A fuzzy fixed-point theorem and its applications to maximal elements and Nash equilibria in noncompact CAT(0) spaces

Abstract

In this paper, based on the KKM method, we prove a new fuzzy fixed-point theorem in noncompact CAT(0) spaces. As applications of this fixed-point theorem, we obtain some existence theorems of fuzzy maximal element points. Finally, we utilize these fuzzy maximal element theorems to establish some new existence theorems of Nash equilibrium points for generalized fuzzy noncooperative games and fuzzy noncooperative qualitative games in noncompact CAT(0) spaces. The results obtained in this paper generalize and extend many known results in the existing literature.

Keywords

CAT(0) space fuzzy fixed point fuzzy maximal element fuzzy noncooperative game Nash equilibrium point

1 Introduction

As a special kind of metric spaces, CAT(0) spaces have played an important role in various fields of mathematics. In 2003, Kirk [1] first paid attention to studying the existence of fixed points in CAT(0) spaces. Since then, fixed-point theorems with applications in CAT(0) spaces have attracted the attention of many researchers. In 2005, in the setting of CAT(0) spaces, Dhompongsa et al. [2] showed that a nonexpansive mapping satisfying suitable conditions possesses at least one fixed point. In 2009, Shahzad [3] gave some existence results of common fixed points for families of nonexpansive mappings with single values and set-valued nonexpansive mappings with certain continuous properties in CAT(0) spaces. In 2011, Abkar and Mohammad [4] promoted the work of the aforementioned authors and under the condition that a generalized set-valued nonexpansive mapping and a single-valued quasi-nonexpansive mapping commute weakly, established common fixed point theorems for them. In the same year, based on a Ky Fan minimax inequality, Shabanian and Vaezpour [5] obtained some fixed-point theorems and best approximation theorems in CAT(0) spaces. In 2014, Lu et al. [6] further generalized the results due to Shabanian and Vaezpour [5]. They proved some Browder type fixed-point theorems in noncompact CAT(0) spaces and gave applications to the existence of solutions to minimax inequalities, the existence of equilibrium points for zero-sum games, best approximation and fixed-point theorems, and the existence of φ-equilibrium points for multiobjective noncooperative games. In 2015, Khatibzadeh and Ranjbar [7] established several existence results of solutions to variational inequalities in CAT(0) spaces and reached a conclusion that the convergence point of Δ-convergent sequence constructed by the inexactproximal point algorithm essentially coincides with a fixed point of the mapping with nonexpansive characteristics. In 2018, Uddin et al. [8] investigated the convergence mechanism of the Mann iteration for a monotone nonexpansive mapping in the setting of ordered CAT(0) spaces and obtained some Δ-convergence and strong convergence theorems with an application to solving integral equations. Based on a different perspective from Uddin et al. [8], in 2018, Pakkaranang et al. [9] gave the algorithm of common fixed points for the mapping with asymptotically quasi-nonexpansive characteristics in the setting of CAT(0) spaces and also discussed the convergence problem of the presented algorithm. Recently, Aremu et al. [10] used the gate conditions added to two k-demicontractive set-valued mappings and obtained a common approximate solution to monotone inclusion and fixed-point problems in complete CAT(0) spaces. Very recently, Uddin et al. [11] established some theorems about the convergence characteristic of a new iteration procedure for total asymptotically nonexpansive mappings in the setting of complete CAT(0) spaces. At the same time, they gave a numerical experiment to show the superiority of this new iteration procedure compared with other existing iteration processes in terms of convergence rate.

Moreover, the fixed-point theory in CAT(0) spaces also has a wide range of applications in many applied disciplines such as medicine, biology, graph theory, computer and image recovery science, and game theory. For details, interested readers can refer to Bartolini et al. [12], Semple and Steel [13], Bestvina [14], Espínola and Kirk [15], Kirk [16], Uddin et al. [17], Lu et al. [6], and references therein.

From the above, we can see that the fixed-point theorems and their applications in the framework of CAT(0) spaces are all almost obtained in deterministic environments. In order to better match and adapt to the fuzzy and uncertain features of the real world, it is necessary to study the fixed-point theorems in CAT(0) spaces and their applications in fuzzy environments. Inspired and motivated by the work of the above-mentioned authors, in this paper, we prove a new fuzzy fixed-point theorem in the framework of noncompact CAT(0) spaces and next, as applications of this fuzzy fixed-point theorem, we obtain some new existence theorems of fuzzy maximal elements and existence theorems of Nash equilibrium points for generalized fuzzy noncooperative games and fuzzy noncooperative qualitative games in noncompact CAT(0) spaces. The results presented in this paper improve and extend the corresponding results of Lu et al. [6], Tarafdar [31], Tan and Yuan [33], Yuan [35], Lu et al. [36], Yang and Pu [37], Borglin and Keiding [42], Toussaint [43], and the references therein.

2 Preliminaries

For a nonempty set X, let 2^X and 〈X〉 denote the family of all subsets of X and the family of nonempty finite subsets of X, respectively. We use int_XZ and cl_XZ to represent the interior of Z in X and the closure of Z in X, respectively, where Z is a nonempty subset of a topological space X. For two nonempty sets X and Y, the graph and inverse mapping H^-1 : Y → 2^X of a set-valued mapping H : X → 2^Y are defined by Graph (H) = {(x, y) ∈ X × Y : y ∈ H (x)} and by H^-1 (y) = {x ∈ X : y ∈ H (x)} for every y ∈ Y, repectively. Let H, G : X → 2^Y be two set-valued mappings, where X and Y are two topological spaces. Subsequently, we define the set-valued mapping H ⋂ G : X → 2^Y by (H ⋂ G) (x) = H (x) ⋂ G (x) for every x ∈ X and the set-valued mapping $\bar{H} : X \to 2^{Y}$ by $\bar{H} (x) = {y \in Y : (x, y) \in {cl}_{X \times Y} (G_{r} (H))}$ for every x ∈ X, where we call cl_X×Y (Graph (H)) the adherence of Graph (H).

Let X be a nonempty set. A fuzzy set on X is a function from X into [0, 1]. The family of all fuzzy sets on X is denoted by $F (X)$ . Let Y be another nonempty set. Then a mapping from Y into $F (X)$ is called a fuzzy mapping. If $F : Y \to F (X)$ is a fuzzy mapping, then for all y ∈ Y, F (y) (denoted it by F_y in the sequel) is a fuzzy set in $F (X)$ and the number F_y (x) is called the degree of membership of point x in F_y. If $A \in F (X)$ is a fuzzy set, then the set (A) _α = {x ∈ X : A (x) > α}, α ∈ [0, 1), is called strong α-cut, and (A) ₀ is called the support of A. For more details of fuzzy set theory, see Zadeh [18-19], Kim and Lee [20], Chang et al. [21], Tian et al. [22], Tang et al. [23], Lu et al. [24], and the references therein.

Let (X, d) be a metric space and s, t ∈ X. A geodesic path from s to t is a mapping $ϱ : [0, d (s, t)] \subseteq ℝ \to X$ such that ϱ (0) = s, ϱ (d (s, t)) = t, and for every . It follows from the definition of a geodesic path that the mapping ϱ is an isometry. We call ρ ([0, d (s, t)]) a geodesic segment connecting s and t. From now on, for simplicity and convenience, we use a geodesic instead of a geodesic path. The metric space (X, d) is said to be a geodesic space if every pair of points in X can be connected by a geodesic, and X is said to be uniquely geodesic if there is only one geodesic that connects s and t for every s, t ∈ X. A subset A ⊆ X is called convex if for each s, t ∈ X and each geodesic $ϱ : [0, d (s, t)] \subseteq ℝ \to X$ connecting s and t, we have A ⊇ ϱ (0, d (s, t)]).

For any three pairwise different points s, t, z ∈ X, a geodesic triangle Δ (s, t, z) in a geodesic space (X, d) is formed by s, t, z and a geodesic segment linked by any two of s, t, z, where s, t, z can be seen as the vertices of Δ (s, t, z) and an edge of Δ (s, t, z) is defined by a geodesic segment connected by any two of these three points. A comparable triangle for Δ (s, t, z) ⊆ X is a triangle $\tilde{Δ} (\tilde{s}, \tilde{t}, \tilde{z})$ in the two-dimensional Euclidean space $ℝ^{2}$ such that $d_{ℝ^{2}} (\tilde{k}, \tilde{l}) = d (k, l)$ for every k, l ∈ {s, t, z}.

Definition 2.1 ([25]). A geodesic space (X, d) is said to be a CAT(0) space if for each pair $(Δ, \tilde{Δ}) \subseteq X \times ℝ^{2}$ , the inequality $d_{ℝ^{2}} (\tilde{e}, \tilde{f}) \geq d (e, f)$ is satisfied for every $\tilde{e}, \tilde{f} \in \tilde{Δ}$ and every e, f ∈ Δ.

Remark 2.1. It is worth pointing out that CAT(0) spaces include Euclidean spaces $ℝ^{n}$ , Hilbert spaces, $ℝ$ -trees (see [15]), classical hyperbolic spaces $ℍ^{n}$ (see [26]), Euclidean buildings (see [27]), Hilbert balls (see [28]), and complete simply connected Riemannian manifolds with nonpositive sectional curvature (see [36]) as particular examples.

Let ∅ ≠ B ⊆ X, where (X, d) is a CAT(0) space. Then we define the convex hull of B by $co (B) = ⋃_{k = 0}^{\infty} B_{k}$ , where B₀ = B and for every k ≥ 1, B_k contains all points in X lying on geodesics, whose start and end points are in B_k-1.

Definition 2.2 ([29]). Let H : Y → 2^Y be a set-valued mapping such that co (A) ⊆ ⋃ _x∈AH (x) for every A ∈ 〈Y〉, then H is called a KKM mapping, where Y is a nonempty subset of a CAT(0) space (X, d).

Let C be a nonempty subset of a CAT(0) space (X, d) and g : C → C be a continuous mapping. If g has at least a fixed point, then we claim that C has the fixed-point property.

Definition 2.3 ([5]). A CAT(0) space (X, d) is said to have the finite convex hull property if the closed convex hull of B has the fixed-point property for every B ∈ 〈X〉.

Lemma 2.1 ([5]). Suppose that C is a nonempty subset of a CAT(0) space (X, d) which has the finite convex hull property. Let H : C → 2^C be a KKM mapping such that H (x) is a closed in X for every x ∈ X and there exists a point x^* ∈ C such that H (x^*) is compact. Then we have ⋂_x∈CH (x)≠ ∅.

Lemma 2.2 ([29]). If a CAT(0) space (X, d) is locally compact, then (X, d) possesses the finite convex hull property.

3 A fuzzy fixed-point theorem

In this section, by using Lemma 2.1, we present the following fuzzy fixed-point theorem in the framework of noncompact CAT(0) spaces as follows:

Theorem 3.1. Let (X, d) be a complete CAT(0) space satisfying the finite convex hull property. Suppose that K is a nonempty compact subset of X, β : X → [0, 1) is a function, and $S, T : X \to F (X)$ are two fuzzy mappings such that the following assumptions hold:

(i) ∀ x ∈ X, (S_x) _β(x) ⊆ (T_x) _β(x) and (T_x) _β(x) is convex;

(ii) ⋃_y∈X {x ∈ X : S_x (y) > β (x)} = ⋃ _y∈Xint_X {x ∈ X : S_x (y) > β (x)};

(iii) either (a) ∀ M ∈ 〈X〉, there is a nonempty subset C_M of X such that C_M is compact convex, M ⊆ C_M, and C_M \ K ⊆ ⋃ _{y∈C_M}int_{C
_M} ({x ∈ X : T_x (y) > β (x)} ⋂ C_M), or (b) there exists y₀ ∈ X such that X \ K ⊆ int_X {x ∈ X : T_x (y₀) > β (x)};

(iv) ∀ x ∈ K, (S_x) _β(x)≠ ∅.

Then there exists at least $\hat{x} \in X$ such that $\hat{x} \in (T_{\hat{x}})_{β (\hat{x})}$ .

Proof. When (a) of (iii) holds, we prove this theorem by contradiction. Suppose that the conclusion of this theorem is false, that is, x ∉ (T_x) _β(x) for every x ∈ X. Define two set-valued mappings $\bar{S}, \bar{T} : X \to 2^{X}$ as follows:

$\bar{S} (y) = {cl}_{X} (X \ {x \in X : S_{x} (y) > β (x)}) ⋂ K, \forall y \in X,$ $\bar{T} (y) = {cl}_{X} (X \ {x \in X : T_{x} (y) > β (x)}) ⋂ K, \forall y \in X .$

We claim that the family ${\bar{T} (y) : y \in X}$ possesses the property that the intersection of any finite number of set-valued mappings is nonempty. Let M ∈ 〈X〉 be any given. Then by (iii), we know that that there there is a nonempty subset C_M of X such that C_M is compact convex and M ⊆ C_M. Further, let us define a set-valued mapping T′ : C_M → 2^{C
_M} by T′ (y) = cl_{C
_M} (C_M \ {x ∈ X : T_x (y) > β (x)}) for every y ∈ C_M. Since C_M is compact and T′ (y) is relatively closed in C_M for every y ∈ C_M, it follows that T′ (y) is compact for every y ∈ C_M. Next, we shall show that the set-valued mapping $\tilde{T} : C_{M} \to 2^{C_{M}}$ defined by $\tilde{T} (y) = C_{M} \ {x \in X : T_{x} (y) > β (x)}, \forall y \in C_{M},$ is a KKM mapping. Assuming the conclusion is opposite, we know that there exist F ∈ 〈C_M〉 and x ∈ co (F) ⊆ C_M such that $x \notin ⋃_{y \in F} \tilde{T} (y) = C_{M} \ ⋂_{y \in F} {x \in X : T_{x} (y) > β (x)}$ . This implies that x ∈ ⋂ _y∈F {x ∈ X : T_x (y) > β (x)} and F ⊆ (T_x) _β(x). Thus, it follows from (i) that x ∈ co (F) ⊆ (T_x) _β(x), which is a contradiction. Hence, $\tilde{T}$ is a KKM mapping. By the fact that $\tilde{T} (y) \subseteq T^{'} (y)$ for every y ∈ C_M, we can see that T′ is also a KKM mapping. By (iii) and Lemma 2.1, we get $\begin{matrix} \emptyset \neq ⋂_{y \in C_{M}} T^{'} (y) = ⋂_{y \in C_{M}} {cl}_{C_{M}} (C_{M} \ {x \in X : T_{x} (y) & > β (x)}) \subseteq C_{M} ⋂ K . \end{matrix}$ Taking any point $\hat{x} \in ⋂_{y \in C_{M}} T^{'} (y)$ , we have the following: $\begin{matrix} \hat{x} & \in & ⋂_{y \in C_{M}} T^{'} (y) \subseteq ⋂_{y \in M} (T^{'} (y) ⋂ K) & \subseteq & ⋂_{y \in M} ({cl}_{X} (X \ {x \in X : T_{x} (y) > β (x)}) ⋂ K) & = & ⋂_{y \in M} \bar{T} (y), \end{matrix}$ which confirms that the family ${\bar{T} (y) : y \in X}$ has the property that the intersection of any finite number of set-valued mappings is nonempty. Since K is compact, we have $⋂_{y \in X} \bar{T} (y) \neq \emptyset$ . By (i), (ii), and the definitions of $\bar{S}$ and $\bar{T}$ , we have $\bar{T} (y) \subseteq \bar{S} (y)$ for every y ∈ X and thus, $\begin{matrix} \emptyset & \neq & ⋂_{y \in X} \bar{S} (y) = ⋂_{y \in X} ({cl}_{X} (X \ {x \in X : S_{x} (y) & > β (x)})) ⋂ K & = & (X \ ⋃_{y \in X} {int}_{X} {x \in X : S_{x} (y) > β (x)}) ⋂ K & = & K \ ⋃_{y \in X} {x \in X : S_{x} (y) > β (x)} . \end{matrix}$

By (iv), (S_x) _β(x)≠ ∅ for all x ∈ K and thus, K ⊆ ⋃ _y∈X {x ∈ X : S_x (y) > β (x)}. This is a contradiction. Therefore, there exists $\hat{x} \in X$ such that $\hat{x} \in (T_{\hat{x}})_{β (\hat{x})}$ .

Now, under the condition that (b) of (iii) is satisfied, we prove that the conclusion of the theorem is true. Similarly, we use the contradiction method. Assume that x ∉ (T_x) _β(x) for every x ∈ X. Furthermore, we define two set-valued mappings $\bar{S}, \bar{T} : X \to 2^{X}$ as follows: $\bar{S} (y) = {cl}_{X} (X \ {x \in X : S_{x} (y) > β (x)}), \forall y \in X,$ $\bar{T} (y) = {cl}_{X} (X \ {x \in X : T_{x} (y) > β (x)}), \forall y \in X .$ Then it easily follows from (i) that $\bar{T} (y) \subseteq \bar{S} (y)$ for all y ∈ X. We check that $\bar{T}$ is a KKM mapping, that is, $co (F) \subseteq ⋃_{y \in F} \bar{T} (y)$ for every F ∈ 〈X〉. Suppose that the conclusion is opposite, and then we can see that there exist F ∈ 〈X〉 and x ∈ co (F) ⊆ X such that $x \notin ⋃_{y \in F} \bar{T} (y) = X \ ⋂_{y \in F} {int}_{X} {x \in X : T_{x} (y) > β (x)}$ , which implies that $\begin{matrix} x \in ⋂_{y \in F} {int}_{X} {x \in X : T_{x} (y) > β (x)} & \subseteq ⋂_{y \in F} {x \in X : T_{x} (y) > β (x)} . \end{matrix}$ From the above formula, we have F ⊆ (T_x) _β(x). By the fact that (T_x) _β(x) is convex for every x ∈ X, one can see that x ∈ co (F) ⊆ (T_x) _β(x). This is a contradiction. Thus, it follows that $\bar{T}$ is a KKM mapping. By (b) of (iii), there is y₀ ∈ X such that X \ K ⊆ int_X ({x ∈ X : T_x (y₀) > β (x)}), which means that $\bar{T} (y_{0}) = {cl}_{X} (X \ {x \in X : T_{x} (y_{0}) > β (x)} \subseteq K$ . Since $\bar{T} (y_{0})$ is closed in X, $\bar{T} (y_{0})$ is a compact subset of X. Using Lemma 2.1, we can conclude that $\emptyset \neq ⋂_{y \in X} \bar{T} (y) \subseteq \bar{T} (y_{0}) \subseteq K .$ This implies that

$\emptyset \neq ⋂_{y \in X} \bar{T} (y) ⋂ K = ⋂_{y \in X} \bar{T} (y) ⋂ K \subseteq ⋂_{y \in X} \bar{S} (y) ⋂ K .$

By (ii), we can deduce the following: $\begin{matrix} \emptyset & \neq & ⋂_{y \in X} \bar{S} (y) ⋂ K = ⋂_{y \in X} ({cl}_{X} (X \ {x \in X : S_{x} (y) & > β (x)})) ⋂ K & = & (X \ ⋃_{y \in X} {int}_{X} {x \in X : S_{x} (y) > β (x)}) ⋂ K & = & (X \ ⋃_{y \in X} {x \in X : S_{x} (y) > β (x)}) ⋂ K & = & ⋂_{y \in X} (X \ {x \in X : S_{x} (y) > β (x)}) ⋂ K . \end{matrix}$ So, we can take x₀ ∈ ⋂ _y∈X (X \ {x ∈ X : S_x (y) > β (x)}) ⋂ K. Then x₀ ∈ K and y ∉ (S_{x
₀}) _β(x₀) for every y ∈ X, which shows that (S_{x
₀}) _β(x₀) =∅. Thus, we can see a contradiction with (iv). Therefore, there must be at least a point $\hat{x} \in X$ such that $\hat{x} \in (T_{\hat{x}})_{β (\hat{x})}$ . The proof of this theorem is finished.

□

Remark 3.1. (1) Theorem 3.1 generalizes Theorem 3.1 of Lu et al. [6] in the following two aspects: (a) from crisp settings to fuzzy settings; (b) (ii) of Theorem 3.1 is weaker than (ii) of Theorem 3.1 due to Lu et al. [6]. In addition, Theorem 3.1 is a new result and extends Theorem 1 of Browder [30] and Tarafdar [31], Theorem 1 obtained by Yannelis [32], Theorem 2.4 due to Tan and Yuan [33], and Theorem 2 due to Metha and Tarafdar [34] to CAT(0) spaces without any linear structure. It is needed to point out that the results of the authors just mentioned above are obtained in the framework of compact linear spaces, or in the framework of noncompact linear spaces with only one coercive condition. And our Theorem 3.1 is established in the setting of noncompact CAT(0) spaces with nonlinear characteristics and two alternative coercive conditions are provided.

(2) (ii) of Theorem 3.1 can be replaced with the following stronger condition: (ii)^′ the set {x ∈ X : S_x (y) > β (x)} is an open subset of X for every y ∈ X.

4 Existence of fuzzy maximal elements

Before obtaining the existence theorems of fuzzy maximal elements in noncompact CAT(0) spaces, we need to give the following useful definitions.

Definition 4.1. Suppose that Y is a topological space and ∅ ≠ X ⊆ Z, where (Z, d) is a CAT(0) space. Further, let ξ : Y → Z be a single-valued mapping, β, σ : Y → [0, 1) be two functions, and $ρ : Y \to F (X)$ be a fuzzy mapping. Then

(1) ρ is said to belong to the family $P_{ξ, C}$ if

(a) ∀ y ∈ Y, ξ (y) ∉ (ρ_y) _β(y) and the set (ρ_y) _β(y) is convex,

(b) there is a fuzzy mapping $γ : Y \to F (X)$ such that (γ_y) _β(y) ⊆ (ρ_y) _β(y) for every y ∈ Y and the set ⋃_x∈X {y ∈ Y : γ_y (x) > β (y)} = ⋃ _x∈Xint_Y {y ∈ Y : γ_y (x) > β (y)},

(2) (ρ^{y
^*}, γ^{y
^*}, N_{y
^*}) is said to be a $P_{ξ, C}$ -matchment of ρ at some y^* ∈ Y if $ρ^{y^{*}}, γ^{y^{*}} : Y \to F (X)$ are two fuzzy mappings and N_{y
^*} is an open subset of Y containing y^* such that

(a) ∀ s ∈ Y, $(ρ_{s}^{y^{*}})_{σ (s)}$ is a convex subset of X,

(b) ∀ s ∈ N_{y
^*}, $(ρ_{s})_{β (s)} \subseteq (ρ_{s}^{y^{*}})_{σ (s)}$ and $ξ (s) \notin (ρ_{s}^{y^{*}})_{σ (s)}$ ,

(d) ∀ x ∈ X, ${s \in Y : γ_{s}^{y^{*}} (x) > σ (s)}$ is an open subset of Y;

(3) ρ is called $P_{ξ, C}$ -matched if for every y ∈ Y with (ρ_y) _β(y)≠ ∅, there is a $P_{ξ, C}$ -matchment (ρ^y, γ^y, N_y) of ρ at y such that for each A ∈ 〈 {y ∈ Y : (ρ_y) _β(y) ≠ ∅} 〉, we have $\begin{matrix} {s \in ⋂_{y \in A} N_{y} : ⋂_{y \in A} (ρ_{s}^{y})_{σ (s)} \neq \emptyset} & = {s \in ⋂_{y \in A} N_{y} : ⋂_{y \in A} (γ_{s}^{y})_{σ (s)} \neq \emptyset} . \end{matrix}$

Remark 4.1. By Definition 4.1, we can see that every fuzzy mapping of class $P_{ω, C}$ is $P_{ω, C}$ -matched. In addition, we emphasis that Definition 4.1 is introduced in a fuzzy environment and Z in Definition 4.1 is a CAT(0) space without linear structure. Therefore, Definition 4.1 is new and extends the corresponding definitions introduced by Tan and Yuan [33] and Yuan [35] to CAT(0) spaces. In addition, Definition 4.1 generalizes Definition 2.10 of Lu et al. [36] from Hadamard manifolds to CAT(0) spaces and from crisp settings to fuzzy settings.

Definition 4.2. Suppose that Y is a topological space and ∅ ≠ X ⊆ Z, where (Z, d) is a CAT(0) space. Let ξ : Y → Z be a single-valued mapping, $ρ : Y \to F (X)$ be a fuzzy mapping, and β, σ : Y → [0, 1) be two functions. Then (ρ^{y
^*}, γ^{y
^*}, N_{y
^*}) is said to be a $P_{ξ, C}$ -strong matchment of ξ at some y^* ∈ Y if $ρ^{y^{*}}, γ^{y^{*}} : Y \to F (X)$ are two fuzzy mappings and N_{y
^*} is an open neighborhood of y^* in Y such that (a) for each s ∈ Y, $(ρ_{s}^{y^{*}})_{σ (s)}$ is convex; (b) for each s ∈ N_{y
^*}, $ξ (s) \notin (ρ_{s}^{y^{*}})_{σ (s)}$ ; (c) for each s ∈ Y, $(γ_{s}^{y^{*}})_{σ (s)} \subseteq (ρ_{y})_{β (y)} \subseteq (ρ_{s}^{y^{*}})_{σ (s)}$ ; (d) for each x ∈ X, the set ${s \in Y : γ_{s}^{y^{*}} (x) > σ (s)}$ is open in Y. We call ρ to be $P_{ξ, C}$ -strongly matched if for each y ∈ Y with (ρ_y) _β(y)≠ ∅, there exists a $P_{ω, C}$ -strong matchment (ρ^y, γ^y, N_y) of ρ at y such that for any nonempty finite subset A of the set {y ∈ Y : (ρ_y) _β(y) ≠ ∅}, we have $\begin{matrix} {s \in ⋂_{y \in A} N_{y} : ⋂_{y \in A} (ρ_{s}^{y})_{σ (s)} \neq \emptyset} & = {s \in ⋂_{y \in A} N_{y} : ⋂_{y \in A} (γ_{s}^{y})_{σ (s)} \neq \emptyset} . \end{matrix}$

Remark 4.2. It is clear that every $P_{ξ, C}$ -strongly matched set-valued mapping is $P_{ξ, C}$ -matched. The reverse is not necessarily true.

Lemma 4.1. Suppose that Y is a regular topological space and ∅ ≠ X ⊆ Z, where (Z, d) is a CAT(0) space. Let ξ : Y → Z be a single-valued mapping and $Q : Y \to F (X)$ be a fuzzy mapping with Q being $P_{ξ, C}$ -matched. If every open subset of Y that contains the set C = {y ∈ Y : (Q_y) _β(y) ≠ ∅}, is paracompact, where β : Y → [0, 1) is a function in touch with Definition 4.1, then there exist a function $\tilde{β} : Y \to [0, 1)$ and a fuzzy mapping $ρ : Y \to F (X)$ of class $P_{ξ, C}$ such that $(Q_{y})_{β (y)} \subseteq (ρ_{y})_{\tilde{β} (y)}$ for every y ∈ Y.

Proof. Since Q is $P_{ξ, C}$ -matched, for every y ∈ B, there exist a real-valued mapping σ : Y → [0, 1), two fuzzy mappings $ρ^{y}, γ^{y} : Y \to F (X)$ , and an open neighborhood N_y ⊆ Y of y such that

(a) ∀ s ∈ Y, $(ρ_{s}^{y})_{σ (s)}$ is convex subset of X;

(b) ∀ s ∈ N_y, $(Q_{s})_{β (s)} \subseteq (ρ_{s}^{y})_{σ (s)}$ and $ξ (s) \notin (ρ_{s}^{y})_{σ (s)}$ ;

(d) ∀ x ∈ X, the set ${s \in Y : γ_{s}^{y} (x) > σ (s)}$ is an open subset of Y;

(e) ∀ A ∈ 〈B〉, we have $\begin{matrix} {s \in ⋂_{y \in A} N_{y} : ⋂_{y \in A} (ρ_{s}^{y})_{σ (s)} \neq \emptyset} & = {s \in ⋂_{y \in A} N_{y} : ⋂_{y \in A} (γ_{s}^{y})_{σ (s)} \neq \emptyset} . \end{matrix}$ Since Y is a regular topological space, it follows that there exists an open neighborhood $V_{y}$ of y in Y such that $y \in V_{y} \subseteq {cl}_{Y} V_{y} \subseteq N_{y}$ for every y ∈ C. Let $V = ⋃_{y \in C} V_{y}$ . Then $V$ is open in Y containing C = {y ∈ Y : (Q_y) _β(y) ≠ ∅} and so, according to the known conditions of this lemma, $V$ is paracompact. By paracompactness, the open cover ${V_{y} : y \in C}$ of $V$ has precise open neighborhood finite refinement ${V_{y}' : y \in C}$ . Let y ∈ C be given arbitrarily. Then we can define two set-valued mappings ${\tilde{γ}}_{y}, {\tilde{ρ}}_{y} : V \to 2^{X}$ by setting, for each $s \in V$ , $\begin{matrix} {\tilde{γ}}_{y} (s) = {\begin{matrix} (γ_{s}^{y})_{σ (s)}, & if s \in V ⋂ {cl}_{Y} V_{y}', X, & if s \in V \ {cl}_{Y} V_{y}', \end{matrix} \end{matrix}$ $\begin{matrix} {\tilde{ρ}}_{y} (s) = {\begin{matrix} (ρ_{s}^{y})_{σ (s)}, & if s \in V ⋂ {cl}_{Y} V_{y}', X, & if s \in V \ {cl}_{Y} V_{y}' . \end{matrix} \end{matrix}$ Then we have

(i) by (c), ∀ $s \in V$ , ${\tilde{γ}}_{y} (s) \subseteq {\tilde{ρ}}_{y} (s)$ ,

(ii) by (e), ${s \in V : {\tilde{γ}}_{y} (s) \neq \emptyset} = {s \in V : {\tilde{ρ}}_{y} (s) \neq \emptyset}$ ,

(iii) ∀ x ∈ X, we have $\begin{matrix} {\tilde{γ}}_{y}^{- 1} (x) & = & {s \in V : x \in {\tilde{γ}}_{y} (s)} & = & {s \in V ⋂ {cl}_{Y} V_{y}' : x \in {\tilde{γ}}_{y} (s)} & ⋃ {s \in V \ {cl}_{Y} V_{y}' : x \in {\tilde{γ}}_{y} (s)} & = & {s \in V ⋂ {cl}_{Y} V_{y}' : x \in {(γ_{s}^{y})}_{σ (s)}} & ⋃ {s \in V \ {cl}_{Y} V_{y}' : x \in X} & = & [(V ⋂ {cl}_{Y} V_{y}') ⋂ {s \in Y : γ_{s}^{y} (x) & > σ (s)}] ⋃ (V \ {cl}_{Y} V_{y}') & = & [V ⋂ {s \in Y : γ_{s}^{y} (x) > σ (s)}] & ⋃ (V \ {cl}_{Y} V_{y}') . \end{matrix}$ By (d), we can see that ${\tilde{γ}}_{y}^{- 1} (x)$ is an open subset of Y for each x ∈ X. Two set-valued mappings γ′, ρ′ : Y → 2^X are defined by setting, for every s ∈ Y, $\begin{matrix} γ^{'} (s) = {\begin{matrix} ⋂_{y \in C} {\tilde{γ}}_{y} (s), & if s \in V, \\ \emptyset, & if s \in Y \ V, \end{matrix} \end{matrix}$ $\begin{matrix} ρ^{'} (s) = {\begin{matrix} ⋂_{y \in C} {\tilde{ρ}}_{y} (s), & if s \in V, \\ \emptyset, & if s \in Y \ V . \end{matrix} \end{matrix}$ Let $\tilde{β} : Y \to [0, 1)$ be defined by $\tilde{β} (y) \equiv 0$ for every y ∈ Y. Further, we define two fuzzy mappings $γ, ρ : Y \to F (X)$ by γ_s = χ_γ′(s) and by ρ_s = χ_ρ′(s) for every s ∈ Y, where χ_E is the characteristic function of the subset E ⊆ X. Now, we check that ρ is of class $P_{ξ, C}$ and $(Q_{s})_{β (s)} \subseteq (ρ_{s})_{\tilde{β} (s)}$ for every s ∈ Y as follows:

(I) By (c), we have $(γ_{s})_{\tilde{β} (s)} \subseteq (ρ_{s})_{\tilde{β} (s)}$ for every s ∈ Y. For each s ∈ Y, when $s \in Y \ V$ , one has $(ρ_{s})_{\tilde{β} (s)} = ρ^{'} (s) = \emptyset$ and so, $ξ (s) \notin (ρ_{s})_{\tilde{β} (s)}$ ; when $s \in V$ , we know that there is y ∈ C such that $s \in V ⋂ {cl}_{Y} V_{y}'$ and thus, ${\tilde{ρ}}_{y} (s) = (ρ_{s}^{y})_{σ (s)}$ . Hence, $(ρ_{s})_{\tilde{β} (s)} = ρ^{'} (s) \subseteq {\tilde{ρ}}_{y} (s)$ . Since $ξ (s) \notin (ρ_{s}^{y})_{σ (s)}$ by (b), we have $ξ (s) \notin (ρ_{s})_{\tilde{β} (s)}$ . Therefore, $ξ (s) \notin (ρ_{s})_{\tilde{β} (s)}$ for every s ∈ Y.

(II) We prove that the set ${s \in Y : γ_{s} (x) > \tilde{β} (s)}$ is an open subset of Y for every x ∈ X. Indeed, let x ∈ X such that ${s \in Y : γ_{s} (x) > \tilde{β} (s)} \neq \emptyset$ and then take an arbitrary point $u \in {s \in Y : γ_{s} (x) > \tilde{β} (s)}$ . By using the fact that ${V_{y}' : y \in C}$ is a neighborhood-finite refinement, one can see that there is an open neighborhood Ω_u of u in $V$ such that ${y \in C : Ω_{u} ⋂ V_{y}' \neq \emptyset} = {y_{1}, \dots, y_{n}}$ . For each y ∈ C with y ∉ {y₁, …, y_n}, we have $Ω_{u} ⋂ V_{y}' = \emptyset$ , and on this basis, we can further prove that that $Ω_{u} ⋂ {cl}_{Y} V_{y}' = \emptyset$ . In fact, if $Ω_{u} ⋂ {cl}_{Y} V_{y}' \neq \emptyset$ , then we can take $s_{0} \in Ω_{u} ⋂ {cl}_{Y} V_{y}'$ . By the fact that Ω_u is an open neighborhood of s₀ and $s_{0} \in {cl}_{Y} V_{y}'$ , we have $Ω_{u} ⋂ V_{y}' \neq \emptyset$ , which is a contradiction. Therefore, we have $Ω_{u} ⋂ {cl}_{Y} V_{y}' = \emptyset$ and so, ${\tilde{γ}}_{y} (s) = X$ for every s ∈ Ω_u. Thus, we have

$(γ_{s})_{\tilde{β} (s)} = γ^{'} (s) = ⋂_{y \in B} {\tilde{γ}}_{y} (s) = ⋂_{i = 1}^{n} {\tilde{γ}}_{y_{i}} (s) for all s \in Ω_{u} .$

Therefore, we get the following: $\begin{matrix} {s \in Y : γ_{s} (x) > \tilde{β} (s)} \\ = {s \in O : x \in ⋂_{y \in B} {\tilde{γ}}_{y} (s)} \\ \supseteq {s \in Ω_{u} : x \in ⋂_{y \in B} {\tilde{γ}}_{y} (s)} \\ = {s \in Ω_{u} : x \in ⋂_{i = 1}^{n} {\tilde{γ}}_{y_{i}} (s)} \\ = Ω_{u} ⋂ {s \in O : x \in ⋂_{i = 1}^{n} {\tilde{γ}}_{y_{i}} (s)} \\ = Ω_{u} ⋂ (⋂_{i = 1}^{n} {\tilde{γ}}_{y_{i}}^{- 1} (x)) . \end{matrix}$ By the fact that the set ${\tilde{γ}}_{y_{i}}^{- 1} (x)$ is an open subset of Y and $u \in {\tilde{γ}}_{y_{i}}^{- 1} (x)$ , we can conclude that the set ${s \in Y : γ_{s} (x) > \tilde{β} (s)}$ is an open subset of Y for every x ∈ X. Thus, we have $⋃_{x \in X} {s \in Y : γ_{s} (x) > \tilde{β} (s)} = ⋃_{x \in X} {int}_{Y} {s \in Y : γ_{s} (x) > \tilde{β} (s)}$ .

(III) We claim that ${s \in Y : (γ_{s})_{\tilde{β} (s)} \neq \emptyset} = {s \in Y : (ρ_{s})_{\tilde{β} (s)} \neq \emptyset}$ . In fact, for each s ∈ Y with $(ρ_{s})_{\tilde{β} (s)} \neq \emptyset$ , we must have $s \in O$ . Since ${O_{y}^{'} : y \in B}$ is a neighborhood-finite refinement, there is an open neighborhood L_s of s in $O$ such that ${y \in B : L_{s} ⋂ O_{y}' \neq \emptyset}$ is a finite subset of B. Since $z \in O$ and ${O_{y}' : y \in B}$ is neighborhood refinement of ${O_{y} : y \in B}$ , it follows that there exists y₀ ∈ B such that $s \in O_{y_{0}}^{'} \subseteq {cl}_{Y} O_{y_{0}}^{'}$ . Thus, we have $\emptyset \neq {y \in B : s \in {cl}_{Y} O_{y}' and L_{s} \cap O_{y}' \neq \emptyset} \subseteq {y \in B : L_{s} \cap O_{y}' \neq \emptyset} .$

Hence, ${y \in B : z \in {cl}_{Y} O_{y}' and L_{s} ⋂ O_{y}' \neq \emptyset}$ is also finite subset of B. Suppose ${y \in B : s \in {cl}_{Y} O_{y}' and L_{s} ⋂ O_{y}' \neq \emptyset} = {y_{1}', \dots, y_{m}'}$ . Thus, for every y ∈ B \ {^{y₁′,…,y_m′}}, $z \notin {cl}_{Y} O_{y}'$ or $L_{s} ⋂ O_{y}' = \emptyset$ . If $s \notin {cl}_{Y} O_{y}'$ , then by the definitions of ${\tilde{γ}}_{y}$ and ${\tilde{ρ}}_{y}$ , we know that ${\tilde{γ}}_{y} (z) = {\tilde{ρ}}_{y} (z) = X$ . If $L_{s} ⋂ O_{y}' = \emptyset$ , then $L_{s} ⋂ {cl}_{Y} O_{y}' = L_{s} ⋂ O_{y}' = \emptyset$ and so, $s \in L_{s} \subseteq O$ and $s \notin {cl}_{Y} O_{y}'$ . Therefore, ${\tilde{γ}}_{y} (s) = {\tilde{ρ}}_{y} (s) = X$ . Thus, in both cases, by the definitions of γ and ρ, we have $(γ_{s})_{\tilde{β} (s)} = γ^{'} (s) = ⋂_{y \in B} {\tilde{γ}}_{y} (s) = ⋂_{i = 1}^{m} {\tilde{γ}}_{y_{i}'} (z),$

$(ρ_{s})_{\tilde{β} (s)} = ρ^{'} (s) = ⋂_{y \in B} {\tilde{ρ}}_{y} (s) = ⋂_{i = 1}^{m} {\tilde{ρ}}_{y_{i}'} (z) .$ Since $s \in ⋂_{i = 1}^{m} {cl}_{Y} O_{y_{i}'} \subseteq ⋂_{i = 1}^{m} N_{y_{i}'}$ , it follows from (e) that $(γ_{s})_{\tilde{β} (s)} \neq \emptyset$ . Hence, ${s \in Y : (γ_{s})_{\tilde{β} (s)} \neq \emptyset} \supseteq {s \in Y : (ρ_{s})_{\tilde{β} (s)} \neq \emptyset}$ . Conversely, by (c), ${s \in Y : (γ_{s})_{\tilde{β} (s)} \neq \emptyset} \subseteq {s \in Y : (ρ_{s})_{\tilde{β} (s)} \neq \emptyset}$ . Therefore, ${s \in Y : (γ_{s})_{\tilde{β} (s)} \neq \emptyset} = {s \in Y : (ρ_{s})_{\tilde{β} (s)} \neq \emptyset}$ .

(IV) To complete the proof of this lemma, we only need to prove that, for each s ∈ Y, $(Q_{s})_{β (s)} \subseteq (ρ_{s})_{\tilde{β} (s)}$ and $(ρ_{s})_{\tilde{β} (s)}$ is convex. Indeed, let s ∈ Y with (Q_s) _β(s)≠ ∅. Then $s \in O$ . For each y ∈ B, if $s \in O \ {cl}_{Y} O_{y}'$ , then ${\tilde{ρ}}_{y} (s) = X \supseteq (Q_{s})_{β (s)}$ ; if $s \in O ⋂ {cl}_{Y} O_{y}'$ , then we have $s \in {cl}_{Y} O_{y}' \subseteq {cl}_{Y} O_{y} \subseteq N_{y}$ . Thus, it follows from (b) that $(Q_{s})_{β (s)} \subseteq (ρ_{s}^{y})_{σ (s)} \subseteq {\tilde{ρ}}_{y} (s)$ . Hence, $(Q_{s})_{β (s)} \subseteq {\tilde{ρ}}_{y} (s)$ for every y ∈ B, and so, we have $(Q_{s})_{β (s)} \subseteq ⋂_{y \in B} {\tilde{ρ}}_{y} (s) = ρ^{'} (s) = (ρ_{s})_{\tilde{β} (s)}$ . In addition, it follows from the definitions of ${\tilde{ρ}}_{y}$ and ρ and the fact that every $(ρ_{s}^{y})_{σ (s)}$ is a convex subset of X, one can see that $(ρ_{s})_{\tilde{β} (s)}$ is convex for every s ∈ Y. The proof of Lemma 4.1 is complete.□

Remark 4.3. Lemma 4.1 generalizes and extends Lemma 2.2 of Lu et al. [36] in the following aspects: (a) from crisp cases to fuzzy cases; (b) Lemma 4.1 involves two spaces, one of which is a regular topological space and the other is a CAT(0) space, while Lemma 2.2 of Lu et al. [36] has only one Hadamard manifold which is a special form of CAT(0) space; (c) the condition that Q is $P_{ξ, C}$ -matched in Lemma 4.1 is weaker than the condition that G possesses an $H_{E}$ combination in Lemma 2.2 of Lu et al. [36]. In addition, Lemma 4.1 is established in the framework of CAT(0) spaces without any linear structure. Therefore, from this point of view, Lemma 4.1 is a new result and extends Theorem 3.1 of Yuan [35] to a nonlinear form.

In what follows, we mainly deal with two situations: (1) X = Z = Y and (X, d) is a nonempty CAT(0) space and ξ = I_X, where I_X : X → X is the identity mapping defined on X; (2) Let I be a finite index set and (X_i, d_i) be a CAT(0) space for every i ∈ I. Let $X = \prod_{i \in I} X_{i}$ and ξ = π_i be the projection from X to X_i for every i ∈ I. For the above two situations, we shall replace $P_{ξ, C}$ with $P_{C}$ .

By means of Theorem 3.1, we have the following existence theorem of fuzzy maximal elements in the framework of noncompact CAT(0) spaces.

Theorem 4.1. Let (X, d) be a complete CAT(0) space satisfying the finite convex hull property. Suppose that K is a nonempty compact subset of X, β : X → [0, 1) is a function, and $G : X \to F (X)$ is a fuzzy mapping which belongs to the family $P_{C}$ . Either (a) ∀ M ∈ 〈X〉, there is a nonempty subset C_M of X such that C_M is compact convex, M ⊆ C_M, and C_M \ K ⊆ ⋃ _{y∈C_M}int_{C
_M} ({x ∈ X : G_x (y) > β (x)} ⋂ C_M), or (b) there exists y₀ ∈ X such that X \ K ⊆ int_X {x ∈ X : G_x (y₀) > β (x)}. Then there exists at least $\hat{x} \in K$ such that $(G_{\hat{x}})_{β (\hat{x})} = \emptyset$ .

Proof. Since G belongs to the family $P_{C}$ , we have

(a) ∀ x ∈ X, (G_x) _β(x) is a convex subset of X and x ∉ (G_x) _β(x);

(b) there is a fuzzy mapping $γ : X \to F (X)$ such that the set ⋃_y∈X {x ∈ X : γ_x (y) > β (x)} = ⋃ _y∈Xint_X {x ∈ X : γ_x (y) > β (x)} and (γ_x) _β(x) ⊆ (G_x) _β(x) for all x ∈ X;

If the conclusion of Theorem 4.1 were not true, then we have (G_x) _β(x)≠ ∅ for every x ∈ K. By (c), we have (γ_x) _β(x)≠ ∅ for every x ∈ K. Thus, all the conditions of Theorem 3.1 are fulfilled. So, it follows from Theorem 3.1 that there exists at least $\hat{x} \in K$ such that $\hat{x} \in (G_{\hat{x}})_{β (\hat{x})}$ , which contradicts (a). Therefore, the conclusion of Theorem 4.1 is correct. The proof is complete.□

Remark 4.4. (1) Theorem 4.1 extends and generalizes Theorem 3.1 due to Lu et al. [36] in the following aspects: (a) from noncompact Hadamard manifolds to noncompact CAT(0) spaces; (b) from crisp cases to fuzzy cases; (c) even if the CAT(0) space in Theorem 4.1 is strengthened to be a Hadamard manifold, it can be seen from the related definitions that the condition that G belongs to the family $P_{C}$ is still weaker than the condition that G is of class $H_{E}$ in Theorem 3.1 of Lu et al. [36].

(2) Theorem 4.1 improves and extends Theorem 3.1 obtained by Yang and Pu [37] in the following aspects: (a) concerns on noncompact CAT(0) space which is more general than the compact Hadamard manifold involved in Theorem 3.1 due to Yang and Pu [37]; (b) from crisp environments to fuzzy environments; (c) the condition that G belongs to the family $P_{C}$ in Theorem 4.1 is weaker than conditions (1) and (2) of Theorem 3.1 in Yang and Pu [37]. In fact, for Theorem 3.1 due to Yang and Pu [37], let us define a set-valued mapping $\tilde{A} : X \to 2^{X}$ by $\tilde{A} (x) = GcoA (x)$ for every x ∈ X, where X is a compact Hadamard manifold. Then we can easily verify that ${\tilde{A}}^{- 1} (y)$ is open in X for every y ∈ X. When (2) of Theorem 3.1 in Yang and Pu [37] is replaced with the condition that ${\tilde{A}}^{- 1} (y)$ is open in X for every y ∈ X, the conclusion still holds, and the new theorem is equivalent to Theorem 3.1 of Yang and Pu [37]. Moreover, it is needed to point out that the proof of our Theorem 4.1 is different from that of Theorem 3.1 obtained by Yang and Pu [37]. The proof of Theorem 3.1 due to Yang and Pu [37] is based on unity partition and fixed pony theory, but essentially, we use the KKM method to prove Theorem 4.1.

Example 4.1. Let X = (0, 1] be endowed the Euclidean topology induced by the metric d = |y - x| for every x, y ∈ X. It is obvious that (X, d) is a complete CAT(0) space which satisfies the finite convex hull property. Further, we define a fuzzy mapping $G : X \to F (X)$ by $\begin{matrix} G_{x} (y) = {\begin{matrix} 0, & if x \in (0, \frac{1}{2}], y \in (0, 1], \\ (\frac{2}{3} x + \frac{1}{3}) y, & if x \in (\frac{1}{2}, 1], y \in (0, 1] . \end{matrix} \end{matrix}$ The function β : X → [0, 1) is defined by $\begin{matrix} β (x) = {\begin{matrix} 0, & if x \in (0, \frac{1}{2}), \\ \frac{2}{3} x^{2} + \frac{1}{3} x, & if x \in [\frac{1}{2}, 1] . \end{matrix} \end{matrix}$ Then we have $\begin{matrix} (G_{x})_{β (x)} = {\begin{matrix} \emptyset, & if x \in (0, \frac{1}{2}] ⋃ {1}, \\ (x, 1], & if x \in (\frac{1}{2}, 1) . \end{matrix} \end{matrix}$ Thus, (G_x) _β(x) is a convex subset of X and x ∉ (G_x) _β(x) for every x ∈ X. Let γ = G. Then by the upper semicontinuity of β and the lower semicontinuity of the function x ↦ G_x (y) of x, it follows that the set {x ∈ X : γ_x (y) > β (x)} is open in X for every y ∈ (0, 1]. Thus, we have ⋃_y∈X {x ∈ X : γ_x (y) > β (x)} = ⋃ _y∈Xint_X {x ∈ X : γ_x (y) > β (x)}. The equation {x ∈ X : (γ_x) _β(x) ≠ ∅} = {x ∈ X : (G_x) _β(x) ≠ ∅} obviously holds. Therefore, from the above discussion, we know that G belongs to the family $P_{C}$ .

Now, for each y ∈ (0, 1], by the definition of G, we have the following: $\begin{matrix} {x \in X : G_{x} (y) > β (x)} = {\begin{matrix} \emptyset, & if y \in (0, \frac{1}{2}], \\ (0, y), & if y \in (\frac{1}{2}, 1] . \end{matrix} \end{matrix}$ Put $K = [\frac{1}{2}, 1]$ , which is a compact subset of X. Choose $y_{0} \in (\frac{1}{2}, 1]$ . Then we get $\begin{matrix} X \ K = (0, \frac{1}{2}) \subseteq {int}_{X} {x \in X : G_{x} (y_{0}) > β (x)} \\ = (0, y_{0}) . \end{matrix}$ Hence, all the hypotheses of Theorem 4.1 are fulfilled, so that there exists at least $\hat{x} \in K$ such that $(G_{\hat{x}})_{β (\hat{x})} = \emptyset$ . In fact, we can find $\hat{x} = \frac{1}{2} \in K$ such that $(G_{\frac{1}{2}})_{β (\frac{1}{2})} = \emptyset$ .

Corollary 4.1. Let (X, d) be a complete CAT(0) space satisfying the finite convex hull property. Suppose that S, T : X → 2^X are two set-valued mappings, K is a nonempty compact subset of X, and the following conditions are fulfilled:

(i) for each x ∈ X, S (x) ⊆ T (x) and T (x) is a convex subset of X;

(ii) ⋃_y∈XS^-1 (y) = ⋃ _y∈Xint_XS^-1 (y);

(iii) ∀ x ∈ X, x ∉ T (x);

(iv) {x ∈ X : S (x) ≠ ∅} = {x ∈ X : T (x) ≠ ∅};

(v) either (a) ∀ M ∈ 〈X〉, there is a nonempty subset C_M of X such that C_M is compact convex, M ⊆ C_M, and C_M \ K ⊆ ⋃ _{y∈C_M}int_{C
_M} (T^-1 (y) ⋂ C_M), or (b) there exists y₀ ∈ X such that X \ K ⊆ int_XT^-1 (y₀).

Then there exists at least $\hat{x} \in K$ such that $T (\hat{x}) = \emptyset$ .

Proof. Define two fuzzy mappings $\tilde{S}, \tilde{T} : X \to F (X)$ by ${\tilde{S}}_{x} = χ_{S (x)}$ and by ${\tilde{T}}_{x} = χ_{T (x)}$ for every x ∈ X, where χ_E denotes the characteristic function of the subset E ⊆ X. Let β : X → [0, 1) be a real-valued function defined by β (x) ≡0 for every x ∈ X. Then by (i)-(v), we have

(1) ∀ x ∈ X, $({\tilde{S}}_{x})_{β (x)} \subseteq ({\tilde{T}}_{x})_{β (x)}$ and $({\tilde{T}}_{x})_{β (x)}$ is a convex subset of X;

(2) $⋃_{y \in X} {x \in X : {\tilde{S}}_{x} (y) > β (x)} = ⋃_{y \in X} {int}_{X} {x \in X : {\tilde{S}}_{x} (y) > β (x)}$ ;

(3) ∀ x ∈ X, $x \notin ({\tilde{T}}_{x})_{β (x)}$ ;

(4) ${x \in X : ({\tilde{S}}_{x})_{β (x)} \neq \emptyset} = {x \in X : ({\tilde{T}}_{x})_{β (x)} \neq \emptyset}$ ;

(5) either (a) ∀ M ∈ 〈X〉, there is a nonempty subset C_M of X such that C_M is compact convex, M ⊆ C_M, and $C_{M} \ K \subseteq ⋃_{y \in C_{M}} {int}_{C_{M}} ({x \in X : {\tilde{T}}_{x} (y) > β (x)} ⋂ C_{M})$ , or (b) there exists y₀ ∈ X such that $X \ K \subseteq {int}_{X} {x \in X : {\tilde{T}}_{x} (y_{0}) > β (x)}$ .

By (1)-(4), we can see that $\tilde{T}$ belongs to the family $P_{C}$ . Coupled with (5), we know that all the conditions of Theorem 4.1 are satisfied. Thus, it follows from Theorem 4.1 that there exists at least $\hat{x} \in K$ such that $({\tilde{T}}_{\hat{x}})_{β (\hat{x})} = \emptyset$ . This implies that $T (\hat{x}) = \emptyset$ . The proof is complete.□

Example 4.2. Let us first give an example of CAT(0) space, which comes from Kristály et al. [38]. Suppose that $X = {x = (s, t) \in ℝ^{2} : t > 0}$ is the two-dimensional Poincaré upper half-plane whose Riemannian metric is defined by setting, for every (s, t) ∈ X, $\begin{matrix} r_{jk} (s, t) = \frac{1}{t^{2}} δ_{k}^{j} = {\begin{matrix} \frac{1}{t^{2}}, & if j = k, \\ 0, & if j \neq k \end{matrix} \end{matrix}$ for j, k ∈ {1, 2}. Then it follows from Udri scedil ;te [39] that (X, r) is a Hadamard manifold with the sectional curvature of X being -1, and the geodesics in X are semilines and semicircles centered on x-axis. Therefore, (X, r) naturally becomes a complete CAT(0) space. By Theorem 3 of Rahimi et al. [40], we can see that the convex hull co (B) of B is compact for every B ∈ 〈X〉, and naturally co (B) is also a closed subset of (X, r) from the Hausdorffness of X. Further, by Lemma 1 due to Németh [41], (X, r) has the finite convex hull property.

Now, we define a set-valued mapping T : X → 2^X by T (x) = {y = (u, v) ∈ X : 4ln² (u² + v²) < ln² (s² + t²)} for every x = (s, t) ∈ X. Since $T (x) = {y = (u, v) \in X : 4 \ln^{2} (u^{2} + v^{2}) < \ln^{2} (s^{2} + t^{2})} = {y = (u, v) \in X : e^{- \frac{1}{2} | \ln (s^{2} + t^{2}) |} < u^{2} + v^{2} < e^{\frac{1}{2} | \ln (s^{2} + t^{2}) |}}$ for every x = (s, t) ∈ X, it follows that x ∉ T (x) and T (x) is convex for every x = (s, t) ∈ X. Let S = T. Then for each y = (u, v) ∈ X, we have $\begin{matrix} S^{- 1} (y) & = & {x = (s, t) \in X : 4 \ln^{2} (u^{2} + v^{2}) \\ < \ln^{2} (s^{2} + t^{2})} \\ = & {x = (s, t) \in X : s^{2} + t^{2} > e^{2 | \ln (u^{2} + v^{2}) |}} \\ ⋃ & {x = (s, t) \in X : s^{2} + t^{2} < e^{- 2 | \ln (u^{2} + v^{2}) |}}, \end{matrix}$ which is open in X. Thus, we have ⋃_y∈XS^-1 (y) = ⋃ _y∈Xint_XS^-1 (y). Let K = {(s, t) ∈ X : s² + t² ≤ 1}, which is a compact subset of X. Let y₀ = (0, 1) ∈ X. Then we have ln² (s² + t²) >0 for every (s, t) ∈ X \ K, which means that X \ K ⊆ int_XT^-1 (y₀). Therefore, all the requirements of Corollary 4.1 are satisfied. By direct observing, we can see that there exists $\hat{x} = (\frac{\sqrt{3}}{2}, \frac{1}{2}) \in K$ such that such that $T (\hat{x}) = \emptyset$ .

Theorem 4.2. Let (X, d) be a complete CAT(0) space satisfying the finite convex hull property. Suppose that K is a nonempty compact subset of X, β : X → [0, 1) is a function, and $T : X \to F (X)$ is a fuzzy mapping which is $P_{C}$ -matched. Either (a) for every M ∈ 〈X〉, there is a nonempty subset C_M of X such that C_M is compact convex, M ⊆ C_M, and C_M \ K ⊆ ⋃ _{y∈C_M}int_{C
_M} ({x ∈ X : T_x (y) > β (x)} ⋂ C_M), or (b) there exists y₀ ∈ X such that X \ K ⊆ int_X {x ∈ X : T_x (y₀) > β (x)}. Then there exists at least $\hat{x} \in K$ such that $(T_{\hat{x}})_{β (\hat{x})} = \emptyset$ .

Proof. We adopt the contradiction method. Assume that (T_x) _β(x)≠ ∅ for all x ∈ X. Then we have {x ∈ X : (T_x) _β(x) ≠ ∅} = X, which is paracompact by the fact that any metric space is a paracompact space. On the other hand, since any metric space is also a regular topological space, by Lemma 4.1, there is a real-valued function $\tilde{β} : X \to [0, 1)$ and a fuzzy mapping $ρ : X \to F (X)$ such that $(T_{x})_{β (x)} \subseteq (ρ_{x})_{\tilde{β} (x)}$ for all x ∈ X, where ρ belongs to the family $P_{C}$ . For each M ∈ 〈X〉, there is a nonempty subset C_M of X such that C_M is compact convex, M ⊆ C_M, and $C_{M} \ K \subseteq ⋃_{y \in C_{M}} {int}_{C_{M}} ({x \in X : T_{x} (y) > β (x)} ⋂ C_{M}) \subseteq ⋃_{y \in C_{M}} {int}_{C_{M}} ({x \in X : ρ_{x} (y) > \tilde{β} (x)} ⋂ C_{M})$ .\!Hence, by Theorem 4.1, we know that there exists at least $\hat{x} \in K$ such that $(ρ_{\hat{x}})_{\tilde{β} (\hat{x})} = \emptyset$ . So, we get $(T_{\hat{x}})_{β (\hat{x})} = \emptyset$ , which contradicts the assumption that (T_x) _β(x)≠ ∅ for all x ∈ X. Therefore, there exists at least $\hat{x} \in X$ such that $(T_{\hat{x}})_{β (\hat{x})} = \emptyset$ . Suppose that $\hat{x} \in X \ K$ . Then by the coercive condition of this theorem, there is a nonempty subset $C_{{\hat{x}}}$ of X such that $C_{{\hat{x}}}$ is compact convex, $x \in C_{{\hat{x}}}$ , and

$C_{{\hat{x}}} \ K \subseteq ⋃_{y \in C_{{\hat{x}}}} {int}_{C_{{\hat{x}}}} ({x \in X : T_{x} (y) > β (x)} ⋂ C_{{\hat{x}}}) .$

It follows from $\hat{x} \in C_{{\hat{x}}} \ K$ that there is such that $\hat{x} \in {int}_{C_{{\hat{x}}}} ({x \in X : T_{x} (y^{'}) > β (x)} ⋂ C_{{\hat{x}}})$ . Thus, there is an open subset $U (\hat{x})$ of X such that $\hat{x} \in U (\hat{x})$ and $U (\hat{x}) ⋂ C_{{\hat{x}}} \subseteq {x \in X : T_{x} (y^{'}) > β (x)} ⋂ C_{{\hat{x}}}$ . Therefore, we have $\hat{x} \in {x \in X : T_{x} (y^{'}) > β (x)}$ , which means that $(T_{\hat{x}})_{β (\hat{x})} \neq \emptyset$ . Therefore, one can see that $\hat{x}$ must be in K such that $(T_{\hat{x}})_{β (\hat{x})} = \emptyset$ . Using a similar method, we can prove that the conclusion is true when (b) is satisfied. The proof of Theorem 4.2 is finished.□

Remark 4.5. (1) Since the condition that $T : X \to F (X)$ is $P_{C}$ -matched in Theorem 4.2 is weaker than the condition that $G : X \to F (X)$ belongs to the family $P_{C}$ in Theorem 4.1, it follows that Theorem 4.2 is a generalization of Theorem 4.1. Therefore, by Remark 4.4, we can see that Theorem 4.2 further generalizes Theorem 3.1 due to Lu et al. [36] and Theorem 3.1 obtained by Yang and Pu [37] in many ways.

(2) Theorem 4.2 extends Corollary 1 in Borglin and Keiding [42], Theorem 2.2 due to Toussaint [43], and Theorem 3.3 obtained by Tan and Yuan [33] to fuzzy environments and CAT(0) spaces without any linear structure. On the other hand, compared with Corollary 1 in Borglin and Keiding [42], Theorem 2.2 due to Toussaint [43], and Theorem 3.3 obtained by Tan and Yuan [33], our Theorem 4.2 is more flexible and it provides two coercive conditions to be available.

5 Generalized fuzzy noncooperative games

Let Ω be a finite (or an infinite) set of players and further, for every i ∈ Ω, let X_i be the set of actions adopted by the i-th player. Let $X = \prod_{i \in Ω} X_{i}$ . A family of ordered quintuple Γ = (X_i, A_i, B_i, P_i, a_i, b_i, p_i) _i∈Ω is called a generalized fuzzy noncooperative game, where $A_{i}, B_{i} : X \to F (X_{i})$ denote fuzzy constraint set-valued mappings and each $P_{i} : X \to F (X_{i})$ is a fuzzy preference set-valued mapping. A Nash equilibrium point of Γ is an $\hat{x} \in X$ such that ${\hat{x}}_{i} \in {\bar{(B_{i_{\hat{x}}})}}_{a_{i} (\hat{x})}$ , and $(A_{i_{\hat{x}}})_{b_{i} (\hat{x})} ⋂ (P_{i_{\hat{x}}})_{p_{i} (\hat{x})} = \emptyset$ for all i ∈ Ω, where a_i, b_i, p_i : X → [0, 1) are real-valued functions and denote fuzzy constraint functions and ${\bar{(B_{i_{x}})}}_{a_{i} (x)} = {y_{i} \in X_{i} : (x, y_{i}) \in {cl}_{X \times X_{i}} ({(x, y_{i}) \in X \times X_{i} : y_{i} \in (B_{i_{x}})_{a_{i} (x)}})$ . What we need to point out is that when Ω contains only a single player, we call the generalized fuzzy noncooperative game Γ = (X, A, B, P, a, b, p) to be a generalized fuzzy noncooperative game with one-player, where X is the set of actions adopted by the player, $A, B : X \to F (X)$ are fuzzy constraint set-valued mappings, a, b, p : X → [0, 1) are fuzzy constraint functions, and $P : X \to F (X)$ is a fuzzy preference set-valued mapping. We call Γ = (X_i, P_i, p_i) _i∈Ω a fuzzy noncooperative qualitative game if X_i is the set of actions adopted by the ith-player, $P_{i} : X \to F (X_{i})$ is a fuzzy preference set-valued mapping, and p_i : X → [0, 1) is a fuzzy constraint function for every i ∈ Ω. A Nash equilibrium point of Γ = (X_i, P_i, p_i) _i∈Ω is an $\hat{x} \in X$ such that for each i ∈ Ω, $(P_{i_{\hat{x}}})_{p_{i} (\hat{x})} = \emptyset$ . If A_i, B_i, and P_i are defined by using the ith characteristic functions,and a_i ≡ b_i ≡ p_i ≡ 0 for every i ∈ Ω, then the Nash equilibrium for generalized fuzzy noncooperative games (respectively, fuzzy noncooperative qualitative games) implies the Nash equilibrium for generalized noncooperative games (respectively, qualitative games). For each i ∈ Ω, let Z_i ⊆ X_i. Then in the case of any given m ∈ Ω, let us define $\prod_{j \in Ω, j \neq m} Z_{j} \otimes Z_{m} = {x = (x_{i})_{i \in Ω} : x_{i} \in Z_{i} for every i \in Ω}$ .

Now, we are ready to use Corollary 4.1 to obtain the following existence theorem of Nash equilibrium points for generalized fuzzy noncooperative games with one-player in the framework of noncompact CAT(0) spaces.

Theorem 5.1. Let Γ = (X, A, B, P, a, b, p) be a generalized fuzzy noncooperative game with one-player, where (X, d) is a CAT(0) space which is complete and satisfies the convex hull finite property. Suppose that K is a nonempty compact subset of X, l : X → [0, 1) is a real-valued function, $γ : X \to F (X)$ is a fuzzy mapping, and the following conditions are fulfilled:

(i) ∀ x ∈ X, (A_x) _b(x), (P_x) _p(x) are convex subsets of X and $(A_{x})_{b (x)} \subseteq {\bar{(B_{x})}}_{a (x)}$ ;

(ii) ∀ x ∈ X, x ∉ (A_x) _b(x) ⋂ (P_x) _p(x) and (γ_x) _l(x) ⊆ (A_x) _b(x) ⋂ (P_x) _p(x);

(iii) ⋃_y∈X {x ∈ X : A_x (y) > b (x)} = ⋃ _y∈Xint_X {x ∈ X : A_x (y) > b (x)};

(iv) ∀ y ∈ X, the set {x ∈ X : γ_x (y) > l (x)} is an open subset of X;

(v) {x ∈ X : (γ_x) _l(x) ≠ ∅} = {x ∈ X : (A_x) _b(x) ⋂ (P_x) _p(x) ≠ ∅};

(vi) either (a) ∀ M ∈ 〈X〉, there is a nonempty subset C_M of X such that C_M is compact convex, M ⊆ C_M, and C_M \ K ⊆ ⋃ _{y∈C_M}int_{C
_M} ({x ∈ X : A_x (y) > b (x) and P_x (y) > p (x)} ⋂ C_M), or (b) there exists y₀ ∈ X such that X \ K ⊆ int_X {x ∈ X : A_x (y₀) > b (x) and P_x (y₀) > p (x)};

(vii) ∀ x ∈ K, (A_x) _b(x)≠ ∅.

Then there exists at least $\hat{x} \in K$ such that $\hat{x} \in {\bar{(B_{\hat{x}})}}_{a (\hat{x})}$ and $(A_{\hat{x}})_{b (\hat{x})} ⋂ (P_{\hat{x}})_{p (\hat{x})} = \emptyset$ .

Proof. Set $V = {x \in X : x \notin {\bar{(B_{x})}}_{a (x)}}$ . Then V is an open subset of X. Further, let us define two set-valued mappings S, T : X → 2^X by setting, ∀ x ∈ X, $\begin{matrix} S (x) = {\begin{matrix} (γ_{x})_{l (x)}, & if x \notin V, \\ (A_{x})_{b (x)}, & if x \in V . \end{matrix} \end{matrix}$ and $\begin{matrix} T (x) = {\begin{matrix} (A_{x})_{b (x)} ⋂ (P_{x})_{p (x)}, & if x \notin V, \\ (A_{x})_{b (x)}, & if x \in V . \end{matrix} \end{matrix}$ By (i) and the definition of T, we can see that T (x) is a convex subset of X for every x ∈ X. By (ii) and the definitions of S and T, one can see that S (x) ⊆ T (x). Obviously, ∀ y ∈ X, we can see that $\begin{matrix} S^{- 1} (y) & = & (V ⋃ {x \in X : γ_{x} (y) > l (x)}) \\ ⋂ {x \in X : A_{x} (y) > b (x)} \\ = & ({x \in X : γ_{x} (y) > l (x)}) \\ ⋃ (V ⋂ {x \in X : A_{x} (y) > b (x)}) . \end{matrix}$ We are ready to show that ⋃_y∈XS^-1 (y) = ⋃ _y∈Xint_XS^-1 (y). Indeed, it is easy to see that ⋃_y∈Xint_XS^-1 (y) ⊆ ⋃ _y∈XS^-1 (y). Thus, it suffices to prove that ⋃_y∈XS^-1 (y) ⊆ ⋃ _y∈Xint_XS^-1 (y). Let x^* be an arbitrary point of ⋃_y∈XS^-1 (y). Then there exists y^* ∈ X such that $\begin{matrix} x^{*} \in S^{- 1} (y^{*}) = ({x \in X : γ_{x} (y^{*}) > l (x)}) \\ ⋃ (V ⋂ {x \in X : A_{x} (y^{*}) > b (x)}) . \end{matrix}$ If x^* ∈ {x ∈ X : γ_x (y^*) > l (x)}, then it follows from (iv) that $\begin{matrix} x^{*} \in {x \in X : γ_{x} (y^{*}) > l (x)} \\ = {int}_{X} {x \in X : γ_{x} (y^{*}) > l (x)} \subseteq {int}_{X} S^{- 1} (y^{*}) . \end{matrix}$ If x^* ∈ V ⋂ {x ∈ X : A_x (y^*) > b (x)}, then by (iii) and the fact that V is an open subset of X, there exists $\tilde{y} \in X$ such that $\begin{matrix} x^{*} \in V ⋂ {int}_{X} {x \in X : A_{x} (\tilde{y}) > b (x)} \\ = {int}_{X} (V ⋂ {x \in X : A_{x} (\tilde{y}) > b (x)}) \subseteq {int}_{X} S^{- 1} (\tilde{y}) . \end{matrix}$ Thus, we can see that in both cases, ⋃_y∈XS^-1 (y) ⊆ ⋃ _y∈Xint_XS^-1 (y) can be obtained. Therefore, we have ⋃_y∈XS^-1 (y) = ⋃ _y∈Xint_XS^-1 (y).

Now, by (v) and the definitions of S and T, we have {x ∈ X : S (x) ≠ ∅} = {x ∈ X : T (x) ≠ ∅}. From the definitions of V and T, along with (ii), we can see that x ∉ T (x) for all x ∈ X. For each y ∈ X, we get

$\begin{matrix} T^{- 1} (y) & = & {x \notin V : A_{x} (y) > b (x) and \\ P_{x} (y) > p (x)} ⋃ {x \in V : A_{x} (y) > b (x)} \\ = & ({x \in X : A_{x} (y) > b (x) and P_{x} (y) > p (x)}) \\ ⋃ (V ⋂ {x \in X : A_{x} (y) > b (x)}) . \end{matrix}$

Then by (vi), we know that either for each M ∈ 〈X〉, there is a nonempty subset C_M of X such that C_M is compact convex, M ⊆ C_M, and $\begin{matrix} C_{M} \ K & \subseteq & ⋃_{y \in C_{M}} {int}_{C_{M}} ({x \in X : A_{x} (y) > b (x) \\ and P_{x} (y) > p (x)} ⋂ C_{M}) \\ \subseteq & ⋃_{y \in C_{M}} {int}_{C_{M}} (T^{- 1} (y) ⋂ C_{M}), \end{matrix}$ or X \ K ⊆ int_X {x ∈ X : A_x (y₀) > b (x) and P_x (y₀) > p (x)} ⊆ int_XT^-1 (y₀).

At this point, we see that all the conditions of Corollary 4.1 have been satisfied. Thus, by virtue of Corollary 4.1, there exists at least $\hat{x} \in K$ such that $T (\hat{x}) = \emptyset$ . Assuming $\hat{x} \in K ⋂ V$ , we have $T (\hat{x}) = (A_{\hat{x}})_{b (\hat{x})} = \emptyset$ . By (vii), (A_x) _b(x)≠ ∅ for all x ∈ K. Then a contradiction occurred. Therefore, we have $\hat{x} \in K \ V$ and thus, $\hat{x} \in {\bar{(B_{\hat{x}})}}_{a (\hat{x})}$ and $T (\hat{x}) = (A_{\hat{x}})_{b (\hat{x})} ⋂ (P_{\hat{x}})_{p (\hat{x})} = \emptyset$ . The proof of Theorem 5.1 is complete.□

If A = B in Theorem 5.1, then we obtain the following corollary.

Corollary 5.1. Let Γ = (X, A, P, a, b, p) be a generalized fuzzy noncooperative game with one-player, where (X, d) is a CAT(0) space which is complete and satisfies the convex hull finite property. Suppose that K is a nonempty compact subset of X, l : X → [0, 1) is a real-valued function, $γ : X \to F (X)$ is a fuzzy mapping, and the following conditions are fulfilled:

(i) ∀ x ∈ X, (A_x) _b(x), (P_x) _p(x) are convex subsets of X;

(ii) ∀ x ∈ X, x ∉ (A_x) _b(x) ⋂ (P_x) _p(x) and (γ_x) _l(x) ⊆ (A_x) _b(x) ⋂ (P_x) _p(x);\\

(iii) ⋃_y∈X {x ∈ X : A_x (y) > b (x)} = ⋃ _y∈Xint_X {x ∈ X : A_x (y) > b (x)};

(iv) ∀ y ∈ X, the set {x ∈ X : γ_x (y) > l (x)} is an open subset of X;

(v) {x ∈ X : (γ_x) _l(x) ≠ ∅} = {x ∈ X : (A_x) _b(x) ⋂ (P_x) _p(x) ≠ ∅};

(vii) ∀ x ∈ K, (A_x) _b(x)≠ ∅.

Then there exists at least $\hat{x} \in K$ such that $\hat{x} \in {\bar{(A_{\hat{x}})}}_{a (\hat{x})}$ and $(A_{\hat{x}})_{b (\hat{x})} ⋂ (P_{\hat{x}})_{p (\hat{x})} = \emptyset$ .

By using Theorem 4.2, we have the following existence theorem of Nash equilibrium points for fuzzy noncooperative qualitative games in noncompact CAT(0) spaces.

Theorem 5.2. Let Ω be a finite index set, Γ = (X_i, P_i, p_i) _i∈Ω be a fuzzy noncooperative qualitative game, and K be a nonempty compact subset of $X = \prod_{i \in Ω} X_{i}$ . Assume that the following conditions are fulfilled:

(i) ∀ i ∈ Ω, (X_i, d_i) is a complete locally compact CAT(0) space;

(ii) ∀ i ∈ Ω, $P_{i} : X \to F (X_{i})$ is $P_{C}$ -matched;

(iii) ⋃_i∈Ω {x ∈ X : (P_{i
_x}) _{p_i(x)} ≠ ∅} = ⋃ _i∈Ωint_X {x ∈ X : (P_{i
_x}) _{p_i(x)} ≠ ∅};

(iv) either (a) ∀ i ∈ Ω and ∀ M_i ∈ 〈X_i〉, there is anonempty subset C_{M
_i} of X_i such that C_{M
_i} is compact convex, M_i ⊂ C_{M
_i}, and C_M \ K ⊆ ⋃ _{x∈C_M}int_{C
_M} (C_M ⋂ (⋂ _i∈Ω {z ∈ X : (P_{i
_z}) (x_i) > p_i (z)})) , where $C_{M} = \prod_{i \in Ω} C_{M_{i}}$ , or (b) there exists x₀ = (x_0i) _i∈Ω ∈ X such that X \ K ⊆ int_X (⋂ _i∈Ω {z ∈ X : (P_{i
_z}) (x_0i) > p_i (z)}).

Then Γ has at least a Nash equilibrium point in K.

Proof. By using the same proof method as Theorem 3.4 of Lu et al. [6], we can prove that (X, d) is a locally compact CAT(0) space which is complete, where the metric function $d : X \times X \to ℝ$ is defined by $d (x, y) = (\sum_{i \in Ω} d_{i}^{2} (x_{i}, y_{i})) \frac{1}{2}$ for every x = (x_i) _i∈Ω, y = (y_i) _i∈Ω ∈ X. By Lemma 2.2, (X, d) possesses the convex hull finite property. Let Ω (x) = {i ∈ Ω : (P_{i
_x}) _{p_i(x)} ≠ ∅} ∀ x ∈ X. On this basis, we define a set-valued mapping P : X → 2^X by setting, for each x ∈ X, $\begin{matrix} P (x) = {\begin{matrix} ⋂_{i \in Ω (x)} P_{i}^{'} (x), & if Ω (x) \neq \emptyset, \\ \emptyset, & if Ω (x) = \emptyset, \end{matrix} \end{matrix}$ where for all x ∈ X. Further, let us define a fuzzy mapping $\tilde{P} : X \to F (X)$ by ${\tilde{P}}_{x} = χ_{P (x)}$ for all x ∈ X, where χ_E denotes the characteristic function of the subset E ⊆ X. Let $\tilde{β} : X \to [0, 1)$ be a real-valued function defined by $\tilde{β} (x) \equiv 0$ for all x ∈ X. Thus, ∀ x ∈ X, Ω (x)≠ ∅ if and only if $({\tilde{P}}_{x})_{\tilde{β} (x)} \neq \emptyset$ . Now, we show that $\tilde{P}$ is $P_{C}$ -matched. In fact, let x ∈ X be any given such that $({\tilde{P}}_{x})_{\tilde{β} (x)} \neq \emptyset$ . Then by (iii), we can see that there is an i (x) ∈ Ω such that x ∈ int_X {z ∈ X : (P_{i(x)_z}) _{p_i(z)} ≠ ∅}. Next, by (ii), there exist an open neighborhood N_x of x in X, a real-valued function σ : X → [0, 1), and two fuzzy mappings $ρ_{i (x)}^{x}, γ_{i (x)}^{x} : X \to F (X_{i (x)})$ such that

(a) ∀ z ∈ X, $(ρ_{i (x)_{z}}^{x})_{σ (z)}$ is an convex subset of X_i(x),

(b) ∀ z ∈ N_x, $(P_{i (x)_{z}})_{p_{i (x)} (z)} \subseteq (ρ_{i (x)_{z}}^{x})_{σ (z)}$ and $z_{i (x)} \notin (ρ_{i (x)_{z}}^{x})_{σ (z)}$ ,

(d) ∀ y ∈ X_i(x), ${z \in X : γ_{i (x)_{z}}^{x} (y) > σ (z)}$ is an open subset of X,

(e) ∀ ${x_{1}, \dots, x_{n}} \in 〈 {x \in X : ({\tilde{P}}_{x})_{\tilde{β} (x)} \neq \emptyset} 〉$ with i (x₁) = i (x₂) = … = i (x_n), $\begin{matrix} {z \in ⋂_{j = 1}^{n} N_{x_{j}} : ⋂_{j = 1}^{n} (γ_{i (x_{j})_{z}}^{x_{j}})_{σ (z)} \neq \emptyset} \\ = {z \in ⋂_{j = 1}^{n} N_{x_{j}} : ⋂_{j = 1}^{n} (ρ_{i (x_{j})_{z}}^{x_{j}})_{σ (z)} \neq \emptyset} . \end{matrix}$ Without loss of generality, let N_x ⊆ int_X {z ∈ X : (P_{i(x)_z}) _{p_i(z)} ≠ ∅}, from which, for all z ∈ N_x, we can deduce that (P_{i(x)_z}) _{p_i(z)}≠ ∅ and so, i (x) ∈ Ω (z). Let x ∈ X such that $({\tilde{P}}_{x})_{\tilde{β} (x)} \neq \emptyset$ . Also, we assume that ρ_i(x)′, γ_i(x)′ : X → 2^X are two set-valued mappings defined by $ρ_{i (x)}' (z) = \prod_{j \in I, j \neq i (x)} X_{j} \otimes (ρ_{i (x)_{z}}^{x})_{σ (z)} \forall z \in X,$ $γ_{i (x)}' (z) = \prod_{j \in I, j \neq i (x)} X_{j} \otimes (γ_{i (x)_{z}}^{x})_{σ (z)} \forall z \in X .$ Let ζ : X → [0, 1) be a real-valued function defined by ς (x) ≡0 for every x ∈ X. Let us define fuzzy mappings $ρ_{i (x)}, γ_{i (x)} : X \to F (X)$ by ρ_{i(x)_z} = χ_{ρ_i(x)′(z)} and by γ_{i(x)_z} = χ_{γ_i(x)′(z)} for every z ∈ X, where χ_E is the characteristic function of the subset E ⊆ X. Then we have

(1) by (a), ∀ z ∈ X, $(ρ_{i (x)_{z}}^{x})_{σ (z)}$ is a convex subset of X_i(x). Therefore, ρ_i(x)′ (z) = (ρ_{i(x)_z}) _ς(z) is naturally a convex subset of X,

(2) by (b), ∀ z ∈ N_x, we get $\begin{matrix} ({\tilde{P}}_{z})_{\tilde{β} (z)} = P (z) & = & ⋂_{i \in I (z)} P_{i}^{'} (z) \subseteq P_{i (x)}^{'} (z) \\ = & \prod_{j \neq i (x), j \in I} X_{j} \otimes (P_{i (x)_{z}})_{p_{i (x)} (z)} \\ \subseteq & \prod_{j \neq i (x), j \in I} X_{j} \otimes (ρ_{i (x)_{z}}^{x})_{σ (z)} \\ = & (ρ_{i (x)_{z}})_{ς (z)} \end{matrix}$ and z = z_{j∈Ω,j≠i(x)} ⊗ z_i(x) ∉ (ρ_{i(x)_z}) _ς(z),

(3) by (c), ∀ z ∈ X, (γ_{i(x)_z}) _ς(z) ⊆ (ρ_{i(x)_z}) _ς(z),

(4) ∀ y ∈ X, since ${z \in X : γ_{i (x)_{z}} (y) > ς (z)} = {z \in X : γ_{i (x)_{z}}^{x} (y_{i (x)}) > σ (z)}$ , it follows from (d) that {z ∈ X : γ_{i(x)_z} (y) > ς (z)} is an open subset of X,

(5) ∀ $A \in 〈 {x \in X : ({\tilde{P}}_{x})_{\tilde{β} (x)} \neq \emptyset} 〉$ , we set ⋃ {i (x) : x ∈ A} = {i₁, …, i_m}, where i₁, …, i_m are different. For every k = 1, 2, …, m, set A_k = {x ∈ A : i (x) = i_k}. Then ∀ z ∈ X, we know that the following relationship holds.

$\begin{matrix} ⋂_{x \in A} (γ_{i (x)_{z}})_{ς (z)} & = & ⋂_{x \in A} \prod_{j \in I, j \neq i (x)} X_{j} \otimes (γ_{i (x)_{z}}^{x})_{σ (z)} \\ = & ⋂_{k = 1}^{m} \prod_{j \in I, j \neq i_{k}} X_{j} \otimes (⋂_{x \in A_{k}} (γ_{i (x)_{z}}^{x})_{σ (z)}) . \end{matrix}$

Thus, for each z ∈ ⋂ _x∈AN_x, if z∈ ⋂ _x∈A (γ_{i(x)_z}) _ς(z) = ∅, then there exists n ∈ {1, …, m} such that $⋂_{x \in A_{n}} (γ_{i (x)_{z}}^{x})_{σ (z)} = \emptyset$ . By (e), we have $⋂_{x \in A_{n}} (ρ_{i (x)_{z}}^{x})_{σ (z)} = \emptyset$ . Therefore, we get $\begin{matrix} ⋂_{x \in A} (ρ_{i (x)_{z}})_{ς (z)} & = & ⋂_{x \in A} \prod_{j \in I, j \neq i (x)} X_{j} \otimes (ρ_{i (x)_{z}}^{x}) (z) \\ = & ⋂_{k = 1}^{n} \prod_{j \in I, j \neq i_{k}} X_{j} \otimes (⋂_{x \in A_{k}} (ρ_{i (x)_{z}}^{x}) (z)) \\ = & \emptyset . \end{matrix}$

This fact, together with (3), leads to the following: $\begin{matrix} {z \in ⋂_{x \in A} N_{x} : ⋂_{x \in A} (γ_{i (x)_{z}})_{ς (z)} \neq \emptyset} \\ = {z \in ⋂_{x \in A} N_{x} : ⋂_{x \in A} (ρ_{i (x)_{z}})_{ς (z)} \neq \emptyset}, \end{matrix}$ which shows that $\tilde{P}$ is $P_{C}$ -matched. It follows from (iv) that ∀ M ∈ 〈X〉, ∀ i ∈ Ω and ∀ π_i (M) = M_i ∈ 〈X_i〉, there is a a nonempty subset C_{M
_i} of X_i such that C_{M
_i} is compact convex and M_i ⊂ C_{M
_i}, where π_i denotes the projection of M onto M_i. Thus, we can conclude that $C_{M} = \prod_{i \in Ω} C_{M_{i}} \subseteq X$ is nonempty compact convex and M ⊆ C_M. Let x ∈ X be any given. If ⋂_i∈Ω {z∈ X : P_{i
_z} (x_i) > p_i (z)} = ∅, then it is easy to see that $⋂_{i \in Ω} {z \in X : P_{i_{z}} (x_{i}) > p_{i} (z)} \subseteq {z \in X : {\tilde{P}}_{z} (x) > \tilde{β} (z)}$ . If ⋂_i∈Ω {z∈ X : P_{i
_z} (x_i) > p_i (z)} ≠ ∅, then we have $\begin{matrix} ⋂_{i \in Ω} {z \in X : P_{i_{z}} (x_{i}) > p_{i} (z)} \\ = {z \in X : P_{i_{z}} (x_{i}) > p_{i} (z), \forall i \in Ω} \\ = {z \in X : P_{i_{z}} (x_{i}) > p_{i} (z), \forall i \in Ω (z)} \\ = {z \in X : x \in P_{i}^{'} (z), \forall i \in Ω (z)} \\ = {z \in X : x \in ⋂_{i \in Ω (z)} P_{i}^{'} (z)} \\ = {z \in X : x \in P (z)} \\ = {z \in X : {\tilde{P}}_{z} (x) > \tilde{β} (z)} . \end{matrix}$ Hence, in both cases, by (iv), either ∀ M ∈ 〈X〉, there is a nonempty subset C_M of X such that C_M is compact convex, M ⊆ C_M, and $\begin{matrix} C_{M} \ K & \subseteq & ⋃_{x \in C_{M}} {int}_{C_{M}} (C_{M} ⋂ (⋂_{i \in Ω} {z \in X : \\ (P_{i_{z}}) (x_{i}) > p_{i} (z)})) \\ \subseteq & ⋃_{x \in C_{M}} {int}_{C_{M}} (C_{M} ⋂ ({z \in X : \\ {\tilde{P}}_{z} (x) > \tilde{β} (z)})), \end{matrix}$ or there exists x₀ = (x_0i) _i∈Ω ∈ X such that $\begin{matrix} X \ K & \subseteq & {int}_{X} (⋂_{i \in Ω} {z \in X : (P_{i_{z}}) (x_{0 i}) > p_{i} (z)}) \\ \subseteq & {int}_{X} {z \in X : {\tilde{P}}_{z} (x_{0}) > \tilde{β} (z)} . \end{matrix}$ Thus, Theorem 4.2 guarantees that there exists at least $\hat{x} \in K$ such that $({\tilde{P}}_{\hat{x}})_{\tilde{β} (\hat{x})} = \emptyset$ , which means that $Ω (\hat{x}) = \emptyset$ . Therefore, $(P_{i_{\hat{x}}})_{p_{i} (\hat{x})} = \emptyset$ for every i ∈ Ω, that is, Γ has at least an equilibrium point in K. The proof of Theorem 5.2 is finished.□

Remark 5.1. Theorem 5.2 generalizes Theorem 3.3 of Lu et al. [36] in the following several aspects: (a) from noncompact Hadamard manifolds to noncompact CAT(0) spaces; (b) from crisp cases to fuzzy cases; (c) (iii) of Theorem 5.2 is weaker than (i) of Theorem 3.3 due to Lu et al. [36]; (d) even if we strengthen the CAT(0) space in Theorem 5.2 to be a Hadamard manifold, it follows from Remark 4.1 that the condition that each P_i is $P_{C}$ -matched is still weaker than the condition that every G_i possesses an $H_{E}$ -combination in Theorem 3.3 of Lu et al. [36]. Moreover, Theorem 5.2 extends Theorem 2.4 due to Toussaint [43] and Theorem 4.2 due to Tan and Yuan [33] to fuzzy settings and noncompact CAT(0) spaces without linear structure.

By Theorem 5.2, we derive the following existence theorem of Nash equilibrium points for generalized fuzzy noncooperative games in noncompact CAT(0) spaces.

Theorem 5.3. Let Ω be a finite index set, Γ = (X_i, A_i, B_i, P_i, a_i, b_i, p_i) _i∈Ω be a generalized fuzzy noncooperative game, and K be a nonempty compact subset of $X = \prod_{i \in Ω} X_{i}$ . Assume that the following conditions are satisfied:

(i) ∀ i ∈ Ω, X_i is a complete locally compact CAT(0) space;

(ii) ∀ i ∈ Ω and ∀ x ∈ X, (A_{i
_x}) _{b_i(x)} is nonempty convex and $(A_{i_{x}})_{b_{i} (x)} \subseteq {\bar{(B_{i_{x}})}}_{a_{i} (x)}$ ;

(iii) ∀ i ∈ Ω, there is a real-valued function α_i : X → [0, 1) and a fuzzy mapping $ϑ_{i} : X \to F (X_{i})$ such that (ϑ_{i
_x}) _{α_i(x)} = (A_{i
_x}) _{b_i(x)} ⋂ (P_{i
_x}) _{p_i(x)} if $x_{i} \in {\bar{(B_{i_{x}})}}_{a_{i} (x)}$ and (ϑ_{i
_x}) _{α_i(x)} = (A_{i
_x}) _{b_i(x)} if $x_{i} \notin {\bar{(B_{i_{x}})}}_{a_{i} (x)}$ ;

(iv) ∀ i ∈ Ω, V_i = {x ∈ X : (A_{i
_x}) _{b_i(x)} ⋂ (P_{i
_x}) _{p_i(x)} ≠ ∅} is an open subset of X;

(v) ∀ i ∈ Ω, there exist a real-valued function β_i : X → [0, 1) and a $P_{C}$ -strongly matched fuzzy mapping $ρ_{i} : X \to F (X_{i})$ such that (A_{i
_x}) _{b_i(x)} ⋂ (P_{i
_x}) _{p_i(x)} = (ρ_{i
_x}) _{β_i(x)} for all x ∈ X;

(vi) ∀ i ∈ Ω and ∀ y ∈ X_i, {x ∈ X : A_{i
_x} (y) > b_i (x)} is an open subset of X;

(vii) either (a)∀ i ∈ Ω and ∀ M_i ∈ 〈X_i〉, there is a nonempty subset C_{M
_i} of X_i such that C_{M
_i} is compact convex, M_i ⊂ C_{M
_i}, and C_M \ K ⊆ ⋃ _{x∈C_M}int_{C
_M} (C_M ⋂ (⋂ _i∈Ω {z ∈ X : (A_{i
_z}) (x_i) > b_i (z) and (P_{i
_z}) (x_i) > p_i (z)})), where $C_{M} = \prod_{i \in Ω} C_{M_{i}}$ , or (b) there exists x₀ = (x_0i) _i∈Ω ∈ X such that X \ K ⊆ int_X (⋂ _i∈Ω {z ∈ X : (A_{i
_z}) (x_0i) > b_i (z) and (P_{i
_z}) (x_0i) > p_i (z)}).

Then Γ has at least an equilibrium point in K.

Proof. ∀ i ∈ Ω, let us set $W_{i} = {x \in X : x_{i} \notin {\bar{(B_{i_{x}})}}_{a_{i} (x)}}$ . Then W_i is an open subset of X. It follows from (ii) and (iii) that we get the following: $\begin{matrix} {x \in X : ϑ_{i} (x) \neq \emptyset} \\ = {x \notin W_{i} : ϑ_{i} (x) \neq \emptyset} ⋃ {x \in W_{i} : ϑ_{i} (x) \neq \emptyset} \\ = W_{i} ⋃ {x \notin W_{i} : (A_{i_{x}})_{b_{i} (x)} ⋂ (P_{i_{x}})_{p_{i} (x)} \neq \emptyset} \\ = W_{i} ⋃ [(X \ W_{i}) ⋂ V_{i}] \\ = W_{i} ⋃ V_{i} . \end{matrix}$ From (iv), combined with the fact that W_i is an open subset of X for every i ∈ Ω, we have ⋃_i∈Ω {x ∈ X : ϑ_i (x) ≠ ∅} = ⋃ _i∈Ωint_X {x ∈ X : ϑ_i (x) ≠ ∅}. By (v), ∀ i ∈ Ω and ∀ x ∈ V_i = {x ∈ X : (ρ_{i
_x}) _{β_i(x)} ≠ ∅}, there exist a real-valued function σ : X → [0, 1) and a $P_{C}$ -strong matchment $(υ_{i}^{x}, τ_{i}^{x}, N_{x})$ of ρ_i at x, where N_x is an open neighborhood of x in X and $υ_{i}^{x}, τ_{i}^{x} : X \to F (X_{i})$ are two fuzzy mappings such that the following holds:

(a) ∀ z ∈ X, $(υ_{i_{z}}^{x})_{σ (z)}$ is convex,

(b) ∀ z ∈ N_x, $z_{i} \notin (υ_{i_{z}}^{x})_{σ (z)}$ ,

(c) ∀ z ∈ X, $(τ_{i_{z}}^{x})_{σ (z)} \subseteq (A_{i_{z}})_{b_{i} (z)} ⋂ (P_{i_{z}})_{p_{i} (z)} \subseteq (υ_{i_{z}}^{x})_{σ (z)}$ ,

(d) ∀ y ∈ X_i, the set ${z \in X : τ_{i_{z}}^{x} (y) > σ (z)}$ is an open subset of X,

(e) ∀ A ∈ 〈V_i〉, we have $\begin{matrix} {z \in ⋂_{x \in A} N_{x} : ⋂_{x \in A} (τ_{i_{z}}^{x})_{σ (z)} \neq \emptyset} \\ = {z \in ⋂_{x \in A} N_{x} : ⋂_{x \in A} (υ_{i_{z}}^{x})_{σ (z)} \neq \emptyset} . \end{matrix}$

Two set-valued mappings Ψ_i,x, Φ_i,x : X → 2X_i are defined by setting, ∀ i ∈ Ω and ∀ z ∈ X, $\begin{matrix} Ψ_{i, x} (z) = {\begin{matrix} (τ_{i_{z}}^{x})_{σ (z)}, & if z \notin W_{i}, \\ (A_{i_{z}})_{b_{i} (z)}, & if z \in W_{i} . \end{matrix} \end{matrix}$ $\begin{matrix} Φ_{i, x} (z) = {\begin{matrix} (υ_{i_{z}}^{x})_{σ (z)}, & if z \notin W_{i}, \\ (A_{i_{z}})_{b_{i} (z)}, & if z \in W_{i} . \end{matrix} \end{matrix}$ Further, let us define two fuzzy mappings ${\tilde{Ψ}}_{i}^{x}, {\tilde{Φ}}_{i}^{x} : X \to F (X_{i})$ by ${\tilde{Ψ}}_{i_{z}}^{x} = χ_{Ψ_{i, x} (z)}$ and by ${\tilde{Φ}}_{i_{z}}^{x} = χ_{Φ_{i, x} (z)}$ for every z ∈ X, where χ_E is the characteristic function of E ⊆ X_i. Let ζ_i : X → [0, 1) be a real-valued function defined by ζ_i (x) ≡0 for every i ∈ Ω and every x ∈ X. Now, ∀ x ∈ X with (ϑ_{i
_x}) _{α_i(x)}≠ ∅, we define $\begin{matrix} U (x) = {\begin{matrix} W_{i}, & if x \in W_{i}, \\ N_{x}, & if x \notin W_{i} . \end{matrix} \end{matrix}$ Then, ∀ x ∈ X with (ϑ_{i
_x}) _{α_i(x)}≠ ∅, we can see that U (x) is an open subset of X and x ∈ U (x). Next, ∀ i ∈ Ω, we prove that ϑ_i is $P_{C}$ -strongly matched as follows.

(I) by (ii) and (a), ∀ z ∈ X, $({\tilde{Φ}}_{i_{z}}^{x})_{ζ_{i} (z)} = Φ_{i, x} (z)$ is a convex subset of X_i,

(II) ∀ z ∈ U (x), $z_{i} \notin ({\tilde{Φ}}_{i_{z}}^{x})_{ζ_{i} (z)} = Φ_{i, x} (z)$ . In fact, if x ∉ W_i, then we have U (x) = N (x) and so, ∀ z ∈ U (x) = N (x) with z ∉ W_i, it follows from (b) and the definition of Φ_i,x that $z_{i} \notin ({\tilde{Φ}}_{i_{z}}^{x})_{ζ_{i} (z)} = Φ_{i, x} (z)$ ; ∀ z ∈ U (x) = N (x) with z ∈ W_i, by the definitions of Φ_i,x and W_i, we have $z_{i} \notin ({\tilde{Φ}}_{i_{z}}^{x})_{ζ_{i} (z)} = Φ_{i, x} (z)$ . If x ∈ W_i, then U (x) = W_i and so, ∀ z ∈ U (x) = W_i, by the definitions of Φ_i,x and W_i, we have $z_{i} \notin ({\tilde{Φ}}_{i_{z}}^{x})_{ζ_{i} (z)} = Φ_{i, x} (z)$ ,

(III) ∀ z ∈ X, $({\tilde{Ψ}}_{i_{z}}^{x})_{ζ_{i} (z)} \subseteq (ϑ_{i_{z}})_{α_{i} (z)} \subseteq ({\tilde{Φ}}_{i_{z}}^{x})_{ζ_{i} (z)}$ by (c),

(IV) ∀ y ∈ X_i, we have the following: $\begin{matrix} {z \in X : {\tilde{Ψ}}_{i_{z}}^{x} (y) > ζ_{i} (z)} \\ = {z \notin W_{i} : τ_{i_{z}}^{x} (y) > σ (z)} \\ ⋃ {z \in W_{i} : A_{i_{z}} (y) > b_{i} (z)} \\ = [W_{i} ⋃ {z \in X : τ_{i_{z}}^{x} (y) > σ (z)}] \\ ⋂ {z \in X : A_{i_{z}} (y) > b_{i} (z)} . \end{matrix}$ Therefore, by (vi), (d) and the fact that W_i is an open subset of X, we can see that the set ${z \in X : {\tilde{Ψ}}_{i_{z}}^{x} (y) > ζ_{i} (z)}$ is an open subset of X for all y ∈ X_i. Hence, $({\tilde{Ψ}}_{i}^{x}, {\tilde{Φ}}_{i}^{x}, U (x))$ forms a $P_{C}$ -strong matchment of ϑ_i at x,

(V) ∀ F ∈ 〈 {x ∈ X : (ϑ_{i
_x}) _{α_i(x)} ≠ ∅} 〉, let F = F₁ ⋃ F₂, where F₁ = F ⋂ W_i and F₂ = F ⋂ (X \ W_i).

Case 1. If F₁ =∅, then by (e), we have $\begin{matrix} {z \in ⋂_{x \in F} U (x) : ⋂_{x \in F} ({\tilde{Ψ}}_{i_{z}}^{x})_{ζ_{i} (z)} \neq \emptyset} \\ = {z \in ⋂_{x \in F_{2}} U (x) ⋂ W_{i} : ⋂_{x \in F_{2}} ({\tilde{Ψ}}_{i_{z}}^{x})_{ζ_{i} (z)} \neq \emptyset} \\ ⋃ {z \in ⋂_{x \in F_{2}} U (x) ⋂ (X \ W_{i}) : ⋂_{x \in F_{2}} ({\tilde{Ψ}}_{i_{z}}^{x})_{ζ_{i} (z)} \neq \emptyset} \\ = {z \in ⋂_{x \in F_{2}} U (x) ⋂ W_{i} : ⋂_{x \in F_{2}} (A_{i_{z}})_{b_{i} (z)} \neq \emptyset} \\ ⋃ {z \in ⋂_{x \in F_{2}} U (x) ⋂ (X \ W_{i}) : ⋂_{x \in F_{2}} (τ_{i_{z}}^{x})_{σ (z)} \neq \emptyset} \\ = {z \in ⋂_{x \in F_{2}} U (x) ⋂ W_{i} : ⋂_{x \in F_{2}} (A_{i_{z}})_{b_{i} (z)} \neq \emptyset} \\ ⋃ {z \in ⋂_{x \in F_{2}} U (x) ⋂ (X \ W_{i}) : ⋂_{x \in F_{2}} (υ_{i_{z}}^{x})_{σ (z)} \neq \emptyset} \\ = {z \in ⋂_{x \in F_{2}} U (x) ⋂ W_{i} : ⋂_{x \in F_{2}} ({\tilde{Φ}}_{i_{z}}^{x})_{ζ_{i} (z)} \neq \emptyset} \\ ⋃ {z \in ⋂_{x \in F_{2}} U (x) ⋂ (X \ W_{i}) : ⋂_{x \in F_{2}} ({\tilde{Φ}}_{i_{z}}^{x})_{ζ_{i} (z)} \neq \emptyset} \\ = {z \in ⋂_{x \in F} U (x) : ⋂_{x \in F} ({\tilde{Φ}}_{i_{z}}^{x})_{ζ_{i} (z)} \neq \emptyset} . \end{matrix}$

Case 2. If F₁≠ ∅, then we have $\begin{matrix} {z \in ⋂_{x \in F} U (x) : ⋂_{x \in F} ({\tilde{Ψ}}_{i_{z}}^{x})_{ζ_{i} (z)} \neq \emptyset} \\ = {z \in ⋂_{x \in F_{1}} U (x) ⋂ (⋂_{x \in F_{2}} U (x)) : ⋂_{x \in F} ({\tilde{Ψ}}_{i_{z}}^{x})_{ζ_{i} (z)} \neq \emptyset} \\ = {z \in ⋂_{x \in F_{2}} U (x) ⋂ W_{i} : ⋂_{x \in F} ({\tilde{Ψ}}_{i_{z}}^{x})_{ζ_{i} (z)} \neq \emptyset} \\ = {z \in ⋂_{x \in F_{2}} U (x) ⋂ W_{i} : ⋂_{x \in F} ({\tilde{Φ}}_{i_{z}}^{x})_{ζ_{i} (z)} \neq \emptyset} \\ = {z \in ⋂_{x \in F} U (x) : ⋂_{x \in F} ({\tilde{Φ}}_{i_{z}}^{x})_{ζ_{i} (z)} \neq \emptyset} \end{matrix}$ by the fact that $({\tilde{Ψ}}_{i_{z}}^{x})_{ζ_{i} (z)} = ({\tilde{Φ}}_{i_{z}}^{x})_{ζ_{i} (z)} = (A_{i_{z}})_{b_{i} (z)}$ for each z ∈ W_i. From this, we can see that for each i ∈ Ω, ϑ_i is $P_{C}$ -strongly matched and so, it is $P_{C}$ -matched. By (iii), we can easily verify that {z ∈ X : A_{i
_z} (x_i) > b_i (z) and P_{i
_z} (x_i) > p_i (z)} ⊆ {z ∈ X : ϑ_{i
_z} (x_i) > α_i (z)} for every i ∈ Ω and every x_i ∈ X_i. Thus, by (vii), we can see that either for every i ∈ Ω and every M_i ∈ 〈X_i〉, there is a nonempty subset C_{M
_i} of X_i such that C_{M
_i} is compact convex, M_i ⊂ C_{M
_i}, and $\begin{matrix} C_{M} \ K \subseteq ⋃_{x \in C_{M}} {int}_{C_{M}} (C_{M} ⋂ (⋂_{i \in Ω} {z \in X : (ϑ_{i_{z}}) (x_{i}) \\ > α_{i} (z)})), \end{matrix}$ where $C_{M} = \prod_{i \in Ω} C_{M_{i}}$ , or there exists x₀ = (x_0i) _i∈Ω ∈ X such that X \ K ⊆ int_X (⋂ _i∈Ω {z ∈ X : (ϑ_{i
_z}) (x_0i) > α_i (z)}). Hence, all the assumptions of Theorem 5.2 are fulfilled. By virtue of Theorem 5.2, there is at least a point $\hat{x} \in K$ such that $(ϑ_{i_{\hat{x}}})_{α_{i} (\hat{x})} = \emptyset$ for every i ∈ Ω. By (ii) and (iii) again, we have ${\hat{x}}_{i} \in {\bar{(B_{i_{\hat{x}}})}}_{a_{i} (\hat{x})}$ and $(A_{i_{\hat{x}}})_{b_{i} (\hat{x})} ⋂ (P_{i_{\hat{x}}})_{p_{i} (\hat{x})} = \emptyset$ for every i ∈ Ω. The proof of Theorem 5.3 is complete.□

Remark 5.2. Theorem 5.3 describes the existence of Nash equilibrium points in generalized fuzzy noncooperative games under fuzzy environments and strategy spaces without any linear structure. Therefore, in this sense, Theorem 5.3 extends Theorem 4.3 of Tan and Yuan [33], Theorem 4.3 of Yuan [35], and Theorem 2.5 due to Toussaint [43].

6 Concluding remarks

In this paper, we have dealt with the existence of fuzzy fixed-points in the setting of noncompact CAT(0) spaces with applications to existence theorems of fuzzy maximal element points and existence theorems of Nash equilibrium points for generalized fuzzy noncooperative games and fuzzy noncooperative qualitative games. Firstly, we utilize the KKM principle in CAT(0) spaces to prove a fuzzy fixed-point theorem. Subsequently, we use the obtained fuzzy fixed-point theorem to obtain several existence theorems of fuzzy maximal element points when the fuzzy mapping satisfies certain matching properties. Finally, as applications of fuzzy maximal element point theorems, we investigate the existence of Nash equilibrium points for generalized fuzzy noncooperative games and fuzzy noncooperative qualitative games, in which, the strategy spaces are noncompact CAT(0) spaces. In our opinion, in the future work, the following issues are worthy of attention: (a) further transform CAT(0) space and give it a uniform convex structure and next then, try to use KKM principles and topological methods to prove a fixed point theorem for fuzzy set-valued mappings in the setting of CAT(0) spaces. By using the degenerate form of this fixed point theorem, we consider studying the existence of the solution of the integral equation; (b) further generalize CAT(0) spaces, for example, whether it can be considered studying related issues in the setting of CAT_p(0) spaces introduced by Shukri [44]; (c) under the same or broader space framework, extend the generalized fuzzy noncooperative games with finite players in this paper to the generalized fuzzy noncooperative games with infinite players, and further analyze the existence mechanism of Nash equilibrium points for these extended generalized fuzzy noncooperative games; (d) study the fuzzy fixed-point theorems for set-valued mappings and their applications to fuzzy maximal elements and the existence of Nash equilibrium points for the generalized fuzzy noncooperative games with finite or infinite players in the framework of FMS-tripled fuzzy metric spaces introduced by Tian et al [22].

Footnotes

Acknowledgments

The authors would like to thank the editor and anonymous referees for their valuable suggestions and comments which improved the exposition of this paper. This work was supported by the Planning Foundation for Humanities and Social Sciences of Ministry of Education of China (No. 18YJA790058).

Conflicts of interest

The authors declare no conflict of interest.

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