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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 2X the family of all non-empty subsets of X and by F x 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 or the complex field .

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 : supxA|〈y, x| < }. Let σF, 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 δF, 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 σF, E〉 or the (strong) topology δF, 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 }αΓin F and y F,

(a) y α → y in σF, E〉 if and only if 〈y α , xy, x〉 for each x E;

(b) y α → y in δF, E〉 if and only if 〈y α , xy, 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 → 2X , T : X → 2F are two set-valued mappings, f : X → F , η : X ×X → E are two single-valued mappings and h : X × X → 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 ŷS ŷ and ŵT ŷ such that

Re ŵ , ŷ - x 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 ŷS ŷ and ŵT ŷ such that

Re ŵ , η ŷ , x + h ŷ , x 0

for all x S(ŷ).

For more results related to the generalized quasi-variational-like inequality problem, refer to [25], 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 σF, E〉-topology, where 〈·, ·〉 : F ×E → Φ is a bilinear functional. Let h : X × X → , η : X × XE, and T : X → 2F 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 }αΓin X converging to y (respectively, weakly to y) with

lim sup α inf u T y α Re u , η y α , y + h y α , y 0 ,

we have

lim sup α inf u T y α Re u , η y α , y + h y α , x inf w T y Re w , η y , x + h y , x

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 : X , h x , y = h x - h y 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 → {-∞, + } be such that

(a) for each A F X and each fixed x co(A), y f(x, y) is lower semi-continuous on co(A);

(b) for each A F X and y co(A), minxAf(x, y) 0;

(c) for each A F X and x, y co(A), every net {y α }αΓin 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 → {±∞} 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 i = 1 n λ i = 1 , we have i = 1 n λ i ϕ y , x i 0 , where y= i = 1 n λ 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 → 2F, η : X × X → E, g : X → E be mappings.

  1. (1)

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

    ϕ x , y = inf w T x Re w , η x , y

is 0-DCV in y;

  1. (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 → 2E 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 defined by f p (y) = supxS(y)Re〈p, xis upper semi-continuous, i.e., for each λ , the set {y X : f p (y) = supxS(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 be non-negative and continuous and g : Y → be lower semi-continuous. Then the mapping F : X × Y → 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 F E and X = co(A). Let F be a vector space over Φ which is equipped with σF, E-topology such that, for each w F, x w, xis continuous. Let η : X × XE be continuous in the first argument. Let T : X 2F \ be upper semi-continuous from X to the σF, E-topology on F such that each T(x) is σF, E-compact. Let f : X × X → be defined by f(x, y) = infwT(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

min y Y sup x X f x , y = sup x X min y Y f x , y .

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 σF, E-topology, where 〈·, ·〉 : F × E → Φ is a bilinear functional such that, for each w F, the function x Re〈w, xis continuous. Let S : X → 2X, T : X → 2F, η : X × X → F, and h : E × E 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 σF, E-topology on F for each A F X such that each T (x) is σF, E- compact and convex and T (X) is δF, 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 F X 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 : supxS(y)[infwT(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 infwT(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. (1)

    ŷ S(ŷ);

  2. (2)

    there exists a point ŵ T (ŷ) with Re 〈ŵ, η, 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

sup x S ŷ inf w T ŷ Re w , η ŷ , x + h ŷ , x 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 infwT(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

Re p , y - sup x S y Re p , x > 0 .

For each y X, set

γ y : = sup x S y inf w T y Re w , η y , x + h y , x , V 0 : = = y X : γ y > 0

and, for each continuous linear functional p on E,

V p : = y X : Re p , y - sup x S y Re p , x > 0 .

Then we have

X = V 0 p L F E 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 F X , 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 → by

ϕ x , y = β 0 y min w T y Re w , η y , x + h y , x + p L F E β p y Re p , η y , x

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

  1. (I)

    Since E is Hausdorff, for each A F X and each fixed x co(A), the mapping

    y inf w T y Re w , η y , x + h y , x

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 β 0 y inf w T y Re w , η y , x + h y , x

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

y p L F E β p y Re p , η y , x

is continuous on X. Hence, for each A F X and each fixed x co(A), the mapping y ϕ (x, y) is lower semi-continuous on co(A).

  1. (II)

    For each A F X and y co ( A ), minxAϕ (x, y) 0. Indeed, if this were false, then, for some A = x 1 , x 2 , . . . , x n F X and some y co(A) (say y = i = 1 n λ i x i , where λ1, λ2, . . ., λ n ≥ 0 with i = 1 n λ i = 1 ), we have min1≤inϕ (x i , y) > 0. Then, for each i = 1, 2, . . ., n,

    β 0 y inf w T y Re w , η y , x i + h y , x i + p L F E β p y Re p , η y , x i > 0

and so

0 = ϕ y , y = β 0 y inf w T y Re w , η ( y , i = 1 n λ i x i ) + h y , i = 1 n λ i x i + p L F E β p y Re p , η ( y , i = 1 n λ i x i ) i = 1 n λ i β 0 y inf w T y Re w , η y , x i + h y , x i + p L F E β p y Re p , η y , x i > 0 ,

which is a contradiction.

  1. (III)

    Suppose that A F X , x, y co(A) and {y α }αΓis 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 }αΓis a bounded net, it follows that

lim sup α β 0 y α min w T y α Re w , η y α , x + h y α , x = 0 .
(3.1)

Also, we have

β 0 y min w T y Re w , η y , x + h y , x = 0 .

Thus it follows from (3.1) that

lim sup α β 0 y α min w T y α Re w , η y α , x + h y α , x + p L F E β p y Re p , η y , x = p L F E β p y Re p , η y , x = β 0 y min w T y Re w , η y , x + h y , x + p L F E β p y Re p , η y , x .
(3.2)

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

β 0 y α min w T y α Re w , η y α , x + h y α , x + p L F E β p y α Re p , η y α , x 0
(3.3)

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

lim sup α β 0 y α min w T y α Re w , η y α , x + h y α , x + lim inf α p L F E β p y α Re p , η y α , x lim sup α β 0 y α min w T y α Re w , η y α , x + h y α , x + p L F E β p y α Re p , η y α , x 0

and so

lim sup α β 0 y α min w T y α Re w , η y α , x + h y α , x + p L F E n β p y Re p , η y , x 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.,

β 0 y α inf w T y α Re w , η y α , y + h y α , y + p L F E β p y α Re p , η y α , y 0

for all α Γ and so

lim sup α β 0 y α inf w T y α Re w , η y α , y + h y α , y + p L F E β p y α Re p , η y α , y 0 .
(3.5)

Hence, by (3.5), we have

lim sup α β 0 y α inf w T y α Re w , η y α , y + h y α , y + lim inf α p L F E β p y α Re p , η y α , y lim sup α β 0 y α inf w T y α Re w , η y α , y + h y α , y + p L F E β p y α Re p , η y α , y 0 .

Since lim inf α [ p L F E β p y α Re p , η y α , y ] = 0 , we have

lim sup α β 0 y α min w T y α Re w , η y α , y + h y α , y 0 .
(3.6)

Since β0(y α ) > 0 for all αλ, it follows that

β 0 y lim sup α min w T y α Re w , η y α , y + h y α , y = lim sup α β 0 y α min w T y α Re w , η y α , y + h y α , y .
(3.7)

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

lim sup α min w T y α Re w , η y α , y + h y α , y 0 .

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

lim sup α min w T y α Re w , η y α , x + h y α , x min w T y Re w , η y , x + h y , x

for all x X. Since β0(y) > 0, we have

β 0 y lim sup α min w T y α Re w , η y α , x + h y α , x β 0 y [ min w T y Re w , η y , x + h y , x

and thus

β 0 y lim sup α min w T y α Re w , η y α , x + h y α , x + p L F E β p y Re p , η y , x β 0 y min w T y Re w , η y , x + h y , x + p L F E β p y Re p , η y , x .
(3.8)

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

β 0 y α min w T y α Re w , η y α , x + h y α , x + p L F E β p y α Re p , η y α , x 0

for all α Γ and so, by (3.8),

0 lim sup α β 0 y α min w T y α Re w , η y α , x + h y α , x + p L F E β p y α Re p , η y α , x lim sup α β 0 y α min w T y α Re w , η y α , x + h y α , x + lim inf α p L F E β p y α Re p , η y α , x = β 0 y lim sup α min w T y α Re w , η y α , x + h y α , x + p L F E β p y Re p , η y , x β 0 y min w T y Re w , η y , x + h y , x + p L F E β p y Re p , η y , x .
(3.9)

Hence we have ϕ (x, y) ≤ 0.

  1. (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 K S y , inf w T y [ Re w , η y , x 0 + h y , x 0 ] > 0

for all y X \ K. Thus, for all y X \ K, β0(y)[infwT(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

ϕ x 0 , y = β 0 y inf w T y Re w , η y , x 0 + h y , x 0 + p L F E β p y Re p , η y , x 0 > 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.,

β 0 ŷ inf w T ŷ Re w , η ŷ , x + h ŷ , x + p L F E β p ŷ Re p , η ŷ , x 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

sup x S ŷ inf w T ŷ Re w , η ŷ , x + h ŷ , x 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 σF, E- topology where 〈·, ·〉 : F × E → Φ is a bilinear functional such that, for each w F, the function x Re〈w, xis continuous. Let S : X → 2X, T : X → 2F, η : X × X → F, and h : E × E → 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 σF, E-topology on F for each A F X such that each T (x) is σF, E-compact and convex and T (X) is δF, 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 F X 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 : supxS(y)[infwT(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 → 2X 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 σF, E-topology, where 〈·, ·〉 : F × E → Φ is a bilinear functional such that, for each w F, the function x Re〈w, xis continuous. Let S : X → 2X , T : X → 2F, η : X × XF and h : E × E 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 σF, E-topology on F for each A F X such that each T (x) is σF, E-compact and convex and T (X) is δF, 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 F X 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 : supxS(y)[infwT(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 δF, 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 K S y , inf w T y Re w , η y , x 0 + h y , x 0 > 0

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

  1. (1)

    ŷ S(ŷ);

  2. (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 σF, E-topology where 〈·, ·〉 : F × E → Φ is a bilinear functional such that for each w F, the function x Re 〈w, xis continuous. Let S : X → 2X, T : X → 2F, η : X × XF, and h : E × E → 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 σF, E-topology on F for each A F X such that each T (x) is σF, E-compact and convex and T (X) is δF, 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 F X 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 : supxS(y)[infwT(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 δF, E-topology on F. Then there exist a point ŷ X such that

  1. (1)

    ŷ S(ŷ);

  2. (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.

  1. (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.

  2. (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]).

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Acknowledgment

This study was supported by the Korea Research Foundation Grant funded by the Korean Government (KRF-2008-313-C00050).

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Correspondence to Yeol Je Cho.

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The authors declare that they have no competing interests.

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All authors read and approved the final manuscript.

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Keywords

  • generalized quasi-variational-like inequalities
  • η-h-pseudo-monotone type I operators
  • locally convex Hausdorff topological vector spaces