Hindawi Publishing Corporation Fixed Point Theory and Applications Volume 2009, Article ID 684304,14 pages doi:10.1155/2009/684304

Research Article

Generalized Levitin-Polyak Well-Posedness of Vector Equilibrium Problems

Jian-Wen Peng,1 Yan Wang,1 and Lai-Jun Zhao2

1 College of Mathematics and Computer Science, Chongqing Normal University, Chongqing 400047, China

2 Management School, Shanghai University, Shanghai 200444, China

Correspondence should be addressed to Lai-Jun Zhao, zhao_laijun@163.com Received 1 July 2009; Revised 19 October 2009; Accepted 18 November 2009 Recommended by Nanjing Jing Huang

We study generalized Levitin-Polyak well-posedness of vector equilibrium problems with functional constraints as well as an abstract set constraint. We will introduce several types of generalized Levitin-Polyak well-posedness of vector equilibrium problems and give various criteria and characterizations for these types of generalized Levitin-Polyak well-posedness.

Copyright © 2009 Jian-Wen Peng et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

1. Introduction

It is well known that the well-posedness is very important for both optimization theory and numerical methods of optimization problems, which guarantees that, for approximating solution sequences, there is a subsequence which converges to a solution. The study of well-posedness originates from Tykhonov [1] in dealing with unconstrained optimization problems. Levitin and Polyak [2] extended the notion to constrained (scalar) optimization, allowing minimizing sequences {xn} to be outside of the feasible set X0 and requiring d(xn,X0) (the distance from xn to X0) to tend to zero. The Levitin and Polyak well-posedness is generalized in [3, 4] for problems with explicit constraint g(x) e K, where g is a continuous map between two metric spaces and K is a closed set. For minimizing sequences {xn}, instead of d(xn,X0), here the distance d(g(xn),K) is required to tend to zero. This generalization is appropriate for penalty-type methods (e.g., penalty function methods, augmented Lagrangian methods) with iteration processes terminating when d(g(xn),K) is small enough (but d(xn,X0) may be large). Recently, the study of generalized Levitin-Polyak well-posedness was extended to nonconvex vector optimization problems with abstract and functional constraints (see [5]), variational inequality problems with abstract and functional constraints (see [6]), generalized variational inequality problems with abstract and functional constraints [7], generalized vector variational inequality problems with abstract

and functional constraints [8], and equilibrium problems with abstract and functional constraints [9]. Most recently, S. J. Li and M. H. Li [10] introduced and researched two types of Levitin-Polyak well-posedness of vector equilibrium problems with variable domination structures. Huang et al. [11] introduced and researched the Levitin-Polyak well-posedness of vector quasiequilibrium problems. Li et al. [12] introduced and researched the Levitin-Polyak well-posedness for two types of generalized vector quasiequilibrium problems. However, there is no study on the generalized Levitin-Polyak well-posedness for vector equilibrium problems and vector quasiequilibrium problems with explicit constraint g(x) e K.

Motivated and inspired by the above works, in this paper, we introduce two types of generalized Levitin-Polyak well-posedness of vector equilibrium problems with functional constraints as well as an abstract set constraint and investigate criteria and characterizations for these two types of generalized Levitin-Polyak well-posedness. The results in this paper generalize and extend some known results in literature.

2. Preliminaries

Let (X,dX), (Z,dZ), and Y be locally convex Hausdorff topological vector spaces, where dX (dZ) is the metric which compatible with the topology of X(Z). Throughout this paper, we suppose that K c Z and X1 c X are nonempty and closed sets, C : X ^ 2Y is a set-valued mapping such that for any x e X, C(x) is a pointed, closed, and convex cone in Z with nonempty interior int C(x), e : X ^ Y is a continuous vector-valued mapping and satisfies that for any x e X, e(x) e int C(x), f : X x X1 ^ Y and g : X1 ^ Z are two vector-valued mappings, and X0 = {x e X1 : g(x) e K}. We consider the following vector equilibrium problem with variable domination structures, functional constraints, as well as an abstract set constraint: finding a point x* e X0, such that

f (x*,y) / - int C(x*), Vy e Xo. (VEP)

We always assume that X0 = and g is continuous on X1 and the solution set of (VEP) is denoted by Q.

Let (P,d) be a metric space, P1 c P, and x e P. We denote by d(x,P1) = inf{d(x,p) : p e P1} the distance function from the point x e P to the set P1.

Definition 2.1. (i) A sequence {xn} c X1 is called a type I Levitin-Polyak (in short LP) approximating solution sequence for (VEP) if there exists {en }c R+ with en ^ 0 such that

d(xn,X0) < en, (2.1)

f(xn,y) + ene(xn) e - int C(xn), Vy e X0. (2.2)

(ii) {xn} c Xi is called type II approximating solution sequence for (VEP) if there exists {en} c R+ with en ^ 0 and {yn} c X0 satisfying (2.1), (2.2), and

f(xn,yn) - ene(xn) e -C(xn).

(iii) {xn} c X1 is called a generalized type I approximating solution sequence for (VEP) if there exists {en }c R| with en —> 0 satisfying

d(g(xn),K) < en (2.4)

and (2.2).

(iv){xn} c X1 is called a generalized type II approximating solution sequence for (VEP) if there exists {en} c R| with en — 0 and {yn} c Xo satisfying (2.2), (2.3), and (2.4).

Definition 2.2. The vector equilibrium problem (VEP) is said to be type I (resp., type II, generalized type I, generalized type II) LP well-posed if Q = 0 and for any type I (resp., type II, generalized type I, generalized type II) LP approximating solution sequence {xn} of (VEP), there exists a subsequence {xH]} of {xn} and x e Q such that xnj — x.

Remark 2.3. (i) If Y = R and C(x) = R| = {r e R : r > 0} for all x e X, then the type I (resp., type II, generalized type I, generalized type II) LP well-posedness of (VEP) defined in Definition 2.2 reduces to the type I (resp., type II, generalized type I, generalized type II) LP well-posedness of the scalar equilibrium problem with abstract and functional constraints introduced by Long et al. [9]. Moreover, if X* is the topological dual space of X, F : X1 — X* is a mapping, (F(x),z) denotes the value of the functional F(x) at z, and f (x,y) = (F(x),y-x) for all x,y e X1, then the type I (resp., type II, generalized type I, generalized type II) LP well-posedness of (VEP) defined in Definition 2.2 reduces to the type I (resp., type II, generalized type I, generalized type II) LP well-posedness for the variational inequality with abstract and functional constraints introduced by Huang et al. [6]. If K = Z, then X1 = X0 and the type I (resp., type II) LP well-posedness of (VEP) defined in Definition 2.2 reduces to the type I (resp., type II) LP well-posedness of the vector equilibrium problem introduced by S.J. Li and M. H. Li [10].

(ii) It is clear that any (generalized) type II LP approximating solution sequence of (VEP) is a (generalized) type I LP approximating solution sequence of (VEP). Thus the (generalized) type I LP well-posedness of (VEP) implies the (generalized) type II LP well-posedness of (VEP).

(iii) Each type of LP well-posedness of (VEP) implies that the solution set Q is nonempty and compact.

(iv) Let g be a uniformly continuous functions on the set

S(60) = {x e X1: d(g(x),K) < 60} (2.5)

for some 60 > 0. Then generalized type I (resp., type II) LP well-posedness implies type I (resp., type II) LP well-posedness.

3. Criteria and Characterizations for Generalized LP Well-Posedness of (VEP)

In this section, we present necessary and/or sufficient conditions for the various types of (generalized) LP well-posedness of (VEP) defined in Section 2.

3.1. Criteria and Characterizations without Using Gap Functions

In this subsection, we give some criteria and characterizations for the (generalized) LP well-posedness of (VEP) without using any gap functions of (VEP).

Now we introduce the Kuratowski measure of noncompactness for a nonempty subset A of X (see [13]) defined by

a(A) = inf|e > 0 : A c (J Ai, for every Ai, diamAi < ej, (3.1)

where diamAi is the diameter of Ai defined by

diamAi = sup(d(x^x2) : xi,x2 e Ai}. (3.2)

Given two nonempty subsets A and B of X, the excess of set A to set B is defined by e(A,B) = sup(d(a,B) : a e A}, (3.3)

and the Hausdorff distance between A and B is defined by

H (A,B) = max(e(A,B),e(B,A)}. (3.4)

For any e > 0, four types of approximating solution sets for (VEP) are defined, respectively, by

T1(e) := (x e X1 : d(g(x),K) < e and f (x,y) + ee(x) / - int C(x), for all y e X0}, T2(e) := (x e X1 : d(x,X0) < e and f (x,y) + ee(x) e - intC(x), for all y e X0}, T3(e) := (x e X1 : d(g(x),K) < e and f (x,y) + ee(x) / - int C(x), for all y e X0 and f (x,y) - ee(x) e -C(x), for some y e X0},

T4(e) := (x e X1 : d(x,X0) < e and f (x,y) + ee(x) / - intC(x), for all y e X0 and f (x,y) - ee(x) e -C(x), for some y e X0}.

Theorem 3.1. Let X be complete.

(i) (VEP) is generalized type I LP well-posed if and only if the solution set Q is nonempty and compact and

e(T1(e), Q) 0 as e 0. (3.5)

(ii) (VEP) is type I LP well-posed if and only if the solution set Q is nonempty and compact

e(T2(e),Q) 0 as e 0. (3.6)

(iii) (VEP) is generalized type II LP well-posed if and only if the solution set Q is nonempty and compact and

e(T3(e),Q) 0 as e 0. (3.7)

(iv) (VEP) is type II LP well-posed if and only if the solution set Q is nonempty and compact

e(T4(e),Q) 0 as e 0. (3.8)

Proof. The proofs of (ii), (iii), and (iv) are similar with that of (i) and they are omitted here. Let (VEP) be generalized type I LP well-posed. Then Q is nonempty and compact. Now we show that (3.5) holds. Suppose to the contrary that there exist l > 0, en> 0 with en ^ 0 and zn € Ti(en) such that

d(zn, Q) > l. (3.9)

Since (zn}c T1(en) we know that {zn} is generalized type I LP approximating solution for (VEP). By the generalized type I LP well-posedness of (VEP), there exists a subsequence {znj} of {zn} converging to some element of Q. This contradicts (3.9). Hence (3.5) holds.

Conversely, suppose that Q is nonempty and compact and (3.5) holds. Let {xn} be a generalized type I LP approximating solution for (VEP). Then there exists a sequence {en} with {en } c R+ and en ^ 0 such that

d(g(xn),K) < en,

(3.i0)

f(xn,y) + ene(xn) € - int C(xn), Vy € X0.

Thus, {xn}c T1(e). It follows from (3.5) that there exists a sequence {zn} cQ such that

d(xn, zn) = d(xn, Q) < e(T1 (e), Q) —> 0.

(3.11)

Since Q is compact, there exists a subsequence {znk} of {zn} converging to x0 € Q. And so the corresponding subsequence {xnk} of {xn} converging to x0. Therefore (VEP) is generalized type I LP well-posed. This completes the proof. □

Theorem 3.2. Let X be complete. Assume that

(i) for any y € X1, the vector-valued function x ^ f (x, y) is continuous;

(ii) the mapping W : X ^ 2r defined by W(x) = Y \- int C(x) is closed.

Then (VEP) is generalized type I LP well-posed if and only if

T1 (e) =, Ve > 0, lim a(T1(e)) = 0. (3.12)

Proof. First we show that for every e > 0, T1(e) is closed. In fact, let {xn} c T1(e) and xn ^ x. Then

d(g(xn),K) < e, f(xn,y) + ee(xn) € - int C(xn), "Vy € X0.

(3.13)

From (3.13), we get

d(g(x),K) < e,

(3.14)

f(xn,y) + ee(xn) e W(xn), Vy e X0.

By assumptions (i), (ii), we have f (x, y) + ee(x) e - int C(x), for all y e X0. Hence x e T1(e). Second, we show that

O = He>oTi(e). (3.15)

It is obvious that

O c He>oTi(e). (3.16)

Now suppose that en> 0 with en ^ 0 and x* e f)n=1T1(en). Then

d(g(x*),K) < en, Vn e N, (3.17)

f (x*,y) + ene(x*) e - int C(x*), Vy e Xo. (3.18)

Since K is closed, g is continuous, and (3.17) holds, we have x* e X0. By (3.18) and closedness of W(x*), we get f (x*,y) e W(x*), for all y e X0, that is, x* e O. Hence (3.15) holds.

Now we assume that (3.12) holds. Clearly, Ti(-) is increasing with e > 0. By the Kuratowski theorem (see [14]), we have

H(T1(e), O) 0, as e 0. (3.19)

Let {xn} be any generalized type I LP approximating solution sequence for (VEP). Then there exists en> 0 with en ^ 0 such that (3.13) holds. Thus, xn e T1(en). It follows from (3.19) that d(xn, O) ^ 0. So there exsist un e O, such that

d(xn,un) 0. (3.20)

Since O is compact, there exists a subsequence {unj} of {un} and a solution x* e O satisfying

uUj x*. (3.21)

From (3.20) and (3.21), we get d(xnj,x*) ^ 0.

Conversely, let (VEP) be generalized type I LP well-posed. Observe that for every

e > 0,

H(T1(e), O) = max{e(T1(e), O),e(O,T1(e))} = e(T1(e), O). (3.22)

Fixed Point Theory and Applications Hence,

«(71(e)) < 2H(T1(e), Q) + a(Q) = 2e(T1(e), Q),

(3.23)

where a(Q) = 0 since Q is compact. From Theorem 3.1 (i), we know that e(T1(e), Q) ^ 0 as e ^ 0. It follows from (3.23) that (3.12) holds. This completes the proof. □

Similar to Theorem 3.2, we can prove the following result.

Theorem 3.3. Let X be complete. Assume that

(i) for any y e X1, the vector-valued function x ^ f (x, y) is continuous;

(ii) the mapping W : X ^ 2r defined by W(x) = Y \- int C(x) is closed;

(iii) the set-valued mapping C : X1 ^ 2Y is closed;

(iv) for any x* e Q, f (x*,y) e -dC,for some y e X0. Then (VEP) is generalized type II LP well-posed if and only if

73(e) =, Ve > 0, lim a(Ts(e)) = 0. (3.24)

Definition 3.4. (VEP) is said to be generalized type I (resp., generalized type II) well-set if Q / 0 and for any generalized type I (resp., generalized type II) LP approximating solution sequence {xn} for (VEP), we have

d(xn, Q) —> 0, as n —> œ. (3.25)

From the definitions of the generalized LP well-posedness for (VEP) and those of the generalized well-set for (VEP), we can easily obtain the following proposition.

Proposition 3.5. The relations between generalized LP well-posedness and generalized well set are

(i) (VEP) is generalized type I LP well-posed if and only if (VEP) is generalized type I well-set and Q is compact.

(ii) (VEP) is generalized type II LP well-posed if and only if (VEP) is generalized type II well-set and Q is compact.

By combining the proof of Theorem 3.3 in [10] and that of Theorem 3.1, we can prove that the following results show that the relations between the generalized LP well-posedness for (VEP) and the solution set Q of (VEP).

Theorem 3.6. Let X be finite dimensional. Assume that

(i) for any y e X1, the vector-valued function x ^ f (x, y) is continuous;

(ii) the mapping W : X ^ 2Y defined by W (x) = Y \- int C(x) is closed;

(iii) there exists e0 > 0 such that T1(e0) (resp., T3(e0)) is bounded.

If Q is nonempty, then (VEP) is generalized type I (resp., generalized type II) LP well-

posed.

Corollary 3.7. Suppose Q /. And assume that

(i) for any y e X1 the vector-valued function x ^ f (x, y) is continuous;

(ii) the mapping W : X ^ 2Y defined by W(x) = Y \- int C(x) is closed;

(iii) there exists e0 > 0 such that T1(e0) (resp., T3(e0)) is compact.

If Q is nonempty, then (VEP) is generalized type I (resp., generalized type II) LP well-

posed.

3.2. Criteria and Characterizations Using Gap Functions

In this subsection, we give some criteria and characterizations for the (generalized) LP well-posedness of (VEP) using the gap functions of (VEP) introduced by S. J. Li and M. H. Li [10].

Chen et al. [15] introduced a nonlinear scalarization function £,e : X x Z ^ R defined

le(x,y) = inf{! e R : y e Xe(x) - C(x)}. (3.26)

Definition 3.8 ([10]). A mapping g : X ^ R is said to be a gap function on X0 for (VEP) if

(i) g (x) > 0, for all x e X0;

(ii) g (x*) = 0 and x* e X0 if and only if x* e Q.

S. J. Li and M. H. Li [10] introduced a mapping $ : X ^ R defined as follows:

#(x) = sup {-£e(x,f (:x,y))}. (3.27)

Lemma 3.9 (see [10]). If for any x e X0, f (x,x) e -dC(x), where dC(x) is the topological boundary of C(x), then the mapping $ defined by (3.27) is a gap function on X0 for (VEP).

Now we consider the following general constrained optimization problems introduced and researched by Huang and Yang [4]:

(P) min $(x)

(3.28)

s.t. x e X1, g(x) e K.

We use argmin $ and v* denote the optimal set and value of (P), respectively.

The following example illustrates that it is useful to consider sequences that satisfy d(g(xn),K) ^ 0 instead of d(xn,X0) ^ for (VEP).

Example 3.10. Let a> 0, X = R1, Z = R1, C(x) = R+, and e(x) = (1,1) for each x e X, K = R—,

X1 = R+,g(x) =

f(x,y) =

x, if x e [0,1],

1 x2 if x > 1,

\xa — -ya, —xa — y — 1), if x e [0,1], Vy e X1,

(1 \ xa 1 1 A ya, xa y )' if x > 1, Vy e X1,

s-h- 1), if x < 0, Vy e X1.

(3.29)

Then, it is easy to verify that X0 = {x e X1 : g(x) e K} and (VEP) is equivalent to the optimization problem (P) with

$(x) =

—xa, if x e [0,1],

-—, if x > 1.

(3.30)

Huang and Yang [4] showed that xn = (2n)1/a is the unique solution to the following penalty problem (PPa(n)):

(PPa(n))min$(x) + n[ma^{0,g(x^]a, n e N,

(3.31)

and d(g(xn),K) ^ 0 and d(xn,X0) ^ +<x>.

Now, we recall the definitions about generalized well-posedness for (P) introduced by Huang and Yang [4] (or [7]) as follows

Definition 3.11. A sequence {xn} c Xi is called a generalized type I (resp., generalized type II) LP approximating solution sequence for (P) if the following (3.32) and (3.33) (resp., (3.32) and (3.34)) hold:

d(g(xn),K) —> 0, as n —, limsup$(xn) < v*,

(3.32)

(3.33)

lim $(xn) = v*.

(3.34)

Definition 3.12. (P) is said to be generalized type I (resp., generalized type II) LP well-posed if

(i) argmin $ /;

(ii) for every generalized type I (resp., generalized type II) LP approximating solution sequence {xn} for (P), there exists a subsequence {xnj} of {xn} converging to some element of argmin

The following result shows the equivalent relations between the generalized LP well-posedness of (VEP) and the generalized LP well-posedness of (P).

Theorem 3.13. Suppose that f (x,x) e -dC(x),for all x e X0. Then

(i) (VEP) is generalized type I well-posed if and only if(P) is generalized type I well-posed;

(ii) (VEP) is generalized type II well-posed if and only if (P) is generalized type II well-posed.

Proof. (i) By Lemma 3.9, we know that $ is a gap function on X0, x e ^ if and only if x e argmin $ with v* = $(x) = 0.

Assume that {xn} is any generalized type I LP approximating solution sequence for (VEP). Then there exists en> 0 with en ^ 0 such that

d(g(xn),K) < en, (3.35)

f(xn,y) + ene(xn) e - int C(xn), Vy e X0. (3.36)

It follows from (3.35) and (3.36) that

d(g(xn),K) 0, as n —> to, (3.37)

le{xn,f{xn,y)) > -en, Vy e X0. (3.38)

Hence, we obtain

$(xn) = supj-ie{xn,f {xn,y))} < en. (3.39)

limsup $(xn) < 0 since en —> 0. (3.40)

The above formula and (3.37) imply that {xn} is a generalized type I LP approximating solution sequence for (P).

Conversely, assume that {xn} is any generalized type I LP approximating solution sequence for (P). Then d(g(xn),K) ^ 0 and limsup^^ $(xn) < 0. Thus, there exists en> 0 with en ^ 0 satisfying (3.35) and

$(xn) = sup j-¿,e(xn,f(xn,y))} < en. (3.41)

From (3.41), we have

¿,e(xn,f(xn,y)) > -en, Vy e Xq.

(3.42)

Equivalently, (3.36) holds. Hence, {xn} is a generalized type I LP approximating solution sequence for (VEP).

(ii) The proof is similar to (i) and is omitted. This completes the proof. □

Now we consider a real-valued function c = c(t, s) defined for t,s > 0 sufficiently small, such that

c(t,s) > 0, Vt,s, c(0,0) = 0,

(3.43)

sn —> 0, tn > 0, c(tn, sn) —> 0, imply tn —> 0.

Lemma 3.14 (see [4, Theorem 2.2]). Suppose that f (x,x) e -dC(x) for any x e X0.

(i) If(P) is generalized type II LP well-posed, then there exists a function c satisfying (3.43) such that

|$(x) - v*| > c(d(x,argmin$),d(g(x),K)), Vx e Xi. (3.44)

(ii) Assume that argmin $ is nonempty and compact, and (3.44) holds for some c satisfying (3.43). Then (P) is generalized type II LP well-posed.

The following theorem follows immediately from Lemma 3.14 and Theorem 3.13 with $(x) defined by (3.27) and v* = 0.

Theorem 3.15. Suppose that f (x,x) e -dC(x) for any x e X0.

(i) If (VEP) is generalized type II LP well-posed, then there exists a function c satisfying (3.43) such that

\$(x)\> c(d(x,Q),d(g(x),K)), Vx e Xi. (3.45)

(ii) Assume that Q is nonempty and compact, and (3.45) holds for some c satisfying (3.43). Then (VEP) is generalized type II LP well-posed.

Definition 3.16 (see [4, 7]). (i) Let Z be a topological space and let Z1 c Z be a nonempty subset. Suppose that G : Z ^ R u{+<x>} is an extend real-valued function. Then the function G is said to be level-compact on Z1 if for any s e R1 the subset {z e Z1 : G(z) < s} is compact.

(ii) Let Z be a finite dimensional normed space and Z1 c Z be nonempty. A function h : Z ^ R1 u {+to} is said to be level-bounded on Z1 if Z1 is bounded or

lim h(z) = (3.46)

Proposition 3.17. Assume that for any y e X1, the vector-valued function x ^ f (x,y) is continuous and the mapping W : X ^ 2r defined by W(x) = Y \ -int C(x) is closed, and Q is nonempty. Then, (VEP) is generalized type I LP well-posed if one of the following conditions holds: (i) there exists 61 > 0 such that S(61) is compact, where

S(61) = {x e X1 : d(g(x),K) < 6i};

(3.47)

(ii) the function $ defined by (3.27) is level-compact on X1; (in) X is a finite-dimensional normed space and

lim max{$(x),d(g(x),K)} =+<x>; (3.48)

xeXitWxW ^ +<x>

(iv) there exists 61 > 0 such that $ is level-compact on S(61) defined by (3.47).

Proof. Let {xn} c X1 be a generalized type I LP approximating solution sequence for (VEP). Then there exists a sequence {en} c R+ with en > 0 such that (3.35) and (3.36) hold. From (3.20), without loss of generality, we assume that {xn} c S(61). Since S(61) is compact, there exists a subsequence {xH]} of {xn} and x0 e S(61) such that xnj ^ x0. This fact combined with (3.35) yields that x0 e X0. Furthermore, it follows from (3.36) and the continuity of f with respect to the first argument and the closedness of W that we have f (x0,y) e - intC(x0), for all y e X0. So x0 e Q. This implies that (VEP) is generalized type I LP well-posed.

It is easy to see that condition (ii) implies condition (iv). Now we show that condition (iii) implies condition (iv). It follows from [10, Proposition 4.2] that the function $ defined by (3.27) is lower semicontinuous, and thus for any t e R1, the set {x e S(61) : $(x) < t} is closed. Since X is a finite dimensional space, we need only to show that for any t e R1, the set {x e S(61) : $(x) < t} is bounded. Suppose to the contrary that there exists t e R1 and {x'n} c S(61) and $(x'n) < t such that \\x'n\\ ^ +<x>. It follows from {x'n} c S(61) that d(g(x'n),K) < 51 and so

max{$(xn),d(g(xn),K)} < max{t,S1}. (3.49)

Which contradicts with (3.48).

Therefore, we only need to prove that if condition (iv) holds, then (VEP) is generalized type I LP well-posed. Suppose that condition (iv) holds and {xn} is a generalized type I LP approximating solution sequence for (VEP). Then there exists {en} c R+ with en> 0 such that (3.35) and (3.36) hold. By (3.35), we can assume without loss of generality that

{xn}c S(61). (3.50)

It follows from (3.36) that ¿,e(xn,f(xn,y)) > -en, for all y e X0. Thus,

$(xn) < en, Vn. (3.51)

From (3.51), without loss of generality, we assume that {xn} c {x e S(61) : $(x) < b} for some b > 0. Since $ is level-compact on S(61), the subset {x e S(61) : $(x) < b} is compact. It follows that there exists a subsequence {xH]} of {xn} and x e S(61) such that xnj ^ x. This together with (3.35) yields x e X0. Furthermore by the continuity of f with respect to the first argument, the closedness of W, and (3.36) we have x0 e Q. This completes the proof. □

Similarly, we can prove Proposition 3.18.

Proposition 3.18. Assume that for any y e Xi, the vector-valued function x ^ f (x,y) is continuous and the mapping W : X ^ 2Y defined by W(x) = Y \ -int C(x) is closed, and Q is nonempty. Then, (VEP) is type I LP well-posed if one of the following conditions holds:

(i) there exists 6i > 0 such that Si(6i) is compact where

Si(6i) = {x e Xi : d(x,Xo) < 6i); (3.52)

(ii) the function $ defined by (3.27) is level-compact on Xi;

(iii)X is a finite-dimensional normed space and

lim max{ $(x),d(x,X0 )} = +<x>; (3.53)

xeXiJxH ^ +<x>

(iv) there exists 6i > 0 such that $ is level-compact on Si(6i) defined by (3.52).

Proposition 3.19. Assume that X is a finite dimensional space, for any y e Xi, the vector-valued function x ^ f (x,y) is continuous and the mapping W : X ^ 2Y defined by W(x) = Y\- int C(x) is closed, and Q is nonempty. Suppose that there exists 6i > 0 such that the function $(x) defined by (3.27) is level-bounded on the set S(6i) defined by (3.47). Then (VEP) is generalized type I LP well-posed.

Proof. Let {xn} be a generalized type I LP approximating solution sequence for (VEP). Then there exists {en} with en> 0 such that (3.35) and (3.36) hold.

From (3.35), without loss of generality, we assume that {xn} c S(6i). Let us show by contradiction that {xn} is bounded. Otherwise we assume without loss of generality that ||xn|| ^ By the level-boundedness of $, we have

lim $(x) = +<x>. (3.54)

||x|| ^ +<x>

It follows from (3.36) and the proof in Proposition 3.i7 that (3.5i) holds. which contradicts with (3.54).

Now we assume without loss of generality that xn ^ x. Furthermore by the continuity of f with respect to the first argument, the closedness of W, and (3.36) we have x0 e Q. This completes the proof. □

Similarly, we can prove the following Proposition 3.20.

Proposition 3.20. Assume that X is a finite dimensional space, for any y e Xi, the vector-valued function x ^ f (x,y) is continuous and the mapping W : X ^ 2Y defined by W(x) = Y\- int C(x) is closed, and Q is nonempty. Suppose that there exists 6i > 0 such that the function $(x) defined by (3.27) is level-bounded on the set Si(6i) defined by (3.52). Then (VEP) is type I LP well-posed.

Remark 3.21. Theorem 3.i generalizes and extends [9, Theorems 3.i-3.6] from scalar-valued case to vector-valued case. Propositions 3.i7-3.20, respectively, generalize and extend [9, Propositions 4.3, 4.2, 4.5, and 4.4] from scalar-valued case to vector-valued case. Theorems 3.2, 3.3, 3.6, 3.i3, and 3.i5, Proposition 3.5 and Corollary 3.7, respectively, extend [i0, Theorems 3.i-3.3, 4.i, and 4.2, Proposition 3.i and Corollary 3.i] from the well-posedness

of (VEP) to the generalized well-posedness of (VEP). It is easy to see that the results in this paper generalize and extende the main results in [6] in several aspects.

Remark 3.22. The generalized Levitin-Polyak well-posedness for vectorquasiequilibrium problems and generalized vector-quasiequilibrium problems with explicit constraint g(x) e K is still an open question and we will do the research in the near future.

Acknowledgments

This study was supported by Grants from the National Natural Science Foundation of China (Project nos. 70673012, 70741028, and 90924030) and the China National Social Science Foundation (Project no. 08CJY026).

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