On Mixed Equilibrium Problems in Hadamard Spaces

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Introduction
Let C be a nonempty set and Ψ be any real-valued function defined on C. e minimization problem (MP) is defined as find x * ∈ C such that Ψ x *  ≤ Ψ(y), ∀y ∈ C. ( In this case, x * is called a minimizer of Ψ and argmin y∈C Ψ(y) denotes the set of minimizers of Ψ. 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.e 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][4][5][6][7] and the references therein), as well as in Hadamard and p-uniformly convex metric spaces (see [8][9][10][11][12][13] and the references therein).
An important generalization of Problem (1) is the following equilibrium problem (EP), defined as find x * ∈ C such that F x * , y  ≥ 0, ∀y ∈ C.
(2) e point x * for which (2) is satisfied is called an equilibrium point of F. e solution set of problem (2) is denoted by EP(C, F). e 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][15][16][17][18]). us, 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 Δ-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 F : C × C ⟶ R be monotone and Δ-upper semicontinuous in the first variable such that D(J F λ )IC for all λ > 0 (where D(J F λ ) means the domain of J F λ ).Suppose that EP(C, F) ≠ ∅ and for an initial guess x 0 ∈ C, the sequence x n   ⊂ C is generated by where λ n   is a sequence of positive real numbers bounded away from 0.
en, x n  Δ-converges to an element of EP(C, F).
In the linear settings (for example, in Hilbert spaces), EPs have been generalized into what is called the mixed equilibrium problem (MEP), defined as find x * ∈ C such that F x * , y  + Ψ(y) − Ψ x *  ≥ 0, ∀y ∈ C.

(4)
The MEP is an important class of optimization problems since it contains many other optimization problems as special cases.For instance, if F ≡ 0 in (3), then the MEP (4) reduces to MP (1).Also, if Ψ ≡ 0 in (3), then the MEP (4) reduces to the EP (2).e 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][23][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 Ψ.We then prove a strong convergence of the resolvent and a Δ-convergence of the PPA to a solution of MEP (4) under some suitable conditions on F and Ψ.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 Ψ.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 Δ-convergence of the PPA to a solution of MEP (4) under some suitable conditions on F and Ψ.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.

Geometry of Geodesic Spaces
Definition 1.Let (X, d) be a metric space, x, y ∈ X and I � [0, d(x, y)] be an interval.A curve c (or simply a geodesic path) joining x to y is an isometry c : I ⟶ X such that c(0) � x, c(d(x, y)) � y, and d(c(t), c(t ′ ) � |t − t ′ |) for all t, t ′ ∈ I. e image of a geodesic path is called a geodesic segment, which is denoted by [x, y] whenever it is unique.Definition 2 (see [28]).A metric space (X, d) 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 x, y ∈ X and t ∈ [0, 1], and we write tx ⊕ (1 − t)y for the unique point z in the geodesic segment joining from x to y such that d(x, z) � (1 − t)d(x, y) and d(z, y) � td(x, y). (5) A geodesic triangle Δ(x 1 , x 2 , x 3 ) in a geodesic metric space (X, d) 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 Let Δ be a geodesic triangle in X and Δ be a comparison triangle for Δ , then Δ is said to satisfy the CAT(0) inequality if for all points x, y ∈Δ and x, y ∈ Δ : 2 Journal of Mathematics Let x, y, and z be points in X and y 0 be the midpoint of the segment [y, z]; 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. e metric projection is a mapping P C : X ⟶ C which assigns to each x ∈ X, the unique point Definition 5 (see [30]).Let X be a CAT(0) space.Denote the pair (a, b) ∈ X × X by ab �→ and call it a vector.en, a mapping 〈., .〉:  [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 R n , Hilbert spaces, simply connected Riemannian manifolds of nonpositive sectional curvature [31], R-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, x, y, z ∈ X, and t, s (see [28]) [28])

Notion of Δ-Convergence
Definition 6.Let x n   be a bounded sequence in a geodesic metric space X. en, the asymptotic center A( x n  ) of x n   is defined by .In this case, we write Δ-lim n⟶∞ x n � v (see [33]).e concept of Δ-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 Δ-convergence.
Lemma 2 (see [28,36]).Let X be an Hadamard space.en, (i) Every bounded sequence in X has a Δ-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 x n   be a sequence in X.If there exists a nonempty subset F in which Lemma 4 ([14], Proposition 4.3).Suppose that x n   is Δ-convergent to q and there exists y ∈ X such that lim sup d(x n , y) ≤ d(q, y), then x n   converges strongly to q.

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 Ψ.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

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Ψ is lower semicontinuous (or upper semicontinuous) at a point x ∈ D(Ψ), if for each sequence x n   in D(Ψ) such that lim n⟶∞ x n � x.We say that Ψ is lower semicontinuous (or upper semicontinuous) on D(Ψ), if it is lower semicontinuous (or upper semicontinuous) at any point in D(Ψ).
For a nonempty subset C of X, we denote by conv(C), the convex hull of C. at is, the smallest convex subset of X containing C. 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], eorem 3.3; see also [14], Lemma 1.8).Let C be a nonempty, closed, and convex subset of an Hadamard space X and G : C ⟶ 2 C be a set-valued mapping with closed values.Suppose that for any finite subset en, the family

Existence of Solution for Mixed Equilibrium Problem
Theorem 3. Let C be a nonempty closed and convex subset of an Hadamard space X.Let Ψ : C ⟶ Rbe a real-valued function and F : C × C ⟶ R be a bifunction such that the following assumptions hold: en, the MEP (4) has a solution.
Proof.For each y ∈ C, define the set-valued mapping G : By (A1), we obtain that, for each y ∈ C, G(y) ≠ ∅ since y ∈ G(y).Also, we obtain from (A2) that G(y) is a closed subset of C for all y ∈ C.
We claim that G satisfies the inclusion (13).Suppose for contradiction that this is not true, then there exist a finite subset y 1 , y 2 , . . ., y m   of C and at is, 0 � F(y * , y * ) + Ψ(y * ) − Ψ(y * ) < 0, which is a contradiction.erefore, G satisfies the inclusion (13).Now, observe that (A3) implies that there exists a compact subset D of C containing y 0 ∈ D such that for any x ∈ C/D, we have which further implies that is implies that there exists x * ∈ C such that at is, MEP (4) has a solution.□

Existence and Uniqueness of Resolvent Operator
Definition 8. Let X be an Hadamard space and C be a nonempty subset of X.Let F : C × C ⟶ R be a bifunction, Ψ : C ⟶ R be a real-valued function, x ∈ X, and λ > 0; then, we define the perturbation In the next theorem, we shall prove the existence and uniqueness of solution of the following auxiliary problem: where  F x is as defined in (19).e proof for existence is similar to the proof of eorem 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 X.Let Ψ : C ⟶ Rbe a convex function and F : C × C ⟶ R be a bifunction such that the following assumptions hold: en, (20) has a unique solution.
Proof.Let x be a point in X.For each y ∈ C, define the setvalued mapping en, it is easy to see that G(y) is a nonempty closed subset of C. As in the proof of eorem 3, we claim that G satisfies the inclusion (13).Suppose for contradiction that this is not true, then there exists By (A3) and the convexity of Ψ, we obtain that which is a contradiction.erefore, G satisfies the inclusion (13).By (A4), we obtain that G(y x ) ⊂ D x .us, G(y x ) is compact and by eorem 2, we get that ∩ y∈C G(y) ≠ ∅. erefore, (20) has a solution.Next, we show that this solution is unique.Suppose that x and x * solve (20).en, Adding both inequalities and noting that F is monotone, we obtain that which implies that x � x * .□ Definition 9. Let X be an Hadamard space and C be a nonempty closed and convex subset of X.Let F : C × C ⟶ R be a bifunction and Ψ : C ⟶ R be a convex function.Assume that (20) has a unique solution for each λ > 0 and x ∈ X. is unique solution is denoted by J Ψ λF x, and it is called the resolvent operator associated with F and Ψ of order λ > 0 and at x ∈ X.In other words, the resolvent operator associated with F and Ψ is the set-valued mapping J Ψ λF : X ⟶ 2 C defined by Under the assumptions of eorem 4, we have the unique existence of J Ψ λF (x).erefore, J Ψ λF is well defined.

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 x ∈ X is called a fixed point of a nonlinear mapping T : X ⟶ X, if Tx � x.We denote the set of fixed points of T by Fix(T).e mapping T is said to be (i) Firmly nonexpansive, if Theorem 5. Let C be a nonempty closed and convex subset of an Hadamard space X.Let Ψ : C ⟶ Rbe a convex function and F : C × C ⟶ R be a bifunction satisfying assumptions (A1)-(A4) of eorem 4. For λ > 0, we have that

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Proof.For each x ∈ D(J Ψ λF ) and λ > 0, let z 1 , z 2 ∈ J Ψ λF x. en from (26), we have Adding both inequalities and using assumption (A2), we obtain that and Adding ( 32) and (33), and noting that F is monotone, we obtain which implies that at is, (ii) It follows from (36) and the definition of quasilinearization that (iii) Let x ∈ C and 0 < λ ≤ μ, then we have that and Adding ( 38) and ( 39), and using the monotonicity of F, we obtain that By quasilinearization, we obtain that Since (λ/μ) ≤ 1, we obtain that which implies that Moreover, we obtain by triangle inequality and (43) that

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 D(J Ψ λF )IC.

Convergence of Resolvent.
In the following theorem, we shall prove that J Ψ λF x   converges strongly to a solution of MEP (4) as λ ⟶ 0. Theorem 6.Let Ψ : C ⟶ Rbe a convex and lower semicontinuous function and λF is nonexpansive (by Remark 1), we obtain that J Ψ λF x   is bounded.Let λ n   be a sequence that converges to 0 as n ⟶ ∞. en, J Ψ λ n F x   is bounded.us, by Lemma 2(i), there exists a subsequence that Δ-converges to q ∈ C. Now, observe that, by the definition of J Ψ λF , the Δ-upper semicontinuity of F, lower semicontinuous of Ψ, and Lemma 5, we obtain that F(q, y) + Ψ(y) − Ψ(q) ≥ 0.

Proximal Point Algorithm.
In this section, we study the Δ-convergence of the sequence generated by the following PPA for approximating solutions of MEP (4): For an initial starting point x 1 in C, define the sequence x n   in C by where Recall that the PPA does not converge strongly in general without additional assumptions even for the case where F ≡ 0. See for example, the question of interest raised by Rockafella as to whether the PPA can be improved from weak convergence (an analogue of Δ-convergence) to strong convergence in Hilbert space settings.Several counterexamples have been constructed to resolve this question in the negative (see [38,39]).erefore, only weak convergence of the PPA is expected without additional assumptions.For this reason, we propose the following Δ-convergence theorem for the PPA (51).  be a sequence in (0, ∞) such that 0 < λ ≤ λ n , ∀n ≥ 1. Suppose that MEP(C, F, Ψ) ≠ ∅, then, the sequence given by (51) Δ-converges to an element of MEP(C, F, Ψ).
en, by Remark 1 and eorem 5(iv), we obtain that which implies that lim n⟶∞ d(x n , v) exists for all v ∈ MEP(C, F, Ψ).Hence x n   is bounded.It then follows from eorem 5(ii) that Since x n   is bounded, then there exists a subsequence x nk   of x n   that Δ-converges to a point, say q ∈ C. From (51) and (26), we obtain that Journal of Mathematics Since 0 < λ ≤ λ nk , x n   is bounded, F is Δ-upper semicontinuous in the first argument and Ψ is lower semicontinuous, we obtained from (54) and (55) that for some M > 0 and for all y ∈ C. is implies that q ∈ MEP(C, F, Ψ).
It then follows from Lemma 3 that x n  Δ-converges to an element of MEP(C, F, Ψ).

Corollary 3. Let Ψ : C ⟶ R be a convex and lower semicontinuous function and λ n
be a sequence in (0, ∞) such that 0 < λ ≤ λ n , ∀n ≥ 1. Suppose that argmin y∈C Ψ(y) ≠ ∅; then, the sequence given for x 1 ∈ C by Δ-converges to an element of argmin y∈C Ψ(y).

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 x 1 , u ∈ C, define the sequence x n   ⊂ C by where α n   is a sequence in (0, 1) and λ n  , F and Ψ are as defined in (51).
We begin by establishing the following lemmas which will be very useful to our study.Lemma 6.Let Ψ : C ⟶ R be a convex and lower semicontinuous function and F : C × C ⟶ R be a bifunction satisfying (A1)-(A4) of eorem 4. If λ, μ > 0 and x, y ∈ C, then the following inequalities hold: Proof.We first prove (60).Let λ, μ > 0 and x, y ∈ C. en, by (26), we obtain that Now, set z � tJ Ψ μF y ⊕ (1 − t)J Ψ λF x for all t ∈ (0, 1) in ( 5).Since Ψ is convex and F satisfies conditions (A1) and (A3) of eorem 4, we obtain that Journal of Mathematics which implies that As t ⟶ 0 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 Proof.From (60), we obtain that which implies that Since lim n⟶∞ λ n � ∞, x n   is bounded and which by Lemma 2(ii) and Theorem 5(iv) implies that v ∈ fix(J Ψ F ) � MEP(C, F, Ψ).
Lemma 8 (Xu,[40]).Let a n   be a sequence of nonnegative real numbers satisfying the following relation:   be a sequence defined by ( 59), where α n   is a sequence in (0, 1) and λ n   is a sequence in (0, ∞) such that lim n⟶∞ λ n � ∞. en, we have the following: From (59) and Lemma 1(i), we obtain that Conversely, let MEP(C, F, Ψ) ≠ ∅. en, we may assume that v ∈ MEP(C, F, Ψ) ≠ ∅. us, by (59) and Lemma 1, we obtain that which implies by induction that is also bounded.
(ii) Since Γ ≔ MEP(C, F, Ψ) ≠ ∅, we obtain from (74) that x n   and J Ψ λ n F x n   are bounded.Furthermore, we obtain from Lemma 1(ii) that where Next, we show that x n   converges to  v.By the Δ-lower semicontinuity of d 2 (u, .),we obtain that

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

Theorem 7 .
Let Ψ : C ⟶ R be a convex and lower semicontinuous function and F : C × C ⟶ R be Δ-upper semicontinuous in the first argument which satisfies assumptions (A1)-(A4) of eorem 4. Let λ n

□ Lemma 7 .
Let Ψ : C ⟶ R be a convex and lower semicontinuous function and F : C × C ⟶ R be a bifunction satisfying (A1)-(A4) of eorem 4. Let λ n  be a sequence in (0, ∞) and v be an element of C. Suppose that lim n⟶∞ λ n � ∞ and A( J Ψ λ n x n  ) � v { } for some bounded sequence x n   in X, then v ∈ MEP(C, F, Ψ).

Theorem 8 .
Let Ψ : C ⟶ R be a convex and lower semicontinuous function and F : C × C ⟶ R be a bifunction satisfying (A1-A4) of eorem 4. Let x n and  v ∈ Γ, we obtain from the definition of P Γ and (76) that
Moreover, lim k⟶∞ λ n k � ∞ and x n k   is bounded.Hence, by Lemma 7, we obtain that  v ∈ MEP(C, F, Ψ).
∀n ≥ 1. en, by the boundedness of v n   and Lemma 2(i), we obtain that there exists a subsequence v n k that Δ-converges to some  v ∈ C.
≤ d 2 (u, v) − lim inf n⟶∞ d 2 u, v n  ≤ d 2 (u,  v) − lim inf n⟶∞ d 2 u, v n  ≤ 0.us, applying Lemma 8 to (75) gives that x n   converges to v � P Γ u.It then follows thatJ Ψ λ n F x n   is convergent to v � P Γ u.By setting Ψ ≡ 0 in eorem 8, we obtain the following new result for equilibrium problem in an Hadamard space.Corollary 4. Let F : C × C ⟶ R be a bifunction satisfying (A1-A3) of eorem 4 and x n  be a sequence defined for u, x 1 ∈ C, byx n+1 � α n u ⊕ 1 − α n J λ n F x n , (78)where α n   is a sequence in (0, 1) and λ n   is a sequence in (0, ∞) such that lim n⟶∞ λ n � ∞.en, we have the following:(i) The sequence J λ n F x n   is bounded if and only if EP(C, F) ≠ ∅ (ii) If lim n⟶∞ α n � 0,  ∞ n�1 α n � ∞ and Γ ≔ EP(C, F) ≠ ∅, then x n   and J λ n F x n   converge to v � P Γ u,where P Γ is the metric projection of X onto Γ By setting F ≡ 0 in eorem 8, we obtain the following result which coincides with([41], eorem 5.1).Let Ψ : C ⟶ C be a proper convex and lower semicontinuous function and x n   be a sequence defined for u, x 1 ∈ C, byx n+1 � α n u ⊕ 1 − α n J   is a sequence in (0, 1) and λ n   is a sequence in (0, ∞) such that lim n⟶∞ λ n � ∞.If lim n⟶∞ α n � 0,  ∞ n�1 α n � ∞ and Γ ≔ argmin y∈C Ψ (y) ≠ ∅,then x n  and J Ψ λ n x n   converge to v � P Γ u, where P Γ is the metric projection of X onto Γ n⟶∞ δ n □ [42]1 + y 2  Example 5.2)) with the geodesic joining x to y given by(1 − t)x ⊕ ty � (1 − t)x 1 + ty 1 , (1 − t)x 1 + ty 1  Now, define Ψ : R 2 ⟶ R by Ψ x 1 , x 2  � 100 x 2 − 2  − x 1 − 2 en, it follows from ([42], Example 5.2) that Ψ is a proper convex and lower semicontinuous function in (R 2 , d X ) but not convex in the classical sense (Figure1).