Existence theorems of generalized quasi-variational-like inequalities for η-h-pseudo-monotone type I operators on non-compact sets

Abstract

In this article, we prove the existence results of solutions for a new class of generalized quasi-variational-like inequalities (GQVLI) for η-h-pseudo-monotone type I operators defined on non-compact sets in locally convex Hausdorff topological vector spaces. In obtaining our results on GQVLI for η-h-pseudo-monotone type I operators, we use Chowdhury and Tan's generalized version of Ky Fan's minimax inequality as the main tool.

1. Introduction

If X is a nonempty set, then we denote by 2^X the family of all non-empty subsets of X and by $\mathcal{F}\left(x\right)$ the family of all non-empty finite subsets of X. Let E be a topological vector space over Φ, F be a vector space over Φ and X be a non-empty subset of E. Let 〈·, ·〉 F × E → Φ be a bilinear functional. Throughout this article, Φ denotes either the real field ℝ or the complex field ℂ.

For each x₀ ∈ E, each nonempty subset A of E and each ∊ > 0, let W (x₀; ∊) := {y ∈ F : |〈y, x₀〉| < ∊} and U(A; ∊) := {y ∈ F : sup_x∈A|〈y, x〉| < ∊}. Let σ〈F, E〉 be the (weak) topology on F generated by the family {W (x; ∊) : x ∈ E, ∊ > 0} as a subbase for the neighborhood system at 0 and δ〈F, E〉 be the (strong) topology on F generated by the family {U(A; ∊) : A is a non-empty bounded subset of E and ∊ > 0} as a base for the neighborhood system at 0. We note then that F, when equipped with the (weak) topology σ〈F, E〉 or the (strong) topology δ〈F, E〉, becomes a locally convex topological vector space which is not necessarily Hausdorff. But, if the bilinear functional 〈·, ·〉 : F × E → Φ separates points in F , i.e., for each y ∈ F with y ≠ 0, there exists x ∈ E such that 〈y, x〉 ≠ 0, then F also becomes Hausdorff. Furthermore, for any net {y _α }_α∈Γin F and y ∈ F,

(a) y _α → y in σ 〈F, E〉 if and only if 〈y _α , x〉 → 〈y, x〉 for each x ∈ E;

(b) y _α → y in δ 〈F, E〉 if and only if 〈y _α , x〉 → 〈y, x〉 uniformly for each x ∈ A, where A is a nonempty bounded subset of E.

Suppose that, for the sets X, E, and F mentioned above, S : X → 2^X , T : X → 2^F are two set-valued mappings, f : X → F , η : X ×X → E are two single-valued mappings and h : X × X → ℝ is a real-valued function. As introduced by Shih and Tan [1], the generalized quasi-variational inequality in infinite dimensional spaces is defined as follows: Find $ŷ\in S\left(ŷ\right)$ and $ŵ\in T\left(ŷ\right)$ such that

$$\mathsf{\text{Re}}〈ŵ,ŷ-x〉\le 0$$

for all x ∈ S(ŷ).

Now, we introduce the following definition:

Definition 1.1. Let X, E, and F be the sets and the mappings S, T, η, and h be as defined above. Then the generalized quasi-variational-like inequality problem is defined as follows: Find $ŷ\in S\left(ŷ\right)$ and $ŵ\in T\left(ŷ\right)$ such that

$$\mathsf{\text{Re}}〈ŵ,\eta \left(ŷ,x\right)〉+h\left(ŷ,x\right)\le 0$$

for all x ∈ S(ŷ).

For more results related to the generalized quasi-variational-like inequality problem, refer to [2–5], and therein.

The following definition is a slight modifications of pseudo-monotone operators defined in [6, Definition 1] and of pseudo-monotone type I operators defined in [7] (see also [8]):

Definition 1.2. Let X be a non-empty subset of a topological vector space E over Φ, F be a vector space over Φ which is equipped with σ〈F, E〉-topology, where 〈·, ·〉 : F ×E → Φ is a bilinear functional. Let h : X × X → ℝ, η : X × X → E, and T : X → 2^F be three mappings. Then T is said to be:

(1) an (η, h)-pseudo-monotone type I operator if, for each y ∈ X and every net {y _α }_α∈Γin X converging to y (respectively, weakly to y) with

$$\underset{\alpha }{\mathsf{\text{lim}}\mathsf{\text{sup}}}\left[\underset{u\in T\left(y_{\alpha }\right)}{\mathsf{\text{inf}}}\mathsf{\text{Re}}〈u,\eta \left(y_{\alpha },y\right)〉+h\left(y_{\alpha },y\right)\right]\le 0,$$

we have

$$\begin{array}{l}\underset{\alpha }{\mathsf{\text{lim}}\mathsf{\text{sup}}}\left[\underset{u\in T\left(y_{\alpha }\right)}{\mathsf{\text{inf}}}\mathsf{\text{Re}}〈u,\eta \left(y_{\alpha },y\right)〉+h\left(y_{\alpha },x\right)\right]\\ \ge \underset{w\in T\left(y\right)}{\mathsf{\text{inf}}}\mathsf{\text{Re}}〈w,\eta \left(y,x\right)〉+h\left(y,x\right)\end{array}$$

for all x ∈ X;

(2) an h-pseudo-monotone type I operator if T is an (η, h)-pseudo-monotone type I operator with η(x, y) = x - y and, for some $h^{\prime }:X\to ℝ,h\left(x,y\right)=h^{\prime }\left(x\right)-h^{\prime }\left(y\right)$ for all x, y∈ X.

Note that, if F = E*, the topological dual space of E, then the notions of h-pseudo-monotone type I operators coincide with those in [6].

Pseudo-monotone type I operators were first introduced by Chowdhury and Tan [6] with a slight variation in the name of this operator. Later, these operators were renamed as pseudo-monotone type I operators by Chowdhury [7]. The pseudo-monotone type I operators are set-valued generalization of the classical (single-valued) pseudo-monotone operators with slight variations. The classical definition of a single-valued pseudo-monotone operator was introduced by Brézis et al. [9].

In this article, we obtain some general theorems on solutions for a new class of generalized quasi-variational-like inequalities for pseudo-monotone type I operators defined on non-compact sets in topological vector spaces. For the main results, we mainly use the following generalized version of Ky Fan's minimax inequality [10] due to Chowdhury and Tan [6].

Theorem 1.1. Let E be a topological vector space, X be a nonempty convex subset of E and f : X × X → ℝ ∪ {-∞, + ∞} be such that

(a) for each $A\in \mathcal{F}\left(X\right)$ and each fixed x ∈ co(A), y ↦ f(x, y) is lower semi-continuous on co(A);

(b) for each $A\in \mathcal{F}\left(X\right)$ and y ∈ co(A), min_x∈Af(x, y) ≤ 0;

(c) for each $A\in \mathcal{F}\left(X\right)$ and x, y ∈ co(A), every net {y _α }_α∈Γin X converging to y with f(tx + (1 - t)y, y _α ) ≤ 0 for all α ∈ Γ and t ∈ [0, 1], we have f(x, y) ≤ 0;

(d) there exist a nonempty closed and compact subset K of X and x₀ ∈ K such that f(x₀, y) > 0 for all y ∈ X \ K.

Then there exists ŷ ∈ K such that f(x, ŷ) ≤ 0 for all x ∈ X.

Definition 1.3. A function ϕ : X × X → ℝ ∪ {±∞} is said to be 0-diagonally concave (in short, 0-DCV) in the second argument [14] if, for any finite set {x₁, . . ., x _n } ⊂ X and λ _i ≥ 0 with $\sum _{i=1}^n{\lambda }_i=1,$ we have ${\sum }_{i=1}^n{\lambda }_i\varphi \left(y,x_i\right)\le 0$, where $y=\sum _{i=1}^n{\lambda }_i{x}_i$.

Now, we state the following definition given in [8]:

Definition 1.4. Let X, E, F be be the sets defined before and T : X → 2^F, η : X × X → E, g : X → E be mappings.

• (1) The mappings T and η are said to have 0-diagonally concave relation (in short, 0-DCVR) if the function ϕ : X × X → ℝ ∪ {±∞} defined by

  $$\varphi \left(x,y\right)=\underset{w\in T\left(x\right)}{\mathsf{\text{inf}}}\mathsf{\text{Re}}〈w,\eta \left(x,y\right)〉$$

is 0-DCV in y;

• (2) The mappings T and g are said to have 0-diagonally concave relation if T and η(x, y) = g(x) - g(y) have the 0-DCVR.

2. Preliminaries

Now, we start with some earlier studies which will be needed for our main results. We first state the following result which is Lemma 1 of Shih and Tan [1]:

Lemma 2.1. Let X be a nonempty subset of a Hausdorff topological vector space E and S : X → 2^E be an upper semi-continuous map such that S(x) is a bounded subset of E for each x ∈ X. Then, for each continuous linear functional p on E, the mapping f _p : X → ℝ defined by f _p (y) = sup_x∈S(y)Re〈p, x〉 is upper semi-continuous, i.e., for each λ ∈ ℝ, the set {y ∈ X : f _p (y) = sup_x∈S(y)Re〈p, x〉 < λ} is open in X.

The following result is Takahashi [[11], Lemma 3] (see also [[12], Lemma 3]):

Lemma 2.2. Let X and Y be topological spaces, f : X → ℝ be non-negative and continuous and g : Y → ℝ be lower semi-continuous. Then the mapping F : X × Y → ℝ defined by F (x, y) = f(x)g(y) for all (x, y) ∈ X × Y is lower semi-continuous.

The following result which follows from slight modification of Chowdhury and Tan [6, Lemma 3]:

Lemma 2.3. Let E be a Hausdorff topological vector space over Φ, $A\in \mathcal{F}\left(E\right)$ and X = co(A). Let F be a vector space over Φ which is equipped with σ〈F, E〉 -topology such that, for each w ∈ F, x ↦ 〈w, x〉 is continuous. Let η : X × X → E be continuous in the first argument. Let T : X ↦ 2F \ ∅ be upper semi-continuous from X to the σ〈F, E〉-topology on F such that each T(x) is σ〈F, E〉 -compact. Let f : X × X → ℝ be defined by f(x, y) = inf_w∈T(y)Re〈w, η(y, x) 〉 for all x, y ∈ X. Then, for each fixed x ∈ X, y ↦ f(x, y) is lower semi-continuous on X.

We need the following Kneser's minimax theorem in [13] (see also Aubin [14]):

Theorem 2.1. Let X be a non-empty 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 X × Y such that for each fixed x ∈ X, the map y ↦ f(x, y), i.e., f(x, ·) is lower semi-continuous and convex on Y and, for each fixed y ∈ Y, the mapping x ↦ f(x, y), i.e., f(·, y) is concave on X. Then

$\underset{y\in Y}{\mathsf{\text{min}}}\underset{x\in X}{\mathsf{\text{sup}}}f\left(x,y\right)=\underset{x\in X}{\mathsf{\text{sup}}}\underset{y\in Y}{\mathsf{\text{min}}}f\left(x,y\right).$

3. Existence theorems for generalized quasi-variational-like inequalities for η-h-pseudo-monotone type I operators

In this section, we prove some existence theorems for the solutions to the generalized quasi-variational-like inequalities for pseudo-monotone type I operators T with non-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 non-empty para-compact convex and bounded subset of E and F be a vector space over Φ with σ〈F, E〉 -topology, where 〈·, ·〉 : F × E → Φ is a bilinear functional such that, for each w ∈ F, the function x ↦ Re〈w, x〉 is continuous. Let S : X → 2^X, T : X → 2^F, η : X × X → F, and h : E × E → ℝ be the mappings such that

(a) S is upper semi-continuous such that each S(x) is compact and convex;

(b) h(X × X) is bounded;

(c) T is an (η, h)-pseudo-monotone type I operator and upper semi-continuous from co(A) to the σ〈F, E〉-topology on F for each $A\in \mathcal{F}\left(X\right)$ such that each T (x) is σ〈F, E〉 - compact and convex and T (X) is δ〈F, E〉 -bounded;

(d) T and η have the 0 - DCV R;

(e) for each fixed y ∈ X, x ↦ η(x, y), i.e., η(·, y) is continuous and x ↦ h(x, y), i.e., h(·, y) is lower semi-continuous on co(A) for each $A\in \mathcal{F}\left(X\right)$ and for each x ∈ X, h(x, ·) and η(x, ·) are concave, and h(x, x) = 0, η(x, x) = 0;

(f) the set Σ = {y ∈ X : sup_x∈S(y)[inf_w∈T(y)Re〈w, η(y, x)〉 + h(y, x)] > 0} is open in X.

Suppose further that there exist a non-empty compact convex subset K of X and a point x₀ ∈ X such that x₀ ∈ K ∩ S(y) and inf_w∈T(y)Re〈w, η(y, x₀)〉 + h(y, x₀) > 0 for all y ∈ X \ K. Then there exist a point ŷ ∈ X such that

• (1) ŷ ∈ S(ŷ);

• (2) there exists a point ŵ∈ T (ŷ) with Re 〈ŵ, η(ŷ, x)〉 + h(ŷ, x) ≤ 0 for all x ∈ S(ŷ).

Proof. Let us first show that there exist a point ŷ ∈ X such that ŷ ∈ S(ŷ) and

$$\underset{x\in S\left(ŷ\right)}{\mathsf{\text{sup}}}\left[\underset{w\in T\left(ŷ\right)}{\mathsf{\text{inf}}}\mathsf{\text{Re}}〈w,\eta \left(ŷ,x\right)〉+h\left(ŷ,x\right)\right]\le 0.$$

Now, we prove this by contradiction. So, we assume that, for each y ∈ X, either y ∉ S(y) or there exists x ∈ S(y) such that inf_w∈T(y)Re〈w, η(y, x) 〉 + h(y, x) > 0, that is, for each y ∈ X, either y ∉ S(y) or y ∈ Σ. If y ∉ 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

$$\mathsf{\text{Re}}〈p,y〉-\underset{x\in S\left(y\right)}{\mathsf{\text{sup}}}\mathsf{\text{Re}}〈p,x〉>0.$$

For each y ∈ X, set

$$\begin{array}{l}\gamma \left(y\right):=\underset{x\in S\left(y\right)}{\mathsf{\text{sup}}}\left[\underset{w\in T\left(y\right)}{\mathsf{\text{inf}}}\mathsf{\text{Re}}〈w,\eta \left(y,x\right)+h\left(y,x\right)\right],\\ V_0:=\sum =\left\{y\in X:\gamma \left(y\right)>0\right\}\end{array}$$

and, for each continuous linear functional p on E,

$$V_p:=\left\{y\in X:\mathsf{\text{Re}}〈p,y〉-\underset{x\in S\left(y\right)}{\mathsf{\text{sup}}}\mathsf{\text{Re}}〈p,x〉>0\right\}.$$

Then we have

$$X=V_0\cup \bigcup _{p\in LF\left(E\right)}{V}_p,$$

where LF (E) denotes the set of all continuous linear functionals on E. Since V₀ is open by hypothesis and each V _p is open in X by Lemma 2.1 ([[12], Lemma 1]), {V₀, V _p : p ∈ LF(E)} is an open covering for X. Since X is para-compact, there exists a continuous partition of unity {β₀, β _p : p ∈ LF(E)} for X subordinated to the open cover {V₀, V _p : p ∈ LF(E)}. Note that, for each y ∈ X and $A\in \mathcal{F}\left(X\right)$, x ↦ h(x, y), i.e., h(·, y) is continuous on co(A) (see [[15], Corollary 10.1.1]). Define a function ϕ : X × X → ℝ by

$$\begin{array}{cc}\varphi \left(x,y\right)=\hfill & {\beta }_0\left(y\right)\left[\underset{w\in T\left(y\right)}{\mathsf{\text{min}}}\mathsf{\text{Re}}〈w,\eta \left(y,x\right)〉+h\left(y,x\right)\right]\hfill \\ +\sum _{p\in LF\left(E\right)}{\beta }_p\left(y\right)\mathsf{\text{Re}}〈p,\eta \left(y,x\right)〉\hfill \end{array}$$

for all x, y ∈ X. Then we have the following:

• (I) Since E is Hausdorff, for each $A\in \mathcal{F}\left(X\right)$ and each fixed x ∈ co(A), the mapping

  $$y\mapsto \underset{w\in T\left(y\right)}{\mathsf{\text{inf}}}\mathsf{\text{Re}}〈w,\eta \left(y,x\right)〉+h\left(y,x\right)$$

is lower semi-continuous on co(A) by Lemma 2.3 and the fact that h is continuous on co(A) and so the mapping

$$y\mapsto {\beta }_0\left(y\right)\left[\underset{w\in T\left(y\right)}{\mathsf{\text{inf}}}\mathsf{\text{Re}}〈w,\eta \left(y,x\right)〉+h\left(y,x\right)\right]$$

is lower semi-continuous on co(A) by Lemma 2.2. Also, for each fixed x ∈ X,

$$y\mapsto \sum _{p\in LF\left(E\right)}{\beta }_p\left(y\right)\mathsf{\text{Re}}〈p,\eta \left(y,x\right)〉$$

is continuous on X. Hence, for each $A\in \mathcal{F}\left(X\right)$ and each fixed x ∈ co(A), the mapping y ↦ϕ (x, y) is lower semi-continuous on co(A).

• (II) For each $A\in \mathcal{F}\left(X\right)$ and y ∈ co ( A ), min_x∈Aϕ (x, y) ≤ 0. Indeed, if this were false, then, for some $A=\left\{x_1,x_2,...,x_n\right\}\in \mathcal{F}\left(X\right)$ and some y ∈ co(A) (say $y={\sum }_{i=1}^n{\lambda }_i{x}_i,$ where λ₁, λ₂, . . ., λ _n ≥ 0 with ${\sum }_{i=1}^n{\lambda }_i=1$), we have min_1≤i≤nϕ (x _i , y) > 0. Then, for each i = 1, 2, . . ., n,

  $${\beta }_0\left(y\right)\left[\underset{w\in T\left(y\right)}{\mathsf{\text{inf}}}\mathsf{\text{Re}}〈w,\eta \left(y,x_i\right)〉+h\left(y,x_i\right)\right]+\sum _{p\in LF\left(E\right)}{\beta }_p\left(y\right)\mathsf{\text{Re}}〈p,\eta \left(y,x_i\right)〉>0$$

and so

$$\begin{array}{cc}0& =\varphi \left(y,y\right)\hfill \\ ={\beta }_0\left(y\right)\left[\underset{w\in T\left(y\right)}{\mathsf{\text{inf}}}\mathsf{\text{Re}}〈w,\eta \left(y,\sum _{i=1}^n{\lambda }_i{x}_i\right)〉+h\left(y,\sum _{i=1}^n{\lambda }_i{x}_i\right)\right]\hfill \\ +\sum _{p\in LF\left(E\right)}{\beta }_p\left(y\right)\mathsf{\text{Re}}〈p,\eta \left(y,\sum _{i=1}^n{\lambda }_i{x}_i\right)〉\hfill \\ \ge \sum _{i=1}^n{\lambda }_i\left({\beta }_0\left(y\right.\right)\left[\underset{w\in T\left(y\right)}{\mathsf{\text{inf}}}\mathsf{\text{Re}}〈w,\eta \left(y,x_i\right)〉+h\left(y,x_i\right)\right]\hfill \\ +\sum _{p\in LF\left(E\right)}{\beta }_p\left(y\right)\mathsf{\text{Re}}〈p,\eta \left(y,x_i\right)〉\hfill \\ >0,\hfill \end{array}$$

which is a contradiction.

• (III) Suppose that $A\in \mathcal{F}\left(X\right)$, x, y ∈ co(A) and {y _α }_α∈Γis a net in X converging to y with ϕ(tx + (1 - t)y, y _α ) ≤ 0 for all α ∈ Γ and all t ∈ [0, 1].

Case 1: β₀(y) = 0. Note that β₀(y _α ) ≥ 0 for each α ∈ Γ and β₀(y _α ) → 0. Since T(X) is strongly bounded and {y _α }_α∈Γis a bounded net, it follows that

$$\underset{\alpha }{\mathsf{\text{lim}}\mathsf{\text{sup}}}\left[{\beta }_0\left(y_{\alpha }\right)\underset{w\in T\left(y_{\alpha }\right)}{\mathsf{\text{min}}}\mathsf{\text{Re}}〈w,\eta \left(y_{\alpha },x\right)〉+h\left(y_{\alpha },x\right)\right]=0.$$

(3.1)

Also, we have

$${\beta }_0\left(y\right)\left[\underset{w\in T\left(y\right)}{\mathsf{\text{min}}}\mathsf{\text{Re}}〈w,\eta \left(y,x\right)〉+h\left(y,x\right)\right]=0.$$

Thus it follows from (3.1) that

$$\begin{array}{l}\underset{\alpha }{\mathsf{\text{lim}}\mathsf{\text{sup}}}\left[{\beta }_0\left(y_{\alpha }\right)\underset{w\in T\left(y_{\alpha }\right)}{\mathsf{\text{min}}}\mathsf{\text{Re}}〈w,\eta \left(y_{\alpha },x\right)〉+h\left(y_{\alpha },x\right)\right]+\sum _{p\in LF\left(E\right)}{\beta }_p\left(y\right)\mathsf{\text{Re}}〈p,\eta \left(y,x\right)〉\\ =\sum _{p\in LF\left(E\right)}{\beta }_p\left(y\right)\mathsf{\text{Re}}〈p,\eta \left(y,x\right)〉\\ ={\beta }_0\left(y\right)\left[\underset{w\in T\left(y\right)}{\mathsf{\text{min}}}\mathsf{\text{Re}}〈w,\eta \left(y,x\right)〉+h\left(y,x\right)\right]+\sum _{p\in LF\left(E\right)}{\beta }_p\left(y\right)\mathsf{\text{Re}}〈p,\eta \left(y,x\right)〉.\end{array}$$

(3.2)

When t = 1, we have ϕ(x, y _α ) ≤ 0 for all α ∈ Γ, i.e.,

$${\beta }_0\left(y_{\alpha }\right)\left[\underset{w\in T\left(y_{\alpha }\right)}{\mathsf{\text{min}}}\mathsf{\text{Re}}〈w,\eta \left(y_{\alpha },x\right)〉+h\left(y_{\alpha },x\right)\right]+\sum _{p\in LF\left(E\right)}{\beta }_p\left(y_{\alpha }\right)\mathsf{\text{Re}}〈p,\eta \left(y_{\alpha },x\right)〉\le 0$$

(3.3)

for all α ∈ Γ. Therefore, by (3.3), we have

$$\begin{array}{l}\underset{\alpha }{\mathsf{\text{lim}}\mathsf{\text{sup}}}\left[{\beta }_0\left(y_{\alpha }\right)\underset{w\in T\left(y_{\alpha }\right)}{\mathsf{\text{min}}}\mathsf{\text{Re}}〈w,\eta \left(y_{\alpha },x\right)〉+h\left(y_{\alpha },x\right)\right]\\ +\underset{\alpha }{\mathsf{\text{lim}}\mathsf{\text{inf}}}\left[\sum _{p\in LF\left(E\right)}{\beta }_p\left(y_{\alpha }\right)\mathsf{\text{Re}}〈p,\eta \left(y_{\alpha },x\right)〉\right]\\ \le \underset{\alpha }{\mathsf{\text{lim}}\mathsf{\text{sup}}}\left[{\beta }_0\left(y_{\alpha }\right)\underset{w\in T\left(y_{\alpha }\right)}{\mathsf{\text{min}}}\mathsf{\text{Re}}〈w,\eta \left(y_{\alpha },x\right)〉+h\left(y_{\alpha },x\right)+\sum _{p\in LF\left(E\right)}{\beta }_p\left(y_{\alpha }\right)\mathsf{\text{Re}}〈p,\eta \left(y_{\alpha },x\right)〉\right]\\ \le 0\end{array}$$

and so

$$\underset{\alpha }{\mathsf{\text{lim}}\mathsf{\text{sup}}}\left[{\beta }_0\left(y_{\alpha }\right)\underset{w\in T\left(y_{\alpha }\right)}{\mathsf{\text{min}}}\mathsf{\text{Re}}〈w,\eta \left(y_{\alpha },x\right)〉+h\left(y_{\alpha },x\right)\right]+\sum _{p\in LF\left(E\right)}^n{\beta }_p\left(y\right)\mathsf{\text{Re}}〈p,\eta \left(y,x\right)〉\le 0.$$

(3.4)

Hence, by (3.2) and (3.4), we have ϕ(x, y) ≤ 0.

Case 2. β₀(y) > 0. Since β₀(y _α ) → β₀(y), there exists λ ∈ Γ such that β₀(y _α ) > 0 for all α ≥ λ. When t = 0, we have ϕ(y, y _α ) ≤ 0 for all α ∈ Γ, i.e.,

$${\beta }_0\left(y_{\alpha }\right)\left[\underset{w\in T\left(y_{\alpha }\right)}{\mathsf{\text{inf}}}\mathsf{\text{Re}}〈w,\eta \left(y_{\alpha },y\right)〉+h\left(y_{\alpha },y\right)\right]+\sum _{p\in LF\left(E\right)}{\beta }_p\left(y_{\alpha }\right)\mathsf{\text{Re}}〈p,\eta \left(y_{\alpha },y\right)〉\le 0$$

for all α ∈ Γ and so

$$\underset{\alpha }{\mathsf{\text{lim}}\mathsf{\text{sup}}}\left[{\beta }_0\left(y_{\alpha }\right)\underset{w\in T\left(y_{\alpha }\right)}{\mathsf{\text{inf}}}\mathsf{\text{Re}}〈w,\eta \left(y_{\alpha },y\right)〉+h\left(y_{\alpha },y\right)+\sum _{p\in LF\left(E\right)}{\beta }_p\left(y_{\alpha }\right)\mathsf{\text{Re}}〈p,\eta \left(y_{\alpha },y\right)〉\right]\le 0.$$

(3.5)

Hence, by (3.5), we have

$$\begin{array}{l}\underset{\alpha }{\mathsf{\text{lim}}\mathsf{\text{sup}}}\left[{\beta }_0\left(y_{\alpha }\right)\underset{w\in T\left(y_{\alpha }\right)}{\mathsf{\text{inf}}}\mathsf{\text{Re}}〈w,\eta \left(y_{\alpha },y\right)〉+h\left(y_{\alpha },y\right)\right]\\ +\underset{\alpha }{\mathsf{\text{lim}}\mathsf{\text{inf}}}\left[\sum _{p\in LF\left(E\right)}{\beta }_p\left(y_{\alpha }\right)\mathsf{\text{Re}}〈p,\eta \left(y_{\alpha },y\right)〉\right]\\ \le \underset{\alpha }{\mathsf{\text{lim}}\mathsf{\text{sup}}}\left[{\beta }_0\left(y_{\alpha }\right)\underset{w\in T\left(y_{\alpha }\right)}{\mathsf{\text{inf}}}\mathsf{\text{Re}}〈w,\eta \left(y_{\alpha },y\right)〉+h\left(y_{\alpha },y\right)+\sum _{p\in LF\left(E\right)}{\beta }_p\left(y_{\alpha }\right)\mathsf{\text{Re}}〈p,\eta \left(y_{\alpha },y\right)〉\right]\\ \le 0.\end{array}$$

Since $\mathsf{\text{lim}}\underset{\alpha }{\mathsf{\text{inf}}}\left[{\sum }_{p\in LF\left(E\right)}{\beta }_p\left(y_{\alpha }\right)\mathsf{\text{Re}}〈p,\eta \left(y_{\alpha },y\right)〉\right]=0,$ we have

$$\underset{\alpha }{\mathsf{\text{lim}}\mathsf{\text{sup}}}\left[{\beta }_0\left(y_{\alpha }\right)\underset{w\in T\left(y_{\alpha }\right)}{\mathsf{\text{min}}}\mathsf{\text{Re}}〈w,\eta \left(y_{\alpha },y\right)〉+h\left(y_{\alpha },y\right)\right]\le 0.$$

(3.6)

Since β₀(y _α ) > 0 for all α ≥ λ, it follows that

$$\begin{array}{l}{\beta }_0\left(y\right)\underset{\alpha }{\mathsf{\text{lim}}\mathsf{\text{sup}}}\left[\underset{w\in T\left(y_{\alpha }\right)}{\mathsf{\text{min}}}\mathsf{\text{Re}}〈w,\eta \left(y_{\alpha },y\right)〉+h\left(y_{\alpha },y\right)\right]\\ =\underset{\alpha }{\mathsf{\text{lim}}\mathsf{\text{sup}}}\left[{\beta }_0\left(y_{\alpha }\right)\underset{w\in T\left(y_{\alpha }\right)}{\mathsf{\text{min}}}\mathsf{\text{Re}}〈w,\eta \left(y_{\alpha },y\right)〉+h\left(y_{\alpha },y\right)\right].\end{array}$$

(3.7)

Since β₀(y) > 0, by (3.6) and (3.7), we have

$$\underset{\alpha }{\mathsf{\text{lim}}\mathsf{\text{sup}}}\left[\underset{w\in T\left(y_{\alpha }\right)}{\mathsf{\text{min}}}\mathsf{\text{Re}}〈w,\eta \left(y_{\alpha },y\right)〉+h\left(y_{\alpha },y\right)\right]\le 0.$$

Since T is an (η, h)-pseudo-monotone type I operator, we have

$$\begin{array}{l}\underset{\alpha }{\mathsf{\text{lim}}\mathsf{\text{sup}}}\left[\underset{w\in T\left(y_{\alpha }\right)}{\mathsf{\text{min}}}\mathsf{\text{Re}}〈w,\eta \left(y_{\alpha },x\right)〉+h\left(y_{\alpha },x\right)\right]\\ \ge \underset{w\in T\left(y\right)}{\mathsf{\text{min}}}\mathsf{\text{Re}}〈w,\eta \left(y,x\right)〉+h\left(y,x\right)\end{array}$$

for all x ∈ X. Since β₀(y) > 0, we have

$$\begin{array}{l}{\beta }_0\left(y\right)\left[\underset{\alpha }{\mathsf{\text{lim}}\mathsf{\text{sup}}}\underset{w\in T\left(y_{\alpha }\right)}{\mathsf{\text{min}}}\mathsf{\text{Re}}〈w,\eta \left(y_{\alpha },x\right)〉+h\left(y_{\alpha },x\right)\right]\\ \ge {\beta }_0\left(y\right)[\underset{w\in T\left(y\right)}{\mathsf{\text{min}}}\mathsf{\text{Re}}〈w,\eta \left(y,x\right)〉+h\left(y,x\right)\end{array}$$

and thus

$$\begin{array}{l}{\beta }_0\left(y\right)\left[\underset{\alpha }{\mathsf{\text{lim}}\mathsf{\text{sup}}}\underset{w\in T\left(y_{\alpha }\right)}{\mathsf{\text{min}}}\mathsf{\text{Re}}〈w,\eta \left(y_{\alpha },x\right)〉+h\left(y_{\alpha },x\right)\right]+\sum _{p\in LF\left(E\right)}{\beta }_p\left(y\right)\mathsf{\text{Re}}〈p,\eta \left(y,x\right)〉\\ \ge {\beta }_0\left(y\right)\left[\underset{w\in T\left(y\right)}{\mathsf{\text{min}}}\mathsf{\text{Re}}〈w,\eta \left(y,x\right)〉+h\left(y,x\right)\right]+\sum _{p\in LF\left(E\right)}{\beta }_p\left(y\right)\mathsf{\text{Re}}〈p,\eta \left(y,x\right)〉.\end{array}$$

(3.8)

When t = 1, we have ϕ (x, y _α ) ≤ 0 for all α ∈ Γ, i.e.,

$$\begin{array}{l}{\beta }_0\left(y_{\alpha }\right)\left[\underset{w\in T\left(y_{\alpha }\right)}{\mathsf{\text{min}}}\mathsf{\text{Re}}〈w,\eta \left(y_{\alpha },x\right)〉+h\left(y_{\alpha },x\right)\right]+\sum _{p\in LF\left(E\right)}{\beta }_p\left(y_{\alpha }\right)\mathsf{\text{Re}}〈p,\eta \left(y_{\alpha },x\right)〉\\ \le 0\end{array}$$

for all α ∈ Γ and so, by (3.8),

$$\begin{array}{l}0\ge \underset{\alpha }{\mathsf{\text{lim}}\mathsf{\text{sup}}}\left[{\beta }_0\left(y_{\alpha }\right)\underset{w\in T\left(y_{\alpha }\right)}{\mathsf{\text{min}}}\mathsf{\text{Re}}〈w,\eta \left(y_{\alpha },x\right)〉+h\left(y_{\alpha },x\right)\right.\\ +\left.\sum _{p\in LF\left(E\right)}{\beta }_p\left(y_{\alpha }\right)\mathsf{\text{Re}}〈p,\eta \left(y_{\alpha },x\right)〉\right]\\ \ge \underset{\alpha }{\mathsf{\text{lim}}\mathsf{\text{sup}}}\left[{\beta }_0\left(y_{\alpha }\right)\underset{w\in T\left(y_{\alpha }\right)}{\mathsf{\text{min}}}\mathsf{\text{Re}}〈w,\eta \left(y_{\alpha },x\right)〉+h\left(y_{\alpha },x\right)\right]\\ +\underset{\alpha }{\mathsf{\text{lim}}\mathsf{\text{inf}}}\left[\sum _{p\in LF\left(E\right)}{\beta }_p\left(y_{\alpha }\right)\mathsf{\text{Re}}〈p,\eta \left(y_{\alpha },x\right)〉\right]\\ ={\beta }_0\left(y\right)\left[\underset{\alpha }{\mathsf{\text{lim}}\mathsf{\text{sup}}}\underset{w\in T\left(y_{\alpha }\right)}{\mathsf{\text{min}}}\mathsf{\text{Re}}〈w,\eta \left(y_{\alpha },x\right)〉+h\left(y_{\alpha },x\right)\right]\\ +\sum _{p\in LF\left(E\right)}{\beta }_p\left(y\right)\mathsf{\text{Re}}〈p,\eta \left(y,x\right)〉\\ \ge {\beta }_0\left(y\right)\left[\underset{w\in T\left(y\right)}{\mathsf{\text{min}}}\mathsf{\text{Re}}〈w,\eta \left(y,x\right)〉+h\left(y,x\right)\right]\\ +\sum _{p\in LF\left(E\right)}{\beta }_p\left(y\right)\mathsf{\text{Re}}〈p,\eta \left(y,x\right)〉.\end{array}$$

(3.9)

Hence we have ϕ (x, y) ≤ 0.

• (IV) By hypothesis, there exists a non-empty compact (and so closed) subset K of X and a point x₀ ∈ X such that

  $$x_0\in K\cap S\left(y\right),\underset{w\in T\left(y\right)}{\mathsf{\text{inf}}}\left[\mathsf{\text{Re}}〈w,\eta \left(y,x_0\right)〉+h\left(y,x_0\right)\right]>0$$

for all y ∈ X \ K. Thus, for all y ∈ X \ K, β₀(y)[inf_w∈T(y)Re〈w, η(y, x₀)〉 + h(y, x₀)] > 0 whenever β₀(y) > 0 and Re〈p, η(y, x₀)〉 > 0, whenever β _p (y) > 0 for any p ∈ LF(E). Consequently, we have

$$\varphi \left(x_0,y\right)={\beta }_0\left(y\right)\left[\underset{w\in T\left(y\right)}{\mathsf{\text{inf}}}\mathsf{\text{Re}}〈w,\eta \left(y,x_0\right)〉+h\left(y,x_0\right)\right]+\sum _{p\in LF\left(E\right)}{\beta }_p\left(y\right)\mathsf{\text{Re}}〈p,\eta \left(y,x_0\right)〉>0$$

for all y ∈ X \ K and so ϕ satisfies all the hypotheses of Theorem 1.1. Hence, by Theorem 1.1, there exists a point ŷ ∈ K such that ϕ(x, ŷ) ≤ 0 for all x ∈ X, i.e.,

$${\beta }_0\left(ŷ\right)\left[\underset{w\in T\left(ŷ\right)}{\mathsf{\text{inf}}}\mathsf{\text{Re}}〈w,\eta \left(ŷ,x\right)〉+h\left(ŷ,x\right)\right]+\sum _{p\in LF\left(E\right)}{\beta }_p\left(ŷ\right)\mathsf{\text{Re}}〈p,\eta \left(ŷ,x\right)〉0$$

(3.10)

for all x ∈ X.

Now, the rest of the proof of this part is similar to the proof in Step 1 of [16, Theorem 2.1]. Hence we have shown that there exist a point ŷ ∈ X such that ŷ ∈ S(ŷ) and

$$\underset{x\in S\left(ŷ\right)}{\mathsf{\text{sup}}}\left[\underset{w\in T\left(ŷ\right)}{\mathsf{\text{inf}}}\mathsf{\text{Re}}〈w,\eta \left(ŷ,x\right)〉+h\left(ŷ,x\right)\right]\le 0.$$

By following the proof of Step 2 in [16, Theorem 2.1] and applying Theorem 2.1 (Keneser's Minimax Theorem) above, we can show that there exist a point ŵ ∈ T(ŷ) such that Re〈ŵ, η(ŷ, x)〉 + h(ŷ, x) ≤ 0 for all x ∈ S(ŷ). This completes the proof.

When X is compact, we obtain the following immediate consequence of Theorem 3.1:

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 σ〈F, E〉 - topology where 〈·, ·〉 : F × E → Φ is a bilinear functional such that, for each w ∈ F, the function x ↦ Re〈w, x〉 is continuous. Let S : X → 2^X, T : X → 2^F, η : X × X → F, and h : E × E → ℝ be the mappings such that

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

(b) h(X × X) is bounded;

(c) T is an (η, h)-pseudo-monotone type I operator and is upper semi-continuous from co(A) to the σ〈F, E〉-topology on F for each $A\in \mathcal{F}\left(X\right)$ such that each T (x) is σ〈F, E〉-compact and convex and T (X) is δ〈F, E〉 -bounded;

(d) T and η have the 0 - DCV R;

(e) for each fixed y ∈ X, x ↦ η(x, y), i.e., η(·, y) is continuous and x ↦ h(x, y), i.e., h(·, y) is lower semi-continuous on co(A) for each $A\in \mathcal{F}\left(X\right)$ and, for each x ∈ X, h(x, ·) and η(x, ·) are concave and h(x, x) = 0, η(x, x) = 0;

(f) the set Σ = {y ∈ X : sup_x∈S(y)[inf_w∈T(y)Re〈w, η(y, x)〉 + h(y, x)] > 0} is open in X.

Then there exist a point ŷ ∈ X such that

ŷ ∈ S(ŷ);

there exists a point ŵ ∈ T (ŷ) with Re〈ŵ, η(ŷ, x)〉 + h(ŷ, x) ≤ 0 for all x ∈ S(ŷ).

Note that, if the mapping S : X → 2^X is, in addition, lower semi-continuous and, for each y ∈ Σ, T is upper semi-continuous at y in X, then the set Σ in Theorem 3.1 is always open in X and so we obtain the following theorem:

Theorem 3.3. Let E be a locally convex Hausdorff topological vector space over Φ, X be a nonempty para-compact convex and bounded subset of E and F be a vector space over Φ with σ 〈F, E〉-topology, where 〈·, ·〉 : F × E → Φ is a bilinear functional such that, for each w ∈ F, the function x ↦ Re〈w, x〉 is continuous. Let S : X → 2^X , T : X → 2^F, η : X × X → F and h : E × E → ℝ be mappings such that

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

(b) h(X × X) is bounded;

(c) T is an (η, h)-pseudo-monotone type I operator and is upper semi-continuous from co(A) to the σ〈F, E〉-topology on F for each $A\in \mathcal{F}\left(X\right)$ such that each T (x) is σ〈F, E〉-compact and convex and T (X) is δ〈F, E〉-bounded;

(d) T and η have the 0 - DCV R;

(e) for each fixed y ∈ X, x ↦ η(x, y), i.e., η(·, y) is continuous and x ↦ h(x, y), i.e., h(·, y) is lower semi-continuous on co(A) for each $A\in \mathcal{F}\left(X\right)$ and, for each x ∈ X, h(x, ·) and η(x, ·) are concave and h(x, x) = 0, η(x, x) = 0.

Suppose that, for each y ∈ Σ = {y ∈ X : sup_x∈S(y)[inf_w∈T(y)Re〈w, η(y, x)〉 + h(y, x)] > 0}, T is upper semi-continuous at y from the relative topology on X to the δ〈F, E〉-topology on F. Further, suppose that there exist a non-empty compact convex subset K of X and a point x₀ ∈ X such that

$$x_0\in K\cap S\left(y\right),\underset{w\in T\left(y\right)}{\mathsf{\text{inf}}}\mathsf{\text{Re}}〈w,\eta \left(y,x_0\right)〉+h\left(y,x_0\right)>0$$

for all y ∈ X \ K. Then there exist a point ŷ ∈ X such that

• (1) ŷ ∈ S(ŷ);

• (2) there exists a point ŵ ∈ T (ŷ) with Re 〈ŵ, η(ŷ, x)〉 + h(ŷ, x) ≤ 0 for all x ∈ S(ŷ).

The proof is exactly similar to the proof of Theorems 2.3 and 3.3 in [16] and so is omitted.

When X is compact, we obtain the following theorem:

Theorem 3.4. Let E be a locally convex Hausdorff topological vector space over Φ, X be a non-empty compact convex subset of E and F be a vector space over Φ with σ〈F, E〉-topology where 〈·, ·〉 : F × E → Φ is a bilinear functional such that for each w ∈ F, the function x ↦ Re 〈w, x〉 is continuous. Let S : X → 2^X, T : X → 2^F, η : X × X → F, and h : E × E → ℝ be mappings such that

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

(b) h(X × X) is bounded;

(c) T is an (η, h)-pseudo-monotone type I operator and is upper semi-continuous from co(A) to the σ〈F, E〉-topology on F for each $A\in \mathcal{F}\left(X\right)$ such that each T (x) is σ〈F, E〉-compact and convex and T (X) is δ〈F, E〉-bounded;

(d) T and η have the 0 - DCV R;

(e) for each fixed y ∈ X, x ↦ η(x, y), i.e., η(·, y) is continuous and x ↦ h(x, y), i.e., h(·, y) is lower semi-continuous on co(A) for each $A\in \mathcal{F}\left(X\right)$ and, for each x ∈ X, h(x, ·) and η(x, ·) are concave and h(x, x) = 0, η(x, x) = 0.

Suppose that, for each y ∈ Σ = {y ∈ X : sup_x∈S(y)[inf_w∈T(y)Re 〈w, η(y, x)〉 + h(y, x)] > 0}, T is upper semi-continuous at y from the relative topology on X to the δ〈F, E〉-topology on F. Then there exist a point ŷ ∈ X such that

• (1) ŷ ∈ S(ŷ);

• (2) there exists a point ŵ ∈ T (ŷ) with Re 〈ŵ, η(ŷ, x)〉 + h(ŷ, x) ≤ 0 for all x ∈ S(ŷ).

Remark 3.1. (1) Theorems 3.1 and 3.3 of this article are further generalizations of the results obtained in [16, Theorems 3.1 and 3.3], respectively, into generalized quasivariational-like inequalities of (η, h)-pseudo-monotone type I operators on non-compact sets.

• (2) 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 semicontinuous. Our present article is another extension of the original study in [1] using (η, h)-pseudo-monotone type I operators on non-compact sets.

• (3) The results in [16] were obtained on non-compact sets where one of the set-valued mappings is a pseudo-monotone type I operators which were defined first in [6] and later renamed by pseudo-monotone type I operators in [7]. Our present results are extensions of the results in [16] using an extension of the operators defined in [7] (and originally in [6]).