The Tikhonov Regularization Method for Set-Valued Variational Inequalities

and Applied Analysis 3 Definition 2.1. Let F : K → 2Rn be a set-valued mapping. F is said to be i monotone on K if for each pair of points x, y ∈ K and for all x∗ ∈ F x and y∗ ∈ F y , 〈y∗ − x∗, y − x〉 ≥ 0, ii maximal monotone on K if, for any u ∈ K, 〈ξ − x∗, u − x〉 ≥ 0 for all x ∈ K and all x∗ ∈ F x implies ξ ∈ F u , iii quasimonotone on K if for each pair of points x, y ∈ K and for all x∗ ∈ F x and y∗ ∈ F y , 〈x∗, y − x〉 > 0 implies that 〈y∗, y − x〉 ≥ 0, iv F is said to be upper semicontinuous at x ∈ K if for every open set V containing F x , there is an open set U containing x such that F y ⊂ V for all y ∈ K ∩U; if F is upper semicontinuous at every x ∈ K, we say F is upper semicontinuous on K, v upper hemicontinuous on K if the restriction of F to every line segment of K is upper semicontinuous. The following result is celebrated; see 17 . Lemma 2.2. Let K be a nonempty convex subset of a Hausdorff topological vector space E, and let G : K → 2 be a set-valued mapping from K into E satisfying the following properties: i G is a KKMmapping: for every finite subsetA ofK, co A ⊂ x∈A G x , where co denotes the convex hull; ii G x is closed in E for every x ∈ K; iii G x0 is compact in E for some x0 ∈ K. Then ⋂ x∈K G x / ∅. 3. Existence of Solutions and Coercivity Conditions Definition 3.1. F is said to have variational inequality property on K if for every nonempty bounded closed convex subset D of K, GVIP F,D has a solution. Proposition 3.2. The following classes of mappings have the variational inequality property: i every upper semicontinuous set-valued mapping with nonempty compact convex values, ii every upper hemicontinuous quasimonotone set-valued mapping with nonempty compact convex values; iii if F is a single-valued continuous mapping and T is upper hemicontinuous and monotone with nonempty compact convex values, then F T has the variational inequality property. Proof. i is well known in the literature. ii is verified in 18 . iii is a consequence of the Debrunner-Flor lemma 19 and 20, Theorem 41.1 . Indeed, let D be a bounded closed convex subset of K, and let ND x : { ξ ∈ R : sup y∈D 〈 ξ, y − x ≤ 0 } , 3.1 4 Abstract and Applied Analysis the normal cone of D at x ∈ D. By 20, Theorem 41.1 , M x : T x ND x is a maximal monotone mapping. By the Debrunner-Flor lemma 19 , there is u ∈ D such that 〈ξ F u , x − u〉 ≥ 0, ∀x ∈ D, ∀ξ ∈ M x . 3.2 Since M is maximal monotone, −F u ∈ M u ≡ T u ND u . By the definition of ND, u ∈ SOL F T,D . Proposition 3.3 below shows that Proposition 3.2 iii can be extended to the case where F is a set-valued mapping. Proposition 3.3. If F : K → 2Rn is upper semicontinuous with nonempty compact convex values and T : K → 2Rn is monotone and upper hemicontinuous on K with nonempty compact convex values, then F T has the variational inequality property on K. Proof. Let D be a bounded closed convex subset of K. Define G,H : D → 2 by G ( y ) : { x ∈ D : sup ξ∈T x 〈 ξ, y − x sup ζ∈F x 〈 ζ, y − x ≥ 0 } , H ( y ) : { x ∈ D : inf ξ∈T y 〈 ξ, y − x sup ζ∈F x 〈 ζ, y − x ≥ 0 } . 3.3 Since T is monotone, G y ⊂ H y . Since F is upper semicontinuous, H y is closed and hence compact in D for every y ∈ D. Now we prove that G is a KKM map. If not, there is {y1, . . . , yn} ⊂ D and x0 ∑ λiyi such that x0 / ∈ ⋃n i 1 G yi 0 sup ξ∈T x0 〈 ξ, ∑ λiyi − x0 〉 sup ζ∈F x0 〈 ζ, ∑ λiyi − x0 〉 ≤ ∑ λi [ sup ξ∈T x0 〈 ξ, yi − x0 〉 sup ζ∈F x0 〈 ζ, yi − x0 〉 ] < 0. 3.4 This contradiction shows thatG is a KKMmap, so isH. By Lemma 2.2, there is x ∈ ∩y∈DH y . Fix any y ∈ D. Let yt : x t y − x . Then for small t ∈ 0, 1 , yt ∈ D and hence 0 ≤ inf ξ∈T yt 〈 ξ, yt − x 〉 sup ζ∈F x 〈 ζ, yt − x 〉 t inf ξ∈T yt 〈 ξ, y − x t sup ζ∈F x 〈 ζ, y − x. 3.5 Dividing t > 0 on both sides, we have sup ξ∈T yt 〈 ξ, y − x sup ζ∈F x 〈 ζ, y − x ≥ 0. 3.6 Abstract and Applied Analysis 5 Letting t → 0 yields that x ∈ G y , as T is upper hemicontinuous. Since y ∈ D is arbitrary, x ∈ y∈D G y :and Applied Analysis 5 Letting t → 0 yields that x ∈ G y , as T is upper hemicontinuous. Since y ∈ D is arbitrary, x ∈ y∈D G y : sup ξ∈T x sup ζ∈F x 〈 ξ ζ, y − x ≥ 0, ∀y ∈ D. 3.7 Since F x and T x are compact and convex, the Sionminimax theorem implies the existence of u ∈ F x and v ∈ T x such that 〈 u v, y − x ≥ 0, ∀y ∈ D. 3.8 Thus x solves GVIP F T,D . Before making further discussion, we need to state some coercivity conditions. The relationships of these coercivity conditions are well known in the literature; however, we provide the proof for completeness. Consider the following coercivity conditions. A There exists r > 0 such that for every x ∈ K \ Kr , there is y ∈ K with ‖y‖ < ‖x‖ satisfying infx∗∈F x 〈x∗, x − y〉 ≥ 0. B There exists r > 0 such that for every x ∈ K \ Kr , there is y ∈ Kr satisfying infx∗∈F x 〈x∗, x − y〉 ≥ 0. C There exists r > 0 such that for every x ∈ K \Kr and every x∗ ∈ F x , there exists some y ∈ Kr such that 〈x∗, x − y〉 > 0. D There exists r > 0 such that for every x ∈ K \Kr , there exists some y ∈ Kr such that supy∗∈F y 〈y∗, x − y〉 > 0. E There exists y0 ∈ K such that the set L ( y0 ) : { x ∈ K : inf x∗∈F x 〈 x∗, x − y0 〉 < 0 } 3.9 is bounded, if nonempty. Proposition 3.4. The following statements hold. i C ⇒ B if F is of convex values. ii D ⇒ B if F is quasimonotone. iii E ⇒ B ⇒ A . Proof. i By C , for every x ∈ K \ Kr , infx∗∈F x supy∈Kr〈x, x − y〉 ≥ 0. Since F x is convex and Kr is compact convex, the Kneser minimax theorem implies that sup y∈Kr inf x∗∈F x 〈 x∗, x − y inf x∗∈F x sup y∈Kr 〈 x∗, x − y ≥ 0. 3.10 6 Abstract and Applied Analysis Since y → infx∗∈F x 〈x∗, x − y〉 is upper semicontinuous and since Kr is compact, there is y x ∈ Kr such that inf x∗∈F x 〈 x∗, x − y x 〉 sup y∈Kr inf x∗∈F x 〈 x∗, x − y ≥ 0. 3.11 This verifies B . ii The implication of D ⇒ B is an immediate consequence of F being quasimonotone. iii E ⇒ B . If L y0 ∅, then for any x ∈ K, infx∗∈F x 〈x∗, x − y0〉 ≥ 0; thus B holds. If L y0 / ∅, then by E , there is r > 0 such that L y0 ∪ {y0} ⊂ Kr . Thus B is still true. B ⇒ A . Let r > 0 be such that B holds. Then x ∈ K \ Kr 1, x / ∈ Kr . By B , there is y ∈ Kr such that infx∗∈F x 〈x∗, x−y〉 ≥ 0. Obviously, ‖y‖ ≤ r < r 1 < ‖x‖. Thus A is verified with r replaced by r 1. Remark 3.5. The coercivity condition A is actually C’ in 3 where it is shown that if F is quasimonotone and upper hemicontinuous with nonempty compact convex values, then A implies that GVIP F,K has a solution. However, it seems unknown whether this assertion still holds if one replaces “quasimonotone and upper hemicontinuous” by “upper semicontinuous.” An affirmative answer is given by Corollary 3.9. Remark 3.6. The coercivity condition B is the condition C in 3 . The condition D appears in 21 . The condition E appears essentially in Corollary 3.1 in 22 ; see also Proposition 2.2.3 in 8 . If F is single valued, then E reduces to Proposition 2.2.3 a in 8 . Remark 3.7. Example 3.1 in 9 shows that A does not necessarily imply B , even if F is single-valued and continuous. From the above discussion, A is the weakest coercivity condition among them. 9 proved if F is single valued and continuous, then A implies that GVIP F,K has a solution. Corollary 3.9 shows that this assertion still holds even if F is a set-valued mapping. Theorem 3.8. LetK ⊂ R be a nonempty closed convex set, and let F : K → 2Rn be a mapping with nonempty compact convex values. Suppose that (A) holds. If F has the variational inequality property on K, then GVIP F,K has a solution. Proof. Let m > r. Since Km is bounded closed convex and F has the variational inequality property, there is xm ∈ Km such that sup x∗∈F xm 〈x∗, y − xm〉 ≥ 0, ∀y ∈ Km. 3.12 i If ‖xm‖ m, then ‖xm‖ > r, and by assumption, there is y0 ∈ K with ‖y0‖ < ‖xm‖ such that sup x∗∈F xm 〈x∗, y0 − xm〉 ≤ 0. 3.13 Fix any y ∈ K. Since ‖y0‖ < ‖xm‖ ≤ m, there is t ∈ 0, 1 such that zt : y0 t y − y0 ∈ Km. Abstract and Applied Analysis 7and Applied Analysis 7