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A vector version of Minty's Lemma and application
PERGAMON
Applied MathematicsLetters 12 (1999) 43-50
Applied Mathematics Letters
A Vector Version of Minty's L e m m a and Application B.-S. LEE* Department of Mathematics Kyungsung University Pusan 608-736, Korea
G,-M. LEE Department of Applied Mathematics Pukyong National University Pusan 608-737, Korea (Received March 1998; revised and accepted September 1998) •A b s t r a c t - - l n this paper, we consider a vector version of Minty's L e m m a and obtain existence theorems for two kinds of vector variational-like inequalities. (~) 1999 Elsevier Science Ltd. All rights reserved.
K e y w o r d s h K K M map, KKM-Fantheorem, Vector variational-likeinequalities, Cone.
1. I N T R O D U C T I O N In the last 20 years, variational inequalities for numerical functions, which was originated from Hartman and Stampacchia [1], have made much developments in the theory and applications. On the other hand, Minty's Lemma [2,3] has been shown to be an important tool in the variational field including variational inequality problems, obstacle problems, confined plasmas, freeboundary problems, elasticity problems, and stochastic optimal control problems when the operator is monotone and the domain is convex. Since Giannessi [4] introduced the vector variational inequalities in a finite-dimensional Euclidean space, recently Chen et al. [5,6], Konnov et al. [7], Lee et al. [8,9], Lee et al. [10-12], Siddiqi et al. [13], and Yu et al. [14] have investigated vector variational inequalities in abstract spaces. In particular, Lee et al. [8,9] and Lee et al. [10-12] introduced and studied vector variational inequalities for multifunctions with vector values. Very recently, Ansari [15] introduced and investigated vector variational-like inequalities for multifunctions with vector values in reflexive Banach spaces under the ~/-pseudomonotonicity and V-hemicontinuity of multifunctions. In 1997, Lee et al. [8] considered vector variational-like inequalities in locally convex Hausdorff topological vector spaces by using the ~/-pseudomonotonicity and V-hemicontinuity of multifunctions or the upper semicontinuity of multi functions under coercivity conditions. *Supported in part by the Basic Science Research Institute Program, Ministry of Education, 1998, Project No. BSRI-98-1405. The authors are most grateful to the referee for his valuable comments. 0893-9659/99/$ - see front matter (~) 1999 Elsevier Science Ltd. All rights reserved. PII: S0893-9659(99)00055-5
Typeset by .A~S-TEX
44
B.-S. LEE AND G . - M . LEE
In this paper, a vector version of Minty's Lemma is obtained, and two kinds of vector variational-like inequalities for multifunctions are considered. We show the existence of solutions for a kind of vector variational-like inequality for a multifunction under certain pseudomonotonicity condition (see Condition (iv) in Theorem 3.2 below) different from that in [8,15]. By using the vector version of Minty's Lemma and the vector variational-like inequality, we prove the existence of solutions for another type of vector variational-like inequality for a compact-valued multifunction under certain hemicontinuity condition (see Condition (v) in Theorem 3.2 below) different from that in [8,15].
2. A V E C T O R V E R S I O N OF M I N T Y ' S L E M M A First, let us give the following lemma. LEMMA 2.1. (See [16].) Let (X, [[. [[) be a normed vector space and H be a Hausdorff metric on the collection C ( X ) of ali closed and bounded subsets of X , induced by a metric d in terms of d(x, y) = [Ix - y[[, which is defined by H ( A , B) = max ( s u p inf [Ix - y[[, sup inf [Ix - y[[) \ x E A yEB
yEB xEA
'
for A and B in C ( X ) . H A and B are compact sets in X , then for each x E A, there exists y E B such that ]]x - Yt[ <- H ( A , B).
Classical Minty's Lemma is stated as the following. THEOREM 2.2. Let X be a reflexive real Banach space, K a nonempty dosed convex subset of X and X * the dual of X . Let T : K ~ X * be a monotone and hemicontinuous operator. Then the following are equivalent: (a) there exists an xo E K such that (T (x0), y - x0) _> 0,
t'or a / / y e K;
(b) there exists an Xo E K such that (T(y), y - Xo) >_ 0,
for a11 y E K.
We obtain the following vector version of Minty's Lemma under conditions different from those in [5-7,11,14]. THEOREM 2.3. Let X and Y be real Banach spaces, K a n o n e m p t y convex subset of X , and {C(x) : x E K} a family of closed convex solid cones of Y . Let T : K --* 2 L(x,g) be a n o n e m p t y compact-valued multifunction such that for any x, y E K , H ( T ( x + A(y - x)), T ( x ) ) ---* O,
as A --* 0 +,
where H is a Hausdorff metric defined on L ( X , Y ) , and rl : K x K -* X an operator. Suppose that the following hold: (i) (t, ~I(Y, Y)) E C(x) for each x, y e K and t e T(y), (ii) the operator x , ' n(Y,X) of K into X is continuous for each y E K ,
Minty's Lemma
45
(iii) the operator =,
, (t,,7(z,y))
of K into Y is affine for each y • K and t • T(y), (iv) for each x, y • K , the existence ors E T(x) such that
(s, ~/(y, x)) ¢ - Int C(x) implies
it,,(=, y)) ¢ Int c(=) for any t • T(y). Then the following are equivalent:
(a) there exists an Xo • K such that/'or each y • K, there exists an So • T(xo) such that (so, ~ (y, xo)) ¢ - Int C (=o) ; (b) there exists an xo • K such that
(t, n (zo, y)) ¢ Int C (=o), for all y • K and t • T(y).
PROOF. Suppose that there exists an xo • K such that for each y • K, there exists So E T(xo) satisfying (So,71(y, xo)) ¢ - I n t C ( x o ) . Then it follows from (iv) that (b) holds. Conversely, suppose that there exists an Xo • K such that (t, ~/(xo, y)) ¢ Int C (x0), for all y • K and t • T(y). For any arbitrary y • K, letting y~ = Ay + (1 - A)Xo, 0 < A < 1, we have Yx • K by the convexity of K. Hence, for all t~ • T(yn)
(tA, 7/(xo, YA)) ¢ Int C (xo).
(2.1)
By the affinity of the operator
x,
, (t,n(x,y)),
we have (tn, 7/(yn, y~)) = (tn, ~/(Ay + (1 - A)xo, y~))
= A (tx, 7/(y, y;~)) + (1 - A) (t~, ~/(xo, Y;0)-
Hence, (t;~, ~/(y, y~)) ¢ - Int C (xo). In fact, suppose to the contrary that (tx, 77(y, yx)) E - Int C (xo). Since - Int C(xo) is a convex cone, A (tx, ~ (y, y~)) E - Int C (xo). Since
(t~, n (y~, y~)) • c (=o)
(2.2)
46
B.-S. L~.E AND G.-M. LEE
by (i), we have
(~ - ~) =
(t~, n (xo, y~)) • Int C (zo), which contradicts (2.1). Hence,
(t~, ~ (y, y~)) ¢ - Int c (~0). Since T(y~) and T(xo) are compact, by Lemma 2.1, for each t~ • T(y~), we can find an s~ • T(xo) such that
IIt~ - sJ, II < H ( T ( y ~ ) , T (xo)) • Since T(xo) is compact, without loss of generality, we can assume that s~ ~ So • T(xo) as )~ ~ 0 +. Moreover, we have
lit~ - soil < IIt~, - s~ll + IIs~ - soil < H(T(y~),T(xo))
+
IIs~ - s o i l .
Since H(T(y,~), T(xo)) ~ 0 as )~ ~ 0 +, t~ --. So. By Assumption (ii)
(y, y~) ---* ~ (y, x o ) ,
as ~ -* o +.
Moreover, we have
II Its, ~(y, y~)> - Iso, ~(y, xo)> II = IIIts, ~ (y, y~)> - + (so, ~ (y, y~)> - II < ll(tx - so,~(y,y~))ll
+ II(so,n(y,y~)
n (y, xo))ll
-
< llt~ - soil lln (Y, ~)11 + llsoll lit (y, y~) - n (y, x o ) l l . Since {~(y,y),)} is bounded and t), --* so as ~ --~ 0 +,
(t~,, ,7 (y, y~,))
> (so, ~ (~, x o ) ) ,
as ~ --+ o +.
So, it follows from (2.2) and the closedness of Y \ ( - Int C(xo)) t h a t (so, 77(y, x0)) ¢ - Int C (x0),
for all y E K.
This completes the proof. REMARK. Let T : K --+ X* be a monotone operator. Taking ~?(x, y) := y - x, C(x) := [0, co), Y := ( - c o , +co), and T ( x ) := (:F(x), .) satisfying the following continuity condition: for any x, y E K,
U ( T ( x + )~(y - x), T ( x ) ) --* O,
as )~ --* 0 +,
(*)
we can obtain the conclusion of the classical Minty's Lemma from Theorem 2.3. But our continuity condition (*) is stronger than the hemicontinuity assumption in Theorem 2.2.
Minty's Lemma
47
3. A P P L I C A T I O N The classical Minty's inequality and Minty's Lemma offered the regularity results of the solution for a generalized nonhomogeneous boundary value problem [2] and, when the operator is a gradient, also a minimum principle for convex optimization problems [17]. On the other hand, vector variational inequality is closely related to vector optimization problem. Giannessi [18] established the equivalence between a differentiable convex vector optimization problem and a vector variational inequality. Lee et al. [19] showed that vector variational inequality can be an efficient tool for studying vector optimization problems. Moreover, using a vector variationallike inequality, Lee et al. [20] proved existence theorems for solutions of nondifferentiable invex optimization problems. Now we will apply our vector version of Minty's Lemma (Theorem 2.3) to show the existence of solutions for a vector variational-like inequality. The following useful and important KKM-Fan theorem will be used in the proof of the existence. DEFINITION 3.1. Let K be a subset of a topological vector space X . Then a multifunction F : K ~ 2 x is called K K M if for each n o n e m p t y fin/re subset N of K , c o N C F ( N ) , where co denotes the convex hull and F ( N ) = U{F(x) : x E N}.
THEOREM 3.1. (See [21].) Let K be an arbitrary n o n e m p t y subset of a Hausdorff topological vector space. Let the set-valued mapping F : K ~ 2 x be a K K M - m a p such that F ( x ) is closed for all x E K and is compact for at least one x E K . Then N F ( x ) # ¢. xEK
THEOREM 3.2. Let X and Y be two real Banach spaces, K a n o n e m p t y compact convex subset of X , and { C ( x ) [ x E K } be a family of dosed convex solid cones of Y such that for any x E K , C ( x ) ~ Y . Let T : K ~ 2 L(x,v) be a multifunction, 77 : K x K -* X an operator and W : K --* 2 r a multifunction, defined by W ( x ) = Y \ ( - I n t C ( x ) ) , such that the graph Gr (W) is closed in X x Y . Suppose that the following condition holds: (i) (t, ~?(y,y)> e C ( z ) , for each x, y E K and t E T ( y ) , (ii) the operator z,
, V(x,y)
is continuous for each y E K ,
(iii) the operator ,
of K into Y is aft/he for each y E K and t E T ( y ) , (iv) for each x, y E K, the existence o f s E T ( x ) such that
(s,
¢ - Int c ( x )
implies
(t, ~?(x, y)) • Int C(x) for any t E T ( y ) . Then there exists a solution xo E K of the following vector variational-like inequality: find xo E K such that
(t, 77(zo, y)) ¢ Int C (xo) for a11 y E K and t E T(V).
48
B.-S. LEE AND G.-M. LEE
Moreover, i[ T is a nonempty compact-valued multifunctio n satisfying the following conditions: (v) for any x, y • K , H ( T ( x + A ( y - x ) ) , T ( x ) ) --+ 0 as A -+ 0 +, where H is a Hausdorff metric
defined on L ( X , Y), (vi) the operator
x ~ ~(y, x)
of K into X is continuous for each y • K , then there exists a solution xo • K of the following vector variational-like inequality: find xo • K such that for each y • K , there exists an So • T(xo) such that ( s o , , (y, x0)) ¢ - Int C (x0). PROOF. Define a multi[unction F1 : K --* 2 K by
FI(y) = {x • K : there exists s • T ( x ) such t h a t (s,n(y,x)) • - I n t C(x)} for each y • K . T h e n FI(y) is n o n e m p t y for each y • K , since y • FI(y). Note t h a t F1 is a K K M multi[unction on K . In fact, suppose t h a t N . {xl,x2, . . . , xn} C K , ~-~=n 1 ai = 1, ai >_ O, n i = 1 , 2 , . . . , n and x = )-~i=la~x~ q~ F I ( N ) . Then for any s • T ( x ) ,
¢ - Int C(x), i = 1,2,...,n.
Thus we have Int C(x), i=l
II
for each j = 1, 2,. .... n. By the affinity of the operator
• ,
, (s,n(~,y)),
it follows t h a t
" )) j=l
i=l
(/( 5=1
" //) i=1
/
• - Int C(x). By (i),
e c ( ~ ) n ( - Int c ( ~ ) ) , and hence 0 6 Int C(x), which contradicts C(x) # Y. Hence, F1 is a K K M multi[unction on K . Define a multi[unction F2 : K --+ 2 g by
F2(y) = {z • K : (t, rl(x,y)) q~ Int C(x), for all t • T(y)}, then by (iv), F I ( y ) C F2(y) for each y • K . Therefore, F2 is also a K K M multi[unction on K . Now we show t h a t F2(y) is closed. Let {xn} be a sequence in F2(y) converging to xo • K . Then we have
(t, 77(xn, y)} q~ Int C (zn),
for all t • T(y).
By (ii)
¢.,y)
,
.(xo,y),
(3.1)
Minty's L e m m a
49
a n d hence, for all t • T ( y ) ,
(t,
y))
, (t,n
y)),
as n
oo.
F r o m (3.1) a n d t h e closedness of G r ( W ) , (t, 7/(x0, y)) ¢ I n t C ( x 0 ) , a n d hence F 2 ( y ) is closed. Since K is c o m p a c t , so is F2(y) for all y • K . Hence, b y t h e K K M - F a n theorem yEK
i.e., t h e r e e x i s t s a n xo • K such t h a t (t, ~/(z0, y)) ¢ I n t C ( x 0 ) , for a n y y • K a n d a n y t • T ( y ) . If T is a n o n e m p t y c o m p a c t - v a l u e d m u l t i f u n c t i o n such t h a t for a n y x, y • K , g(T(x
+ A(y - x ) ) , T ( x ) ) ----* O,
as A --* 0 +
a n d t h e o p e r a t o r x ~ 7}(y, x) of K into X is c o n t i n u o u s for each y • K , t h e n it follows from T h e o r e m 2.3 t h a t t h e r e exists a n x0 • K such t h a t for each y • K , t h e r e exists a n so • T ( x o ) such t h a t
(so, ~ (y, x0)) ~ - Int C (x0). REFERENCES i. P. Hartman and G. Stampacchia, O n some nonlihear ellipticdifferentialfunctional equations, Acta Math. 115, 271-310 (1966). 2. C. Baiocchi and A. Capelo, Variational and Quasivariational Inequalities, Applications to Free Boundary Problems, John Wiley and Sons, New York, (1984). 3. D. Kinderlehrer and G. Stampacchia, An Introduction to Variational Inequalities, Academic Press, New
York, (1980). 4. F. Gianneesi, Theorems of alternative, quadratic programmes and complementarity problems, In Variational Inequalities and Complementarity Problems, (Edited by R.W. Cottle, F. Giannessi and J.L. Lions), pp. 151-186, John Wiley and Sons, Chichester, (1980). 5. G.Y. Chen, Existence of solutions for a vector variational inequality: An extension of the Hartman-Stampacchia theorem, J. Optim. Th. Appl. 74 (3), 445-456 (1992). 6. G.Y. Chen and X.Q. Yang, The vector complementarity problem and its equivalence with the weak minimal element in ordered sets, J. Math. Anal. Appl. 153, 136-158 (1990). 7. I.V. Konnov and J.C. Yao, On the generalized vector variational inequality problem, J. Math. Anal. Appl. 206, 42-58 (1997). 8. B.S. Lee, G.M. Lee and D.S. Kim, Generalized vector variational-like inequalities on locally convex Hausdorff topological vector spaces, Indian J. Pure Appl. Math. 28 (1), 33-41 (1997). 9. B.S. Lee, G.M. Lee and D.S. Kim, Generalized vector-valued variational inequalities and fuzzy extensions, J. Korean Math. Soc. 33, 609-624 (1996). 10. G.M. Lee, D.S. Kim, B.S. Lee and S.J. Cho, Generalized vector variational inequality and fuzzy extension, Appl. Math. Lett. 6 (6), 47-51 (1993). 11. G.M. Lee and S. Kum, Vector variational inequalities in a Hausdorff topological vector space (to be submitted). 12. G.M. Lee, B.S. Lee and S.S. Chang, On vector quasivariational inequalities, J. Math. Anal. Appl. 203, 626-638 (1996). 13. A.H. Siddiqi, Q.H. Ansari and A. Khaliq, On vector variational inequalities, J. Optim. Th. Appl. 84, 171-180 (1995). 14. S.J. Yu and J.C. Yao, On vector variational inequalities, J. Optim. Th. Appl. 89, 749-769 (1996). 15. Q.H. Ansari, On generalized vector variational-like inequalities, Ann. Sci. Math. Qudbec 19, 131-137 (1995). 16. S.B. Nadler, Jr., Multi-valued contraction mappings, Pac. J. Math. 30 (2), 475-488 (1969).
50
B.-S. L~.~ AND G.-M. LEE
17. F. Giannessi, O n connections among separation, penalization and regularization for variational inequalities with point-to-set-operators, Rendiconti del Circolo Matematico di Palermo Series II (Suppl. 48), 11-18 (1997). 18. F. Giannessi, On Minty variational principle, In New 7~ends in Mathematical Programming, Kluwer Academic, Dordrecht, (1997). 19. G.M. Lee, D.S. Kim, B.S. Lee and N.D. Yen, Vector variational inequality as a tool for studying vector optimization problems, Nonlinear Anal. Th. Meth. AppI. 3;4, 745-765 (1998). 20. G.M. Lee, D.S. Kim and H. Kuk, Existence of solutions for vector optimization problems, J. Math. Anal. Appl. 220, 90-98 (1998). 21. K. Fan, A generalization of Tychonoff's fixed point theorem, Math. Ann. 142, 305-310 (1961).
Applied MathematicsLetters 12 (1999) 43-50
Applied Mathematics Letters
A Vector Version of Minty's L e m m a and Application B.-S. LEE* Department of Mathematics Kyungsung University Pusan 608-736, Korea
G,-M. LEE Department of Applied Mathematics Pukyong National University Pusan 608-737, Korea (Received March 1998; revised and accepted September 1998) •A b s t r a c t - - l n this paper, we consider a vector version of Minty's L e m m a and obtain existence theorems for two kinds of vector variational-like inequalities. (~) 1999 Elsevier Science Ltd. All rights reserved.
K e y w o r d s h K K M map, KKM-Fantheorem, Vector variational-likeinequalities, Cone.
1. I N T R O D U C T I O N In the last 20 years, variational inequalities for numerical functions, which was originated from Hartman and Stampacchia [1], have made much developments in the theory and applications. On the other hand, Minty's Lemma [2,3] has been shown to be an important tool in the variational field including variational inequality problems, obstacle problems, confined plasmas, freeboundary problems, elasticity problems, and stochastic optimal control problems when the operator is monotone and the domain is convex. Since Giannessi [4] introduced the vector variational inequalities in a finite-dimensional Euclidean space, recently Chen et al. [5,6], Konnov et al. [7], Lee et al. [8,9], Lee et al. [10-12], Siddiqi et al. [13], and Yu et al. [14] have investigated vector variational inequalities in abstract spaces. In particular, Lee et al. [8,9] and Lee et al. [10-12] introduced and studied vector variational inequalities for multifunctions with vector values. Very recently, Ansari [15] introduced and investigated vector variational-like inequalities for multifunctions with vector values in reflexive Banach spaces under the ~/-pseudomonotonicity and V-hemicontinuity of multifunctions. In 1997, Lee et al. [8] considered vector variational-like inequalities in locally convex Hausdorff topological vector spaces by using the ~/-pseudomonotonicity and V-hemicontinuity of multifunctions or the upper semicontinuity of multi functions under coercivity conditions. *Supported in part by the Basic Science Research Institute Program, Ministry of Education, 1998, Project No. BSRI-98-1405. The authors are most grateful to the referee for his valuable comments. 0893-9659/99/$ - see front matter (~) 1999 Elsevier Science Ltd. All rights reserved. PII: S0893-9659(99)00055-5
Typeset by .A~S-TEX
44
B.-S. LEE AND G . - M . LEE
In this paper, a vector version of Minty's Lemma is obtained, and two kinds of vector variational-like inequalities for multifunctions are considered. We show the existence of solutions for a kind of vector variational-like inequality for a multifunction under certain pseudomonotonicity condition (see Condition (iv) in Theorem 3.2 below) different from that in [8,15]. By using the vector version of Minty's Lemma and the vector variational-like inequality, we prove the existence of solutions for another type of vector variational-like inequality for a compact-valued multifunction under certain hemicontinuity condition (see Condition (v) in Theorem 3.2 below) different from that in [8,15].
2. A V E C T O R V E R S I O N OF M I N T Y ' S L E M M A First, let us give the following lemma. LEMMA 2.1. (See [16].) Let (X, [[. [[) be a normed vector space and H be a Hausdorff metric on the collection C ( X ) of ali closed and bounded subsets of X , induced by a metric d in terms of d(x, y) = [Ix - y[[, which is defined by H ( A , B) = max ( s u p inf [Ix - y[[, sup inf [Ix - y[[) \ x E A yEB
yEB xEA
'
for A and B in C ( X ) . H A and B are compact sets in X , then for each x E A, there exists y E B such that ]]x - Yt[ <- H ( A , B).
Classical Minty's Lemma is stated as the following. THEOREM 2.2. Let X be a reflexive real Banach space, K a nonempty dosed convex subset of X and X * the dual of X . Let T : K ~ X * be a monotone and hemicontinuous operator. Then the following are equivalent: (a) there exists an xo E K such that (T (x0), y - x0) _> 0,
t'or a / / y e K;
(b) there exists an Xo E K such that (T(y), y - Xo) >_ 0,
for a11 y E K.
We obtain the following vector version of Minty's Lemma under conditions different from those in [5-7,11,14]. THEOREM 2.3. Let X and Y be real Banach spaces, K a n o n e m p t y convex subset of X , and {C(x) : x E K} a family of closed convex solid cones of Y . Let T : K --* 2 L(x,g) be a n o n e m p t y compact-valued multifunction such that for any x, y E K , H ( T ( x + A(y - x)), T ( x ) ) ---* O,
as A --* 0 +,
where H is a Hausdorff metric defined on L ( X , Y ) , and rl : K x K -* X an operator. Suppose that the following hold: (i) (t, ~I(Y, Y)) E C(x) for each x, y e K and t e T(y), (ii) the operator x , ' n(Y,X) of K into X is continuous for each y E K ,
Minty's Lemma
45
(iii) the operator =,
, (t,,7(z,y))
of K into Y is affine for each y • K and t • T(y), (iv) for each x, y • K , the existence ors E T(x) such that
(s, ~/(y, x)) ¢ - Int C(x) implies
it,,(=, y)) ¢ Int c(=) for any t • T(y). Then the following are equivalent:
(a) there exists an Xo • K such that/'or each y • K, there exists an So • T(xo) such that (so, ~ (y, xo)) ¢ - Int C (=o) ; (b) there exists an xo • K such that
(t, n (zo, y)) ¢ Int C (=o), for all y • K and t • T(y).
PROOF. Suppose that there exists an xo • K such that for each y • K, there exists So E T(xo) satisfying (So,71(y, xo)) ¢ - I n t C ( x o ) . Then it follows from (iv) that (b) holds. Conversely, suppose that there exists an Xo • K such that (t, ~/(xo, y)) ¢ Int C (x0), for all y • K and t • T(y). For any arbitrary y • K, letting y~ = Ay + (1 - A)Xo, 0 < A < 1, we have Yx • K by the convexity of K. Hence, for all t~ • T(yn)
(tA, 7/(xo, YA)) ¢ Int C (xo).
(2.1)
By the affinity of the operator
x,
, (t,n(x,y)),
we have (tn, 7/(yn, y~)) = (tn, ~/(Ay + (1 - A)xo, y~))
= A (tx, 7/(y, y;~)) + (1 - A) (t~, ~/(xo, Y;0)-
Hence, (t;~, ~/(y, y~)) ¢ - Int C (xo). In fact, suppose to the contrary that (tx, 77(y, yx)) E - Int C (xo). Since - Int C(xo) is a convex cone, A (tx, ~ (y, y~)) E - Int C (xo). Since
(t~, n (y~, y~)) • c (=o)
(2.2)
46
B.-S. L~.E AND G.-M. LEE
by (i), we have
(~ - ~)
(t~, n (xo, y~)) • Int C (zo), which contradicts (2.1). Hence,
(t~, ~ (y, y~)) ¢ - Int c (~0). Since T(y~) and T(xo) are compact, by Lemma 2.1, for each t~ • T(y~), we can find an s~ • T(xo) such that
IIt~ - sJ, II < H ( T ( y ~ ) , T (xo)) • Since T(xo) is compact, without loss of generality, we can assume that s~ ~ So • T(xo) as )~ ~ 0 +. Moreover, we have
lit~ - soil < IIt~, - s~ll + IIs~ - soil < H(T(y~),T(xo))
+
IIs~ - s o i l .
Since H(T(y,~), T(xo)) ~ 0 as )~ ~ 0 +, t~ --. So. By Assumption (ii)
(y, y~) ---* ~ (y, x o ) ,
as ~ -* o +.
Moreover, we have
II Its, ~(y, y~)> - Iso, ~(y, xo)> II = IIIts, ~ (y, y~)> -
+ II(so,n(y,y~)
n (y, xo))ll
-
< llt~ - soil lln (Y, ~)11 + llsoll lit (y, y~) - n (y, x o ) l l . Since {~(y,y),)} is bounded and t), --* so as ~ --~ 0 +,
(t~,, ,7 (y, y~,))
> (so, ~ (~, x o ) ) ,
as ~ --+ o +.
So, it follows from (2.2) and the closedness of Y \ ( - Int C(xo)) t h a t (so, 77(y, x0)) ¢ - Int C (x0),
for all y E K.
This completes the proof. REMARK. Let T : K --+ X* be a monotone operator. Taking ~?(x, y) := y - x, C(x) := [0, co), Y := ( - c o , +co), and T ( x ) := (:F(x), .) satisfying the following continuity condition: for any x, y E K,
U ( T ( x + )~(y - x), T ( x ) ) --* O,
as )~ --* 0 +,
(*)
we can obtain the conclusion of the classical Minty's Lemma from Theorem 2.3. But our continuity condition (*) is stronger than the hemicontinuity assumption in Theorem 2.2.
Minty's Lemma
47
3. A P P L I C A T I O N The classical Minty's inequality and Minty's Lemma offered the regularity results of the solution for a generalized nonhomogeneous boundary value problem [2] and, when the operator is a gradient, also a minimum principle for convex optimization problems [17]. On the other hand, vector variational inequality is closely related to vector optimization problem. Giannessi [18] established the equivalence between a differentiable convex vector optimization problem and a vector variational inequality. Lee et al. [19] showed that vector variational inequality can be an efficient tool for studying vector optimization problems. Moreover, using a vector variationallike inequality, Lee et al. [20] proved existence theorems for solutions of nondifferentiable invex optimization problems. Now we will apply our vector version of Minty's Lemma (Theorem 2.3) to show the existence of solutions for a vector variational-like inequality. The following useful and important KKM-Fan theorem will be used in the proof of the existence. DEFINITION 3.1. Let K be a subset of a topological vector space X . Then a multifunction F : K ~ 2 x is called K K M if for each n o n e m p t y fin/re subset N of K , c o N C F ( N ) , where co denotes the convex hull and F ( N ) = U{F(x) : x E N}.
THEOREM 3.1. (See [21].) Let K be an arbitrary n o n e m p t y subset of a Hausdorff topological vector space. Let the set-valued mapping F : K ~ 2 x be a K K M - m a p such that F ( x ) is closed for all x E K and is compact for at least one x E K . Then N F ( x ) # ¢. xEK
THEOREM 3.2. Let X and Y be two real Banach spaces, K a n o n e m p t y compact convex subset of X , and { C ( x ) [ x E K } be a family of dosed convex solid cones of Y such that for any x E K , C ( x ) ~ Y . Let T : K ~ 2 L(x,v) be a multifunction, 77 : K x K -* X an operator and W : K --* 2 r a multifunction, defined by W ( x ) = Y \ ( - I n t C ( x ) ) , such that the graph Gr (W) is closed in X x Y . Suppose that the following condition holds: (i) (t, ~?(y,y)> e C ( z ) , for each x, y E K and t E T ( y ) , (ii) the operator z,
, V(x,y)
is continuous for each y E K ,
(iii) the operator ,
of K into Y is aft/he for each y E K and t E T ( y ) , (iv) for each x, y E K, the existence o f s E T ( x ) such that
(s,
¢ - Int c ( x )
implies
(t, ~?(x, y)) • Int C(x) for any t E T ( y ) . Then there exists a solution xo E K of the following vector variational-like inequality: find xo E K such that
(t, 77(zo, y)) ¢ Int C (xo) for a11 y E K and t E T(V).
48
B.-S. LEE AND G.-M. LEE
Moreover, i[ T is a nonempty compact-valued multifunctio n satisfying the following conditions: (v) for any x, y • K , H ( T ( x + A ( y - x ) ) , T ( x ) ) --+ 0 as A -+ 0 +, where H is a Hausdorff metric
defined on L ( X , Y), (vi) the operator
x ~ ~(y, x)
of K into X is continuous for each y • K , then there exists a solution xo • K of the following vector variational-like inequality: find xo • K such that for each y • K , there exists an So • T(xo) such that ( s o , , (y, x0)) ¢ - Int C (x0). PROOF. Define a multi[unction F1 : K --* 2 K by
FI(y) = {x • K : there exists s • T ( x ) such t h a t (s,n(y,x)) • - I n t C(x)} for each y • K . T h e n FI(y) is n o n e m p t y for each y • K , since y • FI(y). Note t h a t F1 is a K K M multi[unction on K . In fact, suppose t h a t N . {xl,x2, . . . , xn} C K , ~-~=n 1 ai = 1, ai >_ O, n i = 1 , 2 , . . . , n and x = )-~i=la~x~ q~ F I ( N ) . Then for any s • T ( x ) ,
Thus we have Int C(x), i=l
II
for each j = 1, 2,. .... n. By the affinity of the operator
• ,
, (s,n(~,y)),
it follows t h a t
" )) j=l
i=l
(/( 5=1
" //) i=1
/
• - Int C(x). By (i),
F2(y) = {z • K : (t, rl(x,y)) q~ Int C(x), for all t • T(y)}, then by (iv), F I ( y ) C F2(y) for each y • K . Therefore, F2 is also a K K M multi[unction on K . Now we show t h a t F2(y) is closed. Let {xn} be a sequence in F2(y) converging to xo • K . Then we have
(t, 77(xn, y)} q~ Int C (zn),
for all t • T(y).
By (ii)
¢.,y)
,
.(xo,y),
(3.1)
Minty's L e m m a
49
a n d hence, for all t • T ( y ) ,
(t,
y))
, (t,n
y)),
as n
oo.
F r o m (3.1) a n d t h e closedness of G r ( W ) , (t, 7/(x0, y)) ¢ I n t C ( x 0 ) , a n d hence F 2 ( y ) is closed. Since K is c o m p a c t , so is F2(y) for all y • K . Hence, b y t h e K K M - F a n theorem yEK
i.e., t h e r e e x i s t s a n xo • K such t h a t (t, ~/(z0, y)) ¢ I n t C ( x 0 ) , for a n y y • K a n d a n y t • T ( y ) . If T is a n o n e m p t y c o m p a c t - v a l u e d m u l t i f u n c t i o n such t h a t for a n y x, y • K , g(T(x
+ A(y - x ) ) , T ( x ) ) ----* O,
as A --* 0 +
a n d t h e o p e r a t o r x ~ 7}(y, x) of K into X is c o n t i n u o u s for each y • K , t h e n it follows from T h e o r e m 2.3 t h a t t h e r e exists a n x0 • K such t h a t for each y • K , t h e r e exists a n so • T ( x o ) such t h a t
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B.-S. L~.~ AND G.-M. LEE
17. F. Giannessi, O n connections among separation, penalization and regularization for variational inequalities with point-to-set-operators, Rendiconti del Circolo Matematico di Palermo Series II (Suppl. 48), 11-18 (1997). 18. F. Giannessi, On Minty variational principle, In New 7~ends in Mathematical Programming, Kluwer Academic, Dordrecht, (1997). 19. G.M. Lee, D.S. Kim, B.S. Lee and N.D. Yen, Vector variational inequality as a tool for studying vector optimization problems, Nonlinear Anal. Th. Meth. AppI. 3;4, 745-765 (1998). 20. G.M. Lee, D.S. Kim and H. Kuk, Existence of solutions for vector optimization problems, J. Math. Anal. Appl. 220, 90-98 (1998). 21. K. Fan, A generalization of Tychonoff's fixed point theorem, Math. Ann. 142, 305-310 (1961).