Tykhonov Well-posedness for Quasi-equilibrium Problems

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Tykhonov Well-posedness for Quasi-equilibrium Problems M. Darabi · J. Zafarani

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Abstract We consider an extension of the notion of Tykhonov well-posedness for perturbed vector quasi-equilibrium problems. We establish some necessary and sufficient conditions for verifying these well-posedness properties. As for applications of our results, the Tykhonov well-posedness of vector variational-like inequalities and vector optimization problems are established Keywords Tykhonov well-posedness · Vector quasi-equilibrium problems · Vector quasi-variational inequalities · Vector optimization Mathematics Subject Classification (2000) 26B25 · 49J52 · 90C30 · 49J40

1 Introduction Well-posedness has played a crucial role in the stability analysis for optimization theory. This fact has motivated many authors to study the well-posedness of optimization problems. The first concept of well-posedness is due to Tykhonov [1] dealing with unconstrained optimization problems. Tykhonov well-posedness [1], requires the existence and uniqueness of the solution and convergence of each minimizing sequence to the solution. Levitin and Polyak [2] extended the notion to constrained case. A generalization of Tykhonov well-posedness, which is given for optimization problems having more than one solution, requires the existence and the convergence of some subsequences of every minimizing sequence towards a solution [3]. Another fundamental generalization of Tykhonov well-posedness for an optimization problem (in the scalar case) is the well-posedness by perturbations due to Zolezzi [4- 5]. His idea is to embedding original optimization problems into a family of lightly perturbed optimization problems depending on a parameter. There are various notions of well-posedness, for scalar and vector optimization M. Darabi Department of Mathematics, University of Isfahan, 81745-163 Isfahan, Iran E-mail: [email protected] J. Zafarani (B) Department of Mathematics, Sheikhbahaee University and University of Isfahan, Isfahan, Iran E-mail: [email protected]

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M. Darabi, J. Zafarani

problems [3 - 12], scalar and vector equilibrium problems [13- 15], scalar and vector variational inequality problems [16- 23], Nash equilibrium problems [24 -27] and generalized vector quasi-equilibrium problems [28 -30] and many other problems. For details, we refer the reader to the above mentioned sources and the references therein. Motivated and inspired by the above works, in this paper, we investigate some generalized vector versions of Tykhonov well-posedness for two class of parametric vector quasi-equilibrium problems. We also generalize a characterization of Tykhonov well-posedness given in [3] in terms of neighborhoods around solution set in our context. Furthermore, we conclude as applications of our results the well-posedness of vector variational-like inequalities and vector optimization problems. The outline of the paper is as follows: In Section 2, we introduce two new class of generalized parametric vector quasi-equilibrium problems and some preliminary results which are used in the sequel. In Section 3, we define the notion of uniquely well-posed under perturbations for quasi-equilibrium problems and investigate their basic properties. Section 4 deals with some special cases of wellposedness for vector variational inequalities and vector optimization problems.

2 Preliminaries In this section, we recall some definitions and preliminary results which are used in the next sections. Let X, Y, W, and Z be Hausdorff topological vector spaces and Λ, and P be Hausdorff topological spaces. Let A, B and D be nonempty sets of X, W and Z, respectively and C : X × Λ × P ⇒ Y be a set-valued map such that for any x ∈ X and for any λ ∈ Λ, C(x, λ, p) is a closed and convex pointed cone in Y such that int C(x, λ, p) 6= ∅. Assume that e : X × Λ × P −→ Y is a continuous vector valued map satisfying e(x, λ, p) ∈ intC(x, λ, p). Suppose that K1 : A × Λ ⇒ A, K2 : A × Λ ⇒ B, K3 : A × Λ × P ⇒ D and K4 : B × Λ × P ⇒ D are defined. When, W = X and A = B, we assume that for all x ∈ A, λ ∈ Λ and p ∈ P, we have K3 (x, λ, p) = K4 (x, λ, p). Let the machinery of the problems be expressed by F : A × B × D × P ⇒ Y . For any subsets A and B, we adopt the following notations (u, v) r1 A × B

means

∀u ∈ A, ∀v ∈ B,

(u, v) r2 A × B

means

∀u ∈ A, ∃v ∈ B,

(u, v) r3 A × B

means

∃u ∈ A, ∀v ∈ B,

and β1 (A, B)

means

A ⊆ B,

β2 (A, B)

means

A ∩ B 6= ∅.

For r ∈ {r1 , r2 , r3 }, β ∈ {β1 , β2 } and J ∈ {3, 4}, we consider the following para¯ p¯) ∈ Λ × P : metric vector quasi-equilibrium problems, for each (λ, (PJrβ )

¯ such that, (w, z) r K2 (¯ ¯ × KJ (¯ ¯ p¯), Find x ¯ ∈ clK1 (¯ x, λ) x, λ) x, λ,

Tykhonov Well-posedness . . .

3

¯ p¯)). β(F (¯ x, w, z, p¯), Y \ −intC(¯ x, λ, ¯ p¯). We denote the solution set of the above problem by SJrβ (λ, In this article we focus our attention on the case of J = 3, but most of our results for the case of J = 4 also hold. We mention now some special cases of the Problem (PJrβ ) : (i) For β = β1 , r = r1 , clK1 (x, λ), K1 (x, λ) = K2 (x, λ), C = Y \−intC(x, w, z, p) and F (x, y, µ) = F (x, w, z, p), we obtain (SQEP) in [?]. (ii) For β = β1 , r = r3 , clK1 (x, λ) = S(x) = K3 (x, λ, p), T (x) = K2 (x, λ), C(x) = Y \ −intC(x, w, z, p) and f (x, y, µ) = F (x, w, z, p), we obtain Problem (I) in [?], while for β = β2 we obtain their Problem (II). For β = β1 , , r = r3 , and C(x) = C(x, w, z, p), we obtain Problem (III) in [?], while for β = β2 we deduce their Problem (IV). (iii) For β = β1 , r = r1 , Λ = A = X, clK1 (x, λ) = S(x) = K2 (x, λ), T (x, x) = K3 (x, λ, p), C(x) = Y \ −intC(x, w, z, p) and F (s, x, y) = F (x, w, z, p), we deduce the Problem (GVQEP)(I) in [?], while for β = β2 , r = r2 we get their Problem (GVQEP)(II). For the case that, β = β1 , r = r1 , C(x) = C(x, w, z, p), we have Problem (GVQEP)(III) in [?], while for β = β2 , r = r2 we get their Problem (GVQEP)(IV). Problem are considered in [27 - 29] and [31-34]. Definition 2.1 Let T : X ⇒ Y be a set-valued map. Then, (a) T is said to be upper semi continuous (u.s.c.), iff for each closed set B ⊂ Y , T − (B) = {x ∈ X : T (x) ∩ B 6= ∅} is closed in X. (b) T is said to be lower semi continuous (l.s.c.), iff for each open set B ⊂ Y , T − (B) = {x ∈ X : T (x) ∩ B 6= ∅} is open in X. (c) T is said to be closed, iff the set Gr(T ) = {(x, y) ∈ X × Y : y ∈ T (x)} is closed in X × Y . (d) [35] T is said to be transfer closed valued iff for all (x, y) ∈ X × Y with y 6∈ T (x), there exists x ¯ ∈ X, such that y 6∈ clT (¯ x). It is clear that this definition is equivalent to: ∩x∈X T (x) = ∩x∈X clT (x). We call T is transfer open valued iff the set-valued map x ⇒ Y \ T (x) is transfer closed valued. A characterization of the upper (resp. lower) semicontinuity of a set-valued map in terms of nets is given in the following lemma (see, for example, [36], Theorems 17.16 and 17.19). Lemma 2.1 [36] Let X and Y be topological spaces and T : X ⇒ Y be a setvalued map. (i) If T has compact values, then T is u.s.c. iff for every net xα in X converging to x ∈ X and for any net yα with yα ∈ T (xα ), there exist y ∈ T (x) and a subnet yαi of yα converging to y. (ii) T is l.s.c. iff for any net xα in X converging to x ∈ X and each y ∈ T (x), there exists a net yα converging to y, with yα ∈ T (xα ), for all α. 2

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Let K be a subset of a Hausdorff topological vector space X. Then, we denote by hKi the family of all nonempty finite subsets of K and for each A ∈ hKi, its convex hull is denoted by conv (A). We denote [x, y] the line segment for x, y ∈ Y. The following fixed point theorem will be used in the sequel. Theorem 2.1 [37] Let K be a non-empty convex subset of a Hausdorff topological vector space X. Suppose that T : K ⇒ K is a set-valued map with convex valued such that the following conditions are satisfied: (a) ∀A ∈ hKi, T is transfer open-valued on conv (A); (b) ∀x, y ∈ K, [ [ − − int( T (z)) ∩ [x, y] = ( T (z)) ∩ [x, y]; z∈[x,y]

z∈[x,y]

(c) There exists a non-empty compact convex subset B of K and a non-empty compact subset D of K such that, for each y ∈ K \ D there exists x ∈ B such that x ∈ T (y). Then, T has a fixed point. 2 Let us recall the classical notion of well-posedness proposed by Tykhonov in [1], for a scalar optimization problem: min f (x), s.t. x ∈ X

(1)

where, X is a metric space and f : X → R. A sequence (xn ) ⊆ X is called a minimizing sequence for the optimization problem (1), when f (xn ) → inf X f as n → ∞. Definition 2.2 [1] The optimization problem (1) is called Tykhonov well-posed iff (i) there exists a unique solution x ¯ ∈ X of (1); (ii) every minimizing sequence converges to x ¯. An important characterization of Tykhonov well-posedness is obtained in [3]. Lemma 2.2 [3] The optimization problem (1) is Tykhonov well-posed iff there exists x ¯ ∈ X such that for every neighbourhood U of x ¯ there exists δ > 0 such that f (x)−f (¯ x) < δ =⇒ x ∈ U.

2

3 Well-posedness for Quasi-equilibrium Problems In this section, we define two class of generalized vector versions of Tykhonov well-posedness for parametric vector quasi-equilibrium problems. ¯ p¯). A net Definition 3.1 Let (λα , pα ) ⊆ Λ × P be a net converging to (λ, (xα ) ⊆ clK1 (xα , λα ) is said to be an asymptotically solving net corresponding to (λα , pα ), for Problem (PJrβ ), if there exists a net (εα ) ⊆ R+ such that εα −→ 0 and (w, z) r K2 (xα , λα ) × KJ (xα , λα , pα ), such that β(F (xα , w, z, pα ) + εα e(xα , λα , pα ), Y \ −intC(xα , λα , pα )).

Tykhonov Well-posedness . . .

5

Definition 3.2 The Problem (PJrβ ), is said to be uniquely well-posed iff (i) there exists only one solution for Problem (PJrβ ); ¯ p¯), every asymptotically solving (ii) for any net (λα , pα ) ⊆ Λ × P converging to (λ, ¯ p¯) net for Problem (PJrβ ) corresponding to (λα , pα ), converges to SJrβ (λ, Definition 3.3 The Problem (PJrβ ), is said to be well-posed in the general sense iff (i) there exists one solution for Problem (PJrβ ); ¯ p¯), every asymptotically solving (ii) for any net (λα , pα ) ⊆ Λ × P converging to (λ, net for Problem (PJrβ ) corresponding to (λα , pα ), contains a subnet converges to ¯ p¯). some points of SJrβ (λ, Obviously, if Problem (PJrβ ) is uniquely well-posed, then it is well-posed in the general sense. The above definitions improve Definitions 3.1 and 3.3 in [28] and Definitions 2.2 and 2.3 in [29]. Uniquely well-posedness and well-posedness in the general sense for Problem (PJrβ ), can be investigated and characterized via the notion of some approximate solutions of Problem (PJrβ ). In order to consider this case, let ε ∈ R+ given and let us introduce the set ΠJrβ (ε, λ, p) = {x ∈ clK1 (x, λ) : (w, z) r K2 (x, λ) × KJ (x, λ, p) β(F (x, w, z, p) + εe(x, λ, p), Y \ −intC(x, λ, p))}. The above family of sets are increasing, i.e., for all λ ∈ Λ and p ∈ P if 1 ≤ 2 , then ΠJrβ (ε1 , λ, p) ⊆ ΠJrβ (ε2 , λ, p), and we have the following equalities \ ¯ p¯) = ¯ p¯) = ΠJrβ (0, λ, ¯ p¯), SJrβ (λ, ΠJrβ (ε, λ, >0

In the next theorem and its corollary, we provide some alternative characterization for uniquely well-posedness and well-posedness in the general sense of Problem (PJrβ ). In fact we extend Lemma 2.2 for Problem (PJrβ ) in topological vector spaces. Theorem 3.1 Let X and Y be two topological vector spaces. If Problem (PJrβ ) is well-posed in the general sense, then there exists a nonempty and compact subset ¯ p¯) such that for every neighbourhood U of 0, there exists δ > 0 such H of SJrβ (λ, that ¯ p¯) ⇒ x ∈ H + U. x ∈ ΠJrβ (δ, λ, Conversely, if ¯ p¯) such that for every (i) there exists a nonempty compact subset H of SJrβ (λ, neighbourhood U of 0, there exists δ > 0 such that ¯ p¯) + U x ∈ ΠJrβ (δ, λ,

⇒

x ∈ H + U;

(ii) ΠJrβ , is jointly upper semi continuous in its second and third arguments. Then, Problem (PJrβ ), is well-posed in the general sense.

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M. Darabi, J. Zafarani

Proof Suppose Problem (PJrβ ) is well-posed in the general sense. Let H = ¯ p¯) 6= ∅. First, we show that SJrβ (λ, ¯ p¯) is compact. If (xα ) is a net in SJrβ (λ, ¯ p¯), that is a constant asymptotically solving net corresponding to (λα , pα ), SJrβ (λ, ¯ and pα = p¯. Therefore, there exists x0 ∈ SJrβ (λ, ¯ p¯) and a that for all α: λα = λ ¯ subnet (xβ ) of (xα ) that xβ −→ x0 and hence, SJrβ (λ, p¯) is compact. We prove now, the existence of a δ for every arbitrarily neighborhood of 0. On the contrary, assume there exist a neighborhood U of 0 and a net (δα ) ⊆ R+ such that δα −→ 0 ¯ p¯) and xα 6∈ H + U. and (xα ) ⊆ X such that xα ∈ ΠJrβ (δα , λ, ¯ and pα = p¯, We select the constant net (λα , pα ) such that for all α, λα = λ then, (xα ) is an asymptotically solving net corresponding to (λα , pα ) and by our ¯ p¯) such that xα −→ x0 , which is a contraassumption, there exists x0 ∈ SJrβ (λ, diction. ¯ p¯), every Conversely, we show that for any net (λα , pα ) ⊆ Λ × P converging to (λ, asymptotically solving net for Problem (PJrβ ) corresponding to (λα , pα ) contains a subnet converges to a solution of Problem (PJrβ ). Let (xα ) be an asymptotically solving net for Problem (PJrβ ) corresponding to (λα , pα ), then there exists (εα ) ⊆ R+ such that εα −→ 0 and for all α, xα ∈ ΠJrβ (εα , λα , pα ). To complete the proof, we show that (xα ) contains a subnet converges to a point x0 ∈ H. Otherwise, for all x ∈ H there exists a neighborhood Ux of 0 that {x} + Ux does not contain any subnet of xα . Also, for all S x ∈ H, there exists a neighborhood Vx of 0 such that Vx + Vx ⊆ Ux . Since H ⊆ x∈H ({x} + Vx ) and H is compact, there exists n ∈ N such that n [ H⊆ ({xi } + Vxi ). i=1

Let V =

Tn

i=1 Vxi , then by our assumption there exists δ > 0 such that

¯ p¯) + V x ∈ ΠJrβ (δ, λ,

⇒ x ∈ H + V.

On the other hand, there exists α1 such that for all α ≥ α1 , εα < δ and hence ΠJrβ (εα , λα , pα ) ⊆ ΠJrβ (δ, λα , pα ). Since ΠJrβ is jointly upper semi continuous in ¯ p¯), hence there exists α2 such its second and third arguments and (λα , pα ) −→ (λ, ¯ p¯)+V . Now, if α ≥ max{α1 , α2 }, that for all α ≥ α2 , ΠJrβ (δ, λα , pα ) ⊆ ΠJrβ (δ, λ, ¯ p¯) + V and therefore, xα ∈ H + V. But then xα ∈ ΠJrβ (δ, λ, H +V ⊆ ⊆

n [

({xi } + Vxi ) + V ⊆

n [

i=1 n [

i=1 n [

i=1

i=1

({xi } + Vxi + Vxi ) ⊆

({xi } + Vxi + V ) ({xi } + Uxi ).

S Hence, for those α, xα ∈ n i=1 ({xi } + Uxi ), which is a contradiction, since for all x ∈ H, {x}+Ux does not contain any subnet of xα . 2 Corollary 3.1 Let X and Y be two topological vector spaces. If Problem (PJrβ ), is uniquely well-posed, then there exists x0 ∈ X such that for every neighbourhood U of 0, there exists δ > 0 such that ¯ p¯) x ∈ ΠJrβ (δ, λ,

⇒

x ∈ {x0 } + U.

Tykhonov Well-posedness . . .

7

Conversely, if (i) there exists x0 ∈ X such that for every neighbourhood U of 0, there exists δ > 0 such that ¯ p¯) + U ⇒ x ∈ {x0 } + U ; x ∈ ΠJrβ (δ, λ, ¯ p¯), is nonempty; (ii) SJrβ (λ, (iii) ΠJrβ , is jointly upper semi continuous in its second and third arguments. Then, Problem (PJrβ ) is uniquely well-posed. Here, we consider sufficient conditions for the existence of a solution of Problem (PJrβ ). In the rest of this section, we suppose that X and P are the same. Motivated by an idea of [38], let us define the set-valued map Lβ : X × Λ ⇒ X by Lβ (p, λ) = 2X \ {x ∈ X : (y, z) r K2 (x, λ) × KJ (x, λ, p), β(F (x, y, z, p), Y \ −intC(x, λ, p))}. We say that the pair (Lβ , K1 ) satisfies the coercivity condition if there exist a non-empty compact convex subset B1 × B2 ⊆ X × Λ and a non-empty compact subset D1 × D2 ⊆ X × Λ such that, for each (x, λ) ∈ X × Λ \ (D1 × D2 ), there ¯ exists (x1 , λ1 ) ∈ B1 × B2 such that x1 ∈ clK1 (x, λ) ∩ Lβ (x, λ), if x ∈ E(λ) and ¯ ¯ x1 ∈ clK1 (x, λ), if x 6∈ E(λ), where E(λ) is the closure of the fixed points of K1 (., λ). Agarwal, Balaj and O0 Regan in [39] by using a fixed point theorem unify the existence of solution for variational relation problem. In the following theorem, we improve their result and give a sufficient condition for the existence of solution of (PJrβ ). Theorem 3.2 Suppose that (i) clK1 and convLβ are transfer open valued and K1 is convex valued; ¯ (ii) x 6∈ convLβ (x, λ), for all x ∈ E(λ); (iii) for all x1 , x2 ∈ X [ \ [ \ int( B(´ x)) [x1 , x2 ] = ( B(´ x)) [x1 , x2 ], x ´∈[x1 ,x2 ]

x ´∈[x1 ,x2 ]

T S T ¯ where, B(´ x) = [(clK1 −1 (´ x) convLβ −1 (´ x))] [(X \ E(λ)) (clK1 −1 (´ x)]; (iv) The pair (convLβ , clK1 ) satisfies the coercivity condition. Then, there exists a solution of Problem (PJrβ ). Proof In order to prove Theorem, we will show the existence of an element ¯ 0 ) and clK1 (x0 , λ0 )∩Lβ (x0 , λ0 ) = ∅. Suppose, (x0 , λ0 ) ∈ X ×Λ such that x0 ∈ E(λ this is not the case, then, we have ¯ clK1 (x, λ) ∩ Lβ (x, λ) 6= ∅, ∀x ∈ E(λ), and consequently ¯ clK1 (x, λ) ∩ convLβ (x, λ) 6= ∅, ∀x ∈ E(λ). Let us define the set-valued map ϕ : X × Λ ⇒ X × Λ, by   (clK1 (x, λ) ∩ conv Lβ (x, λ)) × Λ, if ϕ(x, λ) =  clK1 (x, λ) × Λ, if

¯ x ∈ E(λ), ¯ x 6∈ E(λ).

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M. Darabi, J. Zafarani

Since clK1 and convLβ are transfer open valued, then ϕ is transfer open valued. By Theorem 2.1, the set-valued map ϕ has a fixed point, denoted by (x0 , λ0 ). Thus, (x0 , λ0 ) ∈ X × Λ and (x0 , λ0 ) ∈ ϕ(x0 , λ0 ). This proves that λ0 ∈ Λ and x0 ∈ clK1 (x0 , λ0 , ). From the definition of ϕ, this implies that x0 ∈ convLβ (x0 , λ0 ), which contradicts our condition (ii). 2 Theorem 3.2, presents a sufficient condition for existence of solution of Problem (PJrβ ), which is a generalization of Lemma 3.1 of Sach [38]. In fact, in [38] the author assumes that Lβ has open lower sections. In the next example. we justify our claim. Example 3.1 Let K1 , K2 : [0, 1] × [0, 1] ⇒ [0, 1], K3 : [0, 1] × [0, 1] × [0, 1] ⇒ [0, 1] and F : [0, 1] × [0, 1] × [0, 1] × [0, 1] ⇒ R be defined by  1 if λ ∈ Q, x ∈ [0, 14 ] ∪ {1},   ]0, 2 [,    K1 (x, λ) = { 18 }, if x 6= 81 , λ ∈ Qc ,      {1}, o.w., K2 (x, λ) = [0, 1], ∀ x, λ ∈ [0, 1],

K3 (x, λ, p) =

  {x}, 

F (x, y, z, p) =

] 12 , 1[,

  ]0, +∞[, 

] − ∞, 0[,

λ ∈ Q, x ∈ [0, 41 ],

if

o.w., if

x ∈ [0, 41 ], x = z, o.w.,

and for all x, λ, p ∈ [0, 1], C(x, λ, p) =]0, +∞[, where Q is the set of all rational numbers in [0, 1]. Now for β = β1 and r = r2 , one has  1 if λ ∈ Q,  ] 4 , 1], Lβ (p, λ) =  [0,1], o.w., therefore, convLβ (p, λ) = Lβ (p, λ) and for all λ, p ∈ [0, 1] and convLβ (p, λ) is transfer open valued. Obviously,  if λ ∈ Q,  [0, 41 ], ¯ E(λ) =  ∅, o.w., ¯ and for all x ∈ E(λ), we obtain x 6∈ conv Lβ (p, λ). But   Lβ −1 (0) = {(p, λ) ∈ [0, 1] × [0, 1] : 0 ∈ Lβ (p, λ)} = [0, 1] × [0, 1] ∩ Qc , which is not an open set. If 1 1 B1 × B2 = [ , 1] × [0, 1] and D1 × D2 = [ , 1] × [0, 1], 8 40

Tykhonov Well-posedness . . .

9

then, (convLβ , clK1 ) satisfies coercivity condition. If (x, λ) ∈ (X × Λ) \ (D1 × D2 ), 1 ¯ then, x ∈ [0, 40 [. If x ∈ E(λ), then, y1 = 12 ∈ clK1 (x, λ) ∩ Lβ (λ, p), otherwise, 1 we can choose y1 = 8 ∈ clK1 (x, λ). Furtheremore, one can easily deduce that ¯ S3r2 β1 (λ, p) = E(λ).

4 Applications Thorough out this section, we suppose A = X = W = B and D = Z = L(X, Y ), where L(X, Y ) is the space of all continuous linear maps from X into Y provided with the pointwise convergence topology. Here, we will go into further details for two special cases of Problem (PJrβ ). (a) Suppos T : X × Λ × P ⇒ L(X, Y ) is a set-valued map, g : X × P −→ X and η : X × X −→ X. If for any x ∈ X, λ ∈ Λ and p ∈ P define K1 (x, λ) = K2 (x, λ), K3 (x, λ, p) = T (x, λ, p) and F (x, y, z, p) =< z, η(y, g(x, p)) >, then by assuming r = r2 and β = β2 , the Problem (P3rβ ) becomes perturbed vector Stampacchia quasi-variational inequality (in short, (V SQI)(λ, ¯ p) ¯ ) ¯ ¯ ∃z ∈ T (¯ ¯ p¯) : x ¯ ∈ V S (λ, ¯ ∈ clK1 (¯ x, λ), ∀y ∈ K1 (¯ x, λ) x, λ, ¯ p) ¯ (T, K1 ) ⇐⇒ x ¯ p¯). < z, η(y, g(¯ x, p¯)) >6∈ −intC(¯ x, λ, Furthermore, by assuming r = r1 and β = β1 , and K3 (x, λ, p) = T (x, λ, p), the Problem (P3rβ ) becomes perturbed vector Minty quasivariational inequality (in short, (V M QI)(λ, ¯ p) ¯ ) ¯ : ∀y ∈ K1 (¯ ¯ ¯ p¯) x ¯ ∈ V M (λ, ¯ ∈ clK1 (¯ x, λ) x, λ), ∀z ∈ T (y, λ, ¯ p) ¯ (T, K1 ) ⇐⇒ x ¯ p¯). < z, η(y, g(¯ x, p¯)) >6∈ −intC(¯ x, λ, We denote by V S (λ, ¯ p) ¯ (T, K1 ) the solution set of the vector Stampacchia quasivariational inequality and by V M (λ, ¯ p) ¯ (T, K1 ), the solution set of the vector Minty quasi-variational inequality, respectively. ¯ ∈ Λ, (b) If, for a fixed p¯ ∈ P and for all x ¯ ∈ X and λ ¯ = K3 (¯ ¯ p¯), K1 (¯ x, λ) x, λ, and for x, y, z ∈ A, λ ∈ Λ, define F (x, y, z, λ) = K2 (z, λ) − {y}, by assume r = r3 and β = β1 , the Problem (P3rβ ) becomes ¯ : ∃y ∈ K2 (¯ ¯ such that ∃¯ x ∈ clK1 (¯ x, λ) x, λ) ¯ K2 (z, λ) ¯ − {y} ⊆ Y \ −intC(¯ ¯ p¯), ∀z ∈ K1 (¯ x, λ) x, λ, which is perturbed optimization problem (in short, (V OP )(λ, ¯ p) ¯ ) for set-valued map K2 . We denote by (V OP )(λ, (K , K ), the solution set of vector optimiza2 1 ¯ p) ¯ tion problem. The equivalence between solutions of Minty variational inequalities and solutions of optimization problems has been obtained first by Giannessi[40]. Thereafter, this equivalence has been studied and generalized under different conditions

10

M. Darabi, J. Zafarani

by many authors; see [41-45] and references therein. But there are many examples of vector optimization problems whose solutions are not a solution of Minty vector variational-like inequality [46]. The next example shows the importance of perturbed Minty vector variational-like inequalities. Example 4.1 Let Λ = X = R, K = [−1, 1], P = [2, +∞[ and for all x, λ ∈ R, defined C(x, λ) = R+ × R+ and f be defined by f = (f1 , f2 ), where

f1 (x) =

  −x2 + x, 

0,

if

x > 0, f2 (x) =

if

x ≤ 0,

  x + 1,

if

x > 0,

x2 ,

if

x ≤ 0.



The limiting subdifferential of f is    (−2x + 1, 1),    ∂L f (x) = {(k, t) : k ∈ [0, 1], t ∈ [0, +∞[},      (0, 2x),

if

x > 0,

if

x = 0,

if

x < 0.

Let η : X × X −→ X and T : R × P ⇒ R be defined by  if x > 0, y > 0 or x < 0, y < 0,  x − y, η(x, y) =  1 − y − x, o. w., T (y, p) = ∂L f (y) + (p(1 − y), p(1 − y)), then, the perturbed Minty vector variational-like inequalities defined by < ∂L f (y), η(y, x) > +hp(1 − y), η(y, x)i 6⊆ (R × R) \ (R+ × R+ ). It is easy to show that x = 0 is a solution of (V OP ) and perturbed Minty vector variational-like inequality, but is not a solution of Minty vector variational-like inequality. For the perturbed version of the above definitions of (V SV I)(λ, ¯ p) ¯ p) ¯ , (V M V I)(λ, ¯ and (V OP )(λ, , we define ¯ p) ¯ (ε, λ, p) = {¯ x ∈ clK1 (¯ x, λ) : ∀y ∈ K1 (¯ x, λ) ∃z ∈ T (¯ x, λ, p) : ΠV S (λ, ¯ p) ¯ < z, η(y, g(¯ x, p)) > +εe(¯ x, λ, p) 6∈ −intC(¯ x, λ, p)}; (ε, λ, p) = {¯ x ∈ clK1 (¯ x, λ) : ∀y ∈ K1 (¯ x, λ) ∀z ∈ T (y, λ, p) : ΠV M (λ, ¯ p) ¯ < z, η(y, g(¯ x, p)) > +εe(¯ x, λ, p) 6∈ −intC(¯ x, λ, p)}; ΠV OP (λ, (ε, λ, p) = {¯ x ∈ clK1 (¯ x, λ) : ∃¯ z ∈ K2 (¯ x, λ) ∀y ∈ K1 (¯ x, λ) ¯ p) ¯ K2 (y, λ) − z¯ + εe(¯ x, λ, p) ⊆ Y \ −intC(¯ x, λ, p)}.

Tykhonov Well-posedness . . .

11

¯ p¯). A net Definition 4.1 Let (λα , pα ) ⊆ Λ × P be a net converging to (λ, (xα ) ⊆ clK1 (xα , λα ) is said to be an asymptotically solving net corresponding to (λα , pα ), for: + (a) (V SV I)(λ, such that εα −→ 0 and for all ¯ p) ¯ iff there exists a net (εα ) ⊆ R y ∈ K1 (xα , λα ) there exists z ∈ T (xα , λα , pα ) such that < z, η(y, g(xα , pα )) > +εα e(xα , λα , pα )) 6∈ −intC(xα , λα , pα ), + (b) ((V M V I)(λ, ¯ p) ¯ ) iff there exists a net (εα ) ⊆ R such that εα −→ 0 and for all y ∈ K1 (xα , λα ) and for all z ∈ T (y, λα , pα ),

< z, η(y, g(xα , pα )) > +εα e(xα , λα , pα ) 6∈ −intC(xα , λα , pα ), + (c) (V OP )(λ, ¯ p) ¯ iff there exists a net (εα ) ⊆ R such that εα −→ 0 and there exists z ∈ K2 (xα , λα ) such that for all y ∈ K1 (xα , λα )

K2 (y, λα ) − z + εe(xα , λα , pα ) ⊆ Y \ −intC(xα , λα , pα ). Definition 4.2 The (V SV I)(λ, ¯ p) ¯ p) ¯ p) ¯ (resp. (V M V I)(λ, ¯ , (V OP )(λ, ¯ ) is said to be uniquely well-posed if (i) there exists only one solution of (V SV I)(λ, ¯ p) ¯ p) ¯ p) ¯ (resp. (V M V I)(λ, ¯ , (V OP )(λ, ¯ ); ¯ (ii) for any net (λα , pα ) ⊆ Λ × P converging to (λ, p¯), every asymptotically solving net for (V SV I)(λ, ¯ p) ¯ p) ¯ p) ¯ (resp. (V M V I)(λ, ¯ , (V OP )(λ, ¯ ) corresponding to (λα , pα ), converges to a solution of V S (λ, ¯ p) ¯ p) ¯ p) ¯ (T, K1 ) (resp. V M (λ, ¯ (T, K1 ), (V OP )(λ, ¯ (K2 , K1 )). Definition 4.3 The (V SV I)(λ, ¯ p) ¯ p) ¯ p) ¯ (resp. (V M V I)(λ, ¯ , (V OP )(λ, ¯ ), is said to be well-posed in the general sense iff (i) there exists one solution of (V SV I)(λ, ¯ p) ¯ p) ¯ p) ¯ (resp. (V M V I)(λ, ¯ , (V OP )(λ, ¯ ); ¯ p¯), every asymptotically solving (ii) for any net (λα , pα ) ⊆ Λ × P converging to (λ, net for (V SV I)(λ, ¯ p) ¯ p) ¯ p) ¯ (resp. (V M V I)(λ, ¯ , (V OP )(λ, ¯ ) corresponding to (λα , pα ), contains a subnet that converges to some points of V S (λ, ¯ p) ¯ p) ¯ p) ¯ (T, K1 ) (resp. V M (λ, ¯ (T, K1 ), (V OP )(λ, ¯ (K2 , K1 )). Remark 4.1 Let X and Y be two topological vector spaces, from Theorem 3.1 and Corollary 3.1, we can conclude that: (a1) If (V SV I)(λ, ¯ p) ¯ p) ¯ p) ¯ (resp. (V M V I)(λ, ¯ or (V OP )(λ, ¯ ) is uniquely well-posed, then there exists x0 ∈ X such that for every neighbourhood U of 0, there exists δ > 0 such that ¯ p¯) ⇒ x ∈ {x0 } + U ; x ∈ ΠV S (λ, (δ, λ, ¯ p) ¯   ¯ p¯), or x ∈ ΠV OP ¯ (δ, λ, ¯ p¯) ⇒ x ∈ {x0 } + U . resp. x ∈ ΠV M (λ, (δ, λ, ¯ p) ¯ (λ,p) ¯ (a2) If (V SV I)(λ, ¯ p) ¯ p) ¯ p) ¯ (resp.(V M V I)(λ, ¯ or (V OP )(λ, ¯ ), is well-posed in the general sense, then there exists a nonempty compact subset H ⊆ V S (λ, ¯ p) ¯ p) ¯ p) ¯ (T, K1 ) (resp. H ⊆ V M (λ, ¯ (T, K1 ) or H ⊆ (V OP )(λ, ¯ (K2 , K1 ))

12

M. Darabi, J. Zafarani

such that for every neighbourhood U of 0, there exists δ > 0 such that ¯ p¯) ⇒ x ∈ H + U ; x ∈ ΠV S (λ, (δ, λ, ¯ p) ¯ 

 ¯ p¯) or x ∈ ΠV OP ¯ (δ, λ, ¯ p¯) ⇒ x ∈ H + U . resp. x ∈ ΠV M (λ, (δ, λ, ¯ p) ¯ (λ,p) ¯

In the following we establish the well-posedness of (V M V I)(λ, ¯ p) ¯ . Theorem 4.1 Assume solution set for ((V M V I)(λ, ¯ p) ¯ ) is nonempty and the following assumptions are satisfied: (i) η and g are continuous functions; ¯ (ii) K1 is lower semi continuous on A × {λ}; ¯ (iii) T is lower semi continuous on A × {λ} × {¯ p} and has strongly bounded range in L(X, Y ); ¯ and E( ¯ is compact; ¯ is upper semi continuous on λ ¯ λ) (iv) E (v) W (x, λ, p) = Y \ −intC(x, λ, p) is a closed map; (vi) there exists x0 ∈ X such that for every neighbourhood U of 0, there exists δ > 0 such that ¯ p¯) ⇒ x ∈ {x0 } + U. x ∈ ΠV M (λ, (δ, λ, ¯ p) ¯ Then, ((V M V I)(λ, ¯ p) ¯ ) is uniquely well-posed. Proof By Corollary 3.1, it suffices to show that ΠV M (λ, is jointly upper semi ¯ p) ¯ continuous in its second and third arguments. Fix an ε > 0 and suppose the ¯ p¯), a net (λα , pα ) ⊆ Λ × P with existence of an open superset G of ΠV M (λ, (ε, λ, ¯ p) ¯ ¯ (ε, λα , pα ), such that xα 6∈ G, for (λα , pα ) −→ (λ, p¯) and a net (xα ) in ΠV M (λ, ¯ p) ¯ ¯ and compactness of E( ¯ one can ¯ at λ ¯ λ), all α. By the upper semicontinuity of E ¯ ¯ ¯ assume that xα −→ x0 for some x0 ∈ E(λ). If x0 6∈ ΠV M (λ, (ε, λ, p¯), then there ¯ p) ¯ ¯ and z0 ∈ T (y0 , λ, ¯ p¯) such that exist y0 ∈ K1 (x0 , λ) ¯ p¯) 6∈ W (x0 , λ, ¯ p¯). < z0 , η(y0 , g(x0 , p¯)) > +εe(x0 , λ,

(2)

The lower semi continuity of K1 implies the existence of a net yα ∈ K1 (xα , λα ) such that yα −→ y0 and the lower semi continuity of T implies the existence of a net zα ∈ T (yα , λα , pα ) such that zα −→ z0 . On the other hand, xα ∈ ΠV M (λ, (ε, λα , pα ), thus, we have ¯ p) ¯ < zα , η(yα , g(xα , pα )) > +εe(xα , λα , pα ) ∈ W (xα , λα , pα ). Since W is a closed map and η, e and g are continuous, from condition (iii) and Proposition 2.3 in [47], we deduce ¯ p¯) ∈ W (x0 , λ, ¯ p¯), < z0 , η(y0 , g(x0 , p¯)) > +εe(x0 , λ, ¯ p¯) ⊆ G, so there exthat contradicts equation (2). Therefore, x0 ∈ ΠV M (λ, (ε, λ, ¯ p) ¯ ists α0 such that for all α ≥ α0 , we have xα ∈ G, which is a contradiction. 2 Remark 4.2 (a) In the previous theorem, if we replace condition (iii) by the following condition: ¯ (iii)0 T is compact valued and upper semi continuous on A × {λ}. Then, with minor modifications in the proof, one can obtain the well-posedness of (V SV I)(λ, ¯ p) ¯ .

Tykhonov Well-posedness . . .

13

(b) In Theorem 4.1, if we replace condition (vi) by the following condition: (vi)0 there exists a nonempty compact subset H of V M (λ, ¯ p) ¯ (T, K1 ) such that for every neighbourhood U of 0, there exists δ > 0 such that ¯ p¯) + U x ∈ ΠV M (λ, (δ, λ, ¯ p) ¯

⇒ x ∈ H + U.

Then, we can deduce that (V M V I)(λ, ¯ p) ¯ is well-posed in the general sense. (c) When X and Y are Hausdorff locally convex and X is barreled, condition (iii) can be replaced by the boundedness of the range of T in L(X, Y ) with the pointwise convergenc topology (see; [48], Proposition 23.7 and Theorem 24.11). Trivially this will be the case if T is compact. By a similar proof as that of Theorem 4.1, we can obtain the following result for the well-posedness of (V OP )(λ, ¯ p) ¯ . Corollary 4.1 Assume solution set for ((V OP )(λ, ¯ p) ¯ ) is nonempty and ¯ (i) K1 is lower semi continuous on A × {λ}; ¯ (ii) K2 is compact valued and upper semi continuous on A × {λ}; ¯ and E( ¯ is compact; ¯ is upper semi continuous on λ ¯ λ) (iii) E (iv) W (x, λ, p) = Y \ −intC(x, λ, p) is a closed map; (v) there exists x0 ∈ X such that for every neighbourhood U of 0, there exists δ > 0 such that ¯ p¯) ⇒ x ∈ {x0 } + U. (δ, λ, x ∈ ΠV OP (λ, ¯ p) ¯ 2

Then, (V OP )(λ, ¯ p) ¯ , is uniquely well-posed.

Remark 4.3 In the previous result, if we replace condition (v) by the following condition: (v)0 there exists a nonempty compact subset H of (V OP )(λ, ¯ p) ¯ (K2 , K1 ) such that for every neighbourhood U of 0, there exists δ > 0 such that ¯ p¯) + U x ∈ ΠV OP (λ, (δ, λ, ¯ p) ¯

⇒ x ∈ H + U.

Then, (V OP )(λ, ¯ p) ¯ is well-posed in the general sense. In order to obtain a Minty type result for perturbed vector quasi-variational inequality, we need some notions. We give a new version of upper sign continuity due to Pini and Bianchi [49] and Farajzade and Zafarani [50] and a version of monotonicity. Definition 4.4 Let A ⊆ X be convex and ε ∈ R+ . Then a set-valued map T : A × Λ × P ⇒ Y , is said to be (a) (η, ε)-upper sign continuous type B iff, for all x, y ∈ X, λ ∈ Λ and p ∈ P , the following implication holds: ∀u ∈]x, y[, ∀u∗ ∈ T (u, λ, p) < u∗ , η(y, g(x, p)) > +εe(x, λ, p) 6∈ −intC(u, λ, p) =⇒ ∃x∗ ∈ T (x, λ, p) :< x∗ , η(y, g(x, p)) > +εe(x, λ, p) 6∈ −intC(x, λ, p). (b) (η, ε)-pseudomonotone iff for all λ ∈ Λ, p ∈ P , x, y ∈ K1 (x, λ) and for all x∗ ∈ T (x, λ, p), y ∗ ∈ T (y, λ, p), the following implication holds: < x∗ , η(y, g(x, p)) > +εe(x, λ, p) 6∈ −intC(x, λ, p) =⇒

14

M. Darabi, J. Zafarani

< y ∗ , η(y, g(x, p)) > +εe(x, λ, p) 6∈ −intC(x, λ, p). If ε = 0, then the above definition of (η, ε)-pseudomonotonicity reduces to η-pseudomonotonicity of T defined in [45]. Definition 4.5 The set-valued map C : X × Λ × P ⇒ Y is called segmentary nested, iff for all λ ∈ Λ, p ∈ P and t1 , t2 ∈ [0, 1] that t1 ≤ t2 , C(xt2 , λ, p) ⊆ C(xt1 , λ, p). where, xti = (1 − ti )x0 + ti y0 , i ∈ {1, 2} and x0 and y0 are elements in X. When C is a constant convex cone, then it is trivially satisfied the segmentary nested condition. In the following result, we obtain a Minty’s type theorem for uniquely wellposedness of our quasi-variational inequality. Theorem 4.2 Assume that (i) η is affine in the first argument; (ii) for any x ∈ X and p ∈ P , η(x, g(x, p)) = 0; (iii) T is η-upper sign continuous; (iv) T is (η, ε)-pseudomonotone, for all ε ≥ 0; (v) C is segmentary nested; (vi) K1 is convex valued function. Then, uniquely well-posed of vector Minty quasi-variational inequality and the uniquely well-posed of vector Stampacchia quasi-variational inequality coincide. ¯ Proof Suppose x0 is a solution of (V M V I)(λ, ¯ p) ¯ , then x0 ∈ clK1 (x0 , λ) and for all ¯ and y ∈ K1 (x0 , λ) ¯ p¯), < y ∗ , η(y, g(x0 , p¯)) >6∈ −intC(x0 , λ, ¯ p¯). ∀y ∗ ∈ T (y, λ,

(3)

Assume to the contrary that x0 isn0 t a solution of (V SV I)(λ, ¯ p) ¯ , then ¯ : < T (x0 , λ, ¯ p¯), η(y0 , g(x0 , p¯)) >⊆ −intC(x0 , λ, ¯ p¯). ∃y0 ∈ K1 (x0 , λ) ¯ by We set xt = (1 − t)x0 + ty0 for t ∈]0, 1], which is an element of K1 (x0 , λ) assumption (vi). The (η, 0)-upper sign continuity of T , implies that ¯ p¯) : < xt ∗ , η(y0 , g(x0 , p¯)) >∈ −intC(xt , λ, ¯ p¯). ∃t ∈]0, 1[, ∃xt ∗ ∈ T (xt , λ,

(4)

¯ p¯) By using assumptions (i), (v) and equation (4), for xt ∗ ∈ T (xt , λ, < xt ∗ , η(xt , g(x0 , p¯)) >=< xt ∗ , η((1 − t)x0 + ty0 , g(x0 , p¯)) > = (1 − t) < xt ∗ , η(x0 , g(x0 , p¯)) > +t < xt ∗ , η(y0 , g(x0 , p¯)) > ¯ p¯) ⊆ −intC(x0 , λ, ¯ p¯), = t < xt ∗ , η(y0 , g(x0 , p¯)) >∈ −intC(xt , λ, that is in contradiction with (3). Therefore, any solution of (V M V I)(λ, ¯ p) ¯ is a solution of (V SV I)(λ, . ¯ p) ¯ Conversely, since T is (η, ε)-pseudomonotone, the uniquely well-posedness of

Tykhonov Well-posedness . . .

(V SV I)(λ, ¯ p) ¯ p) ¯ implies the uniquely well-posedness of (V M V I)(λ, ¯ .

15

2

Remark 4.4 In the previous theorem, if we replace conditions (iii) and (v) by the following condition: (iii)0 T is (η, ε)-upper hemicontinuous, for all ε ≥ 0, i.e. for all x, y ∈ A, the map t ∈ [0, 1] 7−→< T ((1 − t)x + ty), η(y, x) > +εe(x, λ, p), is upper semi continuous at 0+ . Then, with minor modifications in the proof of Theorem 4.2, one can obtain the result.

5 Concluding Remarks Here we consider the Tykhonov well-posedness for a generalized vector quasiequilibrium problem. As consequences of our results we deduce the Tykhonove well-psedness of vector variational inequalities and vector optimization problems. There are some other interesting kinds of well-posedness, namely B and M-wellposedness, Levitin-Polyak well-posedness and Hadamard well-posedness. It would be interesting to characterize these kinds of well-posedness for our Problem (PJrβ ). Acknowledgement The authors are gratefull to the reviewers for valuable comments and remarks.

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