SEMICONTINUITY OF SOLUTIONS OF MINTY TYPE QUASIVARIATIONAL INEQUALITIES

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QUASIVARIATIONAL INEQUALITIES

DIANA-ELENA STANCIU

We give sufficient conditions ensuring the upper and lower semicontinuity in the sense of Berge of the solutions set and approximate solutions set of Minty type invex quasivariational inequality. Also we study the continuity of the solutions set of an optimization problem with this type of variational inequalities constraints.

AMS 2010 Subject Classification: 49J40, 58E35, 90C33.

Key words: quasivariational inequality, semicontinuity, closed map.

1. INTRODUCTION

Over time several researchers [5–7, 11] have studied the effects of dis- turbance parameters on variational inequalities and optimization problems, introducing the concept of stability. The literature have analyzed several aspects of stability, such as semicontinuity, continuity, Lipschitz continuity or some kind of differentiability of the solutions set for variational inequalities (see [5–9, 12, 14, 20, 22]). Among them, the continuity of the solutions sets is required more often in applications. Since continuity implies the lower and upper semicontinuity, these issues were analyzed separately by authors such as Zhao [22], Kien [9], Khanh and Luu [8], who obtained sufficient conditions for semicontinuity.

Quasivariational inequalities were introduced by Bensoussan and Lions [3] and because of their wide applicability in areas such as economics, en- gineering, mathematical programming problems, complementarity problems have been investigated by several authors [2, 4, 10, 13, 19, 21]. Gong [6] obtained results on upper and lower semicontinuity of the solutions set of some Stampacchia type generalized quasivariational inequalities. Also, Khanh and Luu [8] studied the semicontinuity of the solutions set and approximate solutions set of some Stampacchia type parametric multivalued quasivariational inequalities.

Variational inequalities are closely related to optimization problems. The- se later were intensely studied by many authors. Among them, Preda [15–18]

REV. ROUMAINE MATH. PURES APPL.,56(2011),4, 303–315

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gave optimality conditions for some programming problems under suitable assumptions and studied the nature of solutions of such problems.

Motivated by the works mentioned above, in this paper are studying aspects of stability of the solutions set and of the approximate solutions set of a Minty type perturbed invex quasivariational inequality problem. In Sec- tion 2 will recall the notions of continuity in the sense of Berge and generalized quasimonotony and formulate the parametric invex quasivariational inequality of the Minty type, (M V Iη(x)). In Section 3 we give some examples of such problems and calculate their solutions sets. Making certain assumptions on the set A and on the applicationsK,T and η, in Section 4 we will give sufficient conditions for the upper and lower semicontinuity of the solutions sets.

Section 5 analyzes the stability of the approximate solutions set of the problem (M V Iη(x)). In the last section of the study is extended to the upper and lower semicontinuity of an optimization problem with invex quasivariational inequality constraints.

2. PRELIMINARIES

LetX ⊂Rⁿbe a nonempty, closed set andA⊂Y =R^mbe a nonempty, closed and convex set.

For eachε >0 andx0∈Y, denote byB(x0, ε) the open ball with center in xand radiusε, that is

B(x0, ε) :={x∈Y :kx₀−xk< ε},

and with U(A, ε) an open ε-neighborhood of a subset of A ⊆ Y defined by U(A, ε) :={x∈Y : there existsa∈A such thatka−xk< ε}.

We now recall the notions of upper and lower semicontinuity in the sense of Berge.

Definition 1 ([10]). LetF :X→2^Y be a set-valued map with domF = X, whereX is a nonempty and closed subset ofRⁿ andY =R^m. We say that the application F is:

(i) Upper semicontinuous in the sense of Berge (in short, B-usc) atx0 ∈X if for every open set N satisfying F(x₀) ⊂N, there exists a δ >0, such that for every x∈B(x0, δ), F(x)⊂N.

(ii) Lower semicontinuous in the sense of Berge (in short, B-lsc) atx₀ ∈X if for every open set N satisfying F(x0)∩N 6=∅, there exists a δ > 0, such that for every x∈B(x₀, δ),F(x)∩N 6=∅.

The application F is said to be B-lsc (respectively B-usc) on X ifF is B-lsc (respectively B-usc) at each point x₀ ∈X. F is said to be B-continuous on X if it is both B-lsc and B-usc on X.

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Aubin and Ekeland [1] gave the following equivalent definition for a B- lower semicontinuous function:

Remark 2 ([1]). F is said to be B-lsc at x0 ∈ X if and only if for each sequence {x_n} ⊂ X converging to x₀ and for any y₀ ∈ F(x₀) there exists a sequence {y_n} ⊂F(xn) converging toy0.

Definition 3 ([1]). F is said to be closed at x₀ ∈ X, if for each of the sequences {x_n} ⊂ X converging to x0 and {y_n} ⊂ Y converging to y0 such that y_n ∈ F(x_n), we have y₀ ∈ F(x₀). F is said to be closed on X if it is closed at each x0 ∈X.

Remark 4 ([1]). It is well-known that, ifF is B-usc atx₀ ∈XandF(x₀) is closed, then F is closed at x0.

Definition 5. LetF :X→2^Y be a set-valued map with domF =X and η :Y ×Y →Y. We say that the applicationF is:

(i) η-pseudomonotone on X iff for any x, x₀ ∈ X, hξ, η(x, x₀)i ≥ 0, for someξ ∈F(x0)⇒ hµ, η(x, x₀)i ≥0, for anyµ∈F(x);

(ii)η-quasimonotone onXiff for anyx, x₀∈X,x6=x₀,hξ, η(x, x₀)i>0, for some ξ∈F(x₀)⇒ hµ, η(x, x₀)i ≥0, for anyµ∈F(x).

Next we give the general framework in which we will do the study and the parametric invex quasivariational inequality problem of the Minty type (M V I_η(x₀)) considered here.

LetX ⊂Rⁿbe a nonempty, closed set andA⊂Y =R^mbe a nonempty, closed and convex set, K :X×Y →2^Y be a closed application,T :X×Y → 2^Y and η : Y ×Y → Y. Suppose that domK = domT = X×Y, where domK ={(x, y)∈X×Y :K(x, y)6=∅}.

We consider the following parametric invex quasivariational inequality of the Minty type, corresponding to a parameter x0 ∈X : (M V Iη(x0)) Find u0∈K(x0, u0)∩A, such that

ht, η(u₀, v)i ≤0, ∀v∈K(x₀, u₀), ∀t∈T(x₀, v).

LetM_η(x₀) denote the solution set of (M V I_η(x₀)), that is

Mη(x0) ={u₀ ∈K(x0, u0)∩A:ht, η(u₀, v)i ≤0,∀v∈K(x0, u0),∀t∈T(x0, v)}.

Forη(u, v) =u−v, we obtain the inequality considered by Lalitha and Bhatia in [11].

We also extend the study of upper and lower semicontinuity to approximate solutions set and to modified approximate solutions set of problem (M V Iη(x0)).

For a fixedε≥0, define the set of approximate solutions of (M V I_η(x₀)) as M_η^ε(x₀) = {u₀ ∈ K(x₀, u₀)∩A : ht, η(u₀, v)i ≤ ε, ∀v ∈ K(x₀, u₀), ∀t ∈ T(x0, v)}.

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Remark that forε= 0, we haveM_η^ε(x0) =Mη(x0).

For a fixed ε ≥ 0, define the set of modified approximate solutions of (M V Iη(x0)) as

Mf_η^ε(x) =

M_η(x₀) ifx=x₀ M_η^ε(x) ifx6=x0 . 3. EXAMPLES

In this section we give some examples of problems (M V I_η(x₀)) and we will determine their solutions sets. For the problems (M V Iη(x0)) considered here there are satisfied some hypothesis of the theorems gived in this article.

Some of these examples illustrate the fact that certain conditions cannot be relaxed in the theorems presented here.

Example6. LetX = [−1,1],Y =RandA= [0,2]. Defineη:Y×Y →Y asη(u, v) =u²−v² and the set-valued mapsK :X×Y →2^Y andT :X×Y → 2^Y as follows

K(x, u) =

[u,2] ifu≤2

[2, u] ifu >2 , T(x, u) =







{0} ifx= 0 [0,1] ifx <0 [0,|x+u|] ifx >0

.

We have M_η(x) = [0,2], ∀x∈X.

K(x, u) =

{0} ifx= 1

[0,¹₂] ifx6= 1 , T(x, u) =

{0} ifu≤1 [0, u] ifu >1 . Forx0 = 1, Mη(x0) ={0} and for x6=x0,Mη(x) = [0,¹₂].

Example 8. Let X = [−2,2] and Y =A =R. Define η:Y ×Y → Y as η(u, v) =u³−v³and the set-valued mapsK :X×Y →2^Y andT :X×Y →2^Y as follows

K(x, u) =

[u,0] ifu≤0

[0, u] ifu >0 , T(x, u) =

{0} ifx= 1 [0,^|x−1|₂ ] ifx6= 1 . Forx₀ = 1, M_η(x₀) =Rand for x6=x₀,M_η(x) = (−∞,0].

K(x, u) =

{0, x} ifu= 0

[0,2|u|+ 1] ifu6= 0 , T(x, u) =

[2,|u|+ 2] ifx= 0 {1} ifx6= 0 .

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Here we haveMη(x) ={0},∀x∈X.

Example 10. LetX= [−2,2] andY =A= [0,+∞). Defineη:Y×Y → Y as η(u, v) = u² −v² and the set-valued maps K : X ×Y → 2^Y and T :X×Y →2^Y as follows

K(x, u) =

[u,2 +|x+ 1|] ifu <2 [2, u²+ 2] ifu≥2 , T(x, u) =

[−|x|,2|x|+ 3] ifu <2 {2 +x²} ifu≥2 . Forx0 = 0, Mη(x0) = [0,2] and forx6=x0,Mη(x) ={2}.

Example 11. Let X = [−1,1], Y = R and A = [0,+∞). Define η : Y ×Y → Y as η(u, v) = u³−v³ and the set-valued maps K :X×Y → 2^Y and T :X×Y →2^Y asK(x, u) ={0,1},

T(x, u) =

[−|x|,|x|+a] ifu= 0 {0} ifu6= 0 , where ais a fixed positive real number.

Forx0 = 0, Mη(x0) ={0,1}and for x6=x0,Mη(x) ={1}.

Example 12. LetX= [−2,2], Y =RandA= [0,2]. Defineη:Y ×Y → Y as η(u, v) = u³ −v³ and the set-valued maps K : X ×Y → 2^Y and T :X×Y →2^Y asK(x, u) = [0,|u|],

T(x, u) =

{1} ifx= 0 {0,1} ifx6= 0 . We have Mη(x) ={0},∀x∈X.

4. CONTINUITY OF THE SOLUTIONS SET

In this section we give sufficient conditions for the upper and lower semicontinuity of the application M_η : X → 2^Y, where M_η(x) is the solutions set of problem (M V Iη(x)), making certain assumptions on the applications T and K.

Theorem 13. Suppose that for x0 ∈ X, the following conditions are satisfied:

(i) K is closed and B-lsc on {x₀} ×Y; (ii T is B-lsc on{x₀} ×Y;

(iii) A is a compact subset of Y;

(iv)η(·,·) is continuous in the both arguments.

ThenM_η is B-usc atx₀. Moreover,M_η(x₀) is compact andM_η is closed at x0.

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Proof. Suppose, on the contrary, that Mη be not B-usc at x0. Then there exists an open set N containing M_η(x₀), such that for every sequence xn → x0, there exists un ∈ Mη(xn), un ∈/ N, for every n. Using the line of [11], is sufficient to show that u0 ∈Mη(x0). If u0 ∈/ Mη(x0), then there exist v₀ ∈K(x₀, u₀) and t₀∈T(x₀, v₀), such that

(1) ht₀, η(u₀, v₀)i>0.

Since K is B-lsc at (x0, u0), v0 ∈ K(x0, u0) and (xn, un) → (x0, u0), there exists v_n ∈ K(x_n, u_n), such that v_n → v₀. Similarly, since T is B-lsc at (x₀, v₀), t₀ ∈T(x₀, v₀) and (x_n, v_n) → (x₀, v₀), it follows that there exists tn∈T(xn, vn), such that tn→t0. From un∈Mη(xn) we have

(2) ht, η(u_n, v)i ≤0, ∀v∈K(xn, un),∀t∈T(xn, v).

Taking v = v_n and t =t_n in (2) and taking limits as n → ∞, we have ht₀, η(u0, v0)i ≤0, which contradicts relation (1). ThereforeMη is B-usc atx0. Proceeding like in [11], we can show thatM_η(x₀) is a closed set. More- over, as M_η(x₀)⊂A andA is compact, it follows thatM_η(x₀) is compact. It is well-known that, if Mη is B-usc at x0 ∈X and Mη(x0) is closed, then Mη

is closed at x₀ (see [1]).

In Example 6, are satisfied all the conditions of the above theorem.

Therefore Mη is B-usc atx0,Mη(x0) is compact and Mη is closed at x0. In Example 7, the maps K and T are B-lsc on {x₀} ×Y but K is not closed on{x₀} ×Y. We can see thatMη is neither B-usc atx0 or closed atx0. In Example 8, the first two conditions and fourth are satisfied, butA is not compact. It can be observed thatM_η is B-usc atx₀ and closed atx₀, but Mη(x0) is not compact.

So, the condition of closedness on the mapK and the condition of com- pactness on the set Acannot be relaxed in the above theorem.

(i) K is closed on{x₀} ×Y; (ii)A is a compact subset ofY;

(iii) ∀u₀ ∈K(x0, u0)∩A, ∀(x_n, un)→(x0, u0) and

(3) ht₀, η(u₀, v₀)i>0, for some v₀ ∈K(x₀, u₀), t₀ ∈T(x₀, v₀)

implies that there exists a positive integer n, such that ht, η(u_n, v)i > 0, for some v∈K(x_n, u_n),t∈T(x_n, v);

(iv)η(·,·) is continuous in the both arguments.

ThenM_η is B-usc atx₀. Moreover,M_η(x₀) is compact andM_η is closed at x0.

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Proof. Suppose, on the contrary, that Mη be not B-usc at x0. Then there exists an open set N containing M_η(x₀), such that for every sequence xn→x0, there existsun∈Mη(xn),un∈/ N, for everyn. As in [11], we arrive to a contradiction and hence Mη is B-usc at x0. The fact that Mη(x0) be compact and M_η be closed atx₀, follows as in Theorem 13.

The advantage of assumption (iii) can be illustrated by means of Exam- ple 9, wherein the solutions set is B-usc and compact at x0, even though the map T is not B-lsc on{x₀} ×Y, wherex₀= 0.

If in Example 9 we take T(x, u) =

[2,|u|+ 2] ifx= 0

{0} ifx6= 0 , then it can be verified that the assumption (iii) of Theorem 14 fails to hold. For u0 = 2 and (x_n, u_n) = _n¹,2 +_n¹

, we have (x_n, u_n)→(x₀, u₀) andht₀, η(u₀, v₀)i>0, for some v0 ∈K(x0, u0), t0 ∈ T(x0, v0) butht, η(u_n, v)i= 0, ∀v ∈K(xn, un),

∀t ∈ T(x_n, v). It can be seen that M_η(x₀) = {0} and for x 6= x₀, M_η(x) = [0,2]. Hence Mη is not B-usc at x0. So, if the condition (iii) is not satisfied, then the conclusion of Theorem 14 fails to hold.

Next will be studied the lower semicontinuity of the solutions set of problem (M V I_η(x₀)).

Theorem 15. Suppose that for x₀ ∈ X, the following conditions are satisfied:

(i) K is B-lsc at x0, where K(x) ={u∈A:u∈K(x, u)};

(ii) ∀u₀ ∈K(x₀, u₀)∩A, ∀(x_n, u_n)→(x₀, u₀) and (4) ht, η(u₀, v)i ≤0, ∀v∈K(x0, u0),∀t∈T(x0, v)

implies that there exists a positive integer n, such that ht, η(u_n, v)i ≤0, ∀v∈ K(xn, un), ∀t∈T(xn, v);

(iii) η(·,·) is continuous in the both arguments.

ThenMη is B-lsc at x0.

Proof. Suppose, on the contrary, thatMη be not B-lsc at x0. From Re- mark 2, there exists a sequence {x_n}inX converging tox₀ and u₀ ∈M_η(x₀), such that for every sequence y_n ∈ M_η(x_n), y_n 9 u₀. Since x_n → x₀ and u₀ ∈ K(x₀), from assumption (i) it follows that there exists a sequence u_n∈K(x_n) such thatu_n→u₀. It follows that u_n∈/ M_η(x_n) and then

(5) ht_n, η(un, vn)i>0, for somevn∈K(xn, un), tn∈T(xn, vn).

Since u₀ ∈M_η(x₀), it follows that relation (4) holds and hence by condition (ii) of the hypothesis, there exists n∈N, such that

ht, η(u_n, v)i ≤0, ∀v∈K(x_n, u_n),∀t∈T(x_n, v), which contradicts (5). Therefore, Mη is B-lsc atx0.

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In Example 10 the mapK is B-lsc atx0. It can be seen thatMη is not B- lsc atx₀. It can be easily verified that the condition (ii) of the above theorem fails to hold. As for u0 = 2, (xn, un) = _n¹,2 +_n¹

,n∈N, (xn, un)→(x0, u0) and ht, η(u₀, v)i ≤ 0, ∀v ∈K(x₀, u₀), ∀t ∈ T(x₀, v) but ht, η(u_n, v)i >0, for v = 2∈K(x_n, u_n) and t_n = 2 +_n¹2 ∈T(x_n, v_n), for every n. So, wherein the conclusion of Theorem 15 fails to hold in the absence of condition (ii).

The proofs of the following two results are analogous to those of Theo- rem 15 established above and of Theorems 4.2 and 4.3 from [11].

Theorem 16. Suppose that conditions of Theorem 13 are satisfied and that for x0 ∈X, we have:

(i) for every u0 ∈Mη(x0), ht, η(u₀, v)i <0, ∀v ∈Mη(x0)\{u₀} and for some t∈T(x₀, v);

(ii)−T isη-quasimonotone on {x₀} ×Y. ThenM_η is B-lsc at x₀.

Theorem 17. Suppose that conditions of Theorem 13 are satisfied and that for x₀ ∈X, we have:

(i) for every u0 ∈ Mη(x0), ht, η(u₀, v)i ≤ 0, ∀v ∈Mη(x0) and for some t∈T(x0, v);

(ii)−T isη-pseudomonotone on {x₀} ×Y;

(iii) ht, η(u, v)i= 0, for t∈T(x0, u)∪T(x0, v)⇒u=v;

(iv)η(u, v) =−η(v, u), ∀u, v∈K.

ThenMη is B-lsc at x0.

Therefore, if conditions of Theorem 16 or Theorem 17 are satisfied, the application M_η is B-continuous atx₀.

5. CONTINUITY OF THE APPROXIMATE SOLUTIONS SET This section extends the study of upper and lower semicontinuity to approximate solutions set and to modified approximate solutions set of problem (M V Iη(x0)).

The proofs of the following two theorems are analogous to those of Theo- rems 13, 14 respectively.

Theorem 18. Suppose that for x0 ∈ X the conditions of Theorem 13 hold. Then M_η^ε is B-usc at x₀, for any ε≥0. Moreover, M_η^ε(x₀) is compact and M_η^ε is closed at x0, for anyε≥0.

Theorem19. Suppose that forx0∈X the conditions(i)–(ii)and(iv)of Theorem 13 hold and, in addition, ∀u₀ ∈K(x₀, u₀)∩A, ∀(x_n, u_n)→(x₀, u₀) and ht₀, η(u0, v0)i > ε, for some v0 ∈ K(x0, u0), t0 ∈ T(x0, v0) implies that

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there exists a positive integer n, such that ht, η(u_n, v)i > ε, for some v ∈ K(x_n, u_n), t∈T(x_n, v).

ThenM_η^ε is B-usc atx0. Moreover,M_η^ε(x0) is compact andM_η^ε is closed at x0.

Next we show that if problem (M V Iη(x0)) is well-posed, then Mη is B-upper semicontinuous.

Definition 20. A sequence{u_n}is said to be an approximating sequence for the problem (M V I_η(x₀)) iff there exists a sequence {x_n} in X, such that xn→x0 and there exists a sequence{ε_n}inR,εn>0 withεn→0, such that u_n∈M_η^εⁿ(x_n), ∀n∈N.

Definition 21. We say that the parametric invex quasivariational inequality problem (M V I_η(x₀)) is well-posed iff

(i) the solution setMη(x0) of (M V Iη(x0)) is nonempty;

(ii) every approximating sequence for (M V I_η(x₀)) has a subsequence, which converges to some point of Mη(x0).

Remark 22. Well-posedness of (M V I_η(x₀)) implies that the solution set Mη(x0) is a nonempty compact set.

Theorem 23. If (M V I_η(x₀))is well-posed, then M_η is B-usc at x₀. Proof. Suppose, on the contrary, thatM_η be not B-usc atx₀. Then there exists an open setN containingMη(x0), such that for every sequencexn→x0, there existsun∈Mη(xn) butun∈/ N. Asxn→x0andun∈Mη(xn), it follows that {u_n} is an approximating sequence for (M V I_η(x₀)). Since u_n ∈/ N and Mη(x0)⊂N, none of its subsequences converge to a point ofMη(x0), thereby leading to a contradiction to the fact that (M V I_η(x₀)) is well-posed. So,M_η is B-usc at x0.

The converse of the above result may fail to hold. For the problem (M V Iη(x0)) considered in Example 8, if we choose the sequences {x_n} and {u_n}asx_n= 1+_n¹ andu_n=−nfor everyn, then it can be observed that{u_n} is an approximating sequence for the problem (M V Iη(x0)), but it possesses no convergent subsequence, thereby implying that (M V I_η(x₀)) is not well-posed.

Regarding semicontinuity of the modified approximate solutions set of problem (M V I_η(x₀)), we have the following result:

(i) K is B-usc with compact values on {x₀} ×Y;

(ii)K is B-lsc at x0, where K(x) ={u∈A:u∈K(x, u)};

(iii) T is B-usc with compact values on{x₀} ×Y; (iv)η(·,·) is continuous in the both arguments.

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ThenMf_η^ε is B-lsc atx₀, for each ε >0.

Proof. Suppose, on the contrary, that Mf_η^ε be not B-lsc at x₀. Using the line of [11], there exists a sequence {x_n} inX converging to x₀ and u₀ ∈ Mf_η^ε(x₀), such that for every sequencey_n∈Mf_η^ε(x_n), we havey_n9u₀.

SinceK is B-lsc at x0,xn→x0 and u0 ∈K(x0), there exists a sequence {u_n} ⊂K(xn) converging tou0. It follows that un∈/ Mf_η^ε(xn), that is

(6) ht_n, η(un, vn)i> ε, for somevn∈K(xn, un), tn∈T(xn, vn).

AsK is B-usc and compact valued at (x0, u0) andvn∈K(xn, un), there exists v₀∈K(x₀, u₀), such thatv_n→v₀. Also, since T is B-usc and compact valued at (x0, v0) and tn ∈ T(xn, vn), there exists some t0 ∈ T(x0, v0), such thatt_n→t₀. Taking limit asn→ ∞in relation (6) we haveht₀, η(u₀, v₀)i ≥ε, a contradiction to u₀ ∈Mf_η^ε(x₀).

The significance of introducing a modified approximate solution set can be illustrated by Example 11, wherein the modified approximate solution set is B-lsc at x0, while the approximate solution set is not so. Forx0 = 0,K is B-usc with compact values on {x₀} ×Y and T is B-usc with compact values on {x₀} ×Y. Also, K is B-lsc at x₀. It can be verified that M_η(x₀) ={0,1}

and M_η^ε(x) =

{0,1} if|x| ≤ε−a

{0} if|x|> ε−a ,∀ε >0 and ∀x∈X, where 0< a < ε.

It is easy to see that Mf_η^ε is B-lsc at x0, ∀ε > 0. However, for ε = a, M_η^ε(x0) ={0,1}and M_η^ε(x) ={0}, ∀x6=x0. Therefore, for ε=a, M_η^ε is not B-lsc at x₀.

6. THE CASE OF A SPECIAL OPTIMIZATION PROBLEM Consider the following optimization problem:

(P) minf(x, u), with u∈Mη(x), x∈X,

where f :X×Y → R and the set of feasible points, Mη(x), is the solutions set of the parametric invex quasivariational inequality (M V I_η(x)) defined in Section 2.

We denote by Ω the set of solutions of problem (P), that is Ω :=n

(x, u)∈X×Y :u∈K(x, u)∩A,f(x, u)≤ inf

y∈X,v∈M_η(y)f(y, v) and ht, η(u, v)i ≤0,∀v∈K(x, u), ∀t∈T(x, v)

o .

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Forε≥0, we define a parametric form (P(ε)) of the optimization problem (P), as follows:

(P(ε)) minf(x, u), withu∈M_η^ε(x), x∈X,

where M_η^ε(x) is the approximate solutions set of the parametric invex quasivariational inequality (M V I_η(x)). For ε = 0, the problem reduces to problem (P). This section gives sufficient conditions for continuity of approximate solutions set of problem (P(ε)).

Forδ, ε≥0, define the approximate solutions set for the problem (P(ε)) as Ω^δ(ε) :=

n

(x, u)∈X×Y :u∈K(x, u)∩A, f(x, u)≤ inf

y∈X,v∈Mη(y)f(y, v) +δ and ht, η(u, v)i ≤ε,∀v∈K(x, u),∀t∈T(x, v)o

.

We see that ifδ = 0, then Ω^δ(0) = Ω. Also for anyδ≥0, Ω^δ(0)⊆Ω^δ(ε),

∀ε≥0, from which we deduce that Ω^δ is B-lsc atε= 0. The conditions that ensure the upper semicontinuity of Ω^δ at ε= 0 are similar to those given by Lalitha in [11] for the special case where η(u, v) =u−v, for any u, v∈Y.

Theorem 25. Suppose that the conditions of Theorem 13 hold and (i) X is a bounded subset ofRⁿ;

(ii)f is lower semicontinuous.

Then for every δ≥0, Ω^δ is B-usc at ε= 0.

Corollary 26. Suppose that conditions (i), (ii and (iv) of Theorem 13 hold and

(i) there exist ε⁰, δ⁰>0 such thatΩ^δ⁰(ε⁰) is bounded;

Then for every δ≤δ⁰,Ω^δ is B-usc atε= 0.

Theorem 27. Suppose that the conditions of Theorem 14 hold and (i) X is a bounded subset ofRⁿ;

Then for every δ≥0, Ω^δ is B-usc at ε= 0.

Remark 28. Since Ω^δ is B-usc at ε = 0, if the conditions of one of Theorems 25 or 27 hold, then Ω^δ is B-continuous in ε= 0.

The following optimization problem satisfies the conditions of Theo- rem 25.

Example 29. Consider the problem inff(x, u), with u ∈ Mη(x), where f(x, u) =|x−u|, X = [−2,2] and Y =R. Let M_η(x) be the solutions set of the inequality considered in Example 12.

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It can be verified thatM_η^ε(x) = [0,√³

ε], for anyx∈X and Ω ={(0,0)}.

Also,

Ω^δ(ε) :=

n

(x, u)∈X×Y :u∈K(x, u)∩A, f(x, u)≤ inf

y∈X,v∈Mη(y)f(y, v) +δ,u∈[0,√³ ε]

o

=

=n

(x, u)∈X×Y :u∈K(x, u)∩A,|x−u| ≤δ,u∈[0,√³ ε]o

=

=n

(x, u)∈X×Y :|x−u| ≤δ,u∈[0,min{√³ ε,2}]o

and hence Ω^δ(0) = {(x,0) : |x| ≤ δ, x ∈ X}. It can be observed that Ω^δ is B-usc and hence, B-continuous at ε= 0.

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Received 6 February 2012 University of Bucharest

Faculty of Mathematics and Computer Science Str. Academiei 14

010014 Bucharest, Romania