A generalized mixed vector variational-like inequality problem

Nonlinear Analysis 71 (2009) 5354–5362

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Nonlinear Analysis journal homepage: www.elsevier.com/locate/na

A generalized mixed vector variational-like inequality problem Farhat Usman a , Suhel Ahmad Khan b,∗ a

Department of Mathematics, Aligarh Muslim University, Aligarh-202002, India

b

Department of Mathematics, Amity School of Engineering, Amity University Rajasthan, Rajasthan-303002, India

article

info

Article history: Received 1 November 2008 Accepted 8 April 2009 MSC: 49J40 47H10 47H17

abstract In this paper, we introduce relaxed η-α -P-monotone mapping, and by utilizing KKM technique and Nadler’s Lemma we establish some existence results for the generalized mixed vector variational-like inequality problem. Further, we give the concepts of η-complete semicontinuity and η-strong semicontinuity and prove the solvability for generalized mixed vector variational-like inequality problem without monotonicity assumption by applying the Brouwer’s fixed point theorem. The results presented in this paper are extensions and improvements of some earlier and recent results in the literature. © 2009 Elsevier Ltd. All rights reserved.

Keywords: Hausdorff metric H-hemicontinuous Relaxed η-α -P-monotone η-complete semicontinuity η-strong semicontinuity

1. Introduction Variational inequality theory has played a fundamental and important role in the study of a wide range of problems arising in physics, mechanics, differential equations, contact problems in elasticity, optimization, economics and engineering sciences, etc. As a useful and important branch of variational inequality theory, vector variational inequalities were initially introduced and considered by Giannessi [1] in a finite-dimensional Euclidean space in 1980. Ever since then, vector variational inequalities have been extensively studied and generalized in infinite-dimensional spaces. For details, we refer to [1–19] and references therein. It is known that monotonicity plays an important role in the study of variational inequalities. In recent years, a number of authors have obtained many important generalizations of monotonicity such as quasimonotonicity, pseudomonotonicity, relaxed monotonicity, p-monotonicity, etc.; see for example [3,4,6,7,11,12,16–18] and references therein. In 2003, Huang et al. [13] introduced two classes of variational-like inequalities with generalized monotone mappings in Banach spaces. Using the KKM technique, they obtained the existence of solutions for variational-like inequalities with relaxed η-α -monotone mapping in reflexive Banach spaces. Very recently in 2007, Lee et al. [14] considered generalized vector variational-type inequalities in reflexive Banach spaces and proved the solvability for the class of generalized vector variational-type inequalities in reflexive Banach spaces involving set-valued mappings. Inspired and motivated by Huang et al. [13] and Lee et al. [14], we introduce relaxed η-α -P-monotone mapping for the class of the generalized mixed vector variational-like inequality problem. With the help of KKM technique and Nadler’s Lemma, we establish the existence results of solutions for the generalized mixed vector variational-like inequality problem involving relaxed η-α -P-monotone mapping in reflexive Banach spaces.

∗

Corresponding author. Tel.: +91 9411983025. E-mail address: [email protected] (S.A. Khan).

0362-546X/$ – see front matter © 2009 Elsevier Ltd. All rights reserved. doi:10.1016/j.na.2009.04.023

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Further, motivated and inspired by Fang et al. [10], we define the concepts of η-complete semicontinuity and η-strong semicontinuity for set-valued mapping, and then we prove the solvability for the generalized mixed vector variational-like inequality problem without monotonicity assumption by using these concepts and by applying the Brouwer’s fixed point theorem. The results presented in this paper generalize, unify and improve a number of previously known results, see for example [10,13–15,18,19]. 2. Preliminaries Throughout the paper unless otherwise stated, let X and Y be two real Banach spaces, K ⊂ X be a nonempty closed and convex subset of X and P ⊂ Y be a closed convex and pointed cone with apex at the origin. P is called proper cone, if P 6= Y and P is said to be a closed convex and pointed cone with its apex at the origin, if P is closed and the following conditions hold: (i) λP ⊂ P , for all λ > 0; (ii) P + P ⊂ P; (iii) P ∩ (−P ) = {0}. Given a closed convex and pointed cone P with apex at the origin in Y , we can define relations ‘‘≤P ’’ and ‘‘6≤P ’’ as follows: x ≤P y ⇔ y − x ∈ P

and

x 6≤P y ⇔ y − x 6∈ P .

If ‘‘≤P ’’ is a partial order, then (Y , ‘‘ ≤P ’’) is called an ordered Banach space ordered by P. Let L(X , Y ) denote the space of all continuous linear mappings from X into Y and Lc (X , Y ) be the subspace of L(X , Y ) which consists of all completely continuous linear mappings from X into Y . Let T : K → 2L(X ,Y ) be a vector set-valued mapping. Suppose f : K × K → Y and η : X × X → X are the two bi-mappings, A : L(X , Y ) → L(X , Y ) are the mapping, then we consider the following generalized mixed vector variational-like inequality problem (for short, GMVVLIP): Find x ∈ K and u ∈ T (x) such that

hAu, η(y, x)i + f (x, y) 6≤int P 0,

∀y ∈ K

(2.1)

where a 6≤int P b means b − a 6∈ int P. First, we recall the following concepts and results which are needed in the sequel. Definition 2.1. A mapping f : K → Y is said to be P-convex, if f (tx + (1 − t )y) ≤P tf (x) + (1 − t )f (y),

∀x, y ∈ K , t ∈ [0, 1].

Lemma 2.1 ([5]). Let (Y , P ) be an ordered Banach space with a closed convex and pointed cone P with int P 6= ∅. Then ∀x, y, z ∈ Y , we have (i) z 6≤int P x and x ≥P y ⇒ z 6≤int P y; (ii) z ≥ 6 int P x and x ≤P y ⇒ z ≥ 6 int P y. Definition 2.2. A mapping g : X → Y is said to be completely continuous if and only if the weak convergence of xn to x in X implies that the strong convergence of g (xn ) to g (x) in Y . Lemma 2.2 ([9]). Let E be a subset of a topological vector space X and let F : E → 2X be a KKM mapping. If for each x ∈ E, F (x) is closed and for at least one x ∈ E, F (x) is compact, then

\

F (x) 6= ∅.

x∈E

Lemma 2.3 ([15]). Let (X , k.k) be a normed vector space and H be a Hausdorff metric on the collection CB(X ) of all nonempty closed and bounded subsets of X , induced by a metric d in terms of d(u, v) = ku − vk, defined by H (U , V ) = max(sup inf ku − vk, sup inf ku − vk), u∈U v∈V

v∈V u∈U

for U and V in CB(X ). If U and V are compact sets in X , then for each u ∈ U, there exists v ∈ V such that ku − vk ≤ H (U , V ). Definition 2.3. Let η : X × X → X be a bi-mapping and A : K → L(X , Y ) be a single-valued mapping. Suppose T : K → 2L(X ,Y ) be the nonempty compact set-valued mapping, then (i) A is said to be η-hemicontinuous, if for any given x, y ∈ K , the mapping t → hA(x + t (y − x)), η(y, x)i is continuous at 0+ ;

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(ii) T is said to be H-hemicontinuous, if for any given x, y ∈ K , the mapping t → H (T (x + t (y − x)), T (x)) is continuous at 0+ , where H is the Hausdorff metric defined on CB(L(X , Y )). Definition Pn 2.4. A mapping η : X × X → X is said to be affine in first argument if for any xi ∈ K and λi ≥ 0, (1 ≤ i ≤ n), with i=1 λi = 1 and any y ∈ K ,

η

n X i=1

! λi xi , y =

n X

λi η(xi , y).

i=1

Definition 2.5 ([19]). Let A : L(X , Y ) → L(X , Y ) be a mapping. A vector set-valued mapping T : K → 2L(X ,Y ) is said to be monotone with respect to A, if for each x, y ∈ K ,

hAu − Av, x − yi ≥P 0,

∀u ∈ T (x), v ∈ T (y).

3. GMVVLIP with monotonicity In this section, we shall derive the solvability for the GMVVLIP (2.1) with relaxed η-α -P-monotone under some quite mild conditions by using Ky Fan’s Lemma [9] and Nadler’s Lemma [15]. First, we give the concept of relaxed η-α -P-monotone which is useful for establishing existence theorem for GMVVLIP (2.1). Definition 3.1. Let f : K × K → Y and η : X × X → X are the two bi-mappings, let A : L(X , Y ) → L(X , Y ) be the mapping, then set-valued mapping T : K → 2L(X ,Y ) is said to be relaxed η-α -P-monotone with respect to A, if for any x, y ∈ K ,

hAu − Av, η(x, y)i − α(x − y) ≥P 0,

∀u ∈ T (x), v ∈ T (y)

where α : X → Y is a mapping such that α(tz ) = t p α(z ) for all t > 0, z ∈ X and p > 1 is a constant. Remark 3.1.

(i) If T is single-valued and A the identity mapping of L(X , Y ) and P = R+ , then Definition 3.1 reduces to

hTx − Ty, x − yi ≥ α(x − y),

∀ x, y ∈ K ;

(3.1)

T is said relaxed η-α -monotone, introduced and studied by Fang and Huang [10]. (ii) If we take η(x, y) = x − y, for all x, y ∈ K , then (3.1) reduces to

hTx − Ty, x − yi ≥ α(x − y),

∀ x, y ∈ K ;

(3.2)

and T is said to be α -relaxed monotone. (iii) If we take α(z ) = kkz kp , where k > 0 is constant and ∀z ∈ X , then (3.2) reduces to

hTx − Ty, x − yi ≥ kkx − ykp ,

∀ x, y ∈ K ;

(3.3)

and T is called p-monotone, see for example [11,12,16,17]. Now we prove Minty’s type lemma for GMVVLIP (2.1) with the help of relaxed η-α -P monotone. Lemma 3.1. Let K be a closed and convex subset of a real Banach space X , Y be a real Banach space ordered by a nonempty closed convex pointed cone P with apex at the origin and int P 6= ∅. Let A : L(X , Y ) → L(X , Y ) is a continuous mapping and T : K → 2L(X ,Y ) be a nonempty compact set-valued mapping. Suppose the following conditions hold: (i) f : K × K → Y be a P-convex in second argument with the condition f (x, x) = 0 ∀x ∈ K ; (ii) η : X × X → X is an affine mapping in first argument with the condition η(x, x) = 0, ∀x ∈ K ; (iii) T : K → 2L(X ,Y ) is H-hemicontinuous and relaxed η-α -monotone with respect to A then following two problems are equivalent: (A) there exists x0 ∈ K and u0 ∈ T (x0 ) such that

hAu, η(y, x0 )i + f (x0 , y) 6≤int P 0,

∀y ∈ K .

(3.4)

(B) there exists x0 ∈ K such that

hAv, η(y, x0 )i + f (x0 , y) − α(y − x0 ) 6≤int P 0,

∀y ∈ K , v ∈ T (y).

(3.5)

Proof. Suppose that there exist x0 ∈ K and u0 ∈ T (x0 ) such that

hAu0 , η(y, x0 )i + f (x0 , y) 6≤int P 0,

∀y ∈ K .

Since T is relaxed η-α -P monotone with respect to A, we have

hAv, η(y, x0 )i + f (x0 , y) ≥P hAu0 , η(y, x0 )i + f (x0 , y) + α(y − x),

∀y ∈ K , v ∈ T (y).

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From Lemma 2.1, we have

hAv, η(y, x0 )i + f (x0 , y) − α(y − x0 ) 6≤int P 0,

∀y ∈ K , v ∈ T (y).

Conversely, suppose that there exists x0 ∈ K such that

hAv, η(y, x0 )i + f (x0 , y) − α(y − x0 ) 6≤int P 0,

∀y ∈ K , v ∈ T (y).

For any given y ∈ K , we know that yt = ty + (1 − t )x0 ∈ K , ∀t ∈ (0, 1), as K is convex. Since x0 ∈ K is a solution of problem (3.5), so for each vt ∈ T (yt ) it follows that

hAvt , η(yt , x0 )i + f (x0 , yt ) − α(yt − x0 ) 6≤int P 0, hAvt , η(yt , x0 )i + f (x0 , yt ) − α t (y − x0 ) = hAvt , η(yt , x0 )i + f (x0 , yt ) − t p α(y − x0 ),

for t > 0 and a constant p > 1.

(3.6)

Since f is convex in second argument f (x0 , yt ) ≤P tf (x0 , y) + (1 − t )f (x0 , x0 ) = tf (x0 , y).

(3.7)

From assumption (ii) on η, we have

hAvt , η(yt , x0 )i = hAvt , η(ty + (1 − t )x0 , x0 )i ≤P t hAvt , η(y, x0 )i + (1 − t )hAvt , η(x0 , x0 )i = t hAvt , η(y, x0 )i.

(3.8)

It follows from inclusions (3.6)–(3.8) and Lemma 2.1, that for t > 0 and a constant p > 1

hAvt , η(y, x0 )i + f (x0 , y) − t p−1 α(y − x0 ) 6≤int P 0,

∀vt ∈ T (yt ), t ∈ (0, 1).

(3.9)

Since T (yt ) and T (x0 ) are compact, from Lemma 2.3, it follows that for each fixed vt ∈ T (yt ), there exists an ut ∈ T (x0 ) such that

kvt − ut k ≤ H (T (yt ), T (x0 )). Since T (x0 ) is compact, without loss of generality, we may assume that ut → u0 ∈ T (x0 ) as t → 0+ . Since T is H-hemicontinuous, H (T (yt ), T (x0 )) → 0 as t → 0+ . Thus one has

kvt − u0 k ≤ kvt − ut k + kut − u0 k ≤ H (T (yt ), T (x0 )) + kut − u0 k → 0,

as t → 0+ .

Since A is continuous mapping. Therefore letting t → 0+ , we obtain

khAvt , η(y, x0 )ik − khAu0 , η(y, x0 )ik ≤ khAvt − Au0 , η(y, x0 )ik ≤ kAvt − Au0 kkη(yt , x)k → 0. Also by inclusion (3.9), we deduce that

hAvt , η(y, x0 )i + f (x0 , y) − t p−1 α(y − x0 ) ∈ Y \ (−int C ). Since Y \ (−int C ) is closed and for t > 0 and a constant p > 1, we have

hAu0 , η(y, x0 )i + f (x0 , y) ∈ Y \ (−int C ), and so

hAu0 , η(y, x0 )i + f (x0 , y) 6≤int P 0, This completes the proof.

∀y ∈ K .



Now, with the help of above Lemma 3.1, we have following existence theorem for GMVVLIP (2.1). Theorem 3.1. Let K be a nonempty, bounded closed and convex subset of a real reflexive Banach space X , Y be a real Banach space ordered by a nonempty proper closed convex pointed cone P with apex at the origin and int P 6= ∅. Let A : L(X , Y ) → L(X , Y ) be a continuous mapping and T : K → 2L(X ,Y ) be a nonempty compact set-valued mapping. Suppose the following conditions hold: (i) f : K × K → Y be completely continuous in first argument and affine in second argument with the condition f (x, x) = 0 , ∀x ∈ K ; (ii) η : X × X → X is completely continuous in second argument and affine in first argument with the condition η(x, x) = 0 , ∀x ∈ K ; (iii) α : X → Y is weakly lower semicontinuous; (iv) T : K → 2L(X ,Y ) is H-hemicontinuous and relaxed η-α -monotone with respect to A.

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Then there exist x ∈ K and u ∈ T (x) such that

hAu, η(y, x)i + f (x, y) 6≤int P 0,

∀y ∈ K .

Proof. Define two set-valued mappings F , G : K → 2K as follows: F (y) = {x ∈ K : hAu, η(y, x)i + f (x, y) 6≤int P 0, for some u ∈ T (x)}, ∀y ∈ K , and G(y) = {x ∈ K : hAv, η(y, x)i + f (x, y) − α(y − x) 6≤int P 0, for all v ∈ T (y)}, ∀y ∈ K . Then F (y) and G(y) are nonempty since y ∈ G(y) ∩ F (y). We claim not true, then there Pn that F is a KKM mapping. Pn If this isS n exist a finite set {x1 , . . . , xn } ⊂ K and ti ≥ 0, i = 1, . . . , n with i=1 ti = 1 such that x = i=1 ti xi 6∈ i=1 F (xi ). Hence by the definition of F , we have

hAu, η(xi , x)i + f (x, xi ) ≤int P 0,

i = 1, . . . , n.

From assumptions (i) and (ii), we have 0 =

=

hAu, η(x, x)i + f (x, x) !+ * n X ti xi , x +f Au, η

x,

n X

i =1

=

n X

! ti xi

i=1

ti [hAu, η(xi , x)i + f (x, xi )]

i =1

≤int P 0, which leads to a contradiction since P is a proper cone. Thus our claim is verified. So F is a KKM mapping. Now, we prove F (y) ⊂ G(y) for every y ∈ K . Indeed let x ∈ F (y). Then for some u ∈ T (x) one has

hAu, η(y, x)i + f (x, y) 6≤int P 0. Since T is relaxed η-α -P-monotone with respect to A, we have

hAv, η(y, x)i + f (x, y) ≥P hAu, η(y, x)i + f (x, y) + α(y − x). From Lemma 2.1, we have

hAv, η(y, x)i + f (x, y) − α(y − x) 6≤int P 0, that is x ∈ G(y), which follows F (y) ⊂ G(y), for each y ∈ K and so G is also a KKM mapping. Now we claim that for each y ∈ K , G(y) ⊂ K is closed in the weak topology of X . w Indeed suppose x¯ ∈ G(y) , the weak closure of G(y). Since X is reflexive, there is a sequence {xn } in G(y) such that {xn } converges weakly to x¯ ∈ K . Then for each v ∈ Ty

hAv, η(y, xn )i + f (xn , y) − α(y − xn ) 6≤int P 0, which implies that

hAv, η(y, xn )i + f (xn , y) − α(y − xn ) ∈ Y \ (−int C ). Since Av and f are completely continuous also α is weakly lower semicontinuous and Y \ (−int C ) is closed, therefore

hAv, η(y, x¯ )i + f (¯x, y) − α(y − x¯ ) ∈ Y \ (−int C ) Thus we get

hAv, η(y, x¯ )i + f (¯x, y) − α(y − x¯ ) 6≤int P 0 and so x¯ ∈ G(y). This shows that G(y) is weakly closed for each y ∈ K . Our claim is then verified. Since X is reflexive and K ⊂ X is nonempty, bounded, closed and convex, K is a weakly compact subset of X and so G(y) is also weakly compact. According to Lemma 2.2,

\

G(y) 6= ∅.

y∈K

This implies that there exists x0 ∈ K such that

hAv, η(y, x¯ 0 )i + f (¯x0 , y) − α(y − x¯ 0 ) 6≤int P 0,

∀y ∈ K , v ∈ T (y).

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Therefore by Minty’s type Lemma 3.1, we conclude that there exist x0 ∈ K and u0 ∈ T (x0 ) such that

hAu0 , η(y, x0 )i + f (x0 , y) 6≤int P 0, This completes the proof.

∀y ∈ K .



Remark 3.2. In Theorem 3.1, Lc (X , Y ) and the complete continuity of f cannot be replaced by L(X , Y ) and continuity of f , respectively. Indeed, we can only prove that for each y ∈ K , G(y) is closed in the norm topology of X without convexity of G(y) if A maps L(X , Y ) into L(X , Y ) and f is continuous. So G(y) need not be weakly compact. If the boundedness of K is dropped off, then we have the following theorem under certain coercivity conditions: Theorem 3.2. Let K be a nonempty, closed and convex subset of a real reflexive Banach space X , with 0 ∈ K and Y be a real Banach space ordered by a nonempty proper closed convex pointed cone P with apex at the origin and int P 6= ∅. Let A : L(X , Y ) → Lc (X , Y ) be a continuous mapping and T : K → 2L(X ,Y ) be a nonempty compact set-valued mapping. Suppose the following conditions hold: (i) f : K × K → Y be completely continuous in first argument and affine in second argument with the condition f (x, x) = 0 , ∀x ∈ K ; (ii) η : X × X → X is completely continuous in second argument and affine in first argument with the condition η(x, x) = 0 , ∀x ∈ K ; (iii) α : X → Y is weakly lower semicontinuous; (iv) T : K → 2L(X ,Y ) is H-hemicontinuous and relaxed η-α -monotone with respect to A. If there exists some r > 0 such that

hAv, η(y, 0)i + f (0, y) ≥int P 0,

∀v ∈ T (y), y ∈ K with kyk = r ,

(3.10)

then there exist x ∈ K and u ∈ T (x) such that

hAu, η(y, x)i + f (x, y) 6≤int P 0,

∀y ∈ K .

(3.11)

Proof. Let Br = {x ∈ X : kxk ≤ r }. By Theorem 3.1, there exist xr ∈ K ∩ Br and ur ∈ T (xr ) such that

hAur , η(y, xr )i + f (xr , y) 6≤int P 0,

∀y ∈ K ∩ Br .

(3.12)

Putting y = 0 in the above inclusion, we have

hAur , η(0, xr )i + f (xr , 0) 6≤int P 0,

∀y ∈ K ∩ Br .

(3.13)

Combining (3.10) with (3.13), we know that kxk < r. For any z ∈ K , choose t ∈ (0, 1) enough small such that (1 − t )xr + tz ∈ K ∩ Br . Putting y = (1 − t )xr + tz in (3.12), one has

hAur , η((1 − t )xr + tz , xr )i + f (xr , (1 − t )xr + tz ) 6≤int P 0. Since the mappings f and η are affine in second and first argument, respectively, we have

hAur , η((1 − t )xr + tz , xr )i + f (xr , (1 − t )xr + tz ) = t [hAur , η(z , xr )i + f (xr , z )]. By Lemma 2.1, we have

hAur , η(z , xr )i + f (xr , z ) 6≤int P 0, This completes the proof.

∀z ∈ K .



By Theorems 3.1 and 3.2, we can obtain the following results: Corollary 3.1. Let K be a nonempty, bounded closed and convex subset of a real reflexive Banach space X = Rn , Y be a real Banach space ordered by a nonempty proper closed convex pointed cone P with apex at the origin and int P 6= ∅. Let n A : L(Rn , Y ) → L(Rn , Y ) be a continuous mapping and T : K → 2L(R ,Y ) be a nonempty compact set-valued mapping. Suppose the following conditions hold: (i) f : K × K → Y be completely continuous in first argument and affine in second argument with the condition f (x, x) = 0 , ∀x ∈ K ; (ii) η : X × X → X is completely continuous in second argument and affine in first argument with the condition η(x, x) = 0 , ∀x ∈ K ; (iii) α : X → Y is weakly lower semicontinuous; n (iv) T : K → 2L(R ,Y ) is H-hemicontinuous and relaxed η-α -monotone with respect to A. Then there exist x ∈ K and u ∈ T (x) such that

hAu, η(y, x)i + f (x, y) 6≤int P 0,

∀y ∈ K .

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Corollary 3.2. Let K be a nonempty, closed and convex subset of a real reflexive Banach space X = Rn with 0 ∈ K , Y be a real Banach space ordered by a nonempty proper closed convex pointed cone P with apex at the origin and int P 6= ∅. Let n A : L(Rn , Y ) → L(Rn , Y ) be a continuous mapping and T : K → 2L(R ,Y ) be a nonempty compact set-valued mapping. Suppose the following conditions hold: (i) f : K × K → Y be completely continuous in first argument and affine in second argument with the condition f (x, x) = 0, ∀x ∈ K ; (ii) η : X × X → X is completely continuous in second argument and affine in first argument with the condition η(x, x) = 0, ∀x ∈ K ; (iii) α : X → Y is weakly lower semicontinuous; n (iv) T : K → 2L(R ,Y ) is H-hemicontinuous and relaxed η-α -monotone with respect to A. If there exists some r > 0 such that

hAv, η(y, 0)i + f (0, y) ≥int P 0,

∀v ∈ T (y), y ∈ K with kyk = r

then there exist x ∈ K and u ∈ T (x) such that

hAu, η(y, x)i + f (x, y) ≤int P 0,

∀y ∈ K .

4. GMVVLIP without monotonicity In this section, we shall establish the existence results for GMVVLIP (2.1) without monotonicity in reflexive Banach spaces by using Brouwer’s fixed point theorem. First we recall lemmas due to Brouwer’s [2] and Huang et al. [13]. Lemma 4.1 ([2]). Let X be a nonempty, compact and convex subset of a finite dimensional space and f : X → X be a continuous mapping. Then there exists x ∈ X such that f (x) = x. Lemma 4.2 ([13]). Let X be a real Banach space, K ⊂ X be a nonempty, bounded closed and convex subset, and Y be a real Banach space ordered by a proper closed convex and pointed cone P. Then the following conclusions hold: (i) If T : K → Lc (X , Y ) is completely continuous, then for any given y ∈ K , the map gy : K → Y , defined by gy (x) = hTx, y − xi is completely continuous; (ii) If T : K → L(X , Y ) is continuous, then for any given y ∈ K , the map gy : K → Y defined by gy (x) = hTx, y − xi is continuous. In order to establish the main result in this section, we give the following definition. Definition 4.1. Let K be a nonempty, closed and convex subset of a real Banach space X and Y be a real Banach space ordered by a closed convex and pointed cone P with apex at the origin int P (x) 6= ∅. Let η : X × X → X , f : K × K → Y are the two bi-mappings and A : L(X , Y ) → L(X , Y ) is a mapping. If T : K → 2L(X ,Y ) be a set-valued mapping, then T is said to be (i) η-completely semicontinuous with respect to A and f , if for each y ∈ K ,

{x ∈ K : hAu, η(y, x)i + f (x, y) ≤int P 0, ∀u ∈ Tx} is open in K with respect to the weak topology of X; (ii) η-strongly semicontinuous with respect to A and f , if for each y ∈ K ,

{x ∈ K : hAu, η(y, x)i + f (x, y) ≤int P 0, ∀u ∈ Tx} is open in K with respect to the norm topology of X . Next, we give the existence result for GMVVLIP (2.1). Theorem 4.1. Let K be a nonempty, bounded closed and convex subset of a real reflexive Banach space X , Y be a real Banach space ordered by a nonempty proper closed convex pointed cone P with apex at the origin and int P 6= ∅. Let the mapping A : L(X , Y ) → L(X , Y ) and T : K → 2L(X ,Y ) be a set-valued mapping with nonempty values. Suppose following conditions hold: (i) f : K × K → Y be an affine mapping in second argument with the condition f (x, x) = 0, ∀x ∈ K ; (ii) η : X × X → X be affine mapping in first argument with the condition η(x, x) = 0, ∀x ∈ K ; (iii) T : K → 2L(X ,Y ) is η-completely semicontinuous with respect to A and f . Then there exist x ∈ K and u ∈ T (x) such that

hAu, η(y, x)i + f (x, y) 6≤int P 0,

∀y ∈ K .

F. Usman, S.A. Khan / Nonlinear Analysis 71 (2009) 5354–5362

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Proof. Suppose on contrary that the conclusion is not true. Then for each x0 ∈ K there exists some y ∈ K such that

hAu0 , η(y, x0 )i + f (x0 , y) ≤int P 0,

∀u0 ∈ T (x0 ).

(4.1)

For every y ∈ K , define the set Ny as, Ny = {(x ∈ K ) : hAu, η(y, x)i + f (x, y) ≤int P 0, ∀u ∈ Tx}.

(4.2)

Since T is η-completely semicontinuous with respect to A and f , the set Ny is open in K with respect to the weak topology of X , for every y ∈ K . Now is easy to see S we assert that {Ny : y ∈ K } is an open cover of K with respect to the weak topology of X . Indeed, first itS that y∈K Ny ⊂ K . Second, for each x0 ∈ K by inclusion (4.1) there exists y ∈ K such that x0 ∈ Ny . Hence x0 ∈ y∈K Ny . This S S shows that K ⊂ . So the assertion is valid. The weak compactness of K implies that y∈K Ny . Consequently, K = y∈K NyS there exists a finite set {y1 , . . . , yn } ⊂ K such that K = y∈K Nyi . Hence there exists a continuous (with respect to the weak topology of X ) partition of unity {β1 , . . . , βn } subordinated to {Ny1 , . . . , Nyn } such that βj (x) ≥ 0, ∀x ∈ K , j = 1, . . . , n n X

βj (x) = 1,

∀x ∈ K

j=1

and

βj (x) =



=0, >0,

whenever x 6∈ Nyj , whenever x ∈ Nyj .

Let p : K → X be defined as p(x) = j=1 βj (x)yj , ∀x ∈ K . Since βi is continuous with respect to the weak topology of X for each i, p is continuous with respect to the weak topology of X . Let S = conv{y1 , . . . , yn } ⊂ K . Then S is a simplex of a finite dimensional space and p maps S into S. By Brouwer’s fixed point theorem [2], there exists some x0 ∈ S such that p(x0 ) = x0 . Now for any given x ∈ K , let k(x) = {j : x ∈ Nyj } = {j : βj (x) > 0}. Obviously, k(x) 6= ∅. Since x0 ∈ S ⊂ K is a fixed point of

Pn

p, we have p(x0 ) =

Pn

j=1

βj (x0 )yj and hence from inclusion (4.2) and the convexity of f we derive for each u0 ∈ T (x0 )

0 = hAu0 , η(x0 , x0 )i + f (x0 , x0 ), 0 = hAu0 , η(p(x0 ), x0 )i + f (x0 , p(x0 )),

* =

Au0 , η

n X

!+ βj (x0 )yj , x0

+f

x0 ,

j =1

=

n X

n X

! βj (x0 )yj ,

j =1

βj (x0 )[hAu0 , η(yj , x0 )i + f (x0 , yj )] ≤int P 0

j=1

which leads to a contradiction since P is proper cone. Therefore there exist x ∈ K and u ∈ T (x) such that

hAu, η(y, x)i + f (x, y) 6≤int P 0, This completes the proof.

∀y ∈ K .



Theorem 4.2. Let K be a nonempty, compact and convex subset of a real reflexive Banach space X , Y be a real Banach space ordered by a nonempty proper closed convex and pointed cone P with apex at the origin and int P 6= ∅. Let the mapping A : L(X , Y ) → L(X , Y ) and T : K → 2L(X ,Y ) be a set-valued mapping with nonempty values. Suppose following conditions hold: (i) f : K × K → Y be an affine mapping in second argument with the condition f (x, x) = 0, ∀x ∈ K ; (ii) η : X × X → X is affine in first argument with η(x, x) = 0, ∀x ∈ K ; (iii) T : K → 2L(X ,Y ) is η-strongly semicontinuous with respect to A and f . Then there exist x ∈ K and u ∈ T (x) such that

hAu, η(y, x)i + f (x, y) 6≤int P 0,

∀y ∈ K .

Proof. The proof is similar to that of Theorem 3.1 and so is omitted. If X = Rn , then Lc (Rn , Y ) = L(Rn , Y ), complete continuity is equivalent to continuity and complete semicontinuity is equivalent to strong semicontinuity.  By Theorem 4.1, we can obtain the following result: Corollary 4.1. Let K be a nonempty, bounded closed and convex subset of a real reflexive Banach space Rn , Y be a real Banach space ordered by a nonempty proper closed convex and pointed cone P with apex at the origin and int P 6= ∅. Let the mapping n A : L(Rn , Y ) → L(Rn , Y ) and T : K → 2L(R ,Y ) be a set-valued mapping with nonempty values. Suppose following conditions hold:

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F. Usman, S.A. Khan / Nonlinear Analysis 71 (2009) 5354–5362

(i) f : K × K → Y be an affine mapping in second argument with the condition f (x, x) = 0, ∀x ∈ K ; (ii) η : Rn × Rn → Rn is affine mapping in first argument with the condition η(x, x) = 0, ∀x ∈ K ; n (iii) T : K → 2L(R ,Y ) is strongly semicontinuous with respect to A and f then there exist x ∈ K and u ∈ T (x) such that

hAu, η(y, x)i + f (x, y) 6≤int P 0,

∀y ∈ K .

Acknowledgment The first author is thankful to Junior Research Fellowship, University Grants Commission, India, for supporting this research work. References [1] F. Giannessi, Theorems of alternative, quadratic programs and complementarity problems, in: R.W. Cottle, F. Giannessi, J.L. Lions (Eds.), Variational Inequalities and Complementarity Problems, John Wiley and Sons, New York, 1980, pp. 151–186. [2] L. Brouwer, Zur invarianz des n-dimensional Gebietes, Math. Ann. 71 (1912) 305–313. [3] S.S. Chang, B.S. Lee, Y.Q. Chen, Variational inequalities for monotone operators in nonreflexive banach spaces, Appl. Math. Lett. 8 (1995) 29–34. [4] Y.Q. Chen, On the semimonotone operator theory and applications, J. Math. Anal. Appl. 231 (1999) 177–192. [5] G.Y. Chen, X.Q. Yang, The vector complementarity problem and its equivalences with the weak minimal element, J. Math. Anal. Appl. 153 (1990) 136–158. [6] R.W. Cottle, J.C. Yao, Pseudomonotone complementarity problems in Hilbert spaces, J. Optim. Theory Appl. 78 (1992) 281–295. [7] X.P. Ding, Existence and algorithm of solutions for generalized mixed implicit quasivariational inequalities, Appl. Math. Comput. 113 (2000) 67–80. [8] K. Fan, Some properties of convex sets related to fixed-point theorems, Math. Ann. 266 (1984) 519–537. [9] K. Fan, A generalization of Tychonoff’s fixed-point theorem, Math. Ann. 142 (1961) 305–310. [10] Y.P. Fang, N.J. Huang, Variational-like inequalities with generalized monotone mappings in Banach spaces, J. Optim. Theory Appl. 118 (2) (2003) 327–338. [11] Y.P. Fang, Y.J. Cho, N.J. Huang, S.M. Kang, Generalized nonlinear quasi-variational-like inequalities for set-valued mappings in Banach spaces, Math. Inequal. Appl. 6 (2003) 331–337. [12] D. Goeleven, D. Motreanu, Eigenvalue and dynamic problems for variational and hemivariational inequalities, Commun. Appl. Nonlinear Anal. 3 (1996) 1–21. [13] N.J. Huang, Y.P. Fang, On vector variational inequalities in reflexive Banach spaces, J. Global Optim. 32 (2005) 495–505. [14] B.S. Lee, M.F. Khan, Salahuddin, Generalized vector variational-type inequalities, Comp. Math. Appl. 55 (2008) 1164–1169. [15] J.S.B. Nadler, Multi-valued contraction mappings, Pacific J. Math. 30 (1969) 475–488. [16] R.U. Verma, On monotone nonlinear variational inequality problems, Comment. Math. Univ. Carolin. 39 (1998) 91–98. [17] R.U. Verma, Nonlinear variational inequalities on convex subsets of Banach spaces, Appl. Math. Lett. 10 (1997) 25–27. [18] X.Q. Yang, G.Y. Chen, A class of nonconvex functions and prevariational inequalities, J. Math. Anal. Appl. 169 (1992) 359–373. [19] L.C. Zeng, J.C. Yao, Existence of solutions of generalized vector variational inequalities in reflexive Banach spaces, J. Global Optim. 36 (2006) 483–497.