Bilevel equilibrium problems with lower and upper bounds in locally convex Hausdorff topological vector spaces

Journal Pre-proof Bilevel equilibrium problems with lower and upper bounds in locally convex Hausdorff topological vector spaces

Nguyen Van Hung, Donal O’Regan

PII:

S0166-8641(19)30350-5

DOI:

https://doi.org/10.1016/j.topol.2019.106939

Reference:

TOPOL 106939

To appear in:

Topology and its Applications

Received date:

6 October 2019

Revised date:

4 November 2019

Accepted date:

4 November 2019

Please cite this article as: N. Van Hung, D. O’Regan, Bilevel equilibrium problems with lower and upper bounds in locally convex Hausdorff topological vector spaces, Topol. Appl. (2019), 106939, doi: https://doi.org/10.1016/j.topol.2019.106939.

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Bilevel Equilibrium Problems with Lower and Upper Bounds in Locally Convex Hausdorff Topological Vector Spaces Nguyen Van Hunga,b,∗, Donal O’Reganc a Department

for Management of Science and Technology Development, Ton Duc Thang University, Ho Chi Minh City, Vietnam b Faculty of Mathematics and Statistics, Ton Duc Thang University, Ho Chi Minh City, Vietnam c School of Mathematics, Statistics and Applied Mathematics, National University of Ireland, Galway, Ireland

Abstract In this paper, we introduce a new class of bilevel equilibrium problems with lower and upper bounds in locally convex Hausdorff topological vector spaces and establish some conditions for the existence of solutions to these problems using the Kakutani-Fan-Glicksberg fixed-point theorem. Then, we establish generic stability of set-valued mappings and we show the set of essential points of a map is a dense residual subset of a (Hausdorff) metric space of set-valued maps for bilevel equilibrium problems with lower and upper bounds. The results presented in the paper are new and extend the main results given by some authors in the literature. Keywords: Kakutani-Fan-Glicksberg fixed-point theorem, Equilibrium problems with lower and upper bounds, Bilevel equilibrium problems with lower and upper bounds, Existence conditions, Generic stability. Mathematics Subject Classification 2010: 90C29, 47J20, 49J40.

1. Introduction In 1999, Isac et al. [8] studied an alternative version of a variational inequality in Hausdorff topological vector spaces and gave an open question on whether or not there exists a x ¯ ∈ A such that α ≤ f (¯ x, y) ≤ β, ∀y ∈ A, where A is a subset of a Hausdorff topological vector space X, α, β ∈ R+ , α < β and f : A × A → R is a real function. In 2000, Li [11] introduced the concept of an extremal subset to answer this open question when X is a Hausdorff topological vector space and the author presented a version of a variational inequality using the KKM-technique. Chadli et al. [3] and Zhang [13] obtained some results on this open problem using a fixed point theorem for set-valued mappings and the Fan-KKM theorem with the concept of (α, β)-convexity. In 2005, Ansari et al. [2] introduced weighted variational inequalities over product of sets and gave a relationship between weighted variational inequalities and systems of vector variational inequalities. In 2007, Homidan and Ansari [9] studied the existence of solutions for systems of quasi-equilibrium problems with lower and upper bounds (which are more general than the open problem posed in [8]) using maximal element theorems. Fan [6] considered the existence of solutions for weighted quasi-equilibrium problems with lower and upper bounds using maximal element theorems, a fixed point theorem for set-valued mappings and the Fan-KKM theorem. Finally, Mitrovi´c and Merkle [14] established the existence of a solution to the generalized vector equilibrium problem with $ This research is funded by Ho Chi Minh City University of Technology-VUN-HCM under grant number C201920-02. ∗ Corresponding author. Email addresses: [email protected] (Nguyen Van Hung), [email protected] (Donal O’Regan)

Preprint submitted to Topology and its Applications

November 6, 2019

bounds. However, to the best of our knowledge, there are no papers on existence conditions and generic stability for solutions of bilevel equilibrium problems with lower and upper bounds. Motivated and inspired by the works mentioned above, in this paper, we introduce and study a new class of bilevel equilibrium problems with lower and upper bounds in locally convex Hausdorff topological vector spaces. First, we introduce a new concept of (α, β)-quasiconvexity. Then we establish some conditions for the existence of solutions of bilevel equilibrium problems with lower and upper bounds using the Kakutani-Fan-Glicksberg fixed-point theorem. Finally, we study the generic stability of solutions for bilevel equilibrium problems with lower and upper bounds. The rest of the paper is organized as follows: In Section 2, we introduce some bilevel equilibrium problems with lower and upper bounds and recall some definitions for later uses. In Section 3, we introduce a new concept of (α, β)-quasiconvexity and present some existence conditions for solutions to these problems. The generic stability of solutions for bilevel equilibrium problems with lower and upper bounds is obtained in Section 4. 2. Preliminaries Let X be a real locally convex Hausdorff topological vector space, A be a nonempty compact subset of X and K : A ⇒ A be a multifunction and α, β ∈ R+ , α < β and f : A × A → R be a real function. We consider the following quasi-equilibrium problems with lower and upper bounds (in short, (EPLUB)), which is finding a x ¯ ∈ K(¯ x) such that α ≤ f (¯ x, y) ≤ β, ∀y ∈ K(¯ x). We denote the solution set of the problem (EPLUB) by S(f ). Let a, b ∈ R+ , a < b and h : A × A → R be a real function. Also, we consider the following bilevel equilibrium problems with lower and upper bounds (in short, (BEPLUB)), which is finding ax ¯∗ ∈ S(f ) such that a ≤ h(¯ x∗ , y ∗ ) ≤ b, ∀y ∗ ∈ S(f ); We denote by S(h) the solution (BBEPLUB), i.e., x∗ , y ∗ ) ≤ b, ∀y ∗ ∈ S(f ) S(h) := {¯ x∗ ∈ S(f ) | a ≤ h(¯ and α ≤ f (¯ x, y) ≤ β, ∀y ∈ K(¯ x)}. where S(f ) is the solution set of the quasi-equilibrium problems with lower and upper bounds. First, we recall the following well-known definitions and some results. Definition 2.1. (see [1]) Let X, Y be two topological vector spaces, F : X ⇒ Y be a multifunction and let x0 ∈ X be a given point. (1) F is said to be lower semi-continuous (lsc) at x0 ∈ X if F (x0 ) ∩ U = ∅ for some open set U ⊆ Y implies the existence of a neighborhood N of x0 such that F (x) ∩ U = ∅ for all x ∈ N . (2) F is said to be upper semi-continuous (usc) at x0 ∈ X if, for each open set U ⊇ F (x0 ), there is a neighborhood N of x0 such that U ⊇ F (x) for all x ∈ N . (3) F is said to be continuous at x0 ∈ X if it is both lsc and usc at x0 ∈ X (4) F is said to be closed at x0 if, for each of the nets {xα } in X converging to x0 and {yα } in Y converging to y0 such that yα ∈ F (xα ), we have y0 ∈ F (x0 ). If A ⊂ X, then F is said to be usc (lsc, continuous, closed, respectively) on the set A if F is usc (lsc, continuous, closed, respectively) at all x ∈ domF ∩ A. If A ≡ X, then we omit “on X” in the statement. 2

Lemma 2.1. (see [1]) Let X, Y be two topological vector spaces and F : X ⇒ Y be a multifunction. Then we have the following: (1) If F is upper semi-continuous with closed values, then F is closed. (2) If F is closed and F (X) is compact, then F is upper semi-continuous. Lemma 2.2. (see [1]) Let X, Y be two topological vector spaces and F : X ⇒ Y be a multifunction. Then we have the following: (1) F is lower semi-continuous x0 ∈ X if and only if, for each net {xα } ⊆ X which converges to x0 ∈ X and for each y0 ∈ F (x0 ), there exists {yα } in Y such that yα ∈ F (xα ), yα → y0 . (2) If F has compact values, then F is upper semi-continuous x0 ∈ X if and only if, for each net {xα } ⊆ X which converges to x0 ∈ X and for each net {yα } in Y such that yα ∈ F (xα ), there exist y0 ∈ F (x0 ) and a subnet {yβ } of {yα } such that yβ → y0 . Lemma 2.3. (see [5]) Let A be a nonempty convex compact subset of Hausdorff topological vector space X and N be a subset of A × A such that (i) for each at x ∈ A, (x, x) ∈ N ; (ii) for each at y ∈ A, the set {x ∈ A : (x, y) ∈ N } is open on A; (iii) for each at x ∈ A, the set {y ∈ A : (x, y) ∈ N } is convex or empty. Then there exists x0 ∈ A such that (x0 , y) ∈ N for all y ∈ A. Lemma 2.4. (see [10]) Let A be a nonempty compact convex subset of a locally convex Hausdorff vector topological space X. If F : A ⇒ A is upper semi-continuous and, for any x ∈ A, F (x) is nonempty convex closed, then there exists x∗ ∈ A such that x∗ ∈ F (x∗ ). Now we recall some notions which can be found in [1, 4]. Let (X, d) be a metric space. Let K(X), BC(X) and CK(X) denote all nonempty compact subsets of X, all nonempty bounded closed subsets of X, and all nonempty convex compact subsets of X (if X is a linear metric space), respectively. Let B1 , B2 ⊂ X and define H(B1 , B2 ) := max{H ∗ (B1 , B2 ), H ∗ (B2 , B1 )}, where H ∗ (B1 , B2 ) := supb1 ∈B1 d(b1 , B2 ) and d(b1 , B2 ) := inf b2 ∈B2 ||b1 − b2 ||. Now H is a Hausdorff metric in K(X), BC(X) and CK(X), respectively. Lemma 2.5. Let A be a nonempty compact subset of (X, ||.||X ). Let K : A ⇒ A be a continuous set-valued mapping. Assume that for each x ∈ A, K(x) is a nonempty compact subset. Then in A, when x → x∗ , we have H K(x) −→ K(x∗ ), where H is a Hausdorff metric in K(A). Lemma 2.6. Let A be a nonempty compact subset of (X, ||.||X ). Let K : A ⇒ A be a continuous set-valued mapping with nonempty compact valued. Then K is continuous if and if, for any x∗ ∈ H A, x → x∗ implies K(x) −→ K(x∗ ). Lemma 2.7. Let (X, d) be a metric space and H is a Hausdorff metric in X. Then: 3

(i) (BC(X), H) is complete if and if (X, d) is complete. (ii) (K(X), H) is complete if and if (X, d) is complete. (iii) If X is a linear metric space, then (CK(X), H) is complete if and if (X, d) is complete. Lemma 2.8. (see [12]) Let Z be a metric space and let M, Mn (n = 1, 2, ...) be compact sets in Z. Suppose that for any open set O ⊃ M , there exists n0 such that Mn ⊂ O, ∀n ≥ n0 . Then, any sequence {xn } satisfying xn ∈ Mn has a convergent subsequence with limit in M . 3. Existence of solutions In this section, we first introduce a new concept of (α, β)-quasiconvexity for bilevel equilibrium problems with lower and upper bounds. Definition 3.1. Let X be a topological vector space and α, β ∈ R+ , α < β. Suppose f : X × X → R is a real function. Then f is said to be (α, β)-quasiconvex (in the first variable) in a convex set A ⊂ X if, for each y ∈ X, x1 , x2 ∈ A and λ ∈ [0, 1], α ≤ f (x1 , y) ≤ β and α ≤ f (x2 , y) ≤ β, then α ≤ f (λx1 + (1 − λ)x2 , y) ≤ β. Let X be a topological vector space, f : X × X → R a real function and θ ∈ R. We use the following level-set: Lev≥θ f := {(x, y) ∈ X × X : α ≤ θ + f (x, y) ≤ β}. Now, we establish some existence conditions for solution sets of the equilibrium problem with lower and upper bounds (EPLUB). Lemma 3.1. Assume that, for problem (EPLUB), (i) K is continuous on A with nonempty compact convex values; (ii) for all x ∈ A, α ≤ f (x, x) ≤ β; (iii) the set {y ∈ A : f (·, y) < α or f (·, y) > β} is convex on A; (iv) for all y ∈ A, f (·, y) is (α, β)-quasiconvex on A; (v) for all (x, y) ∈ A × A, Lev≥0 f is closed. Then problem (EPLUB) has a solution, i.e., there exists x ¯ ∈ A such that x ¯ ∈ K(¯ x) and α ≤ f (¯ x, y) ≤ β, ∀y ∈ K(¯ x). Moreover, the solution set of problem (EPLUB) is compact. Proof. For all x ∈ A, we define a multifunction Π : A ⇒ A by Π(x) = {t ∈ K(x) : α ≤ f (t, y) ≤ β, ∀y ∈ K(x)}. First, we show that Π(x) is nonempty. Indeed, K(x) is a nonempty compact convex set. Set Φ = {(t, y) ∈ K(x) × K(x) : f (t, y) < α or f (t, y) > β}. Then we have the following: (a) Condition (ii) implies that, for any t ∈ K(x), (t, t) ∈ Φ; (b) Condition (iii) implies that, for any t ∈ K(x), {y ∈ A : (t, y) ∈ Φ} is convex on K(x); 4

(c) Condition (v) implies that, for any t ∈ K(x), {y ∈ K(x) : (t, y) ∈ Φ} is open on K(x). From Lemma 2.3, there exists a ∈ K(x) such that (t, y) ∈ Φ for all y ∈ K(x), i.e., α ≤ f (t, y) ≤ β for all y ∈ K(x). Thus Π(x) is nonempty. Next we show Π(x) is a convex set. Let t1 , t2 ∈ Π(x), λ ∈ [0, 1] and put t = λt1 + (1 − λ)t2 . Since t1 , t2 ∈ K(x) and K(x) is a convex set, we have t ∈ K(x). Thus it follows that, for any t1 , t2 ∈ Π(x), α ≤ f (t1 , y) ≤ β, α ≤ f (t2 , y) ≤ β, ∀y ∈ K(x). From condition (iv), since f (·, y) is (α, β)-quasiconvex, we have α ≤ f (λt1 + (1 − λ)t2 , y) ≤ β, ∀λ ∈ [0, 1], i.e., t ∈ Π(x). Therefore, Π(x) is convex. Now we prove Π is upper semi-continuous on A with nonempty compact values. Indeed, since A is a compact set, by Lemma 2.1(ii), we need only to show that Π is a closed mapping. Consider a net {xα } ⊂ A with xα → x ∈ A and let tα ∈ Π(xα ) be such that tα → t0 . Now, we need to verify that t0 ∈ Π(x). Since tα ∈ K(xα ) and K is upper semi-continuous on A with nonempty compact values, it follows that K is closed and so we have t0 ∈ K(x). Suppose that t0 ∈ Π(x). Then there exists y0 ∈ K(x) such that α ≤ f (t0 , y0 ) ≤ β.

(3.1)

It follows from the lower semi-continuity of K that there is a net {yα } such that yα ∈ K(xα ) and yα → y0 . Since tα ∈ Π(xα ), we have α ≤ f (tα , yα ) ≤ β.

(3.2)

Condition (v) together with (3.2) yields α ≤ f (t0 , y0 ) ≤ β.

(3.3)

This is the contradiction from (3.1) and (3.3). Therefore, we conclude that a0 ∈ Π(x). Hence Π is upper semi-continuous on A with nonempty compact values. Next, we prove the solution set S(f ) = ∅. Indeed, since Π is upper semi-continuous on A with nonempty compact values, from Lemma 2.4, there exists a point x ˆ ∈ A such that x ˆ ∈ Π(ˆ x). This implies that x ˆ ∈ K(ˆ x) such that α ≤ f (ˆ x, y) ≤ β, ∀y ∈ K(ˆ x), i.e., the problem (EPLUB) has a solution. Finally, we prove that S(f ) is compact. In fact, since A is compact and S(f ) ⊂ A, we need only prove that S(f ) is closed. Consider a net {xα } ⊂ S(f ) with xα → x0 . Now, we prove that x0 ∈ S(f ). If x0 ∈ S(f ), there exists y0 ∈ K(x0 ) such that α ≤ f (x0 , y0 ) ≤ β. By the lower semi-continuity of K, it follows that, for any x0 ∈ K(x0 ), there exists xα ∈ K(xα ) such that xα → x0 . Since xα ∈ S(f ), there exists xα ∈ K(xα ) such that α ≤ f (xα , yα ) ≤ β, ∀yα ∈ K(xα ). It follows from the upper semi-continuity and compactness of K on A that there exists y0 ∈ K(x0 ) such that yα → y0 (taking a subnet if necessary). Condition (v) together with (xα , yα ) → (x0 , y0 ) yields α ≤ f (x0 , y0 ) ≤ β, ∀y0 ∈ K(x0 ), which is a contradiction. Thus x0 ∈ S(f ), so S(f ) is a closed set. Therefore, S(f ) is compact subset of A. This completes the proof.  5

Lemma 3.2. Assume that, for problem (EPLUB), (i) K is continuous on A with nonempty compact convex values; (ii) for all x ∈ A, α ≤ f (x, x) ≤ β; (iii) the set {y ∈ A : f (·, y) < α or f (·, y) > β} is convex on A; (iv) for all y ∈ A, f (·, y) is (α, β)-quasiconvex on A; (v) f is continuous on A × A. Then problem (EPLUB) has a solution, i.e., there exists x ¯ ∈ A such that x ¯ ∈ K(¯ x) and α ≤ f (¯ x, y) ≤ β, ∀y ∈ K(¯ x). Moreover, the solution set of problem (EPLUB) is compact. Proof. It is easy to see that if the mapping f is continuous on A × A, then the set {(x, y) ∈ A × A : α ≤ f (x, y) ≤ β} is closed. Therefore the result follows from Lemma 3.1.  Remark 3.1. It is easy to see that, if f is a continuous function, then the set {(x, y) ∈ A × A : α ≤ f (x, y) ≤ β} is closed. The following example shows that the converse is not true. Example 3.1. Let X = Y = R, A = [0, 1], α = 0, β = 11, K(x) = [0, 1] and f : A × A → R defined by  1 if x = y = 0; 2, f (x, y) = otherwise. x2 + 1, Then {(x, y) ∈ A × A : α ≤ f (x, y) ≤ β} = {(x, y) ∈ [0, 1] × [0, 1] : 0 ≤ f (x, y) ≤ 11} = [0, 1] × [0, 1]. All the assumptions in Lemma 3.1 are satisfied and the solution set of problem (EPLUB) is [0, 1]. However, assumption (v) in Lemma 3.2 is not satisfied since f is not continuous at (0, 0). Lemma 3.1 can be applied, but one cannot apply Lemma 3.2. Remark 3.2. In [13], Zhang also studied existence conditions to quasi-equilibrium problem with lower and upper bounds. Example 3.1 shows that our Lemma 3.1 improves Theorem 4.1 in [13]. Moreover, the proof of Theorem 4.1 in [13] is different from the proof in Lemma 3.1. Remark 3.3. Note that our results, Lemma 3.1 and Lemma 3.2 are different from the main results in [11, 3, 9, 2, 6, 14], since we use the Kakutani-Fan-Glicksberg fixed-point theorem with the (α, β)quasiconvexity to establish existence conditions for solution sets of the quasi-equilibrium problem with lower and upper bounds (EPLUB), while the existence conditions studied in [11, 3, 9, 2, 6, 14] used the KKM-technique. Next we investigate sufficient optimality conditions for problem (BEPLUB). Theorem 3.1. Suppose that all the conditions in Lemma 3.1 are satisfied, S(f ) is convex and the following additional conditions hold (i’) for all x∗ ∈ S(f ), a ≤ h(x∗ , x∗ ) ≤ b; 6

(ii’) the set {y ∗ ∈ S(f ) : h(·, y ∗ ) < a or h(·, y ∗ ) > b} is convex on A; (iii’) for all y ∗ ∈ S(f ), h(·, y ∗ ) is (a, b)-quasiconvex on A; (iv’) for all y ∗ ∈ S(f ), Lev≥0 h(·, y ∗ ) is closed on A. ¯∗ ∈ S(f ) and Then problem (BEPLUB) has a solution, i.e., there exists a x ¯∗ ∈ A such that x a ≤ h(¯ x∗ , y ∗ ) ≤ b, ∀y ∗ ∈ S(f ). Moreover, the solution set of problem (BEPLUB) is compact. Proof. For all x∗ ∈ A, we define a multifunction H : A ⇒ A by H(x∗ ) = {z ∈ S(f ) : a ≤ h(z, y ∗ ) ≤ b, ∀y ∗ ∈ S(f )}. First, we prove that H(x∗ ) is nonempty. Recall S(f ) is a nonempty compact convex set. Set Q = {(z, y ∗ ) ∈ S(f ) × S(f ) : h(z, y ∗ ) < a or h(z, y ∗ ) > b}. Then we have the following: (a) Condition (i’) implies that, for any z ∈ S(f ), (z, z) ∈ Q; (b) Condition (ii’) implies that, for any z ∈ S(f ), {y ∗ ∈ S(f ) : (z, y) ∈ Q} is convex on S(f ); (c) Condition (iv’) implies that, for any z ∈ S(f ), {y ∗ ∈ S(f ) : (z, y ∗ ) ∈ Q} is open on S(f ). From Lemma 2.3, there exists a z ∈ S(f ) such that (z, y ∗ ) ∈ Q for all y ∗ ∈ S(f ), i.e., a ≤ h(z, y ∗ ) ≤ b, for all y ∗ ∈ S(f )}. Thus it follows that H(x∗ ) is nonempty. Next we show that H(x∗ ) is a convex set. Let z1 , z2 ∈ H(x∗ ) and λ ∈ [0, 1] and put z = λz1 + (1 − λ)z2 . Since z1 , z2 ∈ S(f ) and S(f ) is a convex set, we have z ∈ S(f ). Thus it follows that, for all z1 , z2 ∈ H(x∗ ), a ≤ h(z1 , y ∗ ) ≤ b, ∀y ∗ ∈ H(x∗ ). From condition (iii’), since h(·, y ∗ ) is (a, b)-quasiconvex, we have a ≤ h(λz1 + (1 − λ)z2 , y ∗ ) ≤ b, ∀λ ∈ [0, 1], i.e., z ∈ H(x∗ ). Thus, H(x∗ ) is convex. Now we prove that H is upper semi-continuous on A with nonempty compact values. Indeed, since A is a compact set, from Lemma 2.1 (ii), we need only to show that H is a closed mapping. Consider a net {x∗α } ⊂ A with x∗α → x∗ ∈ A and let zα ∈ H(x∗α ) be such that zα → z0 . Now, we need to show that z0 ∈ H(x∗ ). Since zα ∈ S(f ) and S(f ) is compact, we have z0 ∈ S(f ). Suppose that z0 ∈ H(x∗ ). Then there exists a y ∗ ∈ S(f ) such that a ≤ h(z0 , y ∗ ) ≤ b.

(3.4)

On the other hand, since zα ∈ H(x∗α ), we have a ≤ h(zα , y ∗ ) ≤ b, ∀y ∗ ∈ S(f ).

(3.5)

Condition (iii’) together with (3.5) yields a ≤ h(z0 , y ∗ ) ≤ b,

(3.6)

which is a contradiction from (3.4) and (3.6). Thus z0 ∈ H(x∗ ). Hence H is upper semi-continuous on A with nonempty compact values.

7

Next we prove that the solution set S(h) is nonempty. In fact, since H is upper semi-continuous on A with nonempty compact values, from Lemma 2.4, there exists a point x ˆ∗ ∈ A such that ∗ ∗ ∗ x ). Hence there exists x ˆ ∈ S(f ) such that x ˆ ∈ H(ˆ a ≤ h(ˆ x∗ , y ∗ ) ≤ b, ∀y ∗ ∈ S(f ), i.e., problem (BEPLUB) has a solution. Finally, we prove that S(h) is compact. Consider a net {x∗α } ⊂ S(h) with x∗α → x∗0 . Now, we prove that x∗0 ∈ S(h). Indeed, from the closedness of S(f ), there exists x∗α ∈ S(f ) such that x∗α → x∗0 . Since x∗α ∈ S(f ), there exists xα ∈ S(f ) such that a ≤ h(x∗α , y ∗ ) ≤ b, ∀y ∗ ∈ S(f ). Condition (iv’) together with x∗α → x∗0 yields a ≤ h(x∗0 , y ∗ ) ≤ b, ∀y ∗ ∈ S(f ), which means that x∗0 ∈ S(h). Thus S(h) is a closed set. Since S(h) ⊂ S(f ) and S(f ) is compact, S(h) is compact. This completes the proof.  Theorem 3.2. Suppose that all the conditions in Lemma 3.2 are satisfied, S(f ) is convex and the following additional conditions hold (i’) for all x∗ ∈ S(f ), a ≤ h(x∗ , x∗ ) ≤ b; (ii’) the set {y ∗ ∈ S(f ) : h(·, y ∗ ) < a or h(·, y ∗ ) > b} is convex on A; (iii’) for all y ∗ ∈ S(f ), h(·, y ∗ ) is (a, b)-quasiconvex on A; (iv’) for all y ∗ ∈ S(f ), h(·, y ∗ ) is continuous on A. ¯∗ ∈ S(f ) and Then, (BEPLUB) has a solution, i.e., there exists a x ¯∗ ∈ A such that x a ≤ h(¯ x∗ , y ∗ ) ≤ b, ∀y ∗ ∈ S(f ). Moreover, the solution set of (BEPLUB) is compact.

4. Generic Stability Throughout this section, let X be a Banach space, A ⊂ X be a nonempty subset and α, β, a, b ∈ R+ , α < β, a < b. Now, we let Ω := {(K, f, h) : A ⇒ A be continuous in A with nonempty compact convex values, f : A × A → R and h : S(f ) × S(f ) → R are continuous functions and for all y ∈ A, f (·, y) is (α, β)-quasiconvex on A, for all y ∗ ∈ S(f ), h(·, y ∗ ) is (a, b)-quasiconvex on S(f )}. Let E1 , E2 be compact sets in a normed space. Recall that the Hausdorff metric is defined by H(E1 , E2 ) := max{H ∗ (E1 , E2 ), H ∗ (E2 , E1 )}, where H ∗ (E1 , E2 ) := supe1 ∈E1 d(e1 , E2 ) and d(e1 , E2 ) := inf e2 ∈E2 ||e1 − e2 ||. For any u1 = (K1 , f1 , h1 ), u2 = (K2 , f2 , h2 ) ∈ Ω, define ξ(u1 , u2 ) := sup H(K1 (x), K2 (x)) + sup  f1 (x) − f2 (x)  + sup x∈A

x∈A

x∗ ∈S(f )

where H is a Hausdorff metric. Now (Ω, ξ) is a metric space. 8

 h1 (x∗ ) − h2 (x∗ ) ,

Theorem 4.1. (Ω, ξ) is a complete metric space. Proof. Let {un } be any Cauchy sequence in Ω, where un = (Kn , fn , hn ), i = 1, 2, ... Then, for any ε > 0, there exists N > 0 such that ξ(un , um ) <

ε , ∀n, m ≥ N. 3

(4.1)

It follows that, for any x ∈ A and x∗ ∈ S(f ), H(Kn (x), Km (x)) <

ε , 3

(4.2)

and  fn (x) − fm (x) <

ε , 3

 hn (x∗ ) − hm (x∗ ) <

ε . 3

(4.3)

Then, for any fixed point x ∈ A, {Kn (x)} is a Cauchy sequence in CK(A), and {fn (x)}, {hn (x∗ )} are two Cauchy sequences in R. From Lemma 2.7 and the assumptions (CK(A), H), (R,  . ) are complete spaces, it follows that there exist K(x) ∈ CK(A), f (x) ∈ R and h(x∗ ) ∈ R such that H

Kn (x) −→ K(x).

(4.4)

and .

.

fn (x) −−→ f (x), hn (x∗ ) −−→ h(x∗ ).

(4.5)

Since H(., .) and  .  are continuous, from (4.1), (4.2) and (4.3), for any fixed n ≥ N and any x ∈ A, x∗ ∈ S(f ), and let m → +∞, we get H(Kn (x), K(x)) <

ε . 3

(4.6)

and  fn (x) − f (x) <

ε , 3

 hn (x∗ ) − h(x∗ ) <

ε . 3

(4.7)

Now, we will prove that K is continuous. From Lemma 2.6, we need to prove that, for any fixed point x0 ∈ A and any ε > 0, there exists a neighborhood N (x0 ) of x0 in A such that H(K(x), K(x0 )) ≤ ε, ∀x ∈ N (x0 ) ∩ A. Since H(K(x), K(x0 )) ≤ H(K(x), Kn (x)) + H(Kn (x), Kn (x0 )) + H(Kn (x0 ), K(x0 )), from (4.6), there exists N > 0 such that, for any n > N , H(K(x), Kn (x)) <

ε , ∀x ∈ A. 3

Take a fixed n > N , by the continuity of Kn and Lemma 2.6, there exists a neighborhood N (x0 ) of x0 in A such that H(Kn (x), Kn (x0 )) <

ε , ∀x ∈ N (x0 ) ∩ A. 3

Thus, we have H(K(x), K(x0 )) ≤ H(K(x), Kn (x)) + H(Kn (x), Kn (x0 )) + H(Kn (x0 )), K(x0 )) ≤ ε, ∀x ∈ N (x0 ) ∩ A. 9

Hence, K is continuous in A. Similarly, we can show that f, h are continuous. Now, we show that, for all y ∈ A, f (·, y) is (α, β)-quasiconvex on A. Indeed, for any n and for every x1 , x2 ∈ A and ∀λ ∈ [0, 1], α ≤ fn (x1 , y) ≤ β and α ≤ fn (x2 , y) ≤ β. By the (α, β)quasiconvexity, we have α ≤ fn (λx1 + (1 − λ)x2 , y) ≤ β. Since

.

.

fn (x1 , y) −−→ f (x1 , y), fn (x2 , y) −−→ f (x2 , y), and

.

fn (λx1 + (1 − λ)x2 , y) −−→ f (λx1 + (1 − λ)x2 , y), it follows that α ≤ f (λx1 + (1 − λ)x2 , y) ≤ β. Thus, f (·, y) is (α, β)-quasiconvex on A. Similarly, we can prove that h(·, y ∗ ) is (a, b)-quasiconvex on S(f ). From (4.6), (4.7) for any fixed n ≥ N and any x ∈ A, x∗ ∈ S(f ), we have H(Kn (x), K(x)) <

ε . 3

and  fn (x) − f (x) <

ε , 3

 hn (x∗ ) − h(x∗ ) <

ε . 3

Hence, sup H(Kn (x), K(x)) <

x∈A

ε . 3

and sup  fn (x) − f (x) <

x∈A

ε , 3

sup x∗ ∈S(f )

 hn (x∗ ) − h(x∗ ) <

ε . 3

ξ

→ u. Thus, (Ω, ξ) is a For u = (K, f, h), we see that u ∈ Ω and ξ(un , u) < ε, ∀n ≥ N , i.e., un − complete metric space.  Assume that all the conditions of Lemma 3.2 and Theorem 3.2 are satisfied. Then, for each u = (K, f, h) ∈ Ω, (BEPLUB) has solution. For u ∈ Ω, let Ψ(u) := {x¯∗ ∈ A such that x¯∗ ∈ S(f ) satisfies a ≤ h(x¯∗ , y ∗ ) ≤ b, ∀y ∗ ∈ S(f ) and α ≤ f (¯ x, y) ≤ β, ∀y ∈ K(¯ x)}, where S(f ) is the solution set of the quasi-equilibrium problems with lower and upper bounds, which is finding a x ¯ ∈ K(¯ x) such that α ≤ f (¯ x, y) ≤ β, ∀y ∈ K(¯ x). Then Ψ(u) = ∅ and so Ψ(u) defines a set mapping from Ω into S(f ). Theorem 4.2. Ψ : Ω ⇒ S(f ) is upper semicontinuous with compact values. Proof. Since A and S(f ) are compact, we need only show that Ψ is a closed mapping. Let a sequence {(un , xn , x∗n )} ⊂ Graph(Ψ) be given such that (un , xn , x∗n ) → (u, x0 , x∗0 ) ∈ Ω × A × S(f ). We now show that {(u, x0 , x∗0 )} ⊂ Graph(Ψ), where un = (Kn , fn , hn ) and u = (K, f, h).

10

Note, for any n, since xn ∈ Ψ(un ), we have that xn ∈ Kn (xn ) and for all yn ∈ Kn (xn ), α ≤ fn (xn , yn ) ≤ β.

(4.8)

For any open set O ⊃ K(x0 ), since K(x0 ) is a compact set, there exists ε > 0 such that {y ∈ A : d(y, K(x0 )) < ε} ⊂ O, where d(y, K(x0 )) = inf y ∈K(x0 ) ||y − y  ||. Since ξ(un , u) → 0, xn → x0 and K is upper semicontinuous at x0 , ∃n0 such that ε sup H(Kn (x), K(x)) < , 2 x∈A K(xn ) ⊂ {y ∈ A : d(y, K(x0 )) <

ε }, ∀n ≥ n0 . 2

(4.9)

(4.10)

(4.11)

From (4.9), (4.10) and (4.11), we have ε } ⊂ {y ∈ A : d(y, K(x0 )) < ε} ⊂ O, ∀n ≥ n0 . (4.12) 2 Since K(x0 ) ⊂ O and yn ∈ Kn (xn ), we can apply Lemma 2.8 and there exists a subsequence {ynk } of {yn } such that {ynk } converges to y0 , and it follows that y0 ∈ K(x0 ). Next, we show that α ≤ f (x0 , y0 ) ≤ β. Since xn → x0 and K is upper semicontinuous at x0 , K(x0 ) is closed, and there exists y0 ∈ K(x0 ) such that yn → y0 (taking a subsequence if necessary). Since ξ((Kn , K)) → 0, we can choose a subsequence {Knk } of {Kn } such that K(xn ) ⊂ {y ∈ A : d(y, K(x0 )) <

sup H(Knk (x), K(x)) <

x∈X

1 . k

(4.13)

Thus, 1 . k ∈ Knk (xnk ), k = 1, 2, ... such that H(Knk (xnk ), K(xnk )) <

This implies that there exists yn k

||yn k − ynk || <

1 . k

Note 1 + ||ynk − y0 || → 0, k ∈ Knk (xnk ), applying (4.9), we have

||yn k − y0 || ≤ ||yn k − ynk || + ||ynk − y0 || < and so we have yn k → y0 . Since xnk ∈ Knk (xnk ) and yn k

α ≤ fnk (xnk , yn k ) ≤ β. From assumption (v) in Lemma 3.2, we have α ≤ f (x0 , y0 ) ≤ β.

(4.14)

Next, for any n, since x∗n ∈ Ψ(un ), we have that x∗n ∈ S(f ) and for all y ∗ ∈ S(f ), a ≤ hn (x∗n , y ∗ ) ≤ b.

(4.15)

Now x∗n → x∗0 and assumption (iv’) in Theorem 3.2 yield a ≤ h(x∗0 , y ∗ ) ≤ b.

(4.16)

From (4.14) and (4.16), it follows that (u, x0 , x∗0 ) ∈ Graph(Ψ). Therefore, Ψ is closed. Since S(f ) is a compact set and Ψ(K) ⊂ S(f ), then Ψ is upper semicontinuous with compact values.  The following well-known result can be found in [7]: 11

Lemma 4.1. Let X be a complete metric space, Y a space possessing the Baire property (e.g., a Banach space enjoys this property) and F : X ⇒ Y be upper semicontinuous with compact values. Then the set of points, where F is lower semicontinuous, is a dense residual set of X. Theorem 4.3. The set Φ of essential points of Ψ is a dense residual set of Ω. Proof. The theorem follows from Lemma 4.1, Theorem 4.1 and Theorem 4.2.



5. Conclusions In this work, we consider a new class of bilevel equilibrium problems with lower and upper bounds in locally convex Hausdorff topological vector spaces. Using the Kakutani-Fan-Glicksberg fixed-point theorem we present some existence results of solutions for these problems. The generic stability of solutions for these problems is also obtained. References [1] J.P. Aubin and I. Ekeland, Applied Nonlinear Analysis, John Wiley and Sons, New York, 1984. [2] Q.H. Ansari, Z. Khan and A.H. Siddiqi, Weighted variational inequalities, J. Optim. Theory Appl. 127(2005), 263–283. [3] O. Chadli, Y. Chiang and J.C. Yao, Equilibrium problems with lower and upper bounds, Appl. Math. Lett. 15(2002), 327–331. [4] J.C. Chen and X.H. Gong, The stability of set of solutions for symmetric vector quasiequilibrium problems, J. Optim. Theory Appl. 136(2008), 359–374. [5] K. Fan, A generalization of Tychonoff’s fixed point theorem, Math. Ann. 142(1961), 305–310. [6] L. Fan, Weighted quasi-equilibrium problems with lower and upper bounds, Nonlinear Anal. 70(2009), 2280-2287. [7] M.K. Fort, Points of continuity of semicontinuous functions, Publ. Math. Debrecon 2(1951), 100–102. [8] G. Isac, V.M. Sehgal and S.P. Singh, An alternative version of a variational inequality, Indian J. Math. 41(1999), 25–31. [9] S. Al-Homidan and Q.H. Ansari, Systems of quasi-equilibrium problems with lower and upper bounds, Appl. Math. Lett. 20(2007), 323–328. [10] R.B. Holmes, Geometric Functional Analysis and its Application. Springer-Verlag, New York, 1975 [11] J. Li, A lower and upper bounds version of avariational inequality, Appl. Math. Lett. 13(2000), 47–51. [12] J. Yu, Essential weak efficient solution in multiobjective optimization problems. J. Math. Anal. Appl. 166(1992), 230-235. [13] C. Zhang, A class of equilibrium problems with lower and upper bounds, Nonlinear Anal. 15(2005), 2377–2385. [14] Z.D. Mitrovi´c and M. Merkle, On a generalized vector equilibrium problem with bounds, Appl. Math. Lett. 23(2010), 783-787. 12