabanian and Vaezpour [18] in the following aspects: (1) the E in Theorem 4.4 does not need to have the convex hull finite property; (2) the C in Theorem 4.4 does not need to be compact.
By Theorem 4.1, we obtain the following generalized best approximation theorem.
 (i)
for every $y\in C$, the set $\{x\in C:d(G(y),H(y))>d(G(x),H(y))\}$ is convex;
 (ii)one of the following conditions holds:

(ii)_{1} for every $N\in \u3008C\u3009$, there exists a nonempty compact convex subset ${C}_{N}$ of C containing N such that${C}_{N}\setminus K\subseteq \bigcup _{x\in {C}_{N}}\{y\in C:d(G(y),H(y))>d(G(x),H(y))\};$

(ii)_{2} there exists a point ${x}_{0}\in C$ such that$C\setminus \{y\in C:d(G(y),H(y))>d(G({x}_{0}),H(y))\}\subseteq K.$

Then there exists $\stackrel{\u02c6}{y}\in K$ such that $d(G(\stackrel{\u02c6}{y}),H(\stackrel{\u02c6}{y}))={inf}_{x\in C}d(G(x),H(\stackrel{\u02c6}{y}))$.
 (a)
for every $N\in \u3008C\u3009$, there exists a nonempty compact convex subset ${C}_{N}$ of C containing N such that ${C}_{N}\setminus K\subseteq {\bigcup}_{x\in {C}_{N}}\{y\in C:f(x,y)>0\}$;
 (b)
there exists a point ${x}_{0}\in C$ such that $C\setminus \{y\in C:f({x}_{0},y)>0\}\subseteq K$.
By Remark 4.1, all the requirements of Theorem 4.1 with $f=g$ are satisfied. Hence, by Theorem 4.1 with $f=g$, there exists $\stackrel{\u02c6}{y}\in K$ such that $f(x,\stackrel{\u02c6}{y})\le 0$ for every $x\in C$; that is, $d(G(\stackrel{\u02c6}{y}),H(\stackrel{\u02c6}{y}))\le d(G(x),H(\stackrel{\u02c6}{y}))$ for every $x\in C$, which implies that $d(G(\stackrel{\u02c6}{y}),H(\stackrel{\u02c6}{y}))={inf}_{x\in C}d(G(x),H(\stackrel{\u02c6}{y}))$. This completes the proof. □
Remark 4.9 Theorem 4.5 generalizes Theorem 3.4 of Shabanian and Vaezpour [18] in the following aspects: (1) the C in Theorem 4.5 does not need to be compact; (2) the E in Theorem 4.5 does not need to have the convex hull finite property; (3) the setvalued mappings G, H in Theorem 4.5 do not need to have convex values; (4) the condition that the setvalued mapping G in Theorem 3.4 of Shabanian and Vaezpour [18] is quasiconvex is removed. We point out that the proof of Theorem 4.5 is different from that of Theorem 3.4 of Shabanian and Vaezpour [18].
Remark 4.10 Theorem 4.5 can be regarded as a generalization of Theorem 4.3. In fact, let $G(y)=\{y\}$ for every $y\in C$ and H be a singlevalued continuous mapping. Then by using the same method as in the proof Theorem 4.3, we can show that (i) of Theorem 4.5 holds and thus, Theorem 4.5 reduces to Theorem 4.3.
5 Existence of φequilibrium for multiobjective games
In this section, we will consider the multiobjective noncooperative game in its strategic form $\Gamma ={({X}_{i},{V}^{i})}_{i\in I}$, where $I=\{1,2,\dots ,n\}$ is the set of players; every ${X}_{i}$ is the strategy set of the i th player and every ${V}^{i}:X={\prod}_{i\in I}{X}_{i}\to {\mathbb{R}}^{{k}_{i}}$ is the payoff function of the i th player with ${k}_{i}$ being a positive integer. If an action combination $x=({x}_{1},{x}_{2},\dots ,{x}_{n})$ is played, every player i is trying to confirm his/her vector payoff function ${V}^{i}(x):=({f}_{1}^{i}(x),{f}_{2}^{i}(x),\dots ,{f}_{{k}_{i}}^{i}(x))$ and then minimize his/her vector payoff function according to his/her preference.
Before we introduce the equilibrium concepts of multiobjective noncooperative games, we give the following notation.
For every $m\in \mathbb{N}$, let ${\mathbb{R}}_{+}^{m}:=\{q:=({q}_{1},{q}_{2},\dots ,{q}_{m})\in {\mathbb{R}}^{m}:{q}_{j}\ge 0,\mathrm{\forall}j=1,2,\dots ,m\}$ and ${int}_{{\mathbb{R}}^{m}}{\mathbb{R}}_{+}^{m}:=\{q:=({q}_{1},{q}_{2},\dots ,{q}_{m})\in {\mathbb{R}}^{m}:{q}_{j}>0,\mathrm{\forall}j=1,2,\dots ,m\}$ denote the nonnegative orthant of ${\mathbb{R}}^{m}$ and the nonempty interior of ${\mathbb{R}}_{+}^{m}$ with the Euclidian metric topology, respectively. For every $q,r\in {\mathbb{R}}^{m}$, let $q\cdot r$ denote the standard Euclidean inner product.
Now we introduce the following definitions.
Definition 5.2 A strategy $\stackrel{\u02c6}{x}\in X$ is said to be a Pareto φequilibrium (respectively, a weak Pareto φequilibrium) of a game $\Gamma ={({X}_{i},{V}^{i})}_{i\in I}$ if, for every $i\in I$, ${\stackrel{\u02c6}{x}}_{i}\in {X}_{i}$ is a Pareto efficient φstrategy (respectively, a weak Pareto efficient φstrategy) with respect to $\stackrel{\u02c6}{x}$.
Remark 5.1 Definitions 5.15.2 generalize the corresponding definitions of Wang [37], Yuan and Tarafdar [38], and Yu and Yuan [39]. In fact, if ${\phi}_{i}({x}_{i})={x}_{i}$ for every $x={({x}_{i})}_{i\in I}\in X$ and every $i\in I$, then Definitions 5.15.2 coincide with the corresponding definitions of Wang [37], Yuan and Tarafdar [38], and Yu and Yuan [39]. By the above definition, we can see that every Pareto φequilibrium is a weak Pareto φequilibrium, but the converse is not true in general.
 (i)
${Q}_{i}=({Q}_{i,1},{Q}_{i,2},\dots ,{Q}_{i,{k}_{i}})\in {\mathbb{R}}_{+}^{{k}_{i}}\setminus \{0\}$;
 (ii)
${Q}_{i}\cdot {V}^{i}(\stackrel{\u02c6}{x})\le {Q}_{i}\cdot {V}^{i}({\stackrel{\u02c6}{x}}_{\stackrel{\u02c6}{i}},\phi ({x}_{i}))$ for every ${x}_{i}\in {X}_{i}$.
Remark 5.2 If ${\phi}_{i}({x}_{i})={x}_{i}$ for every $x={({x}_{i})}_{i\in I}\in X$ and every $i\in I$, then Definition 5.3 reduces to Definition 2.3 of Wang [37] and Definition 3 of Yuan and Tarafdar [38] and Yu and Yuan [39]. In particular, if ${Q}_{i}\in {\mathbb{R}}_{+}^{{k}_{i}}$ with ${\sum}_{j=1}^{{k}_{i}}{Q}_{i,j}=1$ for every $i\in I$, then the strategy $\stackrel{\u02c6}{x}\in X$ is said to be a normalized weighted Nash φequilibrium with respect to Q.
As an application of Theorem 4.1, we have the following existence theorem of weighted Nash φequilibrium for multiobjective noncooperative games in the setting of noncompact $CAT(0)$ spaces.
 (i)
for every $y\in X$, the set $\{x\in X:{\sum}_{i=1}^{n}{Q}_{i}\cdot [{V}^{i}({y}_{\stackrel{\u02c6}{i}},{y}_{i}){V}^{i}({y}_{\stackrel{\u02c6}{i}},{\phi}_{i}({x}_{i}))]>0\}$ is convex;
 (ii)
for every $x\in X$, the function $y\mapsto {\sum}_{i=1}^{n}{Q}_{i}\cdot [{V}^{i}({y}_{\stackrel{\u02c6}{i}},{y}_{i}){V}^{i}({y}_{\stackrel{\u02c6}{i}},{\phi}_{i}({x}_{i}))]$ is lower semicontinuous on X;
 (iii)
for every $y\in X$, ${\sum}_{i=1}^{n}{Q}_{i}\cdot [{V}^{i}({y}_{\stackrel{\u02c6}{i}},{y}_{i}){V}^{i}({y}_{\stackrel{\u02c6}{i}},{\phi}_{i}({y}_{i}))]\le 0$;
 (iv)one of the following conditions holds:

(iv)_{1} for every $N\in \u3008X\u3009$, there exists a nonempty compact convex subset ${X}_{N}$ of X containing N such that${X}_{N}\setminus K\subseteq \bigcup _{x\in {X}_{N}}\{y\in X:\sum _{i=1}^{n}{Q}_{i}\cdot [{V}^{i}({y}_{\stackrel{\u02c6}{i}},{y}_{i}){V}^{i}({y}_{\stackrel{\u02c6}{i}},{\phi}_{i}({x}_{i}))]>0\};$

(iv)_{2} there exists a point ${x}_{0}={({x}_{0i})}_{i\in I}\in X$ such that$X\setminus \{y\in X:\sum _{i=1}^{n}{Q}_{i}\cdot [{V}^{i}({y}_{\stackrel{\u02c6}{i}},{y}_{i}){V}^{i}({y}_{\stackrel{\u02c6}{i}},{\phi}_{i}({x}_{0i}))]>0\}\subseteq K.$

Then Γ has at least one weight Nash φequilibrium in K with respect to the weight vector Q.
Therefore, ${Q}_{i}\cdot {R}^{i}({\stackrel{\u02c6}{y}}_{\stackrel{\u02c6}{i}},{\stackrel{\u02c6}{y}}_{i})\le {Q}_{i}\cdot {V}^{i}({\stackrel{\u02c6}{y}}_{\stackrel{\u02c6}{i}},{\phi}_{i}({x}_{i}))$ for every $i\in I$ and every ${x}_{i}\in {X}_{i}$; that is, $\stackrel{\u02c6}{y}\in K$ is a weighted Nash φequilibrium of the game Γ with respect to Q. This completes the proof. □
Remark 5.3 Theorem 5.1 is a new result, which is different from Theorem 3.1 of Wang [37], Theorem 1 of Yuan and Tarafdar [38], Theorem 3 of Yu and Yuan [39], and Theorem 1 of Borm et al. [41]. The main difference is that the underlying strategy spaces in Theorem 5.1 are $CAT(0)$ spaces which do not possess any linear structure. In addition, on the basis of an existence theorem for weighted Nash equilibrium for multiobjective noncooperative games in the setting of compact finite dimensional spaces, Lu [42] analyzed the phenomena for the water resources utilizing conflicts among the water users in the lower reaches of Tarim River Basin and revealed the underlying causes of water shortage and water quality deterioration of the lower reaches of Tarim River Basin. We point out that the underlying strategy spaces of multiobjective noncooperative game models in [42] are compact finite dimensional spaces and the payoff functions of players are continuous, which restrict the applicable area of models. In fact, in real world, the situation that the underlying strategy spaces of players are noncompact and nonlinear spaces and the payoff functions of players are discontinuous is very common. So, the multiobjective noncooperative game models in [42] cannot be used to analyze many conflict problems under the situation mentioned above. In contrast with the multiobjective noncooperative game models in [42], the multiobjective noncooperative game model in Theorem 5.1 has two advantages; that is, the strategy spaces of players do not possess any linear and compact structure and the payoff functions of players need not to be continuous. Therefore, by using Theorem 5.1, we can deal with a lot of conflict problems existing in resource utilizing and management under much more mild conditions.
Remark 5.4 (ii) of Theorem 5.1 can be replaced by the following conditions:
(ii)′ for every $x\in X$, the function $y\to {\sum}_{i=1}^{n}{Q}_{i}\cdot {V}^{i}({y}_{\stackrel{\u02c6}{i}},{\phi}_{i}({x}_{i}))$ is upper semicontinuous on X;
(ii)″ the function $(x,y)\to {\sum}_{i=1}^{n}{Q}_{i}\cdot {V}^{i}({x}_{\stackrel{\u02c6}{i}},{y}_{i})$ is jointly lower semicontinuous on $X\times X$.
If ${k}_{i}=1$ for every $i\in I$, then, by Theorem 5.1, we have the following existence result of Nash φequilibrium for noncooperative games.
 (i)
for every $y\in X$, the set $\{x\in X:{\sum}_{i=1}^{n}[{f}^{i}({y}_{\stackrel{\u02c6}{i}},{y}_{i}){f}^{i}({y}_{\stackrel{\u02c6}{i}},{\phi}_{i}({x}_{i}))]>0\}$ is convex;
 (ii)
for every $x\in X$, the function $y\mapsto {\sum}_{i=1}^{n}[{f}^{i}({y}_{\stackrel{\u02c6}{i}},{y}_{i}){f}^{i}({y}_{\stackrel{\u02c6}{i}},{\phi}_{i}({x}_{i}))]$ is lower semicontinuous on X;
 (iii)
for every $y\in X$, ${\sum}_{i=1}^{n}[{f}^{i}({y}_{\stackrel{\u02c6}{i}},{y}_{i}){f}^{i}({y}_{\stackrel{\u02c6}{i}},{\phi}_{i}({y}_{i}))]\le 0$;
 (iv)one of the following conditions holds:

(iv)_{1} for every $N\in \u3008X\u3009$, there exists a nonempty compact convex subset ${X}_{N}$ of X containing N such that${X}_{N}\setminus K\subseteq \bigcup _{x\in {X}_{N}}\{y\in X:\sum _{i=1}^{n}[{f}^{i}({y}_{\stackrel{\u02c6}{i}},{y}_{i}){f}^{i}({y}_{\stackrel{\u02c6}{i}},{\phi}_{i}({x}_{i}))]>0\};$

(iv)_{2} there exists a point ${x}_{0}={({x}_{0i})}_{i\in I}\in X$ such that$X\setminus \{y\in X:\sum _{i=1}^{n}[{f}^{i}({y}_{\stackrel{\u02c6}{i}},{y}_{i}){f}^{i}({y}_{\stackrel{\u02c6}{i}},{\phi}_{i}({x}_{i}))]>0\}\subseteq K.$

Then Γ has a Nash φequilibrium in K.
Remark 5.5 It is interesting to compare Corollary 5.1 with Theorem 4 of Niculescu and Rovenţa [29] in the following aspects: (1) every ${X}_{i}$ in Corollary 5.1 is a nonempty subset of a $CAT(0)$ space $({E}_{i},{d}_{i})$ and it does not need to be compact, where all $({E}_{i},{d}_{i})$ are possibly different; (2) every function ${f}^{i}$ in Corollary 5.1 does not need to be lower semicontinuous and quasiconvex; (3) the mapping φ in Corollary 5.1 does not need to be continuous and affine.
By Theorem 5.1, we can derive an existence theorem of Pareto φequilibrium for multiobjective noncooperative games. In order to do so, we need the following lemma. The proof of this lemma is similar to that of Lemma 2.1 of Wang [37]. For the sake of completeness, we give the proof.
Lemma 5.1 Every normalized weighted Nash φequilibrium $\stackrel{\u02c6}{x}\in X$ with a weight $Q=({Q}_{1},{Q}_{2},\dots ,{Q}_{n})$, ${Q}_{i}\in {\mathbb{R}}_{+}^{{k}_{i}}\setminus \{0\}$ (respectively, ${Q}_{i}\in {int}_{{\mathbb{R}}^{{k}_{i}}}{\mathbb{R}}_{+}^{{k}_{i}}$) and ${\sum}_{j=1}^{{k}_{i}}{Q}_{i,j}=1$ for every $i\in I$, is a weak Pareto φequilibrium (respectively, a Pareto φequilibrium) of the game $\Gamma ={({X}_{i},{V}^{i})}_{i\in I}$.
Since ${Q}_{{i}_{0}}\in {int}_{{\mathbb{R}}^{{k}_{i}}}{\mathbb{R}}_{+}^{{k}_{i}}$, it follows that ${Q}_{{i}_{0}}\cdot [{V}^{i}({\stackrel{\u02c6}{x}}_{{\stackrel{\u02c6}{i}}_{0}},{\stackrel{\u02c6}{x}}_{{i}_{0}}){V}^{i}({\stackrel{\u02c6}{x}}_{{\stackrel{\u02c6}{i}}_{0}},\phi ({x}_{{i}_{0}}))]>0$, which contradicts the assumption that $\stackrel{\u02c6}{x}$ is a normalized weighted Nash φequilibrium with the weight $Q=({Q}_{1},{Q}_{2},\dots ,{Q}_{n})$. Hence, $\stackrel{\u02c6}{x}$ is a Pareto φequilibrium. This completes the proof. □
Remark 5.6 The conclusion of Lemma 5.1 is still true if $\stackrel{\u02c6}{x}\in X$ is a weighted Nash φequilibrium with a weight $Q=({Q}_{1},{Q}_{2},\dots ,{Q}_{n})$ satisfying ${Q}_{i}\in {\mathbb{R}}_{+}^{{k}_{i}}\setminus \{0\}$ (respectively, ${Q}_{i}\in {int}_{{\mathbb{R}}^{{k}_{i}}}{\mathbb{R}}_{+}^{{k}_{i}}$) for every $i\in I$. We point out that a Pareto φequilibrium is not necessarily a weighted Nash φequilibrium.
 (i)
for every $y\in X$, the set $\{x\in X:{\sum}_{i=1}^{n}{Q}_{i}\cdot [{V}^{i}({y}_{\stackrel{\u02c6}{i}},{y}_{i}){V}^{i}({y}_{\stackrel{\u02c6}{i}},{\phi}_{i}({x}_{i}))]>0\}$ is convex;
 (ii)
for every $x\in X$, the function $y\mapsto {\sum}_{i=1}^{n}{Q}_{i}\cdot [{V}^{i}({y}_{\stackrel{\u02c6}{i}},{y}_{i}){V}^{i}({y}_{\stackrel{\u02c6}{i}},{\phi}_{i}({x}_{i}))]$ is lower semicontinuous on X;
 (iii)
for every $y\in X$, ${\sum}_{i=1}^{n}{Q}_{i}\cdot [{V}^{i}({y}_{\stackrel{\u02c6}{i}},{y}_{i}){V}^{i}({y}_{\stackrel{\u02c6}{i}},{\phi}_{i}({y}_{i}))]\le 0$;
 (iv)one of the following conditions holds:

(iv)_{1} for every $N\in \u3008X\u3009$, there exists a nonempty compact convex subset ${X}_{N}$ of X containing N such that${X}_{N}\setminus K\subseteq \bigcup _{x\in {X}_{N}}\{y\in X:\sum _{i=1}^{n}{Q}_{i}\cdot [{V}^{i}({y}_{\stackrel{\u02c6}{i}},{y}_{i}){V}^{i}({y}_{\stackrel{\u02c6}{i}},{\phi}_{i}({x}_{i}))]>0\};$

(iv)_{2} there exists a point ${x}_{0}={({x}_{0i})}_{i\in I}\in X$ such that$X\setminus \{y\in X:\sum _{i=1}^{n}{Q}_{i}\cdot [{V}^{i}({y}_{\stackrel{\u02c6}{i}},{y}_{i}){V}^{i}({y}_{\stackrel{\u02c6}{i}},{\phi}_{i}({x}_{0i}))]>0\}\subseteq K.$

Then Γ has at least one weak Pareto φequilibrium in K. In addition, if $Q=({Q}_{1},{Q}_{2},\dots ,{Q}_{n})$ with each ${Q}_{i}\in {int}_{{\mathbb{R}}^{{k}_{i}}}{\mathbb{R}}_{+}^{{k}_{i}}$, then Γ has at least one Pareto φequilibrium in K.
Proof It follows from Theorem 5.1 that Γ has at least a weighted Nash φequilibrium point $\stackrel{\u02c6}{y}\in K$ with respect to the weighted vector Q. By Lemma 5.1 and by Remark 5.6, we know that $\stackrel{\u02c6}{y}$ is also a weak Pareto φequilibrium point of Γ, and $\stackrel{\u02c6}{y}$ is a Pareto φequilibrium point if ${Q}_{i}\in {int}_{{\mathbb{R}}^{{k}_{i}}}{\mathbb{R}}_{+}^{{k}_{i}}$ for every $i\in I$. This completes the proof. □
Declarations
Acknowledgements
This work was supported by the Planning Foundation for Humanities and Social Sciences of Ministry of Education of China ‘Research on utilizing conflict of water resources and initial water right allocation in a river basin based on game theory’ (No. 12YJAZH084), the Young & MiddleAged Academic Leaders Program of the ‘Qinglan Project’ of Jiangsu Province, and by Jiangsu Overseas Research & Training Program for University Prominent Young & MiddleAged Teachers and Presidents. The corresponding author (the second author) was also supported by the “Six Talent Peaks Project” of Jiangsu Province (No. DZXX028). The authors would like to thank the referees for their many valuable suggestions and comments which improved the exposition of this paper.
Authors’ Affiliations
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