Download EXISTENCE AND STABILITY OF SOLUTIONS FOR GENERALIZED

TAIWANESE JOURNAL OF MATHEMATICS
Vol. 16, No. 3, pp. 941-962, June 2012
This paper is available online at http://journal.taiwanmathsoc.org.tw
EXISTENCE AND STABILITY OF SOLUTIONS FOR GENERALIZED
SYMMETRIC STRONG VECTOR QUASI-EQUILIBRIUM PROBLEMS
Bin Chen, Nan-jing Huang and Ching-Feng Wen*
Abstract. In this paper, a class of generalized symmetric strong vector quasiequilibrium problems in real locally convex Hausdorff topological vector spaces is
studied. By using the Kakutani-Fan-Glicksberg fixed point theorem, an existence
theorem of solutions for the generalized symmetric strong vector quasi-equilibrium
problems is obtained. Moreover, the closedness of the solution set and a stability
result of solutions for such problem are also derived.
1. INTRODUCTION
The equilibrium problem contains many important problems as special cases, such
as optimization problems, problems of Nash equilibrium, fixed point problems, variational inequalities and complementarity problems. In recent years, there has been an
increasing interest in the study of vector equilibrium problems. A lot of existence
results of solutions for vector equilibrium problems and vector variational inequalities
have been established (see, e.g.,[2, 4, 10, 17] and the references therein).
In 1994, Noor and Oettli [34] introduced and studied the symmetric quasi-equilibrium
problem which is a generalization of equilibrium problem proposed by Blum and Oettli
[7]. In 2003, Fu[19] introduced the symmetric vector quasi-equilibrium problem which
is a generalization of the symmetric quasi-equilibrium problem proposed by Noor and
Oettli [34] and gave an existence theorem for weak Pareto solution for the symmetric vector quasi-equilibrium problem in Hausdorff locally convex spaces. Farajzadeh
[16] supplied a further extension to Hausdorff topological vector spaces with several
assumptions being relaxed. Anh and Khanh [2] extended the problem considered in
Noor and Oettli [34], Fu [19] and Farajzadeh [16] from the single-valued case to the
Received December 10, 2010, accepted March 29, 2011.
Communicated by Jen-Chih Yao.
2010 Mathematics Subject Classification: 49J40, 90C29.
Key words and phrases: Generalized symmetric strong vector quasi-equilibrium problems, Stability,
Kakutani-Fan-Glicksberg fixed point theorem.
This work was supported by the Key Program of NSFC (Grant No. 70831005), the National Natural
Science Foundation of China (10671135) and NSC 99-2115-M-037-001.
*Corresponding author.
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Bin Chen, Nan-jing Huang and Ching-Feng Wen
multivalued case in Hausdorff topological vector spaces. Gong [23] introduced the
symmetric strong vector quasi-equilibrium problem and gave an existence theorem of
strong efficient solution for symmetric vector quasi-equilibrium problem in Hausdorff
locally convex spaces. It is well known that a strong efficient solution of a vector
equilibrium problem is an idea solution. It is better than other solutions such as efficient solution, weak efficient solution, proper efficient solution and supper efficient
solution(see [24]). Hence, it is important to study the existence of strong efficient
solution and properties of the strong efficient solution set. Recently, Hou, Gong and
Yang [26] derived an existence theorem of strong efficient solution for generalized
strong vector quasi-equilibrium problem and discussed the stability of strong efficient
solutions. Long, Huang and Teo [33], Yu and Gong [37] extended the main results of
Hou, Gong and Yang [26] from single-valued mappings to set-valued mappings.
On the other hand, one of important problems of vector equilibrium is to investigate
the stability of solutions. For the stability of symmetric equilibrium problems, there
have been limited number of works in the literature. Recently, Anh and Khanh [3]
derived various kinds of semicontinuity for the solution sets of parametric multivalued
symmetric vector quasi-equilibrium problems. Chen and Gong [12] studied the stability
of the solutions set for symmetric vector quasi-equilibrium problems. To the best of
our knowledge, no paper has been devoted to the study of stability for generalized
symmetric strong vector quasi-equilibrium problems.
Motivated and inspired by the research works mentioned above, in this paper, we
consider a class of generalized symmetric strong vector quasi-equilibrium problems
in real locally convex Hausdorff topological vector spaces. We establish an existence
theorem of solutions by using Kakutani-Fan-Glicksberg fixed point theorem and discuss
the closedness of the solution set for the generalized symmetric strong vector quasiequilibrium problems. Moreover, we also show a stability result for such problem.
2. PRELIMINARIES RESULTS
Throughout this paper, unless specified otherwise, we suppose that X, Y and Z be
real locally convex Hausdorff topological vector spaces and C ⊂ Z be a closed convex
cone. The convex cone induces a partially ordering in Z, defined by
z1 ≤ z2 (or z2 ≥ z1 ) if and only if z2 − z1 ∈ C.
Let E be a a nonempty subset of X, and D be a nonempty subset of Y . Let S :
E × D → 2E , T : E × D → 2D , F : E × D × E → 2Z and G : D × E × D → 2Z
be four set-valued mappings.
In this paper, we consider the following generalized symmetric strong vector quasiequilibrium problem(in short, GSSVQEP): finding (x̄, ȳ) ∈ E × D such that x̄ ∈
S (x̄, ȳ), ȳ ∈ T (x̄, ȳ) and
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F (x̄, ȳ, x) ⊂ C, ∀x ∈ S (x̄, ȳ) ,
G(ȳ, x̄, y) ⊂ C, ∀y ∈ T (x̄, ȳ) .
For our main results, we need some definitions and lemmas as follows.
Definition 2.1. ([1]). Let X and Y be two topological vector spaces, and T : X →
2 be a set-valued mapping.
Y
(i) T is said to be upper semicontinuous at x ∈ X if, for any neighborhood U of
T (x), there is a neighborhood V of x such that
T (t) ⊂ U, ∀t ∈ V.
T is said to be upper semicontinuous on X if it is upper semicontinuous at each
x ∈ X.
(ii) T is said to be lower semicontinuous at x ∈ X if, for any y ∈ T (x) and for
any net {xα } converging to x, there exists a net {yα} such that yα ∈ T (xα) and
{yα } converges to y.
T is said to be lower semicontinuous on X if it is lower semicontinuous at each
x ∈ X.
(iii) T is said to be continuous on X if it is both upper semicontinuous and lower
semicontinuous on X.
(iv) T is said to be closed, if Graph(T ) = {(x, y) : x ∈ X, y ∈ T (x)} is a closed
subset in X × Y.
Definition 2.2. Let W be a topological vector space and D ⊂ W be a nonempty
set. A set-valued mapping G : D → 2Z is said to be C-upper semicontinuous on x 0
if, for any neighborhood U of 0 in Z, there exists a neighborhood U (x0 ) of x0 such
that
G (x) ⊂ G (x0 ) + U + C, ∀x ∈ U (x0 ) ∩ D.
Definition 2.3. Let W be a topological vector space and D ⊂ W be a nonempty
set. A set-valued mapping G : D → 2Z is said to be C-lower semicontinuous on x0 if,
for each z ∈ G (x0 ), and any neighborhood U of 0 in Z, there exists a neighborhood
U (x0 ) of x0 such that
G (x) ∩ (z + U − C) = ∅, ∀x ∈ U (x0 ) ∩ D.
Definition 2.4. Let W be a topological vector space and D ⊂ W be a nonempty
set. A set-valued mapping G : D → 2Z is said to be C-quasiconvex if, for any z ∈ W ,
x1 , x2 ∈ D, t ∈ [0, 1], z1 ∈ G (x1 ) , z2 ∈ G (x2 ) and z1 ≤ z, z2 ≤ z, there exist
zt ∈ G(tx1 + (1 − t)x2 ) such that z t ≤ z.
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Bin Chen, Nan-jing Huang and Ching-Feng Wen
Definition 2.5. Let W be a topological vector space and D ⊂ W be a nonempty
set. A set-valued mapping G : D → 2Z is said to be C-properly quasiconvex if, for
any x, y ∈ D, t ∈ [0, 1], u ∈ G(x), v ∈ G(y), there exists z ∈ G(tx + (1 − t)y) such
that either z ≤ u or z ≤ v.
Lemma 2.1. ([1]). Let X and Y be two Hausdorff topological vector spaces and
T : X → 2Y be a set-valued mapping.
(i) If T is upper semicontinuous with closed values, then T is closed.
(ii) If T is closed and Y is compact, then T is upper semicontinuous.
Lemma 2.2. (Kakutani-Fan-Glicksberg [25]). Let X be a locally convex Hausdorff
topological vector space and K be a nonempty compact convex subset of X. Let
T : K → 2K be a upper semicontinuous set-valued mapping with nonempty closed
convex values. Then there exists x̄ ∈ K such that x̄ ∈ T (x̄) .
Lemma 2.3. Let D be a nonempty convex subset of Z and G : D → 2 Z be
C-upper semicontinuous mapping with compact values such that for every x ∈ D,
there exists z 0 ∈ G(x) such that z 0 ≤ z for all z ∈ G(x). Then G is C-properly
quasiconvex if and only if the following conditions hold:
(i) For any x, y ∈ D, there exists t 0 ∈ [0, 1] such that for all u ∈ G(x) and
v ∈ G(y), there exists z ∈ G(t0 x + (1 − t0 )y) with z ≤ u and z ≤ v;
(ii) G is C-quasiconvex.
Proof. The necessity see the Lemma 2.1 of [37]. Next we prove the sufficiency,
i.e. if the conditions (i) and (ii) are satisfied, then G is C-properly quasiconvex.
If t0 = 1, then by (i), for u ∈ G(x), v ∈ G(y), there exists z ∈ G(x) such that
z ≤ u and z ≤ v. By the assumption, there exists z1 ∈ G(y) such that z 1 ≤ v. By
(ii), there exists zt ∈ G(tx + (1 − t)y) such that z t ≤ v.
If t0 = 0, then it follows from (i) that, for u ∈ G(x), v ∈ G(y), there exists
z ∈ G(y) such that z ≤ u and z ≤ v. By the assumption, there exists z2 ∈ G(x) such
that z2 ≤ u. The condition (ii) implies that there exists z t ∈ G(tx + (1 − t)y) such
that zt ≤ u.
If t0 ∈ (0, 1), then for u ∈ G(x), v ∈ G(y) and t ∈ [0, 1], when t ≥ t0 , take
α = (t − t0 )/(1 − t0 ) ∈ [0, 1]. Letting xt0 = t0 x + (1 − t0 ) y, the condition (i)
implies that there exists z ∈ G (x t0 ) such that z ≤ u and z ≤ v. By the assumption,
there exists z1 ∈ G(x) such that z 1 ≤ u. It follows from (ii) that there exists zt ∈
G (αx + (1 − α) xt0 ) = G (tx + (1 − t) y) such that z t ≤ u. When t < t0 , take α =
(t0 − t)/t0 ∈ [0, 1]. Letting xt0 = t0 x+(1 − t0 ) y, by (i), there exists z ∈ G (xt0 ) such
that z ≤ u and z ≤ v. By the assumption, there exists z2 ∈ G(y) such that z 2 ≤ v. The
condition (ii) implies that there exists z t ∈ G (αy + (1 − α) xt0 ) = G (tx + (1 − t) y)
such that z t ≤ v.
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Hence, for u ∈ G(x), v ∈ G(y), t ∈ [0, 1], there exists zt ∈ G (tx + (1 − t) y)
such that either z t ≤ u or zt ≤ v. Thus, G is C-properly quasiconvex.
3. EXISTENCE OF SOLUTIONS
In this section, we establish an existence theorem of solutions by using KakutaniFan-Glicksberg fixed point theorem and discuss the closedness of the solution set for
the generalized symmetric strong vector quasi-equilibrium problems.
Theorem 3.1. Let X, Y, Z be real locally convex Hausdorff topological vector
spaces, E ⊂ X and D ⊂ Y be nonempty compact convex sets. Let S : E × D → 2 E
and T : E × D → 2D be continuous set-valued mappings with nonempty compact
convex values. Let F : E × D × E → 2Z and G : D × E × D → 2Z be C-upper
semicontinuous and C-lower semicontinuous mappings with nonempty compact values.
Assume that
(i) For each (x, y, x ) ∈ E × D × E, there exists z ∈ F (x, y, x) such that
z ≤ z , ∀z ∈ F (x, y, x);
For each (y, x, y ) ∈ D × E × D, there exists z ∈ G(y, x, y ) such that
z ≤ z , ∀z ∈ G(y, x, y );
(ii) For all (x, y) ∈ E × D, F (x, y, x) ⊂ C and G(y, x, y) ⊂ C;
(iii) For all (x, y) ∈ E ×D, F (x, y, u) is C-properly quasiconvex in u and G(y, x, v)
is C-properly quasiconvex in v.
Then GSSVQEP has a solution. Moreover, the solution set of GSSVQEP is closed.
Proof. For any (x, y) ∈ E × D, define A : E × D → 2E and B : E × D → 2D
by
A(x, y) = {v ∈ S(x, y) : for any u ∈ S(x, y), z ∈ F (x, y, u),
there exists w ∈ F (x, y, v), such that w ≤ z} ,
B(x, y) = {a ∈ T (x, y) : for any b ∈ T (x, y), e ∈ G(y, x, b),
there exists d ∈ G(y, x, a), such that d ≤ e} .
(I) For any (x, y) ∈ E × D, A(x, y) is nonempty.
Indeed, for every u ∈ S(x, y), set
H(u) = {v ∈ S(x, y) : for any z ∈ F (x, y, u),
there exists w ∈ F (x, y, v), such that w ≤ z}.
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Bin Chen, Nan-jing Huang and Ching-Feng Wen
Then u ∈ H(u) and so H(u) = ∅.
Now we show the family {H(u) : u ∈ S(x, y)} has the finite intersection property.
Letting u1 , u2 ∈ S(x, y), by the assumptions and Lemma 2.3, there exists t ∈ [0, 1]
such that for any z1 ∈ F (x, y, u1) and z2 ∈ F (x, y, u2 ), there exists z ∈ F (x, y, tu1 +
(1 − t)u2 ) with
z ≤ z1 and z ≤ z2 .
From the convexity of S(x, y), we know that v := tu1 + (1 − t)u2 ∈ S(x, y) and so
n
H(ui) = ∅. Then, there
v ∈ H(u1 ) ∩ H(u2 ). Let u1 , u2, · · · , un ∈ S(x, y) and
exists v ∈
n
i=1
i=1
H(ui). By the definition, v ∈ S(x, y). For any zi ∈ F (x, y, ui), i =
1, 2, · · · , n, there exists wi ∈ F (x, y, v) such that
wi ≤ zi .
(3.1)
By the assumptions, there exists w ∈ F (x, y, v) such that w ≤ w i for i = 1, 2, · · · , n,
It follows from (3.1) that
w ≤ zi , i = 1, 2, · · · , n.
(3.2)
Let un+1 ∈ S(x, y). For any zn+1 ∈ F (x, y, un+1 ), by the assumptions and lemma
2.3, there exists t ∈ [0, 1] and z0 ∈ F (x, y, tv + (1 − t)un+1 ) such that
z0 ≤ w and z0 ≤ zn+1 .
(3.3)
From (3.2) and (3.3), we have
z0 ≤ zi , i = 1.2, · · · , n + 1.
By the convexity of S(x, y), we have tv + (1 − t)un+1 ∈ S(x, y) and so
tv + (1 − t)un+1 ∈
n+1
H(ui).
i=1
Next, we show that H(u) is closed. Let {vα : α ∈ I} ⊂ H(u) be a net such that v α →
v. Then {vα } ⊂ S(x, y) and for any z ∈ F (x, y, u), there exists wα ∈ F (x, y, vα)
such that
(3.4)
wα ≤ z.
It follows from the closedness of S(x, y) that we have v ∈ S(x, y). We claim that, for
any z ∈ F (x, y, u), there exists w ∈ F (x, y, v) such that w ≤ z. If not, there exists
z0 ∈ F (x, y, u) such that for any w ∈ F (x, y, v),
/ C.
z0 − w ∈
Generalized Symmetric Strong Vector Quasi-equilibrium Problems
947
Hence,
(z0 − F (x, y, v)) ∩ C = ∅.
Since C is closed convex cone and F (x, y, v) is compact, there exists some neighborhood U of 0 such that
(z0 − (F (x, y, v) + U + C)) ∩ C = ∅.
(3.5)
Since vα → v and F (x, y, v) is C-upper semicontinuous in v, there exists α 0 ∈ I such
that for all α ≥ α0 ,
F (x, y, vα) ⊂ F (x, y, v) + U + C.
It follows from (3.5) that,
(z0 − F (x, y, vα)) ∩ C = ∅, ∀α ≥ α0 .
(3.6)
Because z0 ∈ F (x, y, u), by (3.4), there exists wα ∈ F (x, y, vα) such that z 0 −wα ∈ C.
This contradicts (3.6) and so H(u) is closed.
Since S(x, y) is closed and E is compact, we know that S(x, y) is compact. Hence,
H(u) = ∅.
u∈S(x,y)
Letting v ∈
u∈S(x,y)
H(u), then v ∈ S(x, y), and for any u ∈ S(x, y), z ∈ F (x, y, u),
there exists w ∈ F (x, y, v) such that w ≤ z. Note that v ∈ A(x, y). Thus, A(x, y) is
nonempty.
(II) For any (x, y) ∈ E × D, A(x, y) is closed subset of E.
In fact, letting {vα : α ∈ I} ⊂ A(x, y) with vα → v ∈ E. Then {vα : α ∈ I} ⊂
S(x, y). For any u ∈ S(x, y) and z ∈ F (x, y, u), there exists wα ∈ F (x, y, vα) such
that
(3.7)
wα ≤ z.
By the closedness of S(x, y), it follows that v ∈ S(x, y). If v ∈
/ A(x, y), then there
/ C.
exists u0 ∈ S(x, y) and z0 ∈ F (x, y, u0 ) such that for any w ∈ F (x, y, v), z0 − w ∈
Hence,
(3.8)
(z0 − F (x, y, v)) ∩ C = ∅.
Since C is closed convex cone and F (x, y, v) is compact, there exists some neighborhood U of 0 such that
(3.9)
(z0 − (F (x, y, v) + U + C)) ∩ C = ∅.
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Bin Chen, Nan-jing Huang and Ching-Feng Wen
Since vα → v and F (x, y, v) is C-upper semicontinuous in v, there exists α 0 ∈ I such
that for all α ≥ α0 ,
(3.10)
F (x, y, vα) ⊂ F (x, y, v) + U + C.
By (3.9) and (3.10), we have
(3.11)
(z0 − F (x, y, vα)) ∩ C = ∅, ∀α ≥ α0 .
On the other hand, by (3.7), there exists wα ∈ F (x, y, vα) such that z 0 − wα ∈ C.
This contradicts (3.11). Hence, v ∈ A(x, y) and so A(x, y) is closed.
(III) For any (x, y) ∈ E × D, A(x, y) is convex.
In fact, letting v1 , v2 ∈ A(x, y), then v1 , v2 ∈ S(x, y) and for any u ∈ S(x, y),
z ∈ F (x, y, u), there exist z1 ∈ F (x, y, v1) and z2 ∈ F (x, y, v2 ) such that
(3.12)
z1 ≤ z and z2 ≤ z.
By Lemma 2.3, F (x, y, u) is C-quasiconvex in u. Thus, for any t ∈ [0, 1], there exists
zt ∈ F (x, y, tv1 + (1 − t)v2 ) such that z t ≤ z. Since S(x, y) is convex, we know that
tv1 + (1 − t)v2 ∈ S(x, y) and so tv1 + (1 − t)v2 ∈ A(x, y). Hence, A(x, y) is convex.
(IV) A(x, y) is upper semicontinuous on E × D.
By the compactness of E × D, we only need to show that A is a closed mapping.
Let {(xα, yα) : α ∈ I} ⊂ E × D be a net such that (x α , yα) → (x, y) ∈ E × D. Let
vα ∈ A(xα, yα ) with vα → v. We will show v ∈ A(x, y).
Since S is upper semicontinuous mapping with nonempty closed values, it follows
that S is a closed mapping. Since vα ∈ S(xα, yα) and (xα, yα , vα) → (x, y, v), then
v ∈ S(x, y). Next we show v ∈ A(x, y), i.e.,
for any u ∈ S(x, y), z ∈ F (x, y, u), there exists w ∈ F (x, y, v), such that w ≤ z.
If not, then there exists u0 ∈ S(x, y) and z0 ∈ F (x, y, u0 ) such that for any w ∈
/ C. Then
F (x, y, v), z0 − w ∈
(z0 − F (x, y, v)) ∩ C = ∅.
Since C is closed convex cone and F (x, y, v) is compact, there exists some neighborhood U of 0 such that
(z0 − (F (x, y, v) + U + C)) ∩ C = ∅.
There exists a balanced neighborhood U1 of 0 such that U1 − U 1 ⊂ U , and so
(3.13)
((z0 + U1 − C) − (F (x, y, v) + U1 + C)) ∩ C = ∅.
Generalized Symmetric Strong Vector Quasi-equilibrium Problems
949
Since u0 ∈ S(x, y), (xα, yα) → (x, y), S is lower semicontinuous, there exists uα ∈
S(xα, yα) such that u α → u0 . Hence, (xα , yα, uα) → (x, y, u0) and (xα, yα, vα) →
(x, y, v). Since F is C-lower semicontinuous and C-upper semicontinuous, for z0 ∈
F (x, y, u0) and U1 , there exist neighborhood U (x, y, u0) of (x, y, u0) such that, when
(x , y , u ) ∈ U (x, y, u0),
F (x , y , u ) ∩ (z0 + U1 − C) = ∅.
Further, there exist neighborhood U (x, y, v) of (x, y, v) such that, when (x , y , v ) ∈
U (x, y, v),
F (x , y , v ) ⊂ F (x, y, v) + U1 + C.
Hence, there exists α0 ∈ I such that, when α ≥ α 0 ,
(3.14)
F (xα , yα, uα) ∩ (z0 + U1 − C) = ∅.
and
(3.15)
F (xα , yα , vα) ⊂ F (x, y, v) + U1 + C.
Let zα ∈ F (xα , yα, uα) ∩ (z0 + U1 − C). (3.13) and (3.15) implies that
(3.16)
((zα − (F (xα, yα, vα)) ∩ C = ∅.
Since vα ∈ A(xα , yα), for uα ∈ S(xα, yα) and zα ∈ F (xα , yα, uα), there exists
wα ∈ F (xα , yα , vα) such that
(3.17)
wα ≤ zα.
This contradicts (3.16) and so A is a closed mapping.
Similarly, we can prove that, for any (x, y) ∈ E × D, B is upper semicontinuous
on E × D with nonempty closed convex values.
(V) Define the set-valued mapping H : E × D → 2E×D by
H(x, y) = (A(x, y), B(x, y)), ∀ (x, y) ∈ E × D.
Then for each (x, y) ∈ E × D, H(x, y) is a nonempty closed convex subset of E × D
and H is upper semicontinuous on E × D. By Lemma 2.2, there exists a point
(x̄, ȳ) ∈ E × D such that (x̄, ȳ) ∈ H(x̄, ȳ), i.e., x̄ ∈ A (x̄, ȳ) and ȳ ∈ B (x̄, ȳ).
It follows from the definition of A that for any u ∈ S (x̄, ȳ) and z ∈ F (x̄, ȳ, u),
there exists w ∈ F (x̄, ȳ, x̄) such that w ≤ z. Similarly, for any v ∈ T (x̄, ȳ) and
e ∈ G(ȳ, x̄, v), there exists d ∈ G(ȳ, x̄, ȳ) such that d ≤ e. By (ii), we have w ∈ C
and d ∈ C. Thus, z ∈ C and e ∈ C. It follows that x̄ ∈ S (x̄, ȳ), ȳ ∈ T (x̄, ȳ) and
F (x̄, ȳ, x) ⊂ C, ∀x ∈ S (x̄, ȳ) ,
G(ȳ, x̄, y) ⊂ C, ∀y ∈ T (x̄, ȳ) .
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Bin Chen, Nan-jing Huang and Ching-Feng Wen
Next we show that the solution set of GSSVQEP is closed. Let {(x α, yα) : α ∈ I}
be a net in the set of solutions of GSSVQEP such that (x α , yα) → (x̄, ȳ). Then
xα ∈ S (xα, yα ), yα ∈ T (xα , yα) and
(3.18)
F (xα , yα, x) ⊂ C, ∀x ∈ S (xα, yα ) ,
(3.19)
G(yα, xα, y) ⊂ C, ∀y ∈ T (xα , yα) .
Since S and T are upper semicontinuous set-valued mappings with nonempty compact
values, by Lemma 2.1, S and T are closed mappings and so x̄ ∈ S (x̄, ȳ), ȳ ∈ T (x̄, ȳ).
Since S and T are lower semicontinuous set-valued mappings, for any z ∈ S(x̄, ȳ),
there exist zα ∈ S(xα, yα) such that z α → z. By (3.18), we have
F (xα , yα, zα) ⊂ C.
(3.20)
We claim that
F (x̄, ȳ, z) ⊂ C.
If not, then there exists m ∈ F (x̄, ȳ, z) such that m ∈
/ C. Thus, there exists some
neighborhood U of 0 such that (m + U ) ∩ C = ∅ and so
(m + U − C) ∩ C = ∅.
(3.21)
Since F : E × D × E → 2Z is C-lower semicontinuous mapping, there exists α0 such
that when α ≥ α0 ,
F (xα , yα, zα) ∩ (m + U − C) = ∅.
/ C.
Taking mα ∈ F (xα , yα , zα) and mα ∈ (m + U − C), by (3.21), we have mα ∈
However, by (3.20), we have mα ∈ C, which is a contradiction. Therefore, by the
arbitrariness of z ∈ S (x̄, ȳ), we have
F (x̄, ȳ, z) ⊂ C, ∀z ∈ S (x̄, ȳ) .
Similarly, we can prove that
G (ȳ, x̄, y) ⊂ C, ∀y ∈ T (x̄, ȳ) .
This shows that (x̄, ȳ) is the solution of GSSVQEP and so the solution set of GSSVQEP
is closed. This completes the proof.
Now we give an example to explain that Theorem 3.1 is applicable.
Example 3.1. Let X = Y = Z = R, C = [0, +∞), and E = D = [0, 1]. For each
x ∈ E, y ∈ D, S(x, y) = [0, 1] and T (x, y) = [0, 1]. Define the set-valued mappings
F and G as follows:
F (x, y, z) = [2x + y − z, 10], ∀(x, y, z) ∈ E × D × E,
Generalized Symmetric Strong Vector Quasi-equilibrium Problems
951
G(y, x, z) = [x + 2y − z, 5], ∀(y, x, z) ∈ D × E × D.
It is easy to check that all conditions in Theorem 3.1 are satisfied. Hence, by Theorem
3.1, GSSVQEP has a solution. Let H be the solution set of GSSVQEP. Then,
H = {x̄, ȳ) ∈ E × D : 2x̄ + ȳ ≥ 1, x̄ + 2ȳ ≥ 1}.
It is easy to see that H is a closed subset of E × D.
4. STABILITY
In this section, we discuss the stability of solutions for generalized symmetric strong
vector quasi-equilibrium problems.
Let (X, d) be a metric space. Denote by K(X), BC(X), and CK(X) all nonempty
compact subsets of X, all nonempty bounded closed subsets of X, and all nonempty
convex compact subsets of X (if X is a linear metric space), respectively. Let B1 , B2 ⊂
X and define
h (B1 , B2 ) = max h0 (B1 , B2 ) , h0 (B2 , B1 ) ,
where h0 (B1 , B2 ) = sup d (b, B2 ), and d (b, B2 ) = inf d (b, b). It is obvious that
b ∈B2
b∈B1
h is a Hausdorff metric on K(X), BC(X), CK(X) respectively.
Lemma 4.1. ([32]). Let E be a nonempty compact subset of (X, · X ) and D be a
nonempty compact subset of (Y, · Y ). Let S : E × D → 2E and T : E × D → 2D be
two set-valued mappings with nonempty compact values. Then S and T are continuous
if and only if, for any (x ∗ , y ∗ ) ∈ E × D, (x, y) → (x∗ , y ∗), implies
h
h
S (x, y) −→ S (x∗ , y ∗ )and T (x, y) −→ T (x∗ , y ∗ ),
where h is a Hausdorff metric on K(E) and h is a Hausdorff metric on K(D).
Lemma 4.2. ([32]). Let (X, d) be a metric space and h be Hausdorff metric on
X. Then
(i) (BC(X), h) is complete if and only if (X, d) is complete;
(ii) (K(X), h) is complete if and only if (X, d) is complete;
(iii) If X is a linear metric space, then (CK(X), h) is complete if and only if (X, d)
is complete.
Lemma 4.3. ([36]). Let W be a metric space, A, A n ⊂ W (n = 1, 2, · · ·) be
compact subsets. If for any open set O with A ⊂ O, there exists n 0 such that
An ⊂ O for all n ≥ n 0 , then any sequence {xn }, satisfying x n ∈ An has a convergent
subsequence with limit in A.
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Bin Chen, Nan-jing Huang and Ching-Feng Wen
Throughout this section, let X, Y and Z be Banach spaces. Let E ⊂ X and
D ⊂ Y be nonempty compact convex sets. Let
M ={(S, T, F, G) : S : E ×D → 2E and T : E × D →2D are continuous set-valued
mappings with nonempty compact convex values, F : E ×D× E → 2Z and G :
D × E × D → 2Z are C-upper semicontinuous and C-lower semicontinuous
mappingswith nonempty compact values such that for every fixed (x, y) ∈ E
× D, F (x, y, u) is C-properly quasiconvex in u and G(y, x, v) is C-properly
quasiconvex in v}.
For any u1 = (S1 , T1 , F1 , G1 ) and u2 = (S2 , T2 , F2 , G2 ) ∈ M , define
ρ (u1 , u2) =
sup
(x,y)∈E×D
+
+
h1(S1 (x, y) , S2 (x, y))+
sup
h2 (T1 (x, y) , T2 (x, y))
(x,y)∈E×D
sup
h3 (F1 (x, y, u) , F2 (x, y, u))
sup
h3 (G1 (y, x, v) , G2 (y, x, v)) .
(x,y,u)∈E×D×E
(y,x,v)∈D×E×D
where h1 , h2 and h3 are Hausdorff metrics on CK(E), CK(D) and C(Z), respectively.
Proposition 4.1. (M, ρ) is a complete metric space.
Proof. It is clear that (M, ρ) is a metric space. Now, we show that (M, ρ) is
complete.
Let {un } be any Cauchy sequence in M , where un = (Sn , Tn, Fn , Gn), n = 1, 2, . . .
Then, for any ε > 0, there exists N such that
(4.1)
ρ (un , um) < ε/4, ∀n, m ≥ N.
It follows that, for any (x, y, u, v) ∈ E × D × E × D,
(4.2)
h1 (Sn (x, y) , Sm (x, y)) < ε/4,
h2 (Tn (x, y) , Tm (x, y)) < ε/4,
and
(4.3)
h3 (Fn (x, y, u) , Fm (x, y, u)) < ε/4,
h3 (Gn (y, x, v) , Gm (y, x, v)) < ε/4
Then, for any fixed (x, y, u, v) ∈ E × D × E × D, {Sn (x, y)} is a Cauchy sequence in CK(E), {Tn(x, y)} is a Cauchy sequence in CK(D), and {Fn (x, y, u)},
{Gn (y, x, v)} are two Cauchy sequences in K(Z). By Lemma 4.2 and assumption, (CK(E), h1), (CK(D), h2 ) and (K(Z), h3) are complete spaces. It follows
Generalized Symmetric Strong Vector Quasi-equilibrium Problems
953
that there exist S(x, y) ∈ CK(E), T (x, y) ∈ CK(D), F (x, y, u) ∈ K(Z) and
G(y, x, v) ∈ K(Z) such that
(4.4)
h
1
S (x, y),
Sn (x, y) −→
h
2
Tn (x, y) −→
T (x, y),
and
(4.5)
h
3
F (x, y, u) ,
Fn (x, y, u) −→
h
3
Gn (y, x, v) −→
G (y, x, v) .
Since h1 (·, ·), h2 (·, ·) and h3 (·, ·) are continuous, by (4.2) and (4.3), for any fixed
n ≥ N and any (x, y, u, v) ∈ E × D × E × D, letting m → ∞, we get
(4.6)
h1 (Sn (x, y) , S (x, y)) ≤ ε/4,
h2 (Tn (x, y) , T (x, y)) ≤ ε/4,
and
(4.7)
h3 (Fn (x, y, u) , F (x, y, u)) ≤ ε/4,
h3 (Gn (y, x, v) , G (y, x, v)) ≤ ε/4
Now we show that S is continuous.
By Lemma 4.1, we need to prove that, for any fixed (x0 , y0 ) ∈ E × D and any
ε > 0, there exists a neighborhood N (x0 , y0 ) of (x0 , y0 ) in E × D such that
h1 (S(x, y), S(x0, y0 )) < ε, ∀(x, y) ∈ N (x0 , y0 ) ∩ E × D.
Since
h1 (S(x, y), S(x0, y0 ))
(4.8)
≤ h1 (S(x, y), Sn(x, y)) + h1 (Sn (x, y), Sn(x0 , y0 ))
+h1 (Sn (x0 , y0 ), S(x0, y0 )),
by (4.6), there exists N such that, for any n > N ,
(4.9)
h1 (S(x, y), Sn(x, y)) ≤ ε/4, ∀(x, y) ∈ E × D
Taking a fixed n > N , by the continuity of Sn and Lemma 4.1, there exists a neighborhood N (x0 , y0 ) of (x0 , y0 ) in E × D such that
(4.10)
h1 (Sn (x, y), Sn(x0 , y0 )) < ε/4, ∀(x, y) ∈ N (x0 , y0 ) ∩ E × D.
By (4.6), (4.8)-(4.10), we have
h1 (S(x, y), S(x0, y0 )) < ε, ∀(x, y) ∈ N (x0 , y0 ) ∩ E × D.
Hence, S is continuous on E × D.
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Bin Chen, Nan-jing Huang and Ching-Feng Wen
Similarly, we can prove T is continuous on E × D.
Now we show that F is C-upper semicontinuous.
Pick any (x0 , y0 , u0 ) ∈ E × D × E. For any neighborhood V of 0 in Z, there
exists r > 0 such that
rBZ ⊂ V,
(4.11)
where BZ is the open unit ball of Z. Taking a fixed n0 ≥ N , by (4.7), we have
(4.12)
h3 (Fn0 (x0 , y0 , u0 ) , F (x0 , y0 , u0 )) ≤ ε/4.
By the C-upper semicontinuity of Fn0 , there exists some neighborhood U (x0 , y0 , u0)
of (x0 , y0 , u0 ) such that
(4.13)
Fn0 (x, y, u) ⊂ Fn0 (x0 , y0 , u0 )
+ε/4BZ + C, ∀ (x, y, u) ∈ U (x0 , y0 , u0) ∩ E × D × E
It follows from (4.7) that, we have
(4.14) F (x, y, u) ⊂ Fn0 (x, y, u) + ε/4B Z , ∀ (x, y, u) ∈ U (x0 , y0 , u0 ) ∩ E × D × E,
where B Z is the closed unit ball of Z. From (4.12), (4.13) and (4.14), we get
(4.15)
F (x, y, u) ⊂ Fn0 (x, y, u) + ε/4B Z
⊂ Fn0 (x0 , y0 , u0 ) + ε/4BZ + ε/4B Z + C
⊂ F (x0 , y0 , u0 ) + ε/4B Z + ε/4BZ + ε/4B Z + C
⊂ F (x0 , y0 , u0 ) + 3ε/4B Z + C.
Due to the arbitrariness of ε, we can pick ε such that 3ε/4 < r. By (4.11) and (4.15),
for all (x, y, u) ∈ U (x0 , y0 , u0 ) ∩ E × D × E, we have
F (x, y, u) ⊂ F (x0 , y0 , u0 ) + rBZ + C ⊂ F (x0 , y0 , u0) + V + C,
and so F is C-upper semicontinuous on (x0 , y0 , u0 ). Since (x0 , y0 , u0 ) is arbitrary, F
is C-upper semicontinuous on E × D × E.
Next we show that F is C-lower semicontinuous.
Pick any (x1 , y1 , u1 ) ∈ E × D × E. For any z ∈ F (x1 , y1 , u1 ) and for any
neighborhood V of 0 in Z, there exists t > 0 such that
(4.16)
tBZ ⊂ V.
Taking a fixed n1 ≥ N , by (4.7), we have
(4.17)
h3 (Fn1 (x1 , y1 , u1 ) , F (x1 , y1 , u1 )) ≤ ε/4.
Generalized Symmetric Strong Vector Quasi-equilibrium Problems
955
Since Fn1 (x1 , y1 , u1 ) and F (x1 , y1 , u1 ) are compact sets, there exists zn1 ∈ Fn1 (x1 ,
y1 , u1 ) such that
zn1 − z ≤ h3 (Fn1 (x1 , y1 , u1 ) , F (x1 , y1 , u1 )) ≤ ε/4.
(4.18)
Since Fn1 is C-lower semicontinuous, there exists some neighborhood U (x1 , y1 , u1)
of (x1 , y1 , u1 ) such that
Fn1 (x, y, u) ∩ zn1 + ε/4BZ − C
(4.19)
= ∅, ∀ (x, y, u) ∈ U (x1 , y1 , u1 ) ∩ E × D × E.
By the arbitrariness of ε, we can pick ε such that 3ε/4 < t. We claim that
F (x, y, u) ∩ (z + tBZ − C)
(4.20)
= ∅, ∀ (x, y, u) ∈ U (x1 , y1 , u1 ) ∩ E × D × E.
Indeed, by (4.19), there exists d ∈ Fn1 (x, y, u) and d ∈ zn1 + ε/4BZ − C. It follows
from (4.7) that
Fn1 (x, y, u) ⊂ F (x, y, u) + ε/4B Z ,
(4.21)
Hence, for d ∈ Fn1 (x, y, u), there exists a ∈ F (x, y, u) and b ∈ ε/4B Z such that
d = a + b. By (4.18),
a = d − b ∈ zn1 + ε/4BZ − C − ε/4B Z
⊂ z + ε/4B Z + ε/4BZ − C − ε/4B Z
(4.22)
⊂ z + 3ε/4B Z − C
⊂ z + tBZ − C.
and so
(4.23)
F (x, y, u) ∩ (z + tBZ − C) = ∅.
From (4.16) and (4.23), we get
F (x, y, u) ∩ (z + V − C) = ∅, ∀ (x, y, u) ∈ U (x1 , y1 , u1 ) ∩ E × D × E.
Hence, F is C-lower semicontinuous on (x1 , y1 , u1 ). Since (x1 , y1 , u1 ) is arbitrary, F
is C-lower semicontinuous on E × D × E.
Next we show that, for any fixed (x, y) ∈ E × D, F (x, y, u) is C-properly quasiconvex in u.
956
Bin Chen, Nan-jing Huang and Ching-Feng Wen
Indeed, for any u1 , u2 ∈ E, z1 ∈ F (x, y, u1), z2 ∈ F (x, y, u2 ), t ∈ [0, 1], since
F (x, y, u1), F (x, y, u2), Fn (x, y, u1) and Fn (x, y, u2) are compact sets, by (4.7),
there exist an ∈ Fn (x, y, u1), bn ∈ Fn (x, y, u2) such that
an − z1 ≤ h3 (Fn (x, y, u1) , F (x, y, u1)) ≤ ε/4
(4.24)
and
bn − z2 ≤ h3 (Fn (x, y, u2) , F (x, y, u2)) ≤ ε/4.
(4.25)
By the C-properly quasiconvexity of Fn , there exists cn ∈ Fn (x, y, tu1 + (1 − t) u2 )
such that
either cn ≤ an or cn ≤ bn ,
(4.26)
Since Fn (x, y, tu1 + (1 − t) u2 ) and F (x, y, tu1 + (1 − t) u2 ) are compact sets, by
(4.7), there exists dn ∈ F (x, y, tu1 + (1 − t) u2 ) such that
cn − dn (4.27)
≤ h3 (Fn (x, y, tu1 + (1 − t) u2 ) , F (x, y, tu1 + (1 − t) u2 )) ≤ ε/4.
By the compactness of F (x, y, tu1 + (1 − t) u2 ), there exist a subsequence {d nk } of
{dn } such that d nk → d ∈ F (x, y, tu1 + (1 − t) u2 ) . Hence, by (4.27), we have
cnk → d. From (4.24), (4.25) and (4.26), we have
either d ≤ z1 or d ≤ z2 .
Hence, F (x, y, u) is C-properly quasiconvex in u.
Similarly, we can prove that G is C-upper semicontinuous and C-lower semicontinuous mappings with nonempty compact values. For every fixed (x, y) ∈ E × D,
G(y, x, v) is C-properly quasiconvex in v. By (4.6) and (4.7), for any fixed n ≥ N
and any (x, y, u, v) ∈ E × D × E × D, we have
sup
(x,y)∈E×D
sup
h1 (Sn (x, y) , S (x, y)) ≤ ε/4,
(x,y,u)∈E×D×E
sup
(x,y)∈E×D
h3 (Fn (x, y, u) , F (x, y, u)) ≤ ε/4,
h2 (Tn (x, y) , T (x, y)) ≤ ε/4,
sup
(y,x,v)∈D×E×D
h3 (Gn (y, x, v) ,
G (y, x, v)) ≤ ε/4.
Set u = (S, T, F, G). We know that u ∈ M and ρ (un , u) ≤ ε for all n ≥ N , i.e.,
ρ
un −→ u. Hence, (M, ρ) is a complete metric space. This completes the proof.
For any u = (S, T, F, G) ∈ M , Theorem 3.1 implies that GSSVQEP has a solution.
Denote by ψ (u) the solution set of GSSVQEP respect to u. Then, ψ defines a setvalued mapping from M to C × D and ψ (u) = ∅ for each u ∈ M .
Generalized Symmetric Strong Vector Quasi-equilibrium Problems
957
Theorem 4.2. ψ : M → 2E×D is a upper semicontinuous mapping with compact
values.
Proof. Since E × D is compact, it suffices to prove that ψ is a closed mapping.
Let a sequence {(un , (xn , yn ))} ⊂ Graph (ψ) be given such that (un , (xn , yn )) →
(u, (x̄, ȳ)) ∈ M × (E × D), where un = (Sn , Tn , Fn , Gn) and u = (S, T, F, G).
For each n, since (xn , yn ) ∈ ψ(un), we have
(4.28)
xn ∈ Sn (xn , yn ) , yn ∈ Tn (xn , yn ) ,
(4.29)
Fn (xn , yn , x) ⊂ C, ∀x ∈ Sn (xn , yn ) ,
and
(4.30)
Gn (yn , xn, y) ⊂ C, ∀y ∈ Tn (xn , yn ) ,
For any open set O with S (x̄, ȳ) ⊂ O, since S (x̄, ȳ) is a compact set, there exists
ε > 0 such that
(4.31)
{x ∈ E : d (x, S (x̄, ȳ)) < ε} ⊂ O,
where d (x, S (x̄, ȳ)) = inf x ∈S(x̄,ȳ) x−x . Since ρ ((Sn , Tn, Fn , Gn) , (S, T, F, G))
→ 0, (xn , yn ) → (x̄, ȳ) and S is upper semicontinuous at (x̄, ȳ), there exists N such
that, for any n ≥ N ,
(4.32)
sup
(x,y)∈E×D
and
(4.33)
h1 (Sn (x, y), S(x, y)) <
1
ε
2
1
S (xn , yn ) ⊂ x ∈ E : d (x, S (x̄, ȳ)) < ε .
2
From (4.31), (4.32) and (4.33), we have
1
Sn (xn , yn ) ⊂ x ∈ E : d (x, S (xn , yn )) < ε
2
(4.34)
⊂ {x ∈ E : d (x, S (x̄, ȳ)) < ε} ⊂ O, ∀n ≥ N.
Noting that S (x̄, ȳ) ⊂ O, (4.34) and xn ∈ Sn (xn , yn ), we can apply Lemma 4.3 to get
a subsequence {xnk } of {xn } which converges to x̄0 ∈ S (x̄, ȳ). Since {xnk } → x̄,
(4.35)
x̄ = x0 ∈ S(x̄, ȳ).
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Bin Chen, Nan-jing Huang and Ching-Feng Wen
Similarly
ȳ ∈ T (x̄, ȳ).
(4.36)
From the lower semicontinuity of S at (x̄, ȳ) and (xn , yn ) → (x̄, ȳ), for any x ∈
S (x̄, ȳ), there exists zn ∈ S (xn , yn ) such that z n → x. Since ρ ((Sn , Tn, Fn , Gn) ,
(S, T, F, G)) → 0, we can choose a subsequence {Snk } of {Sn } such that
h1 (Snk (x, y) , S (x, y)) <
1
.
k
h1 (Snk (xnk , ynk ) , S (xnk , ynk )) <
1
.
k
sup
(x,y)∈E×D
Thus,
This implies that there exist z n k ∈ Snk (xnk , ynk ) such that
zn − zn < 1 .
k
k
k
Since
z − x ≤ z − zn + zn − x < 1 + zn − x → 0,
nk
nk
k
k
k
k
we have zn k → x. Since zn k ∈ Snk (xnk , ynk ), xnk ∈ Snk (xnk , ynk ) and ynk ∈
Tnk (xnk , ynk ), by (4.29), we have
(4.37)
Fnk xnk , ynk , zn k ⊂ C.
We claim that
F (x̄, ȳ, x) ⊂ C.
If not, then there exists z ∈ F (x̄, ȳ, x) such that z ∈
/ C. Hence, there exists some
neighborhood U of 0 in Z such that (z + U ) ∩ C = ∅ and so
(z + U − C) ∩ C = ∅.
(4.38)
Thus, there exists r > 0 such that rB z ⊂ U . Since F is C-lower semicontinuous
mapping, when k is sufficiently large, we have
r
F xnk , ynk , zn k ∩ z + BZ − C = ∅.
3
Let snk ∈ F xnk , ynk , zn k ∩ z + r3 BZ − C . Since ρ ((Sn , Tn, Fn , Gn ) , (S, T,
F, G)) → 0, for any ε > 0(ε < 3r ), there exists N such that, for any n ≥ N ,
(4.39)
sup
(x,y,u)∈E×D×E
h3 (Fn (x, y, u), F (x, y, u)) ≤ ε.
Generalized Symmetric Strong Vector Quasi-equilibrium Problems
959
When k is sufficient large such that n k ≥ N , by (4.39), we have
sup
(x,y,u)∈E×D×E
h3 (Fnk (x, y, u), F (x, y, u)) ≤ ε
and so
(4.40)
For snk
h3 Fnk (xnk , ynk , zn k ), F (xnk , ynk , zn k ) ≤ ε.
∈ F xnk , ynk , zn k , by (4.40), there exists tnk ∈ Fnk (xnk , ynk , zn k ) such that
snk − tnk ≤ ε
and so
r
tnk ∈ snk + εB Z ⊂ z + BZ − C + εB Z ⊂ z + rBZ − C ⊂ z + U − C.
3
From (4.37), we have
tnk ∈ C ∩ (z + U − C),
which contradicts (4.38). By the arbitrariness of x ∈ S (x̄, ȳ), we have
(4.41)
F (x̄, ȳ, x) ⊂ C, ∀x ∈ S (x̄, ȳ) ,
Similarly, we can prove that
(4.42)
G (ȳ, x̄, y) ⊂ C, ∀y ∈ T (x̄, ȳ) .
By (4.35), (4.36), (4.41) and (4.42), we know that ((S, T, F, G), (x̄, ȳ)) ∈ Graph(ψ).
Hence, Graph(ψ) is closed. By Theorem 3.1, we know ψ(u) is closed. By the compactness of E × D, it is easy to see that ψ(u) ⊂ E × D is compact. Therefore, ψ is a
upper semicontinuous mapping with compact values. This completes the proof.
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Bin Chen
Department of Mathematics
Sichuan University
Chengdu, Sichuan 610064
P. R. China
E-mail: [email protected]
962
Bin Chen, Nan-jing Huang and Ching-Feng Wen
Nan-jing Huang
Department of Mathematics
Sichuan University
Chengdu, Sichuan 610064
P. R. China
E-mail: [email protected]
Ching-Feng Wen
Center for General Education
Kaohsiung Medical University
Kaohsiung 807, Taiwan
E-mail: [email protected]