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Generalized vector variational inequalities with star-pseudomonotone and discontinuous operators
Nonlinear Analysis 68 (2008) 2859–2871 www.elsevier.com/locate/na
Generalized vector variational inequalities with star-pseudomonotone and discontinuous operatorsI Bui Trong Kien a , Ngai Ching Wong b , Jen-Chih Yao b,∗ a Department of Information and Technology, National University of Civil Engineering, 55 Giai Phong, Hanoi, Viet Nam b Department of Applied Mathematics, National Sun Yat-Sen University, Kaohsiung, Taiwan
Received 10 January 2006; accepted 22 February 2007
Abstract It is well known that a vector variational inequality can be a very efficient model for use in studying vector optimization problems. By using the Ky Fan fixed point theorem and the scalarization method we will prove some existence theorems for strong solutions for generalized vector variational inequalities where discontinuous and star-pseudomonotone operators are involved. Our results can be applied to the study of the existence of solutions of vector optimal problems. Some examples are given and analyzed. c 2007 Elsevier Ltd. All rights reserved.
Keywords: Generalized vector equilibrium problem; Generalized vector variational inequality; Upper semicontinuity; C-convexity; C-upper semicontinuity; Star-pseudomonotonicity
1. Introduction Throughout this paper, let Z be a (Hausdorff) topological vector space, and let C be a closed and convex cone of Z such that int C 6= ∅ and C 6= Z where int C denotes the interior of C. Let Ac , A and co A denote the complement, the closure and the convex hull of a subset A of Z , respectively. A vector ordering in Z is defined by setting z1 ≤ z2
if and only if
z 2 − z 1 ∈ C.
z1 < z2
if and only if
z 2 − z 1 ∈ int C,
z 1 6< z 2
if and only if
z 2 − z 1 6∈ int C.
Write
and
I This research was partially supported by a grant from the National Science Council of Taiwan, ROC. ∗ Corresponding author. Tel.: +886 7 5253816; fax: +886 7 5253809.
E-mail addresses: [email protected] (B.T. Kien), [email protected] (N.C. Wong), [email protected] (J.-C. Yao). c 2007 Elsevier Ltd. All rights reserved. 0362-546X/$ - see front matter doi:10.1016/j.na.2007.02.032
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Let X , Y be topological vector spaces, and let K ⊆ X and D ⊆ Y be nonempty. Let T : K → 2 D be a multifunction and f : K × D × K → Z be a single-valued mapping. The generalized vector variational inequality GVVI(K , T, f ) is the following problem: (V) Find xˆ ∈ K such that ∃ yˆ ∈ T (x), ˆ ∀z ∈ K ,
we have f (x, ˆ yˆ , z) 6< 0.
Together with (V), the following auxiliary problems are also of interest: (W) Find xˆ ∈ K such that ∀z ∈ K , ∃ yˆ ∈ T (x), ˆ
we have f (x, ˆ yˆ , z) 6< 0,
and the so-called Minty vector variational inequality: (M) Find xˆ ∈ K such that ∀z ∈ K , ∀ yˆ ∈ T (z),
we have f (x, ˆ yˆ , z) 6< 0.
We will denote by SV , SW and S M the solution sets of the problems (V), (W) and (M), respectively. A point xˆ ∈ SV is called a strong solution of GVVI(K , T, f ), while a point x0 in SW is called a weak solution. It is interesting that problem (V) covers several generalized variational inequalities and equilibrium problems. Here are some examples. (GVI) Put Z = R, C = R+ , Y = X ∗ = D and f (x, y, z) = hy, z − xi. Then (V) reduces to the generalized variational inequality problem: Find xˆ ∈ K
and
yˆ ∈ T (x) ˆ
such that h yˆ , z − xi ˆ ≥ 0, ∀z ∈ K .
(1)
(EP) Put Z = R, C = R+ , Y = X = D and f (x, y, z) = g(x, z) + h(z) − h(x) where g is a real valued function on K × K such that g(x, x) = 0 for all x ∈ K . Then (V) reduces to the equilibrium problem: Find xˆ ∈ K
such that g(x, ˆ z) + h(z) − h(x) ˆ ≥ 0, ∀z ∈ K .
(2)
The existence of solutions of (GVI) has been studied by many authors; see for instance Cubiotti [9], Cubiotti and Yen [8], Li [19], Yao [28,29], Yao and Guo [27], Yen [30,31], Aussel and Hadjisavvas [3], Aussel and Luc [4], Bianchi, Hadjisavvas and Shaible [5], Blum and Oettli [6], Cubiotti [10], Cubiotti and Yao [11], Hadjisavvas [14], Kien, Wong and Yao [17], Schaible and Yao [24] and references therein. Also, results about the existence of weak solutions for the generalized vector variational inequality problem were investigated by many authors; see for instance Ansari and Yao [1,2], Chen and Craven [7], Lee et al. [18], Oettli [22]. However, very few papers that have appeared in the literature discuss the existence of a strong solution of GVVI(K , T, f ). The aim of this paper is to derive some results on the existence of strong solutions for GVVI(K , T, f ) with discontinuous operators and star-pseudomonotone operators. Namely, we will prove some existence theorems in which T can be discontinuous and K can be unbounded. Our results extend some preceding results. Besides, an existence theorem involving star-pseudomonotone operators is also established. In order to obtain these results, we utilize the fixed point theorem due to Ky Fan and the scalarization method. In Section 2, we recall some auxiliary results. Section 3 is devoted to presenting existence results for the case of discontinuous operators. Section 4 will be devoted to an existence result for the case where the operators are starpseudomonotone. 2. Preliminaries Let X and Y be topological vector spaces, and let Z be a locally convex space. Let C be a closed and convex proper cone in Z with interior int C that is nonempty. Let K be a nonempty convex subset of X , and D be a nonempty convex subset of Y . Let T : K → 2 D be a multifunction. We denote by ΓT the graph of T , that is, ΓT = {(x, y) ∈ X × Y | y ∈ T (x)}. Definition 2.1. (a) T is said to be lower semicontinuous (l.s.c. for short) at x ∈ K if for any open set V in E such that V ∩ T (x) 6= ∅, there exists a neighborhood U of x such that V ∩ T (x) 6= ∅,
∀x ∈ U ∩ K .
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(b) T is said to be upper semicontinuous (u.s.c. for short) at x ∈ K if for any open set V in E such that T (x) ⊂ V , there exists a neighborhood W of x with the property that T (x) ⊂ V,
∀x ∈ W ∩ K .
(c) T is said to be lower semicontinuous (resp., upper semicontinuous) on K if it is l.s.c. (resp., u.s.c.) at every point x ∈ K. (d) A single-valued mapping g : X → Z is said to be C-upper semicontinuous on X if for every z ∈ Z the set g −1 (z − int C) is open in X . Tanaka [26] proved that g is C-upper semicontinuous on X if and only if for each fixed x ∈ X and for any y ∈ int C, there exists a neighborhood U of x such that g(u) ∈ g(x) + y − int C,
∀u ∈ U.
(e) g is C-lower semicontinuous on X if for each fixed x ∈ X and for any y ∈ int C, there exists a neighborhood U of x such that g(x) ∈ g(u) + y − int C,
∀u ∈ U.
(f) If g is C-lower semicontinuous and C-upper semicontinuous on X then g is called C-continuous on X . (g) A single-valued mapping h : K → Z is called C-convex if for every x, x 0 ∈ K and t ∈ [0, 1] one has th(x) + (1 − t)h(x 0 ) − h(t x + (1 − t)x 0 ) ∈ C. It is called C-strongly convex if for every x, x 0 ∈ K and t ∈ [0, 1] one has th(x) + (1 − t)h(x 0 ) − h(t x + (1 − t)x 0 ) ∈ int C ∪ {0}. If −h is C-convex (resp., C-strongly convex) then h is said to be C-concave (resp., C-strongly concave) on K . We continue by recalling the scalarization method. Let Z ∗ be the topological dual of Z . The set C ∗ := {z ∗ ∈ Z ∗ : hz ∗ , zi ≥ 0 for all z ∈ C} S is the polar cone of C. Note that C ∗ has a weakly* compact base B, that is, C ∗ = t>0 t B, and B is convex and weakly* compact with 0 6∈ B (see Luc [20]). When int C 6= ∅ and z ∈ int C, z 6= 0, the set B = {z ∗ ∈ C ∗ : hz ∗ , zi = 1} ∗ = C ∗ \ {0}. is a weakly* compact convex base for C ∗ . We put C+ Note that from the bipolar theorem (see Peressini [23]) we have
z ∈ C ⇐⇒ [hz ∗ , zi ≥ 0, ∀z ∗ ∈ C ∗ ] ⇐⇒ [hz ∗ , zi ≥ 0, ∀z ∗ ∈ B]
(3)
z ∈ int C ⇐⇒ [hz , zi > 0, ∀z ∈
(4)
∗
∗
∗ C+ ]
⇐⇒ [hz , zi > 0, ∀z ∈ B]. ∗
∗
(see Jeyakumar et al. [15] for developments). The following proposition is a useful tool for the scalarization method. ∗ . Let φ : K → R be a mapping Proposition 2.2. Let g be a single-valued mapping from K into Z and u ∗ ∈ C+ defined by φ(x) = hu ∗ , g(x)i for all x ∈ K . Then the following assertions are valid: (a) If g is C-convex (resp., C-concave) then φ is convex (resp., concave). (b) If g is C-upper semicontinuous (resp., C-lower semicontinuous) then φ u.s.c. (resp., l.s.c.).
Proof. (a) Since g is C-convex, then for all x, x 0 ∈ K and t ∈ [0, 1] one has tg(x) + (1 − t)g(x 0 ) − g(t x + (1 − t)x 0 ) ∈ C. By (5) we have hu ∗ , tg(x) + (1 − t)g(x 0 ) − g(t x + (1 − t)x 0 )i ≥ 0. Hence thu ∗ , g(x)i + (1 − t)hu ∗ , g(x 0 )i ≥ hu ∗ g(t x + (1 − t)x 0 )i. This implies that tφ(x) + (1 − t)φ(x 0 ) ≥ φ(t x + (1 − t)x 0 ) and so φ is convex. By a similar argument we show that if g is C-concave then φ is concave.
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(b) Let xn → x. We want to show that lim supn→∞ φ(xn ) ≤ φ(x). Choose y j ∈ int C such that y j → 0. Then for each j > 0 there exists a neighborhood U j of x such that g(u) ∈ g(x) + y j − int C,
∀u ∈ U j .
Therefore for each j there exists n j > 0 such that g(xn ) ∈ g(x) + y j − int C,
∀n > n j .
By (5) it follows that hu ∗ , g(xn ) − g(x) − y j i < 0. Hence φ(xn ) = hu ∗ , g(xn )i = hu ∗ , g(xn ) − g(x) − y j + g(x) + y j i = hu ∗ , g(xn ) − g(x) − y j i + hu ∗ , g(x) + y j i < hu ∗ , g(x)i + hu ∗ , y j i for all n > n j . This implies that lim sup φ(xn ) ≤ hu ∗ , g(x)i + hu ∗ y j i. n→∞
Letting j → ∞ and noting that hu ∗ , y j i → 0, we obtain lim sup φ(xn ) ≤ hu ∗ , g(x)i = φ(x). n→∞
In the same way, we obtain the second assertion of (b).
One of the tools in deriving our results is the Ky Fan theorem. We state it below for convenience (see Shioji [25]). Theorem 2.3 (Ky Fan). Let K be a nonempty subset of a topological vector space X and G : K → 2 X be a multifunction with closedS values such that the convex hull of every finite subset {x1 , x2 , . . . , xnT } of X is contained in n the corresponding union i=1 G(xi ). If there exists x0 ∈ K such that G(x0 ) is compact, then x∈K G(x) 6= ∅. We recall that the multifunction G satisfying the property in Theorem 2.3 is called a KKM mapping. We also need the following Sion minimax theorem (see Jeyakumar et al. [15] and Zeidler [32]). Theorem 2.4 (Sion). Let A and B be convex subsets of some real topological vector spaces with B compact, and let p : A × B → R. If p(., b) is lower semicontinuous and quasiconvex on A for all b ∈ B, and if p(a, .) is upper semicontinuous and quasiconcave on B for all a ∈ A, then inf max p(a, b) = max inf p(a, b).
a∈A b∈B
b∈B a∈A
3. Solution existence in the case of discontinuous operators In addition to the basic assumptions stated at the beginning of Section 2, we further assume in this section that f : K × D × K → Z is a single-valued mapping satisfying some of the following conditions: (i) the set T (x) is nonempty, weakly compact, and convex for all x ∈ K ; ∗ such that for each z ∈ K , the set (ii) there exists u ∗ ∈ C+ F(z) = {x ∈ K | sup hu ∗ , f (x, y, z)i ≥ 0} y∈T (x)
is closed; (iii) for each x ∈ K and y ∈ T (x) one has f (x, y, x) = 0; (iv) for each x ∈ K and y ∈ T (x), the function f (x, y, ·) is C-convex and C-l.s.c. on K ; (v) for each (x, z) ∈ K × K , the function f (x, ·, z) is C-concave and C-u.s.c. on D. Theorem 3.1. Let X be a dual Banach space and Y be a locally convex space. Assume conditions (i)–(v) hold. Also assume that F(z) is weakly* closed for all z in K and
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(vi) there is a bounded subset K 0 of K such that for each x ∈ K \ K 0 , there exists z ∈ K 0 with f (x, y, z) < 0 for all y ∈ T (x). Then there exists a point xˆ ∈ K 0 such that max hu ∗ , f (x, ˆ y, z)i ≥ 0,
y∈T (x) ˆ
∀z ∈ K .
Moreover, there exists yˆ ∈ T (x) ˆ such that f (x, ˆ yˆ , z) 6< 0,
∀z ∈ K .
(5)
Proof. We choose r > 0 such that int Br contains K 0 , where Br = {x ∈ X : kxk ≤ r }. Set Ω = K ∩ Br . We define a multifunction G : Ω → 2 X by the formula G(z) = {x ∈ Ω | sup hu ∗ , f (x, y, z)i ≥ 0} y∈T (x)
for z ∈ Ω .
By Proposition 2.2, the function y 7→ hu ∗ , f (x, y, z)i is concave and u.s.c. In particular, for each real number δ, the convex set {y ∈ Y : hu ∗ , f (x, y, z)i ≥ δ} is closed, and thus weakly closed, in the locally convex space Y . Hence, this function is also weakly u.s.c. Since T (x) is a weakly compact set, we have G(z) = {x ∈ Ω | max hu ∗ , f (x, y, z)i ≥ 0}. y∈T (x)
We claim that G is a KKM mapping. Indeed, by (iii) we have z ∈ G(z) for all z ∈ Ω . Hence G(z) 6= ∅ for all z ∈ Ω . Since G(z) = F(z) ∩ Br , G(z) is a weak* compact subset of Ω , by the Banach–Alaoglu Theorem. Taking any z 1 , z 2 ∈ Ω we show that co{z 1 , z 2 } ⊂ G(z 1 ) ∪ G(z 2 ). Let z ∈ co{z 1 , z 2 }. Then z = t z 1 + (1 − t)z 2 for some t ∈ [0, 1]. When t = 0 or t = 1, the claim is trivial. Consider t ∈ (0, 1), with z 6∈ G(z i ); i = 1, 2. Since z ∈ Ω , we obtain max hu ∗ , f (z, y, z i )i < 0;
y∈T (z)
i = 1, 2.
By (iv) and Proposition 2.2, the function hu ∗ , f (x, y, ·)i is convex. By (iii) we have 0 = max hu ∗ , f (z, y, z)i y∈T (z)
≤ t max hu ∗ , f (z, y, z 1 )i + (1 − t) max hu ∗ , f (z, y, z 2 )i, y∈T (z)
y∈T (z)
< 0, which is absurd. Thus we must have z ∈ G(z 1 ) ∪ G(z 2 ) and so co{z 1 , z 2 } ⊂ G(z 1 ) ∪ G(z 2 ). By a similar argument we can show that n [ co{z 1 , z 2 , . . . , z n } ⊂ G(z i ) i=1
for any finite subset {x1 , x2 , . . . , xn } in Ω . By Theorem 2.3, \ ∃xˆ ∈ Ω such that xˆ ∈ G(z). z∈Ω
In other words, ∃xˆ ∈ Ω ,
max hu ∗ , f (x, ˆ y, z)i ≥ 0,
y∈T (x) ˆ
∀z ∈ Ω .
(6)
By (vi), we see that xˆ ∈ K 0 . We need to show that max hu ∗ , f (x, ˆ y, z)i ≥ 0,
y∈T (x) ˆ
∀z ∈ K .
(7)
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Take any z ∈ K . Since xˆ ∈ int Br we have xˆ + t (z − x) ˆ ∈ K ∩ int Br for a sufficiently small t ∈ (0, 1). Then (6) implies that max hu ∗ , f (x, ˆ y, x) ˆ + t (z − x)i ˆ = max hu ∗ , f (x, ˆ y, (1 − t)xˆ + t z)i ≥ 0.
y∈T (x) ˆ
y∈T (x) ˆ
(8)
By the convexity of the function hu ∗ , f (x, y, ·)i, we obtain from (8) the following: 0 ≤ max hu ∗ , f (x, ˆ y, (1 − t)xˆ + t z)i y∈T (x) ˆ
≤ t max hu ∗ , f (x, ˆ y, z)i + (1 − t) max hu ∗ , f (x, ˆ y, x)i. ˆ y∈T (x) ˆ
y∈T (x) ˆ
Since f (x, ˆ y, x) ˆ = 0, this implies (7). ˆ y, z)i. By the To complete the proof we will use the result recalled in Theorem 2.4. Put p(z, y) = hu ∗ , f (x, assumptions of the theorem and Proposition 2.2, the function p(·, y) is l.s.c. and convex. Also, the function p(z, ·) is u.s.c. and concave, and thus weakly u.s.c. By Theorem 2.4 and (7) we get 0 ≤ inf max p(z, y) = max inf p(z, y). z∈K y∈T (x) ˆ
y∈T (x) ˆ z∈K
Since y 7→ p(z, y) is weakly u.s.c., the function y 7→ infz∈K p(z, y) is also weakly u.s.c. Hence, there exists yˆ ∈ T (x) ˆ such that inf p(z, yˆ ) = max inf p(z, y) ≥ 0. y∈T (x) ˆ z∈K
z∈K
hu ∗ ,
It follows that f (x, ˆ yˆ , z)i ≥ 0, ∀z ∈ K . This is equivalent to hu ∗ , − f (x, ˆ yˆ , z)i ≤ 0, ∀z ∈ K . By (4) we have − f (x, ˆ yˆ , z) 6∈ int C for all z ∈ K , and (5) is obtained. The proof of the theorem is complete. ∗ . It is difficult to check when such a u ∗ In Theorem 3.1, condition (ii) is checked for a particular element u ∗ ∈ C+ exists. Fortunately, under certain conditions (e.g., T is upper semicontinuous and f (·, ·, z) is C-continuous), we can ∗. choose any u ∗ ∈ C+ The following example illustrates our result.
Example 3.2. Let X = R, K = [0, 1] ⊂ X , Y = R, D = [1, 4], Z = R2 and C = R2+ = {(x, y) | x ≥ 0, y ≥ 0}. Let T and f be defined by [2, 4], if x = 0; T (x) = {1}, if 0 < x ≤ 1, and f (x, y, z) = ( f 1 (x, y, z), f 2 (x, y, z)), where f 1 (x, y, z) = y(z 2 − x 2 )
and
f 2 (x, y, z) = y(z − x).
Then the set {0} × [2, 4] is a solution set of GVVI(K , T, f ). Moreover, T is not upper semicontinuous on [0, 1]. Indeed, first we get C ∗ = {(u, v) | u ≥ 0, v ≥ 0};
−int C = {(x, y) | x < 0, y < 0}.
Taking u ∗ = (1, 0) we want to check all conditions of Theorem 3.1. (i) It is obvious that T (x) is a nonempty compact convex set for each x ∈ [0, 1]. (ii) For each z ∈ [0, 1] we have {x ∈ K | sup hu ∗ , f (x, y, z)i ≥ 0} = {x ∈ [0, 1] | sup y(z 2 − x 2 ) ≥ 0} y∈T (x)
y∈T (x)
= [0, z], which is a closed set.
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(iii) For each x ∈ M and y ∈ T (x) we have f (x, y, x) = (0, 0). (iv) For each x ∈ K and y ∈ T (x) ⊂ [1, 4], the functions f 1 (x, y, ·) and f 2 (x, y, ·) are convex and continuous. Hence, f is C-convex and C-lower semicontinuous. (v) For each x, z ∈ K × K the functions f 1 (x, ., z) and f 2 (x, ·, z) are linear and continuous. (vi) The claim is obvious. Thus all conditions of Theorem 3.1 are fulfilled. Taking xˆ = 0 and yˆ ∈ T (0) = [2, 4], we have f (0, yˆ , z) = (yz 2 , yz 2 ) 6< 0,
∀z ∈ [0, 1].
Hence the set {0} × [2, 4] is a solution set of the problem. It is observed that T is not upper semicontinuous on K . Indeed, xn = 1/n → 0, yn = 1 ∈ T (xn ) but 1 6∈ T (0) = [2, 4]. Theorem 3.1 requires that X is a dual Banach space. This requirement guarantees the weak* compactness of bounded closed convex sets. The following results relax this assumption by strengthening the coercive condition (vi). Theorem 3.3. Let Y be a locally convex space (resp. topological vector space). Assume conditions (i)–(v), T (x) is weakly compact (resp. compact) for each x in K , F(z) is weakly closed (resp. closed) for each z in K and (vi)0 there exists a weakly compact (resp. compact) K 0 of K and z 0 ∈ K such that for each x ∈ K \ K 0 , one has f (x, y, z 0 ) < 0 for all y ∈ T (x). Then there exists a point xˆ ∈ K and yˆ ∈ T (x) ˆ such that f (x, ˆ yˆ , z) 6< 0,
∀z ∈ K .
Proof. Assume that Y is locally convex. We have F(z) = {x ∈ K | max hu ∗ , f (x, y, z)i ≥ 0}. y∈T (x)
We claim that F(z 0 ) ⊂ K 0 . Indeed, suppose that the claim is false. Then there is an x ∈ F(z 0 )\ K 0 and so x ∈ K \ K 0 . By (vi), − f (x, y, z 0 ) ∈ int C for all y ∈ T (x). By (5), it follows that hu ∗ , − f (x, y, z 0 )i > 0 for all y ∈ T (x). Hence max hu ∗ , f (x, y, z 0 )i < 0.
y∈T (x)
This implies that x 6∈ F(z 0 ), which is absurd. Thus F(z 0 ) is a weakly compact set. As in the proof of Theorem 3.1, we can show that F(z) is a KKM mapping, and the desired conclusion follows accordingly. The case where Y is not locally convex is similar. Theorem 3.4. Let X and Y be topological vector spaces. Assume conditions (i)–(v) and (vi)00 there exist compact sets K 1 ⊆ K 2 in K such that K 1 is finite dimensional, and for each x ∈ K \ K 2 there exists z ∈ K 1 with f (x, y, z) < 0 for all y ∈ T (x). Then there exists a point xˆ ∈ K 2 and yˆ ∈ T (x) ˆ such that f (x, ˆ yˆ , z) 6< 0,
∀z ∈ K .
Proof. We denote by F the set of all finite-dimensional subspaces of X containing K 1 . Fix any S ∈ F and put Ω = K ∩ S,
T S = T |Ω ,
f S = f |Ω ×D×Ω .
The task is to check that Theorem 3.1 can be applied to the problem GVVI(Ω , T S , f S ), where Ω plays a role as K in Theorem 3.1. (a1 ) For each x ∈ Ω we have T (x) = T S (x) 6= ∅. Hence condition (i) of Theorem 3.1 is valid.
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(a2 ) For each z ∈ Ω , we have {x ∈ Ω | sup hu ∗ , f (x, y, z)i ≥ 0} = {x ∈ K | sup hu ∗ , f (x, y, z)i ≥ 0} ∩ S. y∈T (x)
y∈T (x)
Since S is finite dimensional, the set {x ∈ Ω | sup hu ∗ , f (x, y, z)i ≥ 0} y∈T (x)
is closed. This implies that condition (ii) of Theorem 3.1 is valid. (a3 ) Conditions (iii)–(vi) are automatically fulfilled. By Theorem 3.1, there exists x S ∈ Ω such that max hu ∗ , f S (x S , yˆ , z)i ≥ 0,
y∈T S (x S )
∀z ∈ Ω .
Since T S (x S ) = T (x S ), f S (x S , yˆ , z) = f (x S , yˆ , z) we have max hu ∗ , f (x S , yˆ , z)i ≥ 0,
y∈T (x S )
∀z ∈ K ∩ S.
(9)
Put Q S = {x ∈ K | x satisfies (9)}. Then Q S 6= ∅ because x S ∈ Q S . Moreover, the family {Q S : S ∈ F} has the finite intersection property. Indeed, taking any V1 , V2 ∈ F and putting M = span{V1 , V2 }, we have Q M ⊆ Q V1 ∩ Q V2 . This implies that Q M ⊆ Q V1 ∩ Q V2 ⊆ Q V1 ∩ Q V2 . T By condition (vi)”, Q S ⊂ K 2 , which is compact. Hence we get S∈F Q S 6= ∅. Thus there exists xˆ ∈ K 2 such that xˆ ∈ Q S for all S ∈ F. Fix any S ∈ F. Then there exists a net {xα }α∈Λ in Q S such that xα → x. ˆ By the definition of Q S one has max hu ∗ , f (xα , y, z)i ≥ 0,
y∈T (xα )
∀z ∈ K ∩ S.
(10)
Fixing z ∈ K ∩ S and using assumption (ii), we obtain from (10) that max hu ∗ , f (x, ˆ yˆ , z)i ≥ 0.
yˆ ∈T (x) ˆ
In summary, we have proved that ∃xˆ ∈ K 2 , ∀z ∈ K ∩ S,
max hu ∗ , f (x, ˆ y, z)i ≥ 0.
y∈T (x) ˆ
(11)
We now take any v ∈ K and put S 0 = span{S, v}. Since xˆ also satisfies (11) for S 0 we have ∃xˆ ∈ K 2 ,
ˆ yˆ , v)i ≥ 0. max hu ∗ , f (x,
y∈T (x) ˆ
Thus we have ∃xˆ ∈ K 2 , ∀v ∈ K ,
max hu ∗ , f (x, ˆ yˆ , v)i ≥ 0.
y∈T (x) ˆ
By the arguments similar to those in the proof of Theorem 3.1, we show that there exists yˆ ∈ T (x) ˆ such that f (x, ˆ yˆ , z) 6< 0 for all z ∈ K . The proof is complete. Theorem 3.5. Let X, Y be locally convex spaces. Assume conditions (i)–(vi), K 0 is weakly compact and F(z) is weakly closed for all z in K . Then there exists a point xˆ ∈ K 0 and yˆ ∈ T (x) ˆ such that f (x, ˆ yˆ , z) 6< 0,
∀z ∈ K .
Proof. For each absolutely convex weakly compact subset S of X , let X S be the subspace of X generated by S. Namely, [ XS = t S. t>0
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Equip X S with the norm defined by the gauge of S; namely, kxk = inf{t > 0 : x ∈ t S}. Then (X S , k · k) is a dual Banach space by the Dixmier–Ng Theorem; see [13,21]. Indeed, the predual of (X S , k · k) consists of those linear functionals on X continuous on S. Hence, the weak* topology of (X S , k · k) is finer than the weak topology σ (X S , X 0 ) of X S inherited from X . As a result, F(z) is weakly* closed in (X S , k·k) for all z in K ∩ X S . Let F be the set of all dual Banach spaces (X S , k · k), where S is an absolutely convex weakly compact subset of X containing K 0 . We can then proceed as in the proof of Theorem 3.4. We remark that in Theorem 3.5 we can assume instead that X is a topological vector space such that X 0 separates points in X . This condition is enough to ensure the conclusion of the Dixmier–Ng Theorem, as one can check by going through the proof of Ng [21] carefully. 4. Solution existence in the case of star-pseudomonotone operators We first introduce some notions related to the star-pseudomonotone operator. ∗ if for Definition 4.1. (a) The operator T : K → 2 D is called star-pseudomonotone with respect to f and u ∗ ∈ C+ any (x1 , y1 ), (x2 , y2 ) ∈ ΓT ,
hu ∗ , f (x1 , y1 , x2 )i ≥ 0
implies hu ∗ , f (x1 , y2 , x2 )i ≥ 0.
∗ if for any (x , y ), (x , y ) ∈ Γ , (b) T is called star-quasimonotone with respect to f and u ∗ ∈ C+ T 1 1 2 2
hu ∗ , f (x1 , y1 , x2 )i > 0
implies hu ∗ , f (x1 , y2 , x2 )i ≥ 0.
∗ if for any finite set {x , x , . . . , x } in K (c) T is called star-properly quasimonotone with respect to f and u ∗ ∈ C+ 1 2 2 and x ∈ co{x1 , x2 , . . . , x2 }, there exists some i ∈ {1, 2, . . . , n} such that
hu ∗ , f (x, y, xi )i ≥ 0,
∀y ∈ T (xi ).
The notions of quasimonotone operators and pseudomonotone operators were first introduced by Karamardian and Schaible [16]. For this case we put Y = D = X ∗ , Z = R, C = R+, f (x, y, z) = hy, z − xi and u ∗ = 1. Then T is a pseudomonotone operator (resp., quasimonotone) in the sense of theirs. While the notion of proper quasimonotonicity was introduced by Daniilidis and Hadjisavvas [12]. We give an example below illustrating the definition. Example 4.2. Let X = Y = R, K = [−1, 1] ⊂ X , D = [−1, 1], Z = R2 and C = R2+ = {(x, y) | x ≥ 0, y ≥ 0}. Let T and f be defined by if x > 0 {1} T (x) = [−1, 1] if x = 0 {−1} if x < 0, and f (x, y, z) = ( f 1 (x, y, z), f 2 (x, y, z)), where f 1 (x, y, z) = f 2 (x, y, z) = y(z − x). The T is star-pseudomonotone with respect to f and u ∗ = (1, 1). Indeed, we first note that C ∗ = {(u, v) | u ≥ 0, v ≥ 0};
−int C = {(x, y) | x < 0, y < 0}.
∗ . Since T is monotone, we get It is obvious that u ∗ ∈ C+
hy2 − y1 , x2 − x1 i ≥ 0
∀(x1 , y1 ), (x2 , y2 ) ∈ ΓT .
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Hence hy2 , x2 − x1 i ≥ hy1 , x2 − x1 i. This implies that f i (x1 , y2 , x2 ) ≥ f i (x1 , y1 , x2 );
i = 1, 2.
By adding these inequalities we have f 1 (x1 , y2 , x2 ) + f 2 (x1 , y2 , x2 ) ≥ f 1 (x1 , y1 , x2 ) + f 2 (x1 , y1 , x2 ). It follows that hu ∗ , f (x1 , y2 , x2 )i ≥ hu ∗ , f (x1 , y1 , x2 )i and we obtain the assertion. The following are some relations among star-monotone operators. Proposition 4.3. (a) If T is star-pseudomonotone with respect to f and u ∗ then T is also star-quasimonotone with respect to f and u ∗ . (b) Suppose T is star-pseudomonotone with respect to f and u ∗ , and the following conditions are fulfilled: (i) for each x ∈ K and each y ∈ T (x), f (x, y, x) = 0; (ii) for each x ∈ K and for each y ∈ T (x), the function f (x, y, ·) is C-convex. Then T is also star-properly quasimonotone with respect to f and u ∗ . Pn Proof. (a) isPobvious. For (b), let {x1 , x2 , . . . , xn } ⊂ K and x ∈ co{x1 , x2 , . . . , xn }. Then x = i=1 ti x i with n ti ≥ 0 and i=1 ti = 1. By (i) we have hu ∗ , f (x, y, x)i = 0 for all y ∈ T (x). Since f (x, y, ·) is C-convex, by Proposition 2.2 the function hu ∗ , f (x, y, ·)i is convex. Hence 0 = hu ∗ , f (x, y, x)i ≤
n X
ti hu ∗ , f (x, y, xi )i.
i=1
It follows that there exists i ∈ {1, 2, . . . , n} satisfying hu ∗ , f (x, y, xi )i ≥ 0 for some y ∈ T (x). Since T is starpseudomonotone, hu ∗ , f (x, y, xi )i ≥ 0 for all y ∈ T (xi ). The proof is complete. Example 4.4. Consider Example 4.2. We see that f satisfies conditions of Proposition 4.3. Hence T is star-properly quasimonotone with respect to f and u ∗ = (1, 1). ∗ if for all x, z in K the following Definition 4.5. T is called upper sign-continuous on K with respect to f and u ∗ ∈ C+ is satisfied:
(∀t ∈ (0, 1), inf hu ∗ , f (x, y, z)i ≥ 0) H⇒ sup hu ∗ , f (x, y, z)i ≥ 0, y∈T (xt )
y∈T (x)
where xt = (1 − t)x + t z. The notion of upper sign-continuous operators was first introduced by Hadjisavvas [14]. In the above definition by putting Y = D = X ∗ , Z = R, C = R+ , u ∗ = 1 and f (x, y, z) = hy, z − xi we also obtain the upper sign-continuous concept in the sense of Hadjisavvas. The following lemma is a type of Minty lemma. ∗ , xˆ ∈ K and the following conditions hold: Lemma 4.6. Suppose that Y is locally convex. Let u ∗ ∈ C+
(i) (ii) (iii) (iv) (v)
for each x ∈ K and y ∈ D one has f (x, y, x) = 0; for each x ∈ K and y ∈ T (x) the function f (x, y, ·) is C-convex and C-lower semicontinuous; for each (x, z) ∈ K × K , the function f (x, ·, z) is C-concave and C-upper semicontinuous; T is upper sign-continuous with respect to f and u ∗ , with weakly compact and convex values; ˆ y, z)i ≥ 0 for all z ∈ K and y ∈ T (z). hu ∗ , f (x,
Then xˆ ∈ SV .
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Proof. Rewrite condition (v) as ∀z ∈ K , ∀y ∈ T (z),
hu ∗ , f (x, ˆ y, z)i ≥ 0.
(12)
Taking any z ∈ K , we have z t = (1 − t)xˆ + t z ∈ K for t ∈ (0, 1). Hence (16) implies that hu ∗ , f (x, ˆ y, z t )i ≥ 0,
∀y ∈ T (z t ).
(13)
We claim that hu ∗ , f (x, ˆ y, z)i ≥ 0,
∀y ∈ T (z t ).
(14)
Suppose, on the contrary, that the assertion is false. Then there exists t ∈ (0, 1) and yt ∈ T (z t ) such that hu ∗ , f (x, ˆ yt , z)i < 0.
(15)
By (ii) and Proposition 2.2 we have thu ∗ , f (x, ˆ yt , z)i + (1 − t)hu ∗ , f (x, ˆ yt , x)i ˆ ≥ hu ∗ , f (x, ˆ yt , z t )i. Combining this and (i) we obtain thu ∗ , f (x, ˆ yt , z)i ≥ hu ∗ , f (x, ˆ yt , z t )i. From here and (15), it follows that hu ∗ , f (x, ˆ yt , z t )i < 0, which contradicts (13). Thus our claim is verified. By (14) we obtain inf hu ∗ , f (x, ˆ y, z)i ≥ 0.
y∈T (z t )
By the upper sign-continuity of T , we have sup hu ∗ , f (x, ˆ y, z)i ≥ 0.
y∈T (x) ˆ
This implies that inf sup hu ∗ , f (x, ˆ y, z)i ≥ 0.
z∈K y∈T (x) ˆ
(16)
Since the function y 7→ hu ∗ , f (x, ˆ y, z)i is upper semicontinuous and concave, it is also upper semicontinuous in the weak topology of Y . According to Theorem 2.4 we obtain from (16) that max inf hu ∗ , f (x, ˆ y, z)i = inf max hu ∗ , f (x, ˆ y, z)i ≥ 0.
y∈T (x) ˆ z∈K
z∈K y∈T (x) ˆ
Hence there exists yˆ ∈ T (x) ˆ such that inf hu ∗ , f (x, ˆ yˆ , z)i ≥ 0.
z∈K
This means that hu ∗ , f (x, ˆ yˆ , z)i ≥ 0,
∀z ∈ K .
By (4) we obtain f (x, ˆ yˆ , z) 6< 0, Hence xˆ ∈ SV .
∀z ∈ K .
On the basis of Lemma 4.6, we have the following existence result for strong solutions for GVVI(K , T, f ).
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Theorem 4.7. Let X be a reflexive Banach space. Let D be a nonempty set in the locally convex space Y , K be a nonempty convex set in X , and K 0 be a weakly compact subset of K . Let T : K → 2 D be a multifunction, ∗ be given. Assume that: f : K × D × K → Z be a single-valued mapping and u ∗ ∈ C+ (i) (ii) (iii) (iv) (v) (vi)
for each x ∈ K and y ∈ D, f (x, y, x) = 0; for each x ∈ K and for each y ∈ T (x), the function f (x, y, ·) is C-strongly convex and lower semicontinuous; for each z ∈ K the function f (·, ·, z) is C-concave and upper semicontinuous; T is star-pseudomonotone and upper sign-continuous with respect to f and u ∗ ; for each x ∈ K , T(x) is a weakly compact and convex set; for each x ∈ K \ K 0 there exists z ∈ K 0 such that f (x, y, z) < 0 for all y ∈ T (x).
Then there exist xˆ ∈ K 0 and yˆ ∈ T (x) ˆ such that f (x, ˆ yˆ , z) 6< 0 for all z ∈ K . Proof. We choose r > 0 such that K 0 ⊂ int Br where Br ⊂ X is the closed ball with radius r . Put Ω = K ∩ Br . We next define the multifunction G : Ω → 2 X by setting G(z) = {x ∈ Ω | hu ∗ , f (x, y, z)i ≥ 0, ∀y ∈ T (z)}. By Proposition 4.3, T is also star-properly quasimonotone with respect to f and u ∗ . Hence for any finite subset {z 1 , z 2 , . . . , z n } in Ω and for every z ∈ co{z 1 , z 2 , . . . , z n } there exists i ∈ {1, 2, . . . , n} such that hu ∗ , f (z, y, z i )i ≥ 0,
∀y ∈ T (z i ).
This implies that z ∈ G(z i ) and so co{z 1 , z 2 , . . . , z n } ⊆
n [
G(z i ).
i=1
By (iii) and Proposition 2.2, the mapping hu ∗ , f (·, y, z)i is concave and u.s.c. on Ω . Hence G(z) is a closed convex set. Consequently, G(z) is a weakly compact subset of Ω . Thus G is a KKM mapping. By Theorem 2.3, there exists xˆ ∈ Ω such that xˆ ∈ G(z) for all z ∈ Ω . This implies that hu ∗ , f (x, ˆ y, z)i ≥ 0,
∀z ∈ Ω , ∀y ∈ T (z).
By Lemma 4.6, xˆ is a strong solution of GVVI(Ω , T, f ). This means that ∃ yˆ ∈ T (x), ˆ ∀z ∈ Ω ,
f (x, ˆ y, z) 6< 0.
(17)
Combining this and (vi) we get xˆ ∈ K 0 ⊂ int Br . It remains to show that ∃ yˆ ∈ T (x), ˆ ∀z ∈ K
f (x, ˆ y, z) 6< 0.
(18)
ˆ ∈ Ω for a sufficiently small t ∈ (0, 1). Hence (17) implies that Taking any z ∈ K we have z t = xˆ + t (z − x) f (x, ˆ yˆ , z t ) 6< 0.
(19)
If f (x, ˆ yˆ , z) < 0 then by (b) we have t f (x, ˆ yˆ , z) + (1 − t) f (x, ˆ yˆ , x) ˆ − f (x, ˆ yˆ , z t ) ∈ int C ∪ {0}. Hence − f (x, ˆ yˆ , z t ) ∈ −t f (x, ˆ yˆ , z) + int C ∪ {0} ⊆ t (int C) + int C ∪ {0} ⊆ int C + int C ∪ {0} ⊆ int C. This implies that f (x, ˆ yˆ , z t ) < 0, which contradicts (19). Thus we must have f (x, ˆ yˆ , z) 6< 0 and so (18) is fulfilled. The proof is complete.
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Acknowledgement This research was partially supported by the National Science Council of Taiwan. References [1] Q.H. Ansari, J.C. Yao, An existence result for the generalized vector equilibrium problem, Applied Mathematics Letters 12 (1999) 53–56. [2] Q.H. Ansari, J.C. Yao, On strong solution of the generalized implicit vector variational inequality problem, Advanced Nonlinear Variational Inequalities 2 (1999) 1–10. [3] D. Aussel, N. Hadjisavvas, On quasimonotone variational inequalities, Journal of Optimization Theory and Applications 121 (2004) 445–450. [4] D. Aussel, D.T. Luc, Existence conditions in general quasimonotone variational inequalities, Bulletin of the Australian Mathematical Society 71 (2005) 285–303. [5] M. Bianchi, N. Hadjisavvas, S. Shaible, Vector equilibrium problems with generalized monotone bifunctions, Journal of Optimization Theory and Applications 92 (1997) 527–542. [6] E. Blum, W. Oettli, From optimization and variational inequalities to equilibrium problems, The Mathematics Student 63 (1996) 123–145. [7] G.Y. Chen, B.D. Craven, A vector variational inequality and optimization over an efficient set, ZOR-Meth, Models Operation Research 3 (1990) 1–12. [8] P. Cubiotti, N.D. Yen, A results related to Ricceri’s conjecture on generalized quasivariational inequalities, Archiv der Mathematik 69 (1997) 507–514. [9] P. Cubiotti, On the discontinuous infinite-dimensional generalized quasivariational inequality problem, Journal of Optimization Theory and Applications 115 (2002) 97–111. [10] P. Cubiotti, Existence theorem for the discontinuous generalized quasivariational inequality problem, Journal of Optimization Theory and Applications 119 (2003) 623–633. [11] P. Cubiotti, J.C. Yao, Discontinuous implicit quasi-variational inequalities with applications to fuzzy mappings, Mathematical Method of Operations Research 46 (1997) 213–328. [12] A. Daniilidis, N. Hadjisavvas, Characterization of nonsmooth semistrictly quasiconvex and strictly quasiconvex functions, Journal of Optimization Theory and Applications 102 (3) (1999) 525–536. [13] J. Dixmier, Sur un theoreme de Banach, Duke Mathematical Journal 15 (1948) 1057–1071. [14] N. Hadjisavvas, Continuity and maximality properties of pseudomonotone operators, Journal of Convex Analysis 10 (2003) 465–475. [15] V. Jeyakumar, W. Oettli, M. Natividad, A solvability theorem for a class of quasiconvex mappings with applications to optimization, Journal of Mathematical Analysis and Applications 179 (1993) 537–546. [16] S. Karamardian, S. Schaible, Seven kinds of monotone maps, Journal of Optimization Theory and Applications 66 (1990) 37–46. [17] B.T. Kien, N.C. Wong, J.C. Yao, On the solution existence of generalized quasivariational inequalities with discontinuous multifunctions, Journal of Optimization Theory and Applications 135 (1) (2007) (in press). [18] B.S. Lee, G.M. Lee, D.S. Kim, Generalized vector variational-like inequalities in locally convex topological vector spaces, India Journal of Pure and Applied Mathematics 28 (1997) 33–41. [19] J. Li, On the existence of solution of variational inequalities in Banach spaces, Journal of Mathematical Analysis and Applications 295 (2004) 115–126. [20] D.T. Luc, Theory of Vector Optimization, Springer, Berlin, 1989. [21] K.F. Ng, On a theorem of Dixmier, Mathematica Scandinavica 29 (1972) 279–280. [22] W. Oettli, A Remark on vector-valued equilibria and generalized monotonicity, Acta Mathematica Vietnamica 22 (1997) 213–221. [23] A.L. Peressini, Ordered Topological Vector Spaces, Harper and Row, New York, 1967. [24] S. Schaible, J.C. Yao, On the equivalence of nonlinear complementarity problem and least element problems, Mathematical Programming 70 (1995) 191–200. [25] N. Shioji, A further generalization of the Knaster–Kuratowski–Mazurkiewicz theorem, Proceedings of the American Mathematical Society 111 (1991) 187–195. [26] T. Tanaka, Generalized semicontinuity and existence theorems for cone saddle point, Applied Mathematics and Optimization 36 (1997) 313–322. [27] J.C. Yao, J.S. Guo, Variational and generalized variational inequalities with discontinuous mappings, Journal of Mathematical Analysis and Applications 182 (1994) 371–382. [28] J.C. Yao, Variational inequalities with generalized monotone operators, Mathematics of Operations Research 19 (1994) 691–705. [29] J.C. Yao, Generalized quasivariational inequality problems with discontinuous mappings, Mathematics of Operations Research 20 (1995) 465–478. [30] N.D. Yen, On a class of discontinuous vector-valued functions and the associated quasivariational inequalities, Optimization 30 (1994) 197–202. [31] N.D. Yen, On an existence theorem for generalized quasivariational inequalities, Set-Valued Analysis 3 (1995) 1–10. [32] E. Zeidler, Nonlinear Functional Analysis and its Application, I Fixed-Point Theorems, Springer-Verlag, 1993.
Generalized vector variational inequalities with star-pseudomonotone and discontinuous operatorsI Bui Trong Kien a , Ngai Ching Wong b , Jen-Chih Yao b,∗ a Department of Information and Technology, National University of Civil Engineering, 55 Giai Phong, Hanoi, Viet Nam b Department of Applied Mathematics, National Sun Yat-Sen University, Kaohsiung, Taiwan
Received 10 January 2006; accepted 22 February 2007
Abstract It is well known that a vector variational inequality can be a very efficient model for use in studying vector optimization problems. By using the Ky Fan fixed point theorem and the scalarization method we will prove some existence theorems for strong solutions for generalized vector variational inequalities where discontinuous and star-pseudomonotone operators are involved. Our results can be applied to the study of the existence of solutions of vector optimal problems. Some examples are given and analyzed. c 2007 Elsevier Ltd. All rights reserved.
Keywords: Generalized vector equilibrium problem; Generalized vector variational inequality; Upper semicontinuity; C-convexity; C-upper semicontinuity; Star-pseudomonotonicity
1. Introduction Throughout this paper, let Z be a (Hausdorff) topological vector space, and let C be a closed and convex cone of Z such that int C 6= ∅ and C 6= Z where int C denotes the interior of C. Let Ac , A and co A denote the complement, the closure and the convex hull of a subset A of Z , respectively. A vector ordering in Z is defined by setting z1 ≤ z2
if and only if
z 2 − z 1 ∈ C.
z1 < z2
if and only if
z 2 − z 1 ∈ int C,
z 1 6< z 2
if and only if
z 2 − z 1 6∈ int C.
Write
and
I This research was partially supported by a grant from the National Science Council of Taiwan, ROC. ∗ Corresponding author. Tel.: +886 7 5253816; fax: +886 7 5253809.
E-mail addresses: [email protected] (B.T. Kien), [email protected] (N.C. Wong), [email protected] (J.-C. Yao). c 2007 Elsevier Ltd. All rights reserved. 0362-546X/$ - see front matter doi:10.1016/j.na.2007.02.032
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Let X , Y be topological vector spaces, and let K ⊆ X and D ⊆ Y be nonempty. Let T : K → 2 D be a multifunction and f : K × D × K → Z be a single-valued mapping. The generalized vector variational inequality GVVI(K , T, f ) is the following problem: (V) Find xˆ ∈ K such that ∃ yˆ ∈ T (x), ˆ ∀z ∈ K ,
we have f (x, ˆ yˆ , z) 6< 0.
Together with (V), the following auxiliary problems are also of interest: (W) Find xˆ ∈ K such that ∀z ∈ K , ∃ yˆ ∈ T (x), ˆ
we have f (x, ˆ yˆ , z) 6< 0,
and the so-called Minty vector variational inequality: (M) Find xˆ ∈ K such that ∀z ∈ K , ∀ yˆ ∈ T (z),
we have f (x, ˆ yˆ , z) 6< 0.
We will denote by SV , SW and S M the solution sets of the problems (V), (W) and (M), respectively. A point xˆ ∈ SV is called a strong solution of GVVI(K , T, f ), while a point x0 in SW is called a weak solution. It is interesting that problem (V) covers several generalized variational inequalities and equilibrium problems. Here are some examples. (GVI) Put Z = R, C = R+ , Y = X ∗ = D and f (x, y, z) = hy, z − xi. Then (V) reduces to the generalized variational inequality problem: Find xˆ ∈ K
and
yˆ ∈ T (x) ˆ
such that h yˆ , z − xi ˆ ≥ 0, ∀z ∈ K .
(1)
(EP) Put Z = R, C = R+ , Y = X = D and f (x, y, z) = g(x, z) + h(z) − h(x) where g is a real valued function on K × K such that g(x, x) = 0 for all x ∈ K . Then (V) reduces to the equilibrium problem: Find xˆ ∈ K
such that g(x, ˆ z) + h(z) − h(x) ˆ ≥ 0, ∀z ∈ K .
(2)
The existence of solutions of (GVI) has been studied by many authors; see for instance Cubiotti [9], Cubiotti and Yen [8], Li [19], Yao [28,29], Yao and Guo [27], Yen [30,31], Aussel and Hadjisavvas [3], Aussel and Luc [4], Bianchi, Hadjisavvas and Shaible [5], Blum and Oettli [6], Cubiotti [10], Cubiotti and Yao [11], Hadjisavvas [14], Kien, Wong and Yao [17], Schaible and Yao [24] and references therein. Also, results about the existence of weak solutions for the generalized vector variational inequality problem were investigated by many authors; see for instance Ansari and Yao [1,2], Chen and Craven [7], Lee et al. [18], Oettli [22]. However, very few papers that have appeared in the literature discuss the existence of a strong solution of GVVI(K , T, f ). The aim of this paper is to derive some results on the existence of strong solutions for GVVI(K , T, f ) with discontinuous operators and star-pseudomonotone operators. Namely, we will prove some existence theorems in which T can be discontinuous and K can be unbounded. Our results extend some preceding results. Besides, an existence theorem involving star-pseudomonotone operators is also established. In order to obtain these results, we utilize the fixed point theorem due to Ky Fan and the scalarization method. In Section 2, we recall some auxiliary results. Section 3 is devoted to presenting existence results for the case of discontinuous operators. Section 4 will be devoted to an existence result for the case where the operators are starpseudomonotone. 2. Preliminaries Let X and Y be topological vector spaces, and let Z be a locally convex space. Let C be a closed and convex proper cone in Z with interior int C that is nonempty. Let K be a nonempty convex subset of X , and D be a nonempty convex subset of Y . Let T : K → 2 D be a multifunction. We denote by ΓT the graph of T , that is, ΓT = {(x, y) ∈ X × Y | y ∈ T (x)}. Definition 2.1. (a) T is said to be lower semicontinuous (l.s.c. for short) at x ∈ K if for any open set V in E such that V ∩ T (x) 6= ∅, there exists a neighborhood U of x such that V ∩ T (x) 6= ∅,
∀x ∈ U ∩ K .
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(b) T is said to be upper semicontinuous (u.s.c. for short) at x ∈ K if for any open set V in E such that T (x) ⊂ V , there exists a neighborhood W of x with the property that T (x) ⊂ V,
∀x ∈ W ∩ K .
(c) T is said to be lower semicontinuous (resp., upper semicontinuous) on K if it is l.s.c. (resp., u.s.c.) at every point x ∈ K. (d) A single-valued mapping g : X → Z is said to be C-upper semicontinuous on X if for every z ∈ Z the set g −1 (z − int C) is open in X . Tanaka [26] proved that g is C-upper semicontinuous on X if and only if for each fixed x ∈ X and for any y ∈ int C, there exists a neighborhood U of x such that g(u) ∈ g(x) + y − int C,
∀u ∈ U.
(e) g is C-lower semicontinuous on X if for each fixed x ∈ X and for any y ∈ int C, there exists a neighborhood U of x such that g(x) ∈ g(u) + y − int C,
∀u ∈ U.
(f) If g is C-lower semicontinuous and C-upper semicontinuous on X then g is called C-continuous on X . (g) A single-valued mapping h : K → Z is called C-convex if for every x, x 0 ∈ K and t ∈ [0, 1] one has th(x) + (1 − t)h(x 0 ) − h(t x + (1 − t)x 0 ) ∈ C. It is called C-strongly convex if for every x, x 0 ∈ K and t ∈ [0, 1] one has th(x) + (1 − t)h(x 0 ) − h(t x + (1 − t)x 0 ) ∈ int C ∪ {0}. If −h is C-convex (resp., C-strongly convex) then h is said to be C-concave (resp., C-strongly concave) on K . We continue by recalling the scalarization method. Let Z ∗ be the topological dual of Z . The set C ∗ := {z ∗ ∈ Z ∗ : hz ∗ , zi ≥ 0 for all z ∈ C} S is the polar cone of C. Note that C ∗ has a weakly* compact base B, that is, C ∗ = t>0 t B, and B is convex and weakly* compact with 0 6∈ B (see Luc [20]). When int C 6= ∅ and z ∈ int C, z 6= 0, the set B = {z ∗ ∈ C ∗ : hz ∗ , zi = 1} ∗ = C ∗ \ {0}. is a weakly* compact convex base for C ∗ . We put C+ Note that from the bipolar theorem (see Peressini [23]) we have
z ∈ C ⇐⇒ [hz ∗ , zi ≥ 0, ∀z ∗ ∈ C ∗ ] ⇐⇒ [hz ∗ , zi ≥ 0, ∀z ∗ ∈ B]
(3)
z ∈ int C ⇐⇒ [hz , zi > 0, ∀z ∈
(4)
∗
∗
∗ C+ ]
⇐⇒ [hz , zi > 0, ∀z ∈ B]. ∗
∗
(see Jeyakumar et al. [15] for developments). The following proposition is a useful tool for the scalarization method. ∗ . Let φ : K → R be a mapping Proposition 2.2. Let g be a single-valued mapping from K into Z and u ∗ ∈ C+ defined by φ(x) = hu ∗ , g(x)i for all x ∈ K . Then the following assertions are valid: (a) If g is C-convex (resp., C-concave) then φ is convex (resp., concave). (b) If g is C-upper semicontinuous (resp., C-lower semicontinuous) then φ u.s.c. (resp., l.s.c.).
Proof. (a) Since g is C-convex, then for all x, x 0 ∈ K and t ∈ [0, 1] one has tg(x) + (1 − t)g(x 0 ) − g(t x + (1 − t)x 0 ) ∈ C. By (5) we have hu ∗ , tg(x) + (1 − t)g(x 0 ) − g(t x + (1 − t)x 0 )i ≥ 0. Hence thu ∗ , g(x)i + (1 − t)hu ∗ , g(x 0 )i ≥ hu ∗ g(t x + (1 − t)x 0 )i. This implies that tφ(x) + (1 − t)φ(x 0 ) ≥ φ(t x + (1 − t)x 0 ) and so φ is convex. By a similar argument we show that if g is C-concave then φ is concave.
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(b) Let xn → x. We want to show that lim supn→∞ φ(xn ) ≤ φ(x). Choose y j ∈ int C such that y j → 0. Then for each j > 0 there exists a neighborhood U j of x such that g(u) ∈ g(x) + y j − int C,
∀u ∈ U j .
Therefore for each j there exists n j > 0 such that g(xn ) ∈ g(x) + y j − int C,
∀n > n j .
By (5) it follows that hu ∗ , g(xn ) − g(x) − y j i < 0. Hence φ(xn ) = hu ∗ , g(xn )i = hu ∗ , g(xn ) − g(x) − y j + g(x) + y j i = hu ∗ , g(xn ) − g(x) − y j i + hu ∗ , g(x) + y j i < hu ∗ , g(x)i + hu ∗ , y j i for all n > n j . This implies that lim sup φ(xn ) ≤ hu ∗ , g(x)i + hu ∗ y j i. n→∞
Letting j → ∞ and noting that hu ∗ , y j i → 0, we obtain lim sup φ(xn ) ≤ hu ∗ , g(x)i = φ(x). n→∞
In the same way, we obtain the second assertion of (b).
One of the tools in deriving our results is the Ky Fan theorem. We state it below for convenience (see Shioji [25]). Theorem 2.3 (Ky Fan). Let K be a nonempty subset of a topological vector space X and G : K → 2 X be a multifunction with closedS values such that the convex hull of every finite subset {x1 , x2 , . . . , xnT } of X is contained in n the corresponding union i=1 G(xi ). If there exists x0 ∈ K such that G(x0 ) is compact, then x∈K G(x) 6= ∅. We recall that the multifunction G satisfying the property in Theorem 2.3 is called a KKM mapping. We also need the following Sion minimax theorem (see Jeyakumar et al. [15] and Zeidler [32]). Theorem 2.4 (Sion). Let A and B be convex subsets of some real topological vector spaces with B compact, and let p : A × B → R. If p(., b) is lower semicontinuous and quasiconvex on A for all b ∈ B, and if p(a, .) is upper semicontinuous and quasiconcave on B for all a ∈ A, then inf max p(a, b) = max inf p(a, b).
a∈A b∈B
b∈B a∈A
3. Solution existence in the case of discontinuous operators In addition to the basic assumptions stated at the beginning of Section 2, we further assume in this section that f : K × D × K → Z is a single-valued mapping satisfying some of the following conditions: (i) the set T (x) is nonempty, weakly compact, and convex for all x ∈ K ; ∗ such that for each z ∈ K , the set (ii) there exists u ∗ ∈ C+ F(z) = {x ∈ K | sup hu ∗ , f (x, y, z)i ≥ 0} y∈T (x)
is closed; (iii) for each x ∈ K and y ∈ T (x) one has f (x, y, x) = 0; (iv) for each x ∈ K and y ∈ T (x), the function f (x, y, ·) is C-convex and C-l.s.c. on K ; (v) for each (x, z) ∈ K × K , the function f (x, ·, z) is C-concave and C-u.s.c. on D. Theorem 3.1. Let X be a dual Banach space and Y be a locally convex space. Assume conditions (i)–(v) hold. Also assume that F(z) is weakly* closed for all z in K and
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(vi) there is a bounded subset K 0 of K such that for each x ∈ K \ K 0 , there exists z ∈ K 0 with f (x, y, z) < 0 for all y ∈ T (x). Then there exists a point xˆ ∈ K 0 such that max hu ∗ , f (x, ˆ y, z)i ≥ 0,
y∈T (x) ˆ
∀z ∈ K .
Moreover, there exists yˆ ∈ T (x) ˆ such that f (x, ˆ yˆ , z) 6< 0,
∀z ∈ K .
(5)
Proof. We choose r > 0 such that int Br contains K 0 , where Br = {x ∈ X : kxk ≤ r }. Set Ω = K ∩ Br . We define a multifunction G : Ω → 2 X by the formula G(z) = {x ∈ Ω | sup hu ∗ , f (x, y, z)i ≥ 0} y∈T (x)
for z ∈ Ω .
By Proposition 2.2, the function y 7→ hu ∗ , f (x, y, z)i is concave and u.s.c. In particular, for each real number δ, the convex set {y ∈ Y : hu ∗ , f (x, y, z)i ≥ δ} is closed, and thus weakly closed, in the locally convex space Y . Hence, this function is also weakly u.s.c. Since T (x) is a weakly compact set, we have G(z) = {x ∈ Ω | max hu ∗ , f (x, y, z)i ≥ 0}. y∈T (x)
We claim that G is a KKM mapping. Indeed, by (iii) we have z ∈ G(z) for all z ∈ Ω . Hence G(z) 6= ∅ for all z ∈ Ω . Since G(z) = F(z) ∩ Br , G(z) is a weak* compact subset of Ω , by the Banach–Alaoglu Theorem. Taking any z 1 , z 2 ∈ Ω we show that co{z 1 , z 2 } ⊂ G(z 1 ) ∪ G(z 2 ). Let z ∈ co{z 1 , z 2 }. Then z = t z 1 + (1 − t)z 2 for some t ∈ [0, 1]. When t = 0 or t = 1, the claim is trivial. Consider t ∈ (0, 1), with z 6∈ G(z i ); i = 1, 2. Since z ∈ Ω , we obtain max hu ∗ , f (z, y, z i )i < 0;
y∈T (z)
i = 1, 2.
By (iv) and Proposition 2.2, the function hu ∗ , f (x, y, ·)i is convex. By (iii) we have 0 = max hu ∗ , f (z, y, z)i y∈T (z)
≤ t max hu ∗ , f (z, y, z 1 )i + (1 − t) max hu ∗ , f (z, y, z 2 )i, y∈T (z)
y∈T (z)
< 0, which is absurd. Thus we must have z ∈ G(z 1 ) ∪ G(z 2 ) and so co{z 1 , z 2 } ⊂ G(z 1 ) ∪ G(z 2 ). By a similar argument we can show that n [ co{z 1 , z 2 , . . . , z n } ⊂ G(z i ) i=1
for any finite subset {x1 , x2 , . . . , xn } in Ω . By Theorem 2.3, \ ∃xˆ ∈ Ω such that xˆ ∈ G(z). z∈Ω
In other words, ∃xˆ ∈ Ω ,
max hu ∗ , f (x, ˆ y, z)i ≥ 0,
y∈T (x) ˆ
∀z ∈ Ω .
(6)
By (vi), we see that xˆ ∈ K 0 . We need to show that max hu ∗ , f (x, ˆ y, z)i ≥ 0,
y∈T (x) ˆ
∀z ∈ K .
(7)
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Take any z ∈ K . Since xˆ ∈ int Br we have xˆ + t (z − x) ˆ ∈ K ∩ int Br for a sufficiently small t ∈ (0, 1). Then (6) implies that max hu ∗ , f (x, ˆ y, x) ˆ + t (z − x)i ˆ = max hu ∗ , f (x, ˆ y, (1 − t)xˆ + t z)i ≥ 0.
y∈T (x) ˆ
y∈T (x) ˆ
(8)
By the convexity of the function hu ∗ , f (x, y, ·)i, we obtain from (8) the following: 0 ≤ max hu ∗ , f (x, ˆ y, (1 − t)xˆ + t z)i y∈T (x) ˆ
≤ t max hu ∗ , f (x, ˆ y, z)i + (1 − t) max hu ∗ , f (x, ˆ y, x)i. ˆ y∈T (x) ˆ
y∈T (x) ˆ
Since f (x, ˆ y, x) ˆ = 0, this implies (7). ˆ y, z)i. By the To complete the proof we will use the result recalled in Theorem 2.4. Put p(z, y) = hu ∗ , f (x, assumptions of the theorem and Proposition 2.2, the function p(·, y) is l.s.c. and convex. Also, the function p(z, ·) is u.s.c. and concave, and thus weakly u.s.c. By Theorem 2.4 and (7) we get 0 ≤ inf max p(z, y) = max inf p(z, y). z∈K y∈T (x) ˆ
y∈T (x) ˆ z∈K
Since y 7→ p(z, y) is weakly u.s.c., the function y 7→ infz∈K p(z, y) is also weakly u.s.c. Hence, there exists yˆ ∈ T (x) ˆ such that inf p(z, yˆ ) = max inf p(z, y) ≥ 0. y∈T (x) ˆ z∈K
z∈K
hu ∗ ,
It follows that f (x, ˆ yˆ , z)i ≥ 0, ∀z ∈ K . This is equivalent to hu ∗ , − f (x, ˆ yˆ , z)i ≤ 0, ∀z ∈ K . By (4) we have − f (x, ˆ yˆ , z) 6∈ int C for all z ∈ K , and (5) is obtained. The proof of the theorem is complete. ∗ . It is difficult to check when such a u ∗ In Theorem 3.1, condition (ii) is checked for a particular element u ∗ ∈ C+ exists. Fortunately, under certain conditions (e.g., T is upper semicontinuous and f (·, ·, z) is C-continuous), we can ∗. choose any u ∗ ∈ C+ The following example illustrates our result.
Example 3.2. Let X = R, K = [0, 1] ⊂ X , Y = R, D = [1, 4], Z = R2 and C = R2+ = {(x, y) | x ≥ 0, y ≥ 0}. Let T and f be defined by [2, 4], if x = 0; T (x) = {1}, if 0 < x ≤ 1, and f (x, y, z) = ( f 1 (x, y, z), f 2 (x, y, z)), where f 1 (x, y, z) = y(z 2 − x 2 )
and
f 2 (x, y, z) = y(z − x).
Then the set {0} × [2, 4] is a solution set of GVVI(K , T, f ). Moreover, T is not upper semicontinuous on [0, 1]. Indeed, first we get C ∗ = {(u, v) | u ≥ 0, v ≥ 0};
−int C = {(x, y) | x < 0, y < 0}.
Taking u ∗ = (1, 0) we want to check all conditions of Theorem 3.1. (i) It is obvious that T (x) is a nonempty compact convex set for each x ∈ [0, 1]. (ii) For each z ∈ [0, 1] we have {x ∈ K | sup hu ∗ , f (x, y, z)i ≥ 0} = {x ∈ [0, 1] | sup y(z 2 − x 2 ) ≥ 0} y∈T (x)
y∈T (x)
= [0, z], which is a closed set.
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(iii) For each x ∈ M and y ∈ T (x) we have f (x, y, x) = (0, 0). (iv) For each x ∈ K and y ∈ T (x) ⊂ [1, 4], the functions f 1 (x, y, ·) and f 2 (x, y, ·) are convex and continuous. Hence, f is C-convex and C-lower semicontinuous. (v) For each x, z ∈ K × K the functions f 1 (x, ., z) and f 2 (x, ·, z) are linear and continuous. (vi) The claim is obvious. Thus all conditions of Theorem 3.1 are fulfilled. Taking xˆ = 0 and yˆ ∈ T (0) = [2, 4], we have f (0, yˆ , z) = (yz 2 , yz 2 ) 6< 0,
∀z ∈ [0, 1].
Hence the set {0} × [2, 4] is a solution set of the problem. It is observed that T is not upper semicontinuous on K . Indeed, xn = 1/n → 0, yn = 1 ∈ T (xn ) but 1 6∈ T (0) = [2, 4]. Theorem 3.1 requires that X is a dual Banach space. This requirement guarantees the weak* compactness of bounded closed convex sets. The following results relax this assumption by strengthening the coercive condition (vi). Theorem 3.3. Let Y be a locally convex space (resp. topological vector space). Assume conditions (i)–(v), T (x) is weakly compact (resp. compact) for each x in K , F(z) is weakly closed (resp. closed) for each z in K and (vi)0 there exists a weakly compact (resp. compact) K 0 of K and z 0 ∈ K such that for each x ∈ K \ K 0 , one has f (x, y, z 0 ) < 0 for all y ∈ T (x). Then there exists a point xˆ ∈ K and yˆ ∈ T (x) ˆ such that f (x, ˆ yˆ , z) 6< 0,
∀z ∈ K .
Proof. Assume that Y is locally convex. We have F(z) = {x ∈ K | max hu ∗ , f (x, y, z)i ≥ 0}. y∈T (x)
We claim that F(z 0 ) ⊂ K 0 . Indeed, suppose that the claim is false. Then there is an x ∈ F(z 0 )\ K 0 and so x ∈ K \ K 0 . By (vi), − f (x, y, z 0 ) ∈ int C for all y ∈ T (x). By (5), it follows that hu ∗ , − f (x, y, z 0 )i > 0 for all y ∈ T (x). Hence max hu ∗ , f (x, y, z 0 )i < 0.
y∈T (x)
This implies that x 6∈ F(z 0 ), which is absurd. Thus F(z 0 ) is a weakly compact set. As in the proof of Theorem 3.1, we can show that F(z) is a KKM mapping, and the desired conclusion follows accordingly. The case where Y is not locally convex is similar. Theorem 3.4. Let X and Y be topological vector spaces. Assume conditions (i)–(v) and (vi)00 there exist compact sets K 1 ⊆ K 2 in K such that K 1 is finite dimensional, and for each x ∈ K \ K 2 there exists z ∈ K 1 with f (x, y, z) < 0 for all y ∈ T (x). Then there exists a point xˆ ∈ K 2 and yˆ ∈ T (x) ˆ such that f (x, ˆ yˆ , z) 6< 0,
∀z ∈ K .
Proof. We denote by F the set of all finite-dimensional subspaces of X containing K 1 . Fix any S ∈ F and put Ω = K ∩ S,
T S = T |Ω ,
f S = f |Ω ×D×Ω .
The task is to check that Theorem 3.1 can be applied to the problem GVVI(Ω , T S , f S ), where Ω plays a role as K in Theorem 3.1. (a1 ) For each x ∈ Ω we have T (x) = T S (x) 6= ∅. Hence condition (i) of Theorem 3.1 is valid.
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(a2 ) For each z ∈ Ω , we have {x ∈ Ω | sup hu ∗ , f (x, y, z)i ≥ 0} = {x ∈ K | sup hu ∗ , f (x, y, z)i ≥ 0} ∩ S. y∈T (x)
y∈T (x)
Since S is finite dimensional, the set {x ∈ Ω | sup hu ∗ , f (x, y, z)i ≥ 0} y∈T (x)
is closed. This implies that condition (ii) of Theorem 3.1 is valid. (a3 ) Conditions (iii)–(vi) are automatically fulfilled. By Theorem 3.1, there exists x S ∈ Ω such that max hu ∗ , f S (x S , yˆ , z)i ≥ 0,
y∈T S (x S )
∀z ∈ Ω .
Since T S (x S ) = T (x S ), f S (x S , yˆ , z) = f (x S , yˆ , z) we have max hu ∗ , f (x S , yˆ , z)i ≥ 0,
y∈T (x S )
∀z ∈ K ∩ S.
(9)
Put Q S = {x ∈ K | x satisfies (9)}. Then Q S 6= ∅ because x S ∈ Q S . Moreover, the family {Q S : S ∈ F} has the finite intersection property. Indeed, taking any V1 , V2 ∈ F and putting M = span{V1 , V2 }, we have Q M ⊆ Q V1 ∩ Q V2 . This implies that Q M ⊆ Q V1 ∩ Q V2 ⊆ Q V1 ∩ Q V2 . T By condition (vi)”, Q S ⊂ K 2 , which is compact. Hence we get S∈F Q S 6= ∅. Thus there exists xˆ ∈ K 2 such that xˆ ∈ Q S for all S ∈ F. Fix any S ∈ F. Then there exists a net {xα }α∈Λ in Q S such that xα → x. ˆ By the definition of Q S one has max hu ∗ , f (xα , y, z)i ≥ 0,
y∈T (xα )
∀z ∈ K ∩ S.
(10)
Fixing z ∈ K ∩ S and using assumption (ii), we obtain from (10) that max hu ∗ , f (x, ˆ yˆ , z)i ≥ 0.
yˆ ∈T (x) ˆ
In summary, we have proved that ∃xˆ ∈ K 2 , ∀z ∈ K ∩ S,
max hu ∗ , f (x, ˆ y, z)i ≥ 0.
y∈T (x) ˆ
(11)
We now take any v ∈ K and put S 0 = span{S, v}. Since xˆ also satisfies (11) for S 0 we have ∃xˆ ∈ K 2 ,
ˆ yˆ , v)i ≥ 0. max hu ∗ , f (x,
y∈T (x) ˆ
Thus we have ∃xˆ ∈ K 2 , ∀v ∈ K ,
max hu ∗ , f (x, ˆ yˆ , v)i ≥ 0.
y∈T (x) ˆ
By the arguments similar to those in the proof of Theorem 3.1, we show that there exists yˆ ∈ T (x) ˆ such that f (x, ˆ yˆ , z) 6< 0 for all z ∈ K . The proof is complete. Theorem 3.5. Let X, Y be locally convex spaces. Assume conditions (i)–(vi), K 0 is weakly compact and F(z) is weakly closed for all z in K . Then there exists a point xˆ ∈ K 0 and yˆ ∈ T (x) ˆ such that f (x, ˆ yˆ , z) 6< 0,
∀z ∈ K .
Proof. For each absolutely convex weakly compact subset S of X , let X S be the subspace of X generated by S. Namely, [ XS = t S. t>0
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Equip X S with the norm defined by the gauge of S; namely, kxk = inf{t > 0 : x ∈ t S}. Then (X S , k · k) is a dual Banach space by the Dixmier–Ng Theorem; see [13,21]. Indeed, the predual of (X S , k · k) consists of those linear functionals on X continuous on S. Hence, the weak* topology of (X S , k · k) is finer than the weak topology σ (X S , X 0 ) of X S inherited from X . As a result, F(z) is weakly* closed in (X S , k·k) for all z in K ∩ X S . Let F be the set of all dual Banach spaces (X S , k · k), where S is an absolutely convex weakly compact subset of X containing K 0 . We can then proceed as in the proof of Theorem 3.4. We remark that in Theorem 3.5 we can assume instead that X is a topological vector space such that X 0 separates points in X . This condition is enough to ensure the conclusion of the Dixmier–Ng Theorem, as one can check by going through the proof of Ng [21] carefully. 4. Solution existence in the case of star-pseudomonotone operators We first introduce some notions related to the star-pseudomonotone operator. ∗ if for Definition 4.1. (a) The operator T : K → 2 D is called star-pseudomonotone with respect to f and u ∗ ∈ C+ any (x1 , y1 ), (x2 , y2 ) ∈ ΓT ,
hu ∗ , f (x1 , y1 , x2 )i ≥ 0
implies hu ∗ , f (x1 , y2 , x2 )i ≥ 0.
∗ if for any (x , y ), (x , y ) ∈ Γ , (b) T is called star-quasimonotone with respect to f and u ∗ ∈ C+ T 1 1 2 2
hu ∗ , f (x1 , y1 , x2 )i > 0
implies hu ∗ , f (x1 , y2 , x2 )i ≥ 0.
∗ if for any finite set {x , x , . . . , x } in K (c) T is called star-properly quasimonotone with respect to f and u ∗ ∈ C+ 1 2 2 and x ∈ co{x1 , x2 , . . . , x2 }, there exists some i ∈ {1, 2, . . . , n} such that
hu ∗ , f (x, y, xi )i ≥ 0,
∀y ∈ T (xi ).
The notions of quasimonotone operators and pseudomonotone operators were first introduced by Karamardian and Schaible [16]. For this case we put Y = D = X ∗ , Z = R, C = R+, f (x, y, z) = hy, z − xi and u ∗ = 1. Then T is a pseudomonotone operator (resp., quasimonotone) in the sense of theirs. While the notion of proper quasimonotonicity was introduced by Daniilidis and Hadjisavvas [12]. We give an example below illustrating the definition. Example 4.2. Let X = Y = R, K = [−1, 1] ⊂ X , D = [−1, 1], Z = R2 and C = R2+ = {(x, y) | x ≥ 0, y ≥ 0}. Let T and f be defined by if x > 0 {1} T (x) = [−1, 1] if x = 0 {−1} if x < 0, and f (x, y, z) = ( f 1 (x, y, z), f 2 (x, y, z)), where f 1 (x, y, z) = f 2 (x, y, z) = y(z − x). The T is star-pseudomonotone with respect to f and u ∗ = (1, 1). Indeed, we first note that C ∗ = {(u, v) | u ≥ 0, v ≥ 0};
−int C = {(x, y) | x < 0, y < 0}.
∗ . Since T is monotone, we get It is obvious that u ∗ ∈ C+
hy2 − y1 , x2 − x1 i ≥ 0
∀(x1 , y1 ), (x2 , y2 ) ∈ ΓT .
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Hence hy2 , x2 − x1 i ≥ hy1 , x2 − x1 i. This implies that f i (x1 , y2 , x2 ) ≥ f i (x1 , y1 , x2 );
i = 1, 2.
By adding these inequalities we have f 1 (x1 , y2 , x2 ) + f 2 (x1 , y2 , x2 ) ≥ f 1 (x1 , y1 , x2 ) + f 2 (x1 , y1 , x2 ). It follows that hu ∗ , f (x1 , y2 , x2 )i ≥ hu ∗ , f (x1 , y1 , x2 )i and we obtain the assertion. The following are some relations among star-monotone operators. Proposition 4.3. (a) If T is star-pseudomonotone with respect to f and u ∗ then T is also star-quasimonotone with respect to f and u ∗ . (b) Suppose T is star-pseudomonotone with respect to f and u ∗ , and the following conditions are fulfilled: (i) for each x ∈ K and each y ∈ T (x), f (x, y, x) = 0; (ii) for each x ∈ K and for each y ∈ T (x), the function f (x, y, ·) is C-convex. Then T is also star-properly quasimonotone with respect to f and u ∗ . Pn Proof. (a) isPobvious. For (b), let {x1 , x2 , . . . , xn } ⊂ K and x ∈ co{x1 , x2 , . . . , xn }. Then x = i=1 ti x i with n ti ≥ 0 and i=1 ti = 1. By (i) we have hu ∗ , f (x, y, x)i = 0 for all y ∈ T (x). Since f (x, y, ·) is C-convex, by Proposition 2.2 the function hu ∗ , f (x, y, ·)i is convex. Hence 0 = hu ∗ , f (x, y, x)i ≤
n X
ti hu ∗ , f (x, y, xi )i.
i=1
It follows that there exists i ∈ {1, 2, . . . , n} satisfying hu ∗ , f (x, y, xi )i ≥ 0 for some y ∈ T (x). Since T is starpseudomonotone, hu ∗ , f (x, y, xi )i ≥ 0 for all y ∈ T (xi ). The proof is complete. Example 4.4. Consider Example 4.2. We see that f satisfies conditions of Proposition 4.3. Hence T is star-properly quasimonotone with respect to f and u ∗ = (1, 1). ∗ if for all x, z in K the following Definition 4.5. T is called upper sign-continuous on K with respect to f and u ∗ ∈ C+ is satisfied:
(∀t ∈ (0, 1), inf hu ∗ , f (x, y, z)i ≥ 0) H⇒ sup hu ∗ , f (x, y, z)i ≥ 0, y∈T (xt )
y∈T (x)
where xt = (1 − t)x + t z. The notion of upper sign-continuous operators was first introduced by Hadjisavvas [14]. In the above definition by putting Y = D = X ∗ , Z = R, C = R+ , u ∗ = 1 and f (x, y, z) = hy, z − xi we also obtain the upper sign-continuous concept in the sense of Hadjisavvas. The following lemma is a type of Minty lemma. ∗ , xˆ ∈ K and the following conditions hold: Lemma 4.6. Suppose that Y is locally convex. Let u ∗ ∈ C+
(i) (ii) (iii) (iv) (v)
for each x ∈ K and y ∈ D one has f (x, y, x) = 0; for each x ∈ K and y ∈ T (x) the function f (x, y, ·) is C-convex and C-lower semicontinuous; for each (x, z) ∈ K × K , the function f (x, ·, z) is C-concave and C-upper semicontinuous; T is upper sign-continuous with respect to f and u ∗ , with weakly compact and convex values; ˆ y, z)i ≥ 0 for all z ∈ K and y ∈ T (z). hu ∗ , f (x,
Then xˆ ∈ SV .
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Proof. Rewrite condition (v) as ∀z ∈ K , ∀y ∈ T (z),
hu ∗ , f (x, ˆ y, z)i ≥ 0.
(12)
Taking any z ∈ K , we have z t = (1 − t)xˆ + t z ∈ K for t ∈ (0, 1). Hence (16) implies that hu ∗ , f (x, ˆ y, z t )i ≥ 0,
∀y ∈ T (z t ).
(13)
We claim that hu ∗ , f (x, ˆ y, z)i ≥ 0,
∀y ∈ T (z t ).
(14)
Suppose, on the contrary, that the assertion is false. Then there exists t ∈ (0, 1) and yt ∈ T (z t ) such that hu ∗ , f (x, ˆ yt , z)i < 0.
(15)
By (ii) and Proposition 2.2 we have thu ∗ , f (x, ˆ yt , z)i + (1 − t)hu ∗ , f (x, ˆ yt , x)i ˆ ≥ hu ∗ , f (x, ˆ yt , z t )i. Combining this and (i) we obtain thu ∗ , f (x, ˆ yt , z)i ≥ hu ∗ , f (x, ˆ yt , z t )i. From here and (15), it follows that hu ∗ , f (x, ˆ yt , z t )i < 0, which contradicts (13). Thus our claim is verified. By (14) we obtain inf hu ∗ , f (x, ˆ y, z)i ≥ 0.
y∈T (z t )
By the upper sign-continuity of T , we have sup hu ∗ , f (x, ˆ y, z)i ≥ 0.
y∈T (x) ˆ
This implies that inf sup hu ∗ , f (x, ˆ y, z)i ≥ 0.
z∈K y∈T (x) ˆ
(16)
Since the function y 7→ hu ∗ , f (x, ˆ y, z)i is upper semicontinuous and concave, it is also upper semicontinuous in the weak topology of Y . According to Theorem 2.4 we obtain from (16) that max inf hu ∗ , f (x, ˆ y, z)i = inf max hu ∗ , f (x, ˆ y, z)i ≥ 0.
y∈T (x) ˆ z∈K
z∈K y∈T (x) ˆ
Hence there exists yˆ ∈ T (x) ˆ such that inf hu ∗ , f (x, ˆ yˆ , z)i ≥ 0.
z∈K
This means that hu ∗ , f (x, ˆ yˆ , z)i ≥ 0,
∀z ∈ K .
By (4) we obtain f (x, ˆ yˆ , z) 6< 0, Hence xˆ ∈ SV .
∀z ∈ K .
On the basis of Lemma 4.6, we have the following existence result for strong solutions for GVVI(K , T, f ).
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Theorem 4.7. Let X be a reflexive Banach space. Let D be a nonempty set in the locally convex space Y , K be a nonempty convex set in X , and K 0 be a weakly compact subset of K . Let T : K → 2 D be a multifunction, ∗ be given. Assume that: f : K × D × K → Z be a single-valued mapping and u ∗ ∈ C+ (i) (ii) (iii) (iv) (v) (vi)
for each x ∈ K and y ∈ D, f (x, y, x) = 0; for each x ∈ K and for each y ∈ T (x), the function f (x, y, ·) is C-strongly convex and lower semicontinuous; for each z ∈ K the function f (·, ·, z) is C-concave and upper semicontinuous; T is star-pseudomonotone and upper sign-continuous with respect to f and u ∗ ; for each x ∈ K , T(x) is a weakly compact and convex set; for each x ∈ K \ K 0 there exists z ∈ K 0 such that f (x, y, z) < 0 for all y ∈ T (x).
Then there exist xˆ ∈ K 0 and yˆ ∈ T (x) ˆ such that f (x, ˆ yˆ , z) 6< 0 for all z ∈ K . Proof. We choose r > 0 such that K 0 ⊂ int Br where Br ⊂ X is the closed ball with radius r . Put Ω = K ∩ Br . We next define the multifunction G : Ω → 2 X by setting G(z) = {x ∈ Ω | hu ∗ , f (x, y, z)i ≥ 0, ∀y ∈ T (z)}. By Proposition 4.3, T is also star-properly quasimonotone with respect to f and u ∗ . Hence for any finite subset {z 1 , z 2 , . . . , z n } in Ω and for every z ∈ co{z 1 , z 2 , . . . , z n } there exists i ∈ {1, 2, . . . , n} such that hu ∗ , f (z, y, z i )i ≥ 0,
∀y ∈ T (z i ).
This implies that z ∈ G(z i ) and so co{z 1 , z 2 , . . . , z n } ⊆
n [
G(z i ).
i=1
By (iii) and Proposition 2.2, the mapping hu ∗ , f (·, y, z)i is concave and u.s.c. on Ω . Hence G(z) is a closed convex set. Consequently, G(z) is a weakly compact subset of Ω . Thus G is a KKM mapping. By Theorem 2.3, there exists xˆ ∈ Ω such that xˆ ∈ G(z) for all z ∈ Ω . This implies that hu ∗ , f (x, ˆ y, z)i ≥ 0,
∀z ∈ Ω , ∀y ∈ T (z).
By Lemma 4.6, xˆ is a strong solution of GVVI(Ω , T, f ). This means that ∃ yˆ ∈ T (x), ˆ ∀z ∈ Ω ,
f (x, ˆ y, z) 6< 0.
(17)
Combining this and (vi) we get xˆ ∈ K 0 ⊂ int Br . It remains to show that ∃ yˆ ∈ T (x), ˆ ∀z ∈ K
f (x, ˆ y, z) 6< 0.
(18)
ˆ ∈ Ω for a sufficiently small t ∈ (0, 1). Hence (17) implies that Taking any z ∈ K we have z t = xˆ + t (z − x) f (x, ˆ yˆ , z t ) 6< 0.
(19)
If f (x, ˆ yˆ , z) < 0 then by (b) we have t f (x, ˆ yˆ , z) + (1 − t) f (x, ˆ yˆ , x) ˆ − f (x, ˆ yˆ , z t ) ∈ int C ∪ {0}. Hence − f (x, ˆ yˆ , z t ) ∈ −t f (x, ˆ yˆ , z) + int C ∪ {0} ⊆ t (int C) + int C ∪ {0} ⊆ int C + int C ∪ {0} ⊆ int C. This implies that f (x, ˆ yˆ , z t ) < 0, which contradicts (19). Thus we must have f (x, ˆ yˆ , z) 6< 0 and so (18) is fulfilled. The proof is complete.
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