0Hindawi Publishing Corporation Journal of Inequalities and Applications Volume 2008, Article ID 435719,14 pages doi:10.1155/2008/435719
Research Article
Painleve-Kuratowski Convergences for the Solution Sets of Set-Valued Weak Vector Variational Inequalities
Z. M. Fang,1 S. J. Li,1 and K. L. Teo2
1 College of Mathematics and Science, Chongqing University, Chongqing, 400044, China
2 Department of Mathematics and Statistics, Curtin University of Technology, P.O. Box U1987, Perth, WA 6845, Australia
Correspondence should be addressed to S. J. Li, lisj@cqu.edu.cn
Received 16 July 2008; Revised 11 November 2008; Accepted 10 December 2008
Recommended by Donal O'Regan
Painleve-Kuratowski convergence of the solution sets is investigated for the perturbed set-valued weak vector variational inequalities with a sequence of mappings converging continuously. The closedness and Painleve-Kuratowski upper convergence of the solution sets are obtained. We also obtain Painleve-Kuratowski upper convergence when the sequence of mappings converges graphically. By virtue of a sequence of gap functions and a key assumption, Painleve-Kuratowski lower convergence of the solution sets is established. Some examples are given for the illustration of our results.
Copyright © 2008 Z. M. Fang 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
Since the concept of vector variational inequality (VVI) was introduced by Giannessi [1] in 1980, many important results on various kinds of vector variational inequality problems have been established, such as existence of solutions, relations with vector optimization, stability of solution set maps, gap function, and duality theories (see, e.g., [2-8] and the references cited therein).
The stability analysis of the solution set maps for the parametric (VVI) problem is of considerable interest amongst researchers in the area. Some results on the semicontinuity of the solution set maps for the parametric (VVI) problem with the parameter perturbed in the space of parameters are now available in the literature. In [4], Khanh and Luu proved the upper semicontinuity of the solution set map for two classes of parametric vector quasivariational inequalities. In [7], Li et al. established the upper semicontinuity property of the solution set map for a perturbed generalized vector quasivariational inequality problem
and also obtained the lower semicontinuity property of the solution set map for a perturbed classical scalar variational inequality. In [9], Cheng and Zhu investigated the upper and lower semicontinuities of the solution set map for a parameterized weak vector variational inequality in a finite dimensional Euclidean space by using a scalarization method. In [6], Li and Chen obtained the closedness and upper semicontinuity of the solution set map for a parametric weak vector variational inequality under weaker conditions than those assumed in [9]. Then, under a key assumption, they proved a lower semicontinuity result of the solution set map in a finite dimensional space by using a parametric gap function.
However, there are few investigations on the convergence of the sequence of mappings. In particular, almost no stability results are available for the perturbed (VVI) problem with the sequence of mappings converging continuously or graphically. It appears that the only relevant paper is [10], where Lignola and Morgan considered generalized variational inequality in a reflexive Banach space with a sequence of operators converging continuously and graphically and obtained the convergence of the solution sets under an assumption of pseudomonotonicity. Since the perturbed (VVI) problem with a sequence of mappings converging is different from the parametric (VVI) problem with the parameter perturbed in a space of parameters, these results do not apply to the parametric (VVI) problem with the parameter perturbed in a space of parameters. Thus, it is important to study Painleve-Kuratowski upper and lower convergences of the sequence of solution sets.
In passing, it is worth noting that some stability results are available for the vector optimization and vector equilibrium problems with a sequence of sets converging in the sense of Painleve-Kuratowski (see [11-13]). It is well known that the vector equilibrium problem is a generalization of (VVI) problem. However, if the results obtained for the vector equilibrium problem are to be applied to the (VVI) problem, the required assumptions are on the (VVI) problem as a whole. There is no information about the conditions that are required on the functions defining the (VVI) problem. Clearly, this is unsatisfactory. Our study of the stability properties for the perturbed (VVI) problem with a sequence of converging mappings is under appropriate assumptions on the function defining the (VVI) problem rather than on the (VVI) problem as a whole.
In this paper, we should establish Painleve-Kuratowski upper and lower convergences of the solution sets of the perturbed set-valued weak variational inequity (SWVVI) with a sequence of converging mappings in a Banach space. We first discuss Painleve-Kuratowski upper convergence and closedness of the solution sets. To obtain Painleve-Kuratowski lower convergence of the solution sets, we introduce a sequence of gap functions based on the nonlinear scalarization function introduced by Chen et al. in [14] and a key assumption (Hg) imposed on the sequence of gap functions. Then, we obtain Painleve-Kuratowski lower convergence of the solution sets for (SWVVI) n.
The rest of the paper is organized an follows. In Section 2, we introduce problems (SWVVI) and (SWVVI)n, and recall some definitions and important properties of these problems. In Section 3, we investigate Painleve-Kuratowski upper convergence and the closedness of the solution sets. In Section 4, we introduce respective gap functions for problems (SWVVI) and (SWVVI) n and then establish Painleve-Kuratowski lower convergence of the solution sets under a key assumption.
2. Preliminaries
Let X and Y be two Banach spaces and let L(X, Y) be the set of all linear continuous mappings from X to Y. The value of a linear mapping t e L(X,Y) at x e X is denoted by (t,x). Let C c Y
be a closed and convex cone with nonempty interior, that is, int C / 0. We define the ordering relations as follows.
For any yi, y2 e Y,
y1 <int c y2 ^^ y2 - y1 e int C,
y1 <int C y2 ^ y2 - y1 e int C
Consider the set-valued weak vector variational inequality (SWVVI) problem for finding x e K and t e T(x) such that
(t,y - x) eY \-int C Vy e K, (2.2)
where K c X is a nonempty subset and T : K ^ 2L(X,Y) is a set-valued mapping.
For a sequence of set-valued mappings Tn : Kn ^ 2L(X,Y), we define a sequence of set-valued weak vector variational inequality (SWVVI) n problems for finding xn e Kn and tn e Tn(xn) such that
(tn,y - xn) e Y \ -int C Vy e Kn, (2.3)
where Kn c X is a sequence of nonempty subsets.
We denote the solution sets of problems (SWVVI) and (SWVVI)n by I(T) and I(Tn), respectively, that is,
I (T) = {x e K | 3t e T (x), s.t. (t,y - x) eY \ -int C Vy e K},
I (Tn) = {xn e Kn | 3t n e Tn ( xn ), s.t. (tn, y - xn) e Y \ -int CVy e Kn}-
Throughout this paper, we assume that I(T) / 0 and I(Tn) / 0. The stability analysis is to investigate the behaviors of the solution sets I(T) and I(Tn).
Now we recall some basic definitions and properties of problems (SWVVI) and (SWVVI)n. For each e > 0 and a subset A c X, let the open ¿-neighborhood of A be defined as U(A,e) = {x e X | 3a e A, s.t. ||a - x|| < e}. The notation B(X,6) denotes the open ball with center X and radius 6 > 0.
In the following, we introduce some concepts of the convergence of set sequences and mapping sequences which will be used in the sequel. Define
{N cN|N \ N finite}
{subsequences of N containing all n eN beyond some n},
{N cN|N infinite} { all subsequences of N},
where N denotes the set of all positive integer numbers and n is an integer in N.
Definition 2.1 (see [11,15]). Let X be a normed space. A sequence of sets {Dn c X : n e N} is
said to converge in the sense of Painleve-Kuratowski (P.K.) to D (i.e., Dn -— D) if
lim sup Dn c D c lim inf Dn (2.6)
lim inf Dn := {x | 3N e Noo, 3xn e Dn (n e N) with xn —> x],
n —> TO
lim sup Dn := {x | 3N e NTO, 3xn e Dn (n e N) with xn —> x}.
It is said that the sequence {Dn} upper converges in the sense of Painleve-Kuratowski to D if limsupn^00Dn c D. Similarly, the sequence {Dn} is said to lower converge in the sense of Painleve-Kuratowski to D if D c liminfn—TODn.
Definition 2.2 (see [15]). A set-valued mapping S : X — 2Y is outer semicontinuous (osc) at x if limsupx —xS(x) c S(x) with limsupx—xS(x) := Ux„ —xlimsupn — TOS(xn).
On the other hand, it is inner semicontinuous (isc) at x if lim infx—xS(x) d S(x) with lim infx—xS(x) := f|xn — xliminfn — ^S(xn).
The set-valued mapping is said to be continuous at x, written as S(x) — S(x) as x — x if it is both outer semicontinuous and inner semicontinuous.
Definition 2.3 (see [15]). Let Sn : X — 2Y be a sequence of set-valued mappings and S : X — 2Y be a set-valued mapping. It is said that the sequence {Sn} converges continuously to S at x if
limsup Sn(xn) c S(x) c liminf Sn(xn) Vsequencexn —> x. (2.8)
n — TO n —
If { S n} converges continuously to S at every x e X, then it is said that {Sn} converges continuously to S on X.
Let S : X — 2Y be a set-valued map, the graph of S is defined as
gph S = {(x,u) | u e S(x)]. (2.9)
Applying set convergence theory to the graphs of the mappings, we obtain the graphical convergence of the sequence of mappings.
Definition 2.4 (see [15]). For a sequence of mappings Sn : X — 2Y, the graphical outer limit, denoted by g - lim supnSn, is the mapping which has as its graph the set lim supn(gph Sn):
gph ( g - lim sup Sn) = lim sup (gph Sn),
^g - limsup Sn^ (x) = ju | 3N e NTO, xn -— x, un -— u, un e Sn(xn)^ .
(2.10)
The graphical inner limit, denote by g - lim infnSn, is the mapping having as its graph the set lim inf„(gph Sn):
gph ( g - lim inf S„) = lim inf (gph Sn),
f \ n N (2.11)
(g - lim inf Snj (x) = ju | 3N e No, xn -— x, un -— u, un e Sn(xn)|.
If the outer and inner limits of the mappings Sn agree, it is said that their graphical limit, g -limnSn, exists. In this case, the notation Sn — S is used, and the sequence {Sn} of mappings is said to converge graphically to S. Clearly, Sn — S & gph Sn ^ gph S.
Proposition 2.5 (see [15]). For any sequence of mappings S n : X —> 2 , it holds that
(g - lim inf Sn )(x)= II lim inf Sn(xn),
\ n I w n — o
X ' {xn — x}
(g - lim sup Sn) (x) = (J lim sup Sn(xn),
\ n / {xn — x} n — o
(2.12)
where the unions are taken over all sequences xn ^ x. Thus, the sequence {Sn} converges graphically to S if and only if, at each point x e X, it holds that
l^j lim sup Sn(xn) c S(x) c y lim inf Sn(xn). (2.13)
{Xn ^ x} n {xn ^ x} n
From Proposition 2.5 and Definition 2.3, the following proposition follows readily.
Proposition 2.6. Let Sn : X ^ 2r be a sequence of set-valued mappings and S : X ^ 2r be a set-valued mapping. Then, the sequence {Sn} outer converges graphically to S if and only if {Sn} outer converges continuously to S, that is,
g - limsup Sn c S ^^ limsup Sn(xn) c S(x) for any x e X, V sequences xn —> x. (2.14)
Definition 2.7 (see [10]). Given a sequence of mappings Sn, {Sn} is said to be uniformly bounded if for any sequence xn contained in a bounded set, there exists a positive number k such that for any sequence un with un e Sn(xn) for all n e N, it holds that
||un|| < k Vn e N. (2.15)
Proposition 2.8 (see [16]). For any fixed e e int C, y e Y, r e R, and the nonlinear scalarization function £ e : Y —> R defined by ¿,e(y) = min{t e R : y e te - C}:
(i) is a continuous and convex function on Y;
(ii) £e(y) <r & y e re - int C;
(iii) £e(y) > r & ye re - int C.
3. Painleve-Kuratowski upper convergence of the solution sets
In this section, our focus is on the Painleve-Kuratowski upper convergence and the closedness of the solution sets.
Theorem 3.1. Suppose that
(i) Tn outer converges continuously to T, that is,
limsup Tn(xn) c T(x) for any sequence {xn} with xn —> x; (3.1)
n —> TO
(ii) Kn K;
(iii) Tn are uniformly bounded.
Then, lim supn(Tn) c I (T), that is to say for any subsequence {xnk} of solutions to (SWVVI)n, if xnk — x, then x is a solution to (SWVVI).
Proof. The proof is listed on contradiction arguments. On a contrary, suppose that 3x e lim supn—TOI(Tn) but x / I(T).
From x e limsup^^I(Tn), we have x = limk— TOxnk, where xnk e I(Tnk) and {nk} is a subsequence of N. Then, there exists tnk e Tnk (xnk) such that
(tnk,z- xnk> e Y \-int C Vz e Knk. (3.2)
Since K c liminfn—TOKn, it is clear that for any zZ e K, there exists a sequence {znk} with {znk} c Knk and znk — z', as k — to. Thus,
{tnk, znk - xnk )e Y \-int C. (3.3)
Since limsup^ooKn c K and xnk e Knk, we have x e K. Now, we note that x e I(T). Thus, for all t e T(x), there exists zt e K such that
(t,zt - x)e-int C. (3.4)
From the uniform boundedness of Tn, we may assume, without loss of generality, that tnk — to (though a subsequence of {tnk} if necessary). By (i), we get t0 e T(x). Thus,
(tnk, znk — xnk )—> (to,z'- x), as k —> +TO. (3.5)
It follows from (3.3) and the closedness of Y \ -int C that
(to,z'- x> e Y \ -int C Vz'e K, (3.6)
which is a contradiction to (3.4). This completed the proof.
Remark 3.2. Let X = E and Y = E*, where E is a reflexive Banach space and E* is its dual. If we take C = R+, (SWVVI)n reduce to the generalized variational inequality problems with perturbed operators (GVI)n considered in [10, Section 3]. The convergence for the solution sets of (GVI)n was studied under the the pseudomonotonicity assumption in [10]. Furthermore, if T and Tn are vector-valued mappings, then (SWVVI)n reduce to (VI)n considered in [10, Section 2]. We also notice that the Painleve-Kuratowski upper convergence of the solution sets of (SWVVI)n is obtained under weaker assumptions than these assumed in [10, Proposition 2.1] for obtaining convergence of the solution sets.
From Proposition 2.5 and Theorem 3.1, we obtain readily the following corollary.
Corollary 3.3. Suppose that
(i) Tn outer converges graphically to T, written as g — limsupnTn c T, that is,
limsup (gph Tn) c gph T; (3.7)
(ii) Kn ^ K;
(iii) {Tn} is uniformly bounded. Then, lim supn^^I(Tn) c I(T).
Remark 3.4. Let X = Y = Rm. Then, problems (SWVVI)n reduce to the generalized variational inequalities with perturbed operators considered in [10, Proposition 3.1] and the convergence was obtained under the assumption that the operators converge graphically.
Theorem 3.5. Suppose that
(i) T is osc on K, that is, for all x e K, limsupn^^T(xn) c T(x) for any sequence xn ^ x;
(ii) K and T(K) are compact sets.
Then, I(T) is a compact set.
Proof. First, we prove that I(T) is a closed set. Take any sequence xn e I(T) with xn ^ x. Then, there exists tn e T(xn) such that
(tn,z- xn) e Y \ —int C Vz e K. (3.8)
It follows from the compactness of K that x e K. Suppose that x e I(T), we have
Vt e T(x), 3z0 e K, s.t. {t, z0 — x) e —int C. (3.9)
Since T(K) is a compact set, without loss of generality, we assume that there exists a t0 such that tn ^ t0. Thus, we have (tn,z — xn) ^ (t0,z — x). By (i), we get a t0 e T(x). It follows from (3.8) and the closedness of Y \ —int C that
{t0,z — x) e Y \— int C Vz e K, (3.10)
which contradicts with (3.9). Hence, x0 e I(T) and I(T) is a closed set. Next, it follows from I(T) c K and the compactness of K that I(T) is a compact set. The proof is completed. □
Similarly, we have the following result.
Theorem 3.6. For any n, suppose that
(i) Tn is osc on Kn, that is, Vx e Kn
limsupTn(xm) c Tn(x) for any sequence xm —> x; (3.11)
(ii) Kn and Tn(Kn) are compact sets. Then, I(Tn) is a compact set.
4. Painleve-Kuratowski lower convergence of the solution sets
In this section, we focus on the lower convergence of the solution sets. Assume that K and Kn are compact sets and that for each x e X, T(x) and Tn(x) are compact sets. Let g : K ^ R and gn : Kn ^ R be functions defined by
g(x) = max min ¿,e((t,y - x)), x e K,
teT(x) yeK
gn(xn) = max min ie({tn,y - xn)), xn e Kn.
tneTn (xn ) yeKn
Since K, K„, T(x), and T„(x) are compact sets and ¡,e(•) is continuous, g(x) and g„(x„) are well defined.
Proposition 4.1. (i) g(x) < 0 for all x e K;
(ii) g„ (x„ ) < 0 for all x„ e K„;
(iii) g (x0) = 0 if and only if x0 e I (T);
(iv) g„(x„) = 0 if and only if x„ e I (T„).
Proof. Define
g(x,t) = min¿,e((t,y - x)), x e K,t e T(x). (4.2)
We first prove that g(x, t) < 0. On a contrary, we suppose that this is false. Then, there exist x e K and t e T(x) such that g(x,t) > 0. Thus,
0 <g(x,t) < £e((i,y - x)) Vy e K, (4.3)
which is impossible when y = x. Therefore,
g(x) = max g(x, t) < 0 Vx e K. (4.4)
teT(x)
By the same taken, we can show that
gn(xn) = max minle{{tn,y — xn}) < 0 Vxn e Kn. (4.5)
tneTn(xn) yeKn
On the other hand, if g(x0) = 0, then there exists a t0 e T(x0) such that g(x0, t0) = 0,
that is,
minle{{h,y — xo)) = 0, x0 e K. (4.6)
From Proposition 2.8, (4.6) is valid if and only if for any y e K,
u{t0,y — xq)) > 0. (4.7)
Clearly, (4.7) holds if and only if for any y e K, (t0,y — x0) e Y \ — intC, that is, xQ e I(T). This proves that (iii) holds.
Similarly, we can show that (iv) holds.
The functions gn are called the gap functions for (SWVVI)n if properties (ii) and (iv) of Proposition 4.1 are satisfied.
In view of hypothesis (Hg) of [6,17,18], we introduce the following key assumption:
(Hg): given the sequence {Tn} for any e > 0, there exist an a > 0 and an n such that gn(xn) < —a for all n>n and for all xn e Kn \ U(I (Tn), e).
Geometrically, the hypothesis (Hg) means that given a sequence of mappings {Tn}, we can find for any small positive number e > 0, a small positive number a > 0 and a large-enough positive number n > 0 such that for all n > n, if a feasible point xn is away from the solution sets I(Tn) by distance of at least e, then the values of all gap functions for (SWVVI)n is less than or equal to at least some "—a."
To illustrate assumption (Hg), we give the following example.
Example 4.2. Let
X = R, Y = R2,
Tn(x) = I 1
1, 1 + - + x2 n
T (x) =(["-1-1 21 \|1,1 + x2J
K = Kn = [0,1], C = R+.
Consider problems (SWVVI)n. From direct computation, we obtain I(Tn) = {0}. To check assumption (Hg), we take e = (1,1)T e int R+. Then,
gn(xn) = max minle{{tn,y - xn))
tneTn (xn ) yeKn
= max min max [(tn, y - xn)l.
tneTn(xn) yeKn 1<i<2 Lx ^ /u (49)
= max min max \y - xn, zn(y - xn)|
z„e[1/1+(1/n)+xn] yeKn
= xn .
For any given 0 < e, we take a = e > 0 and N = 1. Then, for all n > N and for all xn e Kn \ U(I(Tn), e), we have gn(xn) = -xn < -a. Hence, assumption (Hg) is valid. □
Lemma 4.3. Suppose that
(i) Tn inner converges continuously to T, that is,
T(x) c liminf Tn(xn) for any sequence {xn} with xn —> x; (4.10)
(ii) Kn ™ K;
(iii) Un=iKn is a compact set.
Then, for any 6 > 0, x0 e K and sequence {xn} with xn e Kn and xn ^ x0, there exists a subsequence {xni} of {xn} and N > 0 such that gni(xni) > g(x0) - 6 for all l > N.
Proof. Let g : K x L(X,Y) ^ R be a function defined by
g(x, t) = minle((t,y - x)), x e K,t e T(x). (4.11)
From the continuity of ¡,e((t,y - x)) with respect to (x,t,y), the compactness of K and [19, Chapter 3, Section 1, Proposition 23], we have that g(x, t) is continuous with respect to (x, t). Thus, from the compactness of T(x0), there exists a t0 e T(x0) such that
g(x0) = max minlA(t,y - xt))) = max g(x0,t) = minle({h,y - x^). (4.12)
teT(x0) yeK teT(x0) yeK
From assumption (i), there exists a sequence {t„} satisfying t„ e T„(x„) such that
t„ t0. (4.13)
It follows from the compactness of K„ that there exists {y„} with y„ e K„ such that
min ie({tn,y - x^) = ie{{tn,yn - Xn)) .
(4.14)
Since Un=1Kn is compact, we assume, without loss of generality, that yn ^ y0. Thus, it follows from (ii) that y0 e K. Consequently,
lim le{{tn,yn - Xn)) = £,e({to,yo - x0)) > minle{{to,y - xt))) = g(xo). (4.15)
n ^ oo yeK.
So, for any 6 > 0, there exists an N > 0 such that ¿,e ((tn,yn - xn)) > g (xo) - 6 for all n > N .By (4.14), we have
gn(xn) = max minle{{tn,y - xn))
t„eT„(x„) yeKn
> min le{{tn,y - xn))
yeK ^ (4.16)
= yn - xn})
> g(xo) - 6 Vn > N.
Hence, the result holds. □
Set T0 = T and K0 = K. We have the following lemma. Lemma 4.4. Suppose that for n = 0,1,2,...,Tn is osc on Kn, that is, for n = 0,1,2,...,
lim sup Tn(xm) c Tn (x) for any sequence {xm} with xm —> x. (4.17)
Then, I (T) c liminfn (Tn) if and only if for all e> 0, 3N > 0 such that I (T) c U(I (Tn),e) for all n> N.
Proof. We assume I(T) c liminfn(Tn), but there exists an e > 0 such that for all N > 0, there exists an Nn > N satisfying I(T) / U(I(TNn),e). Then, there exists a sequence {xn} with xn e I(T), but xn / U(I(TNn), e). From Theorem 3.5, we note that I(T) is a compact set. Without loss of generality, we assume xn ^ x and x e I(T). Thus, for any sequence {yn} satisfying yn ^ y with yn e I(Tn), we have \\yNn - xn\\ > e > 0. Letting n ^ o, we get \\y - x\ > e > 0. Therefore, there does not exist any sequence yn e I(Tn) satisfying yn ^ x. This is a contradiction to I(T) c liminfn^oI(Tn).
Conversely, suppose that for any e > 0, 3N > 0 such that I(T) c U(I(Tn),e) for all n > N. From Theorem 3.6, we note that I(Tn) is compact for all n. Thus, for any x e I(T), there exists xn e I(Tn) such that \\xn - x\\ = d(x,I(Tn)) < e for all n > N. So, we have xn ^ x and I(T) c liminfn(Tn). Therefore, the result of the lemma follows readily. □
Now, we are in a position to state and prove our main result in the following theorem.
Theorem 4.5. Suppose that assumption (Hg) holds and that the following conditions are satisfied:
(i) Tn is osc on Kn for n = 0,1,2,..., that is, for n = 0,1,2,...,
lim sup Tn(xm) c Tn (x) for any sequence {xm} with xm —> x; (4.18)
(ii) Tn inner converge continuously to T, that is,
T(x) c liminf Tn(xn) for any sequence {xn} with xn —> x; (4.19)
(iii) Un=iKn is a compact set;
(iv) Kn K.
Then, I(T) c liminfn—nI(Tn).
Proof. We prove the result via contradiction. On a contrary, we assume, by Lemma 4.4, that there exists an e > 0 such that for any N > 0, we have Nn > N satisfying
I(T) C U(I(Tn„),e), (4.20)
that is, there exists a sequence {xNn} satisfying
xn„ e I(T) \ U(I(Tn„),e). (4.21)
From the compactness of I(T), we can assume, without loss of generality, that xNn — x e I(T). Then, there exists an N1 > 0 such that \\xn„ - x|| < e/4 forall n > N1. It is clear that B(x, e/Nn) nK = 0 for any positive integer n. Since K c lim infn—nKn, there exist a sequence {yNn} c KNn satisfying yNn — x. Then, there exists an N2 > 0 such that yNn e B(x, e/Nn) n Kn„ for all n > N2.
Now, we note that yNne U(I(Tn„ ),e/4). Otherwise, there would exist a sequence {zNn} with ZNn e I(TNn) such that \\yNn - zn„\\ < e/4. Thus, for N0 = max{N1,N}, we have
IIxNn - ZNn || < ||xNn - x|| + ||x - yNn || + \\yNn - ZNn || < 4 + Nfn + 4 < e Vn > N°. (4.22)
This implies that xNn e U(I(TNn), e), which contradicts with (4.21). Thus,
yNn e Kn„ \ u(i(TNn),|). (4.23)
By hypothesis (Hg), there exist, for any e > 0, an a > 0 and an N such that for all n> N and for all x e Kn \ U(I (Tn), e), gn(x) < -a. In particular, it follows from (4.23) that
gNn(yNn) < -a for n large enough. (4.24)
By virtue of Lemma 4.3, there exists, for any 6 > 0, a subsequence {yN„k} of {yNn} and N > 0 such that
gNnk (yNnk) > g(x) - 6 Vk>N. (4.25)
We can take 6 such that -a + 6 < 0. Thus,
g(x) < gN„k (yN„k) + 6 <-a + 6 < 0, (4.26)
that is,
max min¿e((t,y - x)) < 0. (4.27)
t T(x) y K
So, for any t e T(x), minyeK£e((t,y - x)) < 0. Thus, there exists a y e K such that
¿e((t,y - x)) < 0. (4.28)
Consequently, by Proposition 2.8, we have (t,y - x) e -int C, which shows that x /1(T). This contradicts with x e I(T). Therefore, our result follows readily. □
Now, we explain the applicability of Theorem 4.5 through an example.
Example 4.6. Consider Example 4.2. It follows from a direct computation that I(Tn) = I(T) = {0}. It is easy to testify that assumption (Hg) holds and so are conditions (i)-(v) of Theorem 4.5. Obviously, the solution sets of problem (SWVVI)n lower converge in the sense of Painleve-Kuratowski.
Acknowledgment
This research was partially supported by the National Natural Science Foundation of China (Grants no. 10871216 and no. 60574073).
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