Equilibrium problems on Riemannian manifolds with applications

arXiv:1801.02752v1 [math.OC] 9 Jan 2018 EQUILIBRIUM PROBLEMS ON RIEMANNIAN MANIFOLDS WITH APPLICATIONS CHONG LI∗, XIANGMEI WANG†, GENARO L ´ OPEZ‡,AND JEN-CHIH YAO§ Abstract. We study the equilibrium problem on general Riemannian manifolds. The results on existence of solutions and on the convex structure of the solution set are established. Our approach consists in relating the equilibrium problem to a suitable variational inequality problem on Riemannian manifolds, and is completely different from previous ones on this topic in the literature. As applications, the corresponding results for the mixed variational inequality and the Nash equilibrium are obtained. Moreover, we formulate and analyze the convergence of the proximal point algorithm for the equilibrium problem. In particular, correct proofs are provided for the results claimed in J. Math. Anal. Appl. 388, 61-77, 2012 (i.e., Theorems 3.5 and 4.9 there) regarding the existence of the mixed variational inequality and the domain of the resolvent for the equilibrium problem on Hadamard manifolds. Key words. Riemannian manifold, equilibrium problem, variational inequality problem, proximal point algorithm AMS subject classifications. Primary, 90C25; Secondary, 65K05 1. Introduction. Let Xbe a metric space, Q⊆Xa nonempty subset and F:X× X→(−∞,+∞] a bifunction. The equilibrium problem (introduced by Blum and Oettli in [6]), abbreviated as EP, associated to the pair (F,Q) is to find a point ¯x∈Qsuch that (1.1) F(¯x, y)≥0 for any y∈Q. As shown in [6, 36], EP contains, as special cases, optimization problems, complementarity problems, fixed point problems, variational inequalities and problems of Nash equilibria; and it has been broadly applied in many areas, such as economics, image reconstruction, transportation, network, and elasticity. In recent years, EP has been studied extensively, including the issues regarding existence of solutions and iterative algorithms for finding solutions; see e.g., [5, 6, 12, 15, 21]. Since the classical existence results in EPs work for the case when Qis a convex set and the bifunction Fis convex in the second variable, some authors focused on exploiting ∗School of Mathematical Sciences, Zhejiang University, Hangzhou 310027, P. R. China ([email protected]). Research of this author was supported in part by the National Natural Science Foundation of China (grant 11571308). †College of Mathematics and Statistics, Guizhou University, Guiyang 550025, P. R. China ([email protected]). Research of this author was supported in part by the National Natural Science Foundation of China (grant 11661019) and the Natural Science Foundation of Guizhou Province, China (grant 20161039). ‡Department of Mathematical Analysis, University of Sevilla. 1160, 41080-Sevilla, Spain (glop[email protected]). Research of this author was supported in part by the DGES, grant MTM2015-65242-C2-1P and Junta de Andaluc´ıa, grant FQM-127. §China Medical University, Taichung, Taiwan 40402, R.O.C. and Department of Mathematics, King Abdulaziz University, P.O. Box 80203, Jeddah 21589, Saudi Arabia (yao[email protected]w). Research of this author was supported in part by the Grant MOST 102-2115-M-039-003-MY3. 1 2EQUILIBRIUM PROBLEMS ON RIEMANNIAN MANIFOLD the possible existence without the convexity assumption; see e.g., [3, 4, 9, 20]. One useful approach used in [23] and [24] is to embed the underlying nonconvex and/or nonsmooth Nash/Nash-type equilibrium problems into a suitable Riemannain manifold Mto study the existence and the location problems of the Nash/Nash-type equilibrium points. In particular, Krist´aly established in [23], the existence results for Nash equilibrium points associated to strategy sets {Qi⊆Mi}i∈Iand loss-functions {fi}i∈I(I:= {1,2,...,n}) under the following assumption: (AK): each Qiis a compact and geodesic convex set of Mifor all i∈I (see item (e) in Definition 2.3 for the notion of the geodesic convexity). This class of approaches has also been used extensively in many optimization problems since some nonconvex and/or nonsmooth problems of the constrained optimization in Rncan be reduced to convex and/or smooth unconstrained optimization problems on appropriate Riemannian manifolds; see, for examples, [14, 30, 31, 40, 42]. More about optimization techniques and notions in Riemannian manifolds can be found in [1, 2, 8, 18, 26, 27, 28, 43, 45] and the bibliographies therein. For the equilibrium problem (1.1) on a Riemannian manifold M, Colao et. al, by generalizing the KKM lemma to a Hadamard manifold, established an existence result (i.e., [11, Theorem 3.2]) for solutions of EP under the following assumptions: (AC-1): Mis a Hadamard manifold and Qis closed and convex; (AC-2): the set {y∈Q:F(x, y)≤0}is convex for any x∈Q (which was extensively studied in [47] for the generalized vector equilibrium problem). This existence result was applied there to solve the following problems: (P1). the existence problem of solutions for mixed variational inequality problems; (P2). the well-definedness of the resolvent and the proximal point algorithm for solving EP; (P3). the existence problem of fixed points for set-valued mappings; (P4). the existence problem of solutions for Nash equilibrium problems. However, the applications to problems (P1)-(P3) above rely heavily on the following claim: (1.2) the function y7→ hux,exp−1 xyiis quasi-convex, where x∈Mand ux∈TxM; see the proofs for Theorems 3.5, 3.10 and 4.9 in [11]. Unfortunately, unlike in the linear space setting, claim (1.2) is not true in general as pointed out in [42, Theorem 2.1, p. 299] or [25]. Note that, for any x∈M, the function defined by (1.2) is convex at x(see Definition 2.4 (i)); this motivates us to introduce the new notion of the point-wise (weak) convexity for a bifunction on general manifolds (see Definition 3.1 (c)). Our main purpose in the present paper is to develop a new approach (based on the new notion and the work in [28]) to study the issue on the existence and structure of solutions for equilibrium problems on general Riemannian manifolds, which, in particular, covers problems (1), (2) and (4) as special cases. In our approach, rather than assumptions (AK) or (AC-1)-(AC-2), we make the following ones on the involved Qand F: XIANGMEI WANG, CHONG LI, GENARO L ´ OPEZ AND JEN-CHIH YAO 3 •Qis a closed and weakly convex subset of Riemannian manifold M; •Fis point-wise weakly convex on Q. The technique used in the present paper for proving the main results is completely different from the ones used in [11, 47, 23]. Actually, our technique here is mainly focused on establishing the equivalence between the EP and a suitable variational inequality problem; and then apply the corresponding results in [28] for the variational inequality problem to study the existence of solutions and the convex structure of the solution set of the EP . As applications to problems (P1), (P2) and (P4), we obtained some results on the existence of solutions and convexity of the solution sets for mixed variational inequality problems and Nash equilibrium problems (see Theorem 5.1 and 5.2), as well as the convergence of the proximal point algorithm for solving EP. In particular, the existence result for mixed variational inequality problems and the well-definedness results of the resolvent for solving EP on Hadamard manifolds provide correct proofs for the corresponding ones [11, Theorem 3.5 and 4.9] (see the explanations before Corollaries 4.5 and 5.2 in Section 4 and 5, respectively); while the existence result for the Nash equilibrium on general manifolds relaxes the geodesic convexity assumption made on {Qi}in [23, Theorem 1.1] to the weaker one that each Qiis weakly convex. It is worthwhile to notice that the geodesic convexity assumption for {Qi} in [23] prevents its application to some special but important Riemannian manifolds, such as compact Stiefel manifolds St(p, n) and Grassmann manifolds Grass(p, n) (p < n), in which there is no geodesic convex subset; see Remark 5.1 in Section 5. Moreover, to our best knowledge, the convex structure results on the solution set for mixed variational inequality problems and Nash equilibrium problems are new even in Hadamard manifold settings. The paper is organized as follows. In the next section, we introduce some basic notions and notations on Riemannian manifolds, some properties about the (weakly) convex function and the results about the VIP in [28] which will be used in our approach. In section 3, we establish the existence and the uniqueness result of the solution and the convexity of the solution set of the EP on general Riemannian manifolds. Following these, the formulation of the proximal point algorithm for the equilibrium problem on general Riemnannian manifolds is given and the convergence property about the algorithm is analyzed in section 4. The last section is devoted to the applications to the Nash equilibrium problem and the mixed variational inequality problem. 2. Notations and preliminary results. 2.1. Background of Riemannian manifolds. The notations used in the present paper are standard; and the readers are referred to some textbooks for more details, for example, [13, 39, 42]. Let Mbe a connected n-dimensional Riemannian manifold with the Levi-Civita connection ∇on M. Let x∈M, and let TxMstand for the tangent space at xto Mendowed with the scalar product h,ixand the associated norm k.kx, where the subscript xis sometimes 4EQUILIBRIUM PROBLEMS ON RIEMANNIAN MANIFOLD omitted. Thus the tangent bundle, denoted by T M, is defined by T M := {(x, v) : x∈M, v ∈TxM}. Fix y∈M, and let γ: [0,1] →Mbe a piecewise smooth curve joining xto y. Then, the arc-length of γis defined by l(γ) := R1 0k˙γ(t)kdt, while the Riemannian distance from xto yis defined by d(x, y) := infγl(γ), where the infimum is taken over all piecewise smooth curves γ: [0,1] →Mjoining xto y. We use B(x, r) and B(x, r) to denote, respectively, the open metric ball and the closed metric ball at xwith radius r, that is, B(x, r) := {y∈M: d(x, y)< r}and B(x, r) := {y∈M: d(x, y)≤r}. A vector field Vis said to be parallel along γif ∇˙γV= 0. In particular, for a smooth curve γ, if ˙γis parallel along itself, then γis called a geodesic, that is, a smooth curve γis a geodesic if an only if ∇˙γ˙γ= 0. A geodesic γ: [0,1] →Mjoining xto yis minimal if its arc-length equals its Riemannian distance between xand y. By the Hopf-Rinow theorem [13], if Mis complete, then (M, d) is a complete metric space, and there is at least one minimal geodesic joining xto y. One of the important structures on Mis the exponential map expx:TxM→M, which is defined at x∈Mby expxv=γv(1, x) for each v∈TxM, where γv(·, x) is the geodesic starting at xwith velocity v. Then, expxtv =γv(t, x) for each real number t. Another useful tool is the parallel transport Pγ,·,·on the tangent bundle T M along a geodesic γ, which is defined by Pγ,γ(b),γ(a)(v) = V(γ(b)) for any a, b ∈Rand v∈Tγ(a)M, where Vis the unique vector field satisfying V(γ(a)) = vand ∇˙γ(t)V= 0 for all t. Then, for any a, b ∈R,Pγ,γ(b),γ(a)is an isometry from Tγ(a)Mto Tγ(b)M. We will write Py,x instead of Pγ,y,x in the case when γis a minimal geodesic joining xto yand no confusion arises. The following lemma can be checked easily. Lemma 2.1. Let x0∈Mand {xk} ⊂ Mbe such that limk→∞ xk=x0. Let u0, v0∈ Tx0Mand let {uk},{vk}be sequences with each uk, vk∈TxkMsuch that uk→u0and vk→v0. Then expxkuk→expx0u0and huk, vki → hu0, v0i. The following result is known in any textbook about Riemannain geometry, see e.g., [13, Corollary, p. 73] or [39, Exercise 5, p. 39]. Lemma 2.2. Let γ: [a, b]→Mbe a piecewise differentiable curve. If l(γ) = d(γ(a), γ(b)), then γis a geodesic joining γ(a)and γ(b). Consider a set Q⊆Mand x, y ∈Q. The set of all geodesics γ: [0,1] →Mwith γ(0) = xand γ(1) = ysatisfying γ([0,1]) ⊆Qis denoted by ΓQ xy, that is ΓQ xy := {γ: [0,1] →Q:γ(0) = x, γ(1) = yand ∇˙γ˙γ= 0}. In particular, we write Γxy for ΓM xy, and Γxy is nonempty for all x, y ∈Mprovided that M is complete. Furthermore, for a subset Γ0⊆Γxy, we use min−Γ0to denote the subset of Γ0 XIANGMEI WANG, CHONG LI, GENARO L ´ OPEZ AND JEN-CHIH YAO 5 consisting of all minimal geodesics in Γ0. Thus γxy ∈min−Γ0means that γxy ∈Γ0and γxy is minimal. Recall that a Hadamard manifold is a complete simply connected m-dimensional Riemannian manifold with nonpositive sectional curvatures. In a Hadamard manifold, the geodesic between any two points is unique and the exponential map at each point of Mis a global diffeomorphism; see, e.g., [39, Theorem 4.1, p. 221]. Thus min−Γxy coincides with Γxy in a Hadamard manifold for any x, y ∈M. 2.2. Convex analysis on Riemmanian manifolds. Definition 2.3 below presents the notions of the convexity for subsets in M, where item (e) is known in [23], and see e.g., [29, 43] for the others. As usual, we use Cto stand for the closure of a subset C⊆M. Definition 2.3. Let Q⊆Mbe a nonempty set. The set Qis said to be (a) weakly convex if, for any x, y ∈Q, there is a minimal geodesic of Mjoining xto y and it is in Q; (b) strongly convex if, for any x, y ∈Q, the minimal geodesic in Mjoining xto yis unique and lies in Q; (c) locally convex if, for any x∈Q, there is a positive ε > 0such that Q∩B(x, ε)is strongly convex; (d) r-convex if, for any x, y ∈Qwith d(x, y)≤r, the minimal geodesic in Mjoining x to yis unique and lies in Q; (e) geodesic convex if, for any x, y ∈Q, the geodesic in Mjoining xto yis unique and lies in Q. Remark 2.1. (a) The following implications are obvious: geodesic convexity ⇒strong convexity ⇒r-convexity/weak convexity ⇒local convexity. (b) The intersection of a weakly convex set and a strongly convex set is strongly convex. (c) All convexities (except the local convexity) in a Hadamard manifold coincide and are simply called the convexity. Recall that the convexity radius at xis defined by (2.1) rx:= sup r > 0 : each ball in B(x, r) is strongly convex and each geodesic in B(x, r) is minimal . Then rxis well defined and positive, and rx= +∞for each x∈Min the case when Mis a Hadamard manifold. Moreover, for any compact subset Q⊆M, we have that (2.2) rQ:= inf{rx:x∈Q}>0; see [39, Theorem 5.3, p. 169] or [29, Lemma 3.1]. 6EQUILIBRIUM PROBLEMS ON RIEMANNIAN MANIFOLD Consider now an extended real-valued function f:M→R:= (−∞,∞] and let D(f) denote its domain, that is, D(f) := {x∈M:f(x)6=∞}. Write Γf xy := ΓD(f) xy for simplicity, that is Γf xy stands for the subset consisting of all γxy ∈Γxy such that γ([0,1]]) ⊆ D(f). In the following definition, we introduce the notions of the convexity for functions, where item (c) is known in [27, 28]. Definition 2.4. Let f:M→Rbe a proper function with a weakly convex domain D(f), and let x∈ D(f). Then, fis said to be (a) convex (resp. strictly convex) at xif, for any y∈ D(f)\ {x}and any geodesic γxy ∈Γf xy the composition f◦γxy : [0,1] →Ris convex (resp. strictly convex) on (0,1): (2.3) f◦γxy(t)≤(resp. <)(1 −t)f(x) + tf(y) for all t∈(0,1); (b) weakly convex (resp. weakly strictly convex) at xif, for any y∈ D(f)there exists γxy ∈min−Γf xy such that (2.3) holds; (c) weakly convex (resp. convex, strictly convex, weakly strictly convex) if so is it at each x∈ D(f). Clearly, for a proper function fon M, the convexity implies the weak convexity, and the strict convexity implies the convexity. Let f:M→Rbe proper and weakly convex at x∈ D(f). The directional derivative in direction u∈TxMand the subdifferential of fat xare, respectively, defined by f′(x;u) := lim t→0+ f(expxtu)−f(x) t and ∂f(x) := {v∈TxM:hv, ui ≤ f′(x;u) for any u∈TxM}. Then, by [27, Proposition 3.8(iii)], the following relationship holds between ∂f(x) and clf′(x;·), the lower semi-continuous hull of f′(x;·): (2.4) clf′(x;u) = sup{hu, vi:v∈∂f(x)}for any u∈TxM. Lemma 2.5. Let f:M→Rbe proper with a weakly convex domain D(f). Let x∈ D(f) and v∈TxM. (i) If fis weakly convex (resp. weakly strictly convex) at x, then v∈∂f(x)if and only if, for some or any constant r > 0, and for any y∈ D(f)∩B(x, r), there exists a geodesic γxy ∈min−Γf xy such that (2.5) f(y)≥(resp. >)f(x) + hv, ˙γxy(0)i. (ii) If fis convex (resp. strictly convex) at x, then v∈∂f(x)if and only if, for some or any constant r > 0, the inequality (2.5) holds for any y∈ D(f)∩B(x, r)and any γxy ∈min−Γf xy. XIANGMEI WANG, CHONG LI, GENARO L ´ OPEZ AND JEN-CHIH YAO 7 Proof. We only prove assertion (i) (as the proof for assertion (ii) is similar). To do this, suppose that fis weakly convex (resp. weakly strictly convex) at x. It suffices to verify that the following statements are equivalent: (a) v∈∂f(x). (b) For any r > 0 and any y∈ D(f)∩B(x, r), there exists a geodesic γxy ∈min−Γf xy such that (2.5) holds. (c) There is some r > 0 such that for any y∈ D(f)∩B(x, r), there exists a geodesic γxy ∈min−Γf xy such that (2.5) holds. We shall complete the proof by showing the implications (a) ⇒(b) ⇒(c) ⇒(a). To do this, assume (a). Then v∈∂f(x), and by definition, we have that (2.6) hv, ui ≤ f′(x;u) for any u∈TxM. Let r > 0 and y∈ D(f)∩B(x, r) be arbitrary. Noting that fis weak convex (resp. weakly strictly convex) at x, there exists γxy ∈min−Γf xy such that the composite f◦γxy : [0,1] →R is convex (resp. strictly convex) on [0,1]. Therefore, f′(x; ˙γxy(0)) = inf t>0 f(expxt˙γxy(0)) −f(x) t≤(resp. <)f(y)−f(x). This, together with (2.6), yields that hv, ˙γxy(0)i ≤ f′(x; ˙γxy(0)) ≤(resp. <)f(y)−f(x). Hence (b) holds, and the implication (a) ⇒(b) is checked. Noting that the implication (b) ⇒(c) is evident, it remains to show the implication (c) ⇒(a). To this end, assume (c). Then one can choose r > 0 and γxy ∈min−Γf xy for any y∈ D(f)∩B(x, r) such that (2.5) holds. Without loss of generality, one could assume that r≤rx. Let u∈TxM\ {0}be arbitrary, and set s0:= r kuk. Then, for any s∈(0, s0), y(s) := expx(su)∈B(x, r)⊆B(x, rx). It follows that the geodesic γxy(s)joining xand y(s) is unique. Therefore, γxy(s)is of the form: γxy(s)(t) = expx(t(su)) for each t∈[0,1], and if y(s)∈ D(f), then (2.5) holds with y(s) in place of y: hv, sui=hv, ˙γxy(s)(0)i ≤ f(y(s)) −f(x) = f(expx(su)) −f(x) for any s∈(0, s0). Note that the above inequality holds trivially if y(s)/∈ D(f). Then, by definition, we get that hv, ui ≤ f′(x;u), and so (a) holds as u∈TxMis arbitrary. Thus, the implication (c) ⇒(a) is shown and the proof is complete. Fix ¯x∈ D(f) and recall that fis center Lipschitz continuous at ¯xif there exits a neighborhood Uof ¯xand a constant Lsuch that |f(x)−f(¯x)| ≤ Ld(x, ¯x) for any x∈U. The center Lipschitz constant Lf ¯xat ¯xis defined to be the minimum of all Lsuch that above inequality holds for some neighborhood Uof ¯x. Then it is clear that Lf ¯x= lim δ→0+ sup |f(x)−f(¯x)| d(x, ¯x): 0 <d(x, ¯x)≤δ. 8EQUILIBRIUM PROBLEMS ON RIEMANNIAN MANIFOLD The following properties about the subdifferential of a (weakly) convex function can be found in [28, Proposition 6.2] except (2.7) by definition. Lemma 2.6. Let f:M→Rbe a proper function. Then the following assertions hold: (i) If is weakly convex , then fis continuous on intD(f). (ii) If ¯x∈intD(f)and fis weakly convex at ¯x, then ∂f(¯x)is a nonempty, compact and convex set satisfying (2.7) kvk ≤ Lf ¯xfor any v∈∂f(¯x). The following lemma, which provides some sufficient conditions ensuring the sum rule of subdifferential, was proved in [27, Proposition 4.3]. Lemma 2.7. Let f, g :M→Rbe proper functions such that f, g and f+gare weakly convex at x∈intD(f)∩ D(g). Then the following sum rule for the subdifferential holds: ∂(f+g)(x) = ∂f(x) + ∂g(x). 2.3. VIP: existence and convexity properties of solution sets. Let Q⊆Mbe a nonempty subset and let A:Q⇒T M be a set-valued vector field defined on Q, that is, A(x)⊆TxMis nonempty for each x∈Q. Consider the following variational inequality problem (VIP for short) associated to the pair (A, Q): To find a point ¯x∈Qsuch that (2.8) ∃¯v∈A(¯x) s.t. h¯v, ˙γ¯xy(0)i ≥ 0 for any y∈Qand γ¯xy ∈ΓQ ¯xy. Any point ¯x∈Qsatisfying (2.8) is called a solution of VIP, and the set of all solutions is denoted by VIP(A, Q). Variational inequality problem (2.8) was first introduced in [49], for single-valued vector fields on Hadamard manifolds, and extended respectively in [29] and [28] for single-valued vector fields and multivalued vector fields on general Riemannian manifolds. As we have mentioned previously, our approach to solve the EP is founded strongly on some existence results about the VIP, which are taken from [28]. For this purpose, we recall some notions in the following definition; see, e.g, [26, 28]. Definition 2.8. Let Q⊆Mbe a subset and A:Q⇒T M be a set-valued vector field on Q.Ais said to be (a) upper semi-continuous (usc for short) at x0, if, for any open set Usatisfying A(x0)⊆ U⊆Tx0M, there exists an open neighborhood U(x0)of x0such that Px0,xA(x)⊆Ufor any x∈U(x0)∩Q; (b) upper Kuratowski semi-continuous (uKsc for short) at x0if, for any sequences {xk} ⊂ Qand {uk} ⊂ T M with each uk∈A(xk), relations limk→∞ xk=x0∈Qand limk→∞ uk=u0imply u0∈A(x0); (c) usc (resp., uKsc) on Qif it is usc (resp., uKsc) at each x∈Q. XIANGMEI WANG, CHONG LI, GENARO L ´ OPEZ AND JEN-CHIH YAO 9 By definition, it is evident that upper semi-continuity implies upper Kuratowski semicontinuity. In the following example, we provide two set-valued vector fields which are useful for our study in next sections of the present paper. Example 2.1. Fix a point y∈M, and define vector fields Exp−1 (·)y:⇒T M and exp−1 (·)y:⇒T M respectively by Exp−1 xy:= {u∈TxM: expxu=y}for each x∈M, and (2.9) exp−1 xy:= {u∈Exp−1 xy:kuk= d(x, y)}for each x∈M. Then one can check easily by definition and Lemma 2.1 that Exp−1 (·)yis uKsc on Mand exp−1 (·)yis usc on M. Recall from [28] that a point o∈Qis called a weak pole of Q if for each x∈Q, min−Γox is a singleton and min−Γox ⊆Q. Clearly, any subset with a weak pole is connected. The notions of the monotonicity in the following definition are well known; see for example [11, 28]. Definition 2.9. Let Q⊆Mbe a subset and A:Q⇒T M be a set-valued vector field. The vector field Ais said to be (a) monotone on Qif, for any x, y ∈Qand γxy ∈ΓQ xy the following inequality holds: hvx,˙γxy(0)i − hvy,˙γxy(1)i ≤ 0for any vx∈A(x), vy∈A(y); (b) strictly monotone on Qif it is monotone and, for any x, y ∈Qwith x6=yand γxy ∈ΓQ xy the following inequality holds: hvx,˙γxy(0)i − hvy,˙γxy(1)i<0for any vx∈A(x), vy∈A(y). Let Q⊆Mbe a closed connected and locally convex set. By [39, p. 170], there exists a connected (embedded) k-dimensional totally geodesic sub-manifold Nof Msuch that Q=N. Following [28], the set intRQ:= Nis called the relative interior of Q. Moreover, as in [28], we say that a closed locally convex set Qhas the BCC (bounded convex cover) property if there exists o∈Qsuch that, for any R≥0, there exists a weakly convex compact subset of Mcontaining Q∩B(o, R). Remark 2.2. We remark that the notion of the BCC property defined above is a litter stronger than that defined in [28, Definition 3.9], where it is required that the compact subset containing Q∩B(o, R)is “locally convex” rather than “weakly convex”. From its proof, one sees that the BCC property assumption defined in [28, Definition 3.9] seems insufficient for [28, Theorem 3.10], while the stronger version of the BCC property defined here is sufficient. For the remainder, we use V(Q) to denote the set of all uKsc set-valued vector fields A such that A(x) is compact and convex for each x∈Q. 16 EQUILIBRIUM PROBLEMS ON RIEMANNIAN MANIFOLD This, together with (3.17), yields that F(x0, y)≥F(x0, x0) + hu0,exp−1 x0yi. Thus, Lemma 2.5 is applicable to concluding that u0∈∂F(x0,·)(x0) = AF(x0) as y∈ B(x0,rB 2) is arbitrary, and the upper Kuratowski semi-continuity of AFis proved. Furthermore, by Proposition 3.3, AF(x) is nonempty, compact and convex for each x∈Q. Hence AF∈ V(Q). Thus the proof is complete. 3.3. Existence and convexity properties of the solution set. Let F:M×M→R and Q⊆Msatisfy the conditions assumed at the beginning of the present section. We have the following existence result on the solution of EP associated to the pair (F, Q). Theorem 3.6. Suppose that Qcontains a weak pole o∈intRQand that Fsatisfies (H1)- (H4). Then EP(F, Q)6=∅provided that Qis compact, or assumptions (b) in Proposition 2.10 is satisfied with AFin place of A. Proof. By hypotheses (H1) and (H2), we see from Proposition 3.4 that (3.18) VIP(AF, Q)⊆EP(F, Q). Moreover, by hypotheses (H3) and (H4), we get by Proposition 3.5 (ii) that AF∈ V(Q). Thus, by assumption, Proposition 2.10 is applicable to getting that VIP(AF, Q)6=∅. The result follows immediately from (3.18) and the proof is complete. Remark 3.2. Assumption (b) in Proposition 2.10 is satisfied with AFin place of Aif Qhas the BBC and one of the following assumptions holds (in particular, assumption (b2) was used by Colao et al in [11]): (b1) AFsatisfies the coerciveness condition on Q. (b2) There exists a compact set L⊆Msuch that (3.19) x∈Q\L⇒[∃y∈Q∩Ls.t. F(x, y)<0]. In fact, it is clear from Remark 2.3 in the case of (b1). To check this for the case of (b2), let L⊆M,x∈Q\Land let y∈Q∩Lbe given by (3.19) such that F(x, y)<0. Then, (3.20) F(x, y)−F(x, x)≤F(x, y)<0. By assumption (H2) and the definition of AF, we see that for any v∈AF(x), there exists a minimal geodesic γxy ∈min−ΓQ xy such that F(x, y)≥F(x, x) + hv, ˙γxy(0)i.This, together with (3.20), implies that hv, ˙γxy(0)i<0. Hence condition (2.10) is satisfied as x∈Q\Lis arbitrary, and the proof is complete. The following theorem provides the convexity properties of the solution set EP(F, Q), which is a direct consequence of Propositions 3.4, 3.5 and 2.11 (noting by (3.7) that Qis weakly convex). Theorem 3.7. Suppose that Fsatisfies (H1)-(H3) and EP(F, Q)6=∅. Suppose further that AFis monotone on Qwith (3.1) (e.g., Fis monotone on Q×Q). Then the following assertions hold: XIANGMEI WANG, CHONG LI, GENARO L ´ OPEZ AND JEN-CHIH YAO 17 (i) The solution set EP(F, Q)is locally convex. (ii) If AFis strictly monotone on Q(e.g., Fis strictly monotone on Q×Q), then EP(F, Q)is a singleton. (iii) If Mis of the sectional curvatures bounded above by κ > 0, then EP(F, Q)is Dκ-convex. In particular, in the case when Mis a Hadamard manifold, hypothesis (H1) implies (H2), and every convex subset has both weak poles and the BCC property. Thus the following corollary is immediate from Theorems 3.6 and 3.7. Corollary 3.8. Let Mbe a Hadamard manifold. Suppose that Fsatisfies (H1) and (H3). Then the following assertions hold: (i) If (H4) holds, then EP(F, Q)6=∅provided that Qis compact, or one of (b1) and (b2) in Remark 3.2 holds. (ii) If Fis monotone on Q×Qwith EP(F, Q)6=∅, then EP(F, Q)is convex. Remark 3.3. Assertion (ii) in Corollary 3.8 seems new even in the Hadamard manifold setting; while assertion (i) was established in [11, Theorem 3.2] under the following assumptions: (c1) there exists a compact set L⊆Mand y0∈Q∩Lsuch that F(x, y0)<0∀x∈Q\L; (c2) the set {y∈Q:F(x, y)<0}is convex for each x∈Q. Clearly assumption (c1) implies our assumption (b2) in Remark 3.2. Moreover, as will be seen in the application to the proximal point algorithm in the next section and to the mixed variational inequalities in Subsection 5.2, assumption (c2) is not satisfied, in general (thus [11, Theorem 3.2] is not applicable); while Corollary 3.8 is applicable because our assumptions (H1) and (H2) presented here are satisfied there. 4. Resolvent and proximal point algorithm for EP. As in the previous section, we always assume that F:M×M→Rand Q⊆Msatisfy the conditions assumed at the beginning of Section 3. Recall the equilibrium problem is defines by (1.1) and its solution set is denoted by EP(F, Q). The aim of this section is to introduce the resolvent and the proximal point algorithm for EP (1.1) on general manifolds and show convergence of this algorithm. The applications of the proximal point method to solve many different problems in the Riemannian context could be fond in e.g., [26, 28, 46, 48]. Fix z∈Mand define the bifunction Gz:M×M→Rby Gz(x, y) := sup u∈exp−1 xz,v∈exp−1 xy h−u, vixfor any (x, y)∈M×M. In the following definition, we extend the notion of the resolvent defined in [11, definition 4.6] for the bifunction Fon Hadamard manifolds to the general manifold setting. Let λ > 0. Definition 4.1. The resolvent JF λ:M⇒Qof Fis defined by (4.1) JF λ(z) := EP(Fλ,z, Q)for any z∈M, 18 EQUILIBRIUM PROBLEMS ON RIEMANNIAN MANIFOLD where the bifunction Fλ,z :M×M→Ris defined as Fλ,z(x, y) := λF (x, y) + Gz(x, y)for any (x, y)∈M×M. For the remainder, we always assume that Mis of the sectional curvature bounded above by κ≥0. Recall that Dκ=π √κif κ > 0 and Dκ= +∞if κ= 0. Then, for any z∈M,B(z, Dκ 4) is strongly convex (see, e.g., [39, p. 169]), and so exp−1 (·)zis a singleton on B(z, Dκ 4). Recall that AFis the set valued vector field associated to the bifunction F(see (3.8)). Following [28], the resolvent JAF λ:M⇒Qof AFis defined by (4.2) JAF λ(z) := VIP(AF λ,z, Q) for any z∈M, where AF λ,z :M⇒T M is defined by AF λ,z(x) := λAF(x)−EQ z(x) for any x∈Q, with the set-valued vector field EQ z:M⇒T M defined by EQ z(x) := {u∈exp−1 xz: expxtu ∈Q∀t∈[0,1]}for any x∈Q The following theorem provides an estimate for the domain of the resolvent JF λ. Recall that LF zdenotes the center Lipschitz constant of F(z, ·) at z∈M. Set (4.3) DF λ:= {z∈Q:λLF z<Dκ 4}. Theorem 4.2. Suppose that Fsatisfies hypotheses (H1)-(H3) and is monotone on Q×Q. Then, (i) λd(0, AF(z)) <Dκ 4for each z∈ DF λ. (ii) DF λ⊆ D(JF λ). (iii) JF λ(z)∩B(z, Dκ 4) = JAF λ(z)is a singleton for each z∈ DF λ. Proof. Note that Qis weakly convex by implication (3.7). Moreover, by assumptions made for F, one sees by Propositions 3.3 and 3.5 that AF∈ V(Q) is monotone, and d(0, AF(z)) ≤LF zfor each z∈Q. Thus assertion (i) follows from the definition of DF λin (4.3). Below we show assertions (ii) and (iii). To do this, let z∈ DF λ. Then, λd(0, AF(z)) <Dκ 4by (i), and it follows from [28, Lemma 4.3 and Corollary 5.4] (applied to AF, Q in place of V, A there) that (4.4) JAF λ(z) is a singleton and JAF λ(z)⊆B(z, Dκ 4). Moreover, thanks to hypotheses (H1) and (H2), we have by Proposition 3.2 (i) and (iii) that the bifunction Fλ,z =λF +Gzsatisfies hypotheses (H1) and (H2) (with Fλ,z in place of F). XIANGMEI WANG, CHONG LI, GENARO L ´ OPEZ AND JEN-CHIH YAO 19 Furthermore, for any x∈Q,F(x, x) = 0 by the monotonicity of Fand Gz(x, x) = 0 by of Proposition 3.2 (i); hence Fλ,z(x, x) = 0. Thus one can apply Proposition 3.4 to get that (4.5) EP(Fλ,z, Q) = VIP(AFλ,z , Q). Noting that D(Gz(x, ·)) = Mby Proposition 3.2 (i), we see from Lemma 2.7 and Proposition 3.2 (iv) that, for any x∈Q, (4.6) AFλ,z (x) := ∂(λF(x, ·) + Gz(x, ·))(x) = λAF(x)−coEQ z(x). Hence AF λ,z(x)⊆AFλ,z (x) for any x∈Q, and then VIP(AF λ,z, Q)⊆VIP(AFλ,z , Q). By defintion (see (4.1) and (4.2)) and (4.5), it follows that (4.7) JAF λ(z) = VIP(AF λ,z, Q)⊆VIP(AFλ,z , Q) = EP(Fλ,z, Q) = JF λ(z). In light of (4.4), we see that JF λ(z)6=∅, and so assertion (ii) holds as z∈ DF λis arbitrary. To show assertion (iii), note that AF λ,z(x) = AFλ,z (x) if d(x, z)< Dκby (4.6). It follows from (4.7) that JAF λ(z)∩B(z, Dκ 4) = JF λ(z)∩B(z, Dκ 4). This, together with (4.4), implies that JF λ(z)∩B(z, Dκ 4) = JAF λ(z) is a singleton, and so assertion (iii) holds. The proof is complete. The following theorem provides sufficient conditions for D(JF λ) = M. In particular, in the Hadamard manifold setting, this result was claimed in [11, Theorem 4.9] under the additional assumption (c1) in Remark 3.3 but the proof presented there is not correct. Theorem 4.3. Suppose that Fsatisfies hypotheses (H1)-(H3) and is monotone on Q×Q. Then, D(JF λ) = Mprovided that one of the following assumptions holds: (a) Qis compact and contains a weak pole o∈intRQ; (b) Mis a Hadamard manifold. Proof. Let z∈M. Then by the assumptions made for Fand Proposition 3.2 (i), (iii) and (v), one can checks easily that the bifunction Fλ,z =λF +Gzsatisfies (H1)-(H4). To complete the proof, it suffices to verify that JF λ(z)6=∅, which is true by Theorem 3.6 in case (a). Thus we only consider case (b). To do this, we assume that Mis a Hadamard manifold. Then, for any x, y ∈M, exp−1 xyis a singleton and Fλ,z(x, y) is reduced to Fλ,z(x, y) := λF (x, y)− hexp−1 xz, exp−1 xyi. Recalling from [11, (2.7)] that hexp−1 xz, exp−1 xyi+hexp−1 yw, exp−1 yxi ≥ d2(x, y) for any x, y ∈M and that Fis monotone on Q×Q, we get that (4.8) Fλ,z(x, y) + Fλ,z(y, x)≤ −d2(x, y) for any (x, y)∈Q×Q. 20 EQUILIBRIUM PROBLEMS ON RIEMANNIAN MANIFOLD Below we show that Fλ,z satisfies (b2) in Remark 3.2: there is a compact subset L⊆M such that (4.9) x∈Q\L⇒[∃y∈Q∩L s.t. Fλ,z (x, y)<0]. Granting this, we get JF λ(z) = EP(Fλ,z , Q)6=∅by Corollary 3.8, and the proof is complete. To show (4.9), take y∈Qand set R:= LFλ,z y. Then R < +∞as Fλ,z satisfies (H1), and L:= B(y, R) is as desired. To show this, let x∈Q\Land v∈AFλ,z (y). Then, d(x, y)> R, and kvk ≤ Rby Proposition 3.3. Therefore, we have that Fλ,z(y, x)≥Fλ,z(y, y) + hv, exp−1 yxi ≥ −Rd(x, y) (noting that Fλ,z(y, y) = 0). This, together with (4.8), implies Fλ,z(x, y)≤ −d2(x, y)−Fλ,z(y, x)≤(R−d(x, y)) d(x, y)<0. Thus, (4.9) is shown, and the proof is complete. To define the proximal point algorithm for solving EP (1.1), let x0∈Qand {λk} ⊂ (0,+∞). Thus the proximal point algorithm can be formulated as follows. Algorithm P Letting k= 1,2,... and having xk, choose xk+1 such that xk+1 ∈JF λk(xk)∩B(xk,Dκ 4). Clearly, in the case when Mis a Hadamard manifold, Algorithm P is reduced to the one defined in [11]: xk+1 ∈JF λk(xk) for each k∈N. The convergence result of Algorithm P is as follows. Theorem 4.4. Suppose that Fsatisfies hypotheses (H1)-(H3) and is monotone on Q×Qwith EP(F, Q)6=∅. Let x0∈Qand {λk} ⊂ (0,∞)be such that (4.10) d(x0,EP(F, Q)) <Dκ 8, (4.11) Σ∞ k=0λ2 k=∞and λkLF xk<Dκ 4for all k∈N. Then, Algorithm P is well-defined, and converges to a point in EP(F, Q). Proof. Recall that AF:Q⇒T M is defined by (3.8). By assumption, Propositions 3.4 and 3.5 are applicable; hence AFis monotone, AF∈ V(Q), and (4.12) VIP(AF, Q) = EP(F, Q) (noting that (3.1) hold by the monotonicity assumption). Then, thanks to (4.10), one sees that (4.13) d(x0,VIP(AF, Q)) <Dκ 8. XIANGMEI WANG, CHONG LI, GENARO L ´ OPEZ AND JEN-CHIH YAO 21 Let {˜xk}be a sequence generated by the following proximal algorithm with initial point ˜x0:= x0, which was introduced in [28] for finding a point in VIP(AF, Q): (4.14) ˜xk+1 ∈JA λk(˜xk) for each k∈N. In view of the second assumption in (4.11), and applying Theorem 4.2 (iii), we can check inductively that Algorithm P is well-defined and that the generated sequence {xk}coincides with {˜xk}and satisfies λkd(0, AF(˜xk)) <Dκ 4for each k∈N. This, together with the first assumption in (4.11) and (4.13), implies that [28, Corollary 5.8] (with AF,Qin place of V,A) is applicable, and the sequence {˜xk}and so {xk}converges to a point in VIP(AF, Q). Thus the conclusion follows immediately from (4.12), and the proof is complete. In the special case when Mis a Hadamard manifold, assumption (4.10) and the second one in (4.11) are satisfied automatically. Therefore the following corollary is direct from Theorem 4.4, which was claimed in [11, Theorem 4.9, 4.10] (for constant parameters λk≡ λ > 0) but with an incorrect proof there as we explained in Section 1). Corollary 4.5. Suppose that Mis a Hadamard manifold, and that Fsatisfies hypotheses (H1)-(H3) and is monotone on Q×Qwith EP(F, Q)6=∅. Let {λk} ⊂ (0,∞)be such that Σ∞ k=0λ2 k=∞. Then, Algorithm P is well-defined, and converges to a solution in EP(F, Q). 5. Applications. This section is devoted to two applications of the results regarding the solution set of the EP in the previous sections: One is to the Nash equilibrium and the other to the mixed variational inequality. 5.1. Nash equilibrium. We consider the Nash equilibrium problem (NEP for short) on Riemannian manifolds in this subsection, which is formulated as follow. Let I= {1,2,...,m}be a finite index set which denotes the set of players, and let (Mi,di), i ∈I, be a Riemannian manifold. For each i∈I, let Qi⊆Mibe the strategy set of the i-th player, and fi:M→Rbe his loss-function, where M:= M1×M2× · · · × Mmis the product manifold with the standard Riemannian product metric. The Nash equilibrium problem associated to Q:= Q1×Q2× · · · × Qm⊆Mand {fi}i∈Iconsists of finding a point ¯x= (¯xi)∈Qsuch that (5.1) fi(¯x) = min yi∈Qi fi(¯x1,...,¯xi−1, yi,¯xi+1,...,¯xm) for each i∈I. Any point ¯x∈Qsatisfying (5.1) is called a Nash equilibrium point of the NEP, and we denote the set of all Nash equilibrium points by NEP({fi}i∈I, Q). The most well-known existence results for the classical NEP in the linear space setting is due to Nash [33, 34], where it is assumed that each Qiis compact and convex and each fi is (quasi)convex in the i-th variable. Further extensions and applications of Nash’s original work could be founded in [16, 32, 35, 41] and references therein. Krist´aly seems the first 22 EQUILIBRIUM PROBLEMS ON RIEMANNIAN MANIFOLD one to consider the existence and localization of NEP in the framework of Riemannian manifolds; see [23]. Recently, Krist´aly used in [24] a variational approach to analyzed the NEP with nonconvex strategy sets and nonconvex /nonsmooth payoff functions in Hadamard manifolds. To proceed, we assume for the whole subsection that (HN-a) Qiis closed and weakly convex in Miand Q⊆int Ti∈ID(fi); (HN-b) for each i∈I,fiis continuous on Q; (HN-c) for each i∈I,fiand fi+δQare weakly convex in the i-th variable. To apply our results in the previous sections, we, following [11] and [38], reformulate NEP (5.1) as an EP as follows. Let r:= (ri)∈Rm ++ := {(ri)∈Rm: each ri>0}, and define the bifunction Fr:M×M→Ras the weighted positive sum of the functions {fi}: Fr(x, y) := X i∈I ri(fi(x1,...,xi−1, yi, xi+1,...,xn)−fi(x)) for any x= (xi)i∈I, y = (yi)i∈I∈M. Then, it is easy to check that (5.2) EP(Fr, Q)=NEP({fi}i∈I, Q). In the spirit of the idea in [38] for the NEP in the Euclidean space setting, we introduce the pseudosubgradient mapping gr:M→T M for functions {fi}in the Riemannian manifold setting, which is defined by gr(x) := (r1∂1f1(x), r2∂2f2(x),...,rm∂mfm(x)) for each x∈M, where, for each i∈Iand x∈M,∂ifi(x) stands for the subdifferential of the function fi(x1,...,xi−1,·, xi+1,...,xm) at xi, that is ∂ifi(x) := ∂fi(x1,...,xi−1,·, xi+1,...,xm)(xi). By definition, we check that (5.3) AFr(x) := ∂Fr(x, ·)(x) = gr(x) for any x∈Q. The main theorem in this subsection is as follows, which provides the results on the existence, the uniqueness and the convexity of the Nash equilibrium point. Theorem 5.1. Let r∈Rm ++. Then the following assertions hold: (i) Suppose that Qcontains a weak pole o∈intRQ. Then NEP({fi}i∈I, Q)6=∅provided that Qis compact, or Qhas the BCC property and that there exists a compact subset L⊆M such that (5.4) x∈Q\L⇒[∀v∈gr(x),∃y∈Q∩L, γxy ∈min−ΓQ xy s.t. hv, ˙γxy(0)i<0]. (ii) If gris strictly monotone on Q, then NEP({fi}i∈I, Q)is at most a singleton. (iii) If gris monotone on Qand NEP({fi}i∈I, Q)6=∅, then NEP({fi}i∈I, Q)is locally convex, and NEP({fi}i∈I, Q)is Dκ-convex if Mis additionally assumed to be of the sectional curvatures bounded above by some κ≥0. XIANGMEI WANG, CHONG LI, GENARO L ´ OPEZ AND JEN-CHIH YAO 23 Proof. In view of (5.2), (5.3), (5.4) and thanks to Theorems 3.6 and 3.7 (applied to Fr in place of F), it suffices to show that Frsatisfies hypotheses (H1)-(H4) made in Section 3. Note that (H3) follows trivially from assumption (HN-b); while (H4) is clear as Fr(x, x) = 0 for any x∈Q. Thus we only need to show that Frsatisfies hypotheses (H1) and (H2). To do this, let x= (xi)i∈I∈Q, and write Di:= D(fi(x1,...,xi−1,·, xi+1,...,xm)) for each i∈I. Then (5.5) D(Fr(x, ·)) = D1×...× Di×...× Dm. By assumption (HN-a), each Qi⊆intDiand so x∈Q⊆intD(Fr(x, ·)). Furthermore, in light of assumption (HN-c), one sees that each Diis weakly convex in Mi. This, together with (5.5), implies that D(Fr(x, ·)) is weakly convex in M. We claim that Fr(x, ·) and Fr(x, ·) + δQ×Q(x, ·) are weakly convex in M. Granting this, (H1) and (H2) are checked. In fact, let y= (yi), z = (zi)∈ D(Fr(x, ·)). Then, by assumption (HN-c), for each i∈I, there is a geodesic γi∈min−ΓDi ziyisuch that (5.6) fi(x1,...,xi−1,·, xi+1,...,xm)◦γiis convex on [0,1]. Define γzy[0,1] →Mby γzy(t) := (γ1(t), γ2(t),...,γm(t)) for each t∈[0,1]. Then, γzy ∈ min−ΓFr(x,·) zy (see, e.g., [7]), and Fr(x, ·)◦γzy =X i∈I fi(x1,...,xi−1,·, xi+1,...,xm)◦γi. This means that Fr(x, ·)◦γzy is clearly convex thanks to (5.6), and so Fr(x, ·) is weakly convex in M. Similarly, one can checks that Fr(x, ·) + δQ×Q(x, ·) is also weakly convex in M. Thus the claim stands, and the proof is complete. Remark 5.1. Assertion (i) extends the corresponding one in [23, Theorem 1.1], which was proved under the assumption that each Qiis compact and geodesic convex. It is worthy remarking that the geodesic convexity assumption for Qiprevents its application to some special but important Riemannian manifolds, such as compact Stiefel manifolds St(p, n)and Grassmann manifolds Grass(p, n)(p < n), in which there is no geodesic convex subset (see [1, p. 104 (5.27)]). Example 5.1 below provides the case where our existence result of Theorem 5.1 is applicable but not [23, Theorem 1.1]. Note also that the NEP in Example 5.1 is originally defined on the Euclidean space, and the corresponding existing results in the Euclidean space setting (see, e.g., [16, 32, 33, 34]), to the best our knowledge, are nor applicable because the set Q2 involved is not convex in the usual sense. Example 5.1. Consider the Nash equilibrium problem (5.1) with the associated Q:= Q1×Q2⊆R×R3and {fi}i=1,2defined respectively as follows: Q1:= [−1,1], Q2:= {(t1, t2, t3) : t2 1+t2 2+t2 3= 1, t1>0,|t2| ≤ 1 2, t3>0}, 24 EQUILIBRIUM PROBLEMS ON RIEMANNIAN MANIFOLD f1(x1, x2) = (x1−t3)2and f2(x1, x2) = arccost1for any x1∈R, x2= (t1, t2, t3)∈R3. Clearly Q2⊂R3is not convex, and so the existence results in the Euclidean space setting are not applicable. Below, we shall consider the problem on the Riemannian manifold M:= R×S2, where S2:= (t1, t2, t3)∈R3t2 1+t2 2+t2 3= 1 is the 2-dimensional unit sphere. Denote x:= (0,0,1),y:= (0,0,−1), and consider system of coordinates Φ: (0, π)×[0,2π]⊂R2→S2\ {x,y}around x∈S2\ {x,y}defined by Φ(θ, ϕ) := (sin θcos ϕ, sin θsin ϕ, cos θ)Tfor each (θϕ)∈(0, π)×[0,2π]. Then the Riemannian metric on S2\ {x,y}is given by g11 = 1, g12 = 0, g22 = sin2θfor each θ∈(0, π)and ϕ∈[0,2π], and the geodesics of S2\ {x,y}are great circles or semicircles; see [42, p. 84] for more details. Restricting f1and f2to M=R×S2, one can check by definition that assumptions (HN-a)-(HN-c) are satisfied (noting that arccos t1= d(x2, z0)for each x2= (t1, t2, t3)∈S2, where z0:= (1,0,0)), and that Q⊂Mis compact and has a weak pole in intQ. Thus, Theorem 5.1 is applicable and guarantees NEP({f1, f2}, Q1×Q2)6=∅. Indeed, by a simple calculation, we see that NEP({f1, f2}, Q1×Q2) = {(1,(1,0,0))}. However, the existence result in [23, Theorem 1.1] is not applicable because there is no geodesic subset on S2. As explained before (see the paragraph right before Corollary 3.8), the following corollary is a direct consequence of Theorem 5.1. In particular, assertion (i) was proved in [11, Theorem 3.12] with each Qibeing compact; while assertion (ii) is new even in the Hadamard manifold setting. Corollary 5.2. Suppose that each Miis a Hadamard manifold. Then, the following assertions hold: (i) The solution set NEP({fi}i∈I, Q)6=∅provided Qis compact, or there exists a compact subset L⊆Msuch that (5.4) holds for some r∈Rm ++. (ii) If there exists some r∈Rm ++ such that gris monotone on Q, then NEP({fi}i∈I, Q) is convex. Remark 5.2. In view of (5.3), one checks by Remark 3.2 (applied to gr,Frin place of AF,F) that a compact subset Lexists such that (5.4) holds provided one of the following assumptions holds: (a) grsatisfies the coerciveness condition on Q; (b) there exists a compact set L⊆Msuch that x∈Q\L⇒[∃y∈Q∩Ls.t. Fr(x, y)<0]. XIANGMEI WANG, CHONG LI, GENARO L ´ OPEZ AND JEN-CHIH YAO 25 5.2. Mixed variational inequalities. Let Q⊂Mbe a nonempty closed subset. Given a vector field V:Q→T M and a real-valued function f:M→R. The mixed variational inequality problem (MVIP for short) associated to Vand fis to find ¯x∈Q, called a solution of the MVIP, such that (5.7) hV(¯x),˙γ¯xy(0)i+f(y)−f(¯x)≥0 for any y∈Q,γ¯xy ∈ΓQ ¯xy. The set of all solutions of MVIP (5.7) is denoted by MVIP(V, f, Q). The MVIP has been studied extensively in the linear space setting; see, e.g., [17, 19, 44]; and it seems that [11] is the first paper to explore the MVIP in the Hadamard manifold setting, where only the existence issue of the solution for the MVIP is concerned with. To reformulate the MVIP as an EP considered in the previous sections, we define F: M×M→(−∞,+∞] as follows: (5.8) F(x, y) := sup u∈exp−1 xy hV(x), uix+f(y)−f(x) for any (x, y)∈M×M, where exp−1 xyis defined by (2.9) and we adopt the convention that a−(+∞) = +∞for any a∈R. Proposition 5.3. Let F:M×M→(−∞,+∞]be defined by (5.8). Suppose that fis convex. Then we have (5.9) MVIP(V, f, Q) = EP(F, Q). Proof. It is evident that MVIP(V, f, Q)⊆EP(F, Q). To show the converse inclusion, let ¯x∈EP(F, Q) and it suffices to prove that (5.7) holds. To this end, let y∈Qand γ¯xy ∈ΓQ ¯xy. We have to show that (5.10) hV(¯x),˙γ¯xy(0)i+f(y)−f(¯x)≥0. Take ¯ t∈(0,1] such that d(¯x, γ¯xy(¯ t)) ≤r¯x(note that r¯x>0 by (2.1)). Denote ¯y:= γ¯xy(¯ t). Then ¯y∈Qand ΓQ ¯x¯y={γ¯x¯y}is a singleton, where γ¯x¯y: [0,1] →Mis defined by γ¯x¯y(s) := γ¯xy(¯ ts) for any s∈[0,1]. Then ˙γ¯x¯y(0) = ¯ t˙γ¯xy(0). In view of ¯x∈EP(F, Q) and ¯y∈Q, we see that hV(¯x),˙γ¯x¯y(0)i+f(¯y)−f(¯x) = hV(¯x),¯ t˙γ¯xy(0)i+f(¯y)−f(¯x)≥0. Noting that f(¯y)−f(¯x) ¯ t≤f(y)−f(¯x) by the convexity of f◦γ¯xy (as fis convex), we conclude that (5.10) holds, which completes the proof. We assume in the present subsection that (HM-a) fis convex and Q⊆intD(f) is closed weakly convex. (HM-b) Vis continuous on Q.