Journal of Mathematics
Volume 2019 (2019), Article ID 3210649, 13 pages
https://doi.org/10.1155/2019/3210649

Research Article

On Mixed Equilibrium Problems in Hadamard Spaces

Chinedu Izuchukwu,1,2 Kazeem Olalekan Aremu,¹ Olawale Kazeem Oyewole,1,2 Oluwatosin Temitope Mewomo,¹ and Safeer Hussain Khan³

¹School of Mathematics, Statistics and Computer Science, University of KwaZulu-Natal, Durban, South Africa
²DST-NRF Center of Excellence in Mathematical and Statistical Sciences (CoE-MaSS), Pretoria, South Africa
³Department of Mathematics, Statistics and Physics, Qatar University, Doha, Qatar

Correspondence should be addressed to Safeer Hussain Khan; safeer@qu.edu.qa

Received 14 June 2019; Accepted 17 September 2019; Published 13 October 2019

Guest Editor: Jamshaid Ahmad

Copyright © 2019 Chinedu Izuchukwu et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. The publication of this article was funded by Qatar National Library.

Abstract.

The main purpose of this paper is to study mixed equilibrium problems in Hadamard spaces. First, we establish the existence of solution of the mixed equilibrium problem and the unique existence of the resolvent operator for the problem. We then prove a strong convergence of the resolvent and a -convergence of the proximal point algorithm to a solution of the mixed equilibrium problem under some suitable conditions. Furthermore, we study the asymptotic behavior of the sequence generated by a Halpern-type PPA. Finally, we give a numerical example in a nonlinear space setting to illustrate the applicability of our results. Our results extend and unify some related results in the literature.

1. Introduction

Let C be a nonempty set and be any real-valued function defined on C. The minimization problem (MP) is defined as

In this case, is called a minimizer of and denotes the set of minimizers of . MPs are very useful in optimization theory and convex and nonlinear analysis. One of the most popular and effective methods for solving MPs is the proximal point algorithm (PPA) which was introduced in Hilbert space by Martinet [1] in 1970 and was further extensively studied in the same space by Rockafellar [2] in 1976. The PPA and its generalizations have also been studied extensively for solving MP (1) and related optimization problems in Banach spaces and Hadamard manifolds (see [3–7] and the references therein), as well as in Hadamard and p-uniformly convex metric spaces (see [8–13] and the references therein).

An important generalization of Problem (1) is the following equilibrium problem (EP), defined as

The point for which (2) is satisfied is called an equilibrium point of F. The solution set of problem (2) is denoted by . The EP is one of the most important problems in optimization theory that has received a lot of attention in recent time since it includes many other optimization and mathematical problems as special cases, namely, MPs, variational inequality problems, complementarity problems, fixed point problems, and convex feasibility problems, among others (see, for example, [5, 14–18]). Thus, EPs are of central importance in optimization theory as well as in nonlinear and convex analysis. As a result of this, numerous authors have studied EPs in Hilbert, Banach, and topological vector spaces (see [19, 20] and the references therein), as well as in Hadamard manifolds (see [3, 21]).

Very recently, Kumam and Chaipunya [5] extended these studies to Hadamard spaces. First, they established the existence of an equilibrium point of a bifunction satisfying some convexity, continuity, and coercivity assumptions, and they also established some fundamental properties of the resolvent of the bifunction. Furthermore, they proved that the PPA -converges to an equilibrium point of a monotone bifunction in a Hadamard space. More precisely, they proved the following theorem.

Theorem 1. Let C be a nonempty closed and convex subset of an Hadamard space X and be monotone and -upper semicontinuous in the first variable such that for all (where means the domain of ). Suppose that and for an initial guess , the sequence is generated by

where is a sequence of positive real numbers bounded away from 0. Then, -converges to an element of .
Other authors have also studied EPs in Hadamard spaces (see, for example, [14, 15]).
In the linear settings (for example, in Hilbert spaces), EPs have been generalized into what is called the mixed equilibrium problem (MEP), defined as

The MEP is an important class of optimization problems since it contains many other optimization problems as special cases. For instance, if in (3), then the MEP (4) reduces to MP (1). Also, if in (3), then the MEP (4) reduces to the EP (2). The existence of solutions of the MEP (4) was established in Hilbert spaces by Peng and Yao [22] (see also [23]). More so, different iterative algorithms have been developed by numerous authors for approximating solutions of MEP (4) in real Hilbert spaces (see, for example, [22–24] and the references therein).
Since MEPs contain both MPs and EPs as special cases in Hilbert spaces, it is important to extend their study to Hadamard spaces, so as to unify other optimization problems (in particular, MPs and EPs) in Hadamard spaces. Moreover, Hadamard spaces are more suitable frameworks for the study of optimization problems and other related mathematical problems since many recent results in these spaces have already found applications in diverse fields than they do in Hilbert spaces. For instance, the minimizers of the energy functional (which is an example of a convex and lower semicontinuous functional in a Hadamard space), called harmonic maps, are very useful in geometry and analysis (see [9]). Also, the gradient flow theorem in Hadamard spaces was employed to investigate the asymptotic behavior of the Calabi flow in Kahler geometry (see [25]). Furthermore, the study of the PPA for optimization problems has successfully been applied in Hadamard spaces, for computing medians and means, which are very important in computational phylogenetics, diffusion tensor imaging, consensus algorithms, and modeling of airway systems in human lungs and blood vessels (see [26, 27], for details). It is also worthy to note that many nonconvex problems in the linear settings can be viewed as convex problems in Hadamard spaces (see Section 4 of this paper).
Therefore, it is our interest in this paper to extend the study of the MEP (4) to Hadamard spaces. First, we establish the existence of solution of the MEP (4) and the unique existence of the resolvent operator associated with F and . We then prove a strong convergence of the resolvent and a -convergence of the PPA to a solution of MEP (4) under some suitable conditions on F and . Furthermore, we study the asymptotic behavior of the sequence generated by the Halpern-type PPA. Finally, we give a numerical example in a nonlinear space setting to illustrate the applicability of our results. Our results extend and unify the results of Kumam and Chaipunya [5] and Peng and Yao [22].
The rest of this paper is organized as follows: In Section 2, we recall the geometry of geodesic spaces and some useful definitions and lemmas. In Section 3, we establish the existence of solution for MEP (4) and the unique existence of the resolvent operator associated with F and . Some fundamental properties of the resolvent operator are also established in this section. In Section 4, we prove a strong convergence of the resolvent and a -convergence of the PPA to a solution of MEP (4) under some suitable conditions on F and . In Section 5, we study the asymptotic behavior of the sequence generated by the Halpern-type PPA. In Section 6, we generate some numerical results in nonlinear setting for the PPA and the Halpern-type PPA, to show the applicability of our results.

2. Preliminaries

2.1. Geometry of Geodesic Spaces

Definition 1. Let be a metric space, and be an interval. A curve c (or simply a geodesic path) joining x to y is an isometry such that , , and for all . The image of a geodesic path is called a geodesic segment, which is denoted by whenever it is unique.

Definition 2 (see [28]). A metric space is called a geodesic space if every two points of X are joined by a geodesic path, and X is said to be uniquely geodesic if every two points of X are joined by exactly one geodesic path. A subset C of X is said to be convex if C includes every geodesic segments joining two of its points. Let and , and we write for the unique point z in the geodesic segment joining from x to y such that

A geodesic triangle in a geodesic metric space consists of three vertices (points in X) with unparameterized geodesic segment between each pair of vertices. For any geodesic triangle, there is comparison (Alexandrov) triangle such that for . Let be a geodesic triangle in X and be a comparison triangle for , then is said to satisfy the CAT(0) inequality if for all points and :

Let be points in X and be the midpoint of the segment ; then, the CAT(0) inequality implies

Inequality (7) is known as the CN inequality of Bruhat and Titis [29].

Definition 3. A geodesic space X is said to be a CAT(0) space if all geodesic triangles satisfy the CAT(0) inequality. Equivalently, X is called a CAT(0) space if and only if it satisfies the CN inequality.
CAT(0) spaces are examples of uniquely geodesic spaces, and complete CAT(0) spaces are called Hadamard spaces.

Definition 4. Let C be a nonempty closed and convex subset of a CAT(0) space X. The metric projection is a mapping which assigns to each , the unique point in C such that

Definition 5 (see [30]). Let X be a CAT(0) space. Denote the pair by and call it a vector. Then, a mapping defined by

is called a quasilinearization mapping.
It is easy to check that , , , and for all . A geodesic space X is said to satisfy the Cauchy–Swartz inequality if . It has been established in [30] that a geodesically connected metric space is a CAT(0) space if and only if it satisfies the Cauchy–Schwartz inequality. Examples of CAT(0) spaces include Euclidean spaces , Hilbert spaces, simply connected Riemannian manifolds of nonpositive sectional curvature [31], -trees, and Hilbert ball [32], among others.
We end this section with the following important lemmas which characterize CAT(0) spaces.

Lemma 1. Let X be a CAT(0) space, , and . Then,

(i) (see [28])(ii) (see [28])

2.2. Notion of -Convergence

Definition 6. Let be a bounded sequence in a geodesic metric space X. Then, the asymptotic center of is defined by

A sequence in X is said to be -convergent to a point if for every subsequence of . In this case, we write (see [33]). The concept of -convergence in metric spaces was first introduced and studied by Lim [34]. Kirk and Panyanak [35] later introduced and studied this concept in CAT(0) spaces and proved that it is very similar to the weak convergence in Banach space setting.
We now end this section with the following important lemmas which are concerned with -convergence.

Lemma 2 (see [28, 36]). Let X be an Hadamard space. Then,

(i)Every bounded sequence in X has a -convergent subsequence(ii)Every bounded sequence in X has a unique asymptotic center

Lemma 3 ([37], Opial’s Lemma). Let X be an Hadamard space and be a sequence in X. If there exists a nonempty subset F in which

(i) exists for every (ii)if is a subsequence of which is -convergent to x, then

Then, there is a such that is -convergent to p.

Lemma 4 ([14], Proposition 4.3). Suppose that is -convergent to q and there exists such that then converges strongly to .

3. Existence and Uniqueness of Solution

In this section, we establish the existence of solution for MEP (4). We also establish the unique existence of the resolvent operator associated with the bifunction F and the convex functional . In addition, we study some fundamental properties of this resolvent operator. We begin with the following known results.

Definition 7. Let X be a CAT(0) space. A function (where means the domain of ) is said to be convex, if

is lower semicontinuous (or upper semicontinuous) at a point , if

for each sequence in such that . We say that is lower semicontinuous (or upper semicontinuous) on , if it is lower semicontinuous (or upper semicontinuous) at any point in .

Lemma 5 (See [9]). Let X be a Hadamard space and be a convex and lower semicontinuous function. Then, is -lower semicontinuous.
For a nonempty subset C of we denote by the convex hull of That is, the smallest convex subset of X containing Recall that the convex hull of a finite set is the set of all convex combinations of its points.

Theorem 2 (the KKM principle) (see [5], Theorem 3.3; see also [14], Lemma 1.8). Let C be a nonempty, closed, and convex subset of an Hadamard space X and be a set-valued mapping with closed values. Suppose that for any finite subset of

Then, the family has the finite intersection property. Moreover, if is compact for some then .

3.1. Existence of Solution for Mixed Equilibrium Problem

Theorem 3. Let C be a nonempty closed and convex subset of an Hadamard space X. Let be a real-valued function and be a bifunction such that the following assumptions hold:

(A1)(A2)For every the set is convex(A3)There exists a compact subset containing a point such that whenever

Then, the MEP (4) has a solution.

Proof. For each define the set-valued mapping by

By (A1), we obtain that, for each since . Also, we obtain from (A2) that is a closed subset of C for all .
We claim that G satisfies the inclusion (13). Suppose for contradiction that this is not true, then there exist a finite subset and such that for each . That is, there exists such that for each . By (14), we obtain for each that

Thus, for each which is convex by (A2). Since is the smallest convex set containing , we have that which implies that That is, which is a contradiction. Therefore, G satisfies the inclusion (13).
Now, observe that (A3) implies that there exists a compact subset D of C containing such that for any we have

which further implies that

Thus, is compact. It then follows from Theorem 2 that . This implies that there exists such that

That is, MEP (4) has a solution.

3.2. Existence and Uniqueness of Resolvent Operator

Definition 8. Let X be an Hadamard space and C be a nonempty subset of X. Let be a bifunction, be a real-valued function, , and ; then, we define the perturbation of F and by

In the next theorem, we shall prove the existence and uniqueness of solution of the following auxiliary problem: find such that

where is as defined in (19). The proof for existence is similar to the proof of Theorem 3. But for completeness, we shall give the proof here.

Theorem 4. Let C be a nonempty closed and convex subset of an Hadamard space . Let be a convex function and be a bifunction such that the following assumptions hold:

(A1)(A2)F is monotone, i.e., (A3) is convex (A4)For each and , there exists a compact subset containing a point such that whenever .

Then, (20) has a unique solution.

Proof. Let be a point in . For each define the set-valued mapping by

Then, it is easy to see that is a nonempty closed subset of . As in the proof of Theorem 3, we claim that G satisfies the inclusion (13). Suppose for contradiction that this is not true, then there exists such that

By (A3) and the convexity of , we obtain that

which is a contradiction. Therefore, G satisfies the inclusion (13). By (A4), we obtain that . Thus, is compact and by Theorem 2, we get that . Therefore, (20) has a solution.
Next, we show that this solution is unique. Suppose that x and solve (20). Then,

Adding both inequalities and noting that F is monotone, we obtain that

which implies that .

Definition 9. Let X be an Hadamard space and C be a nonempty closed and convex subset of X. Let be a bifunction and be a convex function. Assume that (20) has a unique solution for each and . This unique solution is denoted by , and it is called the resolvent operator associated with F and of order and at . In other words, the resolvent operator associated with F and is the set-valued mapping defined by

Under the assumptions of Theorem 4, we have the unique existence of . Therefore, is well defined.

3.3. Fundamental Properties of the Resolvent Operator

In the following theorem, we shall study some fundamental properties of the resolvent operator. First, we recall the following definitions which will be needed for our study.

Definition 10. Let X be a CAT(0) space. A point is called a fixed point of a nonlinear mapping , if . We denote the set of fixed points of T by . The mapping T is said to be

(i)Firmly nonexpansive, if

(ii)Nonexpansive, if

Theorem 5. Let C be a nonempty closed and convex subset of an Hadamard space X. Let be a convex function and be a bifunction satisfying assumptions (A1)–(A4) of Theorem 4. For , we have that is single valued. Moreover, if then

(i)is firmly nonexpansive restricted to C(ii)For we have

(iii)For we have which implies that (iv)

Proof. For each and let . Then from (26), we have

Adding both inequalities and using assumption (A2), we obtain that

which implies that . This further implies that . Therefore, is single valued.

(i)Let , then

 and

 Adding (32) and (33), and noting that F is monotone, we obtain

 which implies that

 That is,

(ii)It follows from (36) and the definition of quasilinearization that

(iii)Let and , then we have that

 and

 Adding (38) and (39), and using the monotonicity of F, we obtain that

 By quasilinearization, we obtain that

 Since , we obtain that

 which implies that

 Moreover, we obtain by triangle inequality and (43) that

(iv)Observe that

Remark 1. It follows from Cauchy–Schwartz inequality that firmly nonexpansive mappings are nonexpansive, and it is well known that the set of fixed points of nonexpansive mappings is closed and convex. Thus, by (i) and (iv) of Theorem 5, we have that is closed and convex.

4. Convergence Results

For the rest of this paper, we shall assume that C is a nonempty closed and convex subset of an Hadamard space X and that .

4.1. Convergence of Resolvent

In the following theorem, we shall prove that converges strongly to a solution of MEP (4) as .

Theorem 6. Let be a convex and lower semicontinuous function and be -upper semicontinuous in the first argument which satisfies assumptions (A1)–(A4) of Theorem 4. If , then converges strongly to which is the nearest point of to x as .

Proof. Let since is nonexpansive (by Remark 1), we obtain that is bounded. Let be a sequence that converges to 0 as . Then, is bounded. Thus, by Lemma 2(i), there exists a subsequence of that -converges to .
Now, observe that, by the definition of , the -upper semicontinuity of lower semicontinuous of , and Lemma 5, we obtain that

Therefore, . Hence, we obtain from Theorem 5(ii) that

Since is -lower semicontinuous, we obtain that

which implies that

Thus, where is the metric projection of X onto , and . Therefore, by taking , we have that -converges to as .
Now, observe also that Theorem 5(ii) implies that

It then follows from Lemma 4 that converges strongly to as .
By setting in Theorem 6, we obtain the following result which is similar to ([14], Theorem 4.4).

Corollary 1. Let be -upper semicontinuous in the first argument which satisfies assumptions (A1)–(A4) of Theorem 4. If , then converges strongly to which is the nearest point of to x as .

4.2. Proximal Point Algorithm

In this section, we study the -convergence of the sequence generated by the following PPA for approximating solutions of MEP (4): For an initial starting point in C, define the sequence in C by

where is a sequence in , is a bifunction, and is a convex function.

Recall that the PPA does not converge strongly in general without additional assumptions even for the case where . See for example, the question of interest raised by Rockafella as to whether the PPA can be improved from weak convergence (an analogue of -convergence) to strong convergence in Hilbert space settings. Several counterexamples have been constructed to resolve this question in the negative (see [38, 39]). Therefore, only weak convergence of the PPA is expected without additional assumptions. For this reason, we propose the following -convergence theorem for the PPA (51).

Theorem 7. Let be a convex and lower semicontinuous function and be -upper semicontinuous in the first argument which satisfies assumptions (A1)–(A4) of Theorem 4. Let be a sequence in such that . Suppose that , then, the sequence given by (51) -converges to an element of .

Proof. Let . Then, by Remark 1 and Theorem 5(iv), we obtain that

which implies that exists for all . Hence is bounded. It then follows from Theorem 5(ii) that

That is,

Since is bounded, then there exists a subsequence of that -converges to a point, say . From (51) and (26), we obtain that

Since is bounded, F is -upper semicontinuous in the first argument and is lower semicontinuous, we obtained from (54) and (55) that

for some and for all . This implies that .
It then follows from Lemma 3 that -converges to an element of .
By setting in Theorem 7, we obtain the following result which coincides with ([5], Theorem 7.3).

Corollary 2. Let be -upper semicontinuous in the first argument which satisfies assumptions (A1)–(A4) of Theorem 4 and be a sequence in such that . Suppose that ; then, the sequence given for by

-converges to an element of .
By setting in Theorem 7, we obtain the following corollary which is similar to ([9], Theorem 1.4).

Corollary 3. Let be a convex and lower semicontinuous function and be a sequence in such that . Suppose that ; then, the sequence given for by

-converges to an element of .

5. Asymptotic Behavior of Halpern’s Algorithm

To obtain strong convergence result, we modify the PPA into the following Halpern-type PPA and study the asymptotic behavior of the sequence generated by it: For define the sequence by

where is a sequence in and and are as defined in (51).

We begin by establishing the following lemmas which will be very useful to our study.

Lemma 6. Let be a convex and lower semicontinuous function and be a bifunction satisfying (A1)–(A4) of Theorem 4. If and then the following inequalities hold:

Proof. We first prove (60). Let and . Then, by (26), we obtain that

which implies that

Now, set for all in (5). Since is convex and F satisfies conditions (A1) and (A3) of Theorem 4, we obtain that

which implies that

As in (64), we obtain (60).
Next, we prove (60). From (60), we obtain that

Similarly, we have

Adding both inequalities and noting that F is monotone, we get

Lemma 7. Let be a convex and lower semicontinuous function and be a bifunction satisfying (A1)–(A4) of Theorem 4. Let be a sequence in and be an element of . Suppose that and for some bounded sequence in then .
Proof. From (60), we obtain that

which implies that

Since is bounded and , we obtain that

which by Lemma 2(ii) and Theorem 5(iv) implies that .

Lemma 8. (Xu, [40]). Let be a sequence of nonnegative real numbers satisfying the following relation:

where . Then, as .

Theorem 8. Let be a convex and lower semicontinuous function and be a bifunction satisfying (A1–A4) of Theorem 4. Let be a sequence defined by (59), where is a sequence in and is a sequence in such that . Then, we have the following:

(i)The sequence is bounded if and only if(ii)If and then and converge to whereis the metric projection of X onto

Proof. (i) Suppose that is bounded. Then by Lemma 2(ii), there exists such that . From (59) and Lemma 1(i), we obtain that

which implies that is bounded. Also, since and , we obtain by Lemma 7 that.
Conversely, let . Then, we may assume that . Thus, by (59) and Lemma 1, we obtain that

which implies by induction that

Therefore, is bounded. Consequently, is also bounded.
(ii) Since we obtain from (74) that and are bounded. Furthermore, we obtain from Lemma 1(ii) that

where . Now, set . Then, by the boundedness of and Lemma 2(i), we obtain that there exists a subsequence of that -converges to some . Thus, by Lemma 2(ii), we obtain that . Moreover, and is bounded. Hence, by Lemma 7, we obtain that .
Next, we show that converges to . By the -lower semicontinuity of , we obtain that

Since , , , and , we obtain from the definition of and (76) that

Thus, applying Lemma 8 to (75) gives that converges to . It then follows that is convergent to .
By setting in Theorem 8, we obtain the following new result for equilibrium problem in an Hadamard space.

Corollary 4. Let be a bifunction satisfying (A1–A3) of Theorem 4 and be a sequence defined for , by

where is a sequence in and is a sequence in such that . Then, we have the following:

(i)The sequence is bounded if and only if (ii)If and then and converge to where is the metric projection of X onto

By setting in Theorem 8, we obtain the following result which coincides with ([41], Theorem 5.1).

Corollary 5. Let be a proper convex and lower semicontinuous function and be a sequence defined for , by

where is a sequence in and is a sequence in such that . Then, we have the following:

(i)The sequence is bounded if and only if (ii)If and then and converge to where is the metric projection of X onto

6. Numerical Results

In this section, we generate some numerical results in nonlinear setting for Algorithms (58) and (79).

Let be endowed with a metric defined by

Then, is an Hadamard space (see ([42], Example 5.2)) with the geodesic joining x to y given by

Now, define by

Then, it follows from ([42], Example 5.2) that is a proper convex and lower semicontinuous function in but not convex in the classical sense (Figure¹).

(a)

(b)

(c)

(d)

Figure 1: Errors vs iteration numbers n: Case 1 (a); Case 2 (b); Case 3 (c); Case 4 (d).

Now, take and for all , then all the conditions of Corollaries 4.5 and 5.6 are satisfied. Hence, by considering the following initial vectors, we obtain the numerical results for Algorithms (58) and (79) as shown by the graphs as follows:

 Case 1: and  Case 2: and  Case 3: and  Case 4: and

Data Availability

No data were used to support this study.

Disclosure

Opinions expressed and conclusions arrived are those of the authors and are not necessarily to be attributed to the CoE-MaSS and NRF.

Conflicts of Interest

The authors declare that they have no conflicts of interest.

Acknowledgments

The publication of this article was funded by the Qatar National Library. The first and third authors acknowledge the bursary and financial support from Department of Science and Technology and National Research Foundation, Republic of South Africa Center of Excellence in Mathematical and Statistical Sciences (DST-NRF COE-MaSS) Doctoral Bursary. The fourth author is supported in part by the National Research Foundation (NRF) of South Africa Incentive Funding for Rated Researchers (Grant Number: 119903).

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