On Quasimonotone Variational Inequalities

JOURNAL OF OPTIMIZATION THEORY AND APPLICATIONS: Vol. 121, No. 2, pp. 445–450, May 2004 (Ó 2004) TECHNICAL NOTE On Quasimonotone Variational Inequalities1 D. AUSSEL2 AND N. HADJISAVVAS3 Communicated by S. Schaible Abstract. The purpose of this paper is to prove the existence of solutions of the Stampacchia variational inequality for a quasimonotone multivalued operator without any assumption on the existence of inner points. Moreover, the operator is not supposed to be bounded valued. The result strengthens a variety of other results in the literature. Key Words. Variational inequalities, quasimonotone generalized monotonicity, existence results. operators, 1. Introduction and Definitions Given a Banach space X with topological dual X*, a subset K of X, and
a multivalued operator T: K ! 2X , the Stampacchia variational inequality problem is to find x 2 K such that 8y 2 K; 9x
2 TðxÞ : hx
; y  xi  0: ð1Þ Existence of solutions of (1) under a generalized monotonicity assumption for T has been investigated intensively in recent years. In most cases, T was assumed to be pseudomonotone (in the sense of Karamardian); see e.g. Refs. 1–2. Extension of these results to the broader class of quasimonotone operators has been established also, but only at the cost of restrictive 1 This work was prepared while the second author was visiting the Mathematics Department of the University of Perpignan, Perpignan, France. The author wishes to thank the Mathematics Department for its hospitality. 2 Associate Professor, Département de Mathématiques, Université de Perpignan, Perpignan, France. 3 Professor, Department of Product and Systems Design Engineering, University of the Aegean, Hermoupolis, Syros, Greece. 445 0022-3239/04/0500-0445/0 Ó 2004 Plenum Publishing Corporation 446 JOTA: VOL. 121, NO. 2, MAY 2004 assumptions. For instance, in Ref. 3, K was assumed to contain inner points; in addition, in case T is multivalued, its values were assumed to be compact in the norm topology (Ref. 4); in Ref. 5, T was assumed to be densely pseudomonotone, which is more restrictive than quasimonotone, etc. The purpose of this note is to show existence of solutions of (1) for quasimonotone operators with no additional assumptions apart from those used for pseudomonotone operators (i.e., a kind of continuity along lines and w*-compactness and convexity of the values). In fact, even the latter assumptions will be stated in a very weak form. We recall that an operator T is called quasimonotone (Ref. 6) if, for all ðx; x
Þ, ðy; y
Þ in the graph grT, hx
; y  xi > 0 ) hy
; y  xi  0: The operator T is called properly quasimonotone (Ref.7) if, for all x1 ; . . . ; xn 2 domT, and all x 2 cofx1 ; x2 ; . . . ; xn g, there exists i 2 f1; 2; . . . ; ng such that 8x
2 Tðxi Þ : hx
; xi  xi  0: Finally, T is called pseudomonotone (in the Karamardian sense) (Ref. 6) if, for all ðx; x
Þ; ðy; y
Þ 2 gr T, hx
; y  xi  0 ) hy
; y  xi  0: Pseudomonotone operators are properly quasimonotone and properly quasimonotone operators are quasimonotone. We denote by S(T,K) the set of solutions of the Stampacchia variational inequality x 2 SðT; KÞ () x 2 K and 8y 2 K; 9x
2 TðxÞ : hx
; y  xi  0 and by Sstr ðT; KÞ the set of strong solutions of the same inequality: x 2 Sstr ðT; KÞ () x 2 K and 9x
2 TðxÞ : 8y 2 K; hx
; y  xi  0: Also, we denote by M(T,K) the set of solutions of the Minty variational inequality x 2 MðT; KÞ () x 2 K and 8y 2 K; 8y
2 TðyÞ : hy
; y  xi  0: Finally, we call x 2 K a local solution of the Minty variational inequality if there exists a neighborhood U of x such that x 2 MðT; K \ UÞ. We denote by LM(T,K) the set of these local solutions. Clearly, MðT; KÞ  LMðT; KÞ. In the following lemma, we will clarify the relations between these different sets of solutions. Before this, let us recall (Ref. 8) the definition of a very weak kind of continuity: Given a convex subset K  X and an operator
T : K ! 2X with nonempty values, T is called upper sign-continuous on K if, for every x; y 2 K, the following implication holds: ð8t 20; 1½; inf hx
; y  xi  0Þ ¼) sup hx
; y  xi  0; x
2Tðxt Þ x
2TðxÞ JOTA: VOL. 121, NO. 2, MAY 2004 447 where xt ¼ ð1  tÞx þ ty: For example, if T is upper hemicontinuous (i.e., the restriction of T to every line segment of K is usc with respect to the w*-topology in X*), then T is upper sign-continuous. Any strictly positive real function is upper sign-continuous. 2. Existence Result It is known that a solution of the Minty variational inequality is also a strong solution of the Stampacchia variational inequality, provided that T is upper hemicontinuous with convex, w*-compact values. Using essentially the same argument, we show that the same is true under weaker assumptions. Lemma 2.1. Let K be a nonempty convex subset of the Banach space X
and let T : K ! 2X be an operator. (i) If T is pseudomonotone, then LMðT; KÞ ¼ MðT; KÞ: (ii) If for every x 2 K there exists a convex neighborhood Vx of x and
an upper sign-continuous operator Sx : Vx \ K ! 2X with nonempty, w*-compact values satisfying Sx ðyÞ  TðyÞ, 8y 2 Vx \ K, then LMðT; KÞ  SðT; KÞ. (iii) Additionally to the assumptions of (ii), if the operators Sx are convex valued, then LMðT; KÞ  SðT; KÞ ¼ Sstr ðT; KÞ. Proof. (i) Let x be an element of LM(T,K). Then, there exists a neighborhood U of x such that x 2 MðT; K \ UÞ. For any y 2 K, there exists z ¼ x þ tðy  xÞ, t 20; 1½, such that z 2 K \ U. Then, for any z
2 TðzÞ, hz
; y  zi ¼ ½ð1  tÞ=thz
; z  xi  0: By pseudomonotonicity, hy
; y  xi  0; for all y
2 TðyÞ: Therefore x is an element of M(T,K) (ii) Let x be an element of LM(T,K). Thus, there exists a neighborhood U of x such that x 2 MðSx ; K \ Vx \ UÞ. Let y 2 K \ Vx . Since K \ Vx is ~  ðK \ Vx \ UÞ and thus convex, there exists y~ 2x; y for which ½x; y inf inf hu
; u  xi  0: ~ u
2Sx ðuÞ u2½x;y By the upper sign-continuity of Sx , sup hx
; y  xi  0: x
2Sx ðxÞ 448 JOTA: VOL. 121, NO. 2, MAY 2004 But Sx ðxÞ is w*-compact and we deduce that inf max hx
; y  xi  0; y2Vx \K x
2Sx ðxÞ ð2Þ which means that, for all y 2 Vx \ K, there exists x
2 Sx ðxÞ  TðxÞ such that hx
; y  xi  0. Therefore, x is an element of S(T,K) since, using the convexity of K, one can prove easily that the above relation holds for any y 2 K. (iii) This is a consequence of the Sion minimax theorem applied to the relation (2). h In particular, if T itself is upper sign-continuous and has nonempty, convex, and w*-compact values, then we can take in the lemma Vx ¼ K, Sx ¼ T. However, the lemma in its present form (as well as the forthcoming Theorem 2.1) permits application to operators whose values are unbounded, such as cone-valued operators. We now establish an alternative, valid for every quasimonotone operator. Proposition 2.1. Let K be a nonempty, convex subset of the Banach
space X and let T : K ! 2X be quasimonotone. Then, one of the folowing assertions holds: (i) T is properly quasimonotone (ii) LMðT; KÞ ¼ 6 ;. In addition, if K is weakly compact, then LMðT; KÞ ¼ 6 ; in both cases. Proof. Suppose that T is not properly quasimonotone. Then, there exist x1 ; . . . ; xn 2 K; x
i 2 Tðxi Þ; i ¼ 1; . . . ; n, and x 2 cofx1 ; . . . ; xn g such that hx
i ; x  xi i > 0; i ¼ 1; . . . ; n: By continuity of the functionals x
i , there exists a neighborhood U of x such that, for any y 2 K \ U, one has hx
i ; y  xi i > 0: By quasimonotonicity, for all y
2 TðyÞ; hy
; y  xi i  0: Since x 2 cofx1 ; . . . ; xn g, it follows easily that hy
; y  xi  0: ð3Þ 8y
2 TðyÞ; Thus, x 2 LMðT; KÞ since the previous inequality holds for every y 2 K \ U. It remains to show that LMðT; KÞ ¼ 6 ; whenever K is weakly compact and T is properly quasimonotone. But under such assumptions, it is known JOTA: VOL. 121, NO. 2, MAY 2004 449 (Ref. 7) that MðT; KÞ ¼ 6 ;; since MðT; KÞ  LMðT; KÞ, it follows that LMðT; KÞ ¼ 6 ;. h Combination of the lemma with Proposition 2.1 leads to a result of existence of solutions for the Stampacchia variational inequality without any assumption on the existence of inner points. Theorem 2.1. Let K be a nonempty convex subset of X. Further, let
T : K ! 2X be a quasimonotone operator such that the following coercivity condition holds:  qÞ; 9y 2 K with kyk < kxk 9q > 0; 8x 2 K nBð0; such that 8x
2 TðxÞ; hx
; x  yi  0: ð4Þ 0  Suppose that there exists q > q such that K \ Bð0; q Þ is nonempty weakly compact. Moreover, suppose that, for every x 2 K, there exists a convex neighborhood Vx of x and an upper sign-continuous operator Sx :
Vx \ K ! 2X with nonempty, convex, w*-compact values satisfying Sx ðyÞ  TðyÞ; 8y 2 Vx \ K. Then Sstr ðT; KÞ ¼ 6 ;. 0  q0 Þ is nonempty, convex and weakly Proof. The set Kq0 ¼ K \ Bð0; 6 ;. By Lemma 2.1, the compact. According to Proposition 2.1, LMðT; Kq0 Þ ¼ set Sstr ðT; Kq0 Þ is also nonempty. Choose x0 2 Sstr ðT; Kp0 Þ. Then, 9x
0 2 TðxÞ : 8y 2 Kq0 ; hx
0 ; y  x0 i  0: ð5Þ 0 According to (4), there exists y0 2 Bð0; q Þ \ K such that 8x
2 Tðx0 Þ; hx
; x0  y0 i  0: (If kx0 k < q0 we can take y0 ¼ x0 Þ. From (5) and (6), it follows that hx
0 ; y0  x0 i ¼ 0: ð6Þ ð7Þ Now, for every y 2 K, there exists t 2 ½0; 1½ such that ð1  tÞy þ ty0 2 Kq0 ; hence, hx
0 ; ð1  tÞy þ ty0  x0 i  0: ð8Þ It follows immediately from (7) and (8) that hx
0 ; y  x0 i  0; i.e. x0 2 Sstr ðT; KÞ: (  q0 Þ Note that, in Theorem 2.1, the condition on the compactness of K \ Bð0; is satisfied automatically if K is weakly compact or X is reflexive and K is closed; the coercivity condition is also satisfied automatically if K is bounded. Finally, the condition on the existence of Sx is satisfied if T itself 450 JOTA: VOL. 121, NO. 2, MAY 2004 is upper sign-continuous with nonempty, convex, w*-compact values. Thus, Theorem 2.1 generalizes corresponding results for pseudomonotone operators (Ref. 1), quasimonotone operators where K is assumed to contain inner points (Ref. 3), densely pseudomonotone operators (Ref. 5), etc. Finally, let us compare the results of this paper with Theorem 5.1 of Ref. 7; there, it is established (using no continuity assumption) that, for every properly quasimonotone operator T defined on a weakly compact convex subset K, MðT; KÞ ¼ 6 ; holds. Starting from this, one deduces usually that SðT; KÞ ¼ 6 ; by adjoining some suitable assumptions (for instance, that T is upper hemicontinuous with convex, w*-compact values). If the operator T is quasimonotone, but not properly quasimonotone, then M(T,K) may be empty. However, according to Proposition 2.1, LMðT; KÞ ¼ 6 ;. This last property is again sufficient for proving that SðT; KÞ ¼ 6 ; under the same (or even weaker) additional assumptions, as shown by Theorem 2.1. References 1. YAO, J. C., Multivalued Variational Inequalities with K-Pseudomonotone Operators, Journal of Optimization Theory and Applications, Vol. 83, pp. 391–403, 1994. 2. CROUZEIX, J.-P., Pseudomonotone Variational Inequality Problems: Existence of Solutions, Mathematical Programming, Vol. 78, pp. 305–314, 1997. 3. HADJISAVVAS, N., and SCHAIBLE, S., Quasimonotone Variational Inequalities in Banach Spaces, Journal of Optimization Theory and Applications, Vol. 90, pp. 95–111, 1996. 4. DANIILIDIS, A., and HADJISAVVAS, N., Existence Theorems for Vector Variational Inequalities, Bulletin of the Australian Mathematical Society, Vol. 54, pp. 473– 481, 1996. 5. LUC, D. T., Existence Results for Densely Pseudomonotone Variational Inequalities, Journal of Mathematical Analysis and Applications, Vol. 254, pp. 291–308, 2001. 6. KARAMARDIAN, S., and SCHAIBLE, S., Seven Kinds of Monotone Maps, Journal of Optimization Theory and Applications, Vol. 66, pp. 37–46, 1990. 7. DANIILIDIS, A., and HADJISAVVAS, N., Characterization of Nonsmooth Semistrictly Quasiconvex and Strictly Quasiconvex Functions, Journal of Optimization Theory and Applications, Vol. 102, pp. 525-536, 1999. 8. HADJISAVVAS, N., Continuity and Maximality Properties of Pseudomonotone Operators, Journal of Convex Analysis Vol. 10, pp. 465–475, 2003.