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 f : X × X → R be a bifunction. The scalar equilibrium problem (for short, SEP) is to find x ∈ K such that f ( x, y) ≥  for every y ∈ 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, [-] 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 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 []). Hadamard manifolds have turned out to be a suitable frame- 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 , we introduce notation, definitions, and preliminary results used in the paper. In Section , 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  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 .

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 [-].
Let R denote the set of all real numbers. Let X be a set. Then we let F(X) represent the family of nonempty finite subsets of X. Let A be a subset of a topological space X, and then let int A and cl A denote the interior of A in X and the closure of A in X, respectively. Moreover, if A ⊆ B ⊆ X, then int B A (respectively, 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 set-valued mapping T : X ⇒ Y , the inverse T - : Y ⇒ X of T is defined by T - (y) = {x ∈ X : y ∈ T(x)} for every y ∈ Y . Let X and Y be two topological spaces. Then a set-valued mapping T : X ⇒ Y is said to be upper semicontinuous (respectively, lower semicontinuous) on X iff, for each x ∈ X and each open set . Let δ n be the standard n-dimensional simplex with vertices {e  , e  , . . . , e n } and for each I ⊆ {, , . . . , n}, let δ |I|- denote the simplex with vertices {e j : j ∈ I}, where |I| denotes the cardinality of I.
Let E be a simply-connected m-dimensional manifold. For each x ∈ E, we denote by T x E the tangent space of E at x. Let TE = x∈E T x E denote the tangent bundle of E, which is naturally a manifold. A vector field on E is a mapping σ : E → TE such that σ (x) ∈ T x E for every x ∈ E. Let ·, · x denote the scalar product on T x E with the associated norm · x , 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 γ : [a, b] → E joining x to y (i.e., γ (a) = x and γ (b) = y), we can define the length of γ as follows: Then, for any two points x, y ∈ E, the Riemannian distance d(x, y) is defined by γ is a piecewise smooth curve joining x and y .
A vector field σ is said to be parallel along γ if ∇ γ σ = , where ∇ is the Levi-Civita connection associated with (E, ·, · ) and γ is a smooth curve in E. If γ itself is parallel along γ , then γ is called a geodesic, and in this case, γ is constant. When γ = , γ is said to be normalized. A geodesic which joins x to y in E is said to be minimal if its length equals d(x, y). A Riemannian manifold E is geodesically complete if, for each x ∈ E, all geodesics starting from x are defined for every t ∈ R. By the Hopf-Rinow theorem (see []), 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.
Then exp x : T x E → E is a diffeomorphism, and for any two points x, y ∈ E, there exists a unique minimal normalized geodesic γ x,y = exp x t exp - x y for every t ∈ [, ] joining x to y.
So from now on, a geodesic means the unique minimal normalized one.
Definition . ([]) A set C ⊆ E is said to be convex iff, for any two points x, y ∈ C, the geodesic joining x to y is contained in C; that is, γ x,y = exp x t exp - x y ∈ C for every t ∈ [, ].

Definition . ([])
A real-valued function f : E → R is said to be convex iff, for any two points x, y ∈ E, we have f (exp x t exp - x y) ≤ tf (x) + (t)f (y) for every t ∈ [, ]. f is said to be concave iff -f is convex.
Definition . Let Q be a nonempty subset of V . The set Q is called a convex cone iff Q + Q ⊆ Q and λQ ⊆ Q for λ ≥ . Definition . Let Q ⊆ V be a nonempty convex cone. A set-valued mapping F : E ⇒ V is said to be convex with respect to Q iff, for each x, y ∈ E and each t ∈ [, ], we have Remark . Definition . includes Definition . as a special case. In fact, when F is a single-valued mapping, V = R, and Q = (-∞, ], Definition . coincides with Definition .. Also Definition . generalizes Definition . of Zhou and Huang [] in the following aspects: () from a single-valued mapping to a set-valued mapping; () from the convex cone in Euclidean spaces to the convex cone in Hausdorff topological vector spaces.
Definition . Let Q ⊆ V be a nonempty convex cone. A set-valued mapping F : E ⇒ V is said to be quasiconvex-like with respect to Q iff, for each x, y ∈ E and each t ∈ [, ], we The convex hull of A is defined by the smallest convex subset of E containing A and denoted by conv(A).
Definition . Let X be a topological space. A set-valued mapping F : X ⇒ E is called a Fan-Browder mapping iff the following conditions are fulfilled: Definition . Let G : E ⇒ E be a set-valued mapping. Then G is said to be of class H E iff the following conditions are satisfied: (c) there exists a set-valued mapping F : Then the following statements hold: Then the following statements are equivalent: ) is the minimal geodesic joining q  to q  ; (iv) the sectional curvature of E is identically zero (i.e., E is isometric to the usual Euclidean space).
Let f : E → R be a real-valued function. The subdifferential of f is the set-valued mapping ∂f : E ⇒ TE defined by ∂f (x) = {u ∈ T x E : u, exp - x y ≤ f (y)f (x), ∀y ∈ E} for every x ∈ E and its elements are called subgradients. For each x ∈ E, the subdifferential ∂f (x) is a closed convex subset (possibly empty) of T x E. Let D(∂f ) = {x ∈ E : ∂f (x) = ∅} denote the domain of ∂f .
The following lemma guarantees the existence of subgradients for convex functions. 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 . Let G : E ⇒ E be of class H E and K be a nonempty compact subset of E. Suppose that one of the following conditions holds:
We prove Lemma . by considering the following two cases. Case I. Suppose that (i)  holds. We proceed by contradiction. If the conclusion of Lemma . is false, then G(x) = ∅ for every x ∈ K . By (), we have F(x) = ∅ for every x ∈ K , which implies that K ⊆ y∈E F - (y). Moreover, it follows from () and () that . Consequently, we can define a continuous mapping f : Hence, it follows from the convexity of G(x) that conv( Following the same argument as in the proof of Lemma . in [], we can conclude that there exists a continuous mapping ξ : Since the mapping f • ξ : δ n → δ n is continuous, it follows from the Brouwer fixed point theorem that there exists p * ∈ δ n such that Hence, by the convexity of G(x), we have conv( By using the same method as in the proof of Lemma . in [], we can conclude that there exists a continuous mapping ξ : δ n → D({y  , y  , . . . , y n }) satisfying where D({y  , y  , . . . , y n }) is the same as in the previous case. Now, let g = ξ • f . Then g has the following property: By the Brouwer fixed point theorem, the continuous mapping f • ξ : δ n → δ n has a fixed point p * ∈ δ n ; that is, Remark . (i)  of Lemma . can be replaced by the following equivalent condition. •

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 H : We consider the following AGVQEP as follows: It is worthwhile noting that AGVQEP is motivated by the generalized vector quasiequilibrium problem introduced by Ansari and Flores-Bazán []. In particular, let E = R n , W be a Hausdorff topological vector space, and let P : R n ⇒ W be a set-valued mapping such that, for each x ∈ R n , P(x) is a closed and convex cone with int P(x) = ∅. Moreover, let C : R n ⇒ W be a set-valued mapping defined by C(x) = -int P(x) for every x ∈ R n . Then AGVQEP retrieves a particular instance of the equilibrium problem in []. 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 setvalued mapping must be satisfied; for example, the equilibrium problems of the general- Remark . Some other special cases of AGVQEP are given as follows.
(I) Let H(x) ≡ E for every x ∈ E. Then AGVQEP reduces to the abstract vector quasi-equilibrium problem (for short, AVQEP), which consists in finding x ∈ E such that ψ( x, y) C( x) for every y ∈ E.
Then AGVQEP has at least a solution in K .
Proof Define two set-valued mappings L, Q : E ⇒ E by Moreover, let us define two set-valued mappings F, G : E ⇒ E by is convex by (iv) and each H(x) is convex by the definition of a Fan-Browder mapping, it follows that G(x) is convex for every x ∈ E.
For each y ∈ E, we have We claim that y∈E G - (y) = y∈E int G - (y). In fact, it is clear that y∈E int G - (y) ⊆ y∈E G - (y). In order to prove that y∈E G - (y) ⊆ y∈E int G - (y), we take x  ∈ y∈E G - (y). Then there exists y * ∈ E such that If x  ∈ E * ∩ H - (y * ), 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 Combining these two cases, we can conclude that y∈E G - (y) ⊆ y∈E int G - (y). Therefore, we have y∈E G - (y) = y∈E int G - (y). Now, we show that x / ∈ G(x) for every x ∈ E. Indeed, if x ∈ E * , then by the definition of By (vi) and the above fact, we can see that one of the following conditions holds: • for each N ∈ F(E), there exists a nonempty compact convex subset E N of E containing N such that • there exists a point y  ∈ E such that Therefore, by Lemma . and Remark ., there exists and ψ( x, y) C( x) for every y ∈ H( x). Thus, the conclusion of Theorem . holds and the proof is complete.

Corollary . Let K ⊆ E be a nonempty compact set and W be a nonempty set. Let ψ : E × E ⇒ W , C : E ⇒ W be two set-valued mappings and H : E ⇒ E be a Fan-Browder mapping. Assume that
(v) one of the following conditions holds: (v)  for each N ∈ F(E), there exists a nonempty compact convex subset E N of E containing N such that

Then AGVQEP has at least a solution in K .
Proof Let ς = ψ. It is easy to see that all the conditions of Theorem . are satisfied. Therefore, it follows from Theorem . that AGVQEP has at least a solution in K . Thus, the result holds and the proof of Corollary . is complete.

Corollary .
(v) one of the following conditions holds: (v)  for each N ∈ F(E), there exists a nonempty compact convex subset E N of E containing N such that

Then AGVQEP has at least a solution in K .
Proof By (iv), (v), and the fact that H - (y) is open in E for every y ∈ E, we can see that (iv) and (v) of Corollary . hold. Therefore, by Corollary ., AGVQEP has at least a solution in K . This completes the proof.

Remark .
(iv) one of the following conditions holds: Then AVQEP has at least a solution in K .
Proof The conclusion of Corollary . follows from Corollary . by letting H(x) ≡ E for every x ∈ E. This completes the proof.
Remark . Let us give the following items: () If W is a real Hausdorff topological vector space and {C(x) : y ∈ E} is a family of nonempty convex cones, then (iv) of Theorem ., (iii) of Corollaries .-., and (ii) of Corollary . can be replaced by one of the following conditions: In fact, we consider the first assumption. Let x ∈ E be any given. In order to prove that the set {y ∈ E : ψ(x, y) ⊆ C(x)} is convex, we assume that y  , y  ∈ {y ∈ E : ψ(x, y) ⊆ C(x)} and t ∈ [, ]. Applying this assumption and the fact that each C(x) is a convex cone, we have which implies that exp y  t exp - y  y  ∈ {y ∈ E : ψ(x, y) ⊆ C(x)} for every t ∈ [, ] and so, the set {y ∈ E : ψ(x, y) ⊆ C(x)} is convex for every x ∈ E. Now, we prove that the desired conclusion holds under the condition that the second assumption is satisfied. Indeed, let x ∈ E be fixed and then let y  , y  ∈ {y ∈ E : ψ(x, y) ⊆ C(x)} and t ∈ [, ] be any given. By the definition of a quasiconvex-like mapping, we have Since y  , y  ∈ {y ∈ E : ψ(x, y) ⊆ C(x)}, we have ψ(x, y  ) ⊆ C(x) and ψ(x, y  ) ⊆ C(x). Therefore, given C(x) + C(x) ⊆ C(x), which is obtained by using the property of a convex cone, it follows from the above formulas that ψ(x, exp y  t exp - y  y  ) ⊆ C(x), and this shows that the set {y ∈ E : ψ(x, y) ⊆ C(x)} is convex for every x ∈ E.
() If W is a Hausdorff topological space, then (iv) of Corollary . and (iii) of Corollary . can be replaced by the following conditions: • for each y ∈ E, ψ(·, y) is upper semicontinuous on E with compact values; • the graph G r (C) of C; i.e., Indeed, it suffices to prove that the set {x ∈ E : ψ(x, y) C(x)} is closed in E for every y ∈ E. Let {x α } be a net in {x ∈ E : ψ(x, y) C(x)} such that x α → x  . Since ψ(x α , y) C(x α ), there exists z α ∈ ψ(x α , y) such that z α / ∈ C(x α ). Hence, we have z α ∈ W \C(x α ). By the upper semicontinuity and compact values of ψ on E, it follows from Proposition  in [] that there exists a subnet of {z α } with limit z  and z  ∈ ψ(x  , y). Without loss of generality, let us assume that z α → z  ∈ ψ(x  , y). On the other hand, since the graph G r (C) of C is an open set in E × W , the set-valued mapping x ⇒ W \ C(x) has a closed graph in E × W . It follows that z  ∈ W \ C(x  ) and so, z  / ∈ C(x  ). Thus, x  ∈ {x ∈ E : ψ(x, y) C(x)}, which implies that the set {x ∈ E : ψ(x, y) C(x)} is closed in E for every y ∈ E. Therefore, the set () If (iv)  of Corollary . 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 ., we can see that the above set is nonempty. Furthermore, it follows from (iv)  of Corollary . that Together with (iii) of Corollary ., 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 W = R, C(x) ≡ (-∞, ) for every x ∈ E and F = f , where f : E × E → R is a bifunction, then Corollaries . and . reduce to the following existence results of solutions to GSEP and SEP, respectively. Corollary . Let K ⊆ E be a nonempty compact set, H : E ⇒ E be a Fan-Browder mapping, and f : E × E → R be a bifunction. Assume that (v) one of the following conditions holds: (v)  there exists a point y  ∈ E such that Then GSEP has at least a solution in K .

Corollary . Let K ⊆ E be a nonempty compact set and f : E × E → R be a bifunction.
Assume that (iv) one of the following conditions holds: Then SEP has at least a solution in K . As an application of Corollary ., we have the following perturbed saddle point theorem in noncompact Hadamard manifolds.
Theorem . Let K  , K  ⊆ E be two nonempty compact sets and f , g : E × E → R be two bifunctions. Assume that

y) is continuous;
(v) one of the following conditions holds: Then f has a perturbed saddle point ( x, y) ∈ K  × K  ; i.e., Proof Define a bifunction h  : E ×E → R by h  (x, y) = f (x, y)-g(x, y) for every (x, y) ∈ E ×E. By (i), (ii), (iv) and the first parts of (v)  and (v)  , we can see that all the conditions of Corollary . are satisfied. Thus, by Corollary ., there exists x ∈ K  such that h  ( x, y) ≥  for every y ∈ E. Define a bifunction h  : E × E → R by h  (y, x) = g(x, y)f (x, y) for every (y, x) ∈ E × E. Then it follows from (i), (iii), (iv), and the second parts of (v)  and (v)  that all the hypotheses of Corollary . are fulfilled. Thus, by Corollary . again, there exists y ∈ K  such that h  ( y, x) ≥  for every x ∈ E. Therefore, we have f ( x, y)g( x, y) =  and It follows from the above inequality that This completes the proof.
By setting g(x, y) ≡  for every (x, y) ∈ E × E, we obtain the following saddle point theorem from Theorem ..
Theorem . Let K  , K  ⊆ E be two nonempty compact sets and f : E × E → R be a bifunction. Assume that (v) one of the following conditions holds: Then f has a saddle point ( x, y) ∈ K  × K  ; i.e., In particular, inf y∈E sup x∈E f (x, y) = sup x∈E inf y∈E f (x, y).

Weakly mixed variational inequality problem
In this section, inspired by the idea due to Colao (ii) for each x ∈ E, the function E y → σ (x), exp - x y + ϕ(y) is convex; (iii) one of the following conditions holds: (iii)  for each N ∈ F(E), there exists a nonempty compact convex subset E N of E containing N such that Then WMVIP has at least a solution in K .
Proof Define a bifunction f : It is clear that f (x, x) ≥  for every x ∈ E. Thus, (ii) of Corollary . is satisfied. By Lemma . and the continuous properties of σ and ϕ, one can see that the function E x → f (x, y) is upper semicontinuous. Therefore, the set {x ∈ E : f (x, y) ≥ } is closed in E for every x ∈ E, which implies that (iv) of Corollary . holds. Moreover, it follows from (iii) and the definition of f that one of the following conditions holds: • for each N ∈ F(E), there exists a nonempty compact convex subset E N of E containing N such that • there exists a point y  ∈ E such that Thus, (v) of Corollary . is satisfied. Now, in order to show that (iii) of Corollary . holds, we are ready to prove that {y ∈ E : f (x, y) < } is convex for every x ∈ E. Indeed, let x ∈ E be fixed and let y  , which implies that the set {y ∈ E : f (x, y) < } is convex for every x ∈ E. As a consequence of Corollary ., there exists a point x ∈ K such that x ∈ H( x) and f ( x, y) = σ ( x), exp - x y + ϕ(y)ϕ( x) ≥  for every y ∈ H( x); that is, WMVIP has at least a solution in K . This completes the proof.
Remark . The following two conditions are stronger than (iii)  and (iii)  of Theorem ., respectively.
(iii)  For each N ∈ F(E), there exists a nonempty compact convex subset E N of E containing N such that for all x ∈ E N \ K , there exists y ∈ E N such that y ∈ H(x) and Corollary . Let ϕ : E → R be a convex lower semicontinuous function and σ : E → TE be a continuous vector field such that the function E y → σ (x), exp - x y + ϕ(y) is convex for every x ∈ E. Suppose that one of the following conditions holds: (i)  E is compact; (i)  there exists y  ∈ E such that, for each u ∈ T y  E, the following condition holds: Then MVIP has at least a solution.
Proof Suppose that (i)  holds. For each N ∈ F(E), let E N = E = K . Thus, both (ii)  and (ii)  of Corollary . are satisfied automatically. Therefore, MVIP has at least a solution in E. Now, we suppose that (i)  is satisfied. Let y  ∈ E satisfy (i)  . Since ϕ is convex, it follows from Lemma . that D(∂ϕ) = E. Thus, we may choose u  ∈ ∂ϕ(y  ) ⊆ T y  E; i.e., the subdifferential of ϕ at y  . For each x ∈ E, by the generalized Cauchy-Schwarz inequality, we have It follows from the above relation that σ (y  ), exp - y  x + ϕ(y  )ϕ(x) ≤ σ (y  ) + u  d(y  , x), ∀x ∈ E.
Let K = B(y  , r). 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)  of Corollary . holds. Therefore, by Corollary ., MVIP has at least a solution. This completes the proof.

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.