Abstract generalized vector quasiequilibrium problems in noncompact Hadamard manifolds
 Haishu Lu^{1}Email author and
 Zhihua Wang^{1}
https://doi.org/10.1186/s1366001713752
© The Author(s) 2017
Received: 1 February 2017
Accepted: 21 April 2017
Published: 10 May 2017
Abstract
This paper deals with the abstract generalized vector quasiequilibrium problem in noncompact Hadamard manifolds. We prove the existence of solutions to the abstract generalized vector quasiequilibrium problem under suitable conditions and provide applications to an abstract vector quasiequilibrium 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.
Keywords
Hadamard manifold maximal element abstract generalized vector quasiequilibrium problem variational inequalityMSC
90C33 91E10 65K05 47J251 Introduction
Let K be a nonempty subset of a nonempty set X and \(f:X\times X \rightarrow\mathbb{R}\) be a bifunction. The scalar equilibrium problem (for short, SEP) is to find \(\widehat{x}\in K\) such that \(f( \widehat{x},y)\geq0\) for every \(y\in K\). 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 FloresBazán [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 quasiequilibrium 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 quasiequilibrium 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 quasiequilibrium 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.
2 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 \(\mathbb{R}\) denote the set of all real numbers. Let X be a set. Then we let \({\mathcal {F}}(X)\) 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 \(A\subseteq B\subseteq X\), then \(\operatorname{int}_{B}A\) (respectively, \(\operatorname{cl}_{B}A\)) 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 setvalued mapping \(T:X\rightrightarrows Y\), the inverse \(T^{1}:Y\rightrightarrows X\) of T is defined by \(T^{1}(y)=\{x\in X:y\in T(x)\}\) for every \(y\in Y\). Let X and Y be two topological spaces. Then a setvalued mapping \(T:X\rightrightarrows Y\) is said to be upper semicontinuous (respectively, lower semicontinuous) on X iff, for each \(x\in X\) and each open set \(V\subseteq Y\) with \(T(x)\subseteq V\) (respectively, \(T(x)\cap V \neq\emptyset\)), there exists an open neighborhood \(U(x)\) of x in X such that \(T(z)\subseteq V\) (respectively, \(T(z)\cap V\neq\emptyset \)) for every \(z\in U(x)\). Let \(\delta_{n}\) be the standard ndimensional simplex with vertices \(\{e_{0},e_{1},\ldots,e_{n}\}\) and for each \(I\subseteq\{0,1,\ldots,n\}\), let \(\delta_{\vert I\vert 1}\) denote the simplex with vertices \(\{e_{j}:j\in I\}\), where \(\vert I\vert \) denotes the cardinality of I.
A Riemannian manifold E is geodesically complete if, for each \(x\in E\), all geodesics starting from x are defined for every \(t\in\mathbb{R}\). By the HopfRinow theorem (see [31]), we can see that if a Riemannian manifold is geodesically complete, then \((E,d)\) 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 \(x\in E\). The exponential mapping \(\operatorname{exp}_{x}:T_{x}E\rightarrow E\) at x is defined by \(\operatorname{exp}_{x}v=\gamma_{v}(1,x)\) for every \(v\in T_{x}E\), where \(\gamma_{v}\) is the geodesic starting at x with velocity v (i.e., \(\gamma(0)=x\), \(\gamma^{\prime}(0)=v\)).
Lemma 2.1
[33]
Let \(x\in E\). Then \(\operatorname{exp}_{x}:T _{x}E\rightarrow E\) is a diffeomorphism, and for any two points \(x, y\in E\), there exists a unique minimal normalized geodesic \(\gamma_{x,y}=\operatorname{exp}_{x}t\operatorname{exp}_{x}^{1}y\) for every \(t\in[0,1]\) joining x to y.
So from now on, a geodesic means the unique minimal normalized one.
Definition 2.2
[23]
A set \(C\subseteq E\) is said to be convex iff, for any two points \(x, y\in C\), the geodesic joining x to y is contained in C; that is, \(\gamma_{x,y}=\operatorname{exp}_{x}t \operatorname{exp}_{x}^{1}y\in C\) for every \(t\in[0,1]\).
Definition 2.3
[23]
A realvalued function \(f:E\rightarrow \mathbb{R}\) is said to be convex iff, for any two points \(x, y\in E\), we have \(f(\operatorname{exp}_{x}t\operatorname{exp}_{x}^{1}y)\leq tf(x)+(1t)f(y)\) for every \(t\in[0,1]\). 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 \(Q+Q\subseteq Q\) and \(\lambda Q\subseteq Q\) for \(\lambda\geq0\).
Definition 2.5
Let \(Q\subseteq V\) be a nonempty convex cone. A setvalued mapping \(F:E\rightrightarrows V\) is said to be convex with respect to Q iff, for each \(x, y\in E\) and each \(t\in[0,1]\), we have \(F(\operatorname{exp}_{x}t\operatorname{exp}_{x}^{1}y)\subseteq tF(x)+(1t)F(y)+Q\).
Remark 2.1
Definition 2.5 includes Definition 2.3 as a special case. In fact, when F is a singlevalued mapping, \(V=\mathbb{R}\), and \(Q=(\infty,0]\), 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 singlevalued mapping to a setvalued mapping; (2) from the convex cone in Euclidean spaces to the convex cone in Hausdorff topological vector spaces.
Definition 2.6
Let \(Q\subseteq V\) be a nonempty convex cone. A setvalued mapping \(F:E\rightrightarrows V\) is said to be quasiconvexlike with respect to Q iff, for each \(x, y\in E\) and each \(t\in[0,1]\), we have \(F(\operatorname{exp}_{x}t\operatorname{exp}_{x}^{1}y)\subseteq F(x)+Q\) or \(F(\operatorname{exp}_{x}t\operatorname{exp}_{x}^{1}y)\subseteq F(y)+Q\).
Definition 2.7
[21]
Let \(A\subseteq E\). The convex hull of A is defined by the smallest convex subset of E containing A and denoted by \(\operatorname{conv}(A)\).
Definition 2.8
 (i)
for each \(x\in X\), \(F(x)\) is nonempty and convex;
 (ii)
for each \(y\in E\), \(F^{1}(y)\) is open in X.
Definition 2.9
 (a)
for each \(x\in E\), \(G(x)\) is convex;
 (b)
for each \(x\in E\), \(x\notin G(x)\);
 (c)there exists a setvalued mapping \(F:E\rightrightarrows E\) such that
 (1)
for each \(x\in E\), \(F(x)\subseteq G(x)\);
 (2)
\(\bigcup_{y\in E}F^{1}(y)=\bigcup_{y\in E}\operatorname{int}F^{1}(y)\);
 (3)
\(\{x\in E:F(x)\neq\emptyset\}=\{x\in E:G(x)\neq\emptyset\}\).
 (1)
Lemma 2.2
[34]
 (i)
for each \(y\in E\), \(\operatorname{exp}_{x_{n}}^{1}y\rightarrow \operatorname{exp}_{x_{0}}^{1}y\) and \(\operatorname{exp}_{y}^{1}x_{n}\rightarrow \operatorname{exp}_{y}^{1}x_{0}\);
 (ii)
if \(\{v_{n}\}\) is a sequence such that \(v_{n}\in T_{x _{n}}E\) and \(v_{n}\rightarrow v_{0}\), then \(v_{0}\in T_{x_{0}}E\);
 (iii)
given the sequences \(\{u_{n}\}, \{v_{n}\}\subseteq T _{x_{n}}E\), if \(u_{n}\rightarrow u_{0}\), \(v_{n}\rightarrow v_{0}\) with \(u_{0}, v_{0}\in T_{x_{0}}E\), then \(\langle u_{n},v_{n}\rangle \rightarrow\langle u_{0},v_{0}\rangle\).
Lemma 2.3
[35]
 (i)
the function \(E\ni y\mapsto\langle u,\operatorname{exp}_{x}^{1}y \rangle\) is affine for every \(u\in T_{x}E\);
 (ii)
the mapping \(\operatorname{exp}_{x}:T_{x}E\rightarrow E\) is a global isometry;
 (iii)
for each \(q_{1}, q_{2}\in E\), the curve \([0,1] \ni t\mapsto \operatorname{exp}_{x}((1t)\operatorname{exp}_{x}^{1}(q_{1})+t \operatorname{exp}_{x}^{1}(q_{2}))\) is the minimal geodesic joining \(q_{1}\) to \(q_{2}\);
 (iv)
the sectional curvature of E is identically zero (i.e., E is isometric to the usual Euclidean space).
Let \(f:E\rightarrow\mathbb{R}\) be a realvalued function. The subdifferential of f is the setvalued mapping \(\partial f:E\rightrightarrows TE\) defined by \(\partial f(x)=\{u\in T_{x}E:\langle u,\operatorname{exp}_{x} ^{1}y\rangle\leq f(y)f(x), \forall y\in E\}\) for every \(x\in E\) and its elements are called subgradients. For each \(x\in E\), the subdifferential \(\partial f(x)\) is a closed convex subset (possibly empty) of \(T_{x}E\). Let \(D(\partial f)=\{x\in E:\partial f(x)\neq \emptyset\}\) denote the domain of ∂f.
The following lemma guarantees the existence of subgradients for convex functions.
Lemma 2.4
[36]
Let \(f:E\rightarrow\mathbb{R}\) be a convex function. Then, for any \(x\in E\), the subdifferential \(\partial f(x)\) is a nonempty subset of \(T_{x}E\); i.e., \(D(\partial f)=E\).
The following lemma, which provides the existence of maximal elements for a setvalued mapping, plays a key role in the proof of the existence of solutions to AGVQEP.
Lemma 2.5
 (i)_{1} :

for each \(N\in{ \mathcal {F}}(E)\), there exists a nonempty compact convex subset \(E_{N}\) of E containing N such that \(E_{N}\setminus K\subseteq\bigcup_{y\in E_{N}}\operatorname{int}_{E_{N}}(G ^{1}(y)\cap E_{N})\);
 (i)_{2} :

there exists a point \(y_{0}\in E\) such that \(\operatorname{cl}(E\setminus G^{1}(y_{0}))\subseteq K\).
Proof
 (a)
for each \(x\in E\), \(G(x)\) is convex;
 (b)
for each \(x\in E\), \(x\notin G(x)\);
 (c)there exists a setvalued mapping \(F:E\rightrightarrows E\) such that
 (1)
for each \(x\in E\), \(F(x)\subseteq G(x)\);
 (2)
\(\bigcup_{y\in E}F^{1}(y)=\bigcup_{y\in E}\operatorname{int}F ^{1}(y)\);
 (3)
\(\{x\in E:F(x)\neq\emptyset\}=\{x\in E:G(x)\neq \emptyset\}\).
 (1)
We prove Lemma 2.5 by considering the following two cases.
Now, the mapping \(g=\xi\circ f\) has the property that \(g(x)=\xi(f(x))\in\xi(\delta_{\vert I(x)\vert 1})\subseteq D(\{y_{i}:i \in I(x)\})\subseteq \operatorname{conv}(\{y_{i}:i\in I(x)\})\subseteq G(x)\) for every \(x\in E_{N}\). Since the mapping \(f\circ\xi:\delta_{n}\rightarrow \delta_{n}\) is continuous, it follows from the Brouwer fixed point theorem that there exists \(p^{*}\in\delta_{n}\) such that \(p^{*}=f( \xi(p^{*}))\). Let \(\overline{x}=\xi(p^{*})\). Then we have \(\overline{x}=\xi(p^{*})=\xi(f(\xi(p^{*})))=\xi(f(\overline{x})) \in G(\overline{x})\), which contradicts (b). Therefore, there must exist \(\widehat{x}\in K\) such that \(G(\widehat{x})=\emptyset\). This completes the proof.
Now, let \(g=\xi\circ f\). Then g has the following property: \(g(x)\in\xi(\delta_{\vert I(x)\vert 1})\subseteq D(\{y_{i}:i\in I(x)\}) \subseteq \operatorname{conv}(\{y_{i}:i\in I(x)\})\subseteq G(x)\) for every \(x\in E\). By the Brouwer fixed point theorem, the continuous mapping \(f\circ\xi:\delta_{n}\rightarrow\delta_{n}\) has a fixed point \(p^{*}\in\delta_{n}\); that is, \(p^{*}=f(\xi(p^{*}))\). Let \(\overline{x}=\xi(p^{*})\). Then we have \(\overline{x}=\xi(p^{*})= \xi(f(\xi(p^{*})))=\xi(f(\overline{x}))\in G(\overline{x})\), which contradicts (b). Therefore, there must exist \(\widehat{x}\in K\) such that \(G(\widehat{x})=\emptyset\). 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 setvalued mapping to two setvalued mappings; (c) the condition that \(\bigcup_{y\in E}F^{1}(y)= \bigcup_{y\in E}\operatorname{int}F^{1}(y)\) 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

For each \(N\in{ \mathcal {F}}(E)\), there exists a nonempty compact convex subset \(E_{N}\) of E containing N such that \(E_{N}\setminus K\subseteq\bigcup_{y\in E_{N}}\operatorname{int}G^{1}(y)\).
3 Abstract generalized vector quasiequilibrium 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.
It is worthwhile noting that AGVQEP is motivated by the generalized vector quasiequilibrium problem introduced by Ansari and FloresBazán [7]. In particular, let \(E=\mathbb{R}^{n}\), W be a Hausdorff topological vector space, and let \(P:\mathbb{R} ^{n}\rightrightarrows W\) be a setvalued mapping such that, for each \(x\in\mathbb{R}^{n}\), \(P(x)\) is a closed and convex cone with \(\operatorname{int}P(x)\neq\emptyset\). Moreover, let \(C:\mathbb{R}^{n} \rightrightarrows W\) be a setvalued mapping defined by \(C(x)= \operatorname{int}P(x)\) for every \(x\in\mathbb{R}^{n}\). 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 setvalued 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 setvalued 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 \(H(x)\equiv E\) for every \(x\in E\), W is a Hausdorff topological vector space, and each \(C(x)\) is replaced by \(\operatorname{int}C(x)\), where \(C(x)\) is a closed and convex cone with \(\operatorname{int}C(x)\neq\emptyset\), 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
 (I)
Let \(H(x)\equiv E\) for every \(x\in E\). Then AGVQEP reduces to the abstract vector quasiequilibrium problem (for short, AVQEP), which consists in finding \(\widehat{x}\in E\) such that \(\psi(\widehat{x},y) \nsubseteq C(\widehat{x})\) for every \(y\in E\).
 (II)
If \(W=\mathbb{R}\), \(C(x)\equiv(\infty,0)\) for every \(x\in E\), and \(F=f\), where \(f:E\times E\rightarrow\mathbb{R}\) is a bifunction, then GVQEP reduces to the generalized scalar equilibrium problem (for short, GSEP), which is to find \(\widehat{x}\in E\) such that \(\widehat{x}\in H(\widehat{x})\) and \(f(\widehat{x},y)\geq0\) for every \(y\in H(\widehat{x})\). Furthermore, if \(H(x)\equiv E\) for every \(x\in E\), 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
 (i):

the set \(E^{*}=\{x\in E:x\notin H(x)\}\) is open in E;
 (ii):

for each \((x,y)\in E\times E\), \(\varsigma(x,y)\subseteq C(x)\) implies \(\psi(x,y)\subseteq C(x)\);
 (iii):

for each \(x\in E\), \(\psi(x,x)\nsubseteq C(x)\);
 (iv):

for each \(x\in E\), the set \(\{y\in E:\psi(x,y)\subseteq C(x)\}\) is convex;
 (v):

\(\bigcup_{y\in E}\{x\in H^{1}(y):\psi(x,y)\subseteq C(x)\}= \bigcup_{y\in E}\operatorname{int}\{x\in H^{1}(y):\psi(x,y)\subseteq C(x)\}\);
 (vi):

one of the following conditions holds:
 (vi)_{1} :

for each \(N\in{ \mathcal {F}}(E)\), there exists a nonempty compact convex subset \(E_{N}\) of E containing N such that$$E_{N}\setminus K\subseteq\bigcup_{y\in E_{N}} \operatorname{int}\bigl( \bigl(E^{*}\cap H^{1}(y) \bigr) \cup \bigl(H^{1}(y)\cap \bigl\{ x\in E:\psi(x,y) \subseteq C(x) \bigr\} \bigr) \bigr); $$
 (vi)_{2} :

there exists a point \(y_{0}\in E\) such that$$E\setminus K\subseteq \operatorname{int}\bigl( \bigl(E^{*}\cap H^{1}(y_{0}) \bigr)\cup \bigl(H^{1}(y_{0}) \cap \bigl\{ x\in E:\psi(x,y_{0})\subseteq C(x) \bigr\} \bigr) \bigr). $$
Proof
Now, we show that \(x\notin G(x)\) for every \(x\in E\). Indeed, if \(x\in E^{*}\), then by the definition of \(E^{*}\), we have \(x\notin H(x)=G(x)\); if \(x\notin E^{*}\), then by (iii), \(x\notin Q(x)\) and so, \(x\notin H(x)\cap Q(x)=G(x)\).

for each \(N\in{ \mathcal {F}}(E)\), there exists a nonempty compact convex subset \(E_{N}\) of E containing N such that$$\begin{aligned} E_{N}\setminus K \subseteq&\bigcup_{y\in E_{N}} \operatorname{int}\bigl( \bigl(E^{*} \cap H^{1}(y) \bigr) \cup \bigl(H^{1}(y)\cap \bigl\{ x\in E:\psi(x,y) \subseteq C(x) \bigr\} \bigr) \bigr) \\ =&\bigcup_{y\in E_{N}}\operatorname{int}G^{1}(y); \end{aligned}$$

there exists a point \(y_{0}\in E\) such that$$\begin{aligned} K \supseteq&E\setminus \operatorname{int}\bigl( \bigl(E^{*}\cap H^{1}(y_{0}) \bigr)\cup \bigl(H^{1}(y_{0}) \cap \bigl\{ x\in E:\psi(x,y_{0})\subseteq C(x) \bigr\} \bigr) \bigr) \\ =&\operatorname{cl}\bigl(E\setminus G^{1}(y_{0}) \bigr). \end{aligned}$$
Corollary 3.1
 (i):

the set \(E^{*}=\{x\in E:x\notin H(x)\}\) is open in E;
 (ii):

for each \(x\in E\), \(\psi(x,x)\nsubseteq C(x)\);
 (iii):

for each \(x\in E\), the set \(\{y\in E:\psi(x,y)\subseteq C(x)\}\) is convex;
 (iv):

\(\bigcup_{y\in E}\{x\in H^{1}(y):\psi(x,y)\subseteq C(x)\}= \bigcup_{y\in E}\operatorname{int}\{x\in H^{1}(y):\psi(x,y)\subseteq C(x)\}\);
 (v):

one of the following conditions holds:
 (v)_{1} :

for each \(N\in{ \mathcal {F}}(E)\), there exists a nonempty compact convex subset \(E_{N}\) of E containing N such that$$E_{N}\setminus K\subseteq\bigcup_{y\in E_{N}} \operatorname{int}\bigl( \bigl(E^{*}\cap H^{1}(y) \bigr) \cup \bigl(H^{1}(y)\cap \bigl\{ x\in E:\psi(x,y) \subseteq C(x) \bigr\} \bigr) \bigr); $$
 (v)_{2} :

there exists a point \(y_{0}\in E\) such that$$E\setminus K\subseteq \operatorname{int}\bigl( \bigl(E^{*}\cap H^{1}(y_{0}) \bigr)\cup \bigl(H^{1}(y_{0}) \cap \bigl\{ x\in E:\psi(x,y_{0})\subseteq C(x) \bigr\} \bigr) \bigr). $$
Proof
Let \(\varsigma=\psi\). 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
 (i):

the set \(E^{*}=\{x\in E:x\notin H(x)\}\) is open in E;
 (ii):

for each \(x\in E\), \(\psi(x,x)\nsubseteq C(x)\);
 (iii):

for each \(x\in E\), the set \(\{y\in E:\psi(x,y)\subseteq C(x)\}\) is convex;
 (iv):

for each \(y\in E\), the set \(\{x\in E:\psi(x,y)\subseteq C(x)\}\) is open in E;
 (v):

one of the following conditions holds:
 (v)_{1} :

for each \(N\in{ \mathcal {F}}(E)\), there exists a nonempty compact convex subset \(E_{N}\) of E containing N such that$$E_{N}\setminus K\subseteq\bigcup_{y\in E_{N}} \bigl( \bigl(E^{*}\cap H^{1}(y) \bigr) \cup \bigl(H^{1}(y)\cap \bigl\{ x\in E:\psi(x,y)\subseteq C(x) \bigr\} \bigr) \bigr); $$
 (v)_{2} :

there exists a point \(y_{0}\in E\) such that$$E\setminus K\subseteq \bigl( \bigl(E^{*}\cap H^{1}(y_{0}) \bigr)\cup \bigl(H^{1}(y _{0})\cap \bigl\{ x\in E: \psi(x,y_{0})\subseteq C(x) \bigr\} \bigr) \bigr). $$
Proof
By (iv), (v), and the fact that \(H^{1}(y)\) is open in E for every \(y\in E\), 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 quasiequilibrium 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 \(C(x)\) 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 \(\{x\in E:\psi(x,y) \subseteq C(x)\}\) is open in E for every \(y\in E\), 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
 (i):

for each \(x\in E\), \(\psi(x,x)\nsubseteq C(x)\);
 (ii):

for each \(x\in E\), the set \(\{y\in E:\psi(x,y)\subseteq C(x)\}\) is convex;
 (iii):

for each \(y\in E\), the set \(\{x\in E:\psi(x,y)\subseteq C(x)\}\) is open in E;
 (iv):

one of the following conditions holds:
 (iv)_{1} :

for each \(N\in{ \mathcal {F}}(E)\), there exists a nonempty compact convex subset \(E_{N}\) of E containing N such that \(E_{N}\setminus K\subseteq\bigcup_{y\in E_{N}}\{x\in E:\psi(x,y) \subseteq C(x)\}\);
 (iv)_{2} :

there exists a point \(y_{0}\in E\) such that \(E\setminus K\subseteq\{x\in E:\psi(x,y_{0})\subseteq C(x)\}\).
Proof
The conclusion of Corollary 3.3 follows from Corollary 3.2 by letting \(H(x)\equiv E\) for every \(x\in E\). This completes the proof. □
Remark 3.4
Let us give the following items:

for each \(x\in E\), \(\psi(x,\cdot)\) is convex with respect to \(C(x)\);

for each \(x\in E\), \(\psi(x,\cdot)\) is quasiconvexlike with respect to \(C(x)\).

for each \(y\in E\), \(\psi(\cdot,y)\) is upper semicontinuous on E with compact values;

the graph \(G_{r}(C)\) of C; i.e., \(\{(x,w)\in E\times W:w\in C(x)\}\) is an open set in \(E\times W\).
Indeed, it suffices to prove that the set \(\{x\in E:\psi(x,y)\nsubseteq C(x)\}\) is closed in E for every \(y\in E\). Let \(\{x_{ \alpha}\}\) be a net in \(\{x\in E:\psi(x,y)\nsubseteq C(x)\}\) such that \(x_{\alpha}\rightarrow x_{0}\). Since \(\psi(x_{\alpha},y) \nsubseteq C(x_{\alpha})\), there exists \(z_{\alpha}\in\psi(x _{\alpha},y)\) such that \(z_{\alpha}\notin C(x_{\alpha})\). Hence, we have \(z_{\alpha}\in W\setminus C(x_{\alpha})\). By the upper semicontinuity and compact values of ψ on E, it follows from Proposition 1 in [42] that there exists a subnet of \(\{z_{\alpha}\}\) with limit \(z_{0}\) and \(z_{0}\in\psi(x_{0},y)\). Without loss of generality, let us assume that \(z_{\alpha}\rightarrow z_{0}\in\psi(x_{0},y)\). On the other hand, since the graph \(G_{r}(C)\) of C is an open set in \(E\times W\), the setvalued mapping \(x\rightrightarrows W\setminus C(x)\) has a closed graph in \(E\times W\). It follows that \(z_{0}\in W\setminus C(x_{0})\) and so, \(z_{0}\notin C(x_{0})\). Thus, \(x_{0}\in\{x\in E: \psi(x,y)\nsubseteq C(x)\}\), which implies that the set \(\{x\in E:\psi(x,y)\nsubseteq C(x)\}\) is closed in E for every \(y\in E\). Therefore, the set \(\{x\in E:\psi(x,y)\subseteq C(x)\}\) is open in E for every \(y\in E\).
If \(W=\mathbb{R}\), \(C(x)\equiv(\infty,0)\) for every \(x\in E\) and \(F=f\), where \(f:E\times E\rightarrow\mathbb{R}\) 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
 (i):

the set \(E^{*}=\{x\in E:x\notin H(x)\}\) is open in E;
 (ii):

for each \(x\in E\), \(f(x,x)\geq0\);
 (iii):

for each \(x\in E\), the set \(\{y\in E:f(x,y)< 0\}\) is convex;
 (iv):

for each \(y\in E\), the set \(\{x\in E:f(x,y)\geq0\}\) is closed in E;
 (v):

one of the following conditions holds:
 (v)_{1} :

for each \(N\in{ \mathcal {F}}(E)\), there exists a nonempty compact convex subset \(E_{N}\) of E containing N such that$$E_{N}\setminus K\subseteq\bigcup_{y\in E_{N}} \bigl( \bigl(E^{*}\cap H^{1}(y) \bigr) \cup \bigl(H^{1}(y)\cap \bigl\{ x\in E:f(x,y)< 0 \bigr\} \bigr) \bigr); $$
 (v)_{2} :

there exists a point \(y_{0}\in E\) such that$$E\setminus K\subseteq \bigl( \bigl(E^{*}\cap H^{1}(y_{0}) \bigr)\cup \bigl(H^{1}(y _{0})\cap \bigl\{ x\in E:f(x,y_{0})< 0 \bigr\} \bigr) \bigr). $$
Corollary 3.5
 (i):

for each \(x\in E\), \(f(x,x)\geq0\);
 (ii):

for each \(x\in E\), \(\{y\in E:f(x,y)<0\}\) is convex;
 (iii):

for each \(y\in E\), the set \(\{x\in E:f(x,y)\geq0\}\) is closed in E;
 (iv):

one of the following conditions holds:
 (iv)_{1} :

for each \(N\in{ \mathcal {F}}(E)\), there exists a nonempty compact convex subset \(E_{N}\) of E containing N such that \(E_{N}\setminus K\subseteq\bigcup_{y\in E_{N}}\{x\in E:f(x,y)<0\}\);
 (iv)_{2} :

there exists a point \(y_{0}\in E\) such that \(E\setminus K\subseteq\{x\in E:f(x,y_{0})<0\}\).
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 \(L'=L\cap K\), which is a nonempty compact subset of K. Then it follows from \(K\setminus L=K\setminus L'\) and \(y_{0}\in L\cap K\subseteq K\) 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
 (i):

for each \(x\in E\), \(f(x,x)g(x,x)=0\);
 (ii):

for each \(x\in E\), \(\{y\in E:f(x,y)< g(x,y)\}\) is convex;
 (iii):

for each \(y\in E\), \(\{x\in E:f(x,y)>g(x,y)\}\) is convex;
 (iv):

the bifunction \(E\times E\ni(x,y)\mapsto f(x,y)g(x,y)\) is continuous;
 (v):

one of the following conditions holds:
 (v)_{1} :

for each \(N\in{ \mathcal {F}}(E)\), there exist two nonempty compact convex subsets \(E_{N}\), \(\widetilde{E}_{N}\) of E containing N such that \(E_{N}\setminus K_{1}\subseteq \bigcup_{y\in E_{N}}\{x\in E:f(x,y)< g(x,y)\}\) and \(\widetilde{E}_{N} \setminus K_{2}\subseteq\bigcup_{x\in\widetilde{E}_{N}}\{y\in E:f(x,y)>g(x,y) \}\);
 (v)_{2} :

there exist two points \(y_{0}, x_{0}\in E\) such that \(E\setminus K_{1}\subseteq\{x\in E:f(x,y_{0})< g(x,y_{0})\}\) and \(E\setminus K_{2}\subseteq\{y\in E:f(x_{0},y)>g(x_{0},y)\}\).
Proof
By setting \(g(x,y)\equiv0\) for every \((x,y)\in E\times E\), we obtain the following saddle point theorem from Theorem 3.2.
Theorem 3.3
 (i):

for each \(x\in E\), \(f(x,x)=0\);
 (ii):

for each \(x\in E\), \(\{y\in E:f(x,y)<0\}\) is convex;
 (iii):

for each \(y\in E\), \(\{x\in E:f(x,y)>0\}\) is convex;
 (iv):

the bifunction \(E\times E\ni(x,y)\mapsto f(x,y)\) is continuous;
 (v):

one of the following conditions holds:
 (v)_{1} :

for each \(N\in{ \mathcal {F}}(E)\), there exist two nonempty compact convex subsets \(E_{N}\), \(\widetilde{E}_{N}\) of E containing N such that \(E_{N}\setminus K_{1}\subseteq \bigcup_{y\in E_{N}}\{x\in E:f(x,y)<0\}\) and \(\widetilde{E}_{N}\setminus K_{2}\subseteq\bigcup_{x\in\widetilde{E}_{N}}\{y\in E:f(x,y)>0\}\);
 (v)_{2} :

there exist two points \(y_{0}, x_{0}\in E\) such that \(E\setminus K_{1}\subseteq\{x\in E:f(x,y_{0})<0\}\) and \(E\setminus K _{2}\subseteq\{y\in E:f(x_{0},y)>0\}\).
4 Weakly mixed variational inequality problem
Remark 4.1
By Corollary 3.4, we have the following existence theorem of solutions to WMVIP in noncompact Hadamard manifolds.
Theorem 4.1
 (i):

the set \(E^{*}=\{x\in E:x\notin H(x)\}\) is open in E;
 (ii):

for each \(x\in E\), the function \(E\ni y\mapsto\langle \sigma(x),\operatorname{exp}^{1}_{x}y\rangle+\varphi(y)\) is convex;
 (iii):

one of the following conditions holds:
 (iii)_{1} :

for each \(N\in{ \mathcal {F}}(E)\), there exists a nonempty compact convex subset \(E_{N}\) of E containing N such that$$E_{N}\setminus K\subseteq\bigcup_{y\in E_{N}} \bigl( \bigl(E^{*}\cap H^{1}(y) \bigr) \cup \bigl(H^{1}(y)\cap \bigl\{ x\in E: \bigl\langle \sigma(x),\operatorname{exp}^{1} _{x}y \bigr\rangle < \varphi(x) \varphi(y) \bigr\} \bigr) \bigr); $$
 (iii)_{2} :

there exists a point \(y_{0}\in E\) such that$$E\setminus K\subseteq \bigl( \bigl(E^{*}\cap H^{1}(y_{0}) \bigr)\cup \bigl(H^{1}(y _{0})\cap \bigl\{ x\in E: \bigl\langle \sigma(x), \operatorname{exp}^{1}_{x}y_{0} \bigr\rangle < \varphi(x) \varphi(y_{0}) \bigr\} \bigr) \bigr). $$
Proof

for each \(N\in{ \mathcal {F}}(E)\), there exists a nonempty compact convex subset \(E_{N}\) of E containing N such that$$E_{N}\setminus K\subseteq\bigcup_{y\in E_{N}} \bigl( \bigl(E^{*}\cap H^{1}(y) \bigr) \cup \bigl(H^{1}(y)\cap \bigl\{ x\in E:f(x,y)< 0 \bigr\} \bigr) \bigr); $$

there exists a point \(y_{0}\in E\) such that$$E\setminus K\subseteq \bigl( \bigl(E^{*}\cap H^{1}(y_{0}) \bigr)\cup \bigl(H^{1}(y _{0})\cap \bigl\{ x\in E:f(x,y_{0})< 0 \bigr\} \bigr) \bigr). $$
Remark 4.2
 \((\mathrm{iii})^{\prime}_{1}\) :

For each \(N\in{ \mathcal {F}}(E)\), there exists a nonempty compact convex subset \(E_{N}\) of E containing N such that for all \(x\in E_{N}\setminus K\), there exists \(y\in E_{N}\) such that \(y\in H(x)\) and \(\langle\sigma(x),\operatorname{exp}^{1}_{x}y\rangle< \varphi(x)\varphi(y)\);
 \((\mathrm{iii})^{\prime}_{2}\) :

There exists a point \(y_{0}\in E\) such that \(y_{0}\in H(x)\) and \(\langle\sigma(x),\operatorname{exp}^{1}_{x}y_{0} \rangle<\varphi(x)\varphi(y_{0})\) for every \(x\in E\setminus K\).
If \(H(x)\equiv E\) for every \(x\in E\), then we can obtain the following corollary from Theorem 4.1.
Corollary 4.1
 (i):

for each \(x\in E\), the function \(E\ni y\mapsto\langle \sigma(x),\operatorname{exp}^{1}_{x}y\rangle+\varphi(y)\) is convex;
 (ii):

one of the following conditions holds:
 (ii)_{1} :

for each \(N\in{ \mathcal {F}}(E)\), there exists a nonempty compact convex subset \(E_{N}\) of E containing N such that \(E_{N}\setminus K\subseteq\bigcup_{y\in E_{N}}\{x\in E:\langle \sigma(x),\operatorname{exp}^{1}_{x}y\rangle<\varphi(x)\varphi(y)\}\);
 (ii)_{2} :

there exists \(y_{0}\in E\) such that \(E\setminus K \subseteq\{x\in E:\langle\sigma(x),\operatorname{exp}^{1}_{x}y_{0}\rangle <\varphi(x)\varphi(y_{0})\}\).
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
 (i)_{1} :

E is compact;
 (i)_{2} :

there exists \(y_{0}\in E\) such that, for each \(u\in T_{y_{0}}E\), the following condition holds:$$\lim_{d(y_{0},x)\rightarrow+\infty}\frac{\langle\sigma(y_{0}), \operatorname{exp}_{y_{0}}^{1}x\rangle+\langle\sigma(x),\operatorname{exp}_{x} ^{1}y_{0}\rangle}{d(y_{0},x)}<  \bigl( \bigl\Vert \sigma(y_{0}) \bigr\Vert +\Vert u\Vert \bigr). $$
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].
5 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.
Declarations
Acknowledgements
The authors thank the referees for their many valuable suggestions and helpful comments which improved the exposition of the paper. The first author was supported by the Social Science Fund of Jiangsu Province (16GLB009) and by the Young and MiddleAged Academic Leaders Program of the ‘Qinglan Project’ of Jiangsu Province (SJS201423). The second author was supported by the National Social Science Fund of China (14BGL216).
Open Access This article is distributed under the terms of the Creative Commons Attribution 4.0 International License (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted use, distribution, and reproduction in any medium, provided you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license, and indicate if changes were made.
Authors’ Affiliations
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