A note on implicit multifunction theorems - Semantic Scholar

Optim Lett (2014) 8:329–341 DOI 10.1007/s11590-012-0580-7 ORIGINAL PAPER

A note on implicit multifunction theorems T. T. A. Nghia

Received: 28 February 2012 / Accepted: 11 October 2012 / Published online: 30 October 2012 © Springer-Verlag Berlin Heidelberg 2012

Abstract The paper is mainly devoted to the study of implicit multifunction theorems in terms of Fréchet coderivative in Asplund spaces. It sharpens the well-known implicit multifunction theorem of Ledyaev and Zhu (Set Valued Anal., 7, 209–238, 1999) as well as many recent publications about this significant area. Keywords Implicit multifunction theorem · Metric regularity · Metric subregularity · Fréchet coderivative Mathematics Subject Classification

49J52 · 49J53

1 Introduction The main objective of this paper is to study metric regularity of the generalized equation 0 ∈ F(x, p),

(1.1)

→ Y is a set-valued mapping, X, Y are two Banach spaces, and where F : X × P → P is a metric space. According to [2,24] we say F is locally metrically regular in Robinson’s sense around (x, ¯ p, ¯ 0) ∈ X × P × Y with 0 ∈ F(x, ¯ p) ¯ if there exist constants k, μ > 0 and neighborhoods U of x¯ and V of p¯ such that d(x; G( p)) ≤ kd(0; F(x, p)) whenever x ∈U, p ∈ V with d(0; F(x, p)) < μ, (1.2)

T. T. A. Nghia (B) Department of Mathematics, Wayne State University, Detroit, MI 48202, USA e-mail: nghia@math.wayne.edu

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with implicit multifunction G defined by G( p) := {x ∈ X | 0 ∈ F(x, p)} for p ∈ P and notation d(x; ) denoted by the distance from x ∈ X to the set ⊂ X . This property has been noticed first in two seminal papers [22,23] of Robinson, who considered the stability of generalized inequality systems at that time. In the last two decades, property (1.2) attracted many attentions due to its broad applications to many areas of optimization theory; see, e.g., [1–4,7,11–14,19,20,24]. Many papers in this direction aim to find sufficient conditions for (1.2) by establishing some upper estimates to k in terms of coderivatives. However, the range of μ as well as the constructions of neighborhoods U and V have not been characterized. The more we know about μ, U , and V , the more we understand about the stability of equation (1.1). This motivates us to find some explicit forms of μ, U , and V via k in the sequel. Recently Ngai et al. [20] have introduced a sharp condition, which ensures the property in (1.2). Their approach is fully based on the theory of error bound for single-valued functions, which has been used widely in [1,10,14,19]. Though the constraint “d(0; F(x, p)) < μ” does not appear in their conclusion of the “implicit → F(x, multifunction theorems”, they have to suppose that the multifunction p → ¯ p) is lower semi-continuous at p. ¯ Our approach is very direct, we do not restrict the problem to error bound of single-valued mappings. Not only does it supply the explicit forms of neighborhoods U and V but also it relaxes the assumption about the lower semicontinuity, which is also required in [14,19,20]. Throughout the paper we employ Fréchet subdifferentials and its calculus in Asplund spaces; however, the results are also true for abstract subdifferentials ∂ in ∂ − tr ustwor thy spaces (see [10,20]) by using the same technique. It is worth mentioning further that property (1.2) fully covers two other fundamental properties in variational analysis, which are metric regularity and metric subregularity. Indeed, when P ≡ Y and F(x, p) = H (x) − p for some set-valued mapping H : → P, property (1.2) is clearly equivalent to the following X→ d(x; H −1 ( p)) ≤ kd( p; H (x)) for all x ∈ U, p ∈ V with d( p; H (x)) < μ. (1.3) By choosing smaller neighborhoods U and V , we can ignore the inequality “d( p; H (x)) < μ” in (1.3) easily. This is known as the metric regularity of H around (x, ¯ p) ¯ with modulus k; see, e.g., the books [6,17] as well as the survey [10] for the long history and fruitful discussions about this significant property. If we just omit the space P in the generalized equation (1.1), property (1.2) simply becomes d(x; F −1 (0)) ≤ kd(0; F(x)) whenever x ∈ U with d(0; F(x)) < μ, (1.4) where the inequality “d(0; F(x)) < μ” can be neglected easily via an appropriate neighborhood U . Following Dontchev and Rockafellar [5,6] such feature is called metric subregularity of F at x¯ ∈ F −1 (0). To the best of our knowledge, the most powerful results in seeking sufficient conditions in terms of Fréchet coderivatives for (1.4) have been established recently in [25]. We will prove in Sect. 3 that our approach may provide some improvements to several results of [25].

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A note on implicit multifunction theorems

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The rest of our paper is organized as follows. Section 2 contains the main results of the paper, which devote to the study of refined sufficient conditions in terms of Fréchet coderivatives to obtain (1.2) as well as the range of μ and explicit forms of U, V in (1.2). In Sect. 3 we consider metric subregularity as a specific case of property (1.2). The results in this section are slightly stronger than several ones in [25]. Our notation and terminology are basically standard and conventional in the area of variational analysis and generalized differentials. As usual, · stands for the norm of Banach space X , ·, · stands for the canonical pairing between X and its topologically dual space X ∗ . Moreover, for any x ∈ X and r > 0 we denote the closed ball centered at x with radius r by Br (x) and denote the unit ball and the unit sphere in X by B X , S X respectively. 2 Implicit multifunction theorems Unless otherwise stated in this paper, X and Y are Asplund spaces, which are Banach spaces where every separable subspace has a separable dual. This broad class includes many generalized spaces such as Fréchet smooth spaces, Banach spaces with separable duals, all reflexive spaces, etc; see the book [17] for the more discussions and references therein. Given an extended-real-valued function ϕ : X → B := (−∞, ∞], we always assume that it is proper (i.e., ϕ ≡ ∞) and use the notation dom ϕ := {x ∈ X | ϕ(x) < ∞} for its domain. The regular/Fréchet subdifferential of ϕ at x¯ ∈ dom ϕ is defined by ϕ(x) − ϕ(x) ¯ − x ∗ , x − x ¯ ≥0 ∂ϕ(x) ¯ := x ∗ ∈ X ∗ lim inf x→x¯ x − x ¯

(2.1)

with ∂ϕ(x) ¯ := ∅ if x¯ ∈ / dom ϕ for convenience. Let be a subset of X and x¯ ∈ . The indicator δ(·; ) is defined as 0 if x ∈ and as ∞ otherwise. The Fréchet subdifferential (2.1) of δ(·; ) at x¯ is referred as the (x; so-called regular/Fréchet normal cone to at x¯ and then denoted by N ¯ ). → Given a set-valued mapping H : X → Y between two Asplund spaces, the graph of H is defined by gph H := {(x, y) ∈ X × Y | y ∈ H (x)}. Then we set the regular/Fréchet coderivative of H at (x, y) ∈ gph H as following ((x, y); gph H ) for all y ∗ ∈ Y ∗ , ∗ H (x, y)(y ∗ ) := x ∗ ∈ X ∗ (x ∗ , −y ∗ ) ∈ N D (2.2) which was first introduced by Mordukhovich in [16]. We refer the reader to the twovolume book [17,18] for further discussions and applications of this construction to metric regularity as well as many other areas in variational analysis. Now let P be a metric space with metric ρ. Without confusion we also denote the set {q ∈ P| ρ( p, q) ≤ r } by Br ( p), the closed ball in P with center p ∈ P and radius → Y as in (1.1), the implicit multifunction r > 0. Given the multifunction F : X × P → → G : P → X is defined by G(·) := {x ∈ X | 0 ∈ F(x, ·)}. Set F p (·) := F(·, p) for any p ∈ P. We are ready to construct the main result of the paper in the following theorem.

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Theorem 2.1 (Implicit multifunction theorem) Let (x, ¯ p) ¯ ∈ X × P satisfy 0 ∈ F(x, ¯ p). ¯ Suppose there is r > 0 such that the two following conditions hold: ¯ the set gph F p ∩ Br (x) ¯ × Br (0) is closed in X × Y , (i) for any p ∈ Br ( p), (ii) ∗ F p (x, y)(y ∗ ) with p ∈ Br ( p), κr := lim inf x ∗ x ∗ ∈ D ¯ x ∈ Br (x) ¯ \ G( p), δ↓0 y ∈ δ (0; F p (x)) ∩ Br (0), y ∗ ∈ Jδ (y) > 0, (2.3) where δ (0; F p (x)) := z ∈ F p (x)| z ≤ d(0; F p (x)) + δ and Jδ (y) := y ∗ ∈ ∗ SY ∗ | y − y , y ≤ δ for all (x, y) ∈ X × Y and δ ≥ 0. Then for any μ ∈ 0, min{r κr , r } we have d(x; G( p)) ≤

d(0; F(x, p)) for x ∈ Br − μ (x), ¯ κr κr d(0; F(x, p)) < μ.

p ∈ Br ( p) ¯ with (2.4)

Proof Suppose that both conditions (i) and (ii) hold. Argue by contradiction that the ¯ and p0 ∈ Br ( p) ¯ such inequality in (2.4) is not valid, i.e., there exist x0 ∈ Br − μ (x) κr that d(0; F(x0 , p0 )) < μ and that d(x0 ; G( p0 )) >

d(0; F(x0 , p0 )) . κr

/ G( p0 ) and then 0 ∈ / F(x0 , p0 ). Since gph F p0 ∩ Br (x)×B ¯ It follows that x0 ∈ r (0) is closed, we easily get thatd(0; F(x 0 , p0 )) > 0. Define ε := d(0; F(x0 , p0 )) ∈ (0, μ) and λ := εκ −1 with κ ∈ εκμr , κr satisfying d(x0 ; G( p0 )) >

d(0; F(x0 , p0 )) = λ. κ

(2.5)

By definition of the distance function, for each α ∈ (0, r − μ) there is y0 ∈ F p0 (x0 ) such that r > μ + α > ε + α = d(0; F(x0 , p0 )) + α ≥ y0 .

(2.6)

Let us denote ¯ × Br (0)) for (x, y) ∈ X × Y, ϕ(x, y) := y + δ (x, y); gph F p0 ∩ (Br (x) which is a lower semi-continuous (l.s.c.) function in X × Y due to condition (i). We observe from (2.6) that inf

(x,y)∈X ×Y

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ϕ(x, y) + (ε + α) ≥ ϕ(x0 , y0 ) = y0 .


333

Applying the Ekeland variational principle [8] to the latter via the new norm ||(x, y)||η := x + ηy in the space X × Y for some η > 0 allows us to find ¯ × Br (0)) satisfying ( x, y) ∈ gph F p0 ∩ (Br (x) ⎧ x − x0 + η y − y0 ≤ λ, ⎪ ⎨ y = ϕ( x, y) ≤ ϕ(x0 , y0 ) = y0 , ε+α ⎪ ⎩ inf ϕ(x, y) + (x − x + ηy − y) ≥ ϕ( x, y) = y. (x,y)∈X ×Y λ

(2.7)

It follows from (2.5) and (2.7) that x − x0 ≤ λ < d(x0 ; G( p0 )), which clearly y = 0. Furthermore, we get from (2.7) and the choice of implies that x ∈ G( p0 ) and κ that ¯ ≤ λ + r − μκr−1 =r + εκ −1 −μκr−1 < r. (2.8) x − x ¯ ≤ x −x0 + x0 − x x − x0 , r − y0 , y , employing the For any 0 < β < min d(x0 ; G( p0 )) − “fuzzy sum rule” [9, Lemma 4] for three functions ϕ1 (x, y) := y, ϕ2 (x, y) := ¯ × Br (0)) , and ϕ3 (x, y) := ε+α x + ηy − y) δ (x, y); gph F p0 ∩ (Br (x) λ (x − to the optimization problem in (2.7) allows us to find (x1 , y1 ), (x3 , y3 ) ∈ X × Y , and ¯ × Br (0)) such that (xi , yi ) ∈ Bβ ( x, y), i = 1, 2, 3 and (x2 , y2 ) ∈ gph F p0 ∩ (Br (x) that β ∂ϕ2 (x2 , y2 ) + ∂ϕ3 (x3 , y3 ) + [B X ∗ × ηBY ∗ ] (0, 0) ∈ ∂ϕ1 (x1 , y1 ) + λ ¯ × Br (0)) ⊂ {0} × ∂( · )(y1 ) + N (x2 , y2 ); gph F p0 ∩ (Br (x) ε+α+β [B X ∗ × ηBY ∗ ]. (2.9) + λ Observe further from (2.7) that for any y ∈ F p0 (x2 ) ∩ Br (0) we have ε+α x + ηy − y) ≤ y (x2 − λ ε+α + (x2 − x + ηy + η y) λ ε+α ≤ y + (β + 2ηr ). λ

y ≤ y +

Since the latter holds for all y ∈ F p0 (x2 ) ∩ Br (0) including y2 , we obtain ε+α (β + 2ηr ). λ

(2.10)

y2 ≤ y + y2 − y ≤ y0 + β < r.

(2.11)

y ≤ d(0; F p0 (x2 )) + Note also from (2.7) that

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Moreover, the inequalities in (2.8) give us that x2 − x ¯ ≤ x2 − x + x − x ¯ ≤ β + r + εκ −1 − μκr−1 . Since 0 < μκr−1 − εκ −1 due to the choice of κ, we may appoint β > 0 sufficiently ¯ < r . This together with (2.11) implies small such that x2 − x ((x2 , y2 ); gph F p0 ). (x2 , y2 ); gph F p0 ∩ (Br (x) ¯ × Br (0)) = N (2.12) N Observe that max{x2 − x0 , x1 − x0 } ≤ max{x1 − x , x2 − x } + x − x0 ≤ β + x − x0 < d(x0 ; G( p0 )). It follows that x1 , x2 ∈ G( p0 ) and then y2 = 0. Moreover, note that y1 ≥ y − y − y1 ≥ y − β > 0. Thus we have ∂( · )(y1 ) = y ∗ ∈ SY ∗ | y ∗ , y1 = y1 .

(2.13)

Combining (2.9) with (2.12) allows us to find y1∗ in the latter set satisfying ((x2 , y2 ); gph F p0 ) + ε + α + β [B X ∗ × ηBY ∗ ]. (0, −y1∗ ) ∈ N λ ((x2 , y2 ); gph F p0 ) such that Hence there is (x2∗ , −y2∗ ) ∈ N x2∗ ≤

ε+α+β ε+α+β κ and y1∗ − y2∗ ≤ κη. ε ε

(2.14)

It follows from (2.13) and (2.14) that 1 − ε+α+β κη ≤ y2∗ ≤ 1 + ε+α+β κη and that ε ε y2∗ y2 − y2∗ , y2 ≤ (y2∗ − 1)y2 + y2 − y + y − y1 + y1 − y2∗ , y2 ε+α+β κηr + β + β + y1∗ , y1 − y2∗ , y2 ≤ ε ε+α+β κηr + 2β + y1∗ , y1 − y2 + y1∗ − y2∗ , y2 ≤ ε ε+α+β κηr + 2β + y1∗ (y1 − y + y − y2 ) ≤ ε ∗ ∗ +y1 − y2 y2 ε+α+β ε+α+β ≤ κηr + 2β + 2β + κηr ε ε ε+α+β 2κηr + 4β. = ε

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Define y2∗ := that

y2∗ y2∗

and x2∗ :=

x2∗ y2∗ ,

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diving both sides of the latter by y2∗ gives us

2 ε + α + β κηr + 2β y2∗ ε ε + α + β ε + α + β −1 . ≤2 κηr + 2β 1 − κη ε ε

y2 − y2∗ , y2 ≤

Furthermore, we have x2∗ =

x2∗ ε+α+β ε + α + β −1 ≤ κ 1 − κη . y2∗ ε ε

For any δ > 0 it follows from two preceding inequalities and (2.10) that x2 ∈ Br (x) ¯ \ G( p0 ),

y2 ∈ δ (0; F p0 (x2 )) ∩ Br (0),

y2∗ ∈ Jδ (y2 ), and x2∗ < κ + δ

when α, β, η > 0 are chosen sufficiently small. Taking δ ↓ 0 in the latter gives us a ∗ F p0 (x2 , y2 )( contradiction to the definition of κr in (2.3) since x2∗ ∈ D y2∗ ). The proof of the theorem is completed. As mentioned in Sect. 1, the above result not only gives us a sufficient condition of (1.1) but also supplies information about the neighborhoods U and V as well as μ in (1.1). Letting r → 0 in (2.3) leads us to the following corollary. Corollary 2.2 Let (x, ¯ p) ¯ ∈ X × P satisfy 0 ∈ F(x, ¯ p). ¯ Suppose that the following conditions are satisfied: ¯ 0), (a) for any p near p, ¯ gph F p is locally closed around the point (x, (b) there is κ > 0 satisfying ∗ F p (x, y)(y ∗ ) with p ∈ Br ( p), ¯ x ∈ Br (x) ¯ \ G( p), κ < lim inf x ∗ x ∗ ∈ D r ↓0 y ∈ r (0; F p (x)) ∩ Br (0), y ∗ ∈ Jr (y) . (2.15) Then F is locally metrically regular in Robinson’s sense around (x, ¯ p, ¯ 0) with rank κ −1 , i.e., there exist μ > 0 and a neighborhood U × V ⊂ X × P of the point (x, ¯ p) ¯ such that (1.2) holds with k = κ −1 . Proof Assume that both conditions (a) and (b) of the corollary hold. It is easy to check that there is r > 0 sufficiently small such that (i) and (ii) of Theorem 2.1 are satisfied with κr > κ. Thanks to Theorem 2.1 we can find μ > 0 and two balls U , V with centers x, ¯ p¯ respectively as in (2.4) satisfying (1.2) with k = κ −1 > κr−1 . This clearly ensures the corollary.

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Corollary 2.2 is slightly sharper than [14, Theorem 3.6], whose approach relies on many advanced results in variational analysis such as “smooth variational principle”, “fuzzy sum rule”, “multidirectional mean value theorem” and “decrease principle”. Our approach seems much simpler since we only use the Ekeland variational principle and the fuzzy sum rule. It is worth mentioning further that a similar form of (2.15) has been recently established in [20, Theorem 12] for “abstract coderivatives” by using a different approach via the theory of error bound and the “strong slope”. We notice that our result can be constructed in the same way for such abstract coderivatives. Though the inequality “d(0; F(x, p)) < μ” does not appear in [20, Theorem 12], it somehow lies implicitly in their sufficient conditions (see also [20, Theorem 7]). Furthermore, → F(x, they have to assume that the multifunction p → ¯ p) is lower semi-continuous at p¯ , i.e., the function p → d(0; F(x, ¯ p)) is upper semi-continuous (u.s.c.) at p. ¯ Our approach relaxes this assumption, which is also involved in [14,19]. However, it is observed in [2] that when such property of continuity appears, the inequality “d(0; F(x, p)) < μ” can be eliminated in (1.2). For convenience, we present the proof of this claim in the following corollary. ¯ p). ¯ In addition to the hypothesis Corollary 2.3 Let (x, ¯ p) ¯ ∈ X × P satisfy 0 ∈ F(x, of Corollary 2.2, suppose that → F(x, (c) the multifunction p → ¯ p) is lower semi-continuous at p. ¯ Then there exists a neighborhood U × V ⊂ X × P of (x, ¯ p) ¯ satisfying d(x; G( p)) ≤

d(0; F(x, p)) κ

for all (x, p) ∈ U × V.

Proof Thanks to (a) and (b) in Corollary 2.2 there is some r > 0 such that both (i) and (ii) of Theorem 2.1 holds with κr ≥ κ. It follows from (2.4) that d(x; G( p)) ≤

d(0; F(x, p)) for x ∈ Br − μ (x), ¯ p ∈ Br ( p) ¯ with d(0; F(x, p)) ≤ μ. κr κr

(2.16) By assumption (c) the function p → d(0; F(x, ¯ p)) is u.s.c. at p. ¯ This implies that there μ μ is a neighborhood V ⊂ Br ( p) ¯ of p¯ such that d(0; F(x, ¯ p)) < d(0; F(x, ¯ p))+ ¯ 2 = 2 μ for all p ∈ V . Combining (2.16) with the latter gives us that d(x, ¯ G( p)) ≤ 2κr . Thus we have d(x; G( p)) ≤ x − x ¯ + d(x; ¯ G( p)) ≤

μ κr

for x ∈ U := Br− μ (x) ¯ ∩B κr

μ 2κr

(x), ¯ p ∈ V.

This together with (2.16) clearly ensures that d(x; G( p)) ≤

d(0; F(x, p)) d(0; F(x, p)) for all ≤ κr κ

and then completes the proof of the corollary.

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(x, p) ∈ U × V.


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Another fundamental property of the implicit multifunction G, so-called Lipschitzlike/Aubin property, has been considered widely in many publications [2,3,12,14]. To prove this those papers have to assume that F is partially Lipschitz-like in p with rank L, i.e., there is a neighborhood U × V × O of (x, ¯ p, ¯ 0) in X × P × Y such that F(x, p2 ) ∩ O ⊂ F(x, p1 ) + Lρ( p1 , p2 )BY for all x ∈ U, p1 , p2 ∈ V. However, we observe from the proof of [14, Corollary 3.8] that the above condition can be replaced by a weaker one, that is, there exists a neighborhood U × V of (x, ¯ p) ¯ satisfying 0 ∈ F(x, p1 ) + Lρ( p1 , p2 )BY whenever p1 , p2 ∈ V and x ∈ G( p2 ) ∩ U. (2.17)

This condition certainly covers assumption (c) in Corollary 2.3. Indeed, we get from (2.17) that 0 ∈ F(x, ¯ p) + Lρ( p, p) ¯ for all p ∈ V , which implies d(0; F(x, ¯ p)) ≤ Lρ( p, p). ¯ Therefore p → d(0; F(x, ¯ p)) is u.s.c. at p, ¯ i.e., assumption (c) is satisfied. These discussions together with the proof of [14, Corollary 3.8] lead us to the following result. Corollary 2.4 In addition to the assumptions (a) and (b) of Corollary 2.2, suppose that there is a neighborhood U × V ⊂ X × P of (x, ¯ p) ¯ such that (2.17) holds with ¯ x) ¯ with rank some L > 0. Then the set-valued mapping G is Lipschitz-like around ( p, ×V ⊂ X × P of (x, ¯ p) ¯ satisfying Lκ −1 , i.e., there exists a neighborhood U ⊂ G( p1 ) + Lκ −1 ρ( p1 , p2 )B X for all p1 , p2 ∈ V . G( p2 ) ∩ U This judgement is slightly sharper than the very recent result [3, Theorem 3.1], which is a refinement of the main result in [4]. Let us finish this section by applying Corollary 2.4 to metric regularity. → Y be a multifunction between two Asplund spaces, and Corollary 2.5 Let H : X → let (x, ¯ y¯ ) ∈ gph H . Suppose that gph H is locally closed around (x, ¯ y¯ ) and that there is κ > 0 satisfying ∗ H (x, y)(y ∗ ) with p ∈ Br ( y¯ ), x ∈ Br (x) κ < lim inf x ∗ x ∗ ∈ D ¯ \ H −1 ( p), r ↓0 y ∈ r ( p; H (x)) ∩ Br ( p), y ∗ ∈ Jr (y − p) . Then H is metrically regular around (x, ¯ y¯ ) with modulus κ −1 , i.e., there are neighborhoods U of x¯ and V of y¯ such that d(x; H −1 (y)) ≤

d(y; H (x)) for all x ∈ U, y ∈ V. κ

Proof Let P = Y and F(x, p) := H (x) − p. It is easy to see that all assumptions (a), (b), and (c) in Corollary 2.2 and Corollary 2.3 hold. Applying Corollary 2.3 to the multifunction F fully justifies the corollary.

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3 Metric subregularity of generalized equations As discussed in Sect. 1, property (1.2) becomes (1.4) when we omit the space P. This is known as metric subregularity of the following generalized equation: 0 ∈ F(x) for x ∈ X, → Y is a set-valued mapping between two Asplund spaces. Applying where F : X → Theorem 2.1 or Corollary 2.2 to this circumstance certainly gives us a sufficient condition to (1.4). However, by imitating the proof of Theorem 2.1 with some modifications, we may get a bigger neighborhood U ⊂ X containing x¯ in the following result. For convenience to the reader, we also present the proof but shorten it as possible. Theorem 3.1 (Sufficient conditions for metric subregularity) Let x¯ ∈ X satisfy 0 ∈ F(x). ¯ Suppose that there is r > 0 such that the following conditions hold: (d) the set gph F ∩ Br (x) ¯ × Br (0) is closed in X × Y , ∗ F(x, y)(y ∗ ) with x ∈ Br (x) (e) κr := lim inf x ∗ x ∗ ∈ D ¯ \ F −1 (0) δ↓0 y ∈ δ (0; F(x)) ∩ Br (0), y ∗ ∈ Jδ (y) > 0.

(3.1)

Then for any s ∈ 0, min{ r2 , κrr } we have d(x; F −1 (0)) ≤

d(0; F(x)) for all x ∈ Bs (x). ¯ κr

(3.2)

Proof Suppose that both conditions (d)and (e) hold and argue by contradiction that ¯ such that inequality (3.2) is not true for some s ∈ 0, min{ r2 , κrr } . We find x0 ∈ Bs (x) d(x0 ; F −1 (0)) >

d(0; F(x0 )) . κr

Fix δ > 0 and then define ε := d(0; F(x0 )) and λ := εκ −1 for some κ ∈ (0, κr ) with d(x0 ; F −1 (0)) >

d(0; F(x0 )) = λ. κ

(3.3)

Observe that d(x0 ; F −1 (0)) ≤ x0 − x ¯ ≤ s.

(3.4)

This together with (3.3) implies 0 < ε ≤ κs < κr s < r . Let us pick any y0 ∈ α (0; F(x0 )) for α ∈ (0, r − κs). Similarly to the proof of Theorem 2.1, for any ¯ × Br (0) satisfying η > 0 there is ( x, y) ∈ gph F ∩ Br (x)

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⎧ x − x0 + η y − y0 ≤ λ, ⎪ ⎨ y = ϕ( x, y) ≤ ϕ(x0 , y0 ) = y0 , ε+α ⎪ ⎩ inf (x − x + ηy − y) ≥ ϕ( x, y) ϕ(x, y) + (x,y)∈X ×Y λ

339

(3.5)

¯ × Br (0)) for all (x, y) ∈ X × Y . with ϕ(x, y) := y + δ (x, y); gph F ∩ (Br (x) Thanks to (3.3) and (3.5), we get that x − x0 ≤ λ < d(x0 ; F −1 (0)), which clearly −1 y = 0. Observe further from (3.4) that implies that x ∈ F (0) and x − x ¯ ≤ x − x0 + x0 − x ¯ < d(x0 ; F −1 (0)) + s ≤ 2s < r.

(3.6)

For any 0 < β < r − 2s employing the “fuzzy sum rule” [9, Lemma 4] to the optimization problem in (3.5) and following the proof of Theorem 2.1 allow us to find (x1 , y1 ) ∈ X × Y and (x2 , y2 ) ∈ gph F satisfying x1 − x + y1 − y ≤ β, x + y2 − y ≤ β, y1 > 0, x2 − x ¯ < r , y2 ∈ δ (0; F(x2 )) ∩ Br (0), and x2 − that ε+α+β ((x2 , y2 ); gph F) κ(B X ∗ × ηBY ∗ ) + N (0, 0) ∈ {0} × ∂( · )(y1 ) + ε when α, β, η are chosen sufficiently small. The rest of the proof is quite similar to that of Theorem 2.1. Let us denote subregF(x, ¯ 0) as the infimum of all moduli k over (k, U ) in (1.4). Theorem 3.1 suggests a straightforward upper estimate to this bound as following subregF(x, ¯ 0) ≤ lim sup r ↓0

1 x∗ ∈ D ∗ F(x, y)(y ∗ ) with x ∈ Br (x) ¯ \ F −1 (0), x ∗ y ∈ r (0; F(x)) ∩ Br (0), y ∗ ∈ Jr (y) . (3.7)

This approximation is slightly sharper than [25, Theorem 3.1]. A similar result to Theorem 3.1 is also established in [25, Theorem 3.4]; however, they have to assume that Y is a Hilbert space and use the set J δ (y) := y ∗ ∈ SY ∗ | d(y ∗ , J (y)) ≤ δ} instead of Jδ (y), where J (y) := J0 (y) for y ∈ Y . The proof of this result also relies on some specific properties of Hilbert spaces. We shall prove that it can be extended to the case that Y is a Banach space with Fréchet smooth norm, i.e., the norm · is Fréchet smooth at y ∈ Y \ {0}. It is worth mentioning that this broad class is contained in Asplund spaces but it includes all Hilbert spaces. Corollary 3.2 Let X be an Asplund space and Y be a Banach space with Fréchet smooth norm. Let x¯ ∈ X satisfy 0 ∈ F(x). ¯ Suppose that there is r > 0 such that condition (d) and the following one hold: ∗ F(x, y)(y ∗ ) with x ∈ Br (x) ¯ \ F −1 (0), (e ) κr := lim inf x ∗ x ∗ ∈ D δ↓0 y ∈ δ (0; F(x)) ∩ Br (0), y ∗ ∈ J δ (y) > 0.

(3.8)

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T. T. A. Nghia

Then for any s ∈ 0, min{ r2 , κr } we have d(x; F −1 (0)) ≤

d(0; F(x)) for all x ∈ Bs (x). ¯ κr

(3.9)

Proof Assume that Y is a Banach space with Fréchet smooth norm · and assume to the contrary that (3.9) does not hold. Following the proof of Theorem 3.1 and (3.7) ((x2 , y2 ); gph F) such that (replacing κr by κr ) allows us to find (x2∗ , y2∗ ) ∈ N x2∗ ≤

ε+α+β ε+α+β κ and y2∗ − J (y1 ) ≤ κη, ε ε

which clearly implies that x2∗ ≤ κ + O(α, β, η) and y2∗ = 1 + O(α, β, η). Define y2∗ := y2∗ y2∗ −1 . Similarly to the proof in Theorem 2.1 we x2∗ := x2∗ y2∗ −1 and ∗ derive that x2 ≤ κ + O(α, β, η) and that y2∗ − y2∗ = y2∗ − J (y1 ) + |1 − y2∗ | y2∗ − J (y1 ) ≤ y2∗ − J (y1 ) + = O(α, β, η). (3.10) Since y = 0 and · is Fréchet differentiable around y, it follows from [21, Proposiy, y2 − tion 2.8] that J (·) is norm-to-norm continuous at y. Note that max{y1 − y} ≤ β, thus we may choose β sufficiently small such that y1 , y2 = 0 and that y) + O(β)BY ∗ ⊂ J (y2 ) + O(β)BY ∗ . J (y1 ) ⊂ J ( The latter together with (3.10) yields that x2∗ < κ + δ d( y2∗ ; J (y2 )) ≤ δ and when α, β, η are chosen sufficiently small. Taking δ ↓ 0 leads us to the contradiction to the definition of κr in (3.8) and then completes the proof of the corollary. Let us finish the paper by remarking that the above result has been constructed r recently in [25, Theorem 3.4] when Y is a Hilbert space and s is chosen as 2+κ , r r r which obviously belongs to the interval 0, min{ 2 , κr } in our result. In the recent paper Li–Mordukhovich also establish in [15, Theorem 5.1] a similar condition to (3.8) ensuring the so-called “Hölder metric subregularity” in Fréchet smooth spaces; however, our result gives an explicit estimate to s, which is new in literature. Acknowledgments The author would like to express the gratitude to Prof. Boris Mordukhovich for his helpful discussions and encouragements.

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