Ky Fan

 

J Optim Theory Appl (2011) 149: 540–553 DOI 10.1007/s10957-011-9803 10.1007/s10957-011-9803-9 -9

Hölder Continuity of Solutions to Parametric Weak Generalized Ky Fan Inequality S.J. Li X.B. Li   ·

Published online: 21 January 2011 © Springer Science+Business Media, LLC 2011

Abstract   In this this pa pape perr, by usin using g a scal scalar ariz izat atio ion n tech techni nique que,, we obta obtain in suf suffic ficie ient nt cond condiitions for Hölder continuity of the solution mapping for a parametric weak generalized Ky Fan Inequality in the case where the solution mapping is a general set-valued one.

The result is different from the recent ones in the literature.

 ·

 ·

 ·

Keywords  Ky Fan Inequality  Solution mapping  Hölder continuity  Hausdorff  distance  Scalarization

 ·

1 Intr Introductio oduction n

The Ky Fan Inequality is a very general mathematical format, which embraces the formats of several disciplines, as those for equilibrium problems of Mathematical Physics, those from Game Theory, those from (Vector) Optimization and (Vector) Variational Inequalities, and so on. Since Ky Fan Inequality was introduced in [ 1], it has been extended and generalized to vector-valued mappings. The Ky Fan Inequality for a vector valued mapping is known as the generalized Ky Fan Inequality. In the literature, existence results for various types of generalized Ky Fan Inequalities

This research was partially supported by the National Natural Science Foundation of China (Grant number: 10871216), the Ph.D. Programs Foundation of Ministry of Education of China (Grant number: 20100191120043) and Chongqing University Postgraduates Science and Innovation Fund (Project Number: 201005B1A0010338). The authors thank the anonymous referees for valuable comments and suggestions, which helped to improve the paper. The authors also thank Professor F. Giannessi for helpful comments on a final version of this paper.

 ·

S.J. Li ()  X.B. Li College of Mathematics and Statistics, Chongqing University, Chongqing 401331, China e-mail: [email protected] e-mail:  [email protected] X.B. Li e-mail: [email protected] e-mail:  [email protected]

 

J Optim Theory Appl (2011) 149: 540–553

541

have been investigated intensively; see [2 [2–5] and the references therein, where the generalized Ky Fan Inequalities are called generalized systems or vector equilibrium problems. It is well known that stability analysis of solution mapping for Ky Fan Inequalities is an important topic for optimization theory and its applications. Many papers discussed the semicontinuity of solution mappings (see [6 [ 6–8]); however, only a few results have concerned concerned with the Hölde Hölderr conti continuity nuity of solution solution mappings for perturbed variational inequalities and Ky Fan Inequalities. Yen [9 [ 9] obtained the Hölder continuity of the unique solution of a classic perturbed variational inequality by the metric projection method. Mansour and Riahi [10 [ 10]] proved the Hölder continuity of  the unique solution for a parametric Ky Fan Inequality under the concepts of strong monotonicity and Hölder continuity. Bianchi and Pini [11 [11]] introduced the concept of  strong pseudomonotonicity and got the Hölder continuity of the unique solution of  a parametric Ky Fan Inequality. Anh and Khanh [12 [ 12]] generalized the main results of [11] 11] to two classes of perturbed generalized Ky Fan Inequalities with set-valued mappings. Anh and Khanh [13 [13]] further further discussed discussed uniquenes uniquenesss and Hölder Hölder continuity continuity of the solutions for perturbed Ky Fan Inequalities with set-valued mappings. Anh [13]] to the case of perturbed quasi Ky Fan and Khanh [14 [14]] extended the results of [13 Inequalities with set-valued mappings. Obviously, all these results for Hölder continuity of the solutions are obtained in the case where the solution is unique and under very strong conditions. For general perturbed (generalized) Ky Fan Inequalities, it is well known that a solution mapping is, in general, a set-valued one. Under the Hausdorff distance and the strong quasimonotonicity, Lee et al. [15 [ 15]] first showed that the set-valued solution mapping for a parametric vector variational inequality is Hölder continuous. Recently, by virtue of the strong quasimonotonicity, Mansour and Aussel [16] 16] discu discussed ssed the Hölde Hölderr conti continuity nuity of setset-val valued ued solution mappings for a parametparametric generalized variational inequalities. Then, Li et al. [17 [17]] introduced an assumption, which is weaker than the corresponding ones in the literature, and established the Hölder continuity of the set-valued solution mappings for two classes of parametric generalized Ky Fan Inequalities with set-valued mappings in general metric spaces. Motivated by the work reported in [9 [ 9,  10,  10 ,  12  12– –17], 17], this paper aims at establishing the Hölder continuity of a set-valued solution mapping to a parametric weak generalized Fan Ky Inequality, by using a scalarization technique, which is different from the ones in literature. Our method is based on a scalarization representation of the solution mapping for a parametric Ky Fan Inequality and a property involving the union of a family of Hölder continuous set-valued mappings on a compact set. Of  course, the main consequences of our results are different from corresponding ones [ 17], ], in [17] 17] and overcome the drawback of the assumption (ii) of Theorem 3.1 in [17 which requires the knowledge of detailed values of the solution mapping in a neighborhood of the point under consideration. Moreover, we compare our results with the corresponding ones in [13 [13,,  14  14]] and give some examples to illustrate the application of our results. results. The rest of the paper is organized as follows. In Sect. 2 Sect. 2,, we introduce a parametric weak generalized Ky Fan Inequality, and define a weak solution and a   ξ -solution to the parametric weak generalized Ky Fan Inequality. Then, we introduce a Hölder strongly monotonicity and a Hölder continuity with respect to an interior point of a

 

542

J Optim Theory Appl (2011) 149: 540–553

fixed cone. In Sect. 3 Sect.  3,, we discuss the Hölder continuity of the solution mapping for the parametric weak generalized Ky Fan Inequality.

2 Preli Preliminaries minaries

  ·    and   d (·, ·)  denote the norm and metric in any normed space and metric δspace,   B( 0,interior δ)   denotes  ≥ 0, intrespectively. C stands for the inte rior of  C . closed ball with centre 0 ∈  X   and radius  Cthe In the sequel,

Throug Thro ughou houtt this this pape paperr, if not not othe otherw rwis isee sp spec ecifi ified ed,, X,,M   will will de deno note te th thre reee metr metric ic ∗ spaces, and  Y  a linear normed space. Let  Y   be the topological dual space of  Y   Y . For ∗ any   ξ   Y  ,  we introduce ξ   sup ξ , x  1 ,  where ξ , x  denotes the x value of   ξ   at   y.   Let   C  Y  be a pointed, closed and convex cone with int C . Let   C ∗ ξ   Y ∗ ξ , y  0, y  C  be the dual cone of   C.  Since int C ,  the dual cone   C ∗   of   C  has a weak* compact base. Let   e  int C . Then,   Be∗ ξ   C ∗ ξ, e  1  is a weak* compact base of  C  C ∗ . Let   N (λ0 )     and   N (µ0 )  M  be neighborhoods of considered points   λ0   and µ0 , respectively. Let   K   ⇒ X  be a set-valued mapping and   f   X X M   Y  be a vector-valued mapping. For each   λ  N (λ0 )   and   µ  N (µ0 ), we consider the following parametric weak generalized Ky Fan Inequality: Find  x 0 K(λ), such that

  ∈∈    := {  :   = }  ⊂  := {  ∈  ∈  :   ≥  ∀  ∈ }  ∈  = }  ⊂  ⊂  :  ∈  ∈ f (x0 ,y,µ)

∈   −int C,  ∀y ∈ K(λ).

  

 =   ∅  =   ∅  := {  ∈  ∈  :

  ::  ×  ×  →  →  ∈

 

(1)

∈ N (λ0) and  µ ∈ N (µ0), the weak solution set of (1 ( 1) is    −int C,  ∀y ∈ K(λ)}. S W W  (λ,µ) := {x ∈ K(λ) : f(x,y,µ) ∈  ∈ C ∗ \ {0}, λ ∈ N (λ0) and  µ ∈ N (µ0), the ξ -solution set of (1(1) is For each  ξ  ∈ S(ξ,λ,µ) := {x ∈ K(λ) : ξ,f(x,y,µ)  ≥ 0,  ∀ y ∈ K(λ)}. Definition 2.1   [18 18]] Let   X   and   Y   be two topological spaces, and   G :  X ⇒ Y   be a

For each  λ

set-valued mapping.

 ∈

(i)   G   is said to be upper semicontinuous at   x0  X  iff for every open set   U   with G(x0 )   U , there is a neighborhood   N (x0 )   of   x0   in   X   such that   G(x)  U , x N (x0 ). (ii)   G is said to be lower semicontinuous at  x 0  iff for every open set  U  with  G(x0 ) , there is a neighborhood  N (x0 )  of  x , x  x 0   in  X  such that  G(x) U  U  N (x0 ).

 ⊆

 ⊆

∀  ∈   =   ∅

∩ ∩   =   ∅  ∀  ∈

Definition 2.2   [19] 19] Let   (E,d)  be a metric space and   H   be a Hausdorff metric on the collection   CB (E)  of all nonempty, closed and bounded subsets of   E , which is defined as H(A,B)

:= max sup d(a,B), sup d(A,b) , ∀A, B ∈ C B (E), a A

where  d(  d(a, a, B)

b B

∈ ∈ := inf b∈B d(a,b)  and  d(A,b) := inf a∈A d(a,b).





 

J Optim Theory Appl (2011) 149: 540–553

543

:

Definition 2.3  A set-valued mapping G M  ⇒ Y  is  is said to be .α -Hölder continuous at  µ 0  iff there is a neighborhood  U (µ0 )  of  µ  µ 0 , such that, µ1 , µ2 U (µ0 ), G(µ1 )

where  

 ∀

 ∈

⊆ G(µ2) + B(0, d α (µ1, µ2)),

 

(2)

≥ 0 and α > 0.

 Remark 2.1  G   If   is a set-valued mapping with nonempty, closed and bounded values, then (2 (2) is equivalent to

≤ d α (µ1, µ2). vectorr-va value lued d map mappin ping g   g :   X ×   X →   Y    is Definiti Defi nition on 2.4   [14] 14] A vecto    y, h.β -Hölder strongly monotone on  K  ⊂ X  iff  ∀x, y ∈ K : x = g(x,y) + g(y,x) + hd β (x,y)B(0, 1) ⊂ −C, H(G(µ1 ),G(µ2 ))

call called ed to be

where  h > 0 and  β > 0. From [20 [20], ], we recall the following  h.β -Hölder strong monotonicity property.

 :  ×  →  ∈  ⊂  ∀  ∈  :  =   g(x,y) + g(y,x) + hd β (x,y)e ∈ −C,

Definition 2.5  A vector-valued mapping  g  X X Definition  Y  is called to be   h.β -Hölder strongly monotone with respect to  e  int C   on  K X  iff  x, y K x y,

where  h > 0 and  β > 0.

 :  →  →

Definition 2.6  A vector-valued mapping  g M  Y  is said to be  .α -Hölder continuous with respect to   e  int C   at   µ0  iff, there is a neighborhood   U (µ0 )   of   µ0   such that, µ1 , µ2 U (µ0 ),

 ∀

 ∈

 ∈

g(µ1 )

g(µ2 )

d α (µ1 , µ2 )

e, e ,

∈ + [− ] where   ≥ 0, α > 0 and [−e, e] := {x : x ∈ e − C   and  x  ∈ −e + C }. Whether e ≤ 1 or e ≥ 1, Definitions 2.4 Definitions  2.4 and  and 2.5  2.5 do  do not coincide, as shown by the following example.

1, 2  Example 2.1   Let   X R, Y  R2 , C R2+ . And let   K  X ,   e1  ( 12 ,   43 ) int R2+ , e2 (2, 3)  int R2+ , and f (x,y) (2x (y x ), 3y(x y)). Then, there exist h  4 and  β  2, such that x, y K x y ,

=

 =   ==  = =  =  ∀  ∈  :  =   f(x,y) + f(y,x) + hd β (x,y)e1  = ∈

(2x(y

x ), 3y(x

y))

(2y (x

 = [ ] ⊂ − −

y), 3x (y

x))

4x

 ∈

 =

y 2e

== (−2(x  −− y)2, −3 (x− − y)+2) + 4|x − − y |2e  = −(0, 0) ∈+−|R 2−, | 1 +

1

 

544

J Optim Theory Appl (2011) 149: 540–553

 ∈ int R2+

which implies that   f   is 4.2-Hölder strongly monotone with respect to   e1 on  K . However, it clearly follows that

+ f(y,x) + hd β (x,y)B(0, 1) = (2x(y − x), 3y (x − y)) + (2y(x − y), 3x(y − x)) + 4|x − y |2B(0, 1) = (−2(x − y )2, −3(x − y)2) + 4|x − y |2B(0, 1) ⊂   −R2+.

f(x,y)

Therefore,   f   is not 4.2-Hölder strongly monotone on   K . Moreover,   f   is obviously 2.2-Hölder strongly monotone on   K,   but not 2.2-Hölder strongly monotone with respect to  e 2   on  K . In [14], 14], Anh and Khanh introduced the assumptions   (A2r1 ϕ )  and   (A2r2 ϕ ).  Under the assumptions   (A2r1 ϕ )   and   (A2r2 ϕ ), they obtained the Hölder continuity of the unique solutions in some neighborhood for two classes of parametric generalized quasi Ky Fan Inequalities with set-valued mappings, respectively. When   f    X int C , the X  Y  is a vector-valued mapping without parameter and  ϕ(C)  Y  assumptions (A2r1 ϕ )  and  (A2r2 ϕ )  collapse to the following one: x, y K x y ,

 →

  :  ×

 =  \ \ −

 ∀  ∈  :  =   hd β (x,y) ≤ d(f(x,y),Y  \ −int C) + d(f(y,x),Y  \ −int C).  

(3)

different from the  h.β -Hölder strongly monotonicity Of course, assumption (3 (3) is different of   f   and the   h.β -Hölder strongly monotonicity of   f   with respect to   e  int C , respectively spectiv ely.. Let us show that, when  f   is h.β -Hölder strongly monotone or  h.β -Hölder (3) may not hold. strongly monotone with respect to  e  int C , then assumption (3

 ∈

 ∈

Example  2.1.. Let  e (1, 1)  int R2+   and  Example 2.2   Let X  X,, Y, C,K  be given as in Example 2.1 f(x,y)  (x(y  x), y( y(x x  y)).  For all   x , y  K  x  y ,  taking   h  1, β  2,   we have

 =

 −

 −

 ∈  :  =  

 =

 =

∈

 =

+ f(y,x) + hd β (x,y)B(0, 1) = (x(y − x),y(x − y)) + (y(x − y),x(y − x )) + |x − y |2B(0, 1) = (−(x − y)2, −(x − y)2) + |x − y |2B(0, 1) ⊂ −R2+,

f(x,y)

i.e.,   f    is 1.2-Hö 2-Hölde lderr strongl strongly y monoto monotone ne on   K.   Obviously,   f    is also 1.2-Hölder strongly monotone with respect to  e  on  K  K.. However, assumption (3 (3) does not hold. Indeed, let  x  1 and  y  2.  Direct computation shows that  d(  d(f f (1, 2), Y  int R2+ )  0 and  d(  d(f f (2, 1), Y  int R2+ )  0 . But for any   h > 0 and   α > 0, hd α (1, 2)  h > 0 .  Therefore, assumption (3 (3) is not satisfied.

 \ \ −

From Lemma 3.1 of [8 [8], we have

 =

 =

 =

 =  \ \ −

 =

 

J Optim Theory Appl (2011) 149: 540–553

 ∈

545

 ∈

Lemma 2.1  If for each  each   µ  N (µ0 )   and   x  K(N(λ0 )),  f(x,K(N(λ0 )),µ) )),µ) is  is a Cconvex set ,  i .e.,  f(x,K(N(λ0 )),µ) C  is a convex set ,  then S W W  (λ,µ)

=



+

S(ξ,λ,µ)

 =

S(ξ,λ,µ).

ξ  Be∗

ξ  C ∗

∈

∈ \{0}

From Proposition 3.1 in [8 [ 8], we can easily get the following results, which play an important role in this paper. Lemma 2.2  Suppose that the following conditions hold :

∈  ∈ ;  ∈

 ∈

∈ ;

(i)   For each µ each  µ N (µ0 )  and each x each  x K(N(λ0 )),  f (x,x, µ) C (ii)   F For or each   µ   N (µ0 )   and each each   y  K(N(λ0 )), f( ,y,µ)   is con contin tinuo uous us on K(N(λ0 )) (iii)   For For each   x  K(N(λ0 ))   and ea each ch   µ   N (µ0 ), f(x, , µ)   is C-c C-con onve vexx on 0, 1 , tf (x,y, µ) K(N(λ0 )),   i.e.,   for for an anyy   y, z   K(N(λ0 ))   and  and    t  (1 t)f(x,z,µ) f(x,ty (1 t)z,µ) C (iv)   For each λ each  λ N (λ0 ), K(λ) is K(λ)  is a nonempty,  compact and convex set .

−

·

 ∈

 ∈  + −

∈

∈

 ∈

·  ∈ [ ]

+ ;

 +

Then  (λ,µ) N (λ0 ) N (µ0 ), S(ξ,λ,µ) . (a)   For each ξ  each  ξ  Be∗ C ∗   and  (λ,µ) (b)   For each (λ,µ) each  (λ,µ) N (λ0 ) N (µ0 ), S W (λ,µ) is (λ,µ)  is a nonempty compact set . W  

 ∈  ⊂  ∈ ∈

∈

×

×

=   ∅

(a) From From the the pr proo ooff of Pr Prop opos osit itio ions ns 3.1 3.1 in [8], th this is re resu sult lt hold holds. s. (b (b)) For For Proof   (a) each   λ  N (λ0 ),   µ  N (µ0 )   and   x  K(N(λ0 )),  since  K(λ)   is nonempty, convex and  f(x, , µ)  is C-convex on  K(N(λ0 )), f (x, K(N( K(N(λ λ0 )),µ)  C  is a convex set. Thus, by virtue of Lemma 2.1 Lemma  2.1  and (a), we have   S W W  (λ,µ) ξ ∈C ∗ \{0} S(ξ,λ,µ) [17], ], we can .   Arguing as in the proof of Proposition 3.1 of [17 ξ ∈Be∗ S(ξ,λ,µ)    show that  S W W  (λ,µ)  is compact.

 ∈



 ∈

·

 ∈

 +   =

 =   ∅

 =

3 Hölder Contin Continuity uity

In this section, we mainly discuss the Hölder continuity of the solution sets to (1 ( 1). To obtain the Hölder continuity of the solution mapping around   (λ0 , µ0 ), for each pairs   (λ1 , µ1 ),(λ2 , µ2 )   in   N (λ0 )  N (µ0 )   of   (λ0 , µ0 ), we introduce the following assumptions:

 ×

·  ∈

 ∈

(H 0 ) K ( )  is  .β -Hölder continuous at  λ 0 . (H 1 )   For each µ N (µ0 ), f ( , , µ) is  h.α -Hölder strongly monotone with respect to  e  int C   in  K(N(λ0 )). (H 2 )   For each µ N (µ0 )  and  x K(N(λ0 )) )),, f (x, , µ) is  n.δ -Hölder continuous with respect to  e  int C   in  K(N(λ0 )). (H 3 )   For each x , y K(N(λ0 )),  f (x,y, )  is  m.γ -Hölder continuous with respect to  e  int C   at  µ 0 .

 ∈

∈ ∈

 ∈

 ∈

··

··  ∈

·

·

 ×  

(H 4 )   The function function   f ( , , µ0 )   is bounded on   K(N(λ0 ))  K(N(λ0 )), i.e., there exists  l > 0 such that for any  x , y K(N(λ0 )), one has f( x ,y ,µ0 ) l.

 ∈

≤

 

546

J Optim Theory Appl (2011) 149: 540–553

 ×

Lemma 3.1   Suppose that   that   N (λ0 )  N (µ0 )  is the given neighborhood of   (λ0 , µ0 ). Under assumptio assumptions ns (H   (H 1 )  and  (H   (H 2 ),  the following properties hold :  for each ξ  each  ξ  Be∗ ,

 ∈  ∈

ξ,f(x,y,µ) + ξ,f(y,x,µ) + hd α (x,y) ≤ 0 and 

|ξ , f ( x , y1, µ) − ξ , f ( x , y2, µ)| ≤ nd δ (y1, y2).  ∈ Be∗ ⊂ C ∗ , one has ξ,f(x,y,µ) + f(y,x,µ) + hd α (x,y)e ≤ 0. Proof  For any ξ  ∈ Thus Th us,, by the the line linear arit ity y of   ξ    and   ξ , e =   1,   ξ,f(x,y,µ)  +  ξ,f(y,x,µ)  +  ∈ Be∗ , hd α (x,y) ≤ 0 holds. Arguing the same way, for each  y1 , y2 ∈ K(N(λ0 )) and  ξ  ∈ we can get

−nd δ (y1, y2) ≤ ξ , f ( x , y1, µ) − ξ , f ( x , y2, µ) ≤ nd δ (y1, y2) implying  

ξ , f ( x , y1, µ) − ξ , f ( x , y2, µ)| ≤ nd δ (y1, y2).



Lemma 3. Lemma 3.2 2   Supp Suppose ose that assu assumpti mptions ons (H 3 ) and  and (H  (H 4 ) ar aree sati satisfied  sfied . Then, ther theree exists exists a constant   ll 0 l > 0,  such that  ξ 1 , ξ 2 B ∗ ,

 ≥

 ∈

 ∀

e

|ξ 1, f ( x , y , µ1) − ξ 2, f ( x , y , µ2)| ≤ md γ (µ1, µ2) + l0ξ 1 − ξ 2. Proof   By assumption  (H 4 ), we concl conclude ude that l

 := sup(x,y)∈K(N(λ ))×K(N(λ ))f( x ,y ,µ0) < +∞. 0

0

·

Since  f (x,y, )  is Hölder continuous at  µ 0 , for any fixed    > 0, there exits a neighborhood of  µ  µ 0 , denoted without loss of generality by  N (µ0 ), such that,



 ≤ l +  < +∞.

sup(x,y,µ)∈K(N(λ0 ))×K(N(λ0 ))×N (µ0 ) f(x,y,µ) Take  l 0

 max l, l

 :=

 , which implies that  l 0

l . Then,

{f(x,y,µ)  + }  ≤ l , ∀x, y ∈ K(N(λ  ≥ )), ∀µ ∈ N (µ ). 0 0 0

 

(4)

Now we show that the conclusion holds. Arguing as in the proof of Lemma  3.1,  3.1 , for each ξ  Be∗ ,  we have

  ∈∈ −md γ (µ1, µ2) ≤ ξ , f ( x , y , µ1) − ξ , f ( x , y , µ2) ≤ md γ (µ1, µ2), that is,   |ξ , f ( x , y , µ1 ) − ξ , f ( x , y , µ2 )| ≤   md γ  (µ1 , µ2 ).   Therefore, Therefore, for every every ξ 1 , ξ 2 ∈ Be∗ ,  it follows from (4 (4) that |ξ 1, f ( x , y , µ1) − ξ 2, f ( x , y , µ2)| ξ  , f ( x , y , µ )

ξ  , f ( x , y , µ )

ξ  , f ( x , y , µ )

ξ  , f ( x , y , µ )

1 2 − 2 ≤≤ md  | 1γ (µ , µ ) +1 f( − x ,y1 ,µ )ξ   2−|+| γ  ξ 2  ≤ md  (µ1 , µ2 ) + l0 ξ 1 − ξ 2 . 1 2 2 1

2

 

|



 

J Optim Theory Appl (2011) 149: 540–553

547

 Remark 3.1   If the following conditions are satisfied: (i) there exists a compact set A    such that   N (λ0 )  A,  K ( )  is upper semicontinuou semicontinuouss with compact values values on A; (ii)  f ( , , µ0 )  is continuous on  K(A) K(A), then  (H 4 )  is fulfilled. Indeed, since A  is compact and K  K(( ) is upper semicontinuous with compact values on A , it follows from Proposition 11 in Sect. 1 of Chap. 3 [18 [ 18]] that the sets K(A) and K(A) K(A)  are compact. Noting that  f ( , , µ0 )  is continuous on  K(A) K(A), we have

 ⊂

 ⊂

··

·

·

×

×

··

×

 := sup(x,y)∈K(A)×K(A)f( x ,y ,µ0) < +∞. Obviously, sup(x,y)∈K(N(λ ))×K(N(λ )) f( x ,y ,µ0 ) ≤ l  since N (λ0 ) is a subset of  A. l

0

0

Combing Lemmas 3.1 Lemmas 3.1,, 3.2  3.2 and  and Theorem 2.2.1 in [10 [10], ], we can obtain the following result. Lemma 3.3  Assume that for each ξ  each  ξ  Be∗ ,  the  the ξ   ξ -solution -solution set  S(ξ,   S(ξ, λ,µ) λ,µ) for   for   (1 ( 1)  exists in a neighborhood   N (λ0 )  N (µ0 )   of the considered point   (λ0 , µ0 ).  Furthermore, assume that assumptions   (H 0 ) – (H  (H 4 )   hold .  Then,  for any ξ  ξ   B e∗ ,   there exist open  µ 0 ,  such that , the  the ξ   ξ -solution -solution  λ 0   and  N   N ¯ (µ0 )  of  µ neighborhoods N  neighborhoods  N  (ξ )  of  ξ , N ¯ (λ0 )  of  λ

 ×

 ∈  ∈

  ¯ ∈  ∈

¯  ¯ ξξ   ξ  ξ     set   S (·, ·, ·)   on on   N  (ξ¯ ) × N ¯ (λ0 ) × N ¯ (µ0 )  is a singleton,  and satisfies the following ξξ   ξξ   condition,  for all (ξ  all  (ξ 1 , λ1 , µ1 ),(ξ 2 , λ2 , µ2 ) ∈ N  (ξ¯ ) × N ξξ ¯  (λ0 ) × N ξξ ¯  (µ0 ) : d(x(ξ 1 , λ1 , µ1 ),x(ξ 2 , λ2 , µ2 ))

 ≤

  l0 d  (µ1 , µ2 ) + ξ 1 − ξ 2 h h

m

γ 

 +   1 α

δ

2 n

1 α

h

βδ α

d  (λ1 , λ2 ),

 

(5)

∈ S (ξ i , λi , µi ), i = 1, 2. ¯  ∈  ∈  Be∗,   let   N (ξ¯ ) × N ξ ξ¯ (λ0) × N ξξ ¯ (µ0) ⊂ Be∗ ×  N (λ0) × N (µ0)   be Proof   For any  ξξ  ). Obviously, for each  ( open (where  N ¯ (λ0 ), N ¯ (µ0 ) depend on ¯ξ  ξ  ).  (ξ, ξ, λ,µ) ∈ N  (ξ¯ ) × ξξ   ξξ   N  (λ0 ) N  (µ0 ), S(  S(ξ, ξ, λ,µ) is nonempty. Fix (ξ,  (ξ, λ,µ) N  (ξ¯ ) N  (λ0 ) N  (µ0 ). ξ  ξ  ξ  ξ  If  x  ¯x 0 ∈ S(ξ,λ,µ), × ¯  then ∈ × ¯ × ¯ ξ , f ( x0,y,µ) ≥ 0, ∀y ∈ K(λ).   (6)

where x where  x (ξ i , λi , µi )

By virtue of Lemma 3.1 Lemma  3.1,, one has

ξ , f ( x0,y,µ) + ξ , f ( y , x0, µ) + hd α (x0, y ) ≤ 0, ∀y ∈ K(λ) \ {x0}.

  (7)

(7) together yield that Hence, (6 (6) and (7

ξ , f ( y , x0, µ) < 0, ∀y ∈ K(λ) \ {x0}. Therefore,   y

 K(λ)

 ∈

(1); the   ξ -solution uniqueness folx0  is not a   ξ -solution of (1

 \ { }

lows. Arguing as in the proof of Theorem 2.2.1 in [10 [ 10], ], we have (5 (5) by Lemmas 3.1 Lemmas  3.1    and 3.2 and  3.2..

 

548

J Optim Theory Appl (2011) 149: 540–553

Theorem 3.1  Assume that for each ξ  Be∗ , the ξ -solution ξ -solution set S(ξ,λ,µ) set S(ξ,λ,µ) for   for   (1 (1) exists in a neighborhood   N (λ0 )  N (µ0 )  of the considered point   point   (λ0 , µ0 ). Furthermore ).  Furthermore, assume that assumptio assumptions ns (H   (H 0 ) – (H  (H 4 ) hold . If for each x each  x K(N(λ0 )) )) and   and  µ  µ N (µ0 ), f(x, , µ) µ) is  is C-convex on K(N(λ on K(N(λ0 )), )), then  then there exist neighborhoods N (λ0 ) of   λ λ0  and  such h th that  at , th thee weak weak solu soluti tion on se set  t S  S W N (µ0 ) of  of µ µ0 , suc nonem empty pty, W  ( , ) on N (λ0 ) N (µ0 ) is non and satisfies the following condition,  for all (λ all  (λ 1 , µ1 ),(λ2 , µ2 ) N (λ0 ) N (µ0 )

 ∈  ∈

 ×

 ∈

   · ·   ×  ∈  ×   + 

·



1

S W W  (λ1 , µ1 )

⊂ S W W (λ2, µ2) +

  m h

α

∈

1

δ

d γ α (µ1 , µ2 )

:

α

2 n h

α (λ , λ ) B( 0, 1). d βδ 1 2

(8)

 { { ¯ }

.3,, is an open Lemma  3.3 Proof  Since the system of  N  (ξ ) ξ  ξ¯ ∈Be∗ , which are given by Lemma 3 covering of the weak* compact set  B e∗ ,  there exist  ξ 1 , ξ 2 , . . . , ξn   from  B e∗  such that n

∗ ⊂   =    :=  =

 

(9)

n i 1 N ξ i (µ0 ).

Then N (λ0 )   and

N  (ξ i ).

Be

i 1

   :=  =   ∈ ×{  }    

Hence, let N (λ0 )

n i 1 N ξ i (λ0 )   and



N (µ0 )



 

N (µ0 )   are desired neighborhoods of   λ0   and   µ0 ,   respectively. Indeed, let  (λ,µ) N (λ0 ) N (µ0 )   be given arbitrarily. For any   ξ    Be∗ ,   by virtue of (9 (9), there ex ists   i0 1, 2, . . . , n  such that   ξ   N  (ξ i0 ).  From the construction of the neighborhoods N (λ0 )   and N (µ0 ),   one has   (λ,µ)  N ξ i (λ0 )  N ξ i (µ0 ).  Then, according

 ∈

 ∈  ∈

 ∈  ∈

 ∈

 ×

0

0

3.3,, the   ξ -solution   S(ξ,λ,µ)  is a nonempty singleton. Hence, in view to Lemma   3.3 of Lemma  Lemma   2.1 2.1,,   S W ( 8) W  (λ,µ) ξ ∈B ∗ S(ξ,λ,µ)  is nonempty. Now, we show that (8

 =



e

∈

  ×    +

holds. Indeed, taking any (λ1 , µ1 ),(λ2 , µ2 ) N (λ0 ) N (µ0 ), we need to show that for any  x 1 S W S W W  (λ1 , µ1 ),  there exists  x 2 W  (λ2 , µ2 )  satisfying

 ∈

 ∈

d (x1 , x2 )

Since  x 1

  ≤ m h

S W W  (λ1 , µ1 )

 ∈

1 α

ξ  Be∗

=

γ  α

d  (µ1 , µ2 )

2δ n h

1 α

βδ

 

d  α (λ1 , λ2 ).

S ξ  ξ (λ1 , µ1 ),  there exists ξ 

(10)

Be∗  such that

  :=∈

 ¯ ∈ ¯ x1 x (ξ , λ1 , µ1 ) ∈ S ξ  ξ¯ (λ1 , µ1 ). Thanks to (9 (9), there exists   i0 ∈ {1, 2, . . . , n}   such that  ξ   ∈   N (ξ i ).   Thus, by the ξ¯  ∈ construction of the neighborhoods N (λ0 )   and N (µ0 ),  we have   (λ1 , µ1 ),(λ2 , µ2 ) ∈  that Lemma  3.3 that N ξξ   (λ0 ) × N ξξ   (µ0 ).  Then, it follows from Lemma 3.3 i0

i0

d(x(ξ¯ , λ1 , µ1 ),x(ξ¯ , λ2 , µ2 ))

      ≤ m h

1 α

γ  α

d  (µ1 , µ2 )

0

  + 2δ n h

 := x(ξ¯ , λ2, µ2). Then, (10 (10)) holds and the proof is complete.

Let  x 2

1 α

βδ α

d  (λ1 , λ2 ).  



 Remark 3.2  The Hölder related assumption   (H 3 ), which is different from the corresponding ones in [13 [13,,   14], 14], does not depend on   x, y  as in [11 [11,,   12 12,,   20 20]. ]. Based on

 

J Optim Theory Appl (2011) 149: 540–553

549

comparing our main result with the corresponding ones of [13 [ 13,,   14 14], ], we agree with the opinion that the dependence of   x, y   helps to sharpen the Hölder continuity re3.1 is  is replaced by the sult (concerning Hölder degree). Indeed, if   (H 3 )  in Theorem  Theorem   3.1 following one:  (H 3 ), x, y K(N(λ0 )) x y , µ1 , µ2 N (µ0 ),

 ∀  ∈ :  =    ∀  ∈ f( x ,y ,µ1 ) ∈ f( x ,y ,µ2 ) + md θ (x,y)d γ  (µ1 , µ2 )[−e, e],   where 0 < θ < α, α,

and 3.1 that,  that, (λ1 , µ1 ),(λ2 , µ2 ) then it follows from Theorems 2.1 of [13 [ 13]] and 3.1 N (µ0 ),

 ∀



 ⊂ S WW  (λ2, µ2)

S W W  (λ1 , µ1 )

  + δ

2 n

  + m

  1 α θ 

−

h

1 α

h

γ  α θ 

N (λ0 )

∈

 

×

d  − (µ1 , µ2 )



βδ α

d  (λ1 , λ2 ) B(0, 1),

··

which sharpens remarkably the Hölder degree of   S  S W W  ( , ). Corollary Corolla ry 3.1   Su Supp ppos osee th that at al alll co cond ndit itio ions ns of Le Lemm mma a 2.2 and assu assumpti mptions ons (H 0 ) – (H  (H 4 ) hold .  Then,  there exist neighborhoods N (λ0 )   of   λ0   and  N (µ0 )   of   µ0 ,  such that ,   the

··

  ×    ≤

W  ( , )  on N (λ0 ) weak solution set   S  S W N (µ0 )  is nonempty with compact values ,  and  satisfies the following condition,  for all (λ all  (λ 1 , µ1 ),(λ2 , µ2 ) N (λ0 ) N (µ0 ):

m

H (S W W  (λ1 , µ1 ), S W W  (λ2 , µ2 ))

 ∈

1 α

γ  α

d  (µ1 , µ2 )

h

 ∈  ×    + 2l δ n

1 α

βδ

d  α (λ1 , λ2 ).   (11)

h

 ×

Lemma   2.2 2.2,,   S W Proof   For each  (λ,µ)  N (λ0 )  N (µ0 ), by Lemma  W  (λ,µ)  is a nonempty,    compact set. Then, by Remark  2.1 and  2.1  and Theorem 3.1 Theorem  3.1,, we have (11 (11). ). strong generalized Ky Fan Inequality: Inequality:  Remark 3.3   Consider the following parametric strong Find  x 0 K(λ), such that

 ∈

f (x ,y,µ) 0

0 ,

C

y

∈   − \{ }  ∀  ∈

K(λ).

ξ   Y ∗ ξ , y  >  0, For any  (λ,µ),   S(λ,µ)   stands for its solution set, and   C  3.1 hold,  hold, it follows from  and all assumptions of Corollary   3.1 y  C 0 . If   C  Theorem 2.1 in [21 [21]] that  clS(λ,µ)  S W cl   denotes the closure of  W  (λ,µ), where   cl . Since  H (1 , 2 ) H(cl1 ,cl2 )  for any nonempty set   1 , 2  Y , it follows from (11 (11)) that  S ( , )  satisfies

∀  ∈ \{ }}

··

=   ∅ =

H(S(λ1 , µ1 ),S(λ2 , µ2 ))

:= {  ∈  ∈  :  

 =

  ≤ m h

 ⊂

1 α

γ  α

  + δ

d  (µ1 , µ2 )

2l n h

1 α

βδ α

d  (λ1 , λ2 ).

The following example is to illustrate that the  C -conve -convexity xity of   f  f  is essential.   R+ ,   Example 3.1   Let   X   Y    R, C   M  0, 1 ,K(λ) 1, 2   and f(x,y,λ) λy(x y) . Direct computation shows that  S W ( 0 ) 1 , 2  and  S W W   W  (λ)

=

 −   =   =

 =

 =   = [ ]= [ ] = [ ] =

 

550

J Optim Theory Appl (2011) 149: 540–553

{2} for any λ ∈ (0, 1]. Clearly, we see that  S W W (·) is even not l.s.c. at λ = 0. The main

reason is that   f   is not   C -convex. Hence, the   C -convexity of   f   in Theorem  Theorem   3.1 3.1   (or  3.1)) is essential. Corollary 3.1 Corollary  Remark 3.4  Under the assumptions   (A2r1 ϕ )  and   (A2r2 ϕ )), Anh and Khanh [14 [14]] obtained the Hölder continuity of the unique solutions in some neighborhood for two classes of parametric generalized Ky Fan Inequalities with set-valued mappings, re-

spectively. However, in general, it is well known that a solution mapping for perturbed (generalized) Ky Fan Inequalities is a set-valued one. In Theorem   3.1   and 3.1,, we discussed this case and established the Hölder continuity of the Corollary   3.1 Corollary solution mapping for (1 (1). Thus, Theorem 3.1 Theorem  3.1 (or  (or Corollary 3.1 Corollary  3.1)) is new and different from the results in recent literature. The following example is given to illustrate the case that Theorem  Theorem   3.1 3.1 (or  (or Corollary  Corollary   3.1 3.1)) is applicable, but Theorem 2.1 of [14 [14]] (or [13]) 13]) is not applicable. 1, 2 , C R2+   and   e  ( 1, 1)  int R2+ .  Example 3.2   Let   X  Y  R2+ ,   M  For any given   λ    M ,   let  K(λ) x  (x1 , x2 )  (x1  3)2  (x2  3)2  λ2 , λτ x1   λx2 and for every   x   (x1 , x2 )   X , let   T (x,λ) ,   where the constant   τ  λx   λx

√  −

24 7 21   35

 =   ==  ∈  =  =  ∈

 =  =  = [ ]  =  = {  =  :  −  =



> 1.

 =

 =

For any x (x1 , x2 ), y (y1 , y2 ) ric vector variational inequalities:

  −  =

T (x,λ) x,λ),, y

x

=

1

 =  ∈ +  − ≤ }  =  =



2

∈ X, λ ∈ , we consider the following paramet-

λτ x1   λx2 λx1   λx2



 − x1  − x2

y1 y2

∈−  /

−

int R2+ .

·  :  − × ;

Letting f(x,y,λ) T(x,λ),y x , we sh sho ow th that at:: (i (i)) Ob Obvi viou ousl sly y, K ( ) is 1.1-Hölder continuous with compact convex values on  ,  i.e., assumption  (H 0 )  holds; (ii)     is compact. Direct computation shows that  K() x  (x1 , x2 )  (x1  3)2  (x2 3)2  4 . It is clearly that  f ( , , ) is continuous on  K() K()   f (x,x, λ) (0, 0) R2+  it can be checked that for each   λ    and   x  K(  K(), ), f (x, , λ)  is   R2+ convex on  K(). Therefore, all conditions of Lemma 2.2 Lemma  2.2  are satisfied. Namely, the solution set S W W  (λ) is nonempty; (iii) Assumption (H 1 ) is satisfied. Indeed, there exist h  min τ , 1  1 and  α  2 such that

≤ }  ∈ ;

:=

 = {  = ×  ∈  ∈

···

{ }=

·

+  − =

 =

+ f(y,x,λ) + hd α (x,y)e = ((1 − λτ)(y1 − x1)2 + (1 − λ)(y2 − x2)2, (1 − λ)(y1 − x1 )2 + (1 − λ)(y2 − x2 )2 ) ∈ −int R2+, ∀x , y ∈ K(), x =   y ; √  √   = 1, we have (iv) Taking  m := (8 + 4 2) max{τ , 1} = (8 + 4 2)τ > 0 and  γ  = f( x ,y ,λ1 ) − f( x ,y ,λ2 ) f(x,y,λ)

(λ

λ )(τx (y

x )

x (y

=∈ md(λ − , λ )e −  −, ∀λ ,+λ  ∈ . − 1 2 + 1 2 1

2

1

1 2 R

1

2

2

x )

x ), x (y 2

1

 −

1

1

x ))

x (y

+

2

 −

2

2

 

J Optim Theory Appl (2011) 149: 540–553

551

 ∈

Then, assumption   (H 3 )  holds since the role of   λ1 , λ2    are symmetric; (v) Now we verify that assumption   (H 2 )  is also satisfied. Indeed, let   n  2 2max τ , 1 2 2τ > 0 and  δ   21 .  Then, one has

√ 

 =

√   :=

{ }=

− f( x ,y2, λ) ∈ 2τ (|y1  − y2 | + |y1  − y2 |)e − R2+ ⊂ 2√ 2τ  ((yy1  − y2 )2 + (y1  − y2 )2e − R2+ = nd  (y1, y2)e − R2+, ∀x ∈ K(), ∀λ ∈ .

f( x ,y1 , λ)

1

1 2

 

1

1

2

2

1

2

2

1

Obviously, the relation  f (x ,y1 , λ) f( x ,y2 , λ) nd 2 (y1 , y2 )e R2+  also holds; (vi) Remark   3.1, 3.1, (i), (ii) and (iv) together yield that assumption  (H 4 ). Therefore, all conditions of Theorem 3.1 Theorem  3.1 (or  (or Corollary 3.1 Corollary 3.1)) are satisfied and Theo 3.1 (or  (or Corollary 3.1 Corollary 3.1)) is applicable. Moreover, Moreover, it fo follows llows from direct computation rem 3.1 rem that the weak solution   S W λ2 (x1 3)2 , 3   43 λ x  (x1 , x2 )  x2  3 W  (λ)

−

√ 

 = {  =

 ∈ −

 :

 +

   =  − −

 −  ≤

 −

2  which is not a singleton. Obviously, the solution set   S W x1  3 W  (λ)   is 1.12 λ , Hölder continuous on  .

 ≤  −

}

[13]) ]) is not applicable. The However, Theorem 2.1 in [14 [14]] (or Theorem 2.1 in [13 main mai n reason reason is tha thatt the ass assump umptio tion n   (A2r 1ϕ )   (or   (A2a) ) is vi viol olat ated ed.. In Inde deed ed,, let let 2 2 λ0  2 1, 2 . Hence,   K (λ0 ) x  (x1 , x2 )  (x1  3)  (x2  3)  4 .  Taking x0  ( 3, 3), y0  ( 3.1, 2.9)  K (λ0 ), direct computation shows that   f (x0 , y0 , λ0 ) int R2+  and  f (y0 , x0 , λ0 ) Y  int R2+ ,  which imply that  d(f (x0 , y0 , λ0 ), Y  Y  int R2+ )  0 and d(f (y0 , x0 , λ0 ), Y  int R2+ )  0. But for any h > 0, hd(x0 , y0 )

 =  ∈ [ ]  =  =  \ \ − −√  = 2h   10

 = {  =  ∈ ∈  \−  \ −  \ −

 :  − =

+  −

≤ }

14]] (or  (A2a)  in [13 13]) ]) does not hold. > 0.  Thus, assumption  (A2r 1ϕ )  in [14

 ∈   \\ =

 Remark 3.5   In [17], 17], by virtue of a kind of weak strong pseudo monotonicity (assumption (ii) of Theorem 3.1 of [17 [ 17]), ]), Li et al. discussed the Hölder continuity of  the set-valued solution mappings for parametric generalized Ky Fan Inequalities in

the general metric spaces. values However, assumption (ii) in Theorem 3.1 of [17 [ 17]] requires knowledge of detailed of the solution mapping in a neighborhood of the point under consideration. In many cases, it is not easy to verify assumption (ii) in Theorem 3.1 of [17 [17]] before accessing information on the solution set. In order to overcome 3.1 or  or Corollary  Corollary   3.1 3.1,, we introduce the assumption   (H 1 ), the drawback, in Theorem  Theorem   3.1 which can be verified directly. The following example is given to illustrate the case.

 = R, Y  =  = R2,  = M  =  = [0, 1], C = R2+,K(λ) = [1, 1 + λ], e = (1+λ)(x −y) (1, 1) ∈ int R2+   and  f (x,y, λ) =  (   , (1 +  λ)x(y −  x)).  Obviously, all as1+x  (or Corollary 3.1 Corollary 3.1)) are satisfied and Theorem 3.1 Theorem  3.1 (or  (or Corolsumptions of Theorem 3.1 Theorem 3.1 (or lary   3.1 3.1)) is applicable. In fact, it follows from direct computation that   S W W  (λ) =

 Example 3.3   Let   X

1, 1

λ, λ

, which is 1.1-Hölder continuous on   .

W  (λ) = (1 + λ, 2], ∀λ ∈ . However ver, , we can get  K() = [1, 2],(λ) = K() \ S W [ Howe + ] ∀ ∈ ¯ ) = {1}, there exists   y = 2 ∈ (λλ) ¯ ) = (1, 2], such Let ¯λ =  0. Then, for  xˆ = 1 ∈ S W λ) W  (λ

 

552

J Optim Theory Appl (2011) 149: 540–553

 ∀

that h > 0,

 ˆ  ¯ + hB(0, d (x,y)) xˆ ,y))

f(y, x, λ λ))

  = − + 1 3

,

2

hB(0, 1)

⊆   −R2+,

i.e., assumption (ii) of Theorem 3.1 of [17 [ 17]] does not hold. Thus, Theorem 3.1 of [17 [17]] is not applicable.

4 Con Conclu clusion sionss

In this paper, we consider a parametric weak generalized Ky Fan Inequality in the case where the solution mapping is set-valued. Without using any information of  the solution sets, we derive sufficient conditions for the Hölder continuity of the setvalued mapping for the weak generalized Ky Fan Inequality under the assumptions of  Hölder strongly monotonicity and Hölder continuity with respect to an interior point of a fixed cone. Generally speaking, for a classic variational inequality or Ky Fan Inequality, the solution mapping is not single-valued. Now one open problem arises in a natural way: Are there other ways to establish the Hölder continuity of the set-valued mapping to parametric variational inequalities or Ky Fan Inequalities without using any information of the solution sets? This is a very interesting and valuable topic, and we will investigate it in our future work.

References 1. Fan, K.: Extensions of two fix fixed ed point theorems of F.E. F.E. Browde Browder. r. M Math. ath. Z.  112, 234–240 (1969) 2. Ansari, Q.H.: V Vector ector equilibrium problems and vector variational variational inequalities. In: Giannessi, F. F. (ed.) Vecto ectorr var variatio iational nal inequ inequalit alities ies and vec vector tor equil equilibria ibria.. Math Mathemat ematical ical Theo Theories ries.. Kluw Kluwer er,, Dord Dordrech recht, t, pp. 1–16 (2000) 3. Bianchi, M., Hadjisa Hadjisavvas, vvas, M., Schaible Schaible,, S.: Vector Vector equilibrium problems with generalize generalized d monotone bifunctions. J. Optim. Theory Appl.  92, 527–542 (1997) 4. Fan, K.: A minimax inequality and applica applications. tions. In: Shisha, O. (ed.) Inequality III. III. Academic Press, New York, pp. 103–113 (1972) 5. Li, X.B., Li, S.J. S.J.:: Existences of solutions for generaliz generalized ed vector quasiequili quasiequilibrium brium problems. Optim. Lett. 4, 17–28 (2010) 6. Anh, L.Q., Khanh, P P.Q.: .Q.: Various Various kinds of semicontinuity and solution sets of parametric multivalued multivalued symmetric vector quasiequilibrium problems. J. Glob. Optim.  41, 539–558 (2008) 7. Anh, L.Q., Khanh Khanh,, P.Q.: P.Q.: On the stability of the solution sets of general multivalued multivalued vecto vectorr quasiequilibrium problems. J. Optim. Theory Appl.  135, 271–284 (2007) 8. Chen, C.R., Li, S.J., T Teo, eo, K.L.: Solution semicontinuity of parametric parametric generalized vector vector equilibrium problems. J. Glob. Optim.  45 , 309–318 (2009) 9. Yen, N.D.: Hölder continuity of solutions to parametric parametric variational inequalit inequalities. ies. Appl. Math. Optim. 31, 245–255 (1995) 10. Mansour, M.A M.A., ., Riahi, H.: Sensiti Sensitivity vity analysis for abstract equil equilibrium ibrium problems. J. Math. Anal. Appl. Appl. 306, 684–691 (2005) 11. Bian Bianchi, chi, M., Pini, R.: A note on stability for parame parametric tric equil equilibriu ibrium m prob problems. lems. Oper Oper.. Res. Lett Lett..  31 , 445–450 (2003) 12. Anh, L.Q., Khanh Khanh,, P.Q.: On the Hölder continui continuity ty of solut solutions ions to multi multival valued ued vector equilib equilibrium rium problems. J. Math. Anal. Appl.  321, 308–315 (2006)

 

J Optim Theory Appl (2011) 149: 540–553

553

13. Anh, L.Q., Khanh, P P.Q.: .Q.: Uniqueness and Hölde Hölderr continuity of solution to multivalued multivalued vector equilibrium problems in metric spaces. J. Glob. Optim.  37, 449–465 (2007) 14. Anh, L.Q., Khanh, P.Q.: Sensiti Sensitivity vity analysis for multi multival valued ued quasi quasiequi equilibri librium um problems problems in metr metric ic spaces: Hölder continuity of solutions. J. Glob. Optim.  42, 515–531 (2008) 15. Lee, G.M., Kim Kim,, D.S., Lee, B.S., Y Yen, en, N.D.: V Vector ector variati variational onal inequality as a tool for studying vector optimization problems. Nonlinear Anal.  34, 745–765 (1998) 16. Mansour, M.A., Ausss Ausssel, el, D.: Quasimonotone variat variational ional inequalities and quasiconvex quasiconvex programming: quantitative stability. Pac. J. Optim.  2 , 611–626 (2006) 17. Li, S.J., Li, X.B. X.B.,, Wang, L.N., T Teo, eo, K.L.: The Höld Hölder er continuit continuity y of solutions to gene generali ralized zed vector 18. 19. 20. 21.

equilibrium problems. Eur. J. Oper. Res.  199, 334–338 (2009) Aubin, J.P J.P., ., Ekeland, I.: Applied Nonlinear Analysis. Wiley, Wiley, New Y York ork (1984) Nadler, S.B. S.B.:: Multiv Multivalued alued contraction ma mappings. ppings. Pac. J. Math. Math.  30, 475–488 (1969) Bianchi, M., Pi Pini, ni, R.: Sensitivi Sensitivity ty for parametric v vector ector equilibria. O Optimization ptimization 55 , 221–230 (2006) Gong Gong,, X.H. X.H.,, Yao, J.-C.: Connec Connectedne tedness ss of the set of efficie efficient nt solutions for generali generalized zed systems systems.. J. Optim. Theory Appl.  138, 189–196 (2008)