The Property of the Set of Equilibria of the Equilibrium Problem with Lower and Upper Bounds on Hadamard Manifolds

Abstract

The existence of equilibrium points, and the essential stability of the set of equilibrium points of the equilibrium problem with lower and upper bounds are studied on Hadamard manifolds.

1. Introduction

Let be a given nonempty set, a given function, and and two real numbers satisfying . The equilibrium problem with lower and upper bounds is that of finding such that

If , and , then problem (1) is said to be the scalar equilibrium problem: find such that where is a given function satisfying for all . It is well known that problem (2) is a unified model of several problems, such as variational inequality problems, optimization problems, saddle point problems, complementarity problems, and fixed point problems (e.g., see [1–3]).

In 1999, Isac et al. [4] raised the open problem: if is a nonempty closed subset in a locally convex semireflexive topological vector space, under what conditions does problem (1) have a solution? Since then, some authors begin to study the problem. In 2000, Li [5] gave the answer by using the concept of extremal subsets. In [6], Chadli et al. derived some results by using a fixed point theorem due to Ansari and Yao [7] and Fan lemma [8]. In [9], Zhang also answered the problem by using the concept of -convexity, a fixed point theorem and Fan lemma. The results mentioned above and others in [10–12] are shown in the topological vector space. Therefore, there is a problem: when does problem (1) have a solution in the nonlinear framework of manifolds? On the other hand, as far as we know, there is not a paper in which the essential stability of the set of equilibrium points of the equilibrium problem with lower and upper bounds is given either in topological vector space or on manifolds.

The purpose of this paper is to develop the equilibrium problem with lower and upper bounds in the nonlinear framework of Hadamard manifolds, to study the existence of equilibrium points, and the essential stability of the set of equilibrium points of the equilibrium problem with lower and upper bounds on Hadamard manifolds. Our results extend the corresponding theorems due to Isac et al. [4], Colao et al. [13], and Zhang [9].

2. Preliminaries

In this section we recall some notations, definitions, and basic properties used throughout the paper, which can be found in [14] or [15].

Definition 1. A Hadamard manifold is a complete simply connected Riemannian manifold of nonpositive sectional curvature.

Throughout this paper, let be an -dimensional Hadamard manifold, let be any given point in , and let denote the tangent space at to . We denote by the scalar product on with the associated norm . Let be the distance function; then by the Hopf-Rinow theorem (see [15]), is a complete metric space.

Definition 2. The exponential mapping at is defined by for each , where is the geodesic starting at with velocity (i.e., and ).

Easily, we know that (i) for each real number ; (ii) the exponential mapping and its inverse are continuous on Hadamard manifolds; (iii) for any , the minimal geodesic joining to is (), and is also the minimal geodesic for any with .

Definition 3. A subset is said to be geodesic convex if for any two points and in the geodesic joining to is contained in ; that is, if is a geodesic such that and , then for all .

Definition 4. Let be any given point in . The geodesic convex hull for a set , denoted by , is defined as follows:

Remark 5. If is a geodesic convex subset, then for any .

Definition 6 (see [16]). Let . One says that is a KKM mapping on Hadamard manifolds if, for any , one has

Lemma 7 (see [13, 16]). Let be a nonempty closed geodesic convex subset of and a closed-valued KKM mapping on a Hadamard manifold. If there exists at least one such that is compact in , then

3. Existence of Equilibrium Point

In this section, we show the existence of equilibrium point of the equilibrium problem with lower and upper bounds by using KKM theorem on Hadamard manifolds.

Theorem 8. Let be a nonempty bounded closed and geodesic convex subset of Hadamard manifolds . If the function satisfies the following conditions:(i)for each , the set is closed in ,(ii)for any finite set , ,(iii)there exists , such that is a compact subset of ,then the equilibrium point of the problem (1) exists. That is, there exists , such that

Proof. Let the set-valued mapping be defined by . Then is a KKM mapping. In fact, it follows from Condition (ii) that, for any finite set and any , there exists some such that ; that is, for some . Hence we have and .
By Condition (i), for each , is closed in . By Condition (iii) and the completeness of , there exists such that is compact. By Lemma 7, we have ; that is, for any , there exist . Therefore there exists such that for all . The proof is completed.

Example 9. If for any , the mapping satisfies that the set is geodesic convex and , then satisfies Condition (ii).
In fact, if not, then for any finite set , there exists such that ; that is, . This implies that for any , . By the geodesic convexity of , we have , which contradicts to . Therefore, for any subset , .

Theorem 10. Let be a nonempty bounded closed and geodesic convex subset of Hadamard manifolds . If the function satisfies the following conditions:(i)for each , the set is closed in ,(ii)for any finite set , ,(iii)there exists a compact subset and a point , such that or for all ,then the equilibrium point of problem (1) exists.

Proof. Let the set-valued mapping be defined by . Then by Condition (iii) there exists a point such that . So it follows for Condition (i) and the completeness of that is compact. By Theorem 8, we have that there exists such that for all . This completes the proof.

Theorem 11. Let be a nonempty compact and geodesic convex subset of Hadamard manifolds . If the function satisfies the following conditions: (i)for each , is continuous with respect to in ,(ii)for any finite set , ,then there exists , such that for all .

Proof. From the continuity of , it follows that Condition (i) of Theorem 8 holds. By Theorem 8, we have that there exists such that for all . This completes the proof.

Remark 12. Theorem 8 extends Theorem 3.1 due to Zhang [9] from the topological vector space to Hadamard manifolds.

Next, we show some applications of our results as the following.

Corollary 13. Let be a nonempty compact and geodesic convex subset of Hadamard manifolds , , and . If the function satisfies the following conditions: (i)for each , is continuous with respect to in ,(ii)for any finite set , ,then there exists , such that

Corollary 14. Let be a nonempty bounded closed and geodesic convex subset of Hadamard manifolds . If the mapping satisfies the following conditions: (i)for any , ,(ii)for each , the set is closed in ,(iii)for any , the set is geodesic convex,(iv)there exists a compact subset and a point , such that for all ,then the equilibrium point of the problem (2) exists.

Proof. Define a mapping by , and then By Condition (ii), we have that the is closed. It follows from conditions (i), (iii) and Example 9 that satisfies that for any finite set , . Additionally, by Condition (iv) there exists a point for which . So it follows from Condition (ii) and the completeness of that is compact. By Theorem 8, we have that, for all , there exists such that ; that is, there exists such that . This completes the proof.

Remark 15. When the mapping is upper continuous for all , Condition (i) holds in Corollary 14. If is a compact subset of , then Condition (iv) can be omitted. Therefore, Theorem 3.2 shown in [13] is improved.

4. Essential Stability

In this section, we consider the essential stability of the set of equilibrium points of the equilibrium problem with lower and upper bounds on Hadamard manifolds. We can see the systemic study about the essential stability in the topological vector space in [17].

Let be a nonempty compact and geodesic convex subset of Hadamard manifold , and denotes the set of the function , which is continuous with respect to and satisfies for any finite set .

For any , it follows from Theorem 8 that there exists such that for all , , where is said to be equilibrium points of the equilibrium problem with lower and upper bounds. Let denote the set of equilibrium points ; then So a mapping is well defined, where is the set of all nonempty compact subsets of . For any , we can define a distance as follows: Clearly, is a metric space.

Definition 16. For each , let be a nonempty closed subset of . (i) is called an essential point of if, for any open neighborhood of in , there is a such that for any with , . If all is essential, then is said to be essential.(ii) is called an essential set of if, for any open set , , there is a such that for any with , .(iii) is called a minimal essential set of if it is a minimal element of the family of essential sets ordered by set inclusion.

Lemma 17. The metric space is complete.

Proof. Let be any Cauchy sequence; then for any , there is an such that for any , , or, , which implies that for any , is a Cauchy sequence in . Thus, there is a mapping such that for each . Hence . Then under the metric .
Next we will prove . For any , using we can show is continuous with respect to . For all finite set and all , it follows from the property of that there exists some such that . Since under the metric , holds for some and any . Then that is, .

Lemma 18. The mapping is a usco mapping; that is, is upper semicontinuous on and is nonempty compact for all .

Proof. Since is compact, we need only to prove the closedness of the graph of (); that is, for all with and any with , should be proved.
For any , implies that . Hence, for all , by the continuity of , we have that is closed, and so ; that is, holds for all . It follows from that holds for all . Therefore, . This completes the proof.

Theorem 19. For each , one has that (i)there exists a dense residual subset of such that for each , is essential,(ii)there exists at least one connected minimal essential subset of .

Proof. (i) By Lemmas 17 and 18, we have that the metric space is complete and the mapping is usco. Hence, it follows from Fort theorem (see [18]) that there is a dense residual subset of , such that is lower semicontinuous in . From the definition of lower semicontinuous mapping and Definition 16 (i), it follows that is essential for each .
(ii) Let denote the family of all essential subsets of ordered by set inclusion; then . In fact, the upper semicontinuity of implies that, for each open set with , there exists such that for any with , . Hence is an essential set of itself.
From the compactness of , it follows that the intersection of every decreasing chain of elements in is also in and has a lower bound. Therefore, by Zorns lemma, has a minimal element , which is a minimal essential set of .
Suppose that the minimal essential subset is not connected. Then, there exist two nonempty open subsets and with and two disjoint open subsets and in such that and . From , it follows that there is a such that for any with , .
Since is a minimal essential set of , then neither nor is essential. Hence, for , there exist with and , such that , . Thus, .
Define a mapping as follows: where , , and is the metric of . It is easy to show that, for any , and are continuous, , , and . By the continuity of and , we have that is continuous with respect to .
For any finite set and all , it follows from the property of that there exists some such that , and then . Therefore, .
Since we have . When , let ; then , , , and , which contradicts the fact . Similarly, we can show that results in a contradiction. Therefore, is connected. The proof is completed.

In the sequel, by using the above results, we consider the essential stability of the set of equilibrium points of problem (2) on Hadamard manifolds.

Let denote the set of the function , which is continuous and satisfies that is geodesic convex and for any .

For any , it follows from Remark 15 and Corollary 14 that there exists such that for all , , where is said to be equilibrium points of problem (2). Let denote the set of equilibrium points ; then So a mapping is well defined.

Let , , and ; by Theorem 19, we have the following results.

Corollary 20. For each , one has that (i)there exists a dense residual subset of such that for each , is essential,(ii)there exists at least one connected minimal essential subset of .

Conflict of Interests

The authors declare that there is no conflict of interests regarding the publication of this paper.

Acknowledgments

The authors are grateful to the referees for their suggestions to improve the paper. This work is supported by National Natural Science Foundation of China (no. 11201379), the Fundamental Research Funds for the Central Universities (no. JBK130401), Scientific Research Fund of SiChuan Provincial Education Department (no. 14ZA0362), and the Natural Science Foundation of Hunan Provincial (no. 2014JJ4044).