Relationships between Robinson metric regularity and Lipschitz-like behavior of implicit multifunctions

Nonlinear Analysis 72 (2010) 3594–3601

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Relationships between Robinson metric regularity and Lipschitz-like behavior of implicit multifunctionsI N.H. Chieu a , J.-C. Yao b,∗ , N.D. Yen c a

Department of Mathematics, Vinh University, Vinh, Nghe An, Viet Nam

b

Department of Applied Mathematics, National Sun Yat-Sen University, Kaohsiung 804, Taiwan

c

Institute of Mathematics, Vietnamese Academy of Science and Technology, 18 Hoang Quoc Viet, Hanoi 10307, Viet Nam

article

info

Article history: Received 6 May 2009 Accepted 16 December 2009 MSC: primary 49J53 secondary 54C60 47H04

abstract By constructing some suitable examples, Jeyakumar and Yen (2004) [1] have shown that the Robinson metric regularity (Rmr) and the Lipschitz-like property (Llp) of implicit multifunctions are not equivalent. This paper clarifies relationships between the two properties of implicit multifunctions. It turns out that the (reasonable) sufficient conditions for having (Rmr) ⇒ (Llp) are quite different from those for the validity of the reverse implication. The implicit function theorem due to Yen and Yao (2009) [2] serves as a tool for our analysis of (Rmr) and (Llp). © 2009 Elsevier Ltd. All rights reserved.

Keywords: Implicit multifunction Robinson metric regularity Lipschitz-like property Relationship Normal coderivative

1. Introduction Due to their important roles in set-valued analysis and variational analysis, the Robinson metric regularity (Rmr) and the Lipschitz-like property (Llp) of implicit multifunctions have been studied extensively. Let us recall the definitions of the two properties. Consider a multifunction F : X × Y ⇒ Z from the product X × Y of normed spaces X and Y into a normed space Z . Let (x0 , y0 ) ∈ X × Y be such that 0 ∈ F (x0 , y0 ).

(1.1)

The point-to-set map G : Y → X defined by G(y) := {x ∈ X : 0 ∈ F (x, y)},

(1.2)

is said to be the implicit multifunction defined by the inclusion 0 ∈ F (x, y).

(1.3)

I This work was supported by the Grant NSC 97-2115-M-110-001, the National Sun Yatsen University, and National Foundation for Science & Technology Development (Vietnam). The authors would like to thank the referees for their careful reading and valuable suggestions. ∗ Corresponding author. Tel.: +886 7 5253816; fax: +886 7 5253809. E-mail addresses: [email protected], [email protected] (N.H. Chieu), [email protected], [email protected] (J.-C. Yao), [email protected] (N.D. Yen).

0362-546X/$ – see front matter © 2009 Elsevier Ltd. All rights reserved. doi:10.1016/j.na.2009.12.039

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It is well known that (1.3) can represent either a parametric constraint system (put F (x, y) = f (x, y) + K for all (x, y) from a given subset D ⊂ X × Y and F (x, y) = ∅ whenever (x, y) 6∈ D, where f : D → Z is a single-valued map, K ⊂ Z a closed convex cone) or a parametric variational system (put Z = X ∗ , F (x, y) = f (x, y) + NC (y) (x), where X ∗ is the dual space of X , f : X × Y → X ∗ a single-valued map, C : Y ⇒ X a convex-valued multifunction, and NC (y) (x) ⊂ X ∗ denotes the normal cone to C (y) at x in the sense of convex analysis). Definition 1.1. One says that the implicit multifunction G has the Robinson metric regularity ((Rmr) for brevity)1 around ω0 := (x0 , y0 , 0Z ) if there are constants γ > 0, µ > 0, and neighborhoods U ∈ N (x0 ), V ∈ N (y0 ) such that



dist(x, G(y)) ≤ γ dist(0, F (x, y)) whenever x ∈ U , y ∈ V , dist(0, F (x, y)) < µ,

(1.4)

where N (x0 ) (resp., N (y0 )) stands for the collection of all the neighborhoods of x0 (resp., of y0 ) and dist(x, Ω ) := inf{kx − uk : u ∈ Ω } is the distance from x to Ω . Note that (Rmr) is originated by Robinson [16,3]. A complete characterization of that kind of solution stability of generalized linear inequality systems was given in [16, Theorems 1, 2]. In [3, Theorems 1], Robinson obtained a sufficient condition (called a regularity condition or a constraint qualification) for (Rmr) of the solution map of a generalized differentiable inequality system. In the same paper, it was shown that the sufficient condition is also a necessary one, provided that F (x, y) = f (x) + K − y, f : X → Y is a Fréchet differentiable mapping, and K ⊂ Y = Z is a closed convex cone. Various sufficient conditions for (Rmr) of implicit multifunctions have been obtained; see [17,4,5,11,6,7,1,8–10,12,13,2,29] and the references therein. By usual convention, inf ∅ = +∞. Therefore, (1.4) implies G(y) 6= ∅ for all y ∈ V with dist(0, F (x0 , y)) < µ. If the multifunction F (x0 , ·) is lower semicontinuous at (y0 , 0Z ) (that is, for every W ∈ N (0Z ) there exists V 0 ∈ N (y0 ) such that F (x0 , y) ∩ W 6= ∅ for every y ∈ V 0 ), then from (1.4) it follows that for any U1 ∈ N (x0 ) there exists V1 ∈ N (y0 ) such that G(y) ∩ U1 6= ∅ for all y ∈ V1 . If F is lower semicontinuous at ω0 = (x0 , y0 , 0Z ) in the sense that for every ε > 0 there exists Uε ∈ N (x0 ), Vε ∈ N (y0 ) such that F (x, y) ∩ (ε BX ) 6= ∅ for all (x, y) ∈ Uε × Vε , then the constant µ > 0 and the related inequality in Definition 1.1 can be omitted. Indeed, under this additional assumption, for any µ > 0 there exists U × V ∈ N (x0 ) × N (y0 ) such that dist(0, F (x, y)) < µ for all (x, y) ∈ U × V . Thus, if dist(x, G(y)) ≤ γ dist(0, F (x, y)) for every (x, y) ∈ U × V with dist(0, F (x, y)) < µ then, replacing U ∈ N (x0 ) and V ∈ N (y0 ) by smaller neighborhoods if necessary, we have the inequality for all (x, y) ∈ U × V . (A similar remark was given for inverse multifunctions in [18, Prop. 1.48], where F (x, y) = Φ (x) − y and the lower semicontinuity of Φ : X ⇒ Y is not needed.) In order to deal not only with the parametric constraint systems, but also with parametric variational systems, where the map F may not be lower semicontinuous at ω0 = (x0 , y0 , 0Z ), the definition of (Rmr) is given in the ‘local form’ and in Definition 1.1. Definition 1.2. The implicit multifunction G is said to have the Lipschitz-like property ((Llp) for brevity) around (y0 , x0 ) if there are a constant ` > 0 and neighborhoods U ∈ N (x0 ), V ∈ N (y0 ) such that G(y0 ) ∩ U ⊂ G(y) + `ky0 − ykBX

∀y0 , y ∈ V ,

(1.5)

where BX denotes the closed unit ball in X . If (1.5) is valid, then for any U2 ∈ N (x0 ) there exists V2 ∈ N (y0 ) such that G(y) ∩ U2 6= ∅ for all y ∈ V2 . Thus, like (Rmr) under a slight assumption on the local behavior of F (x0 , ·), (Llp) guarantees that, for every y from a suitable neighborhood of x0 , G(y) meets a given neighborhood of x0 (this can be termed the implicit multifunction existence). Unlike (Rmr), (Llp) is a property of the implicit multifunction G itself around the point (y0 , x0 ) in its graph. The Lipschitzlike property of an arbitrary multifunction between normed spaces around a point in its graph is defined in the same manner: one says that a multifunction Φ : X ⇒ Y has the Lipschitz-like property around (¯x, y¯ ) ∈ gphΦ := {(x, y) ∈ X × Y : y ∈ Φ (x)} if there are a constant ` > 0 and neighborhoods U ∈ N (¯x), V ∈ N (¯y) such that

Φ (x0 ) ∩ V ⊂ Φ (x) + `kx0 − xkBY

∀x0 , x ∈ U .

This concept, which is known also as the Aubin continuity [19], was introduced by Aubin [20] who called it the pseudoLipschitz property. Here we follow Mordukhovich [18] in using the terminology ‘‘the Lipschitz-like property’’. For multifunctions between normed spaces, (Llp) is a fundamental property which guarantees the validity of many calculus rules in variational analysis; see for instance [18]. Sufficient conditions for (Llp) of implicit multifunctions of different types have been given in the literature (see [17,11,21,19,7,1,9,18,22,12,13,2,30] and the papers cited therein).

1 It is worthy to mention that (Rmr) has been known under other names, for instance: stability [3], metric regularity [4–10], metric regularity in Robinson’s sense [11,1,9,12,13], and the local metric regularity in Robinson’s sense of implicit multifunctions [14,15,2].

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In the case Z = Y and F (x, y) = Φ (x) − y, where Φ : X ⇒ Y is a given multifunction, the implicit function G defined by (1.3) coincides with the inverse multifunction Φ −1 (y) := {x ∈ X : y ∈ Φ (x)}. It is clear that the inequality in the definition of (Rmr) can be rewritten equivalently as dist(x, Φ −1 (y)) ≤ γ dist(y, Φ (x)) for all (x, y) ∈ U × V satisfying dist(y, Φ (x)) < µ. If the last property is fulfilled for some γ > 0, µ > 0, U ∈ N (x0 ) and V ∈ N (y0 ), where y0 ∈ Φ (x0 ) then Φ : X ⇒ Y is said to be locally metrically regular around (x0 , y0 ). Borwein and Zhuang [23] and Penot [24] have proved that this local metrical regularity is equivalent to the requirement that Φ −1 has the Lipschitz-like property around (y0 , x0 ) (see [17,25,18,26] and the references therein for further developments of this direction). Thus, for the case of inverse multifunctions, (Rmr) ⇔ (Llp). It is worthy to stress that, in general, the last equivalence does not hold for implicit multifunctions. Example 1.3 ((Llp) ; (Rmr); See [1, Example 3.7]). Let X = Y = Z = R, F (x, y) = {x3 − y3 }, (x0 , y0 ) = (0, 0) ∈ X × Y . We have G(y) = {y} for all y ∈ Y . Obviously, (Llp) holds for G around (y0 , x0 ). If (Rmr) is valid, then there exist γ > 0, µ > 0, U ∈ N (x0 ), and V ∈ N (y0 ) such that

|x − y| ≤ γ |x3 − y3 | whenever x ∈ U , y ∈ V , |x3 − y3 | < µ, which is impossible. Hence G is not Robinson metrically regular around (x0 , y0 , 0Z ). Example 1.4 ((Rmr) ; (Llp); See [1, Example 3.6]). Let X = Y = Z = R, F (x, y) = {x(y + 1) − y1/3 }, (x0 , y0 ) = (0, 0) ∈ X × Y . Clearly, G(y) = {(1 + y)−1 y1/3 } for any y ∈ Y \ {−1} and G(−1) = ∅. The implicit multifunction G is Robinson metrically regular around (x0 , y0 , 0). Indeed, for U = X , V = (−1/2, 1/2), γ = 2 and µ = 1, we have U ∈ N (x0 ), V ∈ N (y0 ) and

γ d(0, F (x, y)) − d(x, G(y)) = γ |x(y + 1) − y1/3 | − |x − (1 + y)−1 y1/3 | = (1 + 2y)|x − (1 + y)−1 y1/3 | ≥0 for any x ∈ U and y ∈ V . Hence (Rmr) holds for G around (x0 , y0 , 0). Note that (Llp) does not hold for G around (y0 , x0 ). Indeed, otherwise there exists ` > 0 such that for any yk , y˜ k → 0, yk 6= y˜ k , it holds 1/3

1/3

|(1 + yk )−1 yk − (1 + y˜ k )−1 y˜ k | ≤ `|yk − y˜ k | for all k large enough. In particular, for y˜ k = 0 and yk = 1/k3 we have

(1 + k−3 )−1 k−1 ≤ `k−3 or, equivalently,

(1 + k−3 )−1 k2 ≤ ` for all k large enough, a contradiction. Hence G is not Lipschitz-like around (y0 , x0 ). The purpose is paper twofold: 1. Find suitable conditions, which ensure that (Rmr) ⇒ (Llp) (resp., (Llp) ⇒ (Rmr)); 2. Find out when (Rmr) and (Llp) are equivalent. It turns out that the (reasonable) sufficient conditions for having (Rmr) ⇒ (Llp) are quite different from those for the validity of the reverse implication. The implicit function theorem due to Yen and Yao [2], which was obtained by means of Mordukhovich’s theory of generalized differentiation and a formula of Levy and Mordukhovich [27] for computing/estimating the normal (limiting) coderivative of implicit multifunctions, will be an effective tool for our analysis of (Rmr) and (Llp). After giving some preliminaries in next section, in Section 3 we find adequate conditions for having the implication (Llp) ⇒ (Rmr). Section 4 presents a simple sufficient condition for the implication (Rmr) ⇒ (Llp) to be valid and derives conditions for the equivalence between (Rmr) and (Llp). 2. Preliminaries Let X be a Banach space, and let Ω ⊂ X . By w ∗ (resp., hx∗ , xi) we denote the weak-star topology in the dual space X ∗ (resp., the canonical pairing between X ∗ and X ). The closed ball centered at x with radius ρ is abbreviated to Bρ (x). For a multifunction Φ : X ⇒ X ∗ , the expression Lim supx→¯x Φ (x) denotes the sequential Kuratowski–Painlevé upper limit of Φ with respect to the norm topology of X , and the weak-star topology of X ∗ , i.e., w∗

Lim sup Φ (x) = {x∗ ∈ X ∗ : ∃ sequences xk → x¯ , x∗k − → x∗ , with x∗k ∈ Φ (xk ) for all k = 1, 2, . . .}. x→¯x

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Normal cones to sets, coderivatives of multifunctions, and subdifferentials of extended-real-valued functions are defined [18] as follows. The set of Fréchet ε -normals to Ω at x ∈ Ω is given by

  h x∗ , u − xi b ≤ε , Nε (x; Ω ) = x ∈ X : lim sup   ku − xk Ω u− →x  

∗

∗

(2.1)

Ω

where u − → x means u → x with u ∈ Ω . For ε = 0, the set in (2.1) is called the Fréchet normal cone to Ω at x and is denoted by b N (x; Ω ). By convention, b Nε (x; Ω ) = ∅ for all ε ≥ 0 whenever x 6∈ Ω . We call N (¯x; Ω ) := Lim sup b Nε (x; Ω )

(2.2)

x→¯x,ε↓0

the normal cone to Ω at x¯ in the sense of Mordukhovich. If x 6∈ Ω , N (x; Ω ) = ∅ by convention. A multifunction Φ : X ⇒ Y between Banach spaces is said to be locally closed around z¯ = (¯x, y¯ ) ∈ gph Φ if there is a ball Bρ (¯z ), ρ > 0, such that gph Φ ∩ Bρ (¯z ) is a closed set in the product space X × Y whose norm is given by k(x, y)k = kxk+kyk. For every (¯x, y¯ ) ∈ gph Φ , the map D∗ Φ (¯x, y¯ ) : Y ∗ ⇒ X ∗ defined by D∗ Φ (¯x, y¯ )(y∗ ) := {x∗ ∈ X ∗ : (x∗ , −y∗ ) ∈ N ((¯x, y¯ ); gph Φ )}

(2.3)

is said to be the normal coderivative (called also the limiting coderivative and the coderivative in the sense of Mordukhovich) of Φ at (¯x, y¯ ). We put D∗ Φ (¯x, y¯ )(y∗ ) = ∅ for any y∗ ∈ Y ∗ when (¯x, y¯ ) 6∈ gph Φ . The Fréchet coderivative b D∗ Φ (¯x, y¯ ) : Y ∗ ⇒ X ∗ of b Φ at (¯x, y¯ ) is defined similarly, provided that N ((¯x, y¯ ); gph Φ ) in (2.3) is replaced by N ((¯x, y¯ ); gph Φ ). Note that the concept of coderivative (2.3), regardless of the normal cone used, is originated by Mordukhovich [28]. If Φ is a single-valued and y¯ = Φ (¯x), it is customary to write D∗ Φ (¯x) for D∗ Φ (¯x, y¯ ) and b D∗ Φ (¯x) for b D∗ Φ (¯x, y¯ ). If Φ : X → Y is strictly differentiable at x¯ with the derivative ∇ Φ (¯x), that is ∇ Φ (¯x) : X → Y is a continuous linear operator and lim

x→¯x,u→¯x

Φ (x) − Φ (u) − ∇ Φ (¯x)(x − u) = 0, k x − uk

then D∗ Φ (¯x)(y∗ ) = b D∗ Φ (¯x)(y∗ ) = {(∇ Φ (¯x))∗ y∗ }

∀y∗ ∈ Y ∗

(2.4)

(see [18, Theorem 1.38]). It is well known that the second equality in (2.4) is valid if Φ is Fréchet differentiable at x¯ . Formula (2.4) and this fact show that the normal coderivative (resp., the Fréchet coderivative) of multifunctions is a natural extension of the adjoint operator of the strict derivative (resp., the Fréchet derivative) of single-valued maps. 3. When does (Llp) imply (Rmr)? Given inclusion (1.3), besides G we can define another implicit multifunction H : X ⇒ Y by setting H (x) := {y ∈ Y : 0 ∈ F (x, y)}

(3.1)

(the unknown y is sought via the variable x). Due to (1.1), y0 ∈ H (x0 ). One can define the Robinson metric regularity (resp., the Lipschitz-like property) for H around (x0 , y0 , 0Z ) (resp., around (x0 , y0 )) similarly as it has been done for G around the points (x0 , y0 , 0Z ) and (y0 , x0 ). Our first result reads as follows. Theorem 3.1. If H is Robinson metrically regular around ω0 := (x0 , y0 , 0Z ) ∈ gphF , then (Llp) implies (Rmr). Proof. By the assumed Robinson metric regularity of H around ω0 , there exist γ1 > 0, µ1 > 0, U1 ∈ N (x0 ), and V1 ∈ N (y0 ) such that



dist(y, H (x)) ≤ γ1 dist(0, F (x, y)) whenever x ∈ U1 , y ∈ V1 , dist(0, F (x, y)) < µ1 .

(3.2)

We have x ∈ G(y) ⇔ 0 ∈ F (x, y)

⇔ y ∈ H (x) ⇔ x ∈ H −1 (y). Hence H −1 (y) = G(y). Suppose that (Llp) holds, i.e., G is Lipschitz-like around (y0 , x0 ). By [18, Theorem 1.49], H is locally metrically regular around (x0 , y0 ), i.e., one can find γ2 > 0, µ2 > 0, U2 ∈ N (x0 ), V2 ∈ N (y0 ) with the property that



dist(x, H −1 (y)) ≤ γ2 dist(y, H (x)) whenever x ∈ U2 , y ∈ V2 , dist(y, H (x)) < µ2 .

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This yields



dist(x, G(y)) ≤ γ2 dist(y, H (x)) whenever x ∈ U2 , y ∈ V2 , dist(y, H (x)) < µ2 .

(3.3)

Let us set U := U1 ∩ U2 , V := V1 ∩ V2 , γ := γ1 γ2 and µ := min{µ2 /γ1 , µ1 }. By (3.2) and (3.3) we can assert that



dist(x, G(y)) ≤ γ dist(0, F (x, y)) whenever x ∈ U , y ∈ V , dist(0, F (x, y)) < µ.

(3.4)

Indeed, given any x ∈ U , y ∈ V with dist(0, F (x, y)) < µ, we observe that x ∈ U1 , y ∈ V1 and dist(0, F (x, y)) < µ1 . According to (3.2), d(y, H (x)) ≤ γ1 dist(0, F (x, y)) < γ1 µ ≤ µ2 . Then we get d(y, H (x)) ≤ γ1 dist(0, F (x, y)) and, by virtue of (3.3), dist(x, G(y)) ≤ γ2 dist(y, H (x)) for every x ∈ U , y ∈ V with dist(0, F (x, y)) < µ. Consequently, dist(x, G(y)) ≤ γ1 γ2 dist(0, F (x, y)) whenever x ∈ U , y ∈ V and dist(0, F (x, y)) < µ; thus (3.4) is established. It follows that G is Robinson metrically regular around ω0 .  Remark 3.2. Let Φ : X ⇒ Y be a multifunction between normed spaces X and Y . Setting F (x, y) := Φ (x) − y for all (x, y) ∈ X × Y , we have H (x) = {y ∈ Y | 0 ∈ F (x, y)} = Φ (x). It is easy to see that (3.2) is satisfied with γ1 = 1, U1 = X , V1 = Y , and any µ1 > 0. Thus H is Robinson metrically regular around ω0 := (x0 , y0 , 0) ∈ gphF . To proceed furthermore, we need the following implicit function theorem. Theorem 3.3 (See [2, Theorem 3.1]). Let X , Y , Z be finite-dimensional Euclidean spaces, F : X × Y ⇒ Z a multifunction, (x0 , y0 ) ∈ X × Y a pair such that (1.1) holds. Let G be the implicit multifunction defined by (1.2). If gph F is locally closed around the point ω0 := (x0 , y0 , 0Z ) and

(C1 ) ker D∗ F (ω0 ) = {0}, (C2 ) {y∗ ∈ Y ∗ : ∃z ∗ ∈ Z ∗ with(0, y∗ ) ∈ D∗ F (ω0 )(z ∗ )} = {0}, then G is Robinson metrically regular around ω0 = (x0 , y0 , 0Z ), i.e., there exist γ > 0, µ > 0, U ∈ N (x0 ), V ∈ N (y0 ) with the property that



dist (x, G(y)) ≤ γ dist (0, F (x, y)) whenever x ∈ U , y ∈ V , dist (0, F (x, y)) < µ.

Interchanging the roles of the variables x and y, from (C2 ) we get a new regularity condition denoted by (C3 ) that, together with (C1 ), ensure the Robinson metric regularity of H around ω0 . Thus, the next statement is a corollary of the preceding two theorems. Theorem 3.4. Let X , Y , Z be finite-dimensional Euclidean spaces, F : X × Y ⇒ Z a multifunction, and ω0 := (x0 , y0 , 0Z ) ∈ gphF . Let G be the implicit multifunction defined by (1.2). If gphF is locally closed around ω0 and

(C1 ) ker D∗ F (ω0 ) = {0}, (C3 ) {x∗ ∈ X ∗ : ∃z ∗ ∈ Z ∗ with (x∗ , 0) ∈ D∗ F (ω0 )(z ∗ )} = {0}, then (Llp) implies (Rmr). Proof. According to Theorem 3.3, from (C1 ), (C3 ) and the assumed local closedness of gphF around ω0 it follows that there exist γ1 > 0, µ1 > 0, U1 ∈ N (x0 ), and V1 ∈ N (y0 ) such that (3.2) is satisfied. This means that H is Robinson metrically regular around ω0 . Hence, applying Theorem 3.1 we get the desired conclusion.  Remark 3.5. Theorem 3.4 gives a set of sufficient conditions for having the implication (Llp) ⇒ (Rmr). There are two restrictions in its assumptions: (a) The spaces under consideration must be finite-dimensional, (b) The graph of the given multifunction must be locally closed around the point of interest. In order to avoid these restrictions, instead of Theorem 3.3 one can try to invoke other implicit function theorems (see for instance [4,11,6–10,3,12]) which assert a property equivalent to the Robinson metric regularity of implicit multifunctions in the sense of Definition 1.1.

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4. When does (Rmr) imply (Llp)? It is natural to hope that the implication (Rmr) ⇒ (Llp) is valid under the assumptions of Theorem 3.4 (hence, roughly speaking, (Rmr) and (Llp) are equivalent under (C1 ), (C3 )). However, the hope is overturned by the next example. Example 4.1. Let X = Y = Z = R, F (x, y) = {x(y + 1) − y1/3 }, ω0 = (x0 , y0 , 0) = (0, 0, 0) ∈ X × Y × Z . Obviously, G(y) = {(1 + y)−1 y1/3 } for every y ∈ Y \ {−1} and G(−1) = ∅. We will show that D F (ω0 )(z ) = ∗

∗



if z ∗ = 0 otherwise.

{0} × R ∅

(4.1)

Indeed, let ϕ(x, y) := x(y + 1) and Φ (x, y) := {−y1/3 }. Clearly, ϕ is strictly differentiable at (x0 , y0 ) with ∇ϕ(x0 , y0 ) = (1, 0) and ∇ϕ(x0 , y0 )∗ (z ∗ ) = (z ∗ , 0) ∈ X ∗ × Y ∗ for all z ∗ ∈ Z ∗ . Since F = ϕ + Φ , by [18, Theorem 1.62], D∗ F (ω0 )(z ∗ ) = ∇ϕ(x0 , y0 )∗ (z ∗ ) + D∗ Φ (x0 , y0 , 0Z − ϕ(x0 , y0 ))(z ∗ )

= (z ∗ , 0) + D∗ Φ (ω0 )(z ∗ ). 1/3

Note that gphΦ = {(x, y, z ) ∈ R |z = −y all (y, z ) ∈ Y × Z . By [18, Prop. 1.2], 3

(4.2)

} = X × Λ (gph(g )), where g : Z → Y , g (z ) = −z and Λ(y, z ) := (z , y) for −1

3

N ((x0 , y0 , 0Z ); gphΦ ) = N (x0 ; X ) × N ((y0 , 0Z ); Λ−1 (gph(g )))

= {0} × N ((y0 , 0Z ); Λ−1 (gph(g ))).

(4.3)

Note that Λ is strictly differentiable at (y0 , 0Z ) with ∇ Λ(y0 , 0Z )(y, z ) = (z , y). Hence ∇ Λ(y0 , 0Z ) is surjective and ∇ Λ(y0 , 0Z )∗ (z ∗ , y∗ ) = (y∗ , z ∗ ). By [18, Theorem 1.17], N ((y0 , 0Z ); Λ−1 (gph(g ))) = ∇ Λ(y0 , 0Z )∗ N ((0Z , y0 ); gph(g )).

(4.4)

We have

(z ∗ , y∗ ) ∈ N ((0Z , y0 ); gph(g )) ⇔ z ∗ ∈ D∗ g (0Z )(−y∗ ), and g is strictly differentiable at 0Z with ∇ g (0Z ) = 0 and ∇ g (0Z )∗ = 0. By [18, Theorem 1.38], D∗ g (0Z )(−y∗ ) = {∇ g (0Z )∗ (−y∗ )} = {0}. Consequently, N ((0Z , y0 ); gph(g )) = {0} × Y ∗ .

(4.5)

Combining (4.4) and (4.5) with the fact that ∇ Λ(y0 , 0Z )∗ (z ∗ , y∗ ) = (y∗ , z ∗ ), we get N ((y0 , 0Z ); Λ−1 (gph(g ))) = Y ∗ × {0}.

(4.6)

From (4.3) and (4.6) it follows that N ((x0 , y0 , 0Z ); gphΦ ) = {0} × Y ∗ × {0}. Hence

(x∗ , y∗ ) ∈ D∗ Φ (ω0 )(z ∗ ) ⇔ (x∗ , y∗ , −z ∗ ) ∈ N ((x0 , y0 , 0Z ); gphΦ ) ⇔ (x∗ , y∗ , −z ∗ ) ∈ {0} × Y ∗ × {0}. Therefore, D∗ Φ (ω0 )(z ∗ ) =

 {0} × Y ∗ ∅

if z ∗ = 0 otherwise.

(4.7)

Combining (4.2) with (4.7) we obtain (4.1). By (4.1), ker D∗ F (ω0 ) = {z ∗ ∈ Z ∗ |(0, 0) ∈ D∗ F (ω0 )(z ∗ )} = {0} and

{x∗ ∈ X ∗ : ∃z ∗ ∈ Z ∗ with (x∗ , 0) ∈ D∗ F (ω0 )(z ∗ )} = {0}. This means that (C1 ) and (C3 ) are satisfied. It has been noted in Example 1.4 (see also [1]) that G is Robinson metrically regular around ω0 but it is not Lipschitz-like around (y0 , x0 ). Consider the following condition:

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(C4 ) There exist a constant ` > 0 and neighborhoods U ∈ N (x0 ), V ∈ N (y0 ), W ∈ N (0Z ) such that F (x, y0 ) ∩ W ⊂ F (x, y) + `ky0 − ykBZ

(4.8)

for all x ∈ U and y , y ∈ V . 0

Remark 4.2. If F (x, y) = Φ (x) − y for all (x, y) ∈ X × Y , where Φ : X ⇒ Y is a multifunction between normed spaces X and Y , then (C4 ) is satisfied. Note that (C4 ) is also fulfilled in many other circumstances. Actually, it is a uniform local Lipschitz property of multifunctions of two variables which has been employed, e.g., in [4,12]. It is worthy observing that to get (Llp), which is a Lipschitz continuity property of the implicit multifunction G(y), one must impose a condition on Lipschitz continuity of the multifunction F (x0 , ·) around y0 . The next theorem and its proof show that (C4 ) is a minimal condition for having the implication (Rmr) ⇒ (Llp). Theorem 4.3. If (C4 ) holds, then (Rmr) implies (Llp). Proof. Let ` > 0, U ∈ N (x0 ), V ∈ N (y0 ), and W ∈ N (0Z ) be such that (4.8) is fulfilled. Suppose that (Rmr) holds, i.e., there exist γ1 > 0, µ1 > 0, U1 ∈ N (x0 ) and V1 ∈ N (y0 ) satisfying



dist(x, G(y)) ≤ γ1 dist(0, F (x, y)) whenever x ∈ U1 , y ∈ V1 , dist(0, F (x, y)) < µ1 .

(4.9)

We need to prove that G is Lipschitz-like around (y0 , x0 ), i.e., there exist `2 > 0, U2 ∈ N (x0 ) and V2 ∈ N (y0 ) with G(y) ∩ U2 ⊂ G(˜y) + `2 ky − y˜ kBX

∀y, y˜ ∈ V2 .

(4.10)

Suppose that (4.10) was false. Then there would exist sequences xk → x0 , yk → y0 , and y˜ k → y0 such that xk ∈ G(yk ) \ [G(˜yk ) + kkyk − y˜ k kBX ] for k = 1, 2, . . . . Hence kkyk − y˜ k k ≤ dist(xk , G(˜yk ))

(4.11)

for k sufficiently large. Since 0 ∈ F (xk , yk ) ∩ W , by (4.8) we have 0 ∈ F (xk , y˜ k ) + `kyk − y˜ k kBZ for all k large enough. Thus dist(0, F (xk , y˜ k )) ≤ `kyk − y˜ k k

(4.12)

for k large enough. In particular, for sufficiently large indices k, it holds dist(0, F (xk , y˜ k )) < µ1 . Combining (4.9) and (4.11) with (4.12) we get k ≤ γ1 ` for k large, a contradiction. We have shown that (Llp) is valid. The proof is complete.  The next example shows that, in general, the condition (C4 ) alone cannot ensure the equivalence between (Rmr) and (Llp). Example 4.4. Let X = Y = Z = R, F (x, y) = {x3 − y3 }, (x0 , y0 ) = (0, 0) ∈ X × Y . For U := X , V := (−1, 1), W = Z and ` = 3, it holds F (x, y) ∩ W ⊂ F (x, y˜ ) + `ky − y˜ kBZ whenever x ∈ U and y, y˜ ∈ V , i.e., (C4 ) is satisfied. By Example 1.3, (Llp) is valid but G is not Robinson metrically regular around (x0 , y0 , 0Z ). This shows that (C4 ) is insufficient for having the implication (Llp) ⇒ (Rmr). Theorem 4.5. If H is Robinson metrically regular around ω0 := (x0 , y0 , 0) ∈ gphF and the condition (C4 ) holds, then the properties (Llp) and (Rmr) are equivalent. Proof. It suffices to combine Theorem 3.1 with Theorem 4.3.



Theorem 4.6. Let X , Y , Z be finite-dimensional Euclidean spaces, F : X × Y ⇒ Z a multifunction, and ω0 := (x0 , y0 , 0Z ) ∈ gphF . Let G be defined by (1.2). If gphF is locally closed around ω0 and the conditions (C1 ), (C3 ) and (C4 ) are satisfied, then (Rmr) is equivalent to (Llp). Proof. The assertion follows from Theorems 3.4 and 4.3.



Using (C4 ) and other sets of conditions for getting the Robinson metric regularity of H around ω0 (see for instance [3,4, 11,6–10,12,13]), we can obtain various sufficient conditions for the equivalence (Rmr) ⇔ (Llp). To specialize (C1 ), (C3 ), (C4 ), and apply the above theorems to parametric smooth constraint systems and parametric affine variational inequalities, one can use some formulae available in [19,14,15,2].

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