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Existence theorems of generalized quasi-variational-like inequalities for pseudo-monotone type II operators
Journal of Inequalities and Applications volume 2014, Article number: 449 (2014)
Abstract
In this paper, we prove the existence results of solutions for a new class of generalized quasi-variational-like inequalities (GQVLI) for pseudo-monotone type II operators defined on compact sets in locally convex Hausdorff topological vector spaces. In obtaining our results on GQVLI for pseudo-monotone type II operators, we use Chowdhury and Tan’s generalized version (Chowdhury and Cho in J. Inequal. Appl. 2012:79, 2012) of Ky Fan’s minimax inequality (Fan in Inequalities, vol. III, pp.103-113, 1972) as the main tool.
1 Introduction
If X is a nonempty set, then we denote by the family of all nonempty subsets of X and by the family of all nonempty finite subsets of X. Let E be a topological vector space over Φ, F be a vector space over Φ and X be a nonempty subset of E. Let be a bilinear functional. Throughout this paper, Φ denotes either the real field ℝ or the complex field ℂ.
For each , each nonempty subset A of E and each , let and . Let be the (weak) topology on F generated by the family as a subbase for the neighborhood system at 0 and be the (strong) topology on F generated by the family as a base for the neighborhood system at 0. We note then that F, when equipped with the (weak) topology or the (strong) topology , becomes a locally convex topological vector space which is not necessarily Hausdorff. But, if the bilinear functional separates points in F, i.e., for each with , there exists such that , then F also becomes Hausdorff. Furthermore, for any net in F and ,
-
(a)
in if and only if for each ;
-
(b)
in if and only if uniformly for each , where A is a nonempty bounded subset of E.
Suppose that, for the sets X, E and F mentioned above, and are two set-valued mappings. We now introduce below a slightly modified definition of the generalized quasi-variational inequality in infinite dimensional spaces given by Shih and Tan in [1]:
Find and such that
for all .
Now, we state the following definition which is a slightly corrected version of the corresponding definition given in [2]. Please note that there were typos in Definition 1.1 in [2].
Definition 1.1 Let the sets X, E and F and the mappings S and T be as defined above. Let be a single-valued mapping and be a real-valued function. Then the generalized quasi-variational-like inequality problem is defined as follows: Find and such that
for all .
For more results related to the generalized quasi-variational-like inequality problems, we refer to [3–6] and the references therein.
The following definition given in [7] is a slight modification of demi-operators defined in [8] and of pseudo-monotone type II operators defined in [9] (see also [10]).
Definition 1.2 Let X be a nonempty subset of a topological vector space E over Φ, F be a vector space over Φ which is equipped with -topology, where is a bilinear functional. Let , and be three mappings. Then T is said to be:
-
(1)
an -pseudo-monotone type II (respectively, a strongly -pseudo-monotone type II) operator if, for each and every net in X converging to y (respectively, weakly to y) with
we have
for all ;
-
(2)
an h-pseudo-monotone type II operator (respectively, a strongly h-pseudo-monotone type II operator) if T is an -pseudo-monotone type II operator with and, for some , for all .
Note that, if , the topological dual space of E, then the notions of h-pseudo-monotone type II operators coincide with those in [8].
Pseudo-monotone type II operators were first introduced by Chowdhury in [8] with a slight variation in the name of this operator. Later, these operators were renamed as pseudo-monotone type II operators by Chowdhury in [9].
Next, we shall state and prove the following lemma which provides a numerous collection of -pseudo-monotone type II and strongly -pseudo-monotone type II operators.
Lemma 1.1 Let E be a topological vector space and X be a nonempty bounded subset of E. Let be an operator such that each is strongly compact. Suppose that is a real-valued function such that, for each , is continuous and is bounded. Let be a continuous mapping. Suppose further that the operator T is a continuous mapping from the relative weak topology on X to the weak∗ topology on . Then T is both an -pseudo-monotone type II and a strongly -pseudo-monotone type II operator.
Proof Suppose that is a net in X and with (respectively, weakly) and that
Let be arbitrarily fixed. Then, using the continuity of , η and T, we obtain the following:
for all . Consequently, T is both an -pseudo-monotone type II and a strongly -pseudo-monotone type II operator. □
The above lemma will, therefore, provide ample examples for our main results in Theorems 3.1 and 3.2 given in Section 3.
In this paper, we obtain some general theorems on solutions for a new class of generalized quasi-variational-like inequalities for pseudo-monotone type II operators defined on compact sets in topological vector spaces. In obtaining our results, we shall mainly use the following generalized version of Ky Fan’s minimax inequality [11] due to Chowdhury and Tan which was stated and proved as Theorem 2.1 in [12] and is a slight modification of Theorem 1 in [13].
Theorem 1.2 Let E be a Hausdorff topological vector space and X be a nonempty convex subset of E. Let and be the mappings such that
-
(a)
for each and fixed , is lower semi-continuous on ;
-
(b)
for each and , ;
-
(c)
for each fixed , is lower semi-continuous and concave on X, and ;
-
(d)
for each and each pair of points such that every net in X converging to y with for all and all , we have ;
-
(e)
there exist a nonempty closed and compact subset K of X and such that for all .
Then there exists such that for all .
Proof For the proof, we refer to [12]. □
Definition 1.3 A function is said to be 0-diagonally concave (in short, 0-DCV) in the second argument [14] if, for any finite set and with , we have , where .
Now, we state the following definition given in [15].
Definition 1.4 Let X, E, F be the sets defined before and , , be mappings.
-
(1)
The mappings T and η are said to have the 0-diagonally concave relation (in short, 0-DCVR) if the function defined by
is 0-DCV in y.
-
(2)
The mappings T and g are said to have the 0-diagonally concave relation if T and have the 0-DCVR.
The following definition of upper hemi-continuity was given in [16]. For a more general definition, we refer to Definition 1 in [17].
Definition 1.5 Let E be a topological vector space, X be a nonempty subset of E and . Then T is said to be upper hemi-continuous on X if and only if, for each , the function defined by
for each is upper semi-continuous on X (if and only if, for each , the function defined by
for each is lower semi-continuous on X).
2 Preliminaries
Now, we present some preliminary results in this section. First, we state the following result which is Lemma 1 of Shih and Tan in [1].
Lemma 2.1 Let X be a nonempty subset of a Hausdorff topological vector space E and be an upper semi-continuous map such that is a bounded subset of E for each . Then, for each continuous linear functional p on E, the mapping defined by
is upper semi-continuous, i.e., for each , the set is open in X.
The following result is Lemma 3 of Takahashi in [18] (see also Lemma 3 in [19]).
Lemma 2.2 Let X and Y be topological spaces, be non-negative and continuous and be lower semi-continuous. Then the mapping defined by for all is lower semi-continuous.
The following result, which was stated and proved as Lemma 2.2 in [12], follows from slight modification of Lemma 3 of Chowdhury and Tan given in [13].
Lemma 2.3 Let E be a Hausdorff topological vector space over Φ, and , where denotes the convex hull of A. Let F be a vector space over Φ and be a bilinear functional such that separates points in F. We equip F with the -topology. Suppose that, for each , is continuous. Let be continuous. Let be upper semi-continuous from X into such that each is -compact. Let be defined by for all . Suppose that is continuous on the (compact) subset of . Then, for each fixed , is lower semi-continuous on X.
For the completeness, we include the proof here given in [12].
Proof Let be given and let be arbitrarily fixed. Let . Suppose that is a net in and such that . Then, for each ,
Since F is equipped with the -topology, for each , the function is continuous. Also, because is continuous. By the -compactness of , there exists such that
Since T is upper semi-continuous from to the -topology on F, X is compact, and each is -compact, is also -compact by Proposition 3.1.11 of Aubin and Ekeland [20]. Thus there is a subnet of and such that in the -topology. Again, as T is upper semi-continuous with the -closed values, .
Suppose that and let with such that . For each , let with such that . Since E is Hausdorff and , we must have for each . Thus
where (2.1) is true since is continuous on X and is continuous on the compact subset of . Hence . Thus is closed in for each . Therefore is lower semi-continuous on X. This completes the proof. □
By the slight modification of Lemma 4.2 in [16], we obtained the following result given in [7] as Lemma 2.3.
Lemma 2.4 Let E be a topological vector over ϕ, X be a nonempty convex subset of E and F be a vector space over ϕ with the -topology such that, for each , the function is continuous. Let be upper hemi-continuous along line segments in X. Let be such that for each fixed , is continuous, and let be a mapping such that, for each fixed , is lower semi-continuous on for each and, for each fixed , is concave and and T, η have the 0-DCVR. Suppose that such that for all . Then
for all .
We need the following Kneser’s minimax theorem in [21] (see also Aubin [14]).
Theorem 2.5 Let X be a nonempty convex subset of a vector space and Y be a nonempty compact convex subset of a Hausdorff topological vector space. Suppose that f is a real-valued function on such that for each fixed , the map , i.e., is lower semi-continuous and convex on Y and, for each fixed , the mapping , i.e., is concave on X. Then
3 Generalized quasi-variational-like inequalities
In this section, we prove some existence theorems for the solutions to the generalized quasi-variational-like inequalities for pseudo-monotone type II operators T with compact domain in locally convex Hausdorff topological vector spaces. Our results extend and/or generalize the corresponding results in [1].
First, we establish the following result.
Theorem 3.1 Let E be a locally convex Hausdorff topological vector space over Φ, X be a nonempty compact convex subset of E and F be a vector space over Φ with -topology, where is a bilinear functional separating points on F such that, for each , the function is continuous. Let , , and be the mappings such that
-
(a)
S is upper semi-continuous such that each is closed and convex;
-
(b)
is bounded;
-
(c)
T is an -pseudo-monotone type II (respectively, a strongly -pseudo-monotone type II) operator and is upper hemi-continuous along line segments in X to the -topology on F such that each is -compact and convex and is -bounded;
-
(d)
T and η have the 0-DCVR and η is continuous;
-
(e)
for each fixed , , i.e., is lower semi-continuous on for each and, for each fixed , and are concave, is affine, and ;
-
(f)
the set is open in X;
-
(g)
for each and each , there exist and such that
for any family of non-negative real-valued functions from X into , where denotes the set of all continuous linear functionals on E;
-
(h)
for each , the bilinear functional is continuous over the compact subset of .
Then there exists a point such that
-
(1)
;
-
(2)
there exists a point with for all .
Proof Step 1. Let us first show that there exists a point such that and
Now, we prove this by contradiction. So, we assume that, for each , either or there exists such that
that is, for each , either or . If , then, by a slight modification of a separation theorem for convex sets in locally convex Hausdorff topological vector spaces, there exists a continuous linear functional p on E such that
For each , set
and, for each continuous linear functional p on E,
Then we have
Since is open by hypothesis and each is open in X by Lemma 2.1 (Lemma 1 in [19]), is an open covering for X. Since X is compact, there exist such that . For the simplicity of notation, let for . Let be a continuous partition of unity on X subordinated to the covering . Then are continuous non-negative real-valued functions on X such that vanishes on for each and for all . Note that, for each and , , i.e., is continuous on (see [22], Corollary 10.1.1). Define a function by
for all . Then we have the following:
(I) Since E is Hausdorff, for each and fixed , the mapping
is continuous on by Lemma 2.3 and the fact that h is continuous on , and so the mapping
is lower semi-continuous on by Lemma 2.2. Also, for each fixed ,
is continuous on X. Hence, for each and fixed , the mapping is lower semi-continuous on .
(II) Since is a family of continuous non-negative real-valued functions on X into , by the hypothesis, for each and each , there exist and such that
Thus we have
i.e.,
Therefore, we have
and so for each and .
(III) Suppose that , and is a net in X converging to y (respectively, weakly to y) with for all and all .
Case 1: . Since is continuous and , we have . Note that for each . Since is strongly bounded and is a bounded net, it follows that
Also, we have
Thus it follows from (3.1) that
When , we have for all , i.e.,
for all . Therefore, by (3.3), we have
and so
Hence, by (3.2) and (3.4), we have .
Case 2: . Since is continuous, . Again since , there exists such that for all .
When , we have for all , i.e.,
for all , and so
Hence, by (3.5), we have
Since , we have
Since for all , it follows that
Since , by (3.6) and (3.7), we have
Since T is an -pseudo-monotone type II (respectively, a strongly -pseudo-monotone type II) operator, we have
for all . Since , we have
and so
When , we have for all , i.e.,
for all and so, by (3.8),
Hence we have .
(IV) Since X is a compact (respectively, weakly compact) subset of the Hausdorff topological vector space E, it is also closed. Now, if we take , then, for any , we have for all (). Thus the hypothesis (d) of Theorem 1.2 is satisfied trivially. (If T is a strongly -quasi-pseudo-monotone type II operator, we equip E with the weak topology.) Thus ϕ satisfies all the hypotheses of Theorem 1.2. Hence, by Theorem 1.2, there exists a point such that for all , i.e.,
for all .
If , then so that . Choose such that
Then it follows that
If for each , then and hence
and so . Then we see that whenever for each . Since or for each , it follows that
which contradicts (3.10). This contradiction proves Step 1. Hence we have shown that there exists a point such that and
Step 2. We need to show that
for all .
From Step 1, we know that which is a convex subset of X and
for all . Hence, applying Lemma 2.4, we obtain
for all .
Step 3. There exists a point with for all . From Step 2, we have
where is a -compact convex subset of the Hausdorff topological vector space and is a convex subset of X.
Now, we define by for each and . Then, for each fixed , the mapping is convex and continuous on and, for each fixed , the mapping is concave on . So, we can apply Keneser’s minimax theorem (Theorem 2.5) and obtain the following:
Hence, by (3.11), we obtain
Since is compact, there exists such that
for all . This completes the proof. □
Note that, if for each open subset U of X and for each , and there exists such that ; and if the mapping is, in addition, lower semi-continuous and, for each , T is upper semi-continuous at some point x in with , then the set Σ in Theorem 3.1 is always open in X, and so we obtain the following result.
Theorem 3.2 Let E be a locally convex Hausdorff topological vector space over Φ, X be a nonempty compact convex subset of E and F be a vector space over Φ with -topology, where is a bilinear functional separating points on F such that, for each , the function is continuous. Let , , and be the mappings such that
-
(a)
S is continuous such that each is closed and convex;
-
(b)
is bounded;
-
(c)
T is an -pseudo-monotone type II (respectively, a strongly -pseudo-monotone type II) operator and is upper hemi-continuous along line segments in X to the -topology on F such that each is -compact and convex and is -bounded;
-
(d)
T and η have the 0-DCVR and η is continuous;
-
(e)
for each fixed , , i.e., is lower semi-continuous on for each and, for each fixed , and are concave, is affine, and ;
-
(f)
for each open subset U of X and , , and there exists such that ;
-
(g)
for each , T is upper semi-continuous at some point in with ;
-
(h)
for each and , there exist and such that
for any family of non-negative real-valued functions from X into ;
-
(i)
for each , the bilinear functional is continuous over the compact subset of .
Then there exists a point such that
-
(1)
;
-
(2)
there exists a point with for all .
The proof is similar to the proof of Theorem 3.2 in [10]. For the completeness, we include the proof here.
Proof The proof follows from Theorem 3.1 if we can show that the set
is open in X. To show that Σ is open in X, we start as follows.
Let be an arbitrary point. We show that there exists an open neighborhood of in X such that . Now, by hypothesis (g), T is upper semi-continuous at some point in with
Let
Thus . Again, let
Then W is a strongly open neighborhood of 0 in F, and so is an open neighborhood of in F. Since T is upper semi-continuous at , there exists an open neighborhood of in X such that for all . Since the mapping is continuous at , there exists an open neighborhood of in X such that
for all . Let . Then is an open neighborhood of in X. Since and S is lower semi-continuous at , there exists an open neighborhood of in X such that for all . Since the mapping is continuous at , there exists an open neighborhood of in X such that
for all .
Let . Then is an open neighborhood of in X such that for each , we have the following:
-
(a)
as ; so we can choose any ;
-
(b)
as ;
-
(c)
as ;
-
(d)
as .
Hence, using property (f) and (b)-(d), we can obtain the following by omitting the details:
Consequently, we have
since . Hence for all . Therefore, . But was arbitrary. Consequently, Σ is open in X. Thus all the hypotheses of Theorem 3.1 are satisfied. Hence, the conclusion follows from Theorem 3.1. This completes the proof. □
Remark 3.1
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(1)
Theorems 3.1 and 3.2 of this paper are further extensions of the results obtained in [[10], Theorem 3.1] and in [[10], Theorem 3.2], respectively, into generalized quasi-variational-like inequalities of -pseudo-monotone type II operators on compact sets.
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(2)
In 1985, Shih and Tan [1] obtained results on generalized quasi-variational inequalities in locally convex topological vector spaces, and their 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 -pseudo-monotone type II operators on compact sets.
-
(3)
The results in [10] were obtained on non-compact sets where one of the set-valued mappings is a pseudo-monotone type II operator which was defined first in [8] and later renamed as pseudo-monotone type II operator in [9]. Our present results are extensions of the results in [10] using an extension of the operators defined in [9] (and originally in [8]).
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Acknowledgements
The second author was supported by the Deanship of Scientific Research (DSR), King Abdulaziz University, under grant No. 31-130-35-HiCi. The third author was supported by Basic Science Research Program through the National Research Foundation of Korea (NRF) funded by the Ministry of Science, ICT and Future Planning (2014R1A2A2A01002100).
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The first author made the first draft of this paper with substantial contributions to conception and design. The second and third authors have been involved equally in drafting the manuscript or revising it critically for important intellectual content. All authors read and approved the final manuscript.
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Chowdhury, M.S., Abdou, A.A. & Cho, Y.J. Existence theorems of generalized quasi-variational-like inequalities for pseudo-monotone type II operators. J Inequal Appl 2014, 449 (2014). https://doi.org/10.1186/1029-242X-2014-449
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DOI: https://doi.org/10.1186/1029-242X-2014-449