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Existence theorems of generalized quasivariationallike inequalities for pseudomonotone 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 quasivariationallike inequalities (GQVLI) for pseudomonotone type II operators defined on compact sets in locally convex Hausdorff topological vector spaces. In obtaining our results on GQVLI for pseudomonotone 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.103113, 1972) as the main tool.
1 Introduction
If X is a nonempty set, then we denote by {2}^{X} the family of all nonempty subsets of X and by \mathcal{F}(x) 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 \u3008\cdot ,\cdot \u3009:F\times E\to \mathrm{\Phi} be a bilinear functional. Throughout this paper, Φ denotes either the real field ℝ or the complex field ℂ.
For each {x}_{0}\in E, each nonempty subset A of E and each \u03f5>0, let W({x}_{0};\u03f5):=\{y\in F:\u3008y,{x}_{0}\u3009<\u03f5\} and U(A;\u03f5):=\{y\in F:{sup}_{x\in A}\u3008y,x\u3009<\u03f5\}. Let \sigma \u3008F,E\u3009 be the (weak) topology on F generated by the family \{W(x;\u03f5):x\in E,\u03f5>0\} as a subbase for the neighborhood system at 0 and \delta \u3008F,E\u3009 be the (strong) topology on F generated by the family \{U(A;\u03f5):A\text{is a nonempty bounded subset of}E\text{and}\u03f50\} as a base for the neighborhood system at 0. We note then that F, when equipped with the (weak) topology \sigma \u3008F,E\u3009 or the (strong) topology \delta \u3008F,E\u3009, becomes a locally convex topological vector space which is not necessarily Hausdorff. But, if the bilinear functional \u3008\cdot ,\cdot \u3009:F\times E\to \mathrm{\Phi} separates points in F, i.e., for each y\in F with y\ne 0, there exists x\in E such that \u3008y,x\u3009\ne 0, then F also becomes Hausdorff. Furthermore, for any net {\{{y}_{\alpha}\}}_{\alpha \in \mathrm{\Gamma}} in F and y\in F,

(a)
{y}_{\alpha}\to y in \sigma \u3008F,E\u3009 if and only if \u3008{y}_{\alpha},x\u3009\to \u3008y,x\u3009 for each x\in E;

(b)
{y}_{\alpha}\to y in \delta \u3008F,E\u3009 if and only if \u3008{y}_{\alpha},x\u3009\to \u3008y,x\u3009 uniformly for each x\in A, where A is a nonempty bounded subset of E.
Suppose that, for the sets X, E and F mentioned above, S:X\to {2}^{X} and T:X\to {2}^{F} are two setvalued mappings. We now introduce below a slightly modified definition of the generalized quasivariational inequality in infinite dimensional spaces given by Shih and Tan in [1]:
Find \stackrel{\u02c6}{y}\in S(\stackrel{\u02c6}{y}) and \stackrel{\u02c6}{w}\in T(\stackrel{\u02c6}{y}) such that
for all x\in S(\stackrel{\u02c6}{y}).
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 \eta :X\times X\to E be a singlevalued mapping and h:X\times X\to \mathbb{R} be a realvalued function. Then the generalized quasivariationallike inequality problem is defined as follows: Find \stackrel{\u02c6}{y}\in S(\stackrel{\u02c6}{y}) and \stackrel{\u02c6}{w}\in T(\stackrel{\u02c6}{y}) such that
for all x\in S(\stackrel{\u02c6}{y}).
For more results related to the generalized quasivariationallike inequality problems, we refer to [3–6] and the references therein.
The following definition given in [7] is a slight modification of demioperators defined in [8] and of pseudomonotone 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 \sigma \u3008F,E\u3009topology, where \u3008\cdot ,\cdot \u3009:F\times E\to \mathrm{\Phi} is a bilinear functional. Let h:X\times X\to \mathbb{R}, \eta :X\times X\to E and T:X\to {2}^{F} be three mappings. Then T is said to be:

(1)
an (\eta ,h)pseudomonotone type II (respectively, a strongly (\eta ,h)pseudomonotone type II) operator if, for each y\in X and every net {\{{y}_{\alpha}\}}_{\alpha \in \mathrm{\Gamma}} in X converging to y (respectively, weakly to y) with
\underset{\alpha}{lim\hspace{0.17em}sup}[\underset{u\in T(y)}{inf}Re\u3008u,\eta ({y}_{\alpha},y)\u3009+h({y}_{\alpha},y)]\le 0,we have
\begin{array}{c}\underset{\alpha}{lim\hspace{0.17em}sup}[\underset{u\in T(x)}{inf}Re\u3008u,\eta ({y}_{\alpha},x)\u3009+h({y}_{\alpha},x)]\hfill \\ \phantom{\rule{1em}{0ex}}\ge \underset{u\in T(x)}{inf}Re\u3008u,\eta (y,x)\u3009+h(y,x)\hfill \end{array}for all x\in X;

(2)
an hpseudomonotone type II operator (respectively, a strongly hpseudomonotone type II operator) if T is an (\eta ,h)pseudomonotone type II operator with \eta (x,y)=xy and, for some {h}^{\prime}:X\to \mathbb{R}, h(x,y)={h}^{\prime}(x){h}^{\prime}(y) for all x,y\in X.
Note that, if F={E}^{\ast}, the topological dual space of E, then the notions of hpseudomonotone type II operators coincide with those in [8].
Pseudomonotone 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 pseudomonotone type II operators by Chowdhury in [9].
Next, we shall state and prove the following lemma which provides a numerous collection of (\eta ,h)pseudomonotone type II and strongly (\eta ,h)pseudomonotone type II operators.
Lemma 1.1 Let E be a topological vector space and X be a nonempty bounded subset of E. Let T:X\to {2}^{{E}^{\ast}} be an operator such that each T(x) is strongly compact. Suppose that h:X\times X\to \mathbb{R} is a realvalued function such that, for each y\in X, h(\cdot ,y) is continuous and h(X\times X) is bounded. Let \eta :X\times X\to E 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 {E}^{\ast}. Then T is both an (\eta ,h)pseudomonotone type II and a strongly (\eta ,h)pseudomonotone type II operator.
Proof Suppose that {\{{y}_{\alpha}\}}_{\alpha \in \mathrm{\Gamma}} is a net in X and y\in X with {y}_{\alpha}\to y (respectively, {y}_{\alpha}\to y weakly) and that
Let x\in X be arbitrarily fixed. Then, using the continuity of h(\cdot ,y), η and T, we obtain the following:
for all x\in X. Consequently, T is both an (\eta ,h)pseudomonotone type II and a strongly (\eta ,h)pseudomonotone 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 quasivariationallike inequalities for pseudomonotone 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 h:X\times X\to \mathbb{R} and \varphi :X\times X\to \mathbb{R}\cup \{\mathrm{\infty},+\mathrm{\infty}\} be the mappings such that

(a)
for each A\in \mathcal{F}(X) and fixed x\in co(A), y\mapsto \varphi (x,y) is lower semicontinuous on co(A);

(b)
for each A\in \mathcal{F}(X) and y\in co(A), {min}_{x\in A}[\varphi (x,y)+h(y,x)]\le 0;

(c)
for each fixed x\in X, y\mapsto h(x,y) is lower semicontinuous and concave on X, and h(x,x)=0;

(d)
for each A\in \mathcal{F}(X) and each pair of points x,y\in co(A) such that every net {\{{y}_{\alpha}\}}_{\alpha \in \mathrm{\Gamma}} in X converging to y with \varphi (tx+(1t)y,{y}_{\alpha})+h({y}_{\alpha},tx+(1t)y)\le 0 for all \alpha \in \mathrm{\Gamma} and all t\in [0,1], we have \varphi (x,y)+h(y,x)\le 0;

(e)
there exist a nonempty closed and compact subset K of X and {x}_{0}\in K such that \varphi ({x}_{0},y)+h(y,{x}_{0})>0 for all y\in X\setminus K.
Then there exists \stackrel{\u02c6}{y}\in K such that \varphi (x,\stackrel{\u02c6}{y})+h(\stackrel{\u02c6}{y},x)\le 0 for all x\in X.
Proof For the proof, we refer to [12]. □
Definition 1.3 A function \varphi :X\times X\to \mathbb{R}\cup \{\pm \mathrm{\infty}\} is said to be 0diagonally concave (in short, 0DCV) in the second argument [14] if, for any finite set \{{x}_{1},\dots ,{x}_{n}\}\subset X and {\lambda}_{i}\ge 0 with {\sum}_{i=1}^{n}{\lambda}_{i}=1, we have {\sum}_{i=1}^{n}{\lambda}_{i}\varphi (y,{x}_{i})\le 0, where y={\sum}_{i=1}^{n}{\lambda}_{i}{x}_{i}.
Now, we state the following definition given in [15].
Definition 1.4 Let X, E, F be the sets defined before and T:X\to {2}^{F}, \eta :X\times X\to E, g:X\to E be mappings.

(1)
The mappings T and η are said to have the 0diagonally concave relation (in short, 0DCVR) if the function \varphi :X\times X\to \mathbb{R}\cup \{\pm \mathrm{\infty}\} defined by
\varphi (x,y)=\underset{w\in T(x)}{inf}Re\u3008w,\eta (x,y)\u3009
is 0DCV in y.

(2)
The mappings T and g are said to have the 0diagonally concave relation if T and \eta (x,y)=g(x)g(y) have the 0DCVR.
The following definition of upper hemicontinuity 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 T:X\to {2}^{{E}^{\ast}}. Then T is said to be upper hemicontinuous on X if and only if, for each p\in E, the function {f}_{p}:X\to \mathbb{R}\cup \{+\mathrm{\infty}\} defined by
for each z\in X is upper semicontinuous on X (if and only if, for each p\in E, the function {g}_{p}:X\to \mathbb{R}\cup \{\mathrm{\infty}\} defined by
for each z\in X is lower semicontinuous 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 S:X\to {2}^{E} be an upper semicontinuous map such that S(x) is a bounded subset of E for each x\in X. Then, for each continuous linear functional p on E, the mapping {f}_{p}:X\to \mathbb{R} defined by
is upper semicontinuous, i.e., for each \lambda \in \mathbb{R}, the set \{y\in X:{f}_{p}(y)={sup}_{x\in S(y)}Re\u3008p,x\u3009<\lambda \} 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, f:X\to \mathbb{R} be nonnegative and continuous and g:Y\to \mathbb{R} be lower semicontinuous. Then the mapping F:X\times Y\to \mathbb{R} defined by F(x,y)=f(x)g(y) for all (x,y)\in X\times Y is lower semicontinuous.
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, A\in \mathcal{F}(E) and X=co(A), where co(A) denotes the convex hull of A. Let F be a vector space over Φ and \u3008\cdot ,\cdot \u3009:F\times E\to \varphi be a bilinear functional such that \u3008\cdot ,\cdot \u3009 separates points in F. We equip F with the \sigma \u3008F,E\u3009topology. Suppose that, for each w\in F, x\mapsto Re\u3008w,x\u3009 is continuous. Let \eta :X\times X\to E be continuous. Let T:X\to {2}^{F} be upper semicontinuous from X into {2}^{F} such that each T(x) is \sigma \u3008F,E\u3009compact. Let f:X\times X\to \mathbb{R} be defined by f(x,y)={inf}_{w\in T(y)}Re\u3008w,\eta (y,x)\u3009 for all x,y\in X. Suppose that \u3008\cdot ,\cdot \u3009 is continuous on the (compact) subset [{\bigcup}_{y\in X}T(y)]\times \eta (X\times X) of F\times E. Then, for each fixed x\in X, y\mapsto f(x,y) is lower semicontinuous on X.
For the completeness, we include the proof here given in [12].
Proof Let \lambda \in \mathbb{R} be given and let x\in X=co(A) be arbitrarily fixed. Let {A}_{\lambda}=\{y\in X:f(x,y)\le \lambda \}. Suppose that {\{{y}_{\alpha}\}}_{\alpha \in \mathrm{\Gamma}} is a net in {A}_{\lambda} and {y}_{0}\in co(A)=X such that {y}_{\alpha}\to {y}_{0}. Then, for each \alpha \in \mathrm{\Gamma},
Since F is equipped with the \sigma \u3008F,E\u3009topology, for each x\in E, the function w\mapsto Re\u3008w,x\u3009 is continuous. Also, \eta ({y}_{\alpha},x)\to \eta ({y}_{0},x) because \eta (\cdot ,x) is continuous. By the \sigma \u3008F,E\u3009compactness of T({y}_{\alpha}), there exists {w}_{\alpha}\in T({y}_{\alpha}) such that
Since T is upper semicontinuous from X=co(A) to the \sigma \u3008F,E\u3009topology on F, X is compact, and each T(z) is \sigma \u3008F,E\u3009compact, {\bigcup}_{z\in X}T(z) is also \sigma \u3008F,E\u3009compact by Proposition 3.1.11 of Aubin and Ekeland [20]. Thus there is a subnet {\{{w}_{{\alpha}^{\prime}}\}}_{{\alpha}^{\prime}\in {\mathrm{\Gamma}}^{\prime}} of {\{{w}_{\alpha}\}}_{\alpha \in \mathrm{\Gamma}} and {w}_{0}\in {\bigcup}_{z\in X}T(z) such that {w}_{{\alpha}^{\prime}}\to {w}_{0} in the \sigma \u3008F,E\u3009topology. Again, as T is upper semicontinuous with the \sigma \u3008F,E\u3009closed values, {w}_{0}\in T({y}_{0}).
Suppose that A=\{{a}_{1},{a}_{2},\dots ,{a}_{n}\} and let {t}_{1},{t}_{2},\dots ,{t}_{n}\ge 0 with {\sum}_{i=1}^{n}{t}_{i}=1 such that {y}_{0}={\sum}_{i=1}^{n}{t}_{i}{a}_{i}. For each {\alpha}^{\prime}\in \mathrm{\Gamma}, let {t}_{1}^{{\alpha}^{\prime}},{t}_{2}^{{\alpha}^{\prime}},\dots ,{t}_{n}^{{\alpha}^{\prime}}\ge 0 with {\sum}_{i=1}^{n}{t}_{i}^{{\alpha}^{\prime}}=1 such that {y}_{{\alpha}^{\prime}}={\sum}_{i=1}^{n}{t}_{i}^{{\alpha}^{\prime}}{a}_{i}. Since E is Hausdorff and {y}_{{\alpha}^{\prime}}\to {y}_{0}, we must have {t}_{i}^{{\alpha}^{\prime}}\to {t}_{i} for each i=1,2,\dots ,n. Thus
where (2.1) is true since \eta (\cdot ,x) is continuous on X and \u3008\cdot ,\cdot \u3009 is continuous on the compact subset [{\bigcup}_{y\in X}T(y)]\times \eta (X\times X) of F\times E. Hence {y}_{0}\in {A}_{\lambda}. Thus {A}_{\lambda} is closed in X=co(A) for each \lambda \in \mathbb{R}. Therefore y\mapsto f(x,y) is lower semicontinuous 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 \sigma \u3008F,E\u3009topology such that, for each w\in F, the function x\mapsto Re\u3008w,x\u3009 is continuous. Let T:X\to {2}^{F} be upper hemicontinuous along line segments in X. Let \eta :X\times X\to E be such that for each fixed y\in X, \eta (\cdot ,y) is continuous, and let h:X\times X\to \mathbb{R} be a mapping such that, for each fixed y\in X, h(\cdot ,y) is lower semicontinuous on co(A) for each A\in \mathcal{F}(X) and, for each fixed x\in X, h(x,\cdot ) is concave and h(x,x)=0 and T, η have the 0DCVR. Suppose that \stackrel{\u02c6}{y}\in X such that {inf}_{u\in T(x)}Re\u3008u,\eta (\stackrel{\u02c6}{y},x)\u3009\le h(x,\stackrel{\u02c6}{y}) for all x\in X. Then
for all x\in X.
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 realvalued function on X\times Y such that for each fixed x\in X, the map y\mapsto f(x,y), i.e., f(x,\cdot ) is lower semicontinuous and convex on Y and, for each fixed y\in Y, the mapping x\mapsto f(x,y), i.e., f(\cdot ,y) is concave on X. Then
3 Generalized quasivariationallike inequalities
In this section, we prove some existence theorems for the solutions to the generalized quasivariationallike inequalities for pseudomonotone 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 \sigma \u3008F,E\u3009topology, where \u3008\cdot ,\cdot \u3009:F\times E\to \mathrm{\Phi} is a bilinear functional separating points on F such that, for each w\in F, the function x\mapsto Re\u3008w,x\u3009 is continuous. Let S:X\to {2}^{X}, T:X\to {2}^{F}, \eta :X\times X\to E and h:E\times E\to \mathbb{R} be the mappings such that

(a)
S is upper semicontinuous such that each S(x) is closed and convex;

(b)
h(X\times X) is bounded;

(c)
T is an (\eta ,h)pseudomonotone type II (respectively, a strongly (\eta ,h)pseudomonotone type II) operator and is upper hemicontinuous along line segments in X to the \sigma \u3008F,E\u3009topology on F such that each T(x) is \sigma \u3008F,E\u3009compact and convex and T(X) is \delta \u3008F,E\u3009bounded;

(d)
T and η have the 0DCVR and η is continuous;

(e)
for each fixed y\in X, x\mapsto h(x,y), i.e., h(\cdot ,y) is lower semicontinuous on co(A) for each A\in \mathcal{F}(X) and, for each fixed x\in X, h(x,\cdot ) and \eta (x,\cdot ) are concave, \eta (x,\cdot ) is affine, h(x,x)=0 and \eta (x,x)=0;

(f)
the set \mathrm{\Sigma}=\{y\in X:{sup}_{x\in S(y)}[{inf}_{u\in T(x)}Re\u3008u,\eta (y,x)\u3009+h(y,x)]>0\} is open in X;

(g)
for each A\in \mathcal{F}(X) and each y\in co(A), there exist \overline{x}\in A and \overline{u}\in T(\overline{x}) such that
{\beta}_{0}(y)[Re\u3008\overline{u},\eta (y,\overline{x})\u3009+h(y,\overline{x})]+\sum _{p\in LF(E)}{\beta}_{p}(y)Re\u3008p,y\overline{x}\u3009\le 0
for any family \{{\beta}_{0},{\beta}_{p}:p\in LF(E)\} of nonnegative realvalued functions from X into [0,1], where LF(E) denotes the set of all continuous linear functionals on E;

(h)
for each A\in \mathcal{F}(X), the bilinear functional \u3008\cdot ,\cdot \u3009 is continuous over the compact subset [{\bigcup}_{y\in co(A)}T(y)]\times \eta (co(A)\times co(A)) of F\times E.
Then there exists a point \stackrel{\u02c6}{y}\in X such that

(1)
\stackrel{\u02c6}{y}\in S(\stackrel{\u02c6}{y});

(2)
there exists a point \stackrel{\u02c6}{w}\in T(\stackrel{\u02c6}{y}) with Re\u3008\stackrel{\u02c6}{w},\eta (\stackrel{\u02c6}{y},x)\u3009+h(\stackrel{\u02c6}{y},x)\le 0 for all x\in S(\stackrel{\u02c6}{y}).
Proof Step 1. Let us first show that there exists a point \stackrel{\u02c6}{y}\in X such that \stackrel{\u02c6}{y}\in S(\stackrel{\u02c6}{y}) and
Now, we prove this by contradiction. So, we assume that, for each y\in X, either y\notin S(y) or there exists x\in S(y) such that
that is, for each y\in X, either y\notin S(y) or y\in \mathrm{\Sigma}. If y\notin S(y), 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 y\in X, set
and, for each continuous linear functional p on E,
Then we have
Since {V}_{0} is open by hypothesis and each {V}_{p} is open in X by Lemma 2.1 (Lemma 1 in [19]), \{{V}_{0},{V}_{p}:p\in LF(E)\} is an open covering for X. Since X is compact, there exist {p}_{1},{p}_{2},\dots ,{p}_{n}\in LF(E) such that X={V}_{0}\cup {\bigcup}_{i=1}^{n}{V}_{{p}_{i}}. For the simplicity of notation, let {V}_{i}={V}_{{p}_{i}} for i=1,2,\dots ,n. Let \{{\beta}_{0},{\beta}_{1},\dots ,{\beta}_{n}\} be a continuous partition of unity on X subordinated to the covering \{{V}_{0},{V}_{1},\dots ,{V}_{n}\}. Then {\beta}_{0},{\beta}_{1},\dots ,{\beta}_{n} are continuous nonnegative realvalued functions on X such that {\beta}_{i} vanishes on X\setminus {V}_{i} for each i=0,1,\dots ,n and {\sum}_{i=0}^{n}{\beta}_{i}(x)=1 for all x\in X. Note that, for each y\in X and A\in \mathcal{F}(X), x\mapsto h(x,y), i.e., h(\cdot ,y) is continuous on co(A) (see [22], Corollary 10.1.1). Define a function \varphi :X\times X\to \mathbb{R} by
for all x,y\in X. Then we have the following:
(I) Since E is Hausdorff, for each A\in \mathcal{F}(X) and fixed x\in co(A), the mapping
is continuous on co(A) by Lemma 2.3 and the fact that h is continuous on co(A), and so the mapping
is lower semicontinuous on co(A) by Lemma 2.2. Also, for each fixed x\in X,
is continuous on X. Hence, for each A\in \mathcal{F}(X) and fixed x\in co(A), the mapping y\mapsto \varphi (x,y) is lower semicontinuous on co(A).
(II) Since {\beta}_{0},{\beta}_{1},\dots ,{\beta}_{n} is a family of continuous nonnegative realvalued functions on X into [0,1], by the hypothesis, for each A\in \mathcal{F}(X) and each y\in co(A), there exist \overline{x}\in A and \overline{u}\in T(\overline{x}) such that
Thus we have
i.e.,
Therefore, we have
and so {min}_{x\in A}\varphi (x,y)\le 0 for each A\in \mathcal{F}(X) and y\in co(A).
(III) Suppose that A\in \mathcal{F}(X), x,y\in co(A) and {\{{y}_{\alpha}\}}_{\alpha \in \mathrm{\Gamma}} is a net in X converging to y (respectively, weakly to y) with \varphi (tx+(1t)y,{y}_{\alpha})\le 0 for all \alpha \in \mathrm{\Gamma} and all t\in [0,1].
Case 1: {\beta}_{0}(y)=0. Since {\beta}_{0} is continuous and {y}_{\alpha}\to y, we have {\beta}_{0}({y}_{\alpha})\to {\beta}_{0}(y)=0. Note that {\beta}_{0}({y}_{\alpha})\ge 0 for each \alpha \in \mathrm{\Gamma}. Since T(X) is strongly bounded and {\{{y}_{\alpha}\}}_{\alpha \in \mathrm{\Gamma}} is a bounded net, it follows that
Also, we have
Thus it follows from (3.1) that
When t=1, we have \varphi (x,{y}_{\alpha})\le 0 for all \alpha \in \mathrm{\Gamma}, i.e.,
for all \alpha \in \mathrm{\Gamma}. Therefore, by (3.3), we have
and so
Hence, by (3.2) and (3.4), we have \varphi (x,y)\le 0.
Case 2: {\beta}_{0}(y)>0. Since {\beta}_{0} is continuous, {\beta}_{0}({y}_{\alpha})\to {\beta}_{0}(y). Again since {\beta}_{0}(y)>0, there exists \lambda \in \mathrm{\Gamma} such that {\beta}_{0}({y}_{\alpha})>0 for all \alpha \ge \lambda.
When t=0, we have \varphi (y,{y}_{\alpha})\le 0 for all \alpha \in \mathrm{\Gamma}, i.e.,
for all \alpha \in \mathrm{\Gamma}, and so
Hence, by (3.5), we have
Since {lim\hspace{0.17em}inf}_{\alpha}[{\sum}_{i=1}^{n}{\beta}_{i}({y}_{\alpha})Re\u3008p,{y}_{\alpha}y\u3009]=0, we have
Since {\beta}_{0}({y}_{\alpha})>0 for all \alpha \ge \lambda, it follows that
Since {\beta}_{0}(y)>0, by (3.6) and (3.7), we have
Since T is an (\eta ,h)pseudomonotone type II (respectively, a strongly (\eta ,h)pseudomonotone type II) operator, we have
for all x\in X. Since {\beta}_{0}(y)>0, we have
and so
When t=1, we have \varphi (x,{y}_{\alpha})\le 0 for all \alpha \in \mathrm{\Gamma}, i.e.,
for all \alpha \in \mathrm{\Gamma} and so, by (3.8),
Hence we have \varphi (x,y)\le 0.
(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 K=X, then, for any {x}_{0}\in K=X, we have \varphi ({x}_{0},y)>0 for all y\in X\setminus K (=X\setminus X=\mathrm{\varnothing}). Thus the hypothesis (d) of Theorem 1.2 is satisfied trivially. (If T is a strongly (\eta ,h)quasipseudomonotone 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 \stackrel{\u02c6}{y}\in K=X such that \varphi (x,\stackrel{\u02c6}{y})\le 0 for all x\in X, i.e.,
for all x\in X.
If {\beta}_{0}(\stackrel{\u02c6}{y})>0, then \stackrel{\u02c6}{y}\in {V}_{0}=\mathrm{\Sigma} so that \gamma (\stackrel{\u02c6}{y})>0. Choose \stackrel{\u02c6}{x}\in S(\stackrel{\u02c6}{y})\subset X such that
Then it follows that
If {\beta}_{i}(\stackrel{\u02c6}{y})>0 for each i=1,2,\dots ,n, then \stackrel{\u02c6}{y}\in {V}_{i} and hence
and so Re\u3008{p}_{i},\stackrel{\u02c6}{y}\stackrel{\u02c6}{x}\u3009>0. Then we see that {\beta}_{i}(\stackrel{\u02c6}{y})Re\u3008{p}_{i},\stackrel{\u02c6}{y}\stackrel{\u02c6}{x}\u3009>0 whenever {\beta}_{i}(\stackrel{\u02c6}{y})>0 for each i=1,2,\dots ,n. Since {\beta}_{0}(\stackrel{\u02c6}{y})>0 or {\beta}_{i}(\stackrel{\u02c6}{y})>0 for each i=1,2,\dots ,n, it follows that
which contradicts (3.10). This contradiction proves Step 1. Hence we have shown that there exists a point \stackrel{\u02c6}{y}\in X such that \stackrel{\u02c6}{y}\in S(\stackrel{\u02c6}{y}) and
Step 2. We need to show that
for all x\in S(\stackrel{\u02c6}{y}).
From Step 1, we know that \stackrel{\u02c6}{y}\in S(\stackrel{\u02c6}{y}) which is a convex subset of X and
for all x\in S(\stackrel{\u02c6}{y}). Hence, applying Lemma 2.4, we obtain
for all x\in S(\stackrel{\u02c6}{y}).
Step 3. There exists a point \stackrel{\u02c6}{w}\in T(\stackrel{\u02c6}{y}) with Re\u3008\stackrel{\u02c6}{w},\eta (\stackrel{\u02c6}{y},x)\u3009\le h(x,\stackrel{\u02c6}{y}) for all x\in S(\stackrel{\u02c6}{y}). From Step 2, we have
where T(\stackrel{\u02c6}{y}) is a \sigma \u3008F,E\u3009compact convex subset of the Hausdorff topological vector space (F,\sigma \u3008F,E\u3009) and S(\stackrel{\u02c6}{y}) is a convex subset of X.
Now, we define f:S(\stackrel{\u02c6}{y})\times T(\stackrel{\u02c6}{y})\to \mathbb{R} by f(x,w)=Re\u3008w,\eta (\stackrel{\u02c6}{y},x)\u3009+h(\stackrel{\u02c6}{y},x) for each x\in S(\stackrel{\u02c6}{y}) and w\in T(\stackrel{\u02c6}{y}). Then, for each fixed x\in S(\stackrel{\u02c6}{y}), the mapping w\mapsto f(x,w) is convex and continuous on T(\stackrel{\u02c6}{y}) and, for each fixed w\in T(\stackrel{\u02c6}{y}), the mapping x\mapsto f(x,w) is concave on S(\stackrel{\u02c6}{y}). So, we can apply Keneser’s minimax theorem (Theorem 2.5) and obtain the following:
Hence, by (3.11), we obtain
Since T(\stackrel{\u02c6}{y}) is compact, there exists \stackrel{\u02c6}{w}\in T(\stackrel{\u02c6}{y}) such that
for all x\in S(\stackrel{\u02c6}{y}). This completes the proof. □
Note that, if for each open subset U of X and for each x,y\in U, \eta (x,y)=xy and there exists {h}^{\prime}:X\to \mathbb{R} such that h(x,y)={h}^{\prime}(x){h}^{\prime}(y); and if the mapping S:X\to {2}^{X} is, in addition, lower semicontinuous and, for each y\in \mathrm{\Sigma}, T is upper semicontinuous at some point x in S(y) with {inf}_{u\in T(x)}Re\u3008u,\eta (y,x)\u3009+h(y,x)>0, 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 \sigma \u3008F,E\u3009topology, where \u3008\cdot ,\cdot \u3009:F\times E\to \mathrm{\Phi} is a bilinear functional separating points on F such that, for each w\in F, the function x\mapsto Re\u3008w,x\u3009 is continuous. Let S:X\to {2}^{X}, T:X\to {2}^{F}, \eta :X\times X\to E and h:E\times E\to \mathbb{R} be the mappings such that

(a)
S is continuous such that each S(x) is closed and convex;

(b)
h(X\times X) is bounded;

(c)
T is an (\eta ,h)pseudomonotone type II (respectively, a strongly (\eta ,h)pseudomonotone type II) operator and is upper hemicontinuous along line segments in X to the \sigma \u3008F,E\u3009topology on F such that each T(x) is \sigma \u3008F,E\u3009compact and convex and T(X) is \delta \u3008F,E\u3009bounded;

(d)
T and η have the 0DCVR and η is continuous;

(e)
for each fixed y\in X, x\mapsto h(x,y), i.e., h(\cdot ,y) is lower semicontinuous on co(A) for each A\in \mathcal{F}(X) and, for each fixed x\in X, h(x,\cdot ) and \eta (x,\cdot ) are concave, \eta (x,\cdot ) is affine, h(x,x)=0 and \eta (x,x)=0;

(f)
for each open subset U of X and x,y\in U, \eta (x,y)=xy, and there exists {h}^{\prime}:X\to \mathbb{R} such that h(x,y)={h}^{\prime}(x){h}^{\prime}(y);

(g)
for each y\in \mathrm{\Sigma}=\{y\in X:{sup}_{x\in S(y)}[{inf}_{u\in T(x)}Re\u3008u,\eta (y,x)\u3009+h(y,x)]>0\}, T is upper semicontinuous at some point {x}_{0} in S(y) with {inf}_{u\in T({x}_{0})}Re\u3008u,\eta (y,{x}_{0})\u3009+h(y,{x}_{0})>0;

(h)
for each A\in \mathcal{F}(X) and y\in co(A), there exist \overline{x}\in A and \overline{u}\in T(\overline{x}) such that
{\beta}_{0}(y)[Re\u3008\overline{u},\eta (y,\overline{x})\u3009+h(y,\overline{x})]+\sum _{p\in LF(E)}{\beta}_{p}(y)Re\u3008p,y\overline{x}\u3009\le 0
for any family \{{\beta}_{0},{\beta}_{p}:p\in LF(E)\} of nonnegative realvalued functions from X into [0,1];

(i)
for each A\in \mathcal{F}(X), the bilinear functional \u3008\cdot ,\cdot \u3009 is continuous over the compact subset [{\bigcup}_{y\in co(A)}T(y)]\times \eta (co(A)\times co(A)) of F\times E.
Then there exists a point \stackrel{\u02c6}{y}\in X such that

(1)
\stackrel{\u02c6}{y}\in S(\stackrel{\u02c6}{y});

(2)
there exists a point \stackrel{\u02c6}{w}\in T(\stackrel{\u02c6}{y}) with Re\u3008\stackrel{\u02c6}{w},\eta (\stackrel{\u02c6}{y},x)\u3009+h(\stackrel{\u02c6}{y},x)\le 0 for all x\in S(\stackrel{\u02c6}{y}).
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 {y}_{0}\in \mathrm{\Sigma} be an arbitrary point. We show that there exists an open neighborhood {N}_{0} of {y}_{0} in X such that {N}_{0}\subset \mathrm{\Sigma}. Now, by hypothesis (g), T is upper semicontinuous at some point {x}_{0} in S({y}_{0}) with
Let
Thus \alpha >0. Again, let
Then W is a strongly open neighborhood of 0 in F, and so {U}_{1}:=T({x}_{0})+W is an open neighborhood of T({x}_{0}) in F. Since T is upper semicontinuous at {x}_{0}, there exists an open neighborhood {V}_{1} of {x}_{0} in X such that T(x)\subset {U}_{1} for all x\in {V}_{1}. Since the mapping x\mapsto {inf}_{u\in T({x}_{0})}Re\u3008u,\eta ({x}_{0},x)\u3009+h({x}_{0},x) is continuous at {x}_{0}, there exists an open neighborhood {V}_{2} of {x}_{0} in X such that
for all x\in {V}_{2}. Let {V}_{0}:={V}_{1}\cap {V}_{2}. Then {V}_{0} is an open neighborhood of {x}_{0} in X. Since {x}_{0}\in {V}_{0}\cap S({y}_{0})\ne \mathrm{\varnothing} and S is lower semicontinuous at {y}_{0}, there exists an open neighborhood {N}_{1} of {y}_{0} in X such that S(y)\cap {V}_{0}\ne \mathrm{\varnothing} for all y\in {N}_{1}. Since the mapping y\mapsto {inf}_{u\in T({x}_{0})}Re\u3008u,\eta (y,{y}_{0})\u3009+h(y,{y}_{0}) is continuous at {y}_{0}, there exists an open neighborhood {N}_{2} of {y}_{0} in X such that
for all y\in {N}_{2}.
Let {N}_{0}:={N}_{1}\cap {N}_{2}. Then {N}_{0} is an open neighborhood of {y}_{0} in X such that for each {y}_{1}\in {N}_{0}, we have the following:

(a)
S({y}_{1})\cap {V}_{0}\ne \mathrm{\varnothing} as {y}_{1}\in {N}_{1}; so we can choose any {x}_{1}\in S({y}_{1})\cap {V}_{0};

(b)
{inf}_{u\in T({x}_{0})}Re\u3008u,\eta ({y}_{1},{y}_{0})\u3009+h({y}_{1},{y}_{0})<\frac{\alpha}{6} as {y}_{1}\in {N}_{2};

(c)
T({x}_{1})\subset {U}_{1}=T({x}_{0})+W as {x}_{1}\in {V}_{1};

(d)
{inf}_{u\in T({x}_{0})}Re\u3008u,\eta ({x}_{0},{x}_{1})\u3009+h({x}_{0},{x}_{1})<\frac{\alpha}{6} as {x}_{1}\in {V}_{2}.
Hence, using property (f) and (b)(d), we can obtain the following by omitting the details:
Consequently, we have
since {x}_{1}\in S({y}_{1}). Hence {y}_{1}\in \mathrm{\Sigma} for all {y}_{1}\in {N}_{0}. Therefore, {y}_{0}\in {N}_{0}\subset \mathrm{\Sigma}. But {y}_{0} 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

(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 quasivariationallike inequalities of (\eta ,h)pseudomonotone type II operators on compact sets.

(2)
In 1985, Shih and Tan [1] obtained results on generalized quasivariational inequalities in locally convex topological vector spaces, and their 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 (\eta ,h)pseudomonotone type II operators on compact sets.

(3)
The results in [10] were obtained on noncompact sets where one of the setvalued mappings is a pseudomonotone type II operator which was defined first in [8] and later renamed as pseudomonotone 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. 3113035HiCi. 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 quasivariationallike inequalities for pseudomonotone type II operators. J Inequal Appl 2014, 449 (2014). https://doi.org/10.1186/1029242X2014449
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DOI: https://doi.org/10.1186/1029242X2014449
Keywords
 generalized quasivariationallike inequalities
 pseudomonotone type II operators
 locally convex Hausdorff topological vector spaces