Compactness Properties, Openness Criteria and ... - Springer Link

Set-Valued Analysis 6: 363–380, 1998. © 1998 Kluwer Academic Publishers. Printed in the Netherlands.

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Compactness Properties, Openness Criteria and Coderivatives JEAN-PAUL PENOT University of Pau, Mathématiques, URA CNRS 1204, Faculté des Sciences, Av. de l’Université, 64000 Pau, France (Received: 16 July 1997; in final form: 14 July 1998) Abstract. We study the local surjectivity and openness properties of mappings and correspondences by using coderivatives. We recall local criteria but concentrate on point criteria. Our study relies on a compactness condition which is of independent interest, for instance, for the study of the behavior of the injectivity constant of a linear map or of a convex process under stabilization procedures. Several known criteria for openness are shown to be a consequences of this new compactness condition. Mathematics Subject Classifications (1991): 26B05, 26B10, 26E25, 46G05, 49J52, 58C20. Key words: coderivative, coderivatively compact, compactness, injectivity, normal compactness, openness, nonsmooth analysis, subdifferential.

1. Introduction The usefulness of local invertibility and of local openness criteria has been stressed in many papers such as [2, 3, 6, 14, 15, 21, 48, 51–54, 60] to cite just a few of the abundant references. Leaving aside the conditions involving generalized Jacobians of mappings which seem to be limited to the finite-dimensional case (see [12] and its references), the first criteria obtained via the techniques of nonsmooth analysis involved primal conditions, i.e. conditions having a bearing on the tangent cone to the graph of the mapping (or the multimapping) F : X → Y as first considered in [2]; see also [3, 4, 8, 44, 46, 48]. However, in the early eighties A. V. Dmitruk, A. A. Milyutin and N. P. Osmolovski [14], J. Borwein [8] and A. D. Ioffe [19] gave dual criteria involving normal cones instead of tangent cones. The last paper uses fans, prederivatives and approximate subdifferentials; the conditions of the first two papers are formulated in terms of subdifferentials and normal cones in the sense of Clarke. More accurate (in view of the observations in [58]) results have been obtained with limiting (or stabilized) normal cones, starting with the work of Kruger (thesis, in Russian) and the note of Mordukhovich [33]; see [17, 21, 22, 24, 27, 28, 32–41]. In particular, a complete characterization of openness in finite dimensions is given in [34] and extensions to spaces with smooth norms have been presented in [27, 28, 47, 48] and in Asplund spaces in [41]. More specialized results pertaining to systems of

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equalities and inequalities have been obtained using dual criteria by Auslender [5] and Ioffe [18]. We take the following result as a starting point: if F is a multimapping with a closed graph (also denoted by F ) between two Asplund spaces X and Y , and if D ∗ F (z) denotes the coderivative of F at z in the sense given in the next section following [32] and numerous contributions, then F is open around some z ∈ F with a linear rate iff some c > 0 exist and some neighborhood W of z in X × Y such that kx ∗ k > cky ∗ k for any x ∗ ∈ D ∗ F (z)(y ∗ ), y ∗ ∈ Y ∗ with z ∈ F ∩ W.

(1)

When X and Y are finite-dimensional, this result is well known (see, for instance, [34] and its references). The axiomatic approach of [23] Th. 2 yields it under various forms. It is somewhat contained in [27, 28, 47, 48] under a quantitative form and the assumption that Y has an equivalent smooth norm. In [36] it is proved with the help of the Bishop–Phelps theorem, using a renorming and a sequential procedure. In [39] Corol. 4.5 (version of August 1995), this characterization has been obtained under the additional assumption that X and Y are weakly compactly generated spaces and a ‘normal compactness’ assumption on the coderivatives. Moreover, the condition on the coderivatives is more restrictive since it bears on the limiting coderivatives instead of the Fréchet coderivatives. Let us note, however, that the limiting coderivatives enjoy useful calculus rules, whereas the Fréchet coderivatives just have fuzzy calculus rules. This is why we also consider the use of limiting coderivatives for such questions (see Section 4). The main thrust of the present paper concerns a new compactness condition which has to be imposed to replace a local condition as the one above with an infinitesimal condition involving limiting coderivatives at one specific point only. We present this condition in a general setting dealing with the study of the injectivity constant of a process. In the case of coderivatives, this constant is the greatest constant c which can be used in (1) and can be related to the openness rate of the multimapping when the process is a coderivative (see Section 2). The compactness condition enables one to control the behavior of this constant under stabilization procedures such as the one involved in taking limiting coderivatives. As with some other compactness conditions introduced by the author for other purposes ([42] for tangential compactness, [30, 43] for asymptotic compactness, [50] for the study of subdifferentials and normal cones), the condition exhibited here is a pointwise condition. It may be satisfied at some point z without being satisfied at nearby points. On the contrary, the partial compactly epi-Lipschitz condition of [26] Definition 3.2 and the partial normal compactness (p.n.c.) condition of [39] Def. 3.6, [41] Def. 2.4? involve a whole neighborhood of the point z. We insist on this feature because, as stressed by several authors [23, 36–41], point criteria are ? We refer here to the original versions; the final versions have been modified after the present

paper was written in July 1995 and disseminated in September and October 1995. Let us note that the proofs of [39 – 41] (and those of [48]) did not require important changes.

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to be distinguished from local criteria such as [39] Def. 3.1 and 3.8. The difference can be seen as analogous to the difference between mappings which are strictly differentiable at some point x0 and mappings which are affine or differentiable on some neighborhood of x0 . This compactness condition is along the lines of Definition 2.2 in [50] which concerns sets and not multimappings. It is in the spirit of the famous condition (S) of Browder (see [11] for instance, and [7] for a recent use in optimization theory). For this reason, it is likely to be more handy than primal conditions of the compact epi-Lipschitz type when dealing with concrete problems of nonlinear analysis and partial differential equations. Without looking for completeness, we make a comparison with several variants of this condition. We refer to [25, 26, 29, 49] for the connection with the important notions of epi-Lipschitzness and compact epi-Lipschitzness (see [9, 57]); applications can be found along the lines of [35]. 2. Preliminaries and Motivations In the sequel, X, Y and Z denote Banach spaces and, unless otherwise stated, → F : X− →Y is a multimapping (or correspondence or relation or multifunction) between X and Y . We identify F with its graph in X ×Y . We always endow a product space Z = X × Y with a norm satisfying max(kxk, kyk) 6 k(x, z)k 6 kxk + kyk

for each (x, y) ∈ Z.

We denote by BX the closed unit ball of X and by SX the unit sphere of X and we use similar notations with X∗ , Y, . . . substituted by X. We set B(x, r) = x + rBX . Throughout, NZ (z) stands for the family of neighborhoods of z in Z and K(Y ∗ ) F

denotes the family of weak∗ compact subsets of Y ∗ . We write (zi )i∈I → z to mean that the net (zi )i∈I converges to z and stays in F . If (xi∗ )i∈I is a net of a dual space ∗ X∗ , we write (xi∗ )i∈I → x ∗ when this net is bounded and converges weakly∗ to x ∗ . The multimapping F is said to be open at z := (x, y) ∈ F if for any r > 0 there exists s > 0 such that B(y, s) ⊂ F (B(x, r)). Then, the canonical gage of openness of F at z = (x, y) is the function γF (z, .) on R+ introduced in [19] and in the work of V. Ptak (see [51–54] and the references of [6]) given by γF (z, r) = sup{s ∈ R+ : B(y, s) ⊂ F (B(x, r))}. The rate of openness of F at z is ωF (z) = lim infr→0+ r −1 γF (z, r). The multimapping F is said to be open with linear rate c on some subset U of Z := X × Y if there exists some r0 > 0 such that, for any r ∈ (0, r0 ) and any z := (x, y) ∈ U ∩F , one has B(y, cr) ⊂ F (B(x, r)). Then, obviously, ωF (z) > c. The rate of openness of F around z is the supremum $ (F, z) of the family of real positive numbers c such that F is open on some neighborhood U of z with linear rate c. Let us now recall some elements about coderivatives of correspondences. In order to define such a notion, one has to dispose of a concept of a normal cone N.

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Then one defines the coderivative D ∗ F (z) of F at some point z ∈ F associated with the normal cone notion N by D ∗ F (z) := {(y ∗ , x ∗ ) ∈ Y ∗ × X∗ : (x ∗ , −y ∗ ) ∈ N(F, z)}. Now the concept of a normal cone can be defined axiomatically. It is closely related to the notion of subdifferential for which a number of authors have introduced various sets of axioms (see [22, 23]). Let us just point out here that while we retain the very natural conditions (i), (ii), (iii) of [23] Section 6, we do not impose condition (iv) of this reference nor conditions (ii) and (iv) of [22] which eliminate some subdifferentials. We consider instead the condition of the following definition which is apparently milder (and is obviously related to the notion of trustworthiness of [20] and to the developments of [49] inspired by this notion). It can serve to establish mean value theorems, but this is not our purpose here. DEFINITION 2.1. Given a class X of Banach spaces and a class C of closed subsets of the members of X, we say that a normal cone notion N is reliable (with respect to C and X) if for any Z in X, C in C, z ∈ C, ε > 0 and any convex Lipschitzian function g on Z such that g attains its infimum on C at z, there exists z0 , z00 ∈ B(z, ε) with z00 ∈ C such that 0 ∈ ∂g(z0 ) + N(C, z00 ) + εBZ ∗ . Here ∂g denotes the usual subdifferential of g in the sense of convex analysis. The advantage of the present axiomatic approach lies in the fact that it includes a number of variants. The choice of the appropriate version may be imposed by the regularity property of the product space X × Y or the regularity property of its subset F . When X × Y is in the class X of Asplund spaces, one can choose the Fréchet subdifferential as in [36–41] in view of the following lemma. For the definition of this class of spaces and for its properties, we refer to [13] for instance. Let us just say that it can be characterized as the class of Banach spaces all of whose separable subspaces have a separable dual space. In particular, reflexive Banach spaces are Asplund spaces. If one takes for X the class of all Banach spaces and for C the class of all closed subsets, one can use the normal cone in the sense of Clarke [12] or in the sense of the approximate subdifferentials of Ioffe [24]. When C is the class of closed convex subsets, one can take for X the class of all Banach spaces and for N the usual normal cone. Then one recovers from Theorem 2.3 below one of the first openness results formulated in dual terms [8]. In order to be precise, let us recall the notion of subdifferential associated with a bornology B: given a function f : Z → R ∪ {∞} finite at z the B-subdifferential of f at z is given by ∂ B f (z) := {z∗ ∈ Z ∗ : ∀B ∈ B lim inf inf t −1 (f (z + tu) − t &0 u∈B

− f (z) − hz∗ , tui) > 0}.

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The B-normal cone to a subset F of Z at z ∈ F is N B (F, z) := ∂ B ιF (z), where ιF is the indicator function of F given by ιF (z) = 0 if z ∈ F, ιF (z) = ∞ otherwise. When B is the family B(z) of bounded subsets (resp. the family K(z) of compact subsets), we speak of the Fréchet (resp. Hadamard) subdifferential ∂ − (resp. ∂ ! ) and normal cone. The first assertion of the following lemma is a consequence of results in [16, 20] and of a personal communication by J. Borwein. As kindly observed by a referee, the second one is a simple consequence of [20] if the Borwein–Preiss variational principle is applied instead of the Ekeland principle. A similar remark holds for Corollary 2.4 below. LEMMA 2.1. For the Fréchet normal, a Banach space is reliable iff it is an Asplund space. For the Hadamard normal, a Banach space X is reliable if it has an Hadamard differentiable Lipschitz bump function b with b0 continuous on X × X. Let us first note a simple necessary condition for openness at a linear rate. PROPOSITION 2.2 ([36, Theorem 4.2]). If F is open at a linear rate c on U ∩ F , where U is an open subset of X × Y , in terms of the Fréchet coderivative, one has for each z ∈ U ∩ F cF (z) := inf{kx ∗ k : x ∗ ∈ D ∗ F (z)(y ∗ ), y ∗ ∈ SY ∗ } > c. Now let us give a sufficient condition for openness. The proof, which is close to the ones in [39, 48], is given for the sake of self-containedness. → THEOREM 2.3. Let X, Y be such that Z := X×Y belongs to X and let F : X− →Y ∗ be a member of the class C. Let D F be the coderivative of F associated with a reliable normal cone N. Suppose that for some b > 0 and some open subset U of Z cF (U ) := inf{kx ∗ k : x ∗ ∈ D ∗ F (z)(y ∗ ), y ∗ ∈ SY ∗ , z ∈ U } > b. Then for each c ∈ (0, b), F is open on U with linear rate c. Proof. Let us endow Z with the sum norm. Let us pick a ∈ (0, 1) such that a(1 − a)−1 ∈ (c, b). Given z0 ∈ F ∩ U and r0 > 0 such that B(z0 , 2r0 ) ⊂ U , we will prove that for each z := (x, y) ∈ F ∩ B(z0 , r0 ) and each r ∈ (0, (1 − a)r0 ] we have B(y, cr) ⊂ F (B(x, r)). Suppose to the contrary that there exist z1 := (x1 , y1 ) ∈ F ∩ B(z0 , r0 ), r ∈ / F (B(x1 , r)). Applying Ekeland’s ]0, (1 − a)r0 ], v1 ∈ B(y1 , cr) such that v1 ∈ theorem to f : (x, y) 7→ ky − v1 k on F , we get some z2 := (x2 , y2 ) ∈ F such that for each (x, y) ∈ F ky2 − v1 k 6 ky − v1 k + a(kx − x2 k + ky − y2 k), ky2 − v1 k 6 ky1 − v1 k − a(kx1 − x2 k + ky1 − y2 k).

(2) (3)

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If we had y2 = v1 this last relation would yield akx1 − x2 k 6 (1 − a)ky1 − v1 k 6 (1 − a)cr and kx1 − x2 k < r as a −1 (1 − a) < c−1 , a contradiction with the facts that v1 ∈ / F (B(x1 , r)), y2 ∈ F (x2 ). Thus, y2 6= v1 . Moreover, we have by (3) kz2 − z1 k < a −1 ky1 − v1 k 6 a −1 cr 6 (1 − a)−1 r 6 r0 . Since g given by g(x, y) = ky − v1 k + ak(x − x2 , y − y2 )k attains its infimum on F at z2 and since N is reliable, one can find some z3 , z4 ∈ B(z2 , γ ) with 0 < γ < min(r0 − kz2 − z1 k, ky2 − v1 k) and some z3∗ , z4∗ , z5∗ ∈ Z ∗ with z3∗ ∈ ∂g(z3), z4∗ ∈ N(F, z4 ), z3∗ + z4∗ + z5∗ = 0, kz5∗ k 6 β, where β > 0 is chosen in such a way that (1 − a − β)−1 (a + β) < b (note that (1 − a)−1 a < b). Then z3 , z4 ∈ B(z1 , r0 ) ⊂ B(z0 , 2r0 ) ⊂ U and z3 6= v1 . The familiar calculus rules for convex subdifferentials provide (u∗3 , v3∗ ) ∈ BZ ∗ and w3∗ ∈ SY ∗ such that z3∗ = (0, w3∗ ) + a(u∗3 , v3∗ ). Then, if z5∗ := (x5∗ , y5∗ ), one has z4∗ = (−x5∗ − au∗3 , −y5∗ − w3∗ − av3∗ ) ∈ N(F, z4 ) or −x5∗ − au∗3 ∈ D ∗ F (z4 )(y5∗ + w3∗ + av3∗ ) and s := ky5∗ + w3∗ + av3∗ k > 1 − a − β. Then y ∗4 := s −1 (y5∗ + w3∗ + av3∗ ) ∈ SY ∗ and x ∗4 := −s −1 (x5∗ + au∗3 ) ∈ D ∗ F (z4 )(y ∗4 ) is such that kx ∗4 k 6 s −1 (a + β) 6 (1 − a − β)−1 (a + β) < b, 2

a contradiction with our assumption.

Besides the convex case and the cases of the Clarke coderivative and the Ioffe coderivative evoked in the preceding observations, we get the following special cases: COROLLARY 2.4. Let X, Y be Banach spaces such that Z = X×Y is an Asplund → space (resp. a space with a bump function as in Lemma 2.1) and let F : X− →Y be a multimapping with closed graph such that for some b > 0 and some open subset U of Z the Fréchet (resp. Hadamard) coderivative of F satisfies for each z ∈ F ∩ U , cF (z) := inf{kx ∗ k : x ∗ ∈ D ∗ F (z)(y ∗ ) : y ∗ ∈ SY ∗ } > b. Then for each c ∈ (0, b), F is open on U with linear rate c.

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3. Injectivity Constant and Stabilization Let us pause to study the properties of the constant cF (z) we used in the preceding → →X∗ is a (posisection. Here we suppose that X and Y are n.v.s. and that H : Y ∗ − ∗ ∗ tively) homogeneous multimapping (i.e. that H (ty ) = tH (y ) for each t > 0 and each y ∗ ∈ Y ∗ ). The injectivity constant c(H ) of H (also called the Banach constant of H ) is defined by ([19] Prop. 3.2) c(H ) = inf{kx ∗ k : x ∗ ∈ H (y ∗ ), y ∗ ∈ SY ∗ } = sup{b ∈ R+ : kx ∗ k > bky ∗ k ∀(y ∗ , x ∗ ) ∈ H }. This constant is related to the norm of the inverse H −1 of H by the relation c(H ) = kH −1 k−1 (with the usual conventions), the norm of a positively homogeneous → multimapping M: V − →W between two n.v.s. being defined by kMk = inf{c > 0 : kwk 6 ckvk ∀(v, w) ∈ M}. Now let us suppose that H is the transpose GT of some positively homogeneous → multimapping G: X− →Y , i.e. that H (y ∗ ) = {x ∗ ∈ X∗ : ∀(x, y) ∈ G hx ∗ , xi 6 hy ∗ , yi}. Then we observe in the following lemma that c(H ) > 0 whenever G is open. The observation of the following lemma shows that primal openness results are consequences of the dual result of Theorem 2.3. → →X∗ is the transpose of a positively homogeneous LEMMA 3.1. Suppose H : Y ∗ − → − multimapping G: X→Y in the sense of [56]. Suppose G is open at (0, 0): for some c > 0 one has B(0, c) ⊂ G(B(0, 1)). Then one has c(H ) > c. Proof. Given q ∈ (0, 1), y ∗ ∈ Y ∗ we pick y ∈ SY such that hy ∗ , yi 6 −qky ∗ k. As for any r > 0, we have B(0, cr) ⊂ G(B(0, r)) and we can find x ∈ B(0, c−1 ) such that y ∈ G(x). Then for each x ∗ ∈ H (y ∗ ), we have hx ∗ , xi 6 hy ∗ , yi 6 −qky ∗ k 6 −qky ∗ kckxk, hence kx ∗ k > qcky ∗ k and, as q is arbitrary in ]0, 1[, we get kx ∗ k > cky ∗ k.

2

A partial converse holds for convex multimappings, i.e. multimappings with convex graphs, or, more generally, for convexiphore multimappings, i.e. multimappings carrying a convex set into a convex set. Here cl S stands for the closure of S. LEMMA 3.2. Suppose H is the transpose of a positively homogeneous multimap→ ping G: X− →Y which is convex, or more generally, convexiphore. Suppose dom ∗ H = Y and c(H ) > c > 0. Then, for each r > 0 one has crBY ⊂ cl G(rBX ). If, moreover, X and Y are complete and if G is closed convex then crBY ⊂ G(rBX ).

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Proof. Given y ∈ / cl G(rBX ), the Hahn–Banach Theorem yields some y ∗ ∈ SY ∗ ∗ such that hy , yi > sup{hy ∗ , yi : y ∈ cl G(rBX )}. Then, given x ∗ ∈ H (y ∗ ), we get hy ∗ , yi > sup{hx ∗ , xi : x ∈ rBX } = rkx ∗ k > crky ∗ k. It follows that kyk > cr and y ∈ / crBY . Thus, crBY ⊂ cl G(BX ). The last assertion follows from [55] Lemma 1. 2 The preceding two results can be combined. → ∗ has a →X COROLLARY 3.3. Suppose X and Y are Banach spaces, H : Y ∗ − ∗ ∗ weak closed convex graph and has Y as its domain. Then c(H ) > 0 iff there exists c > 0 such that the transpose G of H satisfies cBY ⊂ G(BX ). Now let us examine the behavior of the injectivity constant under perturbations. → ∗, →X Given a family (Hz )z∈Z of positively homogeneous multimappings Hz : Y ∗ − where Z is a topological space Z, and given z ∈ Z, we consider two stabilized (or e associated with (Hz )z∈Z and z. The first one, limiting) multimappings H and H ∗ H , is the set of weak limits of bounded nets ((yi∗ , xi∗ ))i∈I such that for some net e, (zi )i∈I with limit z in Z one has xi∗ ∈ Hzi (yi∗ ) for each i ∈ I . The second one, H is defined similarly but we require that (xi∗ )i∈I converges strongly and we call it the mixed stabilized multimapping associated with (Hz )z∈Z and z. We omit the study of obvious sequential variants. Let us observe that although the use of a mixed convergence in Y ∗ × X∗ is not as common as the use of the product convergence of two similar convergences, its appearance here is not new. Besides [48], such a framework is used in the study of maximal monotone operators [1]. Here Kw (Y ∗ ) denotes the set of weak∗ compact subsets of Y ∗ . e) be the stabilized (resp. mixed stabilized) PROPOSITION 3.4. Let H (resp. H e) > c(H ) and c(H e) > multimapping associated with z and (Hz )z∈Z . Then c(H c := lim infz→z c(Hz ). Moreover, lim infz→z c(Hz ) > c(H ) provided the following condition holds: (C0 ) for any r > c there exist W ∈ NZ (z) and K ∈ Kw (Y ∗ ) contained in SY ∗ such that, for each z ∈ W one has Hz (SY ∗ ) ∩ rBX∗ ⊂ Hz (K). e ⊂ H , so that H e(SY ∗ ) ⊂ H (SY ∗ ). Proof. The first assertion is obvious as H Given b < c one can find W ∈ NZ (z) such that c(Hz ) > b for each z ∈ W . Let e(y ∗ ) and let (xi∗ , yi∗ , zi )i∈I be a net as in the definition of H e. For i large x∗ ∈ H ∗ enough we have zi ∈ W , and as the norm of Y is l.s.c. for the weak∗ topology we get kx ∗ k = lim kxi∗ k > lim inf bkyi∗ k > bky ∗ k. i∈I

i∈I

e) > b. Taking the supremum on b we get c(H e) > c. Therefore c(H

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In order to prove the last assertion, let us take r > c := lim infz→z c(Hz ). Then there exists a net (zi )i∈I with limit z such that r > c(Hzi ) for each i ∈ I . Thus, for each i ∈ I one can find yi∗ ∈ SY ∗ and xi∗ ∈ Hzi (yi∗ ) such that kxi∗ k < r. Taking W ∈ NZ (z) and K ∈ Kw (Y ∗ ) as in (C0 ), we may assume that yi∗ ∈ K for each i ∈ I and that (yi∗ ) has a limit y ∗ in K ⊂ SY ∗ . Then any weak∗ cluster point x ∗ of (xi∗ ) satisfies x ∗ ∈ H (y ∗ ) and kx ∗ k 6 lim sup kxi∗ k 6 r so that c(H ) 6 r, hence 2 c(H ) 6 c, r > c being arbitrary. Another compactness condition yields a partial converse. PROPOSITION 3.5. Suppose the following compactness condition is satisfied: ∗

(C) (zi )i∈I → z, (xi∗ )i∈I → 0, (yi∗ )i∈I → 0 with xi∗ ∈ Hzi (yi∗ ) for each i ∈ I ⇒ (yi∗ ) → 0. e := H e−1 (0), one has the following implications: Then, setting Ker H e) > 0 H⇒ Ker H e = {0} H⇒ lim inf c(Hz ) > 0. c(H z→z

Moreover, condition (C) holds whenever lim infz→z c(Hz ) > 0. Proof. The first implication is obvious. Let us suppose c := lim infz→z c(Hz ) = 0. Then we can find a net (zi )i∈I → z such that c(Hzi ) → 0. It follows that there exist a subnet (zj )j ∈J of (zi )i∈I and yj∗ ∈ SY ∗ , xj∗ ∈ Hzi (yj∗ ) for j ∈ J such that (kxj∗ k)j ∈J → 0. Taking another subnet if necessary, we may assume ∗

that (yj∗ )j ∈J → y ∗ for some y ∗ ∈ BY ∗ . In view of condition (C) we cannot have e(y ∗ ) and Ker H e 6= {0}. The last assertion follows from y ∗ = 0. Then 0 ∈ H 0 the fact that condition (C ) of Remark 3.1(a) below is satisfied vacuously when 2 lim infz→z c(Hz ) > 0, and so is the equivalent condition (C). e = H when X is finite dimenThe preceding propositions and the relation H sional yield the following statement. THEOREM 3.6. Under condition (C) one has the following implications. If, moreover, X is finite-dimensional then all these implications are equivalences. c(H ) > 0 H⇒ Ker H = {0} ⇓ ⇓ e e c(H ) > 0 ⇐⇒ Ker H = {0} ⇐⇒ lim inf c(Hz ) > 0. z→z

The following remark explains why condition (C) is weaker than condition (C0 ). Remark 3.1. (a) Condition (C) can be formulated in a slightly different way: 0

(C ) for any net (xi∗ , yi∗ , zi )i∈I with (xi∗ ) → 0, kyi∗ k = 1, (zi ) → z and xi∗ ∈ Hzi (yi∗ ), the net (yi∗ )i∈I has a nonnull weak∗ cluster point.

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(C) implies (C0 ) as the net (yi∗ )i∈I being bounded has necessarily a weak∗ cluster point y ∗ and by (C) we cannot have y ∗ = 0. Conversely, if (xi∗ , yi∗ , zi )i∈I ∗ is a net with (xi∗ )i∈I → 0, (yi∗ )i∈I → 0, (zi ) → z, with xi∗ ∈ H (yi∗ ) and if (yi∗ ) does not converge to 0 we can find r > 0 and a cofinal subset J of I with rj := kyj∗ k > r for each j ∈ J . We may even assume that (rj−1 )j ∈J converges in ∗

[0, r −1 ]. Then (u∗j ) := (rj−1 xj∗ ) → 0, vj∗ := rj−1 yj∗ is such that kvj∗ k = 1, (vj∗ ) → 0, a contradiction with (C0 ). (b) The following strengthening of condition (C0 ) implies (C0 ) (and (C)): (C1 ) there exist r > 0, W ∈ NZ (z) and K ∈ K(Y ∗ ), K ⊂ SY ∗ , such that ∀z ∈ W , Hz−1 (rBX∗ ) ∩ SY ∗ ⊂ K. (c) If c := lim infz→z c(Hz ) > 0 condition (C1 ) is satisfied. It suffices to take r < c and K = ∅. By the preceding observation, condition (C) is satisfied too. (d) The following local compactness condition implies conditions (C1 ) and (C): (LC) there exist a locally compact cone C of Y ∗ and W ∈ NZ (z) such that Hz ⊂ C × X∗ for each z ∈ W . In view of [29], when (LC) satisfied, we say (Hz )z∈Z is a Loewen family. The implication (LC) ⇒ (C1 ) stems from the fact that K = C ∩ SY ∗ is compact when C is locally compact. In order to compare our results with other works, in particular [17, 39], it will be useful to consider some related criteria for condition (C). The following one is a weakening of condition (LC) as it can be seen by taking h = 0, α = 1. (QLC) there exist α > 0, a closed locally compact cone C of Y ∗ , a (positively) homogeneous function h: Y → R+ which is weak∗ continuous on bounded subsets of Y ∗ and W ∈ NZ (z) such that for each z ∈ W the graph of Hz is contained in the cone Q := {(y ∗ , x ∗ ) ∈ Y ∗ × X∗ : d(y ∗ , C) 6 αkx ∗ k + h(y ∗ )}. This condition is also a consequence of the following condition we call the sharp cone condition (or, as in [49], the Bishop–Phelps condition owing to their work on orders determined by such cones): (SC) there exist ρ > 0, a compact subset R of Y and W ∈ NZ (z) such that for any z ∈ W and (y ∗ , x ∗ ) ∈ Hz one has kx ∗ k + maxy∈R |hy ∗ , yi| > ρky ∗ k. LEMMA 3.7. Condition (SC) implies condition (QLC). Proof. As h given by h(y ∗ ) = ρ −1 maxy∈R hy ∗ , yi is weak∗ continuous on 2 bounded subsets of Y ∗ , it suffices to take α = ρ −1 , C = {0}. In turn, condition (SC) is a consequence of an apparently weaker and more sophisticated property considered in [17, 21] and [39], we call the finite codimension condition:

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(FC) there exist σ > 0, γ > 0, a compact subset S of Y , a weak∗ closed subspace L of Y ∗ of finite codimension and W ∈ NZ (z) such that for any z ∈ W , (y ∗ , x ∗ ) ∈ Hz with ky ∗ k = 1 and d(y ∗ , L) 6 γ one has kx ∗ k + maxy∈S |hy ∗ , yi| > σ. It is not our purpose here to compare the various forms of this condition considered in [17, 21], and [39]? . Let us just note that these properties are both local properties and not pointwise, infinitesimal properties. LEMMA 3.8. Condition (FC) implies condition (SC). Proof. Let σ > 0, γ > 0, S, L, W be as in condition (FC) and let M be a finite dimensional subspace of Y complementary to L. We first observe that (FC) can be written z ∈ W, (y ∗ , x ∗ ) ∈ Hz , d(y ∗ , L) 6 γ ky ∗ k H⇒ kx ∗ k + hS (y ∗ ) > σ ky ∗ k with hS (y ∗ ) := maxy∈S |hy ∗ , yi|. Let T be the unit ball of L0 , so that T is compact and d(y ∗ , L) = maxy∈T |hy ∗ , yi| := hT (y ∗ ) by a well-known duality result. Taking ρ = min(σ, γ ), α = ρ −1 , R = S ∪ T we see that condition (SC) is satisfied by considering separately the cases hT (y ∗ ) > γ ky ∗ k and d(y ∗ , L) < γ ky ∗ k. 2 THEOREM 3.9. The following diagram of implications holds: (LC) H⇒ (C1 ) H⇒ (C0 ) ⇓ ⇓ (FC) H⇒ (SC) H⇒ (QLC) H⇒ (C) ⇐⇒ (C0 ) Proof. It remains to prove the implication (QLC) ⇒ (C). Let α > 0, C, h, W be as in condition (QLC) and let ((xi∗ , yi∗ , zi ))i∈I be a net such that (xi∗ ) → 0, ∗ (yi∗ ) → 0, (zi ) → z, xi∗ ∈ Hzi (yi∗ ) for each i ∈ I . We may suppose zi ∈ W for each i ∈ I so that d(yi∗ , C) 6 αkxi∗ k + h(yi∗ ). As (yi∗ ) is bounded and has weak∗ limit 0, the continuity property of h ensures that (h(yi∗ )) → 0. As C is closed we can find vi∗ ∈ C such that kyi∗ − vi∗ k 6 2d(yi∗ , C) (considering the cases d(yi∗ , C) = 0 and d(yi∗ , C) > 0 separately). Then we have (d(yi∗ , C)) → 0, ∗ (kyi∗ −vi∗ k) → 0 so that (vi∗ ) → 0. Then, as C is locally compact, we get (vi∗ ) → 0 2 by [29] Prop. 3.2 (see also [49] Prop. 15). Thus (yi∗ ) → 0. Condition (C) is simple. It is also versatile and convenient to use, as the following shows. PROPOSITION 3.10. Let (Fz )z∈Z , (Gz )z∈Z be families of positively homogeneous relations satisfying condition (C). Then (Hz )z∈Z and (Pz )z∈Z given by Hz (y ∗ ) = Fz (y ∗ ) ∪ Gz (y ∗ ),

Pz (y1∗ , y2∗ ) = Fz (y1∗ ) × Gz (y2∗ )

? (Note added after revision.) A detailed comparison is conducted in the paper ‘Codirectional compactness, metric regularity and subdifferential calculus’ by A. D. Ioffe, preprint Technion, Haifa, April 1996. It is shown there that in fact our condition (C) is equivalent to condition (FC) and thus to the other conditions between them.

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also satisfy condition (C). The proof is immediate. This observation enables one to combine the different criteria implying condition (C) displayed in what precedes. 4. Openness Criteria with Limiting Coderivatives The preceding section has shown the position of condition (C) and justified the introduction of the following definition inspired by [50] Def. 2.12? . Let us note that → this notion pertains to the multimapping F : X− →Y rather than to its graph, so that a change of terminology is justified. It could be called ‘partial normal compactness’, but since this terminology is used in [39] in a different way, we prefer to avoid any confusion. → is said to be coderivatively com− DEFINITION 4.1. The multimapping F : X→Y pact at z = (x, y) ∈ F if its family of coderivatives (D ∗ F (z))z∈Z satisfies condition (C): (C) for any net (zi )i∈I → z with limit z in F , and for any net ((xi∗ , yi∗ ))i∈I with ∗ (xi∗ ) → 0, (yi∗ ) → 0 and xi∗ ∈ D ∗ F (zi )(yi∗ ) for each i ∈ I one has (yi∗ ) → 0. A related property is pointed out in [39] Prop. 3.10, but there the condition (zi ) → z is not required (instead (zi ) belongs to a prescribed neighborhood of z), so that the p.n.c. condition of [39] is more exacting and not directly related to the study of the preceding section. This notion will be crucial for the use of the limiting (or stabilized) coderiv∗ e∗ F (z) of F at z which are the multimappings H and H e atives D F (z) and D associated to the family (Hz )z∈Z := (DF (z))z∈Z according to the stabilization procedures described in the preceding section. Only the first one is widely used (at least in its sequential variant), but as we noticed it, the second stabilization procedure allows a characterization of openness at a linear rate which is an immediate consequence of Theorems 2.3 and 3.6 and Corollary 3.5. → THEOREM 4.1. Let F : X− →Y be a multimapping with closed graph between two Asplund (resp. Hadamard reliable) spaces and let z ∈ F . Then F is open around z e∗ F (z) = {0} and F is coderivatively compact with a linear rate iff (resp. if) Ker D at z for the Fréchet (resp. Hadamard) normal cone. ∗ e∗ F (z), the following criterion follows immediately. As D F (z) contains D ∗

COROLLARY 4.2. Suppose F is as in the preceding theorem and Ker D F (z) = {0}. Then F is open around z with a linear rate. ? In its revised version of December 1994, disseminated in January 1995, the definition was labelled Definition 2.2.

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This criteria is close to [39] Thm. 4.2 but our compactness assumption is milder than the one in [39] as we have seen; in [39] Cor. 4.5 X and Y are supposed to be W CG spaces. Before giving criteria for coderivative compactness, let us make clear some links of this notion with the concept of normal compactness of a set. This last concept has been introduced in [50] Def. 2.2 in the convex case and the definition can be extended to the nonconvex case without any change: a subset C of a n.v.s. Z is said to be normally compact at z ∈ C if, for any net (zi )i∈I of C with limit z, any net (zi∗ )i∈I such that zi∗ ∈ N(C, zi ), kzi∗ k = 1 for each i ∈ I has a nonnull weak∗ cluster point. Remark 2.1(a) shows that this condition amounts to require ∗ that any net (zi∗ )i∈I → 0 with zi∗ ∈ N(C, zi ) for each i ∈ I for some net (zi )i∈I of C with limit z actually converges (strongly) to 0. This reformulation leads to the following implication which is parallel to an observation in [26], remark following Definition 3.2. LEMMA 4.3. If the graph of F is normally compact at z ∈ F , then the multimap→ ping F : X− →Y is coderivatively compact at z. The converse is not true as the following example shows: EXAMPLE. Let A: X → Y be a continuous surjective linear map between two infinite dimensional Banach spaces. Then the multimapping F = A is coderivatively compact at each point z of its graph C as shown below in Propositions 4.6 and 4.8. However C is not normally compact at 0 as we can find bounded nets in N(C, 0) = {(x ∗ , y ∗ ) : x ∗ = −AT (y ∗ )} which converge weakly∗ but not strongly. 2 Other links between the two notions are contained in the following observations. → LEMMA 4.4. Let C, D be subsets of X and Y respectively and let F : X− →Y be the multimapping with graph C × D. If D is normally compact at y ∈ D, then for each x ∈ C the multimapping F is coderivatively compact at (x, y). Proof. The result follows from the relation (valid for any normal cone) N(C × D, (x, y)) = N(C, x) × N(D, y) ∀(x, y) ∈ C × D.

2

LEMMA 4.5. Let C (resp. D) be a subset of X (resp. Y ) and let f : C → Y be a mapping continuous at some x ∈ C. Suppose D is normally compact at some → v ∈ D (for the normal cone associated to some bornology B). Then, F : X− →Y given by F (x) = f (x) + D for x ∈ C, F (x) = ∅ for x ∈ X \ C is coderivatively compact at (x, y) for y = f (x) + v. This criteria is similar to the ones in [26] Proposition 3.3, [39] Proposition 3.9: here f is not supposed to be Lipschitzian but the conclusion is weaker, the p.n.c. condition being changed into coderivative compactness.

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Proof. If zi = (xi , yi ) → z = (x, y) we have vi := yi − f (xi ) → y − f (x) = v by the continuity of f at x. Moreover, it follows from the definition of the normal cone that for any z = (x, y) = (x, f (x) + v) with x ∈ C, v ∈ D one has (x ∗ , −y ∗ ) ∈ N(F, z) H⇒ −y ∗ ∈ N(D, v). Thus condition (C) is implied by the normal compactness assumption of D at v. 2 For convex relations, coderivative compactness is a necessary condition for openness at a linear rate, as the following result shows. → PROPOSITION 4.6. Suppose F : X− →Y has a convex graph and satisfies the condition: (O) there exist r, s, t > 0 such that B(y, s) ⊂ F (B(x, r)) for each (x, y) ∈ F ∩ B(z, t). Then the multimapping F is coderivatively compact at (x, y). Proof. Let F

zi = (xi , yi ) → z = (x, y),

(xi∗ , −yi∗ ) ∈ N(F, zi )

∗

be such that (kxi∗ k) → 0, (yi∗ ) → 0 and let us prove that (kyi∗ k) → 0. For i large enough and for each b ∈ BY we can find xi0 ∈ B(xi , r) such that yi0 := yi + sb ∈ F (xi0 ). Then, hyi∗ , sbi = hyi∗ , yi0 − yi i > hxi∗ , xi0 − xi i > −kxi∗ kr so that kyi∗ k 6 s −1 rkxi∗ k and (kyi∗ k) → 0.

2

The Robinson–Ursescu theorem [55] yields the following consequence. → COROLLARY 4.7. Suppose X and Y are complete and F : X− →Y has a closed convex graph such that R+ (F (X) − y) = Y for some z = (x, y) ∈ F . Then F is coderivatively compact at (x, y). Coderivative compactness is also satisfied under a differentiability assumption and a surjectivity condition. Thus one recovers the classical Graves–Lusternik theorem. It has been kindly pointed out by a referee that it could be deduced from an appropriate modification of [36] Prop. 2.4 Assertion (i); but, as it stands, that assertion simply means that the ε-Fréchet coderivative is locally bounded around x.? PROPOSITION 4.8. Suppose f : X → Y is a mapping which is strictly differentiable at x. Then the graph F of f is Fréchet coderivatively compact at ? During the revision process of the present paper we also have noticed that the result is a special case of [17] Theorem 2.11, taking the closed cylinder of finite codimension S to be the whole space.

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z = (y, f (x)) whenever c(A) > 0 for A := f 0 (x), in particular when X and Y are complete and A is surjective. Proof. Let F

zi := (xi , yi ) → z := (x, y),

(xi∗ , −yi∗ ) ∈ N(F, zi )

∗

with (kxi∗ k) → 0, (yi∗ ) → 0. We will prove that (kxi∗ − AT yi∗ k) → 0, so that (kAT yi∗ k) → 0 and (kyi∗ k) → 0. As f is strictly differentiable at (x, y), the mapping r(·, ·) given by r(x, 0) = 0, f (x + u) = f (x) + Au + r(x, u)kuk for u 6= 0 satisfies r(x, u) → 0 as (x, u) → (x, 0). Now, as (xi∗ , −yi∗ ) ∈ N(F, zi ) for each i ∈ I , given ε ∈ ]0, 1] we can find ρi > 0 such that for (x, y) ∈ F ∩ B(zi , ρi ) we have h(xi∗ , −yi∗ ), (x − xi , y − yi )i 6 εk(x − xi , y − yi )k. Let ζi ∈ ]0, 1] be such that for u ∈ ζi BX one has k(x − xi , y − yi )k 6 ρi for x = xi + u, y = f (xi + u) = yi + Au + r(xi , u)kuk and kr(xi , u)k 6 ε. Then for u ∈ ζi BX one obtains hxi∗ , ui − hyi∗ , Aui 6 εk(u, Au + r(xi , u)kuk)k + hyi∗ , r(xi , u)kuki 6 ε(kAk + 1)kuk + (ε + kyi∗ k)kri (xi , u)kkuk. Let s > supi kyi∗ k and let k ∈ I , α > 0 be such that kr(xi , u)k 6 ε for i > k, kuk 6 α. Then for i > k, u ∈ ζi0 BX , with ζi0 = min(ζi , α) we have |hxi∗ − AT yi∗ , ui| 6 ε(kAk + s + 2)kuk and we have proved that kxi∗ − AT yi∗ k → 0. The last assertion stems from Lemma 3.1. 2 The preceding examples show that coderivative compactness is satisfied in a number of significant cases. These cases can be combined, in view of the following result which is an easy consequence of Proposition 3.10, owing to a decomposition property of normal cones to products which is satisfied by usual notions. → →Yi , PROPOSITION 4.9. Let X = X1 × X2 , Y = Y1 × Y2 , and for Fi : X− → − i = 1, 2, let F : X→Y be given by F (x1 , x2 ) = F1 (x1 ) × F2 (x2 ). Then if Fi is coderivatively compact at zi = (x i , y i ) for i = 1, 2, F is coderivatively compact at z = ((x 1 , x 2 ), (y 1 , y 2 )).

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Acknowledgement The author is grateful to two anonymous referees for their help with references and due credits and for an improvement in the way Theorem 4.1 is stated.

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