Hybrid-type generalized multivalued vector complementarity problems
We introduce a new type of generalized multivalued vector complementarity problems with moving pointed cone. We discuss the existence results for generalized multivalued vector complementarity problems under inclusive assumptions and obtain results on the equivalence between the generalized multival...
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Agarwal, R.P. Ahmad, M.K. Salahuddin 2020-02-11T10:49:41Z 2020-02-11T10:49:41Z 2013 Hybrid-type generalized multivalued vector complementarity problems / R.P. Agarwal, M.K. Ahmad, Salahuddin // Український математичний журнал. — 2013. — Т. 65, № 1. — С. 7-20. — Бібліогр.: 16 назв. — англ. 1027-3190 https://nasplib.isofts.kiev.ua/handle/123456789/164923 517.9 We introduce a new type of generalized multivalued vector complementarity problems with moving pointed cone. We discuss the existence results for generalized multivalued vector complementarity problems under inclusive assumptions and obtain results on the equivalence between the generalized multivalued vector complementarity problems and the generalized multivalued vector variational inequality problems. Введено новий тип узагальнених багатозначних векторних задач доповнюваностi з рухомим загостреним конусом. Розглянуто питання про iснування розв’язкiв узагальнених багатозначних векторних задач доповнюваностi при умовах включення та отримано результати щодо еквiвалентностi мiж узагальненими багатозначними векторними задачами доповнюваностi та узагальненими багатозначними векторними задачами для варiацiйних нерiвностей. en Інститут математики НАН України Український математичний журнал Статті Hybrid-type generalized multivalued vector complementarity problems Узагальненi багатозначнi векторнi задачi доповнюваностi гiбридного типу Article published earlier |
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Hybrid-type generalized multivalued vector complementarity problems |
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Hybrid-type generalized multivalued vector complementarity problems Agarwal, R.P. Ahmad, M.K. Salahuddin Статті |
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Hybrid-type generalized multivalued vector complementarity problems |
| title_full |
Hybrid-type generalized multivalued vector complementarity problems |
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Hybrid-type generalized multivalued vector complementarity problems |
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Hybrid-type generalized multivalued vector complementarity problems |
| title_sort |
hybrid-type generalized multivalued vector complementarity problems |
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Agarwal, R.P. Ahmad, M.K. Salahuddin |
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Agarwal, R.P. Ahmad, M.K. Salahuddin |
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Узагальненi багатозначнi векторнi задачi доповнюваностi гiбридного типу |
| description |
We introduce a new type of generalized multivalued vector complementarity problems with moving pointed cone. We discuss the existence results for generalized multivalued vector complementarity problems under inclusive assumptions and obtain results on the equivalence between the generalized multivalued vector complementarity problems and the generalized multivalued vector variational inequality problems.
Введено новий тип узагальнених багатозначних векторних задач доповнюваностi з рухомим загостреним конусом. Розглянуто питання про iснування розв’язкiв узагальнених багатозначних векторних задач доповнюваностi при умовах включення та отримано результати щодо еквiвалентностi мiж узагальненими багатозначними векторними задачами доповнюваностi та узагальненими багатозначними векторними задачами для варiацiйних нерiвностей.
|
| issn |
1027-3190 |
| url |
https://nasplib.isofts.kiev.ua/handle/123456789/164923 |
| citation_txt |
Hybrid-type generalized multivalued vector complementarity problems / R.P. Agarwal, M.K. Ahmad, Salahuddin // Український математичний журнал. — 2013. — Т. 65, № 1. — С. 7-20. — Бібліогр.: 16 назв. — англ. |
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| fulltext |
UDC 517.9
R. P. Agarwal (Texas A&M University-Kingsville, USA),
Salahuddin, M. K. Ahmad (Aligarh Muslim Univ., India)
HYBRID TYPE GENERALIZED MULTIVALUED VECTOR
COMPLEMENTARITY PROBLEMS
УЗАГАЛЬНЕНI БАГАТОЗНАЧНI ВЕКТОРНI ЗАДАЧI ДОПОВНЮВАНОСТI
ГIБРИДНОГО ТИПУ
We introduce a new type of generalized multivalued vector complementarity problems with moving pointed cone. We
discuss the existence results for generalized multivalued vector complementarity problems under inclusive assumptions and
obtain results on the equivalence between the generalized multivalued vector complementarity problems and the generalized
multivalued vector variational inequality problems.
Введено новий тип узагальнених багатозначних векторних задач доповнюваностi з рухомим загостреним конусом.
Розглянуто питання про iснування розв’язкiв узагальнених багатозначних векторних задач доповнюваностi при
умовах включення та отримано результати щодо еквiвалентностi мiж узагальненими багатозначними векторними
задачами доповнюваностi та узагальненими багатозначними векторними задачами для варiацiйних нерiвностей.
1. Introduction and preliminaries. The purpose of this paper is to introduce and discuss a new
type of generalized multivalued vector complementarity problem with moving pointed cone which is
a variable ordering relation. We derive existence of solutions for this class of generalized multivalued
vector complementarity problems under inclusive type assumptions. This inclusive conditions require
that any two of the family which is closed and convex satisfy an inclusion relation so long as their
corresponding variable satisfy certain conditions. We have also obtained some equivalence results
among a generalized multivalued vector complementarity problem, a generalized multivalued vector
variational inequality problem, a generalized multivalued weak minimal element problem and a
generalized multivalued vector unilateral optimization problem under some monotonicity conditions
and some inclusive type assumptions in ordered Banach spaces. The theorems presented in this paper
improved, extended and developed some earlier and very recent results in the literature including
[2, 4 – 6, 8, 9, 11].
Motivated and inspired by these works [1, 3, 7, 8, 12 – 16], in this paper we initiate three types
of generalized multivalued vector complementarity problems. Let X, Y be the two Banach spaces,
P : D → 2Y a multivalued mapping such that for each x ∈ D,P (x) is a proper, closed, convex
and pointed moving cone with apex at the origin and intP (x) 6= ∅. Let Q(·, ·) : D × L(X,Y ) →
→ L(X,Y ), g : D → Y be the single-valued mappings and A : X → 2L(X,Y ) be the multivalued
mapping, where 2L(X,Y ) is a collection of all nonempty subsets of L(X,Y ). We consider the follow-
ing three kinds of generalized multivalued vector complementarity problems.
Weakly generalized multivalued vector complementarity problem (WGMVCP): finding x ∈ D
and u ∈ A(x) such that
〈Q(x, u), g(x)〉 6≥intP (x) 0, 〈Q(x, u), g(y)〉 6≤intP (x) 0 ∀y ∈ D. (1.1)
Positive generalized multivalued vector complementarity problem (PGMVCP): finding x ∈ D
and u ∈ A(x) such that
c© R. P. AGARWAL, SALAHUDDIN, M. K. AHMAD, 2013
ISSN 1027-3190. Укр. мат. журн., 2013, т. 65, № 1 7
8 R. P. AGARWAL, SALAHUDDIN, M. K. AHMAD
〈Q(x, u), g(x)〉 6>intP (x) 0, 〈Q(x, u), g(y)〉 ≥P (x) 0 ∀y ∈ D. (1.2)
Strong generalized multivalued vector complementarity problem (SGMVCP): finding x ∈ D and
u ∈ A(x) such that
〈Q(x, u), g(x)〉 = 0, 〈Q(x, u), g(y)〉 ≥P (x) 0 ∀y ∈ D. (1.3)
Remark 1.1. If we take g an identity mapping and Q(x, u) = Q(u), then these generalized
multivalued vector complementarity problems reduce to the following types of complementarity
problems in Ceng and Lin [1].
Generalized weak vector complementarity problem (GWVCP): finding x ∈ D and u ∈ A(x)
such that
〈Q(u), x〉 6≥intP (x) 0, 〈Q(u), y〉 6≤intP (x) 0 ∀y ∈ D. (1.4)
Generalized positive vector complementarity problem (GPVCP): finding x ∈ D and u ∈ A(x)
such that
〈Q(u), x〉 6>intP (x) 0, 〈Q(u), y〉 ≥P (x) 0 ∀y ∈ D. (1.5)
Generalized strong vector complementarity problem (GSVCP): finding x ∈ D and u ∈ A(x)
such that
〈Q(u), x〉 = 0, 〈Q(u), y〉 ≥P (x) 0 ∀y ∈ D. (1.6)
Again we remark that, if A : X → L(X,Y ) and Q an identity mapping then aforesaid three kinds
of problems reduce to the problem of Huang et al [8].
(Weak) vector complementarity problem (VCP): finding x ∈ D such that
〈A(x), x〉 6≥intP (x) 0, 〈A(x), y〉 6≤intP (x) 0 ∀y ∈ D. (1.7)
Positive vector complementarity problem (PVCP): finding x ∈ D such that
〈A(x), x〉 6>intP (x) 0, 〈A(x), y〉 ≥P (x) 0 ∀y ∈ D. (1.8)
Strong vector complementarity problem (SVCP): finding x ∈ D such that
〈A(x), x〉 = 0, 〈A(x), y〉 ≥P (x) 0 ∀y ∈ D. (1.9)
We note that if P (x) = P for all x ∈ D, where P is a closed, pointed and convex cone in Y
with nonempty interior intP (x), then all these problems are equivalent to problems considered in
Chen and Yang [3].
2. Existence of a solution for GMVCP. In this section, we extend their results to the cases
involving the multivalued mappings.
Let X be an arbitrary real Hausdorff topological vector space and Y a Banach space. Let L(X,Y )
denotes the space of all continuous linear mappings from X to Y. Let D be a nonempty subset of
X and P : D → 2Y a multivalued mapping such that for each x ∈ D, P (x) is a proper, closed,
convex and pointed moving cone with apex at the origin and intP (x) 6= ∅. Let K be a subset of
Y. For each x ∈ D, a point z ∈ K is called a minimal point of K with respect to the cone P (x)
ISSN 1027-3190. Укр. мат. журн., 2013, т. 65, № 1
HYBRID TYPE GENERALIZED MULTIVALUED VECTOR COMPLEMENTARITY PROBLEMS 9
if K ∩ (z − P (x)) = {z}; MinP (x)K is the set of all minimal points of K with respect to the
cone P (x); a point z ∈ K is called a weakly minimal point of K with respect to the cone P (x) if
K ∩ (z− intP (x)) = ∅; MinwP (x)K is the set of all weakly minimal points of K with respect to the
cone P (x), for details see [11].
Let Q : D × L(X,Y ) → L(X,Y ) and g : D → Y be the single-valued mappings and A : X →
→ 2L(X,Y ) be the multivalued mapping. Consider the generalized multivalued vector complementar-
ity problem (GMVCP): finding x ∈ D and u ∈ A(x) such that
〈Q(x, u), g(x)〉 6≥intP (x) 0, 〈Q(x, u), g(y)〉 6≤intP (x) 0 ∀y ∈ D. (2.1)
A feasible set of (GMVCP) is
Ω =
{
(x, u) ∈ D ×A(D) : u ∈ A(x), 〈Q(x, u), g(y)〉 6≤intP (x) 0 ∀y ∈ D
}
. (2.2)
We consider the following generalized multivalued vector optimization problem (GMVOP):
MinP 〈Q(x, u), g(x)〉 subject to (x, u) ∈ Ω. (2.3)
A point (x, u) ∈ Ω is called a weakly minimal solutions of (GMVOP) with respect to the cone
P (x) if 〈Q(x, u), g(x)〉 is a weakly minimal point of (GMVOP) with respect to the cone P (x),
that is
〈Q(x, u), g(x)〉 ∈ MinwP (x)
{
〈Q(x, u), g(x)〉 : (x, u) ∈ Ω
}
.
We represent the set of all weakly minimal solutions of (GMVOP) with respect to the cone P (x)
by ζwP (x) and the set of all weakly minimal solutions of (GMVOP) by ζw, that is
ζw =
⋃
x∈D
ζwP (x). (2.4)
Theorem 2.1. If ζw 6= ∅ and, for some x ∈ D, there exists (x, u) ∈ ζwP (x) such that
〈Q(x, u), g(x)〉 6≥intP (x) 0,
then generalized multivalued vector complementarity problem (GMVCP) is solvable.
Proof. Let (x, u) ∈ ζwP (x) and
〈Q(x, u), g(x)〉 6≥intP (x) 0, (2.5)
then x ∈ D, u ∈ A(x) and
〈Q(x, u), g(x)〉 6≥intP (x) 0, 〈Q(x, u), g(x)〉 6≤intP (x) 0 ∀y ∈ D.
Therefore, x is a solution of (GMVCP).
Theorem 2.1 is proved.
Remark 2.1. If Q(x, u) = Q(u) and g is an identity mapping, then Theorem 2.1 coincides with
Theorem 2.1 in Ceng and Lin [1]. Again if Q ≡ I, the identity mapping of L(X,Y ) and A is a
single-valued mapping from D = X to L(X,Y ), then Theorem 2.1 coincides with Theorem 2.1 of
Huang et al. [8].
ISSN 1027-3190. Укр. мат. журн., 2013, т. 65, № 1
10 R. P. AGARWAL, SALAHUDDIN, M. K. AHMAD
Definition 2.1. Let A : D → 2L(X,Y ), P : D → 2Y be the two multivalued mappings with
intP (x) 6= ∅ for every x ∈ D, Q : D × L(X,Y ) → L(X,Y ) and g : D → Y be the two single-
valued mappings and Ω a subset of D × A(D). We say that P is inclusive with respect to Ω if for
any (x, u), (y, v) ∈ Ω,
〈Q(x, u), g(x)〉 ≤intP (y) 〈Q(x, v), g(y)〉 implies that P (x) ⊂ P (y). (2.6)
It is easy to see that, if P (x) = P for all x ∈ D, where P is a closed, pointed and convex moving
cone in Y, then P is inclusive with respect to Ω.
Theorem 2.2. Suppose that P is inclusive with respect to Ω. If there exists at most finite num-
ber of solutions for (GMVCP), then (GMVCP) is a solvable if and only if ζw 6= ∅ and there exists
(x, u) ∈ ζwP (x) such that
〈Q(x, u), g(x)〉 6≥intP (x) 0.
Proof. Let ρ1 be a solution of (GMVCP), then there exists u1 ∈ Aρ1 such that
〈Q(ρ1, u1), g(ρ1)〉 6≥intP (ρ1) 0, 〈Q(ρ1, u1), g(y)〉 6≤intP (ρ1) 0 ∀y ∈ D. (2.7)
If (ρ1, u1) ∈ ζwP (ρ1)
, then
〈Q(ρ1, u1), g(ρ1)〉 6≥intP (ρ1) 0, (2.8)
and hence the conclusion holds. If (ρ1, u1) 6∈ ζwP (ρ1)
, by the definition of a weakly minimal solution
there exists (ρ2, u2) ∈ Ω such that
〈Q(ρ2, u2), g(y)〉 6≤intP (ρ2) 0 ∀y ∈ D, (2.9)
〈Q(ρ2, u2), g(ρ2)〉 ≤intP (ρ1) 〈Q(ρ1, u1), g(ρ1)〉 6≥intP (ρ1) 0.
This implies that
〈Q(ρ2, u2), g(ρ2)〉 6≥intP (ρ1) 0. (2.10)
Since
〈Q(ρ2, u2), g(ρ2)〉 ≤intP (ρ1) 〈Q(ρ1, u1), g(ρ1)〉,
and P is inclusive with respect to Ω, it follows that P (ρ2) ⊂ P (ρ1), and this implies that
〈Q(ρ2, u2), g(ρ2)〉 6≥intP (ρ2) 0. (2.11)
Thus ρ2 is a solution of (GMVCP) and ρ2 6= ρ1. Continuing this process there exists (ρn, un) ∈ Ω
such that ρn is a solution of (GMVCP) and (ρn, un) ∈ ζwP (ρn)
, since (GMVCP) has almost a finite
number of solutions. Thus
〈Q(ρn, un), g(ρn)〉 ∈ MinwP (ρn)
{
〈Q(x, u), g(ρn)〉 : (ρn, u) ∈ Ω
}
and
〈Q(ρn, un), g(ρn)〉 6≥intP (ρn) 0. (2.12)
Combining this result and Theorem 2.1, we have the conclusion of the theorem.
ISSN 1027-3190. Укр. мат. журн., 2013, т. 65, № 1
HYBRID TYPE GENERALIZED MULTIVALUED VECTOR COMPLEMENTARITY PROBLEMS 11
Remark 2.2. If Q(x, u) = Q(u) and g is an identity mapping then Theorem 2.2 reduces to the
Theorem 2.4 of Ceng and Lin [1]. If Q ≡ I is an identity mapping, A is single-valued mapping from
X to L(X,Y ) and P (x) = P for all x ∈ X, where P is a closed pointed and convex moving cone in
Y, then P (x) satisfies the inclusive assumption with respect to Ω and Theorem 2.2 is equivalent to the
Theorem 3.2 of Chen and Yang [3]. We note that if Q ≡ I, the identity mapping of L(X,Y ) and A
is a single-valued mapping from D = X to L(X,Y ), then Theorem 2.2 coincides with Theorem 2.2
of Huang et al. [8].
Next we consider the positive generalized multivalued vector complementarity problem (PG-
MVCP): finding x ∈ D,u ∈ A(x) such that
〈Q(x, u), g(x)〉 6≥intP (x) 0, 〈Q(x, u), g(y)〉 ≥P (x) 0, y ∈ D. (2.13)
Let
ψ = {(x, u) ∈ D ×A(D) : u ∈ A(x), 〈Q(x, u), g(y)〉 ≥P (x) 0 ∀y ∈ D}. (2.14)
Consider the following generalized multivalued vector optimization problem (GMVOP) to be
MinP 〈Q(x, u), g(x)〉 subject to (x, u) ∈ ψ. (2.15)
We denote the set of all minimal point of (GMVOP) with respect to the moving cone P (x) by ΓP (x),
that is
ΓP (x) = MinP (x){〈Q(x, u), g(x)〉 : (x, u) ∈ ψ}
and denote the set of all minimal point of (GMVOP) by
Γ =
⋃
x∈D
ΓP (x). (2.16)
Theorem 2.3. If Γ 6= ∅ and there exists (x, u) ∈ ΓP (x) such that
〈Q(x, u), g(x)〉 6≥intP (x) 0,
then (PGMVCP) is solvable.
Theorem 2.4. Suppose that P is inclusive with respect to ψ. If there exists at most a finite
number of solutions of (PGMVCP), then (PGMVCP) is solvable if and only if Γ 6= ∅ and there
exists (x, u) ∈ ΓP (x) such that
〈Q(x, u), g(x)〉 6≥intP (x) 0. (2.17)
We note that if Q(x, u) = Q(u) and g is an identity mapping then Theorem 2.3 and 2.4 coincide
with Theorem 2.6 and 2.7 of Ceng and Lin [1]. Also again if Q ≡ I is an identity mapping of
L(X,Y ) and A is a single-valued mapping from D = X to L(X,Y ), then Theorem 2.3 and 2.4 is
similar to Theorem 2.3 and 2.4 of Huang et al. [8].
3. Equivalence between the generalized multivalued vector complementarity problems and
generalized multivalued weak minimal element problems. LetX, Y be the two Banach spaces and
P : D → 2Y a multivalued mapping such that for each x ∈ D,P (x) is a proper, closed, convex and
pointed moving cone with apex at the origin and intP (x) 6= ∅. Let Q : D × L(X,Y ) → L(X,Y )
ISSN 1027-3190. Укр. мат. журн., 2013, т. 65, № 1
12 R. P. AGARWAL, SALAHUDDIN, M. K. AHMAD
and g : D → D the single-valued mappings, A : X → 2L(X,Y ) the multivalued mapping, where
2L(X,Y ) is a collection of all nonempty subsets of L(X,Y ), and f : X → Y a given operator.
Define the feasible set associated with A and K,
Ω = {x ∈ D : there exists u ∈ A(x) such that 〈Q(x, u), g(y)〉 6≤intP (x) 0 ∀y ∈ D
}
. (3.1)
We consider the following problems:
1. Generalized multivalued vector optimization problem (GMVOP)l for a given l ∈ L(X,Y ):
finding x ∈ Ω such that
l(x) ∈ MinwP (x) l(Ω). (3.2)
2. Generalized multivalued weak minimal element problem (GMWMEP): finding x ∈ Ω such
that
x ∈ MinwD Ω.
3. Generalized multivalued vector complementarity problem (GMVCP): finding x ∈ Ω, u ∈
∈ A(x) such that
〈Q(x, u), g(x)〉 6≥intP (x) 0.
4. Generalized multivalued vector variational inequality problem (GMVVIP): finding x ∈ D
and u ∈ A(x) such that
〈Q(x, u), g(y)− g(x)〉 6≤intP (x) 0 ∀y ∈ D. (3.3)
5. Generalized multivalued vector unilateral optimization problem (GMVUOP): finding x ∈ D
such that
f(x) ∈ MinwP (x) f(D).
Definition 3.1 [3]. A linear operator l : X → Y is called weakly positive if for any x, y ∈ X,
x 6≥intP y implies that
l(x) 6≥intP (x) l(y).
Definition 3.2. Let X and Y be the two Banach spaces, and l be a linear operator from X
to Y. If the image of any bounded set in X is a self-sequentially compact set in Y then l is called
completely continuous. A mapping f : X → Y is said to be convex if
f(λx+ (1− λ)y) ≤P (x) λf(x) + (1− λ)f(y) (3.4)
for all x, y ∈ X and 0 ≤ λ ≤ 1.
Definition 3.3. Let Q : L(X,Y )→ L(X,Y ) and f : X → Y be two mappings. f is said to be
Q-subdifferential at x0 ∈ X if there exists u0 ∈ L(X,Y ) such that
f(x)− f(x0) ≥P (x0) 〈Q(u0), x− x0〉 ∀x ∈ X.
If f is Q-subdifferentiable at x0 ∈ X, then we define the Q-subdifferential of f at x0 as follows:
∂Qf(x0) = {u ∈ L(X,Y ) : f(x)− f(x0) ≥P (x0) 〈Q(u), x− x0〉} ∀x ∈ X.
If f is Q-subdifferentiable at each x ∈ X, then f is Q-subdifferential on X.
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HYBRID TYPE GENERALIZED MULTIVALUED VECTOR COMPLEMENTARITY PROBLEMS 13
Remark 3.1. We note that if X and Y are two Banach spaces, a mapping f : X → Y is Frechet
differentiable at x0 ∈ X if there exists a linear bounded operator Df(x0) such that
lim
x→0
‖f(x0 + x)− f(x0)− 〈Df(x0), x〉‖
‖x‖
= 0,
where Df(x0) is said to be the Frechet derivative of f at x0. The mapping f is said to be Frechet
differentiable at each point of X if f : X → Y is convex and Frechet differentiable on X. Then
f(y)− f(x) ≥P (x) 〈Df(x), y − x〉 ∀x, y ∈ X.
If f is a Frechet differentiable on X, then for each x, y ∈ X we have
f(y)− f(x) ≥P (x) 〈A(u), y − x〉 ∀y ∈ ∂Qf(x).
Definition 3.4. Let X be a Banach space, D ⊂ X be a proper, closed, convex and moving
pointed cone with apex at origin and intD 6= ∅. The norm ‖ · ‖ in X is called strictly monotonically
increasing in D if for each y ∈ D
x ∈ ({y} − intD) ∩D ⇒ ‖x‖ ≤ ‖y‖. (3.5)
Theorem 3.1. Let X, Y be the two Banach spaces, D ⊂ X a proper, closed, convex moving
pointed cone with apex at origin and intD 6= ∅. Let P : D → 2Y be a multivalued mapping with
closed, convex moving pointed cone values such that intP (x) 6= ∅ for all x ∈ D. Suppose that
1) T = ∂Qf is the subdifferential of a convex operator f : X → Y ;
2) l is a weakly positive linear operator;
3) there exists x ∈ Ω such that Q(u) is one to one and completely continuous, where u ∈ A(x)
is associated with x in the definition of Ω;
4) X is a topological dual space of a real normed space and the norm ‖ · ‖ in X is strictly
monotonically increasing on D.
If (GMVVIP) is solvable, then (GMVOP)l, (GMWMEP), (GMVCP) and (GMVUOP) are also
solvable.
We need the following proposition to prove Theorem 3.1.
Proposition 3.1. Let Q : D × L(X,Y ) → L(X,Y ) and f : X → Y be the two single-valued
mappings, A : D → 2L(X,Y ) be the multivalued mapping and let T = ∂Qf be the Q-subdifferential
of f. Then x solves (GMVUOP) which implies that x solves (GMVVIP). If in addition, f is a convex
mapping then conversely x solves (GMVVIP) which implies that x solves (GMVUOP).
Proof. Let x be a solution of (GMVUOP). Then x ∈ D and
f(x) ∈ MinwP (x) f(D)
that is,
f(x) 6≥intP (x) f(y) ∀y ∈ D.
Since D is convex cone, therefore
f(x) 6≥intP (x) f(x+ t(w − x)), 0 < t < 1, w ∈ D. (3.6)
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14 R. P. AGARWAL, SALAHUDDIN, M. K. AHMAD
Also, since f is Q-subdifferential on X, it follows that
f(x) 6≥intP (x) f(x+t(w−x)) ≥P (x) f(x)+〈Q(x, u), t(w−x)〉, 0 < t < 1, w ∈ D. (3.7)
This implies that
〈Q(x, u), t(w − x)〉 6≤intP (x) 0, 0 < t < 1, w ∈ D, (3.8)
and hence
〈Q(x, u), w − x〉 6≤intP (x) 0 ∀w ∈ D. (3.9)
Therefore, x is the solution of (GMVVIP).
Conversely, let x solve (GMVVIP). Then there exists u0 ∈ A(x) = ∂Qf(x) such that
〈Q(x, u0), w − x〉 6≤intP (x) 0 ∀w ∈ D. (3.10)
Since f is Q-subdifferentiable on X, we have for all u ∈ A(x) = ∂Qf(x)
f(w)− f(x) ≥P (x) 〈Q(x, u), w − x〉 ∀w ∈ D. (3.11)
Since f is convex, therefore we have
f(w)− f(x) ≥P (x) 〈Q(x, u0), w − x〉 6≤intP (x) 0 ∀w ∈ D. (3.12)
This implies that
f(w) 6≤intP (x) f(x) ∀w ∈ D. (3.13)
Therefore, x solves (GMVUOP).
Proposition 3.1 is proved.
Proposition 3.2. If x solves (GMVVIP), then x also solves (GMVCP). Conversely, if
〈Q(x, u), g(x)〉 ≤P (x) 0 ∀w ∈ D, u ∈ A(x),
then x solve (GMVCP) which implies that x solve (GMVVIP).
Proof. Let x be a solution of (GMVVIP), then there exists u ∈ A(x) such that
〈Q(x, u), g(y)− g(x)〉 6≤intP (x) 0 ∀y ∈ D. (3.14)
Letting y = 0, we get
〈Q(x, u), g(x)〉 6≥intP (x) 0.
Since g is convex and letting y = w + x such that
g(y) = g(w + x) ≤ g(w) + g(x),
we have
〈Q(x, u), g(w)〉 6≤intP (x) 0 ∀w ∈ D. (3.15)
Thus x is a solution of (GMVCP).
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HYBRID TYPE GENERALIZED MULTIVALUED VECTOR COMPLEMENTARITY PROBLEMS 15
Conversely, let x solve (GMVCP). Then there exists u ∈ Ax such that
〈Q(x, u), g(x)〉 ≤P (x) 0 6≥intP (x) 〈Q(x, u), g(y)〉 ∀y ∈ D. (3.16)
This implies that
〈Q(x, u), g(x)〉 6≥intP (x) 〈Q(x, u), g(y)〉 ∀y ∈ D (3.17)
and so
〈Q(x, u), g(y)− g(x)〉 6≥intP (x) 0 ∀y ∈ D. (3.18)
Proposition 3.2 is proved.
Proposition 3.3. Let l be a weakly positive linear operator. Then x solve (GMWMEP) which
implies that x solve (GMVOP).
Proof. Let x be a solution of (GMWMEP). Then x ∈ Ω and x ∈ MinwD Ω. That is for any
z ∈ Ω,
x 6≥intD z.
Since l is a weakly positive linear operator, it follows that
l(x) 6≥P (x) l(z)
and so
l(x) ∈ MinwP (x) l(Ω), (3.19)
hence x solve (GMVOP).
Proposition 3.3 is proved.
Definition 3.5 [8]. Let X be a Banach space, D ⊂ X be a proper closed convex and moving
pointed cone with apex at the origin and intD 6= ∅, N a nonempty subset of X.
1. If x ∈ X,Nx = ({x} −D) ∩N 6= ∅, then Nx is called a section of the set N.
2. N is called weakly closed if {xn} ⊂ N, x ∈ X,
〈x∗, xn〉 → 〈x∗, x〉 ∀x∗ ∈ X∗,
then x ∈ N.
3. N is called bounded below if there exists a point b ∈ X such that
N ⊂ b+D.
Lemma 3.1 [10]. Let X be a Banach space, D ⊂ X a proper closed convex moving pointed
cone with apex at the origin and intD 6= ∅, N a nonempty subset of X and X be the topological
dual space of a real normed space (Z, ‖ · ‖z). Suppose there exists x ∈ X such that the section Nx is
weakly closed and bounded below and the norm ‖ · ‖ in X is strictly monotonically increasing. Then
the set N has at least one weakly minimal point.
Lemma 3.2. If (GMVVIP) is solvable, then the feasible set Ω is nonempty.
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16 R. P. AGARWAL, SALAHUDDIN, M. K. AHMAD
Proof. Let x be a solution of (GMVVIP), then there exists u ∈ A(x) such that
〈Q(x, u), g(y)− g(x)〉 6≤intP (x) 0 ∀y ∈ D. (3.20)
Since g is convex, taking y = z + x with any z ∈ Ω we know that y ∈ D and
〈Q(x, u), g(z)〉 6≤intP (x) 0 ∀z ∈ D. (3.21)
Thus x ∈ Ω, this completes the proof.
Lemma 3.3 [3]. If the norm ‖ · ‖ in an ordered Banach space X is strictly monotonically in-
creasing, then the order intervals in X are bounded.
Proposition 3.4. Suppose that (GMVVIP) is solvable and
1) there exists x in Ω such that Q(x, u) is one to one and completely continuous, where u ∈ A(x)
is associated with x in the definition of Ω;
2) X is the topological dual space of a real normed space (Z, ‖ · ‖Z) and the norm ‖ · ‖ in X is
strictly monotonically increasing, then (GMWMEP) has at least one solution.
Proof. By assumption and Lemma 3.2, Ω 6= ∅. Let x ∈ Ω be a point such that Q(x, u) is
one-to-one and completely continuous where u ∈ A(x) is associated with x in the definition of Ω
and let {yn} ⊂ Ω with yn → y (weakly). Since
Ωx = ({x} −D) ∩ Ω ⊂ ({x} −D) ∩D = [0, x], (3.22)
by Lemma 3.3. [0, x] is bounded and so is Ωx. Since Q(x, u) is completely continuous, 〈Q(x, u),Ωx〉
is a self-sequentially compact set and{
〈Q(x, u), g(yn)〉
}
⊂ 〈Q(x, u),Ωx〉
implies that there exists a subsequence
{
〈Q(x, u), g(ynk
)〉
}
which converges to z ∈ 〈Q(x, u),Ωx〉.
We get a point y0 ∈ Ωx such that
〈Q(x, u), g(ynk
)〉 → 〈Q(x, u), g(y0)〉 (strongly). (3.23)
On the other hand, since yn → y, g(yn) → g(y) (weakly) and Q(x, u) is completely continuous.
Then
〈Q(x, u), g(yn)〉 → 〈Q(x, u), g(y)〉 (strongly). (3.24)
By the uniqueness of limits, we get
〈Q(x, u), g(y)〉 = 〈Q(x, u), g(y0)〉.
Since Q(x, u) is one-to-one, y = y0 and so y ∈ Ωx. Since Ωx is weakly closed, it follows from
Lemma 3.1 that Ω has a weakly minimal point b such that
b 6≥intP (b) x ∀x ∈ Ω.
Hence (GMWMEP) has at least one solution.
Proposition 3.4 is proved.
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HYBRID TYPE GENERALIZED MULTIVALUED VECTOR COMPLEMENTARITY PROBLEMS 17
Definition 3.6. Let X, Y be the two Banach spaces, D ⊂ X a proper closed convex moving
pointed cone with apex at the origin and intD 6= ∅, and P : D → 2Y a multivalued mapping with
closed convex moving pointed cone such that intP (x) 6= ∅ for all x ∈ D. Let Q : D × L(X,Y )→
→ L(X,Y ), g : D → D be the single-valued mappings and A : X → 2L(X,Y ) the multivalued
mapping. Q is called Q-positive if
〈Q(x, u), g(y)〉 ≥P (x) 0 ∀x, y ∈ D, u ∈ A(x). (3.25)
Now consider the positive generalized multivalued vector complementarity problem (PGMVCP):
Finding x ∈ D and u ∈ A(x) such that
〈Q(x, u), g(y)〉 6≥intP (x) 0, 〈Q(x, u), g(y)〉 ≥P (x) 0, y ∈ D. (3.26)
The feasible set related to (PGMVCP) is defined as Ω = {x ∈ D: there is u ∈ A(x) such that
〈Q(x, u), g(y)〉 ≥P (x) 0 ∀y ∈ D. (3.27)
Let us consider the following problems:
The generalized multivalued vector optimization problem (GMVOP)l0 : finding x ∈ Ω0 such that
l(x) ∈ MinwP l(Ω0).
The generalized multivalued weak minimal element problem (GMWMEP)0: finding x ∈ Ω0
such that
x ∈ MinwD Ω0.
The positive generalized multivalued vector complementarity problem (PGMVCP): finding x ∈
∈ Ω0 such that
〈Q(x, u), g(x)〉 6≥intP (x) 0, (3.28)
where u ∈ A(x) and Q : D × L(X,Y ) → L(X,Y ), g : D → D associated with x in the definition
of Ω0.
The generalized multivalued vector variational inequality problem (GMVVIP): finding x ∈ D
and u ∈ A(x) such that
〈Q(x, u), g(y)− g(x)〉 6≤intP (x) 0 ∀y ∈ D. (3.29)
The generalized multivalued vector unilateral optimization problem (GMVUOP): for a given
mapping f : X → Y finding x ∈ D such that
f(x) ∈ MinwD f(D).
Definition 3.7. Let Q : D × L(X,Y )→ L(X,Y ), g : D → D be the single-valued mappings,
A : D → 2L(X,Y ) be the multivalued mapping. Q is called g-strongly monotone with first variable of
Q, if
〈Q(·, u)−Q(·, v), g(x)− g(y)〉 ≥P (x) 0 ∀x, y ∈ D, x 6= y, u ∈ A(x), v ∈ A(y). (3.30)
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18 R. P. AGARWAL, SALAHUDDIN, M. K. AHMAD
Definition 3.8 [8]. We say that P (x) satisfies an inclusive condition if for any x, y ∈ X, x ≤intP
y only implies that
P (x) ⊂ P (y). (3.31)
Proposition 3.5. Let Q : D × L(X,Y ) → L(X,Y ), g : D → D be the two single-valued
mappings, A : X → 2L(X,Y ) be the multivalued mappings, Q is g-strongly monotone with respect to
the first variable and x is a solution of (PGMVCP). If P satisfies the inclusive condition then x is a
weakly minimal point of Ω0, i.e., x solve (GMWMEP)0.
Proof. It is easy to see that x ∈ Ω0 ⊂ D. If x ∈ bd(D) (where bd(D) denotes the boundary of
D), then x solve (GMWMEP)0, otherwise there exists x0 ∈ Ω0 such that
g(x) ≥intD g(x0),
g(x) = g(x− x0 + x0) ≤ g(x− x0) + g(x) ∈ intD +D ⊂ intD,
which is a contradiction. If x ∈ intD, by the Q-strict monotonicity, we have
〈Q(x, u), g(x)− g(y)〉 ≥intP (x) 〈Q(x′, v′), g(x)− g(y)〉 ∀y ∈ Ω0, y 6= x, v′ ∈ A(y).
Suppose g(x) ≥intD g(y). Since Q is g-positive
〈Q(x, u), g(x)− g(y)〉 ≥P (y) 0
and
〈Q(x, u), g(x)− g(y)〉 ≥intP (x) 〈Q(x, v), g(x)− g(y)〉 ≥P (x) 0.
By the assumption, we get
P (y) ⊂ P (x)
and so
〈Q(x, u), g(x)− g(y)〉 ∈ 〈Q(x, v), g(x)− g(y)〉+ intP (x) ⊂ P (y)+
+ intP (x) ⊂ P (x) + intP (x) = intP (x).
It follows that
〈Q(x, u), g(x)− g(y)〉 ≥intP (x) 0,
and thus
0 6≤intP (x) 〈Q(x, u), g(x)〉 ≥P (x) 〈Q(x, u), g(y)〉+ k
for some k ∈ intP (x). This implies
〈Q(x, u), g(y)〉+ k 6≥intP (x) 0.
Since k ∈ intP (x) and x ∈ Ω0
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HYBRID TYPE GENERALIZED MULTIVALUED VECTOR COMPLEMENTARITY PROBLEMS 19
〈Q(x, u), g(y)〉+ k ∈ P (x) + intP (x) ⊂ intP (x)
and so
〈Q(x, u), g(y)〉+ k ≥intP (x) 0,
which leads to a contradiction. Therefore
g(x) ≥intD g(y)
does not hold, that is
g(x) 6≥intD g(y) ∀y ∈ Ω0.
It follows that x solve (GMWMEP)0.
Proposition 3.5 is proved.
Proposition 3.6. If x solve (PGMVCP), then x also solve (GMVVIP).
Proof. Suppose x solve (PGMVCP). Then x ∈ D and there exists u ∈ A(x) such that
〈Q(x, u), g(x)〉 6≥intP (x) 0, 〈Q(x, u), g(y)〉 ≥P (x) 0 ∀y ∈ D.
If 〈Q(x, u), g(y)− g(x)〉 ≤intP (x) 0, then
〈Q(x, u), g(x)〉 = −〈Q(x, u), g(y)− g(x)〉+ 〈Q(x, u), g(y)〉 ∈ intP (x) + P (x) ⊂ intP (x)
and so
〈Q(x, u), g(x)〉 ≥intP (x) 0,
which is a contradiction. It follows that
〈Q(x, u), g(y)− g(x)〉 6≤intP (x) 0,
and x solve (GMVVIP).
Proposition 3.6 is proved.
Similarly we can obtain other equivalence condition. We have the following theorem.
Theorem 3.2. Let X, Y be two Banach spaces, D ⊂ X a proper closed convex moving pointed
cone with apex at the origin and intD 6= ∅ and {P (x) : x ∈ X} a family of closed moving pointed
cone in Y such that intP (x) 6= ∅ for all x ∈ X. Let g : D → D, Q : D × L(X,Y ) → Y and
A : X → 2L(X,Y ) be the mappings. Suppose that P satisfies the inclusive condition and
1) T = ∂Qf is the Q-subdifferential for the convex operator f : X → Y ;
2) l is a weakly positive operator;
3) A is Q-strictly monotone.
If (GMWMEP)0, (PGMVCP) is solvable, then (GMVOP)l0 , (GMVVIP) and (GMVUOP) have at
least a common solution.
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