R E S E A R C H
Open Access
Well-posedness for parametric strong vector
quasi-equilibrium problems with applications
Qiu-ying Li
1and San-hua Wang
2** Correspondence: wsh_315@163. com
2Department of Mathematics, Nanchang University, Nanchang, Jiangxi 330031, People’s Republic of China
Full list of author information is available at the end of the article
Abstract
In this article, we generalize the concept of well-posedness to the parametric strong vector quasi-equilibrium problem. Under some suitable conditions, we establish some characterizations of well-posedness for parametric strong vector quasi-equilibrium problems. The corresponding concept of well-posedness in the generalized sense is also investigated for the parametric strong vector quasi-equilibrium problem. As applications, we investigate the well-posedness for strong vector quasi-variational inequality problems and strong vector quasi-optimization problems.
2000 Mathematics subject classification:49K40; 90C31.
Keywords:parametric strong vector quasi-equilibrium problem, well-posedness, approximating sequence, upper semicontinuity; lower semicontinuity
1 Introduction
Equilibrium problem was first introduced by Blum and Oettli [1], which includes opti-mization problems, fixed point problems, variational inequality problems, and comple-mentarity problems as special cases. During the recent years, equilibrium problem has been extensively studied and generalized (e.g., [2,3]).
It is well known that well-posedness is very important for both optimization theory and numerical methods of optimization problems, which guarantees that, for every approximating solution sequence, there is a subsequence which converges to a solu-tion. Well-posedness of unconstrained and constrained scalar optimization problems was first introduced and studied by Tykhonov [4] and Levitin and Polyak [5], respec-tively. Since then, various concepts of well-posedness have been introduced and exten-sively studied for minimization problems and vector optimization problems. For details, we refer readers to [6-13] and the references therein.
In recent years, the concept of well-posedness has been generalized to several related problems: variational inequality problems [14-18], saddle point problems [19], Nash equilibrium problems [16,20-23], inclusion problems [24,25], and fixed point problems [24-26].
Recently Fang et al. [27] generalized the concepts of well-posedness to equilibrium problems and to optimization problems with equilibrium constraints and gave some metric characterizations of well-posedness for equilibrium and optimization problems with equilibrium constraints. Subsequently, the well-posedness has been extensively studied by many authors for various equilibrium problems, such as scalar equilibrium
problems [28], vector equilibrium problems [29-31], parametric vector equilibrium pro-blems [32], vector quasi-equilibrium propro-blems [33], generalized vector quasi-equili-brium problems [34], and symmetric quasi-equiliquasi-equili-brium problems [35]. It is worth mentioning that the most of results of well-posedness are for scalar equilibrium pro-blem or for weak vector equilibrium propro-blem which depends on intC≠∅. However,
in many cases, the cone Chas an empty interior. For example, in the classical Banach spaces lpandLp(Ω), where 1< p <∞, the standard ordering cone has an empty inter-ior. In this case, we cannot study the well-posedness for the weak vector equilibrium problem. However, we can study the strong vector equilibrium problem since the strong vector equilibrium problem does not need intC≠∅. On the other hand, to the best of our knowledge, the result of well-posedness for the strong vector equilibrium problem is very few.
On the other hand, the strong vector equilibrium problem is a class of important equilibrium problems since a solution for the strong vector equilibrium problem is an ideal solution, which is better than other solutions such as efficient solution, weak effi-cient solution, Henig effieffi-cient solution and supper effieffi-cient solution (e.g., [36]). Thus, it is important to investigate the well-posedness for strong vector equilibrium problems.
Motivated and inspired by the studies mentioned above, in this article, we introduce and study the well-posedness for parametric strong vector quasi-equilibrium problems. Under suitable conditions, we obtain some necessary and sufficient conditions for the well-posedness of parametric strong vector quasi-equilibrium problems. As applica-tions, we investigate the well-posedness for strong vector quasi-variational inequality problems and strong vector quasi-optimization problems.
2 Preliminaries
Throughout this article, unless otherwise specified, we use the following notations. Let
X be a nonempty closed subset of a metric space (E, d);Ybe a Huasdorff topological vector space; and Λ and P be two nonempty closed subsets of two metric spaces, respectively. Let K: Λ×X ® 2X and C: Λ×X ® 2Ybe set-valued mappings such that, for each (l,x)ÎΛ×X,C(l,x) is a nonempty closed convex cone inY. Lete:X ® Ybe a continuous mapping such that, for each xÎ X and lÎ Λ,e(x)Î C(l, x). Let f:P ×X ×X ®Ybe a vector-valued mapping. For any fixed (l,p)ÎΛ×P, the parametric strong vector quasi-equilibrium problem (for short, (PSVQEP)l,p) is to find
¯
x∈X such that x¯∈K(λ,x¯) and
f(p,x¯,y)∈C(λ,x¯), ∀y∈K(λ,x¯).
Denote by (PSVQEP) the family {(PSVQEP)l,p: (l,p)Î Λ×P}. For each (l,p)ÎΛ
×P, letS(l,p) be the solution set for (PSVQEP)l,p, i.e.,
S(λ,p) ={x∈X:x∈K(λ,x)andf(p,x,y)∈C(λ,x), ∀y∈K(λ,x)}.
Remark 2.1. IfEis a real Banach space,Y=R,f:P×X ×X®R,K(l,x) =Xand C
(l,x) = (0, +∞) for all (l,x)ÎΛ×X, then (PSVQEP)l,preduces to the following
para-metric equilibrium problem (for short, (PEP)p): find ¯x∈X such that
f(p,¯x,y)≥0, ∀y∈X,
Definition 2.1. Let (l,p)Î Λ×P and {(ln,pn)}⊆Λ×Pwith (ln,pn)®(l, p). A
sequence {xn}⊆X is said to be approximating for (PSVQEP)l,pcorresponding to {(ln, pn)} if there exists a sequence {εn}⊆R+ withεn®0 such that, for eachnÎN,
xn∈K(λn,xn)
and
f(pn,xn,y) +εne(xn)∈C(λn,xn), ∀y∈K(λn,xn).
Definition 2.2. (PSVQEP) is said to be well-posed if, for every (l,p)ÎΛ×P,
(i) (PSVQEP)l,phas a unique solutionxl,p, i.e.,S(l,p) = {xl,p};
(ii) for any sequence {(ln,pn)}⊆Λ×P with (ln,pn)®(l,p), every approximating
sequence {xn} for (PSVQEP)l,pcorresponding to {(ln,pn)} converges toxl,p.
Definition 2.3. (PSVQEP) is said to be well-posed in the generalized sense if, for every (l,p)ÎΛ×P,
(i)S(l,p)≠∅;
(ii) for any sequence {(ln,pn)}⊆Λ×P with (ln,pn)®(l,p), every approximating
sequence {xn} for (PSVQEP)l,pcorresponding to {(ln,pn)} has a subsequence which
converges to some point ofS(l,p).
Remark 2.2. When (PSVQEP)l,preduces to (PEP)p, Definition 2.1 coincides with 3.1
of [27], and Definitions 2.2 and 2.3 coincide with 3.2 of [27].
Definition 2.4. Let Λ, P, and Xbe topological spaces andYbe a topological vector space. LetC:Λ×X® 2Ybe a set-valued mapping such that, for each (l,x)Î Λ×X,
C(l, x) is a nonempty closed convex cone inY. Let f: P ×X ×X ® Ybe a vector-valued mapping. For any fixed (l,x)Î Λ×X, fis said to be upper (resp. lower)C(l,
x) continuous if, for any (p,y)Î P×Xand any neighborhood Vof 0 inY, there exist neighborhoodsUp,UxandUyofp,x, andy, respectively, such that
f(p,x,y)∈f(p,x,y) +V+C(λ,x), ∀(p,x,y)∈Up×Ux×Uy.
(resp.f(p,x,y)∈f(p,x,y) +V−C(λ,x), ∀(p,x,y)∈Up×Ux×Uy.)
Remark 2.3. Iff:P×X×X® Yis continuous, then for any fixed (l, x)Î Λ×X, f
is both upperC(l,x) continuous and lowerC(l,x) continuous.
Definition 2.5 [37]. LetXandYbe two topological spaces. A set-valued mappingT:
X ®2Yis said to be
(i) upper semicontinuous (for short, u.s.c.) at x0 Î X if, for each open setVin Y
withT(x0)⊆V, there exists an open neighborhoodU(x0) ofx0such thatT(x)⊆V
for allxÎU(x0);
(ii) lower semicontinuous (for short, l.s.c.) atx0 Î X if, for each open set Vin Y
withT(x0)∩V≠∅, there exists an open neighborhoodU(x0) ofx0 such thatT(x)
∩V≠∅for all xÎU(x0);
(iv) continuous onX if it is both u.s.c. and l.s.c. onX;
(v) closed if the graph Gr(T) = {(x, y)Î X ×Y :y Î T (x)} is a closed subset of
X×Y.
Lemma 2.1[37]. LetXand Ybe two topological spaces, andF:X® 2Ya set-valued mapping.
(i) IfFis u.s.c. and close-valued, thenFis closed;
(ii) IfF(x) is a compact set, thenFis u.s.c. atx ÎX, if and only if for any net {xa} ⊆X withxa®x and any net {ya}⊆YwithyaÎF(xa) for alla, there existyÎ F
(x) and a subnet {yb} of {ya} such thatyb®y;
(iii)Fis l.s.c. at xÎ X, if and only if for anyy ÎF(x) and for any net {xa} withxa ®x, there exists a net {ya} withyaÎF(xa) for allasuch thatya® y.
We also need the concepts of noncompactness measure and Hausdorff metric. Definition 2.6 [38]. LetE be completed. The Kuratowski measure of noncompact-ness of a setA⊆Eis defined by
μ(A) = inf{ε >0 :A⊂ ∪ni=1Ai, diamAi< ε, i= 1, 2,. . .,n},
where diamAdenotes the diameter ofAdefined by diamA= sup{d(x1,x2): x1,x2 Î
A}.
Definition 2.7[38]. LetAand Bbe nonempty subsets ofE. The Hausdorff metricH
(·,·) betweenAandBis defined by
H(A,B) = max{h(A,B),h(B,A)},
whereh(A,B) = supaÎAd(a,B) withd(a,B) = infbÎBd(a,b).
3 Well-posedness for (PSVQEP)
In this section, we shall establish some characterizations of well-posedness for (PSVQEP).
3.1 Continuity characterization of well-posedness for (PSVQEP)
For each (l,p)ÎΛ×Pandε≥0, theε-solution set for (PSVQEP)l,pis defined by (λ,p,ε) :={x∈X:x∈K(λ,x) andf(p,x,y) +εe(x)∈C(λ,x), ∀y∈K(λ,x)}.
We callΠanε-solution mapping.
Clearly, for each (l, p)Î Λ×Pandε≥0,S(l, p)⊆Π(l,p,ε), andS(l, p) = Π(l, p, 0).
Lemma 3.1. For any fixed (l,x)Î Λ×X, iffis lower C(l,x) continuous, then for everyε≥0,f(p,x,y) +εe(x) is lowerC(l,x) continuous.
Proof. Let (l,x)ÎΛ×Xbe such thatfis lowerC(l,x) continuous. For any given ε ≥0, define a vector-valued mapping F:P×X×X®Yas follows:
F(p,x,y) =f(p,x,y) +εe(x), ∀(p,x,y)∈P×X×X.
For any neighborhoodVof 0 inY, there exists a balanced neighborhoodV1of 0 inY
V1+V1⊆V. (3:1)
For any (p,y)Î P×X, sinceeis continuous andfis lowerC(l,x) continuous, there exist neighborhoods Up, Ux andUyofp,x, andy, respectively, such that
f(p,x,y)∈f(p,x,y) +V1−C(λ,x), ∀(p,x,y)∈Up×Ux×Uy. (3:2)
and
εe(x)∈εe(x) +V1, ∀x∈Ux. (3:3)
LetU =Up×Ux×Uy, then by (3.2), (3.3), and (3.1), we have, for each (p’, x’, y’)Î U,
F(p,x,y) =f(p,x,y) +εe(x)
∈f(p,x,y) +V1−C(λ,x) +εe(x) +V1 ⊆f(p,x,y) +εe(x) +V−C(λ,x) =F(p,x,y) +V−C(λ,x).
that is,
F(p,x,y)∈F(p,x,y) +V−C(λ,x), ∀(p,x,y)∈U.
Thus, Fis lowerC(l,x) continuous. This completes the proof. Lemma 3.2. Assume that
(i) for eachlÎΛ,K(l, ·) is continuous and close-valued onX; (ii) for eachl,C(l, ·) is u.s.c. onX;
(iii) for each (l,x)ÎΛ×X,fis lowerC(l, x) continuous.
Then for each (l,p,ε)ÎΛ×P×R+,S(l,p) andΠ(l,p,ε) are closed subsets ofX. Proof. (1) Let (l, p)Î Λ ×P be any given sequence. Let {xn} ⊆ S(l, p) be any
sequence such that xn®xÎX. Then, for each n, we have
xn∈K(λ,xn)
and
f(p,xn,y)∈C(λ,xn), ∀y∈K(λ,xn). (3:4)
Since K(l, ·) is u.s.c. and close-valued,K(l, ·) is closed and soxÎ K(l,x). For eachy Î K(l,x), sinceK(l, ·) is l.s.c. atx, there exists a sequence {yn} with ynÎK(l,xn) such
thatyn®y. Then, by (3.4), we have
f(p,xn,yn)∈C(λ,xn), ∀n∈N. (3:5)
For any neighborhoodVof 0 inY, there exists a balanced neighborhoodV0of 0 inY
such that
V0+V0⊆V. (3:6)
ForV0, sinceC(l, ·) is u.s.c. atxandfis lowerC(l,x) continuous, there exist
C(λ,x)⊆C(λ,x) +V0, ∀x∈Ux. (3:7)
f(p,x,y)∈f(p,x,y) +V0−C(λ,x), ∀(x,y)∈Ux×Uy. (3:8)
Sincexn® xandyn®y, there existsn0 ÎNsuch that, for alln≥n0,xnÎUx, and ynÎUy. Then, by (3.7) and (3.8), for alln≥n0,
C(λ,xn)⊆C(λ,x) +V0, (3:9)
f(p,xn,yn)∈f(p,x,y) +V0−C(λ,x). (3:10)
Noting that V0is balanced, we have -V0=V0. Then, by (3.10), (3.5), (3.9), and (3.6),
we have, for each n≥n0,
f(p,x,y)∈f(p,xn,yn)−V0+C(λ,x)
=f(p,xn,yn) +V0+C(λ,x) ⊆C(λ,xn) +V0+C(λ,x) ⊆C(λ,x) +V0+V0+C(λ,x) ⊆C(λ,x) +V.
By the arbitrary of V, we get
f(p,x,y)∈C(λ,x).
Thus, xÎS(l,p), and this implies thatS(l,p) is closed. (2) For eachε>0, let
F(p,x,y) =f(p,x,y) +εe(x), ∀(p,x,y)∈P×X×X.
Then, by Lemma 3.1, for each (l,x)ÎΛ×X, Fis lowerC(l, x) continuous. It fol-lows from (1) that, for each (l,p)Îl×P, the set
{x∈X:x∈K(λ,x) andF(p,x,y)∈C(λ,x), ∀y∈K(λ,x)}
is closed, i.e., Π(l,p,ε) is closed. This completes the proof.
The following theorem gives a characterization of the well-posedness in the general-ized sense for (PSVQEP).
Theorem 3.1. (PSVQEP) is well-posed in the generalized sense, if and only if for every
(l,p)Î Λ×P, S(l,p)is a nonempty compact subset of X andΠis u.s.c. at (l,p, 0).
Proof. For any (l, p)Î Λ×P, suppose thatS(l,p) is a nonempty compact subset of
X andΠis u.s.c. at (l, p, 0). Let {(ln,pn)}⊆Λ×P be an arbitrary sequence with (ln, pn)® (l, p) and {xn} be an approximating sequence of (PSVQEP)l,pcorresponding to
{(ln,pn)}. Then, there exists a sequence {εn}⊆R+withεn®0 such that, for eachnÎ N,
xn∈K(λn,xn)
and
Thus, xnÎΠ(ln,pn,εn). Note thatΠ(l,p, 0) =S(l,p) is compact. SinceΠis u.s.c. at
(l, p, 0), there exists a subsequence {xnj} of {xn} which converges to some pointx0 Î
Π(l,p, 0) =S(l,p). Therefore, (PSVQEP) is well-posed in the generalized sense. Conversely, suppose that (PSVQEP) is well-posed in the generalized sense. Then, for each (l, p)Î Λ×P,S(l, p) is nonempty and compact, and so Π(l,p, 0) is nonempty and compact. Let {(ln,pn,εn)}⊆Λ×P×R+ be an arbitrary sequence with (ln,pn,εn) ® (l,p, 0) andxnÎΠ(ln,pn,εn). Then, for eachnÎN,
xn∈K(λn,xn)
and
f(pn,xn,y) +εne(xn)∈C(λn,xn), ∀y∈K(λn,xn).
Thus, {xn} is an approximating sequence of (PSQVEP)l,pcorresponding to {(ln,pn)}.
By the well-posedness in the generalized sense of (PSVQEP), {xn} has a subsequence
{xnj} which converges to some point ofS(l,p) = Π(l,p, 0). Thus,Πis u.s.c. at (l,p, 0). This completes the proof.
For the well-posedness of (PSVQEP), we have the following result.
Theorem 3.2.(PSVQEP) is well-posed, if and only if for every(l,p)ÎΛ×P, S(l,p)
has a unique point andΠis u.s.c. at (l,p, 0).
Proof. The necessity is a direct conclusion of Definition 2.2 and Theorem 3.1. For the sufficiency, suppose that for any (l, p)ÎΛ×P, S(l, p) has a unique point
xl,pandΠis u.s.c. at (l, p, 0). Let {(ln,pn)}⊆Λ×P be an arbitrary sequence with
(ln,pn)®(l,p) and {xn} be an approximating sequence of (PSVQEP)l,pcorresponding
to {(ln, pn)}. Then, there exists a sequence {εn}⊆R+withεn®0 such that, for eachn Î N,
xn∈K(λn,xn)
and
f(pn,xn,y) +εne(xn)∈C(λn,xn), ∀y∈K(λn,xn).
Thus,xnÎ Π(ln,pn,εn). Note that Π(l,p, 0) =S(l,p) = {xl,p} is a singleton set and
so is compact. SinceΠis u.s.c. at (l, p, 0), for any neighborhoodVofxl,p, there exists
a neighborhoodUof (l,p, 0) such that
(λ,p,ε)⊆V, ∀(λ,p,ε)∈U.
Since (ln,pn,εn)® (l,p, 0), there existsn0ÎNsuch that
(λn,pn,εn)∈U, ∀n≥n0.
It follows that
xn∈(λn,pn,εn)⊆V, ∀n≥n0.
Hencexn ®xl,p, and this implies that (PSVQEP) is well-posed. This completes the
proof.
Theorem 3.3.Let X be a nonempty compact subset of E. Assume that
(i)K is continuous and close-valued onΛ×X;
(ii)C is u.s.c. on X×X;
(iii)for any fixed(l,x)ÎΛ×X, f is lower C(l,x)continuous;
(iv)for eachlÎΛand pÎP, S(l,p)≠∅.
Then, Πis u.s.c. onΛ×P ×R+.
Proof. Let (l0,p0,ε0)ÎΛ×P ×R+be any fixed point. Then, by (iv) and Lemma 3.2,
Π(l0,p0,ε0) is nonempty and closed. Furthermore,Π(l0,p0,ε0) is nonempty and
com-pact since Π(l0, p0, ε0)⊆X andXis compact. Let (ln,pn,εn)® (l0, p0,ε0) andxnÎ Π(ln, pn,εn). Observe that {xn}⊆ XandX is compact. We may assume that xn® x0
for some x0 Î X. Then, by similar arguments as Lemma 3.2, we can show that x0 Î
Π(l0,p0,ε0). This implies thatΠis u.s.c. at (l0, p0, ε0). Then, by the arbitrary of (l0,
p0,ε0), we know thatΠis u.s.c. onΛ×P×R+. This completes the proof.
Next, we give an example to illustrate Theorem 3.3.
Example 3.1. LetE=R,X= [0, 2],Λ=P= [0, 1]⊆R,Y=R2. Let
e(x) = (1, 0), ∀x∈X,
C(λ,x) =
(ξ,η)∈R2:η≥0,ξ+η
2
πarctan(λx)
≥0
, ∀(λ,x)∈×X,
K(λ,x) = [λx,x], ∀(λ,x)∈×X, f(p,x,y) = (p−x,x−y), ∀(p,x,y)∈P×X×X.
Then, it is easy to see that all the conditions of Theorem 3.3 are satisfied and so Πis u.s.c. on Λ×P×R+. Indeed, by simple computation, we haveΠ(l,p,ε) = [0,p+ε]∩ [0, 2] for all (l,p,ε)ÎΛ×P×R+. Thus,Πis u.s.c. onΛ×P×R+.
3.2 Metric characterization of well-posedness for (PSVQEP)
In order to give metric characterizaton of well-posedness for (PSVQEP), we introduce the following notation.
Let (l,p)ÎΛ×P be given sequence. The approximating solution set of (PSVQEP)l,
pis defined by, for eachδ≥0 andε≥0,
λ,p(δ,ε) = λ∈B(λ,δ)∩ p∈B(p,δ)∩P
(λ,p,ε)
=
λ∈B(λ,δ)∩ p∈B(p,δ)∩P
{x∈X:x∈K(λ,x) andf(p,x,y) +εe(x)∈C(λ,x), ∀y∈K(λ,x)},
whereB(a,r) denotes the closed ball centered atawith radiusr. Clearly, we have, for every (l,p)ÎΛ×P,
(i)Ωl,p(0, 0) =Π(l,p, 0) =S(l,p);
(ii)S(l,p)⊆Π(l,p,ε)⊆Ωl,p(δ, ε),∀δ,ε>0;
(iii) if 0≤δ1 ≤δ2and 0≤ε1≤ε2, thenΩl,p(δ1,ε1)⊆Ωl,p(δ2,ε2).
(i) for eachxÎX,K(·,x) is continuous and close-valued; (ii) for eachxÎX,C(·,x) is u.s.c. and close-valued; (iii) for eachxÎ X,f(·,x, ·) is continuous.
Then, for every (l,p)ÎΛ×P, S(λ,p) =
δ>0,ε>0 λ
,p(δ,ε).
Proof. Let (l,p)Î Λ×Pbe any given sequence. Clearly, S(λ,p)⊆
δ>0,ε>0 λ
,p(δ,ε).
Thus, we only need to show that
δ>0,ε>0 λ
,p(δ,ε)⊆S(λ,p). Indeed, if
x∈
δ>0,ε>0 λ
,p(δ,ε), then, for eachδ> 0, andε>0,xÎΩl,p(δ,ε). Thus, for eachnÎ N, x∈ λ,p(n1,1n) and, so there exists (ln, pn) Î Λ × P with λn∈B(λ,1n) and
pn∈B(p,1n) such that
x∈K(λn,x) (3:11)
and
f(pn,x,y) + 1
ne(x)∈C(λn,x), ∀y∈K(λn,x). (3:12)
Clearly,ln®land pn®pasn®∞. SinceK(·,x) is u.s.c. and closed-valued,K(·,x)
is closed and so xÎ K(l,x). For eachy ÎK(l,x), sinceK(·,x) is l.s.c., there existsynÎ K(ln,x) such thatyn® y. Then, by (3.12), we have
f(pn,x,yn) + 1
ne(x)∈C(λn,x), ∀n∈N.
SinceC(·,x) is u.s.c. and close-valued,C(·,x) is closed. Then, by the continuity off(·,
x, ·), we get
f(p,x,y)∈C(λ,x).
By the arbitrary of y, we know that x Î S(l, p) and this implies that
δ>0,ε>0 λ
,p(δ,ε)⊆S(λ,p). Therefore, S(λ,p) = δ>0,ε>0 λ
,p(δ,ε). This completes the
proof.
Lemma 3.4. Assume that
(i)Λand Pare finite dimensional; (ii)Kis continuous and close-valued; (iii)Cis u.s.c. and close-valued; (iii)fis continuous.
Then, for each δ>0 andε>0,Ωl,p(δ,ε) is closed.
Proof. Let δ≥0 and ε≥ 0 be given. Let {xn}⊆Ωl,p(δ,ε) be any sequence such that xn® x. Then, for eachnÎ N, there exist (ln,pn)ÎΛ×PwithlnÎB(l,δ) and pnÎ B(p,δ) such that
and
f(pn,xn,y) +εe(xn)∈C(λn,xn), ∀y∈K(λn,xn). (3:14)
Noting that Λ and P are finite dimensional, without loss of generality, we may assume that ln® l0andpn®p0for somel0 ÎΛandp0Î P. It follows that
d(λ0,λ) = lim
n→∞d(λn,λ)≤δandd(p0,p) = limn→∞d(pn,p)≤δ.
Hence, l0 ÎB(l, δ) andp0 Î B(p, δ). SinceKis u.s.c. and close-valued, Kis closed
and so xÎ K(l, x). For eachy Î K(l,x), sinceKis l.s.c., there exists yn ÎK(ln,xn)
such thatyn® y. Then, by (3.14), we have
f(pn,xn,yn) +εe(xn)∈C(λn,xn), ∀n∈N.
SinceCis u.s.c. and close-valued,Cis closed. Then, by the continuity offande, we get
f(p0,x,y) +εe(x)∈C(λ0,x).
Thus, xÎΩl,p(δ,ε), and soΩl,p(δ,ε) is closed. This completes the proof.
The following theorem shows that the well-posedness in the generalized sense for (PSVQEP) can be characterized by considering the noncompactness of approximating solution set.
Theorem 3.4.LetΛand P be finite dimensional. Assume that
(i)K is continuous and close-valued;
(ii)C is continuous and close-valued;
(iii)f is continuous.
Then, (PQSVEP) is well-posed in the generalized sense, if and only if for every(l,p)Î
Λ×P
λ,p(δ,ε)= ∅, ∀δ,ε >0, andμ( λ,p(δ,ε))→0 as (δ,ε)→(0, 0). (3:15)
Proof. Suppose that (PSVQEP) is well-posed in the generalized sense. Then, for each (l,p)ÎΛ×P,S(l,p) is nonempty and compact. Clearly,S(l, p)⊆Ωl,p(δ,ε) for allδ,
ε>0. It follows thatΩl,p(δ,ε)≠∅for all δ,ε>0. Now we shall show that μ( λ,p(δ,ε))→0 as (δ,ε)→(0, 0).
Observe that for eachδ,ε>0,
H( λ,p(δ,ε),S(λ,p)) = max{h( λ,p(δ,ε),S(λ,p)),h(S(λ,p), λ,p(δ,ε))}=h( λ,p(δ,ε),S(λ,p)).
Taking into account the compactness of S(l, p), we get
μ( λ,p(δ,ε))≤2H( λ,p(δ,ε),S(λ,p)) +μ(S(λ,p)) = 2h( λ,p(δ,ε),S(λ,p)).
To prove (3.15), it is sufficient to show that
If h(Ωl,p(δ,ε),S(l,p))↛0 as (δ, ε)®(0, 0), then there existl >0 andδn>0,εn>0
withδn®0,εn®0, andxnÎ Ωl,p(δn,εn) such that
xn∈/ S(λ,p) +B(0,l), ∀n∈N. (3:16)
AsxnÎ Ωl,p(δn,εn), {xn} is an approximating sequence for (PSVQEP)l,p. By the
well-posedness in the generalized sense of (PSVQEP), there exists a subsequence {xnk} of {xn} converging to some point ofS(l,p). This contradicts (3.16) and so
h( λ,p(δ,ε),S(λ,p))→0 as (δ,ε)→(0, 0).
Conversely, suppose that (3.15) holds. For each (l, p)Î Λ×P, by Lemma 3.4,Ωl,p
(δ,ε) is closed for allδ,ε>0. By Lemma 3.3, we have S(λ,p) =
δ>0,ε>0 λ
,p(δ,ε). Since
μ( λ,p(δ,ε))→0,
the theorem on p.412 in [38] can be applied and one concludes thatS(l,p) is none-mpty, compact, and
h( λ,p(δ,ε),S(λ,p)) =H( λ,p(δ,ε),S(λ,p))→0 as (δ,ε)→(0, 0). (3:17)
Let (ln, pn) ® (l, p) Î Λ ×P and {xn} ⊆ X be an approximating sequence for
(PSVQEP)l,pcorresponding to {(ln, pn)}. Then there exist εn>0 withεn®0 such that,
for eachnÎN,
xn∈K(λn,xn)
and
f(pn,xn,y) +εne(xn)∈C(λn,xn), ∀y∈K(λn,xn).
For eachnÎ N, letδn= max{d(ln,l), d(pn, p)}. Thenδn®0 andxn ÎΩl,p(δn, εn).
It follows from (3.17) that
d(xn,S(λ,p))≤h( λ,p(δn,εn),S(λ,p))→0.
Since S(l,p) is compact, there exists x¯n∈S(λ,p) such that
d(xn,x¯n) =d(xn,S(λ,p))→0.
Again from the compactness of S(l,p), {¯xn} has a subsequence {¯xnk} converging to ¯
x∈S(λ,p). Hence, the corresponding subsequence {xnk} of {xn} converges to x¯. Therefore, (PSVQEP) is well-posed in the generalized sense. This completes the proof.
Example 3.2. LetE,X,Λ,P,Y,e,K,C, andfbe the same as in Example 3.1. Then, it is easy to see that the conditions (i)-(iii) of Theorem 3.4 are satisfied. Moreover, by simple computation, we have, for each (l, p)Î Λ×Pand each δ, ε≥0 and eachl’Î [l-δ,l+δ]∩Λand eachp’Î[p-δ,p+δ]∩P,
(λ,p,ε) = [0,p+ε]∩[0, 2].
It follows that
λ,p(δ,ε) =
p∈[p−δ,p+δ]∩P
Thus,Ωl,p(δ,ε)≠∅for allδ,ε>0 andμ(Ωl,p(δ,ε)) ®0 as (δ,ε)®(0, 0). By
Theo-rem 3.4, (PSVQEP) is well-posed in the generalized sense.
Indeed, by simple computation, we have S(l,p) = [0,p] for all (l,p)Î Λ×P. Sup-pose that {(ln, pn)} ⊆ Λ× P with (ln, pn) ® (l, p) and {xn} is an approximating
sequence for (PSVQEP)l,pcorresponding to {(ln,pn)}. Then there exists {εn}⊆R+ with
εn®0 such thatxnÎ Π(ln,pn,εn), i.e.,xnÎ[0, pn+εn]∩[0, 2]. Since pn® pÎ P=
[0, 1] andεn®0, there exists n0 ÎNsuch that, for eachn≥n0,pn+εn<2, and soxn Î [0,pn +εn]. Since {xn} is bounded, it has a subsequence which converges to some
point of [0,p] =S(l,p). Therefore, (PSVQEP) is well-posed in the generalized sense.
4 Applications
Since vector quasi-equilibrium problems contain vector quasi-variational inequality problems, vector quasi-optimization problems, and vector quasi-saddle point problems as special cases, we can derive from the result in Section 3, some consequences for such special cases. In this section, we discuss only some applications of our results to strong vector quasi-variational inequality problems and strong vector quasi-optimiza-tion problems.
Let Xand Ybe two real Banach spaces andCbe a closed convex cone inY. LetL(X,
Y) be the set of all the continuous linear operators fromXinto Y. LeteÎCbe a fixed point.
4.1 Strong vector quasi-variational inequality problems
Let F:X ® 2Xbe a set-valued mapping andG:X ® L(X,Y) a vector-valued map-ping. We consider the following strong vector quasi-variational inequality problem (for short, SVQVIP): find xÎXsuch that
x∈F(x) and Gx,y−x ∈C, ∀y∈F(x).
Denote by Sthe set of solutions for (SVQVIP). For eachε≥0, we denote byΠ(ε) the ε-solution set of (SVQVIP), i.e.,
(ε) ={x∈X:x∈F(x) andGx,y−x+εe∈C, ∀y∈F(x)}.
Let f(x,y) =〈Gx,y -x〉. We can obtain the following results for (SVQVIP).
Theorem 4.1. (SVQVIP) is well-posed in the generalized sense, if and only if S is a nonempty compact subset of X and Πis u.s.c. at0.
Theorem 4.2. (SVQVIP) is well-posed, if and only if S has a unique point andΠis u. s.c. at0.
4.2 Strong vector quasi-optimization problems
Let F:X ® 2X be a set-valued mapping and:X ® Ybe a vector-valued mapping. We consider the following strong vector quasi-optimization problem (for short, SVQOP):
minϕ(x) subject tox∈F(x).
A point x0ÎX is called a strongly efficient solution of (SVQOP), if
Denote by Sthe set of all strongly efficient solutions for (SVQOP). For eachε≥ 0, we denote byΠ(ε) theε-strongly efficient solution set of (SVQOP) as follows:
(ε) ={x∈X:x∈F(x) andϕ(y)−ϕ(x) +εe∈C, ∀y∈F(x)}.
Let f(x,y) =(y)-(x). We can obtain the following results for (SVQOP).
Theorem 4.3.(SVQOP) is well-posed in the generalized sense, if and only if S is a nonempty compact subset of X and Πis u.s.c. at0.
Theorem 4.4.(SVQOP) is well-posed, if and only if S has a unique point andΠis u. s.c. at0.
Acknowledgements
This study was supported by the National Natural Science Foundation of China (11061023, 11071108), the Natural Science Foundation of Jiangxi Province (2010GZS0145, 2010GZS0151), the Youth Foundation of Jiangxi Educational Committee (GJJ10086) and the Scientific Research Fund Project of College of Science and Technology of Nanchang University (ZL-2010-01).
Author details
1
College of Science and Technology, Nanchang University, Nanchang, Jiangxi 330029, People’s Republic of China 2Department of Mathematics, Nanchang University, Nanchang, Jiangxi 330031, People’s Republic of China
Authors’contributions
Q-YL carried out the study of well-posedness for parametric strong vector quasi-equilibrium problems, strong vector quasi-variational inequality problems and strong vector quasi-optimization problems, and participated in the sequence alignment and drafted the manuscript. S-HW conceived of the study, and participated in its design and coordination. All authors read and approved the final manuscript.
Competing interests
The authors declare that they have no competing interests.
Received: 13 May 2011 Accepted: 7 October 2011 Published: 7 October 2011
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doi:10.1186/1687-1812-2011-62
Cite this article as:Li and Wang:Well-posedness for parametric strong vector quasi-equilibrium problems with applications.Fixed Point Theory and Applications20112011:62.
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