Đề tài Sensitivity analysis for equilibrium problems and related problems

Chapter 2 Semicontinuity of the approximate solution sets of multivalued quasiequilibrium problems 41 SEMICONTINUITY OF THE APPROXIMATE SOLUTION SETS OF MULTIVALUED QUASIEQUILIBRIUM PROBLEMS Lam Quoc Anh 2 Department of Mathematics, Teacher College, Cantho University, Cantho, Vietnam Phan Quoc Khanh 2 Department of Mathematics, International University of Hochiminh City, Linh Trung, Thu Duc, Hochiminh City, Vietnam 2 We consider two kinds of approximate solutions and approximate solution sets to multivalued quasiequilibrium problems. Sufficient conditions for the lower semicontinuity, Hausdorff lower semicontinuity, upper semicontinuity, Hausdorff upper semicontinuity and closedness of these approximate solution sets are established. Applications in approximate quasivariational inequal- ities, approximate fixed points and approximate quasioptimization problems are provided. Keywords Lower and upper semicontinuity; Hausdorff lower and upper semicontinuity; Closedness of multifunctions; ε-solutions; Quasiequilibrium problems; Quasivariational inequali- ties; ε-fixed points; ε-quasioptimization problems. 1. INTRODUCTION AND PRELIMINARIES Among various meanings of stability and sensitivity of the solutions of a problem the semicontinuity has been increasingly interested recently in the literature. Upper semiconti- nuity is investigated in [12, 15−17] for variational inequalities and in [3, 5, 7−10, 14, 18] for equilibrium problems. Lower semicontinuity is studied in [19] for minimization problems, in [12, 16, 17] for variational inequalities and in [3, 5, 7, 8, 14] for equilibrium problems. Beside semicontinuity we observe [1, 2, 4, 6, 8−10] which deal with stability of equilibrium problems, where Ho¨lder continuity is investigated. On the other hand, in many practical problems, exact solutions do not exist since the data of the problems are not sufficiently “regular”. Moreover, the data of the problems have often been obtained approximately by measure devices or by statistical results. Mathematical models of practical problems describe real situations also approximately and hence the existence of exact solutions of mathematical models may become unimportant. That is why approximate solutions are of real interest. Observing that multivalued quasiequilibrium problems are rather general problems, which include quasivariational inequalities, complementarity problems, fixed- point and coincidence-point problems, optimization, Nash equilibrium problems, etc as special cases, in this note we consider the semicontinuity properties of the approximate solutions of quasiequilibrium problems. In particular, many results of [16] are improved and extended to more general settings. The problems under our consideration are as follows. Throughout the paper, unless speci- fied otherwise, let X, M , N , Λ be Hausdorff topological spaces and let Y be an invariant metric Address corresponding to Phan Quoc Khanh, Department of Mathematics, International University of Hochiminh City, Khu pho 6, Linh Trung, Thu Duc, Hochiminh City, Vietnam; E-mail: pqkhanh@hcmiu.edu.vn 1 linear space, i.e. d(y, z) = d(y + u, z + u),∀y, z, u ∈ Y . Let K : X ×Λ → 2X , G : X ×N → 2X and F : X × X ×M → 2Y be multifunctions. Let C ⊆ Y be a closed subset with nonempty interior. As usual, a problem involving single-valued mappings is split into many generalized problems, while the mappings become multivalued. For the sake of simplicity we adopt the fol- lowing notations. The Roman letters w, m, s are used for weak, middle, and strong problems. For subsets A and B under consideration, we adopt the notations (u, v)wA×B means ∀u ∈ A,∃v ∈ B, (u, v)mA×B means ∃v ∈ B,∀u ∈ A, (u, v) sA×B means ∀u ∈ A,∀v ∈ B, α1(A,B) means A ∩B 6= ∅, α2(A,B) means A ⊆ B, (u, v) w¯A×B means ∃u ∈ A,∀v ∈ B and similarly for m¯, s¯, α¯1 and α¯2. Let r ∈ {w,m, s}, α ∈ {α1, α2}. Our general parametric multivalued vector quasiequilibrium problem is the following, for (λ, µ, η) ∈ Λ×M ×N , (Prα) find x¯ ∈ clK(x¯, λ), such that (y, x¯∗) rK(x¯, λ)×G(x¯, η), α ( F (x¯∗, y, µ), Y \ −intC). Note that (Prα) represents six problems. This statement is not explicit but helps to shorten the presentation of results. Let Srα(λ, µ, η) be the solution set of (Prα) corresponding to λ, µ and η. Recall first some notions. Let X and Y be as above and G : X → 2Y be a multifunction. G is said to be lower semicontinuous (lsc) at x0 if G(x0) ∩ U 6= ∅ for some open set U ⊆ Y implies the existence of a neighborhood N of x0 such that, for all x ∈ N,G(x) ∩ U 6= ∅. An equivalent formulation is that: G is lsc at x0 if ∀xα → x0, ∀y ∈ G(x0),∃yα ∈ G(xα), yα → y. G is called Hausdorff lower semicontinuous (Hlsc) at x0 if for each neighborhood B of the origin in Y , there is a neighborhood N of x0 such that, for all x ∈ N , G(x0) ⊆ G(x) + B. G is called upper semicontinuous (usc) at x0 if for each open set U ⊇ G(x0), there is a neighborhood N of x0 such that U ⊇ G(N). G is termed Hausdorff upper semicontinuous (Husc) at x0 if for each neighborhood B of the origin in Y , there is a neighborhood N of x0 such that G(N) ⊆ G(x0) + B. G is said to be continuous at x0 if it is both lsc and usc at x0 and to be H-continuous at x0 if it is both Hlsc and H-usc at x0. G is called closed at x0 if for each net (xα, yα) ∈ graphG := {(x, y) | y ∈ G(x)}, (xα, yα) → (x0, y0), y0 must belong to G(x0). The closedness is closely related to the upper (and Hausdorff upper) semicontinuity (see e.g. [3], Preposition 3.1). We say that G satisfies a certain property in a subset A ⊆ X if G satisfies it at every points of A. If A = domG := {x ∈ X : G(x) 6= ∅} we omit “in domG” in the statement. We propose the following two definitions of ε-solutions. Let us use the notations comp(−intC)ε1 = { y ∈ Y : d(y, Y \ −intC) ≤ ε}, comp(−intC)ε2 = (Y \ −intC) +BεY , where BεY = { y ∈ Y | d(0, y) ≤ ε} and the notation “comp(.)” is related to the word “comple- ment”. 2 Definition 1.1 (a) x¯ ∈ X is called an ε−solution of type 1 of problem (Prα) if x¯ ∈ clK(x¯, λ) and (y, x¯∗)rK(x¯, λ)×G(x¯, η), α ( F (x¯∗, y, µ), comp(−intC)ε1 ) . (1.1) (b) If (1.1) is replaced by α ( F (x¯∗, y, µ), comp(−intC)ε2 ) , then x¯ is said to be an ε−solution of type 2 of problem (Prα). Remark 1.1 (a) If x¯ is an ε−solution of type 2 then x¯ is an ε−solution of type 1. Indeed, it suffices to show that comp(−intC)ε2 ⊆ comp(−intC)ε1. We check a more general fact that A + BεY ⊆ Aε := {y ∈ Y | d(y,A) ≤ ε}. Let y ∈ A + BεY , i.e. y = a + z for some a ∈ A and z ∈ BεY . Then d(y, a) = d(y − a, 0) = d(z, 0) ≤ ε, i.e. y ∈ Aε. (b) If Y is finite dimensional, then the two types of ε−solutions coincide. Indeed, it suffices to check that Aε ⊆ A+BεY , while A is closed. Let y ∈ Aε be arbitrary. Then, for each n, there is an ∈ A with d(y, an) ≤ ε + 1n . We have some yn ∈ BεY with d(y − an − yn, 0) = d(y − an, yn) ≤ 2n . By the compactness of BεY we can assume that yn → y0 ∈ BεY . Let a = y− y0, un = y−an− y0 = a−an and vn = y−an− yn. Then, 0 ≤ d(an, a) = d(un, 0) ≤ d(un − vn, 0) + d(vn, 0) ≤ d(yn − y0, 0) + 2 n . Therefore, an → a and hence a ∈ A. Thus y = a+ y0 ∈ A+BεY .  The following two examples show that if Y is infinite dimensional, then in general an ε−solution of type 1 is not guaranteed to be an ε−solution type 2. Example 1.1. Let Y = l∞, A = {xk}, where xk1 = 1 + 1k , xkk+1 = 1 and xkj = 0, ∀j 6= 1 and j 6= k + 1. Then ‖xk − xl‖ = 1 if k 6= l and hence A is closed. Taking yk ∈ B1Y with yk1 = −1, ykk+1 = −1 and ykj = 0 if j 6= 1 and j 6= k+1, we have xk + yk = ( 1k , 0, 0, ...) ∈ A+B1Y and xk + yk → 0 /∈ A+B1Y . Thus, A+B1Y is not closed and hence A+B1Y is included properly in A1. The next example is more complicated but provides a case where A has the form Y \−intC with C 6= Y as in Definition 1.1. Example 1.2. Let Y = l∞. Let U1 = { u ∈ l∞ ∣∣∣ 1 2 ≤ u1 ≤ 1, 2 ≤ u2 ≤ 3 and 0 ≤ uk ≤ 1 for k ≥ 3 } , U2 = { u ∈ l∞ ∣∣∣ 1 4 ≤ u1 ≤ 12 , 0 ≤ u2 ≤ 1, 2 ≤ u3 ≤ 3 and 0 ≤ uk ≤ 1 for k ≥ 4 } , · · · Un = { u ∈ l∞ ∣∣∣ 1 2n ≤ u1 ≤ 12n−1 , 2 ≤ un+1 ≤ 3 and 0 ≤ uk ≤ 1 for k 6= 1, k 6= n+ 1 } . 3 Then clearly Un are closed for all n and d(Un, Um) = 1 if n 6= m. Let A = ⋃∞ n=1 Un. We claim that A is closed. Indeed, assume that ai ∈ A, ai → a0. Since d(Un, Um) = 1 for n 6= m, there exist i0 and n0 such that ai ∈ Un0 for all i ≥ i0. Hence a0 ∈ Un0 ⊆ A. To see that A + B1Y is not closed take sequences ak ∈ A and bk ∈ B1Y with components ak1 = 1 2k , akk+1 = 2 and a k j = 1,∀j 6= 1,∀j 6= k + 1, bk1 = −1, bkk+1 = −1 and bkj = 0,∀j 6= 1,∀j 6= k + 1. Then ak + bk = (−1 + 1 2k , 1, 1, ...) → (−1, 1, 1, ...) /∈ A + B1Y . Thus A + B1Y is not closed and contained properly in A1. Now taking −C = cl(l∞ \A) we get C 6= l∞ and A = l∞ \ −intC as wanted. The following example illustrates how the ε-solution may depend on ε and in particular may be empty if ε is too small. Example 1.3. For ε > 0, n ∈ N . Let X = Y = R,Λ ≡ M ≡ N = [0, 1], C = R+ and for λ = 1n , K(x, λ) = [εn, εn+1], G(x, λ) = {x}, F (x, y, λ) = y−x− 1n . In this case the six cases of (Prα) coincide since F and G are single-valued and the two types of the approximate solution sets are equivalent. It is easy to see that for each λ = 1n , S ε(λ) = [εn, εn+ ε− 1n ], if ε ≥ 1n and Sε(λ) = ∅, if ε < 1n . Definition 1.2 (a) The set of ε−solutions of type 1 (of type 2) of problem (Prα) at (λ, µ, η) is denoted by Sε1rα(λ, µ, η) ( Sε2rα(λ, µ, η), respectively ) . (b) Another kind of ε−solution sets is defined by S˜εkrα(λ, µ, η) = { Srα(λ0, µ0, η0) if (λ, µ, η) = (λ0, µ0, η0), Sεkrα(λ, µ, η) otherwise, where (λ0, µ0, η0) is the point under consideration and k = 1, 2. We propose the following weak semicontinuity. Definition 1.3. Let X be a topological space and Y be a topological vector space, and C ⊆ Y with intC 6= ∅ and C 6= Y . (a) A multifunction H : X −→ 2Y is said to be C−upper semicontinuous (C−usc) at x0 if, for any xα → x0, H(x0) ⊆ intC ⇒ ∃α¯,H(xα¯) ⊆ intC. (b) H is called C−lower semicontinuous (C−lsc) at x0 if, for all xα → x0, H(x0)∩intC 6= ∅ ⇒ ∃α¯,H(xα¯) ∩ intC 6= ∅. H is said to be C−continuous at x0, if H is both C−usc and C−lsc at x0. (c) H is termed C− Hausdorff upper semicontinuous (C−Husc) at x0 if, for any xα → x0 and B (open neighborhood of 0 in Y ), H(x0) +B ⊆ intC ⇒ ∃α¯,H(xα¯) ⊆ intC. Remark 1.2 (a) C−upper semicontinuity property is strictly stronger than C−Hausdorff upper semi- continuity property; (b) H is C−usc, C−lsc, C−Husc at x0 if and only if H is intC−usc, intC−lsc, intC−Husc, respectively; (c) H is usc at x0, iff H is C−usc at x0 for all subsets C of Y ; (d) H is lsc at x0, iff H is C−lsc at x0 for all subsets C of Y . 4 Since the solution existence of quasiequilibrium problems has been intensively investigated, we do not study this issue, and instead always assume the existence. The organization of the paper is as follows. In Section 2 we give sufficient conditions for the approximate solution sets to be lsc and Hlsc at the point in question. In Section 3 we investigate sufficient conditions for the approximate solution sets to be usc, Husc and closed. Section 4 is devoted to applications in approximate quasivariational inequalities, approximate fixed points and approximate quasioptimization problems. 2. LOWER SEMICONTINUITY OF THE ε−SOLUTION SETS Considering the lower semicontinuity of approximate solution sets for our quasiequilibrium problem we will see below that the sufficient conditions for this lower semicontinuity are the same for the two types of ε−solutions stated in Definition 1.1. The reason is that an ε−solution of type 1, which is not an ε−solution of type 2, much lie on the boundary of comp(−int)Cε1 , since this set may have more points than comp(−int)Cε2 only in the boundary (see Remark 1.1 and Example 1.2). This difference does not affect the lower semicontinuity. But it affects the upper semicontinuity as the next section makes it clear. Moreover, we will see that under usual assumptions lower semicontinuity holds only for the ε−solution sets of the second kind S˜ε1rα and S˜ε2rα (see Definition 1.2) while upper semicontinuity occurs only for the ε−solution sets of the first kind Sε1rα and S ε2 rα. In the sequel let, for λ ∈ Λ, E(λ) = { x ∈ X | x ∈ clK(x, λ)}. We always assume that Srα(λ, µ, η) 6= ∅ for all (λ, µ, η) in a neighborhood of the considered point (λ0, µ0, η0). In the sequel we will write Srα for the multifunction Srα(., ., .) and similarly for other multifunctions. Theorem 2.1. Assume that K is usc and has compact values in X × {λ0}, E is lsc at λ0 and F is comp(−intC)εk− lsc in clK(X,Λ)× clK(X,Λ)× {µ0}, k = 1, 2. (i) For r ∈ {w,m}, if G is lsc in clK(X,Λ)×{η0}, then S˜εkrα1 is lsc at (λ0, µ0, η0) for ε > 0. (ii) If G is usc and compact-valued in clK(X,Λ) × {η0}, then S˜εksα1 is lsc at (λ0, µ0, η0) for ε > 0. Proof. Since the six assertions for the two types of ε - solutions can be proved by similar techniques we present a proof only for S˜ε1wα1 of problem (Pwα1). Let ε > 0 be fixed. Suppose that S˜ε1wα1 is not lsc at (λ0, µ0, η0), i.e., ∃x0 ∈ S˜ε1wα1(λ0, µ0, η0), ∃(λγ , µγ , ηγ) → (λ0, µ0, η0), ∀xγ ∈ S˜ε1wα1(λγ , µγ , ηγ), xγ 6→ x0. Since x0 ∈ S˜ε1wα1(λ0, µ0, η0) = Swα1(λ0, µ0, η0), ∀y0 ∈ K(x0, λ0), ∃x∗0 ∈ G(x0, η0), F (x∗0, y0, µ0)∩ (Y \− intC) 6= ∅. (2.1) By the lower semicontinuity of E at λ0, there is a net x¯γ ∈ E(λγ), x¯γ → x0. By the contradiction assumption, there must be a subnet x¯β such that, ∀β, x¯β /∈ S˜ε1wα1(λβ , µβ , ηβ), i.e., ∃yβ ∈ K(x¯β , λβ), ∀x¯∗β ∈ G(x¯β , ηβ), F (x¯∗β , yβ , µβ)∩comp(−intC)ε1 = ∅. (2.2) Since K is usc and K(x0, λ0) is compact, one can assume that yβ → y0, for some y0 ∈ K(x0, λ0). By the lower semicontinuity of G at (x0, η0), there exists x¯∗β ∈ G(x¯β , ηβ), x¯∗β → x∗0. In view of the comp(−intC)ε1− lower semicontinuity of F at (x∗0, y0, µ0) and of the inclusion Y \ intC ⊆ int(comp(−intC)ε1), we see a contradiction between (2.1) and (2.2).  5 The following example shows that Theorem 2.1 is no longer true if we replace the ε−solution sets of the second kind by that of the first kind. Example 2.1. Let X = Y = R,Λ ≡ M ≡ N = [0, 1], C = R+, K(λ) = [0, 1], G(x, λ) = {x}, F (x, y, λ) = {y − x − ε − λ, x − y} and λ0 = 0. Since G is single-valued, (Pwα1), (Pmα1) and (Psα1) coincide. From now on if Y is finite dimensional (then the two types of ε−solutions coincide), instead of writing ε1 or ε2 in the index of ε−solution sets, we write simply ε, e.g. Sεrα, S˜ ε rα. It is easy to see that the assumptions of Theorem 2.1 are fulfilled and hence S˜εrα1 is lsc at 0 for r ∈ {w,m, s}. In fact Srα1(0) = {1} and Sεrα1(λ) = [1 − ε, 1], ∀λ ∈ (0, 1], but Sεrα1(0) = {0, 1}. So Sεwα1 is not lsc at 0, for r ∈ {w,m, s}. Theorem 2.2. Assume that K is usc and has compact values in clK(X,Λ)×{λ0}, E is lsc at λ0 and F is comp(−intC)εk− usc in clK(X,Λ)× clK(X, Λ)× {µ0}, k = 1, 2. (i) For r ∈ {w,m}, if G is lsc in clK(X,Λ)×{η0}, then S˜εkrα2 is lsc at (λ0, µ0, η0) for ε > 0. (ii) If G is usc and has compact values in clK(X,Λ) × {η0}, then S˜εksα2 is lsc at (λ0, µ0, η0) for ε > 0. Proof. As an example we demonstrate only for problem (Psα2). Let ε > 0 be fixed. Suppose to the contrary that S˜ε2sα2 is not lsc at (λ0, µ0, η0), i.e., ∃x0 ∈ S˜ε2sα2(λ0, µ0, η0), ∃λγ → λ0,∃µγ → µ0,∃ηγ → η0, ∀xγ ∈ S˜ε2sα2(λγ , µγ , ηγ), xγ 6→ x0. Since x0 ∈ S˜ε2sα2(λ0, µ0, η0) = Ssα2(λ0, µ0, η0), ∀y0 ∈ K(x0, λ0), ∀x∗0 ∈ G(x0, η0), F (x∗0, y0, µ0) ⊆ Y \− intC. (2.3) By the lower semicontinuity of E at λ0, there is a net x¯γ ∈ E(λγ), x¯γ → x0. By virtue of the contradiction assumption, there must be a subnet x¯β such that, ∀β, x¯β /∈ S˜ε2sα2(λβ , µβ , ηβ), i.e., ∃yβ ∈ K(x¯β , λβ), ∃x¯∗β ∈ G(x¯β , ηβ), F (x¯∗β , yβ , µβ) 6⊆ comp(−intC)ε2. (2.4) Since K is usc and K(x0, λ0) is compact, one can assume that yβ → y0, for some y0 ∈ K(x0, λ0). By the upper semicontinuity and the compactness of G at (x0, η0), one can assume that x¯∗β → x∗0 for some x∗0 ∈ G(x0, η0). The comp(−intC)ε2− upper semicontinuity of F at (x∗0, y0, µ0) and (2.3) together imply the existence of β¯ such that F (x¯∗¯β , yβ¯ , µβ¯) ⊆ comp(−intC)ε2, which contradicts (2.4).  The following two examples ensure that Theorem 2.2 is no longer true, for r ∈ {w,m, s} and k = 1, 2, if we replace S˜εkrα2 by S εk rα2 . Example 2.2. Let ε > 0 be fixed and small. Let X,Y,Λ,M,N and C be as in Example 2.1, K(x, λ) = [λ, λ + 1], G(x, λ) = {−x + 1 − ε, x}, F (x, y, λ) = {x − 1 − λ} and λ0 = 0. Then, it is clear that the assumption of Theorem 2.2 (i) holds. Direct calculations yield Swα2(0) = Smα2(0) = {1}, Sεwα2(0) = Sεmα2(0) = {0}∪ [1−ε, 1] and Sεwα2(λ) = Sεmα2(λ) = [1+λ−ε, 1+λ], ∀λ ∈ (0, 1]. So, S˜εwα2 and S˜εmα2 are lsc at 0, while Sεwα2 and Sεmα2 are not. Example 2.3. Let ε > 0 be fixed and small. Let X,Y,Λ,M,N and C be as in Example 2.1, K(x, λ) = [ 0, 1+ε+ √ (1+ε)2+4λ 2 ] , G(x, λ) = {x}, F (x, y, λ) = {x2 − (1 − ε)x − ε − λ} and λ0 = 0. Then, it is not hard to see that the assumption (ii) of Theorem 2.2 is satisfied and Ssα2(0) = {1 + ε}, Sεsα2(0) = {0, 1 + ε} and Sεsα2(λ) = { 1+ε+ √ (1+ε)2+4λ 2 } , ∀λ ∈ (0, 1]. Hence, 6 S˜εsα2 is lsc at 0, while S ε sα2 is not. Passing to Hausdorff lower semicontinuity we have Theorem 2.3. Let X be additionally a Hausdorff topological vector space. Assume that E is lsc at λ0, E(λ0) is compact, K is usc and compact-valued in clK(X,Λ)× {λ0}, K(., λ0) is lsc and closed in clK(X,Λ), F is comp(−intC)εk- lsc in clK(X,Λ) × clK(X,Λ) × {µ0}, k = 1, 2, and F (., ., µ0) is −C -usc in clK(X,Λ)× clK(X,Λ). (i) For r ∈ {w,m}, if G is lsc in clK(X,Λ)×{η0} and G(., η0) is usc and has compact values in clK(X,Λ), then S˜εkrα1 is Hlsc at (λ0, µ0, η0) for ε > 0. (ii) If G is usc and have compact values in clK(X,Λ)×{η0} and G(., η0) is lsc in clK(X,Λ), then S˜εksα1 is Hlsc at (λ0, µ0, η0) for ε > 0. Proof. We demonstrate only for S˜ε1wα1 . We first show that Swα1(λ0, µ0, η0) is compact. Let xγ ∈ Swα1(λ0, µ0, η0), xγ → x0. If x0 /∈ Swα1(λ0, µ0, η0), then there exists y0 ∈ K(x0, λ0) such that, ∀x∗0 ∈ G(x0, η0), F (x∗0, y0, µ0) ⊆ −intC. (2.5) Since K(., λ0) is lsc in clK(X,Λ), there is yγ ∈ K(xγ , λ0) such that yγ → y0. As xγ ∈ Swα1(λ0, µ0, η0), there exists x ∗ γ ∈ G(xγ , η0) with F (x∗γ , yγ , µ0) 6⊆ −intC. (2.6) Since G(., η0) is usc and G(x0, η0) is compact, we can assume that x∗γ → x∗0, for some x∗0 ∈ G(x0, η0). By the −C−upper semicontinuity of F (., ., µ0) in clK(X,Λ) × clK(X,Λ), we see a contradiction between (2.5) and (2.6). Hence, Swα1(λ0, µ0, η0) is closed and then compact, as E(λ0) is compact. We show next that ∀(λγ , µγ , ηγ) → (λ0, µ0, η0), ∀x¯0 ∈ Swα1(λ0, µ0, η0), ∃x¯γ ∈ S˜ε1wα1(λγ , µγ , ηγ), x¯γ → x¯0. Suppose to the contrary that there exist (λγ , µγ , ηγ) → (λ0, µ0, η0) and x¯0 ∈ S˜ε1wα1(λ0, µ0, η0) such that ∀xγ ∈ S˜ε1wα1(λγ , µγ , ηγ), xγ 6→ x¯0. Since E is lsc at λ0, there is x¯γ ∈ E(λγ), x¯γ → x¯0. By the contradiction assumption, there exists a subnet x¯β /∈ S˜ε1wα1(λβ , µβ , ηβ),∀β. The further argument to see a contradiction is similar to that of Theorem 2.1. Now suppose that S˜ε1wα1 is not Hlsc at (λ0, µ0, η0), i.e. ∃B (a neighborhood of the origin in X), ∃(λγ , µγ , ηγ) → (λ0, µ0, η0) such that ∀γ,∃x0γ ∈ S˜ε1wα1(λ0, µ0, η0) = Swα1(λ0, µ0, η0), x0γ /∈ S˜ε1wα1(λγ , µγ , ηγ) + B. Since Swα1(λ0, µ0, η0) is compact, we assume that x0γ → x0 for some x0 ∈ Swα1(λ0, µ0, η0). So there are γ1, a neighborhood B1 of 0 in X with B1 + B1 ⊆ B and bγ ∈ B1 such that, ∀γ ≥ γ1, x0γ = x0 + bγ . By the preceding part of the proof there is x˜γ ∈ S˜ε1wα1(λγ , µγ , ηγ), x˜γ → x0 and hence, one can assume that there is γ2,∀γ ≥ γ2, x˜γ ∈ x0 −B1, i.e., there exists b′γ ∈ B1, x˜γ = x0 − b′γ . Hence ∀γ ≥ γ0 = max{γ1, γ2}, x0γ = x0 + bγ = x˜γ + b′γ + bγ ∈ x˜γ +B. This is impossible due to the fact that x0γ /∈ S˜ε1wα1(λγ , µγ , ηγ) + B. Thus, S˜ε1wα1 is Hlsc at (λ0, µ0, η0).  7 The following example shows that the compactness of E(λ0) is essential. Example 2.4. Let ε > 0 be fixed and small. Let X = R2, Y = R, Λ ≡ M ≡ N = [0, 1], C = R+, K(x, λ) = {(x1, λx1)} (for x = (x1, x2)), G(x, λ) = {x}, λ0 = 0 and F (x, y, λ) = { {1} if λ = 0, {λ− ε2} otherwise. Then our six problems coincide and E(λ) = {x ∈ R2 | x2 = λx1}. So the assumptions of Theorem 2.3, but the compactness of E(λ0), are satisfied. However, S˜εrα(λ) = {x ∈ R2 | x2 = λx1} is not Hlsc at 0 (although S˜εrα is lsc at 0). The reason is that E(0) = {x ∈ R2 | x2 = 0} is not compact (but E and K are continuous and closed). Theorem 2.4. Assume that E is lsc at λ0, E(λ0) is compact, K is usc and compact-valued in clK(X,Λ) × {λ0}, K(., λ0) is lsc and closed in clK(X,Λ), F is comp(−intC)εk - usc in clK(X,Λ)× clK(X,Λ)× {µ0}, k = 1, 2, and F (., ., µ0) is −C -lsc in clK(X,Λ)× clK(X,Λ). (i) For r ∈ {w,m}, if G is lsc in clK(X,Λ)×{η0} and G(., η0) is usc and has compact values in clK(X,Λ), then S˜εkrα2 is Hlsc at (λ0, µ0, η0) for ε > 0. (ii) If G is usc and have compact values in clK(X,Λ)×{η0} and G(., η0) is lsc in clK(X,Λ), then S˜εksα2 is Hlsc at (λ0, µ0, η0) for ε > 0. We omit the proof since it is similar to that of Theorem 2.3. Example 2.4 shows also that the compactness of E(λ0) in Theorem 2.4 cannot be dropped. Moreover, Examples 2.1 - 2.3 also present that Theorems 2.3 and 2.4 are no longer true if we replace S˜εkrα by S εk rα (for r ∈ {w,m, s} and k = 1, 2.) 3. UPPER SEMICONTINUITY OF THE ε−SOLUTION SETS In this section we consider upper semicontinuity, Hausdorff upper semicontinuity and closedness. We will establish sufficient conditions for the ε−solutions of the two types to have these upper semicontinuity properties. We will also see that these results hold only for the ε−solution sets Sεkrα not for S˜εkrα , r ∈ {w,m, s}, α ∈ {α1, α2} and k = 1, 2. Theorem 3.1. Assume that K is lsc in clK(X,Λ)× {λ0}, E is usc, E(λ0) is compact and F is Y \comp(−intC)εk−usc in clK(X,Λ)× clK(X,Λ)× {µ0}, k = 1, 2. (i) For r ∈ {w,m}, if G is usc and has compact values in clK(X,Λ) × {η0}, then Sεkrα1 is both usc and closed at (λ0, µ0, η0) for ε ≥ 0. (ii) If G is lsc in clK(X,Λ)×{η0}, then Sε1sα1 is both usc and closed at (λ0, µ0, η0) for ε ≥ 0. Proof. By the similarity we present only a proof for problem (Psα1) and k = 1. Let ε ≥ 0 be fixed. Reasoning ab absurdo suppose the existence of an open neighborhood U of Sε1sα1(λ0, µ0, η0), of nets λγ → λ0, µγ → µ0, ηγ → η0 and xγ ∈ Sε1sα1(λγ , µγ , ηγ) such that xγ /∈ U,∀γ. By the upper semicontinuity of E and the compactness of E(λ0), we assume that xγ → x0 for some x0 ∈ E(λ0). If x0 /∈ Sε1sα1(λ0, µ0, η0), there are y0 ∈ K(x0, λ0) and x∗0 ∈ G(x0, η0) such that F (x∗0, y0, µ0) ∩ comp(−intC)ε1 = ∅. (3.1) Since K and G are lsc at (x0, λ0) and (x0, η0), respectively, there are nets yγ ∈ K(xγ , λγ) and x∗γ ∈ G(xγ , ηγ) such that yγ → y0 and x∗γ → x∗0. Since xγ ∈ Sε1sα1(λγ , µγ , ηγ), one has F (x∗γ , yγ , µγ) ∩ comp(−intC)ε1 6= ∅. (3.2) 8 By the closedness of comp(−intC)ε1 and the Y \ comp(−intC)ε1− upper semicontinuity of F at (x∗0, y0, µ0) one has a contradiction between (3.1) and (3.2). Thus, x0 ∈ Sε1sα1(λ0, µ0, η0), which is again a contradiction, since xγ /∈ U,∀γ. The closedness of Sε1sα1 can be proved similarly.  Theorem 3.1 is no longer true if we replace Sεkrα1 by S˜ εk rα1 , r ∈ {w,m, s}, respectively, as shown by the example below. Example 3.1. Let ε > 0 be fixed and small. Let X = Y = R, Λ ≡ M ≡ N = [0, 1], C = R+, K(x, λ) = [0, 1], G(x, λ) = {x}, F (x, y, λ) = {y − x+ λ} and λ0 = 0. Since G is single-valued, (Pwα1), (Pmα1) and (Psα1) coincide. It is easy to see that the conditions of Theorem 3.1 hold and accordingly, Sεkrα1 is usc at 0 (for all r ∈ {w,m, s}). In fact Srα1(0) = {0}, Sεkrα1(0) = [0, ε] and Sεkrα1(λ) = [0, ε+ λ]. Thus, S˜ εk rα1 is not usc at 0. Theorem 3.2. Assume that K is lsc in clK(X,Λ)×{λ0}, E is Husc and compact-valued at λ0, and that F is Y \ comp(−intC)εk - Husc in clK(X,Λ)× clK(X,Λ)×{µ0}. Assume further that ∀BX (open neighborhood of 0 in X), ∀x /∈ Sε1rα1(λ0, µ0, η0) + BX , ∃ρ > 0, (y, x∗) r¯K(x, λ0) × G(x, η0), α¯1 ( F (x∗, y, µ0)+intB ρ Y , comp(−intC)ε1 ) (remember that BρY = {y ∈ Y | d(0, y) ≤ ρ}). (i) For r ∈ {w,m}, if G is Husc and compact-valued in clK(X,Λ)×{η0}, then Sεkrα1 is Husc at (λ0, µ0, η0) for ε ≥ 0. (ii) If G is lsc in clK(X,Λ)× {η0}, then Sεksα1 is Husc at (λ0, µ0, η0) for ε ≥ 0. Proof. We demonstrate only for Sε1wα1 . Let ε ≥ 0 be fixed. Suppose that ∃BX (open neighborhood of 0 in X), ∃(λγ , µγ , ηγ) → (λ0, µ0, η0), ∃xγ ∈ Sε1wα1(λγ , µγ , ηγ) such that xγ /∈ Sε1wα1(λ0, µ0, η0)+BX ,∀γ. By the Hausdorff upper semicontinuity of E and the compactness of E(λ0), we can assume that xγ → x0 for some x0 ∈ E(λ0). If x0 /∈ Sε1wα1(λ0, µ0, η0) +BX , there is some ρ > 0 and some y0 ∈ K(x0, λ0) such that, ∀x∗0 ∈ G(x0, η0), [F (x∗0, y0, µ0) + intB ρ Y ]∩ comp(−intC)ε1 = ∅. (3.3) Since K is lsc at (x0, λ0), there is yγ ∈ K(xγ , λγ), yγ → y0. As xγ ∈ Sε1wα1(λγ , µγ , ηγ), ∃x∗γ ∈ G(xγ , ηγ), F (x∗γ , yγ , µγ) ∩ comp(−intC)ε1 6= ∅. (3.4) As G is H-usc and G(x0, η0) is compact, one has a subnet x∗β such that x ∗ β → x∗0 for some x∗0 ∈ G(x0, η0). By the Y \comp(−intC)ε1− Hausdorff upper semicontinuity of F at (x∗0, y0, µ0), we see a contradiction between (3.3) and (3.4). Hence, x0 ∈ Sε1wα1(λ0, µ0, η0) + BX , which is another contradiction, since xβ /∈ Sε1wα1(λ0, µ0, η0) +BX ,∀β.  We see that to ensure the Hausdorff upper semicontinuity of Sεrα1 , r ∈ {w,m, s}, the upper semicontinuity assumed in Theorem 3.2 is reduced to Hausdorff upper semicontinuity. However, we have to add some assumption in Theorem 3.2. The following example shows that this additional assumption is essential. Example 3.2. Let ε = 0, X = Y = R, Λ ≡ M ≡ N = [0, 1], C = R+, K(x, λ) = [0, 1], G(x, λ) = {x}, F (x, y, λ) = (−∞, λx) and λ0 = 0. As G is single-valued, (Pwα1), (Pmα1) and (Psα1) coincide. It is clear that S 01 rα1(0) = {0}, S01rα1(λ) = [0, 1], ∀r ∈ {w,m, s}, ∀λ ∈ (0, 1]. So, S01rα1 is not Husc at 0. The reason is that the additional assumption is violated. Indeed, take BX = (−1, 1) and x = 1. Then, for each ρ > 0 and each y ∈ [0, 1], one has F (1, y, 0)+ intBρX = (−∞, ρ). So, [F (1, y, 0) + intBρX ] ∩ [−ε,+∞) 6= ∅. Theorem 3.3. Assume that K is lsc in clK(X,Λ)× {λ0}, E is usc, E(λ0) is compact and F is Y \comp(−intC)εk−lsc in clK(X,Λ)× clK(X,Λ)× {µ0}, k = 1, 2. 9 (i) For r ∈ {w,m}, if G is usc and compact-valued in clK(X,Λ) × {η0}, then Sεkrα2 is both usc and closed at (λ0, µ0, η0) for ε ≥ 0. (ii) If G is lsc in clK(X,Λ)×{η0}, then Sεksα2 is both usc and closed at (λ0, µ0, η0) for ε ≥ 0. We omit the proof since it is similar to the previous ones. Example 3.1 ensures also that Theorem 3.3 is no longer true if we replace Sε1rα2 by S˜ ε1 rα2 , since F and G are single-valued. 4. APPLICATIONS Quasiequilibrium problems include as special cases many important problems such as qua- sivariational inequalities, complementarity problems, fixed-point and coincidence-point prob- lems, optimization problems, etc. Therefore, applying the results presented in the preceding sections we obviously obtain sufficient conditions for semicontinuity of approximate solution sets of these particular cases. In this section we derive some consequences of the theorems of Sections 2 and 3 as examples. 4.1. Quasivariational inequalities If Y = R, F (x, y, µ) = 〈T (x, µ), y − g(x, µ)〉 and G(x, η) = {x}, where T : X ×M → 2X∗ and g : X × M → X is a continuous mapping, then (Prα1) coincides with (QVI) in [15, 16], and (Prα2) becomes (SQVI) in [15, 16], r ∈ {w,m, s}. We state three direct consequences of Theorems 2.1, 3.1 and 3.3 as examples. Corollary 4.1. The ε−solution set S˜ε1 of (QVI) is lsc at (λ0, µ0) for ε > 0, if K is usc and compact-valued in clK(X,Λ) × {λ0}, E is lsc at λ0 and (x, y, µ) 7→ 〈T (x, µ), y − g(x, µ)〉 is [−ε,+∞)−lsc in clK(X,Λ)× clK(X,Λ)× {µ0}. Corollary 4.2. Let K be lsc in clK(X,Λ)× {λ0}, E be usc and E(λ0) be compact. Let (x, y, µ) 7→ 〈T (x, µ), y − g(x, µ)〉 be (−∞,−ε)−usc in clK(X,Λ) × clK(X,Λ) × {µ0}. Then Sε1 of (QVI) is both usc and closed at (λ0, µ0) for ε ≥ 0. Corollary 4.3. The ε−solution set Sε2 of (SQVI) is both usc and closed at (λ0, µ0) for ε ≥ 0, if K and E are as in Corollary 4.2, (x, y, µ) 7→ 〈T (x, µ), y − g(x, µ)〉 is (−∞,−ε)−lsc in clK(X,Λ)× clK(X,Λ)× {µ0}. These three corollaries improve Theorems 5.1, 6.1 and 6.3, respectively, in [16]. Now we give an example, where the assumptions of our corollaries are strictly weaker than the corre- sponding ones imposed in the mentioned paper. Example 4.1. Let ε > 0 be fixed. Let X = Y = R, Λ ≡ M = [0, 1], C = R+, K(x, λ) = [0, 1], λ0 = 0 and T (x, λ) = { {ε} if λ = 0, [ ε4 , ε 2 ] otherwise. Then 〈T (x, 0), y − g(x, 0)〉 = [−ε, ε] and 〈T (x, λ), y − g(x, λ)〉 = [− ε2 , ε2 ],∀λ ∈ (0, 1]. Hence all the assumptions of Corollaries 4.1 and 4.2 are satisfied. Applying these corollaries we know that S˜ε1 is lsc at 0 and S ε 1 is both usc and closed at 0. In fact, direct calculations give S˜ε1(λ) = S ε 1(λ) = [0, 1],∀λ ∈ Λ. However, 〈T (., .), .〉 is not lsc and T (., .) is not usc in X×X×{0} as required in assumption (ii) of Theorems 5.1 and 6.1 of [16]. Now we pass to quasivariational inequalities with operator solutions introduced in [13]. 10 Let X,M,N,Λ and Y be as defined in Section 1. Let C ⊆ Y be closed convex cone with intC 6= ∅ and C 6= Y . Let K : L(X,Y ) × Λ → 2L(X,Y ), T : L(X,Y ) × L(X,Y ) ×M → 2X be multifunctions. Our quasivariational inequalities with operator solutions are (QVI’) Find f¯ ∈ clK(f¯ , λ) such that, for each f ∈ K(f¯ , λ),( f − f¯ , T (f¯ , µ)) ∩ (Y \ −intC) 6= ∅; (SQVI’) Find f¯ ∈ clK(f¯ , λ) such that, for each f ∈ K(f¯ , λ),( f − f¯ , T (f¯ , µ)) ⊆ Y \ −intC. We have not observed stability results for these problems in the literature. As examples we state now three consequences, which are easily derived from Theorems 2.3, 2.4 and 3.2. Corollary 4.4. Assume for (QVI’) that K is usc and compact-valued in clK(L(X, Y ), Λ) × {λ0}, K(., λ0) is lsc and closed in clK(L(X,Y ), Λ), E is lsc at λ0 and E(λ0) is compact, (f, g, µ) 7→ (f − g, T (g, µ)) is comp(−intC)ε1-lsc or comp(−intC)ε2-lsc in clK(L(X,Y ), Λ) × clK(L(X,Y ), Λ)×{µ0}, and (f, g) 7→ ( f−g, T (x, µ0) ) is −C-usc in clK(L(X,Y ), Λ)×clK(L(X, Y ), Λ). Then the ε−solution set S˜ε1 is Hlsc at (λ0, µ0) for ε > 0. Corollary 4.5. Assume that K and E are as in Corollary 4.4 and (f, g, µ) 7→ (f − g, T (g, µ)) is comp(−intC)ε1−usc or comp(−intC)ε2−usc in clK(L(X,Y ), Λ) × clK(L(X,Y ), Λ) × {µ0}. Assume further that E(λ0) is compact, K(., λ0) is lsc and closed in clK(L(X,Y ), Λ) and (f, g) 7→ (f−g, T (x, µ0)) is −C-lsc in clK(L(X,Y ), Λ)×clK(L(X,Y ), Λ). Then the ε−solution set S˜ε2 of (SQVI’) is Hlsc at (λ0, µ0) for ε > 0. Corollary 4.6. Let K be lsc in clK(L(X,Y ), Λ)×{λ0}, E be Husc and E(λ0) be compact and (f, g, µ) 7→ (f − g, T (g, µ)) be Y \comp(−intC)ε1− Husc in clK(L(X,Y ), Λ)×clK(L(X,Y ), Λ)× {µ0}. If ∀B (open neighborhood of 0 in L(X,Y )), ∀f0 /∈ Sε1(λ0, µ0) +B, ∃BY (open neighbor- hood of 0 in Y ), ∃f ∈ K(f0, λ0), ( (f − f0, T (f0, µ0)) + BY ) ∩ comp(−intC)ε1 = ∅, then Sε1 of (QVI’) is Husc at (λ0, µ0) for ε ≥ 0. 4.2. Approximate fixed points Let X be a Hilbert space, Λ and M be as in Section 1, ε > 0, K : Λ → 2X and T : X ×M → 2X . The problem of finding ε - fixed points of T at (λ, µ) is defined as (see, e.g. a resent paper [16]) (Pε) Find x¯ ∈ K(λ) such that x¯ ∈ T ε(x¯, µ) := {x ∈ X | d(x, T (x¯, µ)) ≤ ε}. Assume that for each λ ∈ Λ, K(λ) is closed and contains 0. The problem of finding ε−fixed points is related to the following problem of finding ε−solutions of a quasiequilibrium problem: (QEPε) Find x¯ ∈ K(λ) such that ∃t¯ ∈ clT (x¯, µ), ∀y ∈ K(λ), 〈x¯− t¯, y + t¯ − x¯〉 ≥ −ε. (4.1) Proposition 4.1. If x¯ is a solution of (QEPε) then x¯ is a solution of (P √ ε). Proof. By the assumption we have (4.1). Taking y = 0 we get ‖x¯ − t¯‖2 ≤ ε and hence d ( x¯, T (x¯, µ) ) ≤ √ε.  Corollary 4.7. If K is lsc at λ0 and T is lsc in K(Λ) × {µ0}, then the ε−solution set S˜ε of (QEPε) is lsc at (λ0, µ0) for ε > 0. 11 Proof. By the lower semicontinuity of T , (x, y, µ) 7→ 〈x − clT (x, µ), y + clT (x, µ) − x〉 is lsc too. Applying Theorem 2.1 yields the corollary.  The following two results are easily derived from Theorems 2.3 and 3.1. Corollary 4.8. Assume the assumptions of Corollary 4.7. Assume further that K(λ0) is compact and T (., µ0) is usc and has compact values in K(Λ). Then the ε−solution set S˜ε of (QEPε) is Hlsc at (λ0, µ0) for ε > 0. Corollary 4.9. If K is usc, K(λ0) is compact and T is usc and has compact values in K(Λ)× {µ0}, then the ε−solution set Sε of (QEPε) is usc and closed at (λ0, µ0) for ε ≥ 0. By Proposition 4.1, Corollary 4.5 yields the semicontinuity with respect to (λ, µ) of the part of the √ ε−fixed points of T (x, µ). To deal with the whole set of ε - fixed points we modify problem (QEPε) as follows. (QEPε1) Find x¯ ∈ K(λ) such that ∃t¯ ∈ T ε(x¯, µ), ∀y ∈ K(λ), 〈x¯ − t¯, y + t¯ − x¯〉 ≥ 0. (4.2) Proposition 4.2. x¯ is a solution of (Pε) if and only if x¯ is a solution of (QEPε1). Proof. Being a solution of (QEPε1), x¯ yields t¯ ∈ T ε(x¯, µ) satisfying (4.2). Taking y = 0 we see that ‖x¯− t¯‖ = 0, and hence x¯ ∈ T ε(x¯, µ). Conversely, if x¯ is an ε−fixed point of T (., µ), i.e. x¯ ∈ T ε(x¯, µ). Taking t¯ = x¯ we see that x¯ satisfies (4.2).  Let us denote the fixed-point set and the ε - fixed-point set of T at (λ, µ) by P (λ, µ) and P ε(λ, µ), respectively. Similarly as for quasiequilibrium problems we consider also the following second kind of ε - fixed-point set P˜ ε(λ, µ) = { P (λ0, µ0) if (λ, µ) = (λ0, µ0), P ε(λ, µ) otherwise. Proposition 4.3. If K is lsc at λ0 and T is lsc in K(Λ)× {µ0}, then P˜ ε is lsc at (λ0, µ0) for ε > 0. Proof. Suppose to the contrary that there are λγ → λ0, µγ → µ0 and x0 ∈ P˜ ε(λ0, µ0) such that, ∀xγ ∈ P˜ ε(λγ , µγ), xγ 6→ x0. As K is lsc at λ0, there exists x¯γ ∈ K(λγ) such that x¯γ → x0. Then there must be a subnet x¯β with x¯β /∈ P˜ ε(λβ , µβ) for all β. Since x0 ∈ P˜ ε(λ0, µ0), x0 ∈ T (x0, µ0). By the lower semicontinuity of T at (x0, µ0), there is tβ ∈ T (x¯β , µβ) such that tβ → x0. Since, for all µβ 6= µ0, x¯β /∈ P˜ ε(λβ , µβ), x¯β /∈ T ε(x¯β , µ), i.e. ‖x¯β − tβ‖ ≥ ε. This is impossible since x¯β → x0 and tβ → x0.  Proposition 4.4. If K is usc, K(λ0) is compact and T is usc and has compact values in K(Λ)× {µ0}, then P ε is usc and closed at (λ0, µ0) for ε ≥ 0. Proof. Arguing by contradiction suppose the existence of a neighborhood U of P ε(λ0, µ0), of nets (λγ , µγ) → (λ0, µ0) and xγ ∈ P ε(λγ , µγ), xγ /∈ U,∀γ. By the assumption on K we have a point x0 ∈ K(λ0) and a subnet xβ → x0. Since xβ ∈ P ε(λβ , µβ), there is tβ ∈ clT (xβ , µβ), ‖tβ − xβ‖ ≤ ε. By the assumption on T we can assume that tβ → t0, for some t0 ∈ T (x0, µ0). Hence ‖x0 − t0‖ ≤ ε and then x0 ∈ P ε(λ0, µ0), a contradiction, since xβ /∈ U,∀β. The closedness of P ε at (λ0, µ0) is similarly verified.  12 4.3. Approximate quasioptimization problems Let X,Y,M,Λ,K and C be as in Section 1. Let T : X ×M → 2Y be a multifunction. We consider the following quasioptimization problem, for (λ, µ) ∈ Λ×M , (QOP) Find x¯ ∈ clK(x¯, λ) such that T (x¯, µ) ∩ wMin{T (K(x¯, λ), µ) ∣∣C} 6= ∅, where wMin { H ∣∣C} denotes the set of all weakly efficient points y∗ of the set H ⊆ Y , with respect to the ordering set C, i.e. points y∗ ∈ H such that, ∀y ∈ H, y − y∗ ∈ Y \ −intC. Note that unlike for the usual weak efficiency here C needs not be a cone. We will show now that (QOP) can be expressed as a case of problem (Pmα1). Set X1 = X × Y and define K1 : X1 ×Λ → 2X1 , G1 : X1 ×M → 2X1 and F1 : X1 ×X1 ×M → 2Y by, for x1 = (x1, y1), x2 = (x2, y2) in X1, K1(x1, λ) = K1 ( (x1, y1), λ ) = K(x, λ)× {0Y }, G1(x2, µ) = G1 ( (x2, y2), µ ) = {0X} × T (x2, µ), F1(x1, x2, µ) = F ( (x1, y1), (x2, y2), µ ) = T (x2, µ)− y2. We consider the following problem (Pmα1): (MQEP) Find x¯ ∈ clK(x¯, λ) such that ∃y¯ ∈ T (x¯, µ), ∀x ∈ K(x¯, λ), F1 ( (0, y¯), (x, 0), µ ) ≡ T (x, µ)− y¯ ⊆ Y \ −intC. Proposition 4.5. x¯ is a solution of (QOP) if and only if x¯ is a solution of (MQEP). The proof is direct and so is omitted. The following two approximate problems of (QOP) and (MQEP) are also equivalent (QOPε) Find x¯ ∈ clK(x¯, λ) such that T (x¯, µ) ∩ wMin{T (K(x¯, λ), µ)∣∣comp(−intC)ε1} 6= ∅; (MQEPε) Find x¯ ∈ clK(x¯, λ) such that ∃y¯ ∈ T (x¯, µ), ∀x ∈ K(x¯, λ), T (x, µ)− y¯ ⊆ comp(−intC)ε1. The following corollaries about the approximate solution sets S˜ε and Sε of (QOP) are direct consequences of Theorems 2.2, 2.4 and 3.3, respectively. Corollary 4.10. Assume that K is usc and has compact values in clK(X,Λ) ×{λ0}, E is lsc at λ0, T is lsc in clK(X,Λ) × {µ0} and (x, y, µ) 7→ T (x, µ) − y is comp(−intC)ε1−usc in clK(X,Λ)× clK(X,Λ)× {µ0}. Then S˜ε is lsc at (λ0, µ0) for ε > 0. Corollary 4.11. Assume the assumptions of Corollary 4.10. Assume further that E(λ0) is compact, K(., λ0) is lsc and closed in clK(X,Λ), T (., µ0) is usc and has compact values in clK(X,Λ) and (x, y) 7→ T (x, µ0)− y is −C-lsc in clK(X,Λ)× clK(X,Λ). Then S˜ε is Hlsc at (λ0, µ0) for ε > 0. Corollary 4.12. Sε is usc and closed at (λ0, µ0) for ε ≥ 0, if K is lsc in clK(X,Λ) × {λ0}, E is usc at λ0 and E(λ0) is compact, T is usc, Y \ comp(−intC)ε1- lsc and compact-valued in clK(X,Λ)× {µ0}. 13 REFERENCES 1. M. Ait Mansour, and L. Scrimali (2007). Ho¨lder continuity of solutions to elastic traffic network models. J. Glob. Optim., online. 2. M. Ait Mansour and H. Riahi (2005). Sensitivity analysis for abstract equilibrium prob- lems. J. Math. Anal. Appl. 306: 684-691. 3. L.Q. Anh and P.Q. Khanh (2004). Semicontinuity of the solution set of parametric multivalued vector quasiequilibrium problems. J. Math. Anal. Appl. 294: 699-711. 4. L.Q. Anh and P.Q. Khanh (2006). On the Ho¨lder continuity of solutions to parametric multivalued vector equilibrium problems. J. Math. Anal. Appl. 321: 308-315. 5. L.Q. Anh and P.Q. Khanh (2007). On the stability of the solution sets of general multi- valued vector quasiequilibrium problems. J. Optim. Theory Appl. 135: 271-284. 6. L.Q. Anh and P.Q. Khanh (2007). Uniqueness and Ho¨lder continuity of the solution to multivalued equilibrium problems in metric spaces. J. Glob. Optim. 37: 449-465. 7. L.Q. Anh and P.Q. Khanh (2007). Various kinds of semicontinuity and the solution sets of parametric multivalued symmetric vector quasiequilibrium problems. J. Glob. Optim., to appear. 8. L.Q. Anh and P.Q. Khanh. Sensitivity analysis for multivalued quasiequilibrium prob- lems in metric spaces: Ho¨lder continuity of solutions. J. Glob. Optim.,to appear. 9. M. Bianchi and R. Pini (2003). A note on stability for parametric equilibrium problems. Oper. Res. Lett. 31: 445-450. 10. M. Bianchi and R. Pini (2006). Sensitivity for parametric vector equilibria. Optimization 55: 221-230. 11. R. Braˆnzei, J. Morgan, V. Scalzo, and S. Tijs. (2003). Approximate fixed point theorems in Banach spaces with applications in game theory. J. Math. Anal. Appl. 285: 619-628. 12. Y.H. Cheng and D.L. Zhu (2005). Global stability results for the weak vector variational inequality. J. Glob. Optim. 32: 543-550. 13. A. Domokos and J. Kolumba´n (2002). Variational inequalities with operator solutions. J. Glob. Optim. 23: 99-110. 14. N.J. Huang, J. Li, and H.B. Thompson (2006). Stability for parametric implicit vector equilibrium problems. Math. Comput. Model. 43: 1267-1274. 15. P.Q. Khanh and L.M. Luu (2005). Upper semicontinuity of the solution set of parametric multivalued vector quasivariational inequalities and applications. J. Glob. Optim. 32: 569-580. 16. P.Q. Khanh and L.M. Luu (2007). Lower and upper semicontinuity of the solution sets and approximate solution sets to parametric multivalued quasivariational inequalities. J. Optim. Theory Appl. 133: no. 1, in press. 17. S.J. Li, G.Y. Chen, and K.L. Teo (2002). On the stability of generalized vector quasi- variational inequality problems. J. Optim. Theory Appl. 113: 283-295. 18. L.D. Muu (1984). Stability property of a class of variational inequalities. Math. Opera- tionsforsch. Statist., ser: Optim. 15: 347-351. 19. J. Zhao (1977). The lower semicontinuity of optimal solution sets. J. Math. Anal. Appl. 207: 240-254. 14