Abstract generalized vector quasi-equilibrium problems in noncompact Hadamard manifolds.

This paper deals with the abstract generalized vector quasi-equilibrium problem in noncompact Hadamard manifolds. We prove the existence of solutions to the abstract generalized vector quasi-equilibrium problem under suitable conditions and provide applications to an abstract vector quasi-equilibrium problem, a generalized scalar equilibrium problem, a scalar equilibrium problem, and a perturbed saddle point problem. Finally, as an application of the existence of solutions to the generalized scalar equilibrium problem, we obtain a weakly mixed variational inequality and two mixed variational inequalities. The results presented in this paper unify and generalize many known results in the literature.

Introduction

Let K be a nonempty subset of a nonempty set X and be a bifunction. The scalar equilibrium problem (for short, SEP) is to find such that for every . It is well known that SEP contains a broad class of problems arising in pure and applied mathematics, such as fixed point, minimax and variational inequality, Nash equilibrium, complementarity, and convex optimization problems (see, for example, [1-5] and the references therein). Recently, in a linear setting, many authors focused on the existence of solutions to equilibrium problems for vector mappings; see, for example, Ansari and Yao [6], Ansari and Flores-Bazán [7], Fu [8], Fu and Wan [9], Khan [10], Khan et al. [11], Khan and Chen [12], Hou et al. [13], Iusem and Sosa [14], Chen et al. [15], Kassay et al. [16], and the authors referenced by their works.

On the other hand, Riemannian manifolds provide a useful framework for the research of the related problems in optimization and equilibrium. Actually, many concepts and techniques fitting in Euclidean spaces have been extended to Riemannian manifolds. Most of the generalized methods require the sectional curvature of Riemannian manifold to be nonpositive. In fact, a large class of Riemannian manifolds, including Hadamard manifolds, possesses this important property which implies tight topological restrictions and rigidity phenomena (see [17]). Hadamard manifolds have turned out to be a suitable framework for diverse disciplines (see, for example, [18-20]). Applying the KKM principle allowed Colao et al. [21] to confirm the existence of solutions to the SEP in Hadamard manifolds. Approximately at the same time, Yang and Pu [22] studied the existence and stability of solutions to the SEP in Hadamard manifolds. By applying the KKM principle, Zhou and Huang [23] proved the existence of solutions to the vector optimization problem in Hadamard manifolds. Similarly, Li and Huang [24] presented some existence results of solutions to the generalized vector quasi-equilibrium problems in Hadamard manifolds. Recently, Batista et al. [25] introduced and studied the generalized vector equilibrium problem in Hadamard manifolds. Their results generalize the corresponding results of Colao et al. [21], Zhou and Huang [23], and Németh [26].

Motivated by the recent work mentioned above, the main purpose of this paper is to introduce and study the abstract generalized vector quasi-equilibrium problem (for short, AGVQEP) in noncompact Hadamard manifolds. The rest of this paper is organized as follows. In Section 2, we introduce notation, definitions, and preliminary results used in the paper. In Section 3, we apply an existence result of maximal elements in noncompact Hadamard manifolds in order to prove an existence theorem of solutions to AGVQEP under some suitable conditions. Applications to an abstract vector quasi-equilibrium problem (for short, AVQEP), a generalized scalar equilibrium problem (for short, GSEP), SEP, and a perturbed saddle point problem are provided. Section 4 is devoted to investigating the weakly mixed variational inequality problem (for short, WMVIP) in noncompact Hadamard manifolds. Our methods are based on a result concerning the existence of solutions to GSEP. Conclusions are presented in Section 5.

Preliminaries

In this section, we recall some notation, definitions, and auxiliary results, which are intended to be used throughout this paper. These can be found in [27-33].

Let denote the set of all real numbers. Let X be a set. Then we let represent the family of nonempty finite subsets of X. Let A be a subset of a topological space X, and then let intA and clA denote the interior of A in X and the closure of A in X, respectively. Moreover, if , then (respectively, ) stands for the interior (respectively, closure) with respect to the topology of B, induced by that of X. Given two nonempty sets X, Y and a set-valued mapping , the inverse of T is defined by for every . Let X and Y be two topological spaces. Then a set-valued mapping is said to be upper semicontinuous (respectively, lower semicontinuous) on X iff, for each and each open set with (respectively, ), there exists an open neighborhood of x in X such that (respectively, ) for every . Let be the standard n-dimensional simplex with vertices and for each , let denote the simplex with vertices , where denotes the cardinality of I.

Let E be a simply-connected m-dimensional manifold. For each , we denote by the tangent space of E at x. Let denote the tangent bundle of E, which is naturally a manifold. A vector field on E is a mapping such that for every . Let denote the scalar product on with the associated norm , where the subscript x will be omitted in the sequel when no confusion arises. If E is a differentiable manifold equipped with a scalar product that varies smoothly from point to point, then we say that E is a Riemannian manifold. The family of scalar products is called a Riemannian metric. We always assume that E can be endowed with a Riemannian metric to become a Riemannian manifold. Given a piecewise smooth curve joining x to y (i.e., and ), we can define the length of γ as follows: Then, for any two points , the Riemannian distance is defined by A vector field σ is said to be parallel along γ if , where ∇ is the Levi-Civita connection associated with and γ is a smooth curve in E. If itself is parallel along γ, then γ is called a geodesic, and in this case, is constant. When , γ is said to be normalized. A geodesic which joins x to y in E is said to be minimal if its length equals .

A Riemannian manifold E is geodesically complete if, for each , all geodesics starting from x are defined for every . By the Hopf-Rinow theorem (see [31]), we can see that if a Riemannian manifold is geodesically complete, then is a complete metric space and each bounded closed subset is compact. In addition, for any two points in E, there exists a minimal geodesic joining these two points.

A Hadamard manifold E is a complete simply connected Riemannian manifold of nonpositive sectional curvature. Unless explicitly stated otherwise, throughout the remainder of this paper, we assume that E is a finite dimensional Hadamard manifold and V is a Hausdorff topological vector space.

Definition 2.1

[31]

Let . The exponential mapping at x is defined by for every , where is the geodesic starting at x with velocity v (i.e., , ).

Lemma 2.1

[33]

Let . Then is a diffeomorphism, and for any two points , there exists a unique minimal normalized geodesic for every joining x to y.

So from now on, a geodesic means the unique minimal normalized one.

Definition 2.2

[23]

A set is said to be convex iff, for any two points , the geodesic joining x to y is contained in C; that is, for every .

Definition 2.3

[23]

A real-valued function is said to be convex iff, for any two points , we have for every . f is said to be concave iff −f is convex.

Definition 2.4

Let Q be a nonempty subset of V. The set Q is called a convex cone iff and for .

Definition 2.5

Let be a nonempty convex cone. A set-valued mapping is said to be convex with respect to Q iff, for each and each , we have .

Remark 2.1

Definition 2.5 includes Definition 2.3 as a special case. In fact, when F is a single-valued mapping, , and , Definition 2.5 coincides with Definition 2.3. Also Definition 2.5 generalizes Definition 3.1 of Zhou and Huang [23] in the following aspects: (1) from a single-valued mapping to a set-valued mapping; (2) from the convex cone in Euclidean spaces to the convex cone in Hausdorff topological vector spaces.

Definition 2.6

Let be a nonempty convex cone. A set-valued mapping is said to be quasiconvex-like with respect to Q iff, for each and each , we have or .

Definition 2.7

[21]

Let . The convex hull of A is defined by the smallest convex subset of E containing A and denoted by .

Definition 2.8

Let X be a topological space. A set-valued mapping is called a Fan-Browder mapping iff the following conditions are fulfilled:

for each , is nonempty and convex;

for each , is open in X.

Definition 2.9

Let be a set-valued mapping. Then G is said to be of class iff the following conditions are satisfied:

for each , is convex;

for each , ;

there exists a set-valued mapping such that

for each , ;

;

.

Lemma 2.2

[34]

Let and such that . Then the following statements hold:

for each , and ;

if is a sequence such that and , then ;

given the sequences , if , with , then .

Lemma 2.3

[35]

Let . Then the following statements are equivalent:

the function is affine for every ;

the mapping is a global isometry;

for each , the curve is the minimal geodesic joining to ;

the sectional curvature of E is identically zero (i.e., E is isometric to the usual Euclidean space).

Let be a real-valued function. The subdifferential of f is the set-valued mapping defined by for every and its elements are called subgradients. For each , the subdifferential is a closed convex subset (possibly empty) of . Let denote the domain of ∂f.

The following lemma guarantees the existence of subgradients for convex functions.

Lemma 2.4

[36]

Let be a convex function. Then, for any , the subdifferential is a nonempty subset of ; i.e., .

The following lemma, which provides the existence of maximal elements for a set-valued mapping, plays a key role in the proof of the existence of solutions to AGVQEP.

Lemma 2.5

Let be of class and K be a nonempty compact subset of E. Suppose that one of the following conditions holds: Then there exists such that .

for each , there exists a nonempty compact convex subset of E containing N such that ;

there exists a point such that .

Proof

Since G is of class , it follows that

for each , is convex;

for each , ;

there exists a set-valued mapping such that

for each , ;

;

.

We prove Lemma 2.5 by considering the following two cases.

Case I. Suppose that (i)1 holds. We proceed by contradiction. If the conclusion of Lemma 2.5 is false, then for every . By (3), we have for every , which implies that . Moreover, it follows from (1) and (2) that . Because K is a nonempty compact subset of E, we may choose such that By (i)1, there exists a nonempty compact convex subset of E containing N such that By (2.1), we get . Together with (2.2), we have . Applying the compactness of , there exists such that is a finite open cover of . By Theorem 5.1 in [37], there is, for every i, a continuous function such that for every and such that implies . Consequently, we can define a continuous mapping by for every with , where is defined by Hence, it follows from the convexity of that .

Following the same argument as in the proof of Lemma 3.1 in [21], we can conclude that there exists a continuous mapping satisfying where , and, for each , is defined by . Then is a closed subset of .

Now, the mapping has the property that for every . Since the mapping is continuous, it follows from the Brouwer fixed point theorem that there exists such that . Let . Then we have , which contradicts (b). Therefore, there must exist such that . This completes the proof.

Case II. Assume that (i)2 holds. Suppose, by way of contradiction, that for every . By (3), we have for every . Thus, by (1) and (2), we have . Since K is compact, there exists such that By (i)2, we have . Together with (2.3), we get where . This implies that is a finite open cover of E. Since E is a normal space, it follows that there exists a finite collection of functions subordinated to the open cover , which implies that Consequently, the function on E defined by for every with is continuous, where is defined by Hence, by the convexity of , we have .

By using the same method as in the proof of Lemma 3.1 in [21], we can conclude that there exists a continuous mapping satisfying where is the same as in the previous case.

Now, let . Then g has the following property: for every . By the Brouwer fixed point theorem, the continuous mapping has a fixed point ; that is, . Let . Then we have , which contradicts (b). Therefore, there must exist such that . This completes the proof. □

Remark 2.2

Lemma 2.5 generalizes Theorem 3.1 of Yang and Pu [22] in the following aspects: (a) from compact Hadamard manifolds to noncompact Hadamard manifolds; (b) from one set-valued mapping to two set-valued mappings; (c) the condition that is weaker than (2) of Theorem 3.1 of Yang and Pu [22]; (d) concerns the more general Hadamard manifold with its sectional curvature being nonpositive instead of the Hadamard manifold with its sectional curvature being identically zero. This fact can be deduced from Lemma 2.3.

Remark 2.3

(i)1 of Lemma 2.5 can be replaced by the following equivalent condition.

For each , there exists a nonempty compact convex subset of E containing N such that .

Abstract generalized vector quasi-equilibrium problem

In this section, we introduce AGVQEP in Hadamard manifolds and present a sufficient condition for the existence of solutions to AGVQEP. As applications, we obtain results to solve AVQEP, GSEP, SEP, and the perturbed saddle point problem in noncompact Hadamard manifolds.

Let K be a nonempty subset of E, W be a nonempty set, and let , , be set-value mappings. We consider the following AGVQEP as follows: find such that

It is worthwhile noting that AGVQEP is motivated by the generalized vector quasi-equilibrium problem introduced by Ansari and Flores-Bazán [7]. In particular, let , W be a Hausdorff topological vector space, and let be a set-valued mapping such that, for each , is a closed and convex cone with . Moreover, let be a set-valued mapping defined by for every . Then AGVQEP retrieves a particular instance of the equilibrium problem in [7]. Here we would like to point out that the feasible set of AGVQEP is controlled by a set-valued mapping. In the real world, there are important problems which can be regarded as AGVQEPs in which the condition that the feasible set of AGVQEP is controlled by a set-valued mapping must be satisfied; for example, the equilibrium problems of the generalized games in Dasgupta and Maskin [38], Smeers et al. [39], Krawczyk [40], and Ansink and Houba [41].

Remark 3.1

If for every , W is a Hausdorff topological vector space, and each is replaced by , where is a closed and convex cone with , then AGVQEP reduces to the generalized vector equilibrium problem investigated by Batista et al. [25]. By the arguments in [25], we can see that AGVQEP also includes the equilibrium problems in [21, 23, 26] as its special cases.

Remark 3.2

Some other special cases of AGVQEP are given as follows.

Let for every . Then AGVQEP reduces to the abstract vector quasi-equilibrium problem (for short, AVQEP), which consists in finding such that for every .

If , for every , and , where is a bifunction, then GVQEP reduces to the generalized scalar equilibrium problem (for short, GSEP), which is to find such that and for every . Furthermore, if for every , then GSEP reduces to SEP.

Now, we are ready, by using Lemma 2.5, to present the following existence theorem of solutions to AGVQEP in noncompact Hadamard manifolds.

Theorem 3.1

Let be a nonempty compact set and W be a nonempty set. Let , be three set-valued mappings and be a Fan-Browder mapping. Assume that Then AGVQEP has at least a solution in K.

the set is open in E;

for each , implies ;

for each , ;

for each , the set is convex;

;

one of the following conditions holds:

for each , there exists a nonempty compact convex subset of E containing N such that

there exists a point such that

Define two set-valued mappings by Moreover, let us define two set-valued mappings by and By (ii), we have for every . Since each is convex by (iv) and each is convex by the definition of a Fan-Browder mapping, it follows that is convex for every .

For each , we have We claim that . In fact, it is clear that . In order to prove that , we take . Then there exists such that If , then by (i) and the definition of a Fan-Browder mapping, we have We notice that, according to (v) and the definition of Q, one has So, if , then there exists such that Combining these two cases, we can conclude that . Therefore, we have .

Now, we show that for every . Indeed, if , then by the definition of , we have ; if , then by (iii), and so, .

By (vi) and the above fact, we can see that one of the following conditions holds: Therefore, by Lemma 2.5 and Remark 2.3, there exists such that . Since for every , it follows that and ; i.e., and for every . Thus, the conclusion of Theorem 3.1 holds and the proof is complete. □

for each , there exists a nonempty compact convex subset of E containing N such that

there exists a point such that

Corollary 3.1

Let be a nonempty compact set and W be a nonempty set. Let , be two set-valued mappings and be a Fan-Browder mapping. Assume that Then AGVQEP has at least a solution in K.

the set is open in E;

for each , ;

for each , the set is convex;

;

one of the following conditions holds:

for each , there exists a nonempty compact convex subset of E containing N such that

there exists a point such that

Let . It is easy to see that all the conditions of Theorem 3.1 are satisfied. Therefore, it follows from Theorem 3.1 that AGVQEP has at least a solution in K. Thus, the result holds and the proof of Corollary 3.1 is complete. □

Corollary 3.2

Let be a nonempty compact set and W be a nonempty set. Let , be two set-valued mappings and be a Fan-Browder mapping. Assume that Then AGVQEP has at least a solution in K.

the set is open in E;

for each , ;

for each , the set is convex;

for each , the set is open in E;

one of the following conditions holds:

for each , there exists a nonempty compact convex subset of E containing N such that

there exists a point such that

By (iv), (v), and the fact that is open in E for every , we can see that (iv) and (v) of Corollary 3.1 hold. Therefore, by Corollary 3.1, AGVQEP has at least a solution in K. This completes the proof. □

Remark 3.3

Corollary 3.2 extends Theorem 3.1 of Batista et al. [25] in the following aspects: (a) concerns the more general abstract generalized vector quasi-equilibrium problems instead of the generalized vector equilibrium problems; (b) from one coercivity condition to two alternative coercivity conditions; (c) since W in Corollary 3.2 does not need to be a real Hausdorff topological vector space, it is not required for each to be a closed and convex cone; (d) (iii) is weaker than h3 of Theorem 3.1 of Batista et al. [25]; (e) by the fact that W in Corollary 3.2 may be any nonempty set without topological structure, we adopt the assumption that the set is open in E for every , which is weaker than h2 of Theorem 3.1 of Batista et al. [25]. In addition, the proof of Corollary 3.2 originates from the existence of maximal elements in noncompact Hadamard manifolds, while the authors of [25] used the KKM property to prove their result. Therefore, the proof technique of Corollary 3.2 is different from that of Theorem 3.1 of Batista et al. [25].

By using Corollary 3.2, we can prove the existence of an equilibrium for the generalized water market game model under the condition that there are a river structure, water balances, and heterogeneous water users via a water delivery infrastructure. We would like to point out that our convex and continuous conditions are weaker than the corresponding conditions in Proposition 2.1 due to Ansink and Houba [41].

Corollary 3.3

Let be a nonempty compact set, W be a nonempty set, and let , be two set-valued mappings. Assume that Then AVQEP has at least a solution in K.

for each , ;

for each , the set is convex;

for each , the set is open in E;

one of the following conditions holds:

for each , there exists a nonempty compact convex subset of E containing N such that ;

there exists a point such that .

The conclusion of Corollary 3.3 follows from Corollary 3.2 by letting for every . This completes the proof. □

Remark 3.4

Let us give the following items:

(1) If W is a real Hausdorff topological vector space and is a family of nonempty convex cones, then (iv) of Theorem 3.1, (iii) of Corollaries 3.1-3.2, and (ii) of Corollary 3.3 can be replaced by one of the following conditions:

for each , is convex with respect to ;

for each , is quasiconvex-like with respect to .

In fact, we consider the first assumption. Let be any given. In order to prove that the set is convex, we assume that and . Applying this assumption and the fact that each is a convex cone, we have which implies that for every and so, the set is convex for every . Now, we prove that the desired conclusion holds under the condition that the second assumption is satisfied. Indeed, let be fixed and then let and be any given. By the definition of a quasiconvex-like mapping, we have or Since , we have and . Therefore, given , which is obtained by using the property of a convex cone, it follows from the above formulas that , and this shows that the set is convex for every .

(2) If W is a Hausdorff topological space, then (iv) of Corollary 3.2 and (iii) of Corollary 3.3 can be replaced by the following conditions:

for each , is upper semicontinuous on E with compact values;

the graph of C; i.e., is an open set in .

Indeed, it suffices to prove that the set is closed in E for every . Let be a net in such that . Since , there exists such that . Hence, we have . By the upper semicontinuity and compact values of ψ on E, it follows from Proposition 1 in [42] that there exists a subnet of with limit and . Without loss of generality, let us assume that . On the other hand, since the graph of C is an open set in , the set-valued mapping has a closed graph in . It follows that and so, . Thus, , which implies that the set is closed in E for every . Therefore, the set is open in E for every .

(3) If (iv)2 of Corollary 3.3 holds, then the solution set of AVQEP is a nonempty compact set, which can be written as follows: In fact, by the conclusion of Corollary 3.3, we can see that the above set is nonempty. Furthermore, it follows from (iv)2 of Corollary 3.3 that Together with (iii) of Corollary 3.3, we can see that the solution set of AVQEP is a nonempty closed subset of K. Therefore, the solution set of AVQEP is a nonempty compact set.

If , for every and , where is a bifunction, then Corollaries 3.2 and 3.3 reduce to the following existence results of solutions to GSEP and SEP, respectively.

Corollary 3.4

Let be a nonempty compact set, be a Fan-Browder mapping, and be a bifunction. Assume that Then GSEP has at least a solution in K.

the set is open in E;

for each , ;

for each , the set is convex;

for each , the set is closed in E;

one of the following conditions holds:

for each , there exists a nonempty compact convex subset of E containing N such that

there exists a point such that

Corollary 3.5

Let be a nonempty compact set and be a bifunction. Assume that Then SEP has at least a solution in K.

for each , ;

for each , is convex;

for each , the set is closed in E;

one of the following conditions holds:

for each , there exists a nonempty compact convex subset of E containing N such that ;

there exists a point such that .

Remark 3.5

Corollary 3.5 improves Theorem 3.2 of Colao et al. [21] because there are two alternative coercivity conditions in Corollary 3.5, while there is only one coercivity condition in Theorem 3.2 of Colao et al. [21]. Furthermore, the coercivity condition (iv)2 of Corollary 3.5 is weaker than the coercivity condition (iv) of Theorem 3.2 of Colao et al. [21]. To see this, we can consider K in Theorem 3.2 of Colao et al. [21] as a Hadamard submanifold, and then let , which is a nonempty compact subset of K. Then it follows from and that (iv) of Theorem 3.2 of Colao et al. [21] implies (iv)2 of Corollary 3.5.

As an application of Corollary 3.5, we have the following perturbed saddle point theorem in noncompact Hadamard manifolds.

Theorem 3.2

Let be two nonempty compact sets and be two bifunctions. Assume that Then f has a perturbed saddle point ; i.e., In particular, .

for each , ;

for each , is convex;

for each , is convex;

the bifunction is continuous;

one of the following conditions holds:

for each , there exist two nonempty compact convex subsets , of E containing N such that and ;

there exist two points such that and .

Define a bifunction by for every . By (i), (ii), (iv) and the first parts of (v)1 and (v)2, we can see that all the conditions of Corollary 3.5 are satisfied. Thus, by Corollary 3.5, there exists such that for every . Define a bifunction by for every . Then it follows from (i), (iii), (iv), and the second parts of (v)1 and (v)2 that all the hypotheses of Corollary 3.5 are fulfilled. Thus, by Corollary 3.5 again, there exists such that for every . Therefore, we have and It follows from the above inequality that and Since is always true, we get This completes the proof. □

By setting for every , we obtain the following saddle point theorem from Theorem 3.2.

Theorem 3.3

Let be two nonempty compact sets and be a bifunction. Assume that Then f has a saddle point ; i.e., In particular, .

for each , ;

for each , is convex;

for each , is convex;

the bifunction is continuous;

one of the following conditions holds:

for each , there exist two nonempty compact convex subsets , of E containing N such that and ;

there exist two points such that and .

Weakly mixed variational inequality problem

In this section, inspired by the idea due to Colao et al. [21], we introduce WMVIP in Hadamard manifolds and present a sufficient condition for the existence of solutions to WMVIP. Let be a set-valued mapping, be a vector field, and be a real-valued function. We consider the following WMVIP: find such that

Remark 4.1

WMVIP contains the mixed variational inequality problem (for short, MVIP) considered by Colao et al. [21]. In fact, let for every . Then WMVIP collapses to MVIP in [21]. Furthermore, if for every , then WMVIP coincides with the following variational inequality problem. which was introduced and studied by Németh [26].

By Corollary 3.4, we have the following existence theorem of solutions to WMVIP in noncompact Hadamard manifolds.

Theorem 4.1

Let be a nonempty compact set, be a Fan-Browder mapping, be a lower semicontinuous function, and be a continuous vector field. Assume that Then WMVIP has at least a solution in K.

the set is open in E;

for each , the function is convex;

one of the following conditions holds:

for each , there exists a nonempty compact convex subset of E containing N such that

there exists a point such that

Define a bifunction by It is clear that for every . Thus, (ii) of Corollary 3.4 is satisfied. By Lemma 2.2 and the continuous properties of σ and φ, one can see that the function is upper semicontinuous. Therefore, the set is closed in E for every , which implies that (iv) of Corollary 3.4 holds. Moreover, it follows from (iii) and the definition of f that one of the following conditions holds: Thus, (v) of Corollary 3.4 is satisfied. Now, in order to show that (iii) of Corollary 3.4 holds, we are ready to prove that is convex for every . Indeed, let be fixed and let , . By (ii), we get which implies that the set is convex for every . As a consequence of Corollary 3.4, there exists a point such that ; that is, WMVIP has at least a solution in K. This completes the proof. □

for each , there exists a nonempty compact convex subset of E containing N such that

there exists a point such that

Remark 4.2

The following two conditions are stronger than (iii)1 and (iii)2 of Theorem 4.1, respectively.

For each , there exists a nonempty compact convex subset of E containing N such that for all , there exists such that and ;

There exists a point such that and for every .

If for every , then we can obtain the following corollary from Theorem 4.1.

Corollary 4.1

Let be a nonempty compact set, be a lower semicontinuous function, and be a continuous vector field. Assume that Then MVIP has at least a solution in K.

for each , the function is convex;

one of the following conditions holds:

for each , there exists a nonempty compact convex subset of E containing N such that ;

there exists such that .

Remark 4.3

Corollary 4.1 extends Theorem 3.5 of Colao et al. [21] in the following aspects: (a) from one coercivity condition to two coercivity conditions; (b) the coercivity condition (ii)2 of Corollary 4.1 is weaker than the coercivity condition (C) of Theorem 3.5 of Colao et al. [21] in view of the same argument as in Remark 3.5; (c) by Lemma 2.3 and the proof of Theorem 3.5 of Colao et al. [21], we can see that the sectional curvature of the Hadamard manifold in Theorem 3.5 of Colao et al. [21] is identically zero, while it is not required for the sectional curvature of the Hadamard manifold in Corollary 4.1 to be identically zero; (d) the convexity of the function f in Theorem 3.5 of Colao et al. [21] is dropped.

Corollary 4.2

Let be a convex lower semicontinuous function and be a continuous vector field such that the function is convex for every . Suppose that one of the following conditions holds: Then MVIP has at least a solution.

E is compact;

there exists such that, for each , the following condition holds:

Suppose that (i)1 holds. For each , let . Thus, both (ii)1 and (ii)2 of Corollary 4.1 are satisfied automatically. Therefore, MVIP has at least a solution in E. Now, we suppose that (i)2 is satisfied. Let satisfy (i)2. Since φ is convex, it follows from Lemma 2.4 that . Thus, we may choose ; i.e., the subdifferential of φ at . For each , by the generalized Cauchy-Schwarz inequality, we have It follows from the above relation that By (i)2, there exists such that We can choose large enough such that, for each , we have . Then, for each , we have the following: Let . Since K is a bounded closed set in E, it follows from the Hopf-Rinow theorem that K is a compact subset of E. Thus, by the above inequality, one can see that (ii)2 of Corollary 4.1 holds. Therefore, by Corollary 4.1, MVIP has at least a solution. This completes the proof. □

Remark 4.4

Corollary 4.2 generalizes Corollary 3.6 of Colao et al. [21] in the following two aspects: (a) from the Hadamard manifold with its sectional curvature being identically zero to the Hadamard manifold with its sectional curvature being nonpositive. This fact can be deduced from Lemma 2.3; (b) (ii)2 of Corollary 4.2 is weaker than (ii) of Corollary 3.6 of Colao et al. [21].

Conclusions

In this paper, we introduce and study AGVQEP in noncompact Hadamard manifolds. By means of a maximal element theorem, we establish an existence theorem for a solution to AGVQEP in noncompact Hadamard manifolds. Moreover, we provide applications to AVQEP, GSEP, SEP, and the perturbed saddle point problem. Finally, WMVIP in noncompact Hadamard manifolds is introduced, and by applying our results, a weakly mixed variational inequality and two mixed variational inequalities are established.