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Existence theorems of generalized quasi-variational-like inequalities for pseudo-monotone type II operators

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 2 X the family of all nonempty subsets of X and by 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 ,:F×EΦ be a bilinear functional. Throughout this paper, Φ denotes either the real field or the complex field .

For each x 0 E, each nonempty subset A of E and each ϵ>0, let W( x 0 ;ϵ):={yF:|y, x 0 |<ϵ} and U(A;ϵ):={yF: sup x A |y,x|<ϵ}. Let σF,E be the (weak) topology on F generated by the family {W(x;ϵ):xE,ϵ>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;ϵ):A is a nonempty bounded subset of E and ϵ>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×EΦ separates points in F, i.e., for each yF with y0, there exists xE such that y,x0, then F also becomes Hausdorff. Furthermore, for any net { y α } α Γ in F and yF,

  1. (a)

    y α y in σF,E if and only if y α ,xy,x for each xE;

  2. (b)

    y α y in δF,E if and only if y α ,xy,x uniformly for each xA, where A is a nonempty bounded subset of E.

Suppose that, for the sets X, E and F mentioned above, S:X 2 X and T:X 2 F 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 y ˆ S( y ˆ ) and w ˆ T( y ˆ ) such that

Re w ˆ , y ˆ x0

for all xS( y ˆ ).

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 η:X×XE be a single-valued mapping and h:X×XR be a real-valued function. Then the generalized quasi-variational-like inequality problem is defined as follows: Find y ˆ S( y ˆ ) and w ˆ T( y ˆ ) such that

Re w ˆ , η ( y ˆ , x ) +h( y ˆ ,x)0

for all xS( y ˆ ).

For more results related to the generalized quasi-variational-like inequality problems, we refer to [36] 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 σF,E-topology, where ,:F×EΦ is a bilinear functional. Let h:X×XR, η:X×XE and T:X 2 F be three mappings. Then T is said to be:

  1. (1)

    an (η,h)-pseudo-monotone type II (respectively, a strongly (η,h)-pseudo-monotone type II) operator if, for each yX and every net { y α } α Γ in X converging to y (respectively, weakly to y) with

    lim sup α [ inf u T ( y ) Re u , η ( y α , y ) + h ( y α , y ) ] 0,

    we have

    lim sup α [ inf u T ( x ) Re u , η ( y α , x ) + h ( y α , x ) ] inf u T ( x ) Re u , η ( y , x ) + h ( y , x )

    for all xX;

  2. (2)

    an h-pseudo-monotone type II operator (respectively, a strongly h-pseudo-monotone type II operator) if T is an (η,h)-pseudo-monotone type II operator with η(x,y)=xy and, for some h :XR, h(x,y)= h (x) h (y) for all x,yX.

Note that, if F= E , 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 (η,h)-pseudo-monotone type II and strongly (η,h)-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 T:X 2 E be an operator such that each T(x) is strongly compact. Suppose that h:X×XR is a real-valued function such that, for each yX, h(,y) is continuous and h(X×X) is bounded. Let η:X×XE 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 . Then T is both an (η,h)-pseudo-monotone type II and a strongly (η,h)-pseudo-monotone type II operator.

Proof Suppose that { y α } α Γ is a net in X and yX with y α y (respectively, y α y weakly) and that

lim sup α [ inf u T ( y ) Re u , η ( y α , y ) + h ( y α , y ) ] 0.

Let xX be arbitrarily fixed. Then, using the continuity of h(,y), η and T, we obtain the following:

lim sup α [ inf u T ( x ) Re u , η ( y α , x ) + h ( y α , x ) ] lim sup α [ inf u T ( x ) Re u , η ( y α , x ) ] + lim inf α h ( y α , x ) = inf u T ( x ) Re u , η ( y , x ) + h ( y , x )

for all xX. Consequently, T is both an (η,h)-pseudo-monotone type II and a strongly (η,h)-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 h:X×XR and ϕ:X×XR{,+} be the mappings such that

  1. (a)

    for each AF(X) and fixed xco(A), yϕ(x,y) is lower semi-continuous on co(A);

  2. (b)

    for each AF(X) and yco(A), min x A [ϕ(x,y)+h(y,x)]0;

  3. (c)

    for each fixed xX, yh(x,y) is lower semi-continuous and concave on X, and h(x,x)=0;

  4. (d)

    for each AF(X) and each pair of points x,yco(A) such that every net { y α } α Γ in X converging to y with ϕ(tx+(1t)y, y α )+h( y α ,tx+(1t)y)0 for all αΓ and all t[0,1], we have ϕ(x,y)+h(y,x)0;

  5. (e)

    there exist a nonempty closed and compact subset K of X and x 0 K such that ϕ( x 0 ,y)+h(y, x 0 )>0 for all yXK.

Then there exists y ˆ K such that ϕ(x, y ˆ )+h( y ˆ ,x)0 for all xX.

Proof For the proof, we refer to [12]. □

Definition 1.3 A function ϕ:X×XR{±} is said to be 0-diagonally concave (in short, 0-DCV) in the second argument [14] if, for any finite set { x 1 ,, 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 [15].

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

  1. (1)

    The mappings T and η are said to have the 0-diagonally concave relation (in short, 0-DCVR) if the function ϕ:X×XR{±} 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 the 0-diagonally concave relation if T and η(x,y)=g(x)g(y) 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 T:X 2 E . Then T is said to be upper hemi-continuous on X if and only if, for each pE, the function f p :XR{+} defined by

f p (z)= sup u T ( z ) Reu,p

for each zX is upper semi-continuous on X (if and only if, for each pE, the function g p :XR{} defined by

g p (z)= inf u T ( z ) Reu,p

for each zX 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 S:X 2 E be an upper semi-continuous map such that S(x) is a bounded subset of E for each xX. Then, for each continuous linear functional p on E, the mapping f p :XR defined by

f p (y)= sup x S ( y ) Rep,x

is upper semi-continuous, i.e., for each λR, the set {yX: f p (y)= sup x S ( y ) Rep,x<λ} 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:XR be non-negative and continuous and g:YR be lower semi-continuous. Then the mapping F:X×YR defined by F(x,y)=f(x)g(y) for all (x,y)X×Y 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, AF(E) and X=co(A), where co(A) denotes the convex hull of A. Let F be a vector space over Φ and ,:F×Eϕ be a bilinear functional such that , separates points in F. We equip F with the σF,E-topology. Suppose that, for each wF, xRew,x is continuous. Let η:X×XE be continuous. Let T:X 2 F be upper semi-continuous from X into 2 F such that each T(x) is σF,E-compact. Let f:X×XR be defined by f(x,y)= inf w T ( y ) Rew,η(y,x) for all x,yX. Suppose that , is continuous on the (compact) subset [ y X T(y)]×η(X×X) of F×E. Then, for each fixed xX, yf(x,y) is lower semi-continuous on X.

For the completeness, we include the proof here given in [12].

Proof Let λR be given and let xX=co(A) be arbitrarily fixed. Let A λ ={yX:f(x,y)λ}. Suppose that { y α } α Γ is a net in A λ and y 0 co(A)=X such that y α y 0 . Then, for each αΓ,

λf(x, y α )= inf w T ( y α ) Re w , η ( y α , x ) .

Since F is equipped with the σF,E-topology, for each xE, the function wRew,x is continuous. Also, η( y α ,x)η( y 0 ,x) because η(,x) is continuous. By the σF,E-compactness of T( y α ), there exists w α T( y α ) such that

λ inf w T ( y α ) Re w , η ( y α , x ) =Re w α , η ( y α , x ) .

Since T is upper semi-continuous from X=co(A) to the σF,E-topology on F, X is compact, and each T(z) is σF,E-compact, z X T(z) is also σF,E-compact by Proposition 3.1.11 of Aubin and Ekeland [20]. Thus there is a subnet { w α } α Γ of { w α } α Γ and w 0 z X T(z) such that w α w 0 in the σF,E-topology. Again, as T is upper semi-continuous with the σF,E-closed values, w 0 T( y 0 ).

Suppose that A={ a 1 , a 2 ,, a n } and let t 1 , t 2 ,, t n 0 with i = 1 n t i =1 such that y 0 = i = 1 n t i a i . For each α Γ, let t 1 α , t 2 α ,, t n α 0 with i = 1 n t i α =1 such that y α = i = 1 n t i α a i . Since E is Hausdorff and y α y 0 , we must have t i α t i for each i=1,2,,n. Thus

λ Re w α , η ( y α , x ) = Re w α , η ( i = 1 n t i α a i , x ) Re w 0 , η ( i = 1 n t i a i , x ) = Re w 0 , η ( y 0 , x ) inf w T ( y 0 ) Re w , η ( y 0 , x ) = f ( x , y 0 ) ,
(2.1)

where (2.1) is true since η(,x) is continuous on X and , is continuous on the compact subset [ y X T(y)]×η(X×X) of F×E. Hence y 0 A λ . Thus A λ is closed in X=co(A) for each λR. Therefore yf(x,y) 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 σF,E-topology such that, for each wF, the function xRew,x is continuous. Let T:X 2 F be upper hemi-continuous along line segments in X. Let η:X×XE be such that for each fixed yX, η(,y) is continuous, and let h:X×XR be a mapping such that, for each fixed yX, h(,y) is lower semi-continuous on co(A) for each AF(X) and, for each fixed xX, h(x,) is concave and h(x,x)=0 and T, η have the 0-DCVR. Suppose that y ˆ X such that inf u T ( x ) Reu,η( y ˆ ,x)h(x, y ˆ ) for all xX. Then

inf w T ( y ˆ ) Re w , η ( y ˆ , x ) h(x, y ˆ )

for all xX.

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 X×Y such that for each fixed xX, the map yf(x,y), i.e., f(x,) is lower semi-continuous and convex on Y and, for each fixed yY, the mapping xf(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 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 σF,E-topology, where ,:F×EΦ is a bilinear functional separating points on F such that, for each wF, the function xRew,x is continuous. Let S:X 2 X , T:X 2 F , η:X×XE and h:E×ER be the mappings such that

  1. (a)

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

  2. (b)

    h(X×X) is bounded;

  3. (c)

    T is an (η,h)-pseudo-monotone type II (respectively, a strongly (η,h)-pseudo-monotone type II) operator and is upper hemi-continuous along line segments in X to the σF,E-topology on F such that each T(x) is σF,E-compact and convex and T(X) is δF,E-bounded;

  4. (d)

    T and η have the 0-DCVR and η is continuous;

  5. (e)

    for each fixed yX, xh(x,y), i.e., h(,y) is lower semi-continuous on co(A) for each AF(X) and, for each fixed xX, h(x,) and η(x,) are concave, η(x,) is affine, h(x,x)=0 and η(x,x)=0;

  6. (f)

    the set Σ={yX: sup x S ( y ) [ inf u T ( x ) Reu,η(y,x)+h(y,x)]>0} is open in X;

  7. (g)

    for each AF(X) and each yco(A), there exist x ¯ A and u ¯ T( x ¯ ) such that

    β 0 (y) [ Re u ¯ , η ( y , x ¯ ) + h ( y , x ¯ ) ] + p LF ( E ) β p (y)Rep,y x ¯ 0

for any family { β 0 , β p :pLF(E)} of non-negative real-valued functions from X into [0,1], where LF(E) denotes the set of all continuous linear functionals on E;

  1. (h)

    for each AF(X), the bilinear functional , is continuous over the compact subset [ y co ( A ) T(y)]×η(co(A)×co(A)) of F×E.

Then there exists a point y ˆ X such that

  1. (1)

    y ˆ S( y ˆ );

  2. (2)

    there exists a point w ˆ T( y ˆ ) with Re w ˆ ,η( y ˆ ,x)+h( y ˆ ,x)0 for all xS( y ˆ ).

Proof Step 1. Let us first show that there exists a point y ˆ X such that y ˆ S( y ˆ ) and

sup x S ( y ˆ ) [ inf u T ( x ) Re u , η ( y ˆ , x ) + h ( y ˆ , x ) ] 0.

Now, we prove this by contradiction. So, we assume that, for each yX, either yS(y) or there exists xS(y) such that

inf u T ( x ) Re u , η ( y , x ) +h(y,x)>0,

that is, for each yX, either yS(y) or yΣ. If yS(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

Rep,y sup x S ( y ) Rep,x>0.

For each yX, set

γ ( y ) : = sup x S ( y ) [ inf u T ( x ) Re u , η ( y , x ) + h ( y , x ) ] , V 0 : = Σ = { 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 LF ( E ) V p .

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 :pLF(E)} is an open covering for X. Since X is compact, there exist p 1 , p 2 ,, p n LF(E) such that X= V 0 i = 1 n V p i . For the simplicity of notation, let V i = V p i for i=1,2,,n. Let { β 0 , β 1 ,, β n } be a continuous partition of unity on X subordinated to the covering { V 0 , V 1 ,, V n }. Then β 0 , β 1 ,, β n are continuous non-negative real-valued functions on X such that β i vanishes on X V i for each i=0,1,,n and i = 0 n β i (x)=1 for all xX. Note that, for each yX and AF(X), xh(x,y), i.e., h(,y) is continuous on co(A) (see [22], Corollary 10.1.1). Define a function ϕ:X×XR by

ϕ ( x , y ) = β 0 ( y ) [ min u T ( x ) Re u , η ( y , x ) + h ( y , x ) ] + i = 1 n β i ( y ) Re p i , y x

for all x,yX. Then we have the following:

(I) Since E is Hausdorff, for each AF(X) and fixed xco(A), the mapping

y inf u T ( x ) Re u , η ( y , x ) +h(y,x)

is 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) [ min u T ( x ) Re u , η ( y , x ) + h ( y , x ) ]

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

y i = 1 n β i (y)Re p i ,yx

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

(II) Since β 0 , β 1 ,, β n is a family of continuous non-negative real-valued functions on X into [0,1], by the hypothesis, for each AF(X) and each yco(A), there exist x ¯ A and u ¯ T( x ¯ ) such that

β 0 (y) [ Re u ¯ , η ( y , x ¯ ) + h ( y , x ¯ ) ] + i = 1 n β i (y)Re p i ,y x ¯ 0.

Thus we have

min u T ( x ) [ β 0 ( y ) ( Re u , η ( y , x ¯ ) + h ( y , x ¯ ) ) ] + i = 1 n β i (y)Re p i ,y x ¯ 0,

i.e.,

β 0 (y) [ min u T ( x ) ( Re u , η ( y , x ¯ ) + h ( y , x ¯ ) ) ] + i = 1 n β i (y)Re p i ,y x ¯ 0.

Therefore, we have

min x A [ β 0 ( y ) ( min u T ( x ) ( Re u , η ( y , x ) + h ( y , x ) ) ) + i = 1 n β i ( y ) Re p i , y x ] 0

and so min x A ϕ(x,y)0 for each AF(X) and yco(A).

(III) Suppose that AF(X), x,yco(A) and { y α } α Γ is a net in X converging to y (respectively, weakly to y) with ϕ(tx+(1t)y, y α )0 for all αΓ and all t[0,1].

Case 1: β 0 (y)=0. Since β 0 is continuous and y α y, we have β 0 ( y α ) β 0 (y)=0. Note that β 0 ( y α )0 for each αΓ. Since T(X) is strongly bounded and { y α } α Γ is a bounded net, it follows that

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

Also, we have

β 0 (y) [ min u T ( x ) Re u , η ( y , x ) + h ( y , x ) ] =0.

Thus it follows from (3.1) that

lim sup α [ β 0 ( y α ) min u T ( x ) Re u , η ( y α , x ) + h ( y α , x ) ] + i = 1 n β i ( y ) Re p i , y x = i = 1 n β i ( y ) Re p , y x = β 0 ( y ) [ min u T ( x ) Re u , η ( y , x ) + h ( y , x ) ] + i = 1 n β i ( y ) Re p , y x .
(3.2)

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

β 0 ( y α ) [ min u T ( x ) Re u , η ( y α , x ) + h ( y α , x ) ] + i = 1 n β i ( y α ) Re p , y α x 0
(3.3)

for all αΓ. Therefore, by (3.3), we have

lim sup α [ β 0 ( y α ) ( min u T ( x ) Re u , η ( y α , x ) + h ( y α , x ) ) ] + lim inf α [ i = 1 n β i ( y α ) Re p , y α x ] lim sup α [ β 0 ( y α ) ( min u T ( x ) Re u , η ( y α , x ) + h ( y α , x ) ) + i = 1 n β i ( y α ) Re p , y α x ] 0 ,

and so

lim sup α [ β 0 ( y α ) ( min u T ( x ) Re u , η ( y α , x ) + h ( y α , x ) ) ] + i = 1 n β i ( y ) Re p , y x 0 .
(3.4)

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

Case 2: β 0 (y)>0. Since β 0 is continuous, β 0 ( y α ) β 0 (y). Again since β 0 (y)>0, there exists λΓ such that β 0 ( y α )>0 for all αλ.

When t=0, we have ϕ(y, y α )0 for all αΓ, i.e.,

β 0 ( y α ) [ min u T ( y ) Re u , η ( y α , y ) + h ( y α , y ) ] + i = 1 n β i ( y α )Rep, y α y0

for all αΓ, and so

lim sup α [ β 0 ( y α ) ( min u T ( y ) Re u , η ( y α , y ) + h ( y α , y ) ) + i = 1 n β i ( y α ) Re p , y α y ] 0 .
(3.5)

Hence, by (3.5), we have

lim sup α [ β 0 ( y α ) ( min u T ( y ) Re u , η ( y α , y ) + h ( y α , y ) ) ] + lim inf α [ i = 1 n β i ( y α ) Re p , y α y ] lim sup α [ β 0 ( y α ) ( min u T ( y ) Re u , η ( y α , y ) + h ( y α , y ) ) + i = 1 n β i ( y α ) Re p , y α y ] 0 .

Since lim inf α [ i = 1 n β i ( y α )Rep, y α y]=0, we have

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

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

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

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

lim sup α [ min u T ( y ) Re u , η ( y α , y ) + h ( y α , y ) ] 0.

Since T is an (η,h)-pseudo-monotone type II (respectively, a strongly (η,h)-pseudo-monotone type II) operator, we have

lim sup α [ min u T ( x ) Re u , η ( y α , x ) + h ( y α , x ) ] min u T ( x ) Re u , η ( y , x ) + h ( y , x )

for all xX. Since β 0 (y)>0, we have

β 0 ( y ) [ lim sup α min u T ( x ) Re u , η ( y α , x ) + h ( y α , x ) ] β 0 ( y ) [ min u T ( x ) Re u , η ( y , x ) + h ( y , x ) ] ,

and so

β 0 ( y ) [ lim sup α min u T ( x ) Re u , η ( y α , x ) + h ( y α , x ) ] + i = 1 n β i ( y ) Re p , y x β 0 ( y ) [ min u T ( x ) Re u , η ( y , x ) + h ( y , x ) ] + i = 1 n β i ( y ) Re p , y x .
(3.8)

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

β 0 ( y α ) [ min u T ( x ) Re u , η ( y α , x ) + h ( y α , x ) ] + i = 1 n β i ( y α )Rep, y α x0

for all αΓ and so, by (3.8),

0 lim sup α [ β 0 ( y α ) ( min u T ( x ) Re u , η ( y α , x ) + h ( y α , x ) ) + i = 1 n β i ( y α ) Re p , y α x ] lim sup α [ β 0 ( y α ) ( min u T ( x ) Re u , η ( y α , x ) + h ( y α , x ) ) ] + lim inf α [ i = 1 n β i ( y α ) Re p , y α x ] = β 0 ( y ) [ lim sup α ( min u T ( x ) Re u , η ( y α , x ) + h ( y α , x ) ) ] + i = 1 n β i ( y ) Re p , y x β 0 ( y ) [ min u T ( x ) Re u , η ( y , x ) + h ( y , x ) ] + i = 1 n β i ( y ) Re p , y x .
(3.9)

Hence we have ϕ(x,y)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 K=X, we have ϕ( x 0 ,y)>0 for all yXK (=XX=). Thus the hypothesis (d) of Theorem 1.2 is satisfied trivially. (If T is a strongly (η,h)-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 y ˆ K=X such that ϕ(x, y ˆ )0 for all xX, i.e.,

β 0 ( y ˆ ) [ min u T ( x ) Re u , η ( y ˆ , x ) + h ( y ˆ , x ) ] + i = 1 n β i ( y ˆ )Re p i , y ˆ x0
(3.10)

for all xX.

If β 0 ( y ˆ )>0, then y ˆ V 0 =Σ so that γ( y ˆ )>0. Choose x ˆ S( y ˆ )X such that

min u T ( x ˆ ) Re u , η ( y ˆ , x ˆ ) +h( y ˆ , x ˆ )γ( y ˆ )/2>0.

Then it follows that

β 0 ( y ˆ ) [ min u T ( x ˆ ) Re u , η ( y ˆ , x ˆ ) + h ( y ˆ , x ˆ ) ] >0.

If β i ( y ˆ )>0 for each i=1,2,,n, then y ˆ V i and hence

Re p i , y ˆ > sup x S ( y ˆ ) Re p i ,xRe p i , x ˆ

and so Re p i , y ˆ x ˆ >0. Then we see that β i ( y ˆ )Re p i , y ˆ x ˆ >0 whenever β i ( y ˆ )>0 for each i=1,2,,n. Since β 0 ( y ˆ )>0 or β i ( y ˆ )>0 for each i=1,2,,n, it follows that

ϕ( x ˆ , y ˆ )= β 0 ( y ˆ ) [ min u T ( x ˆ ) Re u , η ( y ˆ , x ˆ ) + h ( y ˆ , x ˆ ) ] + i = 1 n β i ( y ˆ )Re p i , y ˆ x ˆ >0,

which contradicts (3.10). This contradiction proves Step 1. Hence we have shown that there exists a point y ˆ X such that y ˆ S( y ˆ ) and

sup x S ( y ˆ ) [ inf u T ( x ) Re u , η ( y ˆ , x ) + h ( y ˆ , x ) ] 0.

Step 2. We need to show that

inf w T ( y ˆ ) Re w , η ( y ˆ , x ) h(x, y ˆ )

for all xS( y ˆ ).

From Step 1, we know that y ˆ S( y ˆ ) which is a convex subset of X and

inf u T ( x ) Re u , η ( y ˆ , x ) h(x, y ˆ )

for all xS( y ˆ ). Hence, applying Lemma 2.4, we obtain

inf w T ( y ˆ ) Re w , η ( y ˆ , x ) h(x, y ˆ )

for all xS( y ˆ ).

Step 3. There exists a point w ˆ T( y ˆ ) with Re w ˆ ,η( y ˆ ,x)h(x, y ˆ ) for all xS( y ˆ ). From Step 2, we have

sup x S ( y ˆ ) [ inf w T ( y ˆ ) Re w , η ( y ˆ , x ) + h ( y ˆ , x ) ] 0,
(3.11)

where T( y ˆ ) is a σF,E-compact convex subset of the Hausdorff topological vector space (F,σF,E) and S( y ˆ ) is a convex subset of X.

Now, we define f:S( y ˆ )×T( y ˆ )R by f(x,w)=Rew,η( y ˆ ,x)+h( y ˆ ,x) for each xS( y ˆ ) and wT( y ˆ ). Then, for each fixed xS( y ˆ ), the mapping wf(x,w) is convex and continuous on T( y ˆ ) and, for each fixed wT( y ˆ ), the mapping xf(x,w) is concave on S( y ˆ ). So, we can apply Keneser’s minimax theorem (Theorem 2.5) and obtain the following:

min w T ( y ˆ ) sup x S ( y ˆ ) [ Re w , η ( y ˆ , x ) + h ( y ˆ , x ) ] = sup x S ( y ˆ ) min w T ( y ˆ ) [ Re w , η ( y ˆ , x ) + h ( y ˆ , x ) ] .

Hence, by (3.11), we obtain

min w T ( y ˆ ) sup x S ( y ˆ ) [ Re w , η ( y ˆ , x ) + h ( y ˆ , x ) ] 0.

Since T( y ˆ ) is compact, there exists w ˆ T( y ˆ ) such that

Re w ˆ , η ( y ˆ , x ) +h( y ˆ ,x)0

for all xS( y ˆ ). This completes the proof. □

Note that, if for each open subset U of X and for each x,yU, η(x,y)=xy and there exists h :XR such that h(x,y)= h (x) h (y); and if the mapping S:X 2 X is, in addition, lower semi-continuous and, for each yΣ, T is upper semi-continuous at some point x in S(y) with inf u T ( x ) Reu,η(y,x)+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 σF,E-topology, where ,:F×EΦ is a bilinear functional separating points on F such that, for each wF, the function xRew,x is continuous. Let S:X 2 X , T:X 2 F , η:X×XE and h:E×ER be the mappings such that

  1. (a)

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

  2. (b)

    h(X×X) is bounded;

  3. (c)

    T is an (η,h)-pseudo-monotone type II (respectively, a strongly (η,h)-pseudo-monotone type II) operator and is upper hemi-continuous along line segments in X to the σF,E-topology on F such that each T(x) is σF,E-compact and convex and T(X) is δF,E-bounded;

  4. (d)

    T and η have the 0-DCVR and η is continuous;

  5. (e)

    for each fixed yX, xh(x,y), i.e., h(,y) is lower semi-continuous on co(A) for each AF(X) and, for each fixed xX, h(x,) and η(x,) are concave, η(x,) is affine, h(x,x)=0 and η(x,x)=0;

  6. (f)

    for each open subset U of X and x,yU, η(x,y)=xy, and there exists h :XR such that h(x,y)= h (x) h (y);

  7. (g)

    for each yΣ={yX: sup x S ( y ) [ inf u T ( x ) Reu,η(y,x)+h(y,x)]>0}, T is upper semi-continuous at some point x 0 in S(y) with inf u T ( x 0 ) Reu,η(y, x 0 )+h(y, x 0 )>0;

  8. (h)

    for each AF(X) and yco(A), there exist x ¯ A and u ¯ T( x ¯ ) such that

    β 0 (y) [ Re u ¯ , η ( y , x ¯ ) + h ( y , x ¯ ) ] + p LF ( E ) β p (y)Rep,y x ¯ 0

for any family { β 0 , β p :pLF(E)} of non-negative real-valued functions from X into [0,1];

  1. (i)

    for each AF(X), the bilinear functional , is continuous over the compact subset [ y co ( A ) T(y)]×η(co(A)×co(A)) of F×E.

Then there exists a point y ˆ X such that

  1. (1)

    y ˆ S( y ˆ );

  2. (2)

    there exists a point w ˆ T( y ˆ ) with Re w ˆ ,η( y ˆ ,x)+h( y ˆ ,x)0 for all xS( y ˆ ).

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

Σ= { y X : sup x S ( y ) [ inf u T ( x ) Re u , η ( y , x ) + h ( y , x ) ] > 0 }

is open in X. To show that Σ is open in X, we start as follows.

Let y 0 Σ be an arbitrary point. We show that there exists an open neighborhood N 0 of y 0 in X such that N 0 Σ. Now, by hypothesis (g), T is upper semi-continuous at some point x 0 in S( y 0 ) with

inf u T ( x 0 ) Re u , η ( y 0 , x 0 ) +h( y 0 , x 0 )>0.

Let

α:= inf u T ( x 0 ) Re u , η ( y 0 , x 0 ) +h( y 0 , x 0 ).

Thus α>0. Again, let

W:= { w F : sup z 1 , z 2 X | w , z 1 z 2 | < α / 6 } .

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 semi-continuous at x 0 , there exists an open neighborhood V 1 of x 0 in X such that T(x) U 1 for all x V 1 . Since the mapping x inf u T ( x 0 ) Reu,η( x 0 ,x)+h( x 0 ,x) is continuous at x 0 , there exists an open neighborhood V 2 of x 0 in X such that

| inf u T ( x 0 ) Re u , η ( x 0 , x ) +h( x 0 ,x)|< α 6

for all x V 2 . Let V 0 := V 1 V 2 . Then V 0 is an open neighborhood of x 0 in X. Since x 0 V 0 S( y 0 ) and S is lower semi-continuous at y 0 , there exists an open neighborhood N 1 of y 0 in X such that S(y) V 0 for all y N 1 . Since the mapping y inf u T ( x 0 ) Reu,η(y, y 0 )+h(y, y 0 ) is continuous at y 0 , there exists an open neighborhood N 2 of y 0 in X such that

| inf u T ( x 0 ) Re u , η ( y , y 0 ) +h(y, y 0 )|< α 6

for all y N 2 .

Let N 0 := N 1 N 2 . Then N 0 is an open neighborhood of y 0 in X such that for each y 1 N 0 , we have the following:

  1. (a)

    S( y 1 ) V 0 as y 1 N 1 ; so we can choose any x 1 S( y 1 ) V 0 ;

  2. (b)

    | inf u T ( x 0 ) Reu,η( y 1 , y 0 )+h( y 1 , y 0 )|< α 6 as y 1 N 2 ;

  3. (c)

    T( x 1 ) U 1 =T( x 0 )+W as x 1 V 1 ;

  4. (d)

    | inf u T ( x 0 ) Reu,η( x 0 , x 1 )+h( x 0 , x 1 )|< α 6 as x 1 V 2 .

Hence, using property (f) and (b)-(d), we can obtain the following by omitting the details:

inf u T ( x 1 ) Re u , η ( y 1 , x 1 ) + h ( y 1 , x 1 ) inf [ u T ( x 0 ) + W ] Re u , η ( y 1 , x 1 ) + h ( y 1 , x 1 ) inf u T ( x 0 ) Re u , η ( y 1 , x 1 ) + h ( y 1 , x 1 ) + inf u W Re u , η ( y 1 , x 1 ) inf u T ( x 0 ) Re u , y 1 y 0 + h ( y 1 ) h ( y 0 ) + inf u T ( x 0 ) Re u , y 0 x 0 + h ( y 0 ) h ( x 0 ) + inf u T ( x 0 ) Re u , x 0 x 1 + h ( x 0 ) h ( x 1 ) + inf u W Re u , y 1 x 1 α 6 + α α 6 α 6 = α 2 > 0 .

Consequently, we have

sup x S ( y 1 ) [ inf u T ( x ) Re u , η ( y 1 , x ) + h ( y 1 , x ) ] >0,

since x 1 S( y 1 ). Hence y 1 Σ for all y 1 N 0 . Therefore, y 0 N 0 Σ. 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. (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 (η,h)-pseudo-monotone type II operators on compact sets.

  2. (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 (η,h)-pseudo-monotone type II operators on compact sets.

  3. (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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Keywords

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