In this paper, we investigate the
α
wellposedness and
α
Lwellposedness for a mixed vector quasivariationallike inequality using bifunctions. Some characterizations are derived for the above mentioned wellposedness concepts. The concepts of
α
wellposedness and
α
Lwellposedness in the generalized sense are also given and similar characterizations are derived.
AMS Mathematics Subject Classification : 49J40, 49K40, 90C31.
1. Introduction
The notion of wellposedness is significant as it plays a crucial role in the stability theory for optimization problems and has been studied in different areas of optimization such as mathematical programming, calculus of variations and optimal control. Such a study becomes important for problems wherein, we may not be able to find the exact solution of the problem. Under these circumstances, the wellposedness of an optimization problem is pivotal in the sense that it ensures the convergence of the sequence of approximate solutions obtained through iterative techniques to the exact solution of the problem.
Wellposedness of a minimization problem was first considered by Tykhonov
[23]
according to which every minimizing sequence converges towards the unique minimum solution. Practically, a problem may have more than one solution. Hence, the notion of wellposedness in the generalized sense was introduced. The nonemptiness of the set of minimizers and the convergence of subsequence of the minimizing sequence towards a member of this set guarantees wellposedness in the generalized sense. Zolezzi
[26
,
27]
introduced and studied the extended wellposedness for an optimization problem by embedding the original problem into a parametric optimization problem. For further details, one may refer to the text by Lucchetti
[17]
.
Variational inequality provides suitable mathematical models for a wide range of practical problems and have been intensively studied in
[7
,
14]
. Since, a minimization problem is closely related to a variational inequality, hence it is important to study the wellposedness of variational inequality. Lucchetti and Patrone
[15]
introduced the notion of wellposedness for a variational inequality by means of Ekeland’s Variational Principle. Lignola and Morgan
[16]
introduced parametric wellposedness for variational inequalities whereas in
[18]
, Lignola introduced the notions of wellposedness and Lwellposedness for quasivariational inequalities and derived some metric characterizations. The corresponding results of Lignola and Morgan
[16]
were extended to the vector case by Fang and Huang
[8]
. Prete et al.
[21]
introduced the concept of
α
wellposedness for the classical variational inequality. Fang, Huang and Yao
[10]
introduced the notion of wellposedness for a mixed variational inequality and studied its relationship with the wellposedness of corresponding inclusion and fixed point problems which was further generalized by Ceng and Yao
[2]
for generalized mixed variational inequality. Parametric variational inequalities are problems where a parameter is allowed to vary in a certain subset of a metric space. It has been shown that the parametric variational inequality is a central ingredient in the class of Mathematical Programs with Equilibrium Constraints which appear in many applied contexts and have been studied by many authors
[16
,
21]
.
A quasivariational inequality is an extension of the classical variational inequality in which the defining set of the problem varies with a variable. The interest in quasivariational inequalities lies in the fact that many economic or engineering problems are modeled through them. Very recently, Ceng et al.
[3]
studied the concepts of wellposedness and Lwellposedness for mixed quasivariationallike inequality problems (MQVLI). Fang and Hu
[9]
and Hu, Fang and Huang
[12]
studied wellposedness for parametric variational inequality and quasivariational inequality respectively using bifunctions.
Motivated by the above mentioned research work,in this paper, we generalize the concepts of
α
wellposedness and
α
Lwellposedness for parametric mixed vector quasivariationallike inequality (MVQVLI
_{p}
) having a unique solution and in the generalized sense if (MVQVLI
_{p}
) has more than one solution. Necessary and sufficient conditions for
α
wellposedness and
α
Lwellposedness are formulated in terms of the diameters of the approximate solution sets. In a similar way,
α
wellposedness and
α
Lwellposedness in the generalized sense is shown to be equivalent to a condition involving a regular measure of non compactness of the approximate solution sets.
The paper is organized as follows: In Section 2, necessary notations, definitions and lemmas have been recalled. Section 3 establishes necessary and sufficient conditions for
α
wellposedness and
α
Lwellposedness for (MVQVLI
_{p}
) using bifunctions, while in Section 4, necessary and sufficient conditions are obtained for
α
wellposedness and
α
Lwellposedness in the generalized sense. Finally in Section 5,
α
wellposedness and
α
Lwellposedness of (MVQVLI
_{p}
) are shown to be equivalent to the existence and uniqueness of their respective solutions.
2. Preliminaries
Throughout this paper, we suppose that
α
≥ 0,
K
is a nonempty closed subset of a real Banach space
X
. Let
η
:
K
×
K
→
X
be a map. Let
P
be a parametric norm space,
S
:
P
×
K
→ 2
^{K}
be a setvalued map. Let
Y
be a real Banach space endowed with a partial order induced by a pointed, closed and convex cone
C
with
intC
nonempty;
Let
h
:
P
×
K
×
X
→
Y
be a function. Let
ϕ
:
K
×
K
→
Y
be a bifunction. We consider the following parametric mixed vector quasivariationallike inequality using bifunctions;
MVQVLI
_{p}
(
h, S
) Find
x
∈
K
such that
x
∈
S
(
p, x
),
It is observed that MVQVLI
_{p}
(
h, S
) provides very general formulations of variational inequalities which include the classical Stampacchia variational inequality as a special case (see
[14]
), mixed quasivariationallike inequalities (see
[3]
), variational inequalities defined using bifunctions (see
[9]
), parametric quasivariational inequality (see
[19]
) and parametric quasivariational inequality defined using bifunctions (see
[12]
).
In particular, we observe that, if
ϕ
(
x, y
) =
ϕ
(
x
) −
ϕ
(
y
) and
Y
=
then MVQVLI
_{p}
(
h, S
) reduces to mixed quasivariationallike inequality studied in
[3]
. If
ϕ
(
x, y
) = 0,
η
(
x, y
) =
x
−
y
, ∀
x, y
∈
K
and
Y
=
then MVQVLI
_{p}
(
h, S
) reduces to the parametric Stampacchia quasivariational inequality using bifunctions SQVI
_{p}
(
h, S
) which has been dealt in
[12]
. If further,
S
(
p, x
) =
K
, ∀
x
∈
K
, then it reduces to the parametric Stampacchia variational inequality using bifunctions studied in
[9]
. The solution set of MVQVLI
_{p}
(
h, S
) is denoted by T
_{p}
. In the sequel, we introduce some notions of wellposedness for MVQVLI
_{p}
(
h, S
).
Definition 2.1.
Let
α
≥ 0. Let
p
∈
P
and {
p_{n}
} ⊂
P
be a sequence converging to
p
. A sequence {
xn
} ⊂
X
is said to be an
α

approximating sequence
[respectively an
α

Lapproximating sequence
] for MVQVLI
_{p}
(
h, S
) corresponding to {
p_{n}
} if and only if:
(i)
x_{n}
∈
K
, ∀
n
∈ ℕ.
(ii) there exists a sequence of positive numbers {
ϵ_{n}
} with
ϵ_{n}
↓ 0 such that:
[respectively if:
(i)
x_{n}
∈
K
, ∀
n
∈ ℕ.
(ii) there exists a sequence of positive numbers {
ϵ_{n}
} with
ϵ_{n}
↓ 0 such that:
where
e
is any fixed point in
intC
.
Remark 2.2.
Definition 2.1 generalizes Definition 2.3 of Lignola
[18]
, Definition 2.2 of Ceng et al.
[3]
and Definition 1 of Hu et al.
[12]
.
Definition 2.3.
The family {MVQVLI
_{p}
(
h, S
) :
p
∈
P
} is said to be
α
wellposed [respectively
α

Lwellposed
] if ∀
p
∈
P
, MVQVLI
_{p}
(
h, S
) has a unique solution
x_{p}
and for all sequences {
p_{n}
} →
p
, every
α
approximating sequence [respectively
α
Lapproximating sequence] corresponding to {
p_{n}
} converges to
x_{p}
.
Remark 2.4.
Definition 2.3 generalizes Definition 2.4 of
[18]
, Definition 2.3 of
[3]
and Definition 2 of
[12]
.
Definition 2.5.
The family {MVQVLI
_{p}
(
h, S
) :
p
∈
P
} is said to be
α

wellposed in the generalized sense
[respectively
α

L

well

posed in the generalized
sense] if ∀
p
∈
P
, MVQVLI
_{p}
(
h, S
) has a nonempty solution set and for all sequences {
p_{n}
} →
p
, every αapproximating sequence [respectively
α
Lapproximating sequence] corresponding to {
p_{n}
} has a subsequence which converges to some point of the solution set.
In order to characterize the wellposedness of the quasivariational inequality, Lignola
[18]
defined some concepts of approximate solutions for quasivariational inequalities. Motivated by these concepts, for every
α
≥ 0,
ϵ
≥ 0,
δ
≥ 0, we consider the following
α
approximate and
α
Lapproximate solution sets;
To investigate the
α
wellposedness and
α
Lwellposedness of MVQVLI
_{p}
(
h, S
), we need the following concepts and results.
Definition 2.6
(
[13]
). Let
H
be a non empty subset of
X
. The measure of noncompactness
μ
of the set
H
is defined by
where diam
H_{i}
= sup{
d
(
α
_{1}
,
α
_{2}
) :
α
_{1}
,
α
_{2}
∈
H_{i}
}.
Definition 2.7
(
[13]
). The
Hausdorff Distance
between two nonempty bounded subsets
A
and
B
of a metric space (
X, d
) is
where
Definition 2.8
(
[3
,
8]
). Let
h
:
P
×
K
×
K
→
Y
be a function and let
ϕ
:
K
×
K
→
Y
be a bifunction. Let
η
:
K
×
K
→
X
be a map. Then
h
is said to be
(i)
C

η

monotone
if for any
x, y
∈
K
,
(ii)
C

η

pseudomonotone
with respect to
ϕ
if for any
x, y
∈
K
,
Definition 2.9
(
[13]
). Let (
E, τ
) and (
F, σ
) be two 1st countable topological spaces. A set valued map
G
:
E
→ 2
^{F}
is said to be,
(i) (
τ, σ
)
closed
if for all
x
∈
E
, for all sequences {
x_{n}
}
τ
converging to
x
and for all sequences {
y_{n}
}
σ
converging to
y
such that
y_{n}
∈
G
(
x_{n}
), ∀
n
∈ ℕ, one has
y
∈
G
(
x
), that is,
G
(
x
) ⊃ lim supn
G
(
x_{n}
).
(ii) (
τ, σ
)
lower semicontinuous
if for all
x
∈
E
, for all sequences {
x_{n}
}
τ
 converging to
x
and for all
y
∈
G
(
x
), there exists a sequence {
y_{n}
}
σ
converging to
y
such that
y_{n}
∈
G
(
x_{n}
) for sufficiently large
n
, that is,
G
(
x
) ⊂ lim inf
_{n}
G
(
x_{n}
).
(iii) (
τ, σ
)
subcontinuous
if for all
x
∈
E
, for all sequences {
x_{n}
}
τ
converging to
x
and for all sequences {
y_{n}
} with
y_{n}
∈
G
(
x_{n}
),
y_{n}
has a
σ
convergent subsequence.
Definition 2.10.
A function
g
:
X
→ ℝ is said to be
positively homogeneous
if
g
(
λx
) =
λg
(
x
), ∀
x
∈
X
, ∀
λ
> 0.
Lemma 2.11.
Let K be convex and x
∈
K be a given point. Let h
:
P
×
K
×
X
→
Y be a positively homogeneous function in 3rd variable, y
↦
h
(
p, x, η
(
x, y
))
be concave and η
(
x, x
) = 0,
ϕ be a bifunction with ϕ
(
x, x
) = 0
for fixed x and y
↦
ϕ
(
x, y
)
concave. Then,
Proof
. Obviously, necessary condition holds true.
For sufficient condition, let
For any
v
∈
K
, let
y
(
t
) =
y
+
t
(
v
−
y
) ∈
K
, ∀
t
∈ [0, 1] with
y
(
t
) ≠
x
. Hence,
Now,
y
↦
h
(
p, x, η
(
x, y
)) is concave and
η
(
x, x
) = 0. So,
h
(
p, x, η
(
x, y
(
t
))) ≥
h
(
p, x, tη
(
x, v
)). Since,
h
is positively homogeneous in the 3rd variable, we obtain,
As
ϕ
(
x
, .) is concave and
ϕ
(
x, x
) = 0, we get that
Therefore, using the fact that
t
∈ [0, 1], we have,
Thus,
Dividing by
t
> 0 and taking limit as
t
→ 0, we get the required sufficient condition.
Remark 2.12.
Lemma 2.11 is a generalization of Lemma 2 of
[12]
.
3. Mixed Vector Quasivariationallike Inequality Having a Unique Solution
In this section, we give some metric characterizations of
α
wellposedness and
α
Lwellposedness for MVQVLI
_{p}
(
h, S
).
Theorem 3.1.
Let K be a closed and convex subset of a real Banach space X. Let ϕ
:
K
×
K
→
Y
be a continuous bifunction with ϕ
(
x, x
) = 0
for fixed x and ϕ
(
x
, .)
concave. Let η
:
K
×
K
→
X
be a continuous mapping with y
↦
h
(
p, x, η
(
x, y
))
concave and η
(
x, x
) = 0.
Let S
:
P
×
K
→ 2
^{K}
be a nonempty set

valued map which is convex valued
, (
s,w
)
closed
, (
s,w
)
subcontinuous and
(
s, s
)
lower semicontinuous. Let h
:
P
×
K
×
K
→
Y
be a continuous function which is positively homogeneous in the 3rd variable. Then, MVQVLI_{p}
is α

well

posed if and only if,
∀
p
∈
P
,
Proof
. Let MVQVLI
_{p}
be αwellposed. Then,
T_{p}
≠ ∅ and
T_{p}
⊂ Q
_{p}
(
δ, ϵ
). Thus, Q
_{p}
(
δ, ϵ
) ≠ ∅, ∀
δ, ϵ
> 0. Assume on the contrary, diam Q
_{p}
(
δ, ϵ
)
0 as (
δ, ϵ
) → (0, 0). Then, there exist sequences {
ϵ_{n}
}, {
δ_{n}
}, {
u_{n}
}, {
v_{n}
} with
ϵ_{n}
→ 0,
δ_{n}
→ 0,
u_{n}
,
v_{n}
∈ Q
_{p}
(
δ_{n}
,
ϵ_{n}
) and a positive number
l
such that
As
u_{n}
,
v_{n}
∈ Q
_{p}
(
δ_{n}
,
ϵ_{n}
), there exist
such that
Thus, {
u_{n}
} and {
v_{n}
} are
α
approximating sequences for MVQVLI
_{p}
corresponding to {
p_{n}
} and
respectively. As MVQVLI
_{p}
is
α
wellposed, both the
α
approximating sequences converge to the unique solution of MVQVLI
_{p}
, which is a contradiction to (3.2).
Conversely, suppose ∀
p
∈
P
, (3.1) holds. We will first show that MVQVLI
_{p}
cannot have more than one solution. Assume
z
_{1}
and
z
_{2}
are its solutions with
z
_{1}
≠
z
_{2}
. Then,
z
_{1}
,
z
_{2}
∈ Q
_{p}
(
δ, ϵ
), ∀
δ, ϵ
≥ 0. Taking (3.1) into account, we get
z
_{1}
=
z
_{2}
, which is a contradiction.
Now, let {
p_{n}
} →
p
∈
P
and {
x_{n}
} be an
α
approximating sequence for MVQVLI
_{p}
. Then, there exists sequence {
ϵ_{n}
} with
ϵ_{n}
↓ 0 such that
x_{n}
∈
K
,
Take
δ_{n}
= ∥
p_{n}
−
p
∥ then
x_{n}
∈ Q
_{p}
(
δ_{n}
,
ϵ_{n}
). By the given condition, {
x_{n}
} is a Cauchy sequence and it strongly converges to a point say
x
_{0}
∈
K
. We will now prove that
x
_{0}
is the unique solution of MVQVLI
_{p}
by two steps.
(i) We show that
x
_{0}
∈
S
(
p
,
x
_{0}
). Since,
d
(
x_{n}
,
S
(
p_{n}
,
x_{n}
))
. Therefore, there exists
y_{n}
∈
S
(
p_{n}
,
x_{n}
) such that
As,
S
is (
s,w
)subcontinuous and (
s,w
)closed, the sequence {
y_{n}
} has a subsequence {
y_{nk}
} weakly converging to
y
∈
S
(
p
,
x
_{0}
). Hence,
Thus,
x
_{0}
∈
S
(
p
,
x
_{0}
).
(ii) Let
z
∈
S
(
p
,
x
_{0}
) be an arbitrary element. Since,
S
is (
s, s
)lower semicontinuous, there exists
z_{n}
∈
S
(
p_{n}
,
x_{n}
) :
z_{n}
→
z
. Thus, there exists sequence {
ϵ_{n}
} ↓ 0 such that
h, η
and
ϕ
being continuous, we get
Thus, by Lemma 2.11, we have
which shows that
x
_{0}
is a solution of MVQVLI
_{p}
.
Hence, MVQVLI
_{p}
is αwellposed.
Theorem 3.2.
Let S
:
P
×
K
→ 2
^{K}
be convex valued. Then MVQVLI_{p} is α

well

posed if and only if its solution set T_{p}
≠ ∅, ∀
p
∈
P and diam Q_{p}
(
δ, ϵ
) → 0
as
(
δ, ϵ
) → (0, 0).
Proof
. The necessary condition has been proved in Theorem 3.1.
For sufficiency, let {
p_{n}
} be a sequence such that
p_{n}
→
p
∈
P
and {
x_{n}
} be an
α
approximating sequence for MVQVLI
_{p}
corresponding to {
p_{n}
}. Then, there exists sequence {
ϵ_{n}
} with
ϵ_{n}
↓ 0 such that
Thus,
x_{n}
∈ Q
_{p}
(
δ_{n}
,
ϵ_{n}
) with
δ_{n}
= ∥
p_{n}
−
p
∥. Let
x
_{0}
be the unique solution of MVQVLI
_{p}
. Then,
x
_{0}
∈ Q
_{p}
(
δ_{n}
,
ϵ_{n}
) ∀
n
. Thus, ∥
x_{n}
−
x
_{0}
∥ ≤ diam Q
_{p}
(
δ_{n}
,
ϵ_{n}
) → 0, that is,
x_{n}
→
x
_{0}
and hence, MVQVLI
_{p}
is
α
wellposed.
We now have analogous results for
α
Lwellposedness.
Theorem 3.3.
Suppose that the hypothesis of Theorem 3.1 hold and let h be C

η

pseudomonotone with respect to ϕ. Then MVQVLI_{p} is α

L

well

posed if and only if L_{p}
(
δ, ϵ
) ≠ ∅, ∀
δ, ϵ
> 0
and diam L_{p}
(
δ, ϵ
) → 0
as
(
δ, ϵ
) → (0, 0).
Proof
. Let MVQVLI
_{p}
be
α
Lwellposed. As
h
is C
η
pseudomonotone with respect to
ϕ
, L
_{p}
(
δ, ϵ
) ≠ ∅. On the same lines of Theorem 3.1, we get diam L
_{p}
(
δ, ϵ
) → 0 as (
δ, ϵ
) → (0, 0).
Conversely, let the given condition hold. As
h
is C
η
pseudomonotone with respect to
ϕ
, every solution of MVQVLI
_{p}
belongs to L
_{p}
(
δ, ϵ
), ∀
δ, ϵ
> 0 which would also be unique. Also, an
α
Lapproximating sequence exists. Let {
x_{n}
} be an
α
Lapproximating sequence which converges to
x
_{0}
, as in Theorem 3.1.
x
_{0}
would then be the solution of MVQVLI
_{p}
. Thus, MVQVLI
_{p}
is
α
Lwellposed.
Theorem 3.4.
Let S
:
P
×
K
→ 2
^{K}
be convex valued and h be C

η

pseudomonotone with respect to ϕ. Then, MVQVLI_{p} is α

L

well

posed if and only if its solution set T_{p}
≠ ∅
and diam L_{p}
(
δ, ϵ
) → 0
as
(
δ, ϵ
) → (0, 0).
Proof
. The necessary condition holds as in Theorem 3.3.
For sufficient condition, let {
p_{n}
} be a sequence converging to
p
∈
P
and {
x_{n}
} be an
α
Lapproximating sequence for MVQVLI
_{p}
corresponding to {
p_{n}
}. Then, there exists sequence {
ϵ_{n}
} with
ϵ_{n}
↓ 0 such that,
Thus,
x_{n}
∈ L
_{p}
(
δ_{n}
,
ϵ_{n}
) with
δ_{n}
= ∥
p_{n}
−
p
∥. Let
x
_{0}
be the unique solution of MVQVLI
_{p}
. Then,
x
_{0}
∈ L
_{p}
(
δ_{n}
,
ϵ_{n}
), ∀
n
. Thus, ∥
x_{n}
−
x
_{0}
∥ ≤ diam L
_{p}
(
δ_{n}
,
ϵ_{n}
) → 0, that is,
x_{n}
→
x
_{0}
and the problem is
α
Lwellposed.
4. Mixed Vector Quasivariationallike Inequality having more than One Solution
In this section, we give some metric characterizations of
α
wellposedness and
α
Lwellposedness in the generalized sense for MVQVLI
_{p}
(
h, S
).
Theorem 4.1.
Let all the assumptions of Theorem 3.1 be true and let P be finite dimensional. Then, MVQVLI_{p} is α

well

posed in the generalized sense if and only if
, ∀
p
∈
P
,
Proof
. Let MVQVLI
_{p}
be
α
wellposed in the generalized sense. Then,
T_{p}
≠ ∅ and
T_{p}
⊂ Q
_{p}
(
δ, ϵ
). Thus, Q
_{p}
(
δ, ϵ
) ≠ ∅ ∀
δ, ϵ
> 0. Also,
T_{p}
is compact as when {
x_{n}
} is any sequence in
T_{p}
, then {
x_{n}
} would be an
α
approximating sequence for MVQVLI
_{p}
which is
α
wellposed in the generalized sense, therefore {
x_{n}
} would have a subsequence converging strongly to some point of
T_{p}
. Now,
Also,
μ
(Q
_{p}
(
δ, ϵ
)) ≤
(Q
_{p}
(
δ, ϵ
),
T_{p}
) +
μ
(
T_{p}
) = 2
e
(Q
_{p}
(
δ, ϵ
),
T_{p}
). It is now sufficient to show that
e
(Q
_{p}
(
δ, ϵ
),
T_{p}
) → 0 as (
δ, ϵ
) → (0, 0). If
e
(Q
_{p}
(
δ, ϵ
),
T_{p}
)
( 0 as (
δ, ϵ
) → (0, 0), there exists
τ
> 0 and sequences {
δ_{n}
}, {
ϵ_{n}
} with
δ_{n}
↓ 0,
ϵ_{n}
↓ 0,
x_{n}
∈
K
with
x_{n}
∈ Q
_{p}
(
δ_{n}
,
ϵ_{n}
) such that
Since,
x_{n}
∈ Q
_{p}
(
δ_{n}
,
ϵ_{n}
), {
x_{n}
} is an
α
approximating sequence of MVQVLI
_{p}
which is
α
wellposed in the generalized sense. Hence, {
x_{n}
} has a subsequence
converging to some point of
T_{p}
which is a contradiction to (4.1).
Conversely, let Q
_{p}
(
δ, ϵ
) ≠ ∅ and lim
_{δ→0,ϵ→0}
μ
(Q
_{p}
(
δ, ϵ
)) = 0. We first show that Q
_{p}
(
δ, ϵ
) is closed, ∀
δ, ϵ
> 0. Let
x_{n}
∈ Q
_{p}
(
δ, ϵ
) such that
x_{n}
→
x
. Then, there exists
p_{n}
∈
B
(
p, δ
) such that
d
(
x_{n}
,
S
(
p_{n}
,
x_{n}
)) ≤
ϵ
and
P
being finite dimensional,
As,
d
(
x_{n}
,
S
(
p_{n}
,
x_{n}
))
there exists
y_{n}
∈
S
(
p_{n}
,
x_{n}
) such that ∥
x_{n}
−
y_{n}
∥
S
being (
s,w
)closed and (
s,w
)subcontinuous, {
y_{n}
} has a subsequence
which converges weakly to
Thus,
that is,
S
being (
s, s
)lower semicontinuous, there exists
z_{n}
∈
S
(
p_{n}
,
x_{n}
) such that
z_{n}
→
z
. Thus,
h
(
p_{n}
,
x_{n}
,
η
(
x_{n}
,
z_{n}
)) +
By continuity of
h, η
and
ϕ
, we have,
Hence,
x
∈ Q
_{p}
(
δ, ϵ
) which shows that Q
_{p}
(
δ, ϵ
) is nonempty and closed. Also,
T_{p}
= ∩
_{δ>0,ϵ>0}
Q
_{p}
(
δ, ϵ
). Since,
μ
(Q
_{p}
(
δ, ϵ
)) → 0 as (
δ, ϵ
) → (0, 0), by the Theorem on Page 412 of
[13]
, we conclude that
T_{p}
is nonempty, compact and
Let
p_{n}
→
p
and {
x_{n}
} be an
α
approximating sequence for MVQVLI
_{p}
. There exists
ϵ_{n}
↓ 0 such that
d
(
x_{n}
,
S
(
p_{n}
,
x_{n}
)) ≤
ϵ_{n}
and
Take
δ_{n}
= ∥
p_{n}
−
p
∥,
x_{n}
∈ Q
_{p}
(
δ_{n}
,
ϵ_{n}
). There exists a sequence
such that
Since,
T_{p}
is compact,
has a subsequence
converging to
Hence, the corresponding sequence
converges strongly to
proving that MVQVLI
_{p}
is
α
wellposed in the generalized sense.
Theorem 4.2.
Let the assumptions be as in Theorem 3.3 and let P be finite dimensional. Then, MVQVLI_{p} is α

L

well

posed in the generalized sense if and only if,
∀
p
∈
P
,
L_{p}
(
δ, ϵ
) ≠ ∅, ∀
δ, ϵ
> 0
and
Proof
. Let MVQVLI
_{p}
be
α
Lwellposed in the generalized sense. As
h
is C
η
pseudomonotone with respect to
ϕ
, L
_{p}
(
δ, ϵ
) ≠ ∅, ∀
δ, ϵ
> 0. To show that
the proof is similar as in Theorem 4.1.
Conversely, let
As done in the previous theorem, we get that every
α
Lapproximating sequence has a convergent subsequence and this limit is a solution of MVQVLI
_{p}
, proving MVQVLI
_{p}
is
α
Lwellposed in the generalized sense.
Corollary 4.3.
MVQVLI_{p} is αwellposed in the generalized sense (respectively αLwellposed in the generalized sense) if and only if,
∀
p
∈
P, the solution set of MVQVLI_{p}, that is, T_{p} is nonempty compact and e
(
Q_{p}
(
δ, ϵ
),
T_{p}
) → 0
as
(
δ, ϵ
) → (0, 0)
(respectively if and only if T_{p} is nonempty compact and e
(
L_{p}
(
δ, ϵ
),
T_{p}
) → 0
as
(
δ, ϵ
) → (0, 0).)
Proof
. Let MVQVLI
_{p}
be
α
wellposed in the generalized sense. Thus,
T_{p}
≠ ∅ and compact. If
e
(Q
_{p}
(
δ, ϵ
),
T_{p}
)
0 as (
δ, ϵ
) → (0, 0) then there exists
τ
> 0, sequences {
δ_{n}
}, {
ϵ_{n}
} with
δ_{n}
→ 0,
ϵ_{n}
→ 0,
x_{n}
∈
K
with
x_{n}
∈ Q
_{p}
(
δ_{n}
,
ϵ_{n}
) such that
x_{n}
∉
T_{p}
+
B
(0,
τ
). Since,
x_{n}
∈ Q
_{p}
(
δ_{n}
,
ϵ_{n}
), {
x_{n}
} is an
α
approximating sequence of MVQVLI
_{p}
which is
α
wellposed in the generalized sense. Hence, {
x_{n}
} has a subsequence
converging to some point of
T_{p}
, which is a contradiction.
Conversely, let
T_{p}
be nonempty compact and
e
(Q
_{p}
(
δ, ϵ
),
T_{p}
) → 0 as (
δ, ϵ
) → (0, 0). Let
p_{n}
→
p
and {
x_{n}
} be an
α
approximating sequence for MVQVLI
_{p}
. There exists
ϵ_{n}
↓ 0 such that
d
(
x_{n}
,
S
(
p_{n}
,
x_{n}
)) ≤
ϵ_{n}
and
Take
δ_{n}
= ∥
p_{n}
−
p
∥,
x_{n}
∈ Q
_{p}
(
δ_{n}
,
ϵ_{n}
).
There exists a sequence
such that
Since,
T_{p}
is compact,
has a subsequence
converging to
Hence, the corresponding sequence
converges strongly to
proving that MVQVLI
_{p}
is
α
wellposed in the generalized sense.
On the similar lines as above, we can show that MVQVLI
_{p}
is
α
Lwellposed in the generalized sense if and only if
T_{p}
is nonempty compact and
e
(
L_{p}
(
δ, ϵ
),
T_{p}
) → 0 as (
δ, ϵ
) → (0, 0).
5. Conditions for αwellposedness and αLwellposedness
In the following section, we will show that
α
wellposedness and
α
Lwellposedness of MVQVLI
_{p}
is equivalent to the existence and uniqueness of its solution.
Theorem 5.1.
Let K be a nonempty compact and convex subset of a real Banach space X. Let ϕ
:
K
×
K
→
Y be a continuous bifunction with ϕ
(
x
, .)
concave and ϕ
(
x, x
) = 0
for fixed x. Let η
:
K
×
K
→
X be a continuous mapping with y
↦
h
(
p, x, η
(
x, y
))
concave and η
(
x, x
) = 0.
Let S
:
P
×
K
→ 2
^{K}
be a nonempty
setvalued map which is convex valued,
(
s,w
)
closed
, (
s,w
)
subcontinuous and
(
s, s
)
lower semicontinuous. Let h
:
P
×
K
×
X
→
Y
be a continuous function which is positively homogeneous in the 3rd variable. Then, MVQVLI_{p} is
α

L

well

posed if and only if it has a unique solution.
Proof
. Let MVQVLI
_{p}
be
α
Lwellposed. Then, by definition, it has a unique solution.
Conversely, let MVQVLI
_{p}
has a unique solution say
z
_{0}
and {
x_{n}
} be an
α
Lapproximating sequence. Let
p_{n}
→
p
∈
P
. Since,
K
is compact, {
x_{n}
} has a subsequence still denoted by {
x_{n}
} converging to
x
_{0}
∈
K
. It is sufficient to show that
x
_{0}
is a solution of MVQVLI
_{p}
. Then,
x
_{0}
=
z
_{0}
and the whole sequence {
x_{n}
} would then converge to
z
_{0}
. Following the proof of Theorem 3.1, we get that
x
_{0}
is a solution of MVQVLI
_{p}
.
Theorem 5.2.
Let the assumptions be as in Theorem 5.1. Further, assume that h is C

η

pseudomonotone with respect to ϕ. Then, MVQVLI_{p} is α

well

posed if and only if it has a unique solution.
Proof
. Necessary condition holds obviously.
For sufficient condition, let MVQVLI
_{p}
has a unique solution say
x
_{0}
. Let {
p_{n}
} be a sequence such that
p_{n}
→
p
∈
P
and {
x_{n}
} be an
α
approximating sequence for MVQVLI
_{p}
. As
h
is C
η
pseudomonotone with respect to
ϕ
. Then, {
x_{n}
} is also an
α
Lapproximating sequence. By Theorem 5.1, MVQVLI
_{p}
is
α
Lwellposed. Hence,
x_{n}
→
x
_{0}
and so, MVQVLI
_{p}
is
α
wellposed.
6. Conclusion
A mixed vector quasivariationallike inequality is considered and various results characterizing (generalized)
α
wellposedness and (generalized)
α
Lwellposedness for this problem have been given. For further research, Levitin–Polyak wellposedness can be investigated for the same problem.
Acknowledgements
The research of the corresponding author was supported by The Council of Scientific and Industrial Research, India Grant No. 09/045(1036)/2010EMRI. We also wish to thank the reviewers for their constructive suggestions.
BIO
Garima Virmani is a research scholar at Delhi University. Her research interests focus on Optimization and Variational inequality.
Department of Mathematics, University of Delhi, Delhi 110007, India.
email: garimavirmani86@gmail.com
Manjari Srivastava received Ph.D. from Delhi University. Since 1987, she has been in Miranda House, Delhi University. Her research interest focus on Multiobjective optimization, Continuous time programming, and Variational inequality.
Miranda House, Department of Mathematics, University of Delhi, Delhi 110007, India.
email: garimavirmani86@gmail.com
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