In this section, we will obtain and prove some existence theorems for the solutions to the generalized bi-quasi-variational inequalities of quasi-pseudo-monotone type II and strongly quasi-pseudo-monotone type II operators with noncompact domain in locally convex Hausdorff topological vector spaces. Our results extend and/or generalize the corresponding results in [1].
Before we establish our main results, we state the following result which is Lemma 3.1 in [3].
Lemma 2.1.
Let
be a Hausdorff topological vector space over
, let
be a vector space over
, and let
be a nonempty compact subset of
. Let
be a bilinear functional such that
separates points in
. Suppose that the
equips with the
-topology; for any
,
is continuous on
and
,
are upper semi-continuous maps such that
and
are compact for any
. Let
and
be continuous. Define a mapping
by
Suppose that
is continuous over the
compact
subset
of
. Then
is lower semi-continuous on
.
Now, we establish our first main result as follows.
Theorem 2.2.
Let
be a locally convex Hausdorff topological vector space over
, let
be a nonempty paracompact convex and bounded subset of
, and let
be a Hausdorff topological vector space over
. Let
be a bilinear functional which is continuous over compact subsets of
. Suppose that
(a)
is upper semi-continuous such that each
is compact and convex,
(b)
is convex and
is bounded,
(c)
is an
-quasi-pseudo-monotone type II
resp., strongly
-quasi-pseudo-monotone type II
operator and is upper semi-continuous such that each
is compact
resp., weakly compact
and convex and
is strongly bounded,
(d)
is an upper semi-continuous mapping such that each
is weakly compact and convex,
(e)the set
is open in
.
Suppose further that there exist a nonempty closed and compact
resp., weakly closed and weakly compact
subset
of
and a point
such that
and
Then there exists a point
such that
(1)
,
(2)there exist a point
and a point
such that
Moreover, if
for all
, then
is not required to be locally convex, and if
, then the continuity assumption on
can be weakened to the assumption that, for any
, the mapping
is continuous
resp., weakly continuous
on
.
Proof.
We need to show that there exists a point
such that
and
Suppose the contrary. Then, for any
, either
or there exists
such that
that is, for any
, either
or
. If
, then, by a separation theorem for convex sets in locally convex Hausdorff topological vector spaces, there exists
such that
Let
and, for any
, set
Then
Since each
is open in
by Lemma 1.7 and
is open in
by hypothesis,
is an open covering for
. Since
is paracompact, there exists a continuous partition of unity
for
subordinated to the covering
(see Dugundji [22, Theorem VIII, 4.2]); that is, for any
,
and
are continuous functions such that, for any
,
for all
and
for all
and
,
is locally finite and
for any
. Note that, for any
,
is continuous on
(see [23, Corollary 10.1.1]). Define a mapping
by
Then we have the following.
-
(i)
Since
is Hausdorff, for any
and fixed
, the mapping
is lower semi-continuous (resp., weakly lower semi-continuous) on
by Lemma 2.1 and so the mapping
is lower semi-continuous (resp., weakly lower semi-continuous) on
by Lemma 1.8. Also, for any fixed
,
is continuous on
. Hence, for any
and fixed
, the mapping
is lower semi-continuous (resp., weakly lower semi-continuous) on
.
-
(ii)
For any
and
,
. Indeed, if this is false, then, for some
and
(say
where
with
, we have
. Then, for any
,
which implies that
which is a contradiction.
-
(iii)
Suppose that
,
, and
is a net in
converging to
(resp., weakly to
) with
for all
and
.
Case 1 (
).
Note that
for any
and
. Since
is strongly bounded and
is a bounded net, it follows that
Also, we have
Thus, from (2.15), it follows that
When
, we have
for all
, that is,
Therefore, by (2.18), we have
which implies that
Hence, by (2.17) and (2.20), we have
.
Case 2 (
).
Since
, there exists
such that
for any
. When
, we have
for all
, that is,
Thus it follows that
Hence, by (2.22), we have
Since
, we have
Since
for all
, it follows that
Since
, by (2.24) and (2.25), we have
Since
is an
-quasi-pseudo-monotone type II (resp., strongly
-quasi-pseudo-monotone type (II) operator, we have
Since
, we have
and so
When
, we have
for all
, that is,
and so, by (2.29),
Hence we have
.
-
(iv)
By the hypothesis, there exists a nonempty compact and so a closed (resp., weakly closed and weakly compact) subset
of
and a point
such that
and
Thus it follows that
whenever
and
whenever
for all
. Consequently, we have
(If
is a strongly
-quasi-pseudo-monotone type II operator, then we equip
with the weak topology.) Thus
satisfies all the hypotheses of Theorem 1.6 and so, by Theorem 1.6, there exists a point
such that
for all
, that is,
Now, the rest of the proof is similar to the proof in Step 1 of Theorem 1 in [24]. Hence we have shown that
Then, by applying Theorem 1.9 as we proved in Step 3 of Theorem 1 in [24], we can show that there exist a point
and a point
such that
We observe from the above proof that the requirement that
is locally convex is needed if and only if the separation theorem is applied to the case
. Thus, if
is the constant mapping
for all
, the
is not required to be locally convex.
Finally, if
, in order to show that, for any
,
is lower semi-continuous (resp., weakly lower semi-continuous), Lemma 2.1 is no longer needed and the weaker continuity assumption on
that, for any
, the mapping
is continuous (resp., weakly continuous) on
is sufficient. This completes the proof.
We will now establish our last result of this section.
Theorem 2.3.
Let
be a locally convex Hausdorff topological vector space over
, let
be a nonempty paracompact convex and bounded subset of
, and let
be a vector space over
. Let
be a bilinear functional such that
separates points in
,
is continuous over compact subsets of
, and, for any
, the mapping
is continuous on
. Suppose that
equips with the strong topology
and
(a)
is a continuous mapping such that each
is compact and convex,
(b)
is convex and
is bounded,
(c)
is an
-quasi-pseudo-monotone type II
resp., strongly
-quasi-pseudo-monotone type II
operator and is an upper semi-continuous mapping such that each
is strongly, that is,
-compact and convex
resp., weakly, i.e.,
-compact and convex
,
(d)
is an upper semi-continuous mapping such that each
is
-compact convex and, for any
,
is upper semi-continuous at some point
in
with
, where
Suppose further that there exist a nonempty closed and compact
resp., weakly closed and weakly compact
subset
of
and a point
such that
and
Then there exists a point
such that
(1)
,
(2)there exist a point
and a point
with
Moreover, if
for all
, then
is not required to be locally convex.
Proof.
The proof is similar to the proof of Theorem 2 in [24] and so the proof is omitted here.
Remark 2.4.
(
) Theorems 2.2 and 2.3 of this paper are generalizations of Theorems 3.2 and 3.3 in [3], respectively, on noncompact sets. In Theorems 2.2 and 2.3,
is considered to be a paracompact convex and bounded subset of locally convex Hausdorff topological vector space
whereas, in [3],
is just a compact and convex subset of
. Hence our results generalize the corresponding results in [3].
(
) The first paper on generalized bi-quasi-variational inequalities was written by Shih and Tan in 1989 in [1] and the results were obtained on compact sets where the set-valued mappings were either lower semi-continuous or upper semi-continuous. Our present paper is another extension of the original work in [1] using quasi-pseudo-monotone type II operators on noncompact sets.
-
(3)
The results in [4] were obtained on compact sets where one of the set-valued mappings is a quasi-pseudo-monotone type I operators which were defined first in [4] and extends the results in [1]. The quasi-pseudo-monotone type I operators are generalizations of pseudo-monotone type I operators introduced first in [17]. In all our results on generalized bi-quasi-variational inequalities, if the operators
and the operators
are replaced by
, then we obtain results on generalized quasi-variational inequalities which generalize the corresponding results in the literature (see [18]).
(
) The results on generalized bi-quasi-variational inequalities given in [5] were obtained for set-valued quasi-semi-monotone and bi-quasi-semi-monotone operators and the corresponding results in [2] were obtained for set-valued upper-hemi-continuous operators introduced in [6]. Our results in this paper are also further extensions of the corresponding results in [2, 5] using set-valued quasi-pseudo-monotone type II operators on noncompact sets.