Existence and stability of solutions for maximal element theorem on Hadamard manifolds with applications

Nonlinear Analysis 75 (2012) 516–525

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Existence and stability of solutions for maximal element theorem on Hadamard manifolds with applications✩ Zhe Yang ∗ , Yong Jian Pu College of Economics and Business Administration, Chongqing University, Chongqing, 400044, China

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Article history: Received 1 June 2011 Accepted 26 August 2011 Communicated by Enzo Mitidieri Keywords: Maximal element theorem Hadamard manifolds Essential set Essential component Variational relation problems Weakly Pareto–Nash equilibrium

abstract In this paper, maximal element theorem on Hadamard manifolds is established. First, we prove the existence of solutions for maximal element theorem on Hadamard manifolds. Further, we prove that most of problems in maximal element theorem on Hadamard manifolds (in the sense of Baire category) are essential and that, for any problem in maximal element theorem on Hadamard manifolds, there exists at least one essential component of its solution set. As applications, we study existence and stability of solutions for variational relation problems on Hadamard manifolds, and existence and stability of weakly Pareto–Nash equilibrium points for n-person multi-objective games on Hadamard manifolds. © 2011 Elsevier Ltd. All rights reserved.

1. Introduction By establishing the existence of selection functions for set-valued mappings with open fibers in product spaces, Deguire and Lassonde gave some fixed-point theorems in product spaces for both compact and non-compact domains (see [1]). It is well known that each fixed-point theorem has an equivalent version of a maximal element, we recall that a point x ∈ X is said to be a maximal element of a mapping F : X ⇒ Y if F (x) = ∅, where X and Y are two topological spaces. The existence of maximal elements for mappings in topological (vector) spaces and its important applications to mathematical models of economy have been studied by many authors in both mathematics and economics, for example; see [2–7]. On the other hand, in the last few years, several important concepts of nonlinear analysis have been extended from Euclidean space to a Riemannian manifold setting in order to go further in the study of the convex theory, the fixed point theory, the variational inequality and related topics. In fact, a manifold is not a linear space. In this setting the linear space is replaced by a Hadamard manifold and the line segment by a geodesic, see [8,9]. In 2003, Nemeth [10] introduced and studied the variational inequality on Hadamard manifolds. Some existence theorems of solutions for the variational inequality are proved, see [11,12]. Readers may consult [13–15], which are closely related to the present work. In this paper, we study maximal element theorem on Hadamard manifolds from the viewpoint of the existence and stability. The method of essential solutions has been widely used in various fields recently. It plays a crucial role in the study of stability of solutions including optimal solutions, Nash equilibria, fixed points, etc. (see [16–31]). The notation of an essential solution for fixed points was first introduced in [16], which means that, for a fixed point x of a mapping f , if each mapping sufficiently near to f has a fixed point arbitrarily near to x, x is said to be essential. However, it is not true that any continuous mapping has at least one essential fixed point, even though the space has fixed point property. Instead of

✩ Supported by NSFC (70661001) and Chongqing University Postgraduates, Science and Innovation Fund (200911B0A0050321).

∗

Corresponding author. Tel.: +86 13629723516; fax: +86 02365102864. E-mail address: [email protected] (Z. Yang).

0362-546X/$ – see front matter © 2011 Elsevier Ltd. All rights reserved. doi:10.1016/j.na.2011.08.053

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considering the essential solution, Kinoshita introduced the notion of essential components of the set of fixed points and proved that, for any continuous mapping of the Hilbert cube into itself, there exists at least one essential component of the set of its fixed points (see [19]). The principal stimulus for a paper of this sort is the influential work of Kohlberg and Mertens for noncooperative games. Kohlberg and Mertens introduced the notions of stable set and essential components of Nash equilibria and proved that every finite n-person noncooperative game has at least one essential connected component of the set of its Nash equilibrium points (see [20]). Later, Yu and Xiang brought forward the notion of essential components of the set of Ky Fan points and deduced that every infinite n-person noncooperative game with concave payoff functions has at least one essential component of the set of its equilibrium points by method of essential solutions (see [27]). Motivated and inspired by research works mentioned above, we study existence and stability of maximal element theorem on Hadamard manifolds. As applications, we study existence and stability of solutions for variational relation problems on Hadamard manifolds, and existence and stability of weakly Pareto–Nash equilibrium points for n-person multiobjective games on Hadamard manifolds. 2. Preliminaries First we recall some definitions in [32]. Definition 2.1. A Hadamard manifold M is a simply connected complete Riemannian manifold of non-positive sectional curvature. The exponential mapping expp : Tp (M ) −→ M is defined by expp v = rv (1), where rv is the geodesic defined by its position p and velocity v at p. Easily, we know that (i) the exponential mapping and its inverse are continuous on Hadamard manifolds. (ii) For any 1 p, q ∈ M, the minimal geodesic joining p to q is expp t exp− p q for t ∈ [0, 1]. For any given o ∈ M, t1 , t2 ∈ [0, 1] with 1 −1 t1 + t2 = 1, it is easy to see that expo (t1 exp− o p + t2 expo q) is also a minimal geodesic.

Definition 2.2. A set K ⊂ M is said to be geodesic convex if for any p, q ∈ K , the minimal geodesic joining p to q is contained in K . Definition 2.3. Let o be any given point in M. The geodesic convex hull for a set S ⊂ M, denoted by GcoS, is defined as follows:

 GcoS =

expo



n − i =1

λi expo

−1

  n  −  λi = 1 . xi  ∀x1 , . . . , xn ∈ S ; λ1 , . . . , λn ∈ [0, 1],  i =1

Definition 2.4. Let (X , d) and (Y , ρ) be two metric spaces, a set-valued F : X ⇒ Y is said to be a usco mapping, if F is upper semicontinuous on each X , and F (x) is nonempty compact for all x ∈ X . We also need the following result, which are due to Nemeth [10]. Lemma 2.1 ([10]). If K ⊂ M is nonempty, compact and geodesic convex, then every continuous function f : K −→ K has a fixed point. We also need the following three results, which are due to Fort [16, Theorem 2], where condition (c) is unnecessary and Yang et al. [23, Lemma 3.3], respectively: Lemma 2.2 ([16]). If X is complete and F : X ⇒ Y is a usco mapping, then the set of points, where F is lower semicontinuous, is a dense residual set in X . Lemma 2.3 ([23]). Let X , Y , Z be three metric spaces, S1 : Y ⇒ X and S2 : Z ⇒ X be two set-valued mappings. Suppose that there exists at least one essential component of S1 (y) for each y ∈ Y and there exists a continuous single-valued mapping T : Z −→ Y such that S2 (z ) ⊃ S1 (T (z )) for each z ∈ Z . Then, there exists at least one essential component of S2 (z ) for each z ∈ Z . Lemma 2.4 ([33]). Let X and Y be two topological spaces with Y compact. If F is a closed set-valued mapping from X to Y , then F is upper semicontinuous. The following result is an inequality concerning the Hausdorff metric between two nonempty compact subsets in a metric space, which is due to Yu and Zhou [30, Lemma 3.1], whose applications can be seen in Section 3. Lemma 2.5 ([30]). Let (Y , ρ) be a metric space, K1 and K2 be two nonempty compact subsets of Y , and V1 , V2 be two nonempty, disjoint and open subsets of Y . If h(K1 , K2 ) < ρ(V1 , V2 ), then h (K1 , (K1 \ V2 ) ∪ (K2 \ V1 )) ≤ h(K1 , K2 ), where h is the Hausdorff metric defined on Y .

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3. Existence Throughout this paper, let M be a Hadamard manifold and o be any given point in M. Let x ∈ M and let Tx M denote the tangent space at x to M. We denote by ⟨·, ·⟩x the scalar product on Tx M with the associated norm ‖ · ‖x , where the subscript x is sometimes omitted. By the Hopf–Rinow theorem (see [32]), (M , d) is a complete metric space. Theorem 3.1. Let X be a nonempty, geodesic convex and compact subset of M. Suppose that A : X ⇒ X is a set-valued mapping with the following conditions: (1) For each x ∈ X , x ̸∈ GcoA(x). (2) For each y ∈ X , A−1 (y) = {x ∈ X | y ∈ A(x)} is open in X . Then, there exists x∗ ∈ X such that A(x∗ ) = ∅. Proof. Suppose the contrary, i.e., for any x ∈ X , there exists y ∈ A(x). Then x ∈ A−1 (y). Thus, we have X =



A−1 (y).

y∈X

Since X is nonempty and compact, and A−1 (y) = {x ∈ X | y ∈ A(x)} is open in X for any y ∈ X , then there exists a finite number of A−1 (y1 ), . . . , A−1 (yn ) such that X =

n 

A−1 (yi ).

i=1

Let {αi | i = 1, . . . , n} be the partition of unity subordinate to open covering {A−1 (yi ) | i = 1, . . . , n} of X , i.e., 0 ≤ αi (x) ≤ 1,

n −

αi (x) = 1,

∀x ∈ X , i = 1, . . . , n;

i=1

and if x ̸∈ A−1 (yj ) for some j, then αj (x) = 0. Now we consider a function f : X −→ X , defined by 1 −1 f (x) = expo (α1 (x) exp− o y1 + · · · + αn (x) expo yn ),

∀x ∈ X .

Then f is continuous, and by Lemma 2.1, there∑ exists x ∈ X such that f (x∗ ) = x∗ . ∗ Let I = {i ∈ {1, . . . , n} | αi (x ) > 0}, then i∈I αi (x∗ ) = 1 and yi ∈ A(x∗ ) for all i ∈ I. Thus, we have ∗

 x = f (x ) = expo ∗

∗

 −

αi (x ) expo yi ∗

−1

∈ GcoA(x∗ ),

i∈I

which contradict the fact x ̸∈ GcoA(x), ∀x ∈ X . This completes the proof.



4. Essential stability Let X be a non-empty, compact and geodesic convex subset of a Hadamard manifolds M, Φ be the set

{A : X ⇒ X | A satisfies conditions (1) and (2) of Theorem 3.1}. For any A ∈ Φ , we can define a mapping B : X ⇒ X by B(y) = {x ∈ X | y ̸∈ A(x)}. By Theorem 3.1,we know B(y) is closed in X and there exists x∗ ∈ X such that A(x∗ ) = ∅. We define the set of x∗ by S (A). Since S (A) = y∈X B(y), then it is nonempty and, in fact, compact. So a solution mapping S : Φ −→ K (X ) is well defined, where K (X ) is the set of all nonempty compact subsets of X . For each A, A′ ∈ Φ , we define

ρ(A, A′ ) = sup h(B(y), B′ (y)), y∈X

where h is the Hausdorff distance on K (X ). Clearly, (Φ , ρ) is a metric space. Next, we give the definition of essential stability of solutions for maximal element theorem on Hadamard manifolds. Definition 4.1. For each A ∈ Φ , let x ∈ S (A) if, for any open neighborhood N (x) of x in X , there is a δ > 0 such that for any A′ ∈ Φ with ρ(A, A′ ) < δ , N (x) S (A′ ) ̸= ∅, x is called an essential point of S (A). If all x ∈ S (A) is essential, then A is said to be essential. Definition 4.2. For each A ∈ Φ , let e(A) be a non-empty  closed subset of S (A) if, for any open set U, e(A) ⊂ U, there is a δ > 0 such that for any A′ ∈ Φ with ρ(A, A′ ) < δ , U S (A′ ) ̸= ∅, e(A) is called an essential set of S (A).

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Definition 4.3. For each A ∈ Φ , let an essential subset m(A) ⊂ S (A), m(A) is called a minimal essential set of S (A) if it is a minimal element of the family of essential sets ordered by set inclusion. A component C (A) is called an essential component of S (A) if C (A) is essential. Remark 4.1. It is easy to see that the problem A ∈ Φ is essential if and only if the mapping S : Φ ⇒ X is lower semicontinuous at A. Next we will prove the following lemma. Lemma 4.1. (Φ , ρ) is a complete metric space. Proof. Let {An ∈ Φ } be any Cauchy sequence, then for any ε > 0, there is an n0 ∈ N such that for any n, m > n0 , ρ(An , Am ) < ε, or, supy∈X h(Bn (y), Bm (y)) < ε , which implies that for any y ∈ X , {Bn (y)} is a Cauchy sequence in K (X ). Since X is nonempty compact, (K (X ), h) is complete. Thus, there is a set-valued mapping B : X −→ K (X ) such that h(Bn (y), B(y)) −→ 0 for each y ∈ X . Hence supy∈X h(Bn (y), B(y)) −→ 0. We denote A(x) = {y ∈ X | x ̸∈ B(y)}, then An −→ A under the metric ρ . Next we will prove A ∈ Φ . Since A−1 (y) = {x ∈ X | y ∈ A(x)} = X \ B(y), then A−1 (y) is open in X for all y ∈ X . Now we prove x ̸∈ GcoA(x) for all x ∈ X. that x ∈ GcoA  (x), then there are {y1 , . . . , ym } ⊂ A(x) and tj ≥ 0, j = 1, . . . , m with ∑mSuppose that there exists x ∈X∑such m −1 j=1 tj = 1 such that x = expo j=1 tj expo yj . Since yj ∈ A(x) for all j = 1, . . . , m, then x ̸∈ B(yj ) for all j = 1, . . . , m. Since supy∈X h(Bn (y), B(y)) −→ 0, then x ̸∈ Bn (yj ) for all j = 1, . . . , m and for all n large enough. Therefore, yj ∈ An (x) for −1 n all j = 1, . . . , m and for all n large enough, which implies that x = expo j=1 tj expo yj ∈ GcoA (x). It contradicts the n n fact that A ∈ Φ , i.e., x ̸∈ GcoA (x) for all x ∈ X . Hence A ∈ Φ and (Φ , ρ) is a complete metric space. 

∑m



Lemma 4.2. The solution mapping S : (Φ , ρ) −→ K (X ) is a usco mapping. Proof. By Lemma 2.4, we only prove Graph(S ) is closed. For any {An ∈ Φ } with An −→ A, any xn ∈ S (An ) with xn −→ x, we will prove x ∈ S (A). For any n, xn ∈ S (An ) implies that xn ∈ Bn (y) for any y ∈ X . Since An −→ A, i.e., supy∈X h(Bn (y), B(y)) −→ 0, then x ∈ B(y) for all y ∈ X , i.e., A(x) = ∅, which implies x ∈ S (A). This complete the proof.  Theorem 4.1. There exists a dense residual subset G of Φ such that for each A ∈ G, A is essential. In other words, there are most of the problems in maximal element theorem on Hadamard manifolds, whose solutions are all essential. Proof. Since the metric space (Φ , ρ) is complete by Lemma 4.1, and the solution mapping S : Φ −→ K (X ) is usco by Lemma 4.2, by Lemma 2.2, there is a dense residual subset G of Φ , where S is lower semicontinuous, thus A is essential for each A ∈ G.  Theorem 4.2. For each A ∈ Φ , there exists at least one minimal essential subset of S (A). Proof. By Lemma 4.2, S : (Φ , ρ) −→ K (X ) is a usco mapping, that is, for each open set O ⊃ S (A), there exists δ > 0 such that for any A′ ∈ Φ with ρ(A, A′ ) < δ , O ⊃ S (A′ ). Hence S (A) is an essential set of itself. Let Θ denote the family of all essential sets of S (A) ordered by set inclusion. Then Θ is non-empty and every decreasing chain of elements in Θ has a lower bound (because by the compactness the intersection is in Θ ); therefore, by Zorn’s lemma, Θ has a minimal element and this minimal element is a minimal essential set of S (A).  Theorem 4.3. For each A ∈ Φ , there exists at least one connected minimal essential subset of S (A). Proof. For each A ∈ Φ , let m(A) be a minimal essential subset of S (A). Suppose that m(A) were not connected. Then, there exist two non-empty compact subsets c1 (A), c2 (A) with m(A) = c1 (A) ∪ c2 (A), and there exist two disjoint open subsets V1 , V2 in X such that V1 ⊃ c1 (A), V2 ⊃ c2 (A) and inf {d(x, y) | x ∈ V1 , y ∈ V2 } = ε > 0.  Since m(A) is essential, then, for V1 ∪ V2 ⊃ m(A), there exists 0 < δ ∗ < ε such that S (A′ ) (V1 ∪ V2 ) ̸= ∅ for any A′ ∈ Φ ∗ with ρ(A, A′ ) < δ ∗ . Since m(A) is a minimal essential set of S (A), then neither c1 (A) nor c2 (A) is essential. Thus for δ4 > 0, there exist two A1 , A2 ∈ Φ such that S (A1 ) ∩ V1 = ∅,

S (A2 ) ∩ V2 = ∅,

ρ(A1 , A) <

δ∗ 4

, ρ(A2 , A) <

δ∗

∗ Thus ρ(A1 , A2 ) < δ2 . Since

B1 (y) = {x ∈ X | y ̸∈ A1 (x)}; B2 (y) = {x ∈ X | y ̸∈ A2 (x)},

∀y ∈ X ,

4

.

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we define the mapping A′ : X ⇒ X by B′ (y) = (B1 (y) \ V2 ) ∪ (B2 (y) \ V1 ), A (x) = {y ∈ X | x ̸∈ B (y)}, ′

′

∀y ∈ X ;

∀x ∈ X .

Now we will prove A ∈ Φ and ρ(A, A′ ) < δ ∗ . (I) We obtain that B′ (y) is closed in X for each y ∈ X , then ′

A′−1 (y) = {x ∈ X | y ∈ A′ (x)} = X \ B′ (y) is open in X for all y ∈ X . (II) Now we prove x ̸∈ GcoA′ (x) for all x ∈ X . Suppose that exists x ∈ X such that x ∑ ∈ GcoA′ (x), then  there ∑there m m −1 ′ exist yj ∈ A (x), j = 1, . . . , m and tj ≥ 0, j = 1, . . . , m with j=1 tj expo yj . Since j=1 tj = 1 such that x = expo yj ∈ A′ (x), j = 1, . . . , m, then x ̸∈ B′ (yj ) for all j = 1, . . . , m. Since V1 ∩ V2 = ∅, then X = V1c ∪ V2c , which implies x ̸∈ V1 or x ̸∈ V2 . Without loss of generality, we may assume that x ̸∈ V1 . From x ̸∈ B′ (yj ) = (B1 (yj ) \ V2 ) ∪ (B2 (yj ) \ V1 ) for any j = 1, . . . , m, it follows that x ̸∈ B2 (yj ) \ V1 for any j =  1, . . . , m. Therefore, x ̸∈ B2 (yj ) for any j = 1, . . . , m, i.e., yj ∈ A2 (x) ∑m −1 for any j = 1, . . . , m. Then x = expo j=1 tj expo yj ∈ GcoA2 (x), which contradicts the fact that A2 ∈ Φ , i.e., x ̸∈ GcoA2 (x) for all x ∈ X . Thus A′ ∈ Φ . (III) By Lemma 2.5, for any y ∈ X , we have h(B′ (y), B1 (y)) = h((B1 (y) \ V2 ) ∪ (B2 (y) \ V1 ), B1 (y)) ≤ h(B1 (y), B2 (y)) <

δ∗ 2

.

Hence h(B′ (y), B(y)) ≤ h(B′ (y), B1 (y)) + h(B1 (y), B(y)) <

δ∗ 2

+

δ∗ 4

< δ∗ .

Thus ρ(A, A′ ) < δ ∗ .    ′ ′ ′ Thus (S (A ) V1 ) (S (A′ ) V (V1 ∪ V2 ) ̸= ∅. We assume Then 2 ) = S (A )  S (A ) V1 ̸= ∅ without loss of generality. ′ there exists x ∈ V1 such that x ∈ y∈X B (y), which implies that x ∈ y∈X B1 (y) and x ∈ V1 by the definition of B′ . Hence   S (A1 ) V1 ̸= ∅, which contradicts the fact that S (A1 ) V1 = ∅. Thus m(A) is connected. This complete the proof.  Theorem 4.4. For each A ∈ Φ , there exists at least one essential component of S (A). Proof. By Theorem 4.3, there exists at least one connected minimal essential subset m(A) of S (A). So there is a component C of S (A) such that m(A) ⊂ C . It is obvious that C is essential. Thus C is an essential component.  5. Application (I): variational relation problems on Hadamard manifolds In this section, we study variational relation problems on Hadamard manifolds. Khanh and Luc [34,35] introduced a more general model of equilibrium problems which is called a variational relation problem (VRP for short). Let A, B and C be nonempty sets, S1 : A ⇒ A, S2 : A ⇒ B, T : A × B ⇒ C be three set-valued mappings with nonempty values and R(a, b, c ) be a relation linking elements a ∈ A, b ∈ B and c ∈ C . (VRP) Find a∗ ∈ A such that (i) a∗ ∈ S1 (a∗ ); (ii) R(a∗ , b, c ) holds for any b ∈ S2 (a∗ ) and any c ∈ T (a∗ , b). For variational relation problems, readers may consult [36–41]. Definition 5.1 ([34,35]). Let A and B be nonempty subsets of topological spaces E1 and E2 , respectively, and R(a, b) be a relation linking a ∈ A and b ∈ B. For each fixed b ∈ B, we say that R(·, b) is closed in the first variable, if for every net {aα } converges to some a and R(aα , b) holds for any α , then the relation R(a, b) holds. Next, we prove the existence of solutions for variational relation problems on Hadamard manifolds. Theorem 5.1. Let X , Y be two non-empty, compact and geodesic convex subsets of two Hadamard manifolds M1 , M2 . S1 : X ⇒ X , S2 : X ⇒ X and S3 : X × X ⇒ Y are three set-valued mappings with nonempty values. Let R(x, y, z ) be a relation linking elements x ∈ X , y ∈ X and z ∈ Y . Assume that (i) F := {x ∈ X | x ∈ S1 (x)} is closed; (ii) Gco(S2 (x)) ⊂ S1 (x) for any x ∈ X , and S2−1 (y) is open in X for any y ∈ X ; (iii) for any fixed y ∈ X , S3 (·, y) is lower semicontinuous;

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(iv) for any fixed y ∈ X , R(·, y, ·) is closed; (v) for any finite subset {x1 , . . . , xn } of X and any x ∈ Gco{x1 , . . . , xn }, there exists i ∈ {1, . . . , n} such that R(x, xi , z ) holds for any z ∈ S3 (x, xi ). Then there exists x∗ ∈ X such that x∗ ∈ S1 (x∗ ) and R(x∗ , y, z ) for any y ∈ S2 (x∗ ) and for any z ∈ S3 (x∗ , y). Proof. (I) Define P : X ⇒ X and A : X ⇒ X by P (y) = [X \ S2−1 (y)] ∪ {x ∈ X | x ∈ S1 (x) and R(x, y, z ) holds ∀z ∈ S3 (x, y)}, and A(x) = X \ P −1 (x). (II) It easy to see X \ S2−1 (y) is closed for any y ∈ X by the condition (ii). For any net {xα } with xα ∈ S1 (xα ), R(xα , y, z ) holds for any z ∈ S3 (xα , y) and xα −→ x, then x ∈ S1 (x) by the condition (i). If there exists z ∈ S3 (x, y) such that R(x, y, z ) does not hold, since S3 (·, y) is lower semicontinuous, then there exists zα ∈ S3 (xα , y) with zα −→ z. And, since R(·, y, ·) is closed, then

{(x, z ) ∈ X × Y | R(x, y, z ) does not hold} is open. Hence there exists α0 such that R(xα , y, zα ) does not hold for any α ≻ α0 , which contradicts the fact that R(xα , y, z ) holds for any z ∈ S3 (xα , y). Thus, x ∈ S1 (x) and R(x, y, z ) holds for any z ∈ S3 (x, y). Then P (y) is closed for any y ∈ X . Thus, for any y ∈ X , A−1 (y) = X \ P (y) is open in X . ∑n(III) Suppose that there exists x ∈ X such that x ∈ GcoA(x), then there exist yi ∈ A(x), i = 1, . . . , n and ti ≥ 0 with i=1 ti = 1, i = 1, . . . , n such that

 x = expo

n −

 .

1 ti exp− o yi

i =1

Since yi ∈ A(x) for all i = 1, . . . , n, then x ̸∈ P (yi ) for all i = 1, . . . , n. By the definition of P (y), we have x ̸∈ X \ S2−1 (yi ),

∀i ∈ {1, . . . , n};

and x ̸∈ {x ∈ X | x ∈ S1 (x) and R(x, yi , z ) holds ∀z ∈ S3 (x, yi )}, ∀i ∈ {1, . . . , n}. From x ̸∈ X \ S2−1 (yi ), ∀i ∈ {1, . . . , n}, it follows that yi ∈ S2 (x), by the condition (ii), which implies that

 x = expo

n −

 −1

ti expo yi

∈ GcoS2 (x) ⊂ S1 (x).

i =1

Thus x ∈ S1 (x). And, since x ̸∈ {x ∈ X | x ∈ S1 (x) and R(x, yi , z ) holds ∀z ∈ S3 (x, yi )}, ∀i ∈ {1, . . . , n}, then, for any i ∈ {1, . . . , n}, there exists z ∈ S3 (x, yi ) such that R(x, yi , z ) does not hold, which contradicts the condition (v). Thus, x ̸∈ GcoA(x) for all x ∈ X . (IV) Hence A satisfies all conditions of Theorem 3.1. By Theorem 3.1, there exists x∗ ∈ X such that A(x∗ ) = ∅, i.e., there exists x∗ ∈ X such that x∗ ∈ S1 (x∗ ) and R(x∗ , y, z ) for any y ∈ S2 (x∗ ) and for any z ∈ S3 (x∗ , y). Let X , Y be two nonempty, compact and geodesic convex subsets of two Hadamard manifolds M1 , M2 and Φ1 be the collection of variational relation problems q satisfying all conditions of Theorem 5.1. For any q ∈ Φ1 , in the proof of Theorem 5.1, we define Pq : X ⇒ X and Aq : X ⇒ X by Pq (y) = [X \ S2−1 (y)] ∪ {x ∈ X | x ∈ S1 (x) and R(x, y, z ) holds ∀z ∈ S3 (x, y)}, and Aq (x) = X \ Pq−1 (x). S1 (q) is the set of solutions for variational relation problem q. In the proof of Theorem 5.1, we know that Aq ∈ Φ for any q ∈ Φ1 . So there exists a single-valued mapping T1 : Φ1 −→ Φ such that T1 (q) = Aq . For q, q′ ∈ Φ1 , we define ρ1 (q, q′ ) = ρ(Aq , Aq′ ).  Theorem 5.2. For each q ∈ Φ1 , there exists at least one essential component of the set S1 (q). Proof. Since T1 : Φ1 −→ Φ is an isometric mapping such that T1 (q) = Aq , it is continuous. By Theorem 4.4, there exists at least one essential component of S (A) for each A ∈ Φ , and by Lemma 2.3, there exists at least one essential component of S1 (q) for each q ∈ Φ1 . 

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Next, we define the geodesic concave function on Hadamard manifolds. Definition 5.2. Let K ⊂ M be a nonempty and ∑n geodesic convex subset, a function f : K −→ R is said to be geodesic concave if, for any xi ∈ K , ti ≥ 0, i = 1, . . . , n with i=1 ti = 1, we have

 f

 expo

n −

 −1

≥

ti expo xi

i=1

n −

ti f (xi );

i=1

f is said to be geodesic quasi-concave if, for any xi ∈ K , ti ≥ 0, i = 1, . . . , n with

 f

 expo

n −

∑n

i =1 t i

= 1, we have

 1 ti exp− o xi

≥ min {f (xi )}.

i=1

i∈{1,...,n}

Remark 5.1 (Equilibrium Problem on Hadamard Manifolds). Let f : X × X −→ R is a function. In the Theorem 5.1, set S1 (x) = X , S2 (x) = X and S3 (x, y) = {y}, for all x ∈ X and y ∈ X . The variational relation R is defined as follows: R(x, y, z ) holds iff f (x, y) ≤ 0. Then, (VR) becomes: Find x∗ ∈ X such that f (x∗ , y) ≤ 0 for all y ∈ X . Theorem 5.3. Let X be a nonempty, compact and geodesic convex subset of a Hadamard manifold M, and let f : X × X −→ R be a real-valued function on X × X such that (1) for each y ∈ X , x −→ f (x, y) is lower semicontinuous on X ; (2) for each x ∈ X , y −→ f (x, y) is geodesic quasi-concave on X ; (3) f (x, x) ≤ 0 for all x ∈ X . Then there exists x∗ ∈ X such that f (x∗ , y) ≤ 0 for all y ∈ X . Proof. Since the conditions (1)–(3) imply the conditions (iv) and (v), then it is proved easily.



Remark 5.2. In [42], there are established KKM-type results on compact subsets of acyclic spaces, which are related to the Ky Fan-type inequality Theorem 5.3. Next, we study the relation between Theorems 5.3 and 2.1 of [42]. Theorem 2.1 of [42] is neither stronger nor weaker than our Theorem 5.3. In the Theorem 5.3, since for each x ∈ X , y −→ f (x, y) is geodesic quasi-concave on X , then ∀x ∈ X , {y ∈ X | f (x, y) > 0} is a geodesic convex subset of Hadamard manifold M. Due to Proposition 2.2 of [43] and Hanner’s theorem (see Bessage and Pelczyński [44, Theorem 5.1]), X and ∀x ∈ X , {y ∈ X | f (x, y) > 0} are contractible and ANRs. Since every contractible space is acyclic and there are examples showing the converse is not true (see Brown [45, p. 31]), then X and ∀x ∈ X , {y ∈ X | f (x, y) > 0} are acyclic finite-dimensional ANRs, which satisfy some conditions of Theorem 2.1 in [42]. But the condition (1) of Theorem 5.3 does not imply that {(x, y) ∈ X × X | f (x, y) > 0} is open. Thus our Theorem 5.3 is not deduced from Theorem 2.1 of [42]. Remark 5.3. Assume that (i) (ii) (iii) (iv)

Let X be a nonempty, compact and geodesic convex subset of a Hadamard manifold M; f is lower semicontinuous on X × X or {(x, y) ∈ X × X | f (x, y) > 0} is open; for each x ∈ X , y −→ f (x, y) is geodesic quasi-concave on X ; f (x, x) ≤ 0 for all x ∈ X .

Then Theorem 5.3 is deduced from Theorem 2.1 of [42]. Remark 5.4 (Variational Inclusion Problem on Hadamard Manifolds). Let F and G be set-valued maps on X × X × Y with values in the space Z . In the Theorem 5.1, the variational relation R is defined as follows: R(x, y, z ) holds iff F (x, y, z ) ⊂ G(x, y, z ). Then, (VR) becomes: Find x∗ ∈ X such that F (x∗ , y, z ) ⊂ G(x∗ , y, z ) for all y ∈ S2 (x∗ ) and all z ∈ S3 (x∗ , y). Theorem 5.4. Let X , Y be two non-empty, compact and geodesic convex subsets of two Hadamard manifolds M1 , M2 . S1 : X ⇒ X , S2 : X ⇒ X and S3 : X × X ⇒ Y are three set-valued mappings with nonempty values. Let Z be a topological vector space. Let F and G be set-valued maps on X × X × Y with values in the space Z . Assume that (i)–(iii) of Theorem 5.1 hold and (1) for any finite {x1 , . . . , xn } of X and for any x ∈ Gco{x1 , . . . , xn }, there exists i ∈ {1, . . . , n} such that F (x, xi , z ) ⊂ G(x, xi , z ) for all z ∈ S3 (x, xi ); (2) for any y ∈ X , {x ∈ X | F (x, y, z ) ⊂ G(x, y, z ), ∀z ∈ S3 (x, y)} is closed in X . Then there exists x∗ ∈ X such that x∗ ∈ S1 (x∗ ) and F (x∗ , y, z ) ⊂ G(x∗ , y, z ) for any y ∈ S2 (x∗ ) and for any z ∈ S3 (x∗ , y). Proof. Since the conditions (1) and (2) imply the conditions (iv) and (v), then it is proved easily.



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523

6. Application (II): multi-objective games on Hadamard manifolds Consider the n-person multi-objective game Γ {I , Xi , F i } on Hadamard manifolds. Assume that (1) I = {1, . . . , n} is the set of players; (2) for each i ∈ I, the nonempty set X ∏i is the strategy set of ith player; (3) for each i ∈ I, F i = (f1i , . . . , fki ) : i∈I Xi −→ Rk is the vector payoff function of ith player. We shall note X−i = j∈I \{i} Xj , x−i = (x1 , . . . , xi−1 , xi+1 , . . . , xn ) ∈ X−i , x = (xi , x−i ) ∈ X . x∗ = (x∗i , x∗−i ) ∈ X is called a weakly Pareto–Nash equilibrium point if, for each i ∈ I,

∏

F i (yi , x∗−i ) − F i (x∗i , x∗−i ) ̸∈ intRk+ , ∀yi ∈ Xi ; where Rk+ = {(u1 , . . . , uk ) ∈ Rk | ui ≥ 0, ∀i = 1, . . . , n}, intRk+ = {(u1 , . . . , uk ) ∈ Rk | ui > 0, ∀i = 1, . . . , n}. Define the set of all weakly Pareto–Nash equilibrium points by WN (Γ ). Next, we define the vector geodesic C -concave function on Hadamard manifolds. Definition 6.1. Let K ⊂ M be a geodesic convex subset, and Z be a topological vector space with nonempty convex closed point cone C with∑ intC ̸= ∅. A vector function f : K −→ Z is said to be geodesic C -concave if, for any xi ∈ K , ti ≥ 0, n i = 1, . . . , n with i=1 ti = 1, we have

 f

 expo

n −

 −1

∈

ti expo xi

i=1

n −

ti f (xi ) + C ;

i=1

f is said to be geodesic quasi-C -concave if, for any xi ∈ K , ti ≥ 0, i = 1, . . . , n with such that

 f

 expo

n −

∑n

i=1 ti

= 1, there exists i ∈ {1, . . . , n}

 −1

ti expo xi

∈ f (xi ) + C .

i=1

Definition 6.2. Let X be a Hausdorff topological space, and Z be a topological vector space with nonempty convex closed point cone C with intC ̸= ∅. Let f : X −→ Z be a vector-valued function. Iff, for any open neighborhood V of the zero element in Z , there exists an open neighborhood U of x0 in X such that, for all x ∈ U, f ( x) ∈ f ( x0 ) + V + C , f is called C -continuous at x0 ∈ X and iff it is C -continuous at every point of X , f is called C -continuous on X . Theorem 6.1. Consider the n-person multi-objective game Γ {I , Xi , F i } on Hadamard manifolds satisfying the following conditions (1) (2) (3) (4)

for each i ∈ I, Xi is a nonempty, compact and geodesic convex subset of a Hadamard manifold Mi ; for each i ∈ I, −F i is Rk+ -continuous on X ; for each i ∈ I and each fixed ui ∈ Xi , F i (ui , ·) is Rk+ -continuous on X−i ; for each fixed u−i ∈ X−i , F i (·, u−i ) is geodesic quasi-Rk+ -concave on Xi . Then there exists one weakly Pareto–Nash equilibrium point x∗ at least.

Proof. (I) We define the set-valued mapping A =

∏

i∈I

Ai (x), where

Ai (x) = {yi ∈ Xi | F (yi , x−i ) − F (xi , x−i ) ∈ intR+ }, i

i

k

∀x ∈ X .

By the conditions (2) and (3), for any fixed y ∈ X , x −→ F i (yi , x−i ) − F i (xi , x−i ) is Rk+ -continuous on X . For any x ∈ A−1 (y), there exists an open neighborhood Vi of zero such that F i (yi , x−i ) − F i (xi , x−i ) + Vi ⊂ intRk+ . Since x −→ F i (yi , x−i ) − F i (xi , x−i ) is Rk+ -continuous on X , then there exists an open neighborhood Ui of x in X such that, for any x′ ∈ Ui , F i (yi , x′−i ) − F i (x′i , x′−i ) ∈ F i (yi , x−i ) − F i (xi , x−i ) + Vi + Rk+ ⊂ intRk+ + Rk+ ⊂ intRk+ .

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Z. Yang, Y.J. Pu / Nonlinear Analysis 75 (2012) 516–525

Then U =

n 

Ui ⊂ A−1 (y).

i=1

Thus A−1 (y) is open for any y ∈ X . j (II) Suppose that there exists x ∈ X such that x ∈ GcoA(x), then  there exist y ∈ A(x), j = 1, .i . . , m and tj ≥ 0, ∑m ∑m −1 j j = 1, . . . , m with j=1 tj expo y . Since, for each fixed u−i ∈ X , F (·, u−i ) is geodesic j=1 tj = 1 such that x = expo quasi-Rk+ -concave on Xi , then there exists j ∈ {1, . . . , m} such that

 F (xi , x−i ) = F i

i

 expoi

m −

 −1 j

tj expoi yi

 , x−i

∈ F i (yji , x−i ) + Rk+ ,

j =1 j yi

F ( , x−i ) − F (xi , x−i ) ∈ −Rk+ , i

i

which contradict the fact that yj ∈ A(x) for all j = 1, . . . , m, i.e., j

F i (yi , x−i ) − F i (xi , x−i ) ∈ intRk+ ,

∀j = 1, . . . , m.

By Theorem 3.1, there exists x ∈ X such that A(x∗ ) = ∅, i.e., ∗

F i (yi , x∗−i ) − F i (x∗i , x∗−i ) ̸∈ intRk+ , for all yi ∈ Xi and for each i ∈ I.



Let Φ2 be the collection of all n-person multi-objective games Γ satisfying all conditions of Theorem 6.1. For any Γ ∈ Φ2 , ∏ we denote AΓ (x) = i∈I AΓi (x), where AΓi (x) = {yi ∈ Xi | F i (yi , x−i ) − F i (xi , x−i ) ∈ intRk+ },

∀x ∈ X .

In the proof of Theorem 6.1, we know that AΓ ∈ Φ for any Γ ∈ Φ2 . So there exists a single-valued mapping T2 : Φ2 −→ Φ ′ such that T2 (Γ ) = AΓ . For Γ , Γ ′ ∈ Φ2 , we define ρ2 (Γ , Γ ′ ) = ρ(AΓ , AΓ ). It is similar to the proof of Theorem 5.2 on the existence of essential component, we obtain the following result: Theorem 6.2. For each Γ ∈ Φ2 , there exists at least one essential component of the set WN (Γ ). Remark 6.1. If k = 1, n-person multi-objective game Γ {I , Xi , F i } is a general n-person noncooperative game. Existence and stability of Nash equilibrium points are obtained, which is similar to Theorems 6.1 and 6.2. Thus, when k = 1, we have Theorem 6.3. Consider the n-person game Γ {I , Xi , fi } on Hadamard manifolds satisfying the following conditions (1) (2) (3) (4)

for each i ∈ I, Xi is a nonempty, compact and geodesic convex subset of a Hadamard manifold Mi ; for each i ∈ I, fi is upper semicontinuous on X ; for each i ∈ I and each fixed ui ∈ Xi , fi (ui , ·) is lower semicontinuous on X−i ; for each fixed u−i ∈ X−i , fi (·, u−i ) is geodesic quasi-concave on Xi . Then there exists one Nash equilibrium point x∗ at least.

Remark 6.2. In [43], Kristaly proved the existence of Nash equilibrium points on generic Riemannian manifolds with applications in Hadamard manifods. Further, Kristaly studied Nash critical points on completely finite-dimensional Riemannian manifolds, and a useful relation between Nash equilibrium points and Nash critical points is established. Remark 6.3. . The Theorem 1.1 of [43] is different from our Theorem 6.3. Theorem 1.1 of [43] is neither stronger nor weaker than our Theorem 6.3. Because (1) In Theorem 1.1 of [43], fi is continuous, which is different from the conditions (2) and (3) of Theorem 6.3; (2) In Theorem 1.1 of [43], for each fixed u−i ∈ X−i , fi (·, u−i ) is geodesic concave on Xi , which is different from the condition (4) of Theorem 6.3; (3) In Theorem 1.1 of [43], Mi is a finite-dimensional Riemannian manifold, but Mi is a finite-dimensional Hadamard manifold in the Theorem 6.3. Remark 6.4. Assume that (i) for each i ∈ I, Xi is a nonempty, compact and geodesic convex subset of a Hadamard manifold Mi ; (ii) fi is continuous and for each fixed u−i ∈ X−i , fi (·, u−i ) is geodesic concave on Xi . Then Theorem 6.3 is deduced from Theorem 1.1 of [43].

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