In this section, we will obtain and prove some existence theorems for the solutions to the generalized biquasivariational inequalities of quasipseudomonotone type II and strongly quasipseudomonotone 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 semicontinuous 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 semicontinuous 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 semicontinuous such that each is compact and convex,
(b) is convex and is bounded,
(c) is an quasipseudomonotone type II resp., strongly quasipseudomonotone type II operator and is upper semicontinuous such that each is compact resp., weakly compact and convex and is strongly bounded,
(d) is an upper semicontinuous 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 semicontinuous (resp., weakly lower semicontinuous) on by Lemma 2.1 and so the mapping
is lower semicontinuous (resp., weakly lower semicontinuous) on by Lemma 1.8. Also, for any fixed ,
is continuous on . Hence, for any and fixed , the mapping is lower semicontinuous (resp., weakly lower semicontinuous) 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 quasipseudomonotone type II (resp., strongly quasipseudomonotone 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 quasipseudomonotone 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 semicontinuous (resp., weakly lower semicontinuous), 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 quasipseudomonotone type II resp., strongly quasipseudomonotone type II operator and is an upper semicontinuous mapping such that each is strongly, that is, compact and convex resp., weakly, i.e., compact and convex,
(d) is an upper semicontinuous mapping such that each is compact convex and, for any , is upper semicontinuous 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 biquasivariational inequalities was written by Shih and Tan in 1989 in [1] and the results were obtained on compact sets where the setvalued mappings were either lower semicontinuous or upper semicontinuous. Our present paper is another extension of the original work in [1] using quasipseudomonotone type II operators on noncompact sets.

(3)
The results in [4] were obtained on compact sets where one of the setvalued mappings is a quasipseudomonotone type I operators which were defined first in [4] and extends the results in [1]. The quasipseudomonotone type I operators are generalizations of pseudomonotone type I operators introduced first in [17]. In all our results on generalized biquasivariational inequalities, if the operators and the operators are replaced by , then we obtain results on generalized quasivariational inequalities which generalize the corresponding results in the literature (see [18]).
() The results on generalized biquasivariational inequalities given in [5] were obtained for setvalued quasisemimonotone and biquasisemimonotone operators and the corresponding results in [2] were obtained for setvalued upperhemicontinuous operators introduced in [6]. Our results in this paper are also further extensions of the corresponding results in [2, 5] using setvalued quasipseudomonotone type II operators on noncompact sets.