Cent. Eur. J. Math. • 6(4) • 2008 • 526-536 DOI: 10.2478/s11533-008-0049-1

Central European Journal of Mathematics

Generalized variational-like inequalities for pseudo-monotone type III operators

Research Article

Mohammad S.R. Chowdhury1∗, Kok-Keong Tan2

1 Department of Mathematics, Lahore University of Management Sciences (LUMS), Phase II, Opposite Sector U, DHA, Lahore Cantt., Lahore - 54792, Pakistan 2 Department of Mathematics & Stattistics, Dalhousie University, Halifax, Nova Scotia, Canada, B3H 3J5

Received 3 April 2008; accepted 31 July 2008

Abstract: Our aim in this paper is mainly to prove some existence results for solutions of generalized variational-like inequal- ities with (η; h)-pseudo-monotone type III operators defined on non-compact sets in topological vector spaces. Keywords: generalized variational-like inequalities • 0-diagonally concave relation • pseudo-monotone type III operators © Versita Warsaw and Springer-Verlag Berlin Heidelberg.

1. Introduction

Browder [4] and Hartman and Stampacchia [20] first introduced variational inequalities and Minty [26, 27] first introduced the theory of monotone nonlinear operators. Since then, there have been many generalizations on the theory of monotone nonlinear operators as well as in variational inequality problems. In 1996, Chowdhury and Tan first obtained generalized variational inequalities for pseudo-monotone type I operators [8].

Moreover, Chowdhury and Tan’s (1997) results on generalized variational inequalities for quasi-monotone operators were obtained in [9]. The recent results of Chowdhury and Tarafdar on generalized variational inequalities for pseudo-monotone type III operators were established in [10].

There are wide applications of variational inequalities in nonlinear elliptic boundary value problems, obstacle problems, complementarity problems, mathematical programming, mathematical economics and in many other areas. There are too many citations of papers concerning variational inequalities. Some references are [20], [23] and [31] where citations of many further references can be found. The topic on monotone variational inequalities is discussed in [35, 36]. In [22], Karamardian showed that the complementarity problems can be reduced to the variational inequality problems. In [25], Mancino and Stampacchia showed the relationship between mathematical programming and the variational inequalities.

∗ E-mail: [email protected]

526 Mohammad S.R. Chowdhury, Kok-Keong Tan

In [28] and [34], Moreau and Rockafellar respectively showed the relationship between variational inequalities and convex functions. The relationship between variational inequalities and optimization is given by Peng in [33] and by Yamashita, Taji and Fukushima in [37]. Moreover, the relationship between the variational inequalities and the equilibrium point of Walrasian economy in Riesz spaces was shown by Aliprantis and Brown in [1].

In the year 2000, Ding and Tarafdar [16] obtained generalized variational-like inequalities (see definition below) for (η; h)-pseudo-monotone set-valued mappings, i.e., for (η; h)-pseudo-monotone type I operators. These operators are simply generalizations of Chowdhury and Tan’s pseudo-monotone type I operators (see [8] and also [6]). Note that in [8] pseudo-monotone type I operators were named as set-valued pseudo-monotone (and demi-monotone) operators given below.

Definition 1.1. E∗ ∗ Let X be a nonempty subset of a topological vector space E, and let T : X → 2 (where E denotes the topological E∗ ∗ dual space of E and 2 denotes the family of all nonempty subsets of E ). If h : X → R, then T is said to be an h-pseudo-monotone (respectively, h-demi-monotone) operator y ∈ X {y } X if for each and every net α α∈Γ in converging to y (respectively, weakly to y) with

Rehu; yα − yi h yα − h y ≤ ; lim sup[u∈ infT yα + ( ) ( )] 0 α∈Γ ( ) we have

Rehu; yα − xi h yα − h x lim sup[u∈ infT yα + ( ) ( )] α∈Γ ( ) ≥ inf Rehw; y − xi + h(y) − h(x) for all x ∈ X; w∈T (y)

T is said to be pseudo-monotone (respectively, demi-monotone) if T is h-pseudo-monotone (respectively, h-demi- monotone) with h ≡ 0.

Later in [6], the above set-valued pseudo-monotone operators were renamed by M.S.R. Chowdhury as pseudo-monotone type I operators.

There are many classes of problems where η(x; y) =6 x − y. As an example we can mention the classes of problems of system of generalized mixed implicit quasi-variational inclusions of Ding-Lee-Yu in [12].

In 2005, we introduced generalized variational-like inequalities for (η; h)-pseudo-monotone type II operators [11]. These operators are also generalizations of pseudo-monotone type II operators introduced by M.S.R. Chowdhury (see [5] and also [6]). Note that in [5] pseudo-monotone type II operators were named as demi operators. Later in [6], the demi operators were renamed by M.S.R. Chowdhury as pseudo-monotone type II operators.

In this paper we shall define (η; h)-pseudo-monotone type III operators. These operators are generalizations of pseudo-monotone type III operators introduced by Chowdhury and Tarafdar in [10]. Note that in [10] pseudo-monotone type III operators were named as hemi-continuous operators. Later in [6], hemi-continuous operators were renamed by M.S.R. Chowdhury as pseudo-monotone type III operators.

In this paper we have used mainly the concepts of pseudo-monotone type III operators which are derived by slight modification of the concepts of pseudo-monotone type I operators given in Definition 1.1. Our classical definition of single-valued pseudo-monotone type I operators was introduced by Brézis, Nirenberg and Satmpacchia in [3]. So, our concept of psedo-monotonocity does not contradict or confuse other classical concepts of pseudo-monotonocity available in the literature.

Our aim in this paper is to establish some results on existence theorems for a new class of generalized variational-like inequalities with (η; h)-pseudo-monotone type III operators defined on non-compact sets in topological vector spaces.

527 Generalized variational-like inequalities for pseudo-monotone type III operators

We shall begin with some preliminary concepts related to this paper:

F Suppose that E and F are two vector spaces over the real or the complex field Φ. Let 2 denote the family of all nonempty subsets of F and let F(E) denote the family of all nonempty finite subsets of E and co(A) denotes the convex hull of A for any subset A ⊂ E.

Let X be a nonempty subset of E and h ; i : F × E → Φ be a bilinear functional satisfying the following property: (P) the family {h·; xi}x∈E separates the points of F; i.e., for each y ∈ F with y =6 0, there exists x ∈ E such that hy; xi= 6 0.

Throughout this paper, unless otherwise stated, E, F and h·; ·i will be assumed to satisfy the property (P).

F With E, F and X as above, let T : X → 2 be a set-valued mapping, f : X → F and η : X × X → E be two single-valued mappings, and h : X × X → R be a real-valued function.

The classical variational inequality (in short, VI(f; X; F)) is: find a point yˆ ∈ X such that

Rehf(yˆ); yˆ − xi ≤ 0; ∀x ∈ X: (1)

The generalized variational-like inequality (in short, GV LI(T ; η; h; X; F)) is: find yˆ ∈ X and wˆ ∈ T (yˆ) such that

Rehw;ˆ η(y;ˆ x)i + h(y;ˆ x) ≤ 0; ∀x ∈ X: (2)

The GV LI(T ; η; h; X; F) given in (2) was defined in [16] by Ding and Tarafdar and includes various variational inequalities studied in [13–15, 18, 19, 21, 29, 30, 32, 33] as special cases. In particular, when η(y;ˆ x) = yˆ − x and there 0 0 0 exists h : X → R such that h(y;ˆ x) = h (yˆ) − h (x), then the GVLI (2) reduces to Chowdhury and Tarafdar’s GVLI in [10].

E F X x ∈ E  > W x  {w ∈ F |hw; x i| < } σ F;E Let , and as above. For each 0 and each 0, let ( 0; ) = : 0 . Let ( ) be the topology on F generated by the family {W (x; ): x ∈ E and  > 0} as a subbase for the neighbourhood system at 0. Then F becomes a locally convex topological vector space which is not necessarily Hausdorff. Since the bilinear functional h ; i : F × E → Φ satisfies the property (P) above, F also becomes Hausdorff. Throughout this paper, the underlying space F in the problem (2) is equipped with the σ(F;E)-topology.

F Let E be also a topological vector space and T : X → 2 be a set-valued mapping. T is said to be upper semi-continuous (in short, u.s.c.) at x ∈ X if for each open set U in F containing T (x), there is an open neighbourhood V of x in X such that T (y) ⊂ U for all y ∈ V . T is called u.s.c. on X if T is u.s.c. at each x ∈ X.

Definition 1.2. Let X be a nonempty subset of a topological vector space E over Φ and F be a vector space over Φ which is equipped F with the σ(F;E)-topology. Let T : X → 2 , η : X × X → E and h : X × X → R. Then T is said to be a (η; h)-pseudo- monotone type III (respectively, a strong (η; h)-pseudo-monotone type III) operator if for each x; y ∈ X and every net {y } X y y α α∈Γ in converging to (respectively, weakly to ) with

Rehu; η yα ; x i h yα ; x ≤ ; lim sup[u∈ infT yα ( ) + ( )] 0 α∈Γ ( )

we have

Rehu; η yα ; y i h yα ; y lim sup[u∈ infT yα ( ) + ( )] α∈Γ ( ) ≥ inf Rehw; η(y; x)i + h(y; x): w∈T (y)

T is said to be an h-pseudo-monotone (respectively, a strong h-pseudo-monotone) type III operator, if T is an (η; h)- pseudo-monotone type III (respectively, a strong (η; h)-pseudo-monotone type III) operator with η(x; y) = x − y and for 0 0 0 some h : X → R, h(x; y) = h (x) − h (y) for all x; y ∈ X.

528 Mohammad S.R. Chowdhury, Kok-Keong Tan

∗ Note that in Definition 1.2 if F = E , the topological dual space of E, then the notions of the h-pseudo-monotone type III operators and the strong h-pseudo-monotone type III operators coincide with those in [7].

Definition 1.3. φ X × X → R ∪ {±∞} -diagonally concave A function : is said to be 0 n (in short,n 0-DCV) in the second argumentn [38], {x ; :::; x } ⊂ X λ ≥ P λ P λ φ y; x ≤ y P λ x if for any finite set 1 n and any i 0, with i = 1, we have i ( i) 0 where = i i. i=1 i=1 i=1

Now, we state the following definition given in [16], p.304:

Definition 1.4. F Let T : X → 2 , η : X × X → E and g : X → E. 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

φ(x; y) = inf Rehw; η(x; y)i (3) w∈T (x) is 0-DCV in the second argument. 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. Preliminary results

We shall first state the following result which is Lemma 2.1 in [16]:

Lemma 2.1. F Let T : X → 2 and η : X × X → E be such that for each fixed x ∈ X, inf hu; η(x; x)i = 0 and η(x; ·) is an affine u∈T (x) mapping. Then T and η have 0-DCVR.

We need the following result which is a slight modification of Lemma 3 of Chowdhury and Tan in [8].

Lemma 2.2. Let E be a Hausdorff topological vector space over Φ, A ∈ F(E) and X = co(A) where co(A) denotes the convex hull of A. Let F be a vector space over Φ and h ; i : F × E → Φ be a bilinear functional which satisfies the property (P). Equip F with the σhF;Ei-topology. Suppose that for each w ∈ F, x 7→ Rehw; xi is continuous. Let η : X × X → E F F be continuous. Let T : X 7→ 2 be upper semi-continuous from X into 2 such that each T (x) is σhF;Ei-compact. Let f X × X → be defined by f x; y Rehw; η y; x i for all x; y ∈ X. Suppose that h ; i is continuous over the : R ( ) = infw∈T (y) ( ) (compact) subset [∪y∈X T (y)] × η(X × X) of F × E. Then for each fixed x ∈ X, y 7→ f(x; y) is lower semi-continuous on X.

Proof. Let λ ∈ R be given and let x ∈ X = co(A) be arbitrarily fixed. Let Aλ = {y ∈ X : f(x; y) ≤ λ}. Suppose that {y } A y ∈ co A X y → y α ∈ α α∈Γ is a net in λ and 0 ( ) = such that α 0. Then for each Γ,

λ ≥ f(x; yα ) = inf Rehw; η(yα ; x)i: w∈T (yα )

Since F is equipped with the σhF;Ei-topology, for each x ∈ E, the function w 7→ Rehw; xi is continuous. Also, η y ; x → η y ; x η ·; x σhF;Ei T y w ∈ T y ( α ) ( 0 ) because ( ) is continuous. By the -compactness of ( α ), there exists α ( α ) such that λ ≥ inf Rehw; η(yα ; x)i = Rehwα ; η(yα ; x)i: w∈T (yα )

529 Generalized variational-like inequalities for pseudo-monotone type III operators

Since T is upper semi-continuous from X = co(A) to the σhF;Ei-topology on F, X is compact, and each T (z) is σhF;Ei-compact, ∪z∈X T (z) is also σhF;Ei-compact by Proposition 3.1.11 of Aubin and Ekeland [2]. Thus there is a {w 0 } 0 0 {w } w ∈ ∪ T z w 0 → w σhF;Ei T subnet α α ∈Γ of α α∈Γ and 0 z∈X ( ) such that α 0 in the -topology. Again, as is upper σhF;Ei w ∈ T y semi-continuous with the -closed values, 0 ( 0).

n n A {a ; a ; ··· ; a } t ; t ; ··· ; t ≥ P t y P t a α0 ∈ Suppose that = 1 2 n and let 1 2 n 0 with i = 1 such that 0 = i i. For each Γ, let i=1 i=1 n n α0 α0 α0 P α0 P α0 α0 t ; t ; ··· ; t ≥ t yα0 t ai E yα0 → y t → ti 1 2 n 0 with i = 1 such that = i . Since is Hausdorff and 0, we must have i i=1 i=1 for each i = 1; 2; ··· ; n. Thus

n n X α0 X λ ≥ Rehw 0 ; η y 0 ; x i Rehw 0 ; η t a ; x i → Rehw ; η t a ; x i α ( α ) = α ( i i ) 0 ( i i ) i=1 i=1 Rehw ; η y ; x i ≥ Rehw; η y ; x i f x; y = 0 ( 0 ) w∈infT y ( 0 ) = ( 0) (4) ( 0)

where (4) is true since η(·; x) is continuous on X and h ; i is continuous on the compact subset [∪y∈X T (y)] × η(X × X) of F × E. y ∈ A A X co A λ ∈ y 7→ f x; y X Hence 0 λ. Thus λ is closed in = ( ) for each R. Therefore ( ) is lower semi-continuous on .

We shall use the following generalized version of Ky Fan’s minimax inequality [17] which is a slight modification of Theorem 1 in [8] due to M.S.R. Chowdhury and K.-K. Tan.

Theorem 2.1. Let E be a Hausdorff topological vector space, X be a nonempty convex subset of E. Let h : X × X → R and φ : X × X → R ∪ {−∞; +∞} be such that

(a) for each A ∈ F(X) and each fixed x ∈ co(A), y 7→ φ(x; y) is lower semi-continuous on co(A);

(b) for each A ∈ F(X) and each y ∈ co(A), minx∈A[φ(x; y) + h(y; x)] ≤ 0;

(c) for each fixed x ∈ X, y 7→ h(x; y) is lower semi-continuous and concave on X, and h(x; x) = 0; (d) for each A ∈ F X and each pair of points x; y ∈ co A such that every net {y } in X converging to y with ( ) ( ) α α∈Γ φ(tx + (1 − t)y; yα ) + h(yα ; tx + (1 − t)y) ≤ 0 for all α ∈ Γ and all t ∈ [0; 1], we have φ(x; y) + h(y; x) ≤ 0; (e) there exist a nonempty closed and compact subset K of X and x ∈ K such that φ x ; y h y; x > for all 0 ( 0 ) + ( 0) 0 y ∈ X \ K.

Then there exists yˆ ∈ K such that φ(x; yˆ) + h(y;ˆ x) ≤ 0 for all x ∈ X.

The proof of this theorem is similar to the proof of Theorem 1 in [8]. For completeness we shall include the proof here:

X Proof. Define F : X → 2 by

F(x) = {y ∈ X : φ(x; y) + h(y; x) ≤ 0} for each x ∈ X:

F KKM {x ; ··· ; x } X α ≥ i ; ··· ; n Pn α If is not a -map, then for some finite subset 1 n of and i 0 for = 1 with i i = 1, we Pn Sn =1 y αixi 6∈ F xi : φ xi; y h y; xi > i ; ··· ; n have ¯ = i=1 i=1 ( ) Thus ( ¯) + (¯ ) 0 for = 1 so that

min [φ(xi; y¯) + h(y;¯ xi)] > 0; 1≤i≤n

X which contradicts the assumption (b). Hence F : X → 2 is a KKM-map. Moreover we have,

530 Mohammad S.R. Chowdhury, Kok-Keong Tan

F x ⊂ K e c` F x ⊂ c` K K c` F x X (i) ( 0) by ( ), so that X ( 0) X = and hence X ( 0) is compact in ; A ∈ F X x ∈ A x ∈ A (ii) for each ( ) with 0 and each co( ),

F(x) ∩ co(A) = {y ∈ co(A): φ(x; y) + h(y; x) ≤ 0}

is closed in co(A) by (a) and the fact that y 7→ h(x; y) is lower semicontinuous on co(A) for each x ∈ co(A); A ∈ F X x ∈ A y ∈ c` T F x ∩ A y ∈ A {y } (iii) for each ( ) with 0 , if ( X ( x∈ A ( ))) co( ), then co( ) and there is a net α α∈Γ T co( ) F x yα → y: x ∈ A tx − t y ∈ A t ∈ ; in x∈co(A) ( ) such that For each co( ), since + (1 ) co( ) for all [0 1], we have yα ∈ F(tx +(1−t)y) for all α ∈ Γ and all t ∈ [0; 1]. This implies that φ(tx +(1−t)y; yα )+h(yα ; tx +(1−t)y) ≤ 0 T for all α ∈ Γ and all t ∈ [0; 1] so that by (d), φ(x; y) + h(y; x) ≤ 0; it follows that y ∈ ( x∈ A F(x)) ∩ co(A). T T co( ) c`X F x ∩ A F x ∩ A Hence, ( ( x∈co(A) ( ))) co( ) = ( x∈co(A) ( )) co( ). T T Hence by Lemma 2 in [8], we have x∈X F(x) =6 ∅. Then there exists yˆ ∈ x∈X F(x), so that φ(x; yˆ) + h(y;ˆ x) ≤ 0 for all x ∈ X.

3. Main results on generalized variational-like inequalities for pseudo- monotone type III operators

In this section, we state and prove some existence theorems for the solutions to the generalized variational-like inequal- ities involving a (η; h)-pseudo-monotone type III operator T with non-compact domain in Hausdorff topological vector spaces.

Theorem 3.1. Let X be a nonempty convex subset of a Hausdorff topological vector space E over Φ and F be a vector space over Φ which is equipped with the σ(F;E)-topology such that for each w ∈ F, the function x 7→ Rehw; xi is continuous. Let F T : X → 2 , η : X × X → E and h : E × E → R be mappings such that

(i) for each x ∈ X, T (x) is σ(F;E)-compact; (ii) T and η have the 0-DCVR, and η is continuous;

(iii) for each fixed y ∈ E, x 7→ h(x; y) (i.e., h(·; y)) is lower semi-continuous on E, and for each fixed x ∈ E, y 7→ h(x; y) (i.e., h(x; ·)) is concave on E and h(x; x) = 0; F F (iv) T : co(A) → 2 is upper semi-continuous from co(A) into 2 for each A ∈ F(X);

(v) T is an (η; h)-pseudo-monotone type III (respectively, a strong (η; h)-pseudo-monotone type III) operator. (vi) for each A ∈ F X , the bilinear functional h ; i is continuous over the compact subset ∪ T y × η co A × ( ) [ y∈co(A) ( )] ( ( ) co(A)) of F × E. Suppose further that there exist a nonempty compact (respectively, weakly closed and weakly compact) subset K of X and x ∈ K such that for each y ∈ X \ K, Rehw; η y; x i h y; x > : 0 infw∈T (y) ( 0) + ( 0) 0 Then there exists a point yˆ ∈ K such that

inf Rehw; η(y;ˆ x)i + h(y;ˆ x) ≤ 0; ∀x ∈ X: (5) w∈T (yˆ)

If, in addition, T (yˆ) is convex and for each x ∈ X, y 7→ η(x; y) (i.e., η(x; ·)) is affine, then there exists a point wˆ ∈ T (yˆ) such that

Rehw;ˆ η(y;ˆ x)i + h(y;ˆ x) ≤ 0; ∀x ∈ X: (6)

531 Generalized variational-like inequalities for pseudo-monotone type III operators

φ X × X → φ x; y Rehw; η y; x i x; y ∈ X Proof. Define : R by ( ) = minw∈T (y) ( ) for each . Then we have the following:

(a) Clearly, for each A ∈ F(X) and each fixed x ∈ co(A), since E is Hausdorff and co(A) is compact, and the relative weak topology on co(A) coincides with its relative topology; it follows that y 7→ φ(x; y) is lower semi-continuous (respectively, weakly lower semi-continuous) on co(A) by Lemma 2.2;

(b) We need to show that for each A ∈ F(X) and each y ∈ co(A),

min[ min Rehw; η(y; x)i + h(y; x)] ≤ 0: x∈A w∈T (y)

A {x ; x ; ··· ; x } ∈ F X y ∈ co A y Pn λ x Suppose the contrary. Then there exist = 1 2 n ( ) and ¯ ( ) with ¯ = i i i (where Pn =1 λ ; λ ; ··· ; λn ≥ λi 1 2 0 with i=1 = 1) such that

min[ min Rehw; η(y;¯ xj ) + h(y;¯ xj )] > 0: xj ∈A w∈T (y¯)

Rehw; η y; x i h y; x > j ; ; ··· ; n iii h y; y Thus minw∈T (y¯) (¯ j ) + (¯ j ) 0 for all = 1 2 . From hypothesis ( ) we see that (¯ ¯) = 0 and h(x; ·) is concave. Thus we have

n n n n X X X X 0 < λj [ min Rehw; η(y;¯ xj )i] + λj h(y;¯ xj ) ≤ λj min Rehw; η(y;¯ xj )i + h(y;¯ λj xj ) w∈T (y¯) w∈T (y¯) j=1 j=1 j=1 j=1 n n n X X X = λj min Rehw; η(y;¯ xj )i + h(y;¯ y¯) = λj min Rehw; η(y;¯ xj )i = λj φ(xj ; y¯) w∈T (y¯) w∈T (y¯) j=1 j=1 j=1

which contradicts the hypothesis (ii) that T and η have the 0-DCVR. A ∈ F X x; y ∈ co A {y } X y → y (c) Suppose that ( ), ( ) and α α∈Γ is a net in with α in the relative topology (respectively, relatively weak topology) such that

φ(tx + (1 − t)y; yα ) + h(yα ; tx + (1 − t)y) ≤ 0

for all α ∈ Γ and all t ∈ [0; 1]. For t = 1, we have φ(x; yα ) + h(yα ; x) ≤ 0 for all α ∈ Γ. Thus

min Rehu; η(yα ; x)i + h(yα ; x) ≤ 0; ∀α ∈ Γ: u∈T (yα )

Hence Rehu; η yα ; x i h yα ; x ≤ : lim sup[u∈ minT yα ( ) + ( )] 0 α∈Γ ( )

Since T is a (η; h)-pseudo-monotone (respectively, a strong (η; h)-pseudo-monotone) type III operator, we have

Rehu; η yα ; y i h yα ; y lim sup[u∈ minT yα ( ) + ( )] α∈Γ ( ) ≥ inf Rehw; η(y; x)i + h(y; x): (7) w∈T (y)

Next for t = 0 we also have φ(y; yα ) + h(yα ; y) ≤ 0 for all α ∈ Γ so that

min Rehu; η(yα ; y)i + h(yα ; y) ≤ 0 u∈T (yα )

532 Mohammad S.R. Chowdhury, Kok-Keong Tan

for all α ∈ Γ. It follows that

Rehu; η yα ; y i h yα ; y ≤ : lim sup[u∈ minT yα ( ) + ( )] 0 (8) α∈Γ ( )

Hence by (7) and (8), we obtain inf Rehw; η(y; x)i + h(y; x) ≤ 0: w∈T (y)

Consequently, φ(x; y) + h(y; x) ≤ 0.

(d) By assumption, K is a compact and therefore closed (respectively, weakly closed and weakly compact) subset of X x ∈ K y ∈ X \ K and 0 such that for each ,

Rehw; η y; x i h y; x > inf ( 0) + ( 0) 0; w∈T (y)

φ x ; y h y; x > i.e., ( 0 ) + ( 0) 0. (If T is a strong (η; h)-pseudo-monotone type III operator, we equip E with the weak topology, i.e., σhF;Ei topology). Then φ satisfies all the hypotheses of Theorem 2.1. Hence by Theorem 2.1 there exists a point yˆ ∈ K with φ(x; yˆ) + h(y;ˆ x) ≤ 0; ∀x ∈ X:

In other words, min Rehw; η(y;ˆ x)i + h(y;ˆ x) ≤ 0; ∀x ∈ X: w∈T (yˆ)

Now, suppose that T (yˆ) is convex and for each x ∈ X, η(x; ·) is affine. Define ψ : X × T (yˆ) → R by

ψ(x; w) = Rehw; η(y;ˆ x)i + h(y;ˆ x)

for all x ∈ X and for all w ∈ T (yˆ). Note that T (yˆ) is σhF;Ei-compact convex, and by the definition of the σhF;Ei-topology on F, for each x ∈ E, the function w 7→ Rehw; xi is continuous. It follows that for each x ∈ X, the function w 7→ ψ(x; w) is σhF;Ei- continuous and affine and for each w ∈ T (yˆ), the function x 7→ ψ(x; w) is concave since η(y;ˆ ·) is affine and h(y;ˆ ·) is concave. Hence by Kneser’s minimax Theorem in [24], we obtain

min sup[Rehw; η(y;ˆ x)i + h(y;ˆ x)] w∈T (yˆ) x∈X = sup min [Rehw; η(y;ˆ x)i + h(y;ˆ x)] x∈X w∈T (yˆ) = sup[ inf Rehw; η(y;ˆ x)i + h(y;ˆ x)] ≤ 0: x∈X w∈T (yˆ)

Hence, by σ(F;E)-compactness of T (yˆ), there exists a point wˆ ∈ T (yˆ) such that

sup[Rehw;ˆ η(y;ˆ x)i + h(y;ˆ x)] ≤ 0 x∈X

and hence Rehw;ˆ η(y;ˆ x)i + h(y;ˆ x) ≤ 0; ∀x ∈ X;

that is, the pair yˆ ∈ K and wˆ ∈ T (yˆ) is a solution of the generalized variational-like inequality given in (2).

533 Generalized variational-like inequalities for pseudo-monotone type III operators

0 0 Now, suppose that for each x ∈ X, T (x) is also convex and that η(x; y) = x − y and h(x; y) = h (x) − h (y) for all 0 x; y ∈ X where h : E → R is a convex function. Then we obtain the following result from Theorem 3.1:

Theorem 3.2. Let X be a nonempty convex subset of a Hausdorff topological vector space E over Φ and F be a vector space over Φ which is equipped with the σ(F;E)-topology such that for each w ∈ F, the function x 7→ Rehw; xi is continuous. Let F 0 T : X → 2 and h : E → R be such that

(i) for each x ∈ X, T (x) is σ(F;E)-compact convex; (ii) h0 is a convex function;

F F (iii) T : co(A) → 2 is upper semi-continuous from co(A) into 2 for each A ∈ F(X); (iv) T is an h0-pseudo-monotone type III (respectively, a strong h0-pseudo-monotone type III) operator;

(v) for each A ∈ F X , the bilinear functional h ; i is continuous over the compact subset ∪ T y × co A ×co A ( ) [ y∈co(A) ( )] ( ( ) ( )) of F × E.

(vi) there exist a nonempty compact (respectively, weakly closed and weakly compact) subset K of X and x0 ∈ K such that for each y ∈ X \ K, Rehw; y − x i h0 y − h0 x > : inf 0 + ( ) ( 0) 0 w∈T (y)

Then there exist yˆ ∈ K and wˆ ∈ T (yˆ) such that

0 0 Rehw;ˆ yˆ − xi ≤ h (x) − h (yˆ); ∀x ∈ IX (yˆ);

where IX (yˆ) = {z ∈ E : z = yˆ + r(x − yˆ) for some x ∈ X and r > 0}.

0 Proof. For each A ∈ F(X), h is continuous on co(A) (see, [34], Corollary 10.1.1, p.83). Let η(x; y) = x − y and 0 0 h(x; y) = h (x) − h (y) for all x; y ∈ X. Clearly, all hypotheses of Theorem 3.1 are satisfied. Hence there exist yˆ ∈ K and wˆ ∈ T (yˆ) such that

0 0 Rehw;ˆ yˆ − xi + h (yˆ) − h (x) ≤ 0; ∀x ∈ X: (9)

Let z ∈ IX (yˆ). Then there exist x ∈ X and r > 0 such that z = yˆ + r(x − yˆ).

Case 1. Suppose 0 < r ≤ 1, then z = rx + (1 − r)yˆ ∈ X as X is convex and x; yˆ ∈ X. Thus we have 0 0 Rehw;ˆ yˆ − zi + h (yˆ) − h (z) ≤ 0 by (9).

1 1 Case 2. Suppose r > 1, then x = (1 − r )yˆ + r z. Since x ∈ X, by (9) again, we have

1 Rehw; y − zi Rehw; y − xi ≤ h0 x − h0 y ≤ − 1 h0 y 1 h0 z − h0 y 1 h0 z − h0 y r ˆ ˆ = ˆ ˆ ( ) (ˆ) (1 r ) (ˆ) + r ( ) (ˆ) = r ( ( ) (ˆ))

0 0 so that Rehw;ˆ yˆ − zi ≤ h (z) − h (yˆ):

Thus in either case, we have 0 0 Rehw;ˆ yˆ − zi ≤ h (z) − h (yˆ); ∀z ∈ IX (yˆ):

Remark 3.1. ∗ If F = E , the dual space of E, then Theorem 3.2 reduces to Theorem 1 of M.S.R. Chowdhury and E. Tarafdar in [10].

534 Mohammad S.R. Chowdhury, Kok-Keong Tan

Acknowledgements

We gratefully acknowledge the referees’ careful review and comments which helped in improving this paper. The project was initiated while one of the authors M.S.R. Chowdhury visited Dalhousie university recently. For this project the second author K.-K. Tan was partially supported by NSERC of Canada under Grant A-8096.

References

[1] Aliprantis C., Brown D., Equilibria in markets with a Riesz space of commodities, J. Math. Econom., 1983, 11, 189–207 [2] Aubin J.P., Ekeland I., Applied nonlinear analysis, John Wiley & Sons, Inc., New York, 1984 [3] Brézis H., Nirenberg L., Stampacchia G., A remark on Ky Fan’s minimax principle, Boll. Un. Mat. Ital., 1972, 6, 293–300 [4] Browder F.E., Nonlinear monotone operators and convex sets in Banach spaces, Bull. Amer. Math. Soc., 1965, 71, 780–785 [5] Chowdhury M.S.R., Generalized variational inequalities for upper hemicontinuous and demi operators with appli- cations to fixed point theorems in Hilbert spaces, Serdica Math. J., 1998, 24, 163–178 [6] Chowdhury M.S.R., The surjectivity of upper-hemi-continuous and pseudo-monotone type II operators in reflexive Banach Spaces, Ganit, 2000, 20, 45–53 [7] Chowdhury M.S.R., Haque A.L., Thompson H.B., Quasem G.M.A., The surjectivity of pseudo-monotone type III operators in reflexive Banach Spaces, Nonlinear Funct. Anal. Appl., 2004, 9, 37–44 [8] Chowdhury M.S.R., Tan K.-K., Generalization of Ky Fan’s minimax inequality with applications to generalized variational inequalities for pseudo-monotone operators and fixed point theorems, J. Math. Anal. Appl., 1996, 204, 910–929 [9] Chowdhury M.S.R., Tan K.-K., Generalized variational inequalities for quasi-monotone operators and applications, Bull. Polish Acad. Sci. Math., 1997, 45, 25–54 [10] Chowdhury M.S.R., Tarafdar E., Hemi-continuous operators and some applications, Acta Math. Hungar., 1999, 83, 251–261 [11] Chowdhury M.S.R., Thompson H.B., Generalized variational-like inequalities for pseudomonotone type II operators, Nonlinear Analysis, 2005, 63, 321–330 [12] Ding X.P., Lee C., Yu S.-J., Algorithm of solutions for a system of generalized mixed implicit quasi-variational inclusions involving (h − η)-maximal monotone mappings, Taiwanese J. Math., 2007, 11, 577–593 [13] Ding X.P., Tarafdar E., Generalized nonlinear variational inequalities with nonmonotone set-valued mappings, Appl. Math. Lett., 1994, 7, 5–11 [14] Ding X.P., Tarafdar E., Existence and uniqueness of solutions for a general nonlinear variational inequality, Appl. Math. Lett., 1995, 8, 31–36 [15] Ding X.P., Tarafdar E., Monotone generalized variational inequalities and generalized complementarity problems, J. Optim. Theory Appl., 1996, 88, 107–122 [16] Ding X.P., Tarafdar E., Generalized variational-like inequalities with pseudomonotone set-valued mappings, Arch. Math. (Basel), 2000, 74, 302–313 [17] Fan K., A minimax inequality and applications, In: Shisha O. (Ed.), Inequalities III, Proc. Third Sympos. (1969 University of California, Los Angeles), Academic Press, New York, 1972, 103–113 [18] Fang S.C., Peterson E.L., Generalized variational inequalities, J. Optim. Theory Appl., 1982, 38, 363–383 [19] Harker P.T., Pang J.S., Finite-dimensional variational inequality and nonlinear complementarity problems: a survey of theory, algorithms and applications, Math. Programming, 1990, 48, 161–220 [20] Hartman P., Stampacchia G., On some non-linear elliptic differential-functional equations, Acta Math., 1966, 115, 271–310 [21] Jou C.R., Yao J.C., Extension of generalized multi-valued variational inequalities, Appl. Math. Lett., 1993, 6, 21–25 [22] Karamardian S., The complementarity problem, Math. Programming, 1972, 2, 107–129 [23] Kinderlehrer D., Stampacchia G., An introduction to variational inequalities and their applications, Academic Press, New York-London, 1980

535 Generalized variational-like inequalities for pseudo-monotone type III operators

[24] Kneser H., Sur un théorème fondamental de la théorie des jeux, C. R. Acad. Sci. Paris, 1952, 234, 2418–2420 (in French) [25] Mancino O., Stampacchia G., Convex programming and variational inequalities, J. Optimization Theory Appl., 1972, 9, 3–23 [26] Minty G.J., Monotone (nonlinear) operators in Hilbert space, Duke Math. J., 1962, 29, 341–346 [27] Minty G.J., On a “monotonicity” method for the solution of nonlinear equations in Banach spaces, Proc. Nat. Acad. Sci. U.S.A., 1963, 50, 1038–1041 [28] Moreau J.J., Principes extrémaux pour le problème de la naissance de la cavitation, J. Mécanique, 1966, 5, 439–470 (in French) [29] Noor M.A., General variational inequalities, Appl. Math. Lett., 1988, 1, 119–122 [30] Noor M.A., Nonconvex functions and variational inequalities, J. Optim. Theory Appl., 1995, 87, 615–630 [31] Parida J., Sahoo M., Kumar A., A variational-like inequality problem, Bull. Austral. Math. Soc., 1989, 39, 225–231 [32] Pascali D., Sburlan S., Nonlinear mappings of monotone type, Sijthoff & Noordhoff International Publishers, Alphen aan den Rijn, 1978 [33] Peng J.M., Equivalence of variational inequality problems to unconstrained optimization, Math. Programming, 1997, 78, 347–355 [34] Rockafellar R.T., Convex analysis, Princeton University Press, Princeton, N.J., 1970 [35] Rockafellar R.T., Wets R.J.-B., Variational analysis, Springer-Verlag, Springer-Verlag, Berlin, 1998 [36] Stefanov S.M., A Lagrangian dual method for solving variational inequalities, Math. Inequal. Appl., 2002, 5, 597–608 [37] Yamashita N., Taji K., Fukushima M., Unconstrained optimization reformulations of variational inequality problems, J. Optim. Theory Appl., 1997, 92, 439–456 [38] Zhou J.X., Chen G., Diagonal convexity conditions for problems in convex analysis and quasi-variational inequalities, J. Math. Anal. Appl., 1988, 132, 213–225

536