Journal of Convex Analysis Volume 13 (2006), No. 2, 225–252
A Unified Theory for Metric Regularity of Multifunctions
D. Az´e UMR CNRS MIP, Universit´ePaul Sabatier, 118 Route de Narbonne, 31062 Toulouse Cedex, France
Dedicated to Jean-Pierre Aubin on the occasion of his 65th birthday.
Received: March 30, 2005 Revised manuscript received: June 16, 2005
We survey a large bunch of results on metric regularity of multifunctions that appeared during the last 25 years. The tools used for this survey rely on a new variational method introduced in [9]1 and further developped in [10] and independently in [35, 36] which provides a characterization of metric regularity. It allows us to give a unified point of view both for primal results (based on some notion of tangent cones) and dual ones (based on some notion of normal cones). For most known results on metric regularity, a simple proof is given along with some slight improvements for some of them.
1. Introduction The theory of metric regularity, whose origin dates back to [44] and [28], for mappings and further for multimappings is one of the cornerstones of nonsmooth analysis. A de- cisive step has been done in this theory by the systematic use of Ekeland’s variational principle. However, there are some cases such as the Ursescu-Robinson Theorem and the finite dimensional case in which one can give metric regularity results without using the variational principle. We refer to [36] for its accurate bibliographical and historical comments. The first time in which these kinds of techniques has been used for metric regularity of mappings and multimappings seems to be [31] and [1]. Another important contribution is [33] in which the use of the Ekeland principle leads to sufficient conditions for metric regularity based both on tangent and normal cones. Recently, the use in [9] of the so-called strong slope of De Giorgi, Marino and Tosques opened the way to a unified theory of metric regularity involving both criteria invoking tangent and normal cones. This unified theory which is purely metric was developped in [10] and independently in the deep and complete survey [36]. The powerfullness of this unifying theory lies in the fact that it provides a purely metric characterization of metric regularity which allows Ioffe to write in [36] Ô. . . the theory of metric regularity and the parallel Lipschitz the- ory of set-valued maps do not need anything like subdifferentials, directional derivatives, coderivatives and tangent conesÔ. The common feeling of mathematicians is that the met- ric regularity problem is linked to the error bounds one. In fact these two problems are equivalent (see [9, 10]) and the obtained characterization of metric regularity is a con- sequence of the characterizations of local and global error bounds obtained for the first time in [9, Remark 3.2] and in [8, Theorem 2.4]. The paper is organized as follows. After recalling in Sections 2 and 3 the characterization of metric regularity in complete metric 1The paper [9] appeared in 2002 but the corresponding preprint dates back to 1998-99 and was quoted and used in [35, 36].
ISSN 0944-6532 / $ 2.50 c Heldermann Verlag 226 D. Az´e/ A Unified Theory for Metric Regularity of Multifunctions
spaces, we review in Section 4 a large sample of results on metric regularity among those which appeared during the last 25 years. Let us begin with some notations that will be used throughout this paper. We let X be a metric space endowed with the metric d, and f : X R + be a lower semicontinuous function. For U X and r (0; + ] (resp., r→ (0;∪+ { [),∞} we denote by ⊂ ∈ ∞ ∈ ∞ Br(U) (resp., B¯r(U)) the open (resp., closed) r-neighborhood of U: B (U) := x X : d(x; U) < r ; B¯ (U) := x X : d(x; U) r ; r { ∈ } r { ∈ ≤ } where d(x; U) := inf d(x; y): y U ; { ∈ } with the convention that d(x; ) = + (according to the general convention inf = + ). U x B∅ x B∞¯ x B B B¯ B¯ «∅ ∞ If = , we simply write r( ), r( ) and X = 1(0), X = 1(0). For R and { } ∈ ‹ R + , we let ∈ ∪ { ∞} [f «] := x X : f(x) « ; [f<‹] := x X : f(x) < ‹ ≤ { ∈ ≤ } { ∈ } denote respectively the closed and open sublevel sets of f, and if « < ‹, we further let [« 2. The strong slope In this section we shall give a characterization of metric regularity of multifunctions in the complete metric space setting. This characteristic conditions will be given in terms of the strong slope of De Giorgi, Marino and Tosques introduced in [20]. The first issue in which strong slope is used aimed at error bounds and metric regularity results seems do be [9]. The method initated in the quoted paper was further used and developped in [35, 36, 10]. We stress the fact that the strong slope allows a complete characterization of metric regularity. As we recall in this section, the strong slope can be estimated both in terms of suitable abstract subdifferential or directional derivatives. When applied to metric regularity, we shall also check in Section 4 that our abstract results encompass most of the results on this topic. 2.1. The strong slope The notion of strong slope was introduced by De Giorgi, Marino, and Tosques in [20] aimed at finding solutions of abstract evolution equations in metric spaces. D. Az´e/ A Unified Theory for Metric Regularity of Multifunctions 227 2.1.1. Defining the slope Definition 2.1. Let f : X R + be a lower semicontinuous function, and let x dom f. Set: → ∪ { ∞} ∈ 0 if x is a local minimum of f; f (x) := f(x) f(y) |∇ | lim sup − otherwise. = d(x; y) y 6 x −→ For x = dom f, let f(x) := + . The nonnegative extended real number f (x) is called∈ the strong slope|∇ of| f at x. ∞ |∇ | The main tool which is needed for our purposes is, of course, Ekeland’s variational princi- ple [24], of which we now recall an appropriate version, as well as an essential consequence in terms of the strong slope. Theorem 2.2. Let X be complete, f : X R + be a (proper) lower semicontin- uous function, and let x¯ X, » > 0, and r→ > 0,∪ be { such∞} that: ∈ f(¯x) inf f + »r : ≤ X Then, there exists x B¯ (¯x) such that f(x) f(¯x) and ∈ r ≤ f(x) < f(y) + »d(x; y) for every y X x : (1) ∈ \{ } Corollary 2.3. Let X be complete, f : X R + be a lower semicontinuous function, and let x¯ X, » > 0, and r > 0, be→ such that:∪ { ∞} ∈ f(¯x) < inf f + »r : B¯r(¯x) Then, there exists x B (¯x) such that f(x) f(¯x) and f (x) < ». ∈ r ≤ |∇ | Proof. Let 0 < »0 < » and 0 < r0 < r be such that f(¯x) inf f + »0r0 : ≤ B¯r(¯x) X B¯ x x B¯ x B x f x f x Applying Theorem 2.2 with := r(¯), we find r0 (¯) r(¯) with ( ) (¯) f x f x » < » ∈ ⊂ ≤ and B¯r(¯x) ( ) = ( ) 0 , as follows from (1) and the definition of the strong slope.|∇ | | |∇ | ≤ The next simple proposition illustrates the role played by the strong slope in existence results. Proposition 2.4. Let X be complete, f : X R + be a lower semicontinuous → ∪ { ∞} function, U be a subset of X, « R, and »; ” > 0. Assume that U [f<« + »”] = and that: ∈ ∩ 6 ∅ inf f »: Bρ(U) [α Proof. Assume, for a contradiction, that [f «] = , and letx ¯ U [« 2.1.2. Computing the slope In this subsection, we consider a Banach space X endowed with a norm , with topolog- k·k ical dual X∗, d denoting the metric associated with the norm of X∗. In the case where ∗ the function f is Fr´echet differentiable at x, it is readily seen that f (x) = Df(x) (2) |∇ | k k∗ We further consider an “abstractÔ subdifferential operator @, which associates to any lower semicontinuous function f : X R + , and any point x X, a subset @f(x) of the → ∪ { ∞} ∈ (topological) dual X∗ of X, in such a way that @f(x) = if x = dom f, and the following two properties are satisfied: ∅ ∈ (P1) if f is convex, then @f(x) ‚ X∗ : f(y) f(x) + ‚; y x for all y X ; (3) ⊂ { ∈ ≥ h − i ∈ } (P2) wheneverx ¯ dom f is a local minimum point of f + g + h where f is lower semi- continuous and∈ g, h are Lipschitz continuous nearx ¯ with h convex then, for every " > 0 there exist x, y, z X, ‚ @f(x), and @g(y), ¿ @h(z) such that ∈ ∈ ∈ ∈ x x¯ "; y x¯ "; z x¯ " ; f(x) f(¯x) + "; and ‚ + + ¿ ": k − k ≤ k − k ≤ k − k ≤ ≤ k k∗ ≤ It follows that if the subdifferential satisisfies (P1) and if the Banach space X is @- trustworthy in the sense of Ioffe, then (P2) is satisfied on X. Given a subdifferential @, one is also interested in the associated limiting subdifferen- @f¯ x ‚ X x tial ( ) defined as the set of ∗ for which there exist sequences ( n)n N with x ; f x x; f x ‚ ∈ X ‚ ‚ ∈@f x (( n ( n)) ( ( )) and ( n)n N ∗ -weakly converging to such that n ( n) → ∈ ⊂ ∗ ∈ for all n N. It is clear that @¯ satisfies (P2) on X whenever it is the case for @. ∈ Example 2.5. Let us list some basic subdifferential. Each of them clearly satisfies (P1). 1. The Clarke-Rockafellar subdifferential satisfies (P2) on every Banach space as easily shown by its definition. Let us recall that the Clarke-Rockafellar subdifferential @f(x) at x dom f of a lower semicontinuous function f : X R + is the ∈ −→ ∪ { ∞} set of ‚ X∗ such that ‚; u f ↑(x; u) for all u X where ∈ h i ≤ ∈ 1 f ↑(x; u) = sup inf sup inf t− (f(z + tv) f(z)): η>0 ε>0 (z,f(z),t) Bη(x) Bη(f(x)) (0,η) v Bε(u) − ∈ × × ∈ 2. The Dini subdifferential @− is defined at x dom f by 0 ∈ @−f(x ) = ‚ X∗ : ‚; u f 0(x ; u) for all u X ; (4) 0 { ∈ h i ≤ 0 ∈ } D. Az´e/ A Unified Theory for Metric Regularity of Multifunctions 229 f x tv f x f x u ( 0 + ) ( 0) where 0( 0; ) = lim inf t − . This subdifferential satisfies (P2) t 0 ↓ v u → on every finite dimensional X (see e.g. [32, Theorem 2]). 3. The Fr´echet subdifferential @fÝ (x), that is the set of ‚ X∗ such that ∈ 1 lim inf z x − (f(z) f(x) ‚; z x ) 0 z x → k − k − − h − i ≥ satisfies (P2) on every Asplund space (see [25]). It follows that when X is asplund, the Mordukhovich subdifferential which in that case is the limiting Fr´echet subdif- ferential, also satisfies (P2). One also uses, for " > 0, the "-Fr´echet subdifferential ε @Ý f(x) defined as the set of ‚ X∗ such that ∈ 1 lim inf z x − (f(z) f(x) ‚; z x ) ": z x → k − k − − h − i ≥ − Then it is noteworthy that 0 @Ýεf(x) if and only if f (x) ": (5) ∈ |∇ | ≤ 4. The Ioffe subdifferential @A satisfies (P2) on every Banach space (see e.g. [36, The- orem 1]). It is defined for Lipschitz functions by @ f x @ f x A ( 0) = Lim sup L− ( ) (6) f L x x \∈F → 0 where is the family of finite dimensional subspaces of X, F @−f (x ) = ‚ X∗ : ‚; u f 0(x ; u) for all u L ; L 0 { ∈ h i ≤ 0 ∈ } f the Lim sup being taken with respect to the weak∗ topology and x x means → 0 (x; f(x)) (x0; f(x0)). If f is lower semicontinuous and x0 dom f, this definition extends to→ ∈ @ f(x ) = ‚ X∗ :(‚; 1) @ d (x; f(x)) : A 0 { ∈ − ∈ A epi f } L> [0 5. A bornology ‹ is a collection of bounded convex symmetric subsets of X which is X B closed under multiplication by scalars and is such that = B β and such that the union of two elements of ‹ is contained in some element of∈ ‹. A function f β S is said to be ‹-differentiable at x0 whenever there is f(x0) X∗ such that for 1 ∇ β ∈ any B ‹, we have limt 0 t− (f(x0 + tu) f(x0) t f(x0); u ) = 0 uniformly ∈ → − − h∇ i β for u B. We say that the function f is ‹-smooth at x0 whenever f exists ∈ β ∇ near x and the mapping f is continuous when X∗ is endowed with the uniform 0 ∇ convergence on elements of the bornology ‹. Then the ‹ subdifferential @βf(x0) is the set of ‚ X∗ such that for all " > 0 and for all B ‹, there exists – > 0 such 1 ∈ ∈ that t− (f(x0 + tu) f(x0)) ‚; u > " for all t (0;–) and for all u B. This subdifferential satisfies− (P2) whenever− h i there− exists a∈‹-differentiable Lipschitz∈ bump function, that is a Lipschitz function ’ with values on [0; 1] with ’(0) = 1 whose support is contained in B¯ and such that each maximizing sequence converges to 0 (see e.g. [36, 2, Theorem 1]). 230 D. Az´e/ A Unified Theory for Metric Regularity of Multifunctions β 6. Given a bornology on X, the ‹-viscosity subdifferential D f(x0) of a lower semicon- f X x f ‚ X tinuous function : R + at 0 dom is the set of ∗ for which → ∪ { ∞} ∈ ∈ there exists a locally Lipschitz function g which is ‹-smooth at x0 and such that β f g attains a local minimum at x0 along with g(x0) = ‚. Such a subdifferential satifies− (P2) whenever X admits an equivalent norm∇ which is ‹-smooth away from 0 (see e.g. [13, Theorem 2.9]). 7. If 0 < s 1, a function g : X R is of class C1,s if it is (GÝateaux-)differentiable on X and≤ its differential Dg is →s-H¨oldercontinuous, i.e., there is a constant C 0 such that ≥ Dg(x) Dg(y) C x y s for all x; y X: k − k ≤ k − k ∈ The Banach space X has a s+1-power modulus of smoothness if X has an equivalent g 1 s+1 C1,s differentiable norm for which the function := s+1 is of class . The Banach spaces withk·k a power modulus of smoothness arek·k the superreflexive spaces. Equipped with their natural norm, Hilbert spaces have a power modulus of smooth- 2 p min p,2 ness t , and L spaces, p > 1, have a power modulus of smoothness t { }. Then s the s-H¨oldersmooth subdifferential is of f at x dom f is the set @ f(x) of ‚ X∗ such that there exists –, » > 0 such that ∈ ∈ f(y) f(x) ‚; y x » y x 1+s for all y B (x): − ≥ h − i − k − k ∈ η In other words f y f x ‚; y x s ( ) ( ) @ f(x) := ‚ X∗ : lim inf − − h − i > ; { ∈ y x y x 1+s −∞} → k − k and the associated viscosity subdifferential is the set @˜sf(x) of those D’(x) such that ’ is of class C1,s and x is a local minimum of f ’. It is clear that @˜sf(x) @sf(x). It is easily seen, by using [22, Lemma 1.2.5], that− the viscosity subdifferential⊂ @˜sf(x) satisfies (P2) whenever X has a power modulus of smoothness ts+1. When s = 1 and X is Hilbert, then @s coincides with Rockafellar’s proximal subdifferential @π [55]. The two following propositions provide useful information for estimating the slope from below. Proposition 2.6. Let X be a Banach space and @ be a subdifferential operator which satisfies (P1) and (P2) on X. Then, for every lower semicontinuous function h : X → R + , for every function g Lipschitz continuous near x dom f, we have, setting f =∪ {g +∞}h: ∈ f (x) lim inf d (0; @g(z) + @h(y)) : |∇ | ≥ (z,y,f(z),f(y)) (x,x,f(x),f(x)) ∗ → Proof. Let x X. We may assume that x is not a local minimum point of f (for, otherwise, the∈ result readily follows from (P2)), and that f (x) < + (so that x dom f). Given » > f (x) and " > 0, let r > 0 be such that|∇ | ∞ ∈ |∇ | f(x) f(y) + » y x for every y B (x) ; ≤ k − k ∈ r so that the function h + g + » x attains a finite local minimum at x. From property (P2), we find y, z u X and k‚ · −@hk(y), @(» x )(u) and ¿ @g(z) such that ∈ ∈ ∈ k · − k ∈ y x " ; h(y) h(x) + "; and ‚ + ¿ + ": k − k ≤ ≤ k k∗ ≤ D. Az´e/ A Unified Theory for Metric Regularity of Multifunctions 231 From property (P1), », so that ‚ + ¿ » + ", and the conclusion follows since ∗ ∗ " > 0 is arbitrary, takingk k into≤ account thek lowerk ≤ semicontinuity of f. Applying the previous proposition with g = 0 we get: Proposition 2.7. Let X be a Banach space and @ be a subdifferential operator such that properties (P1) and (P2) are satisfied on X. Then, for every lower semicontinuous function f : X R + and every x X, we have: → ∪ { ∞} ∈ f (x) lim inf d (0; @f(y)) : |∇ | ≥ (y,f(y)) (x,f(x)) ∗ → Proposition 2.7 tells us that if infy Ω d (0; @f(y)) … where Ω is an open subset of X for ∈ ∗ ≥ some … > 0, then infx Ω f (x) …. ∈ |∇ | ≥ The strong slope can also be estimated, just using the definitions, with various notions of directional derivative. For example, for x dom f, let us consider ∈ f(x + tv) f(x) f 0(x; u) := lim inf − t 0 t v&u → the contingent derivative. Proposition 2.8. Let f : X R + be a proper lower semicontinuous function defined on a Banach space X −→and let∪x { ∞}dom f. We have ∈ f (x) sup ( f 0(x; u)): (7) |∇ | ≥ u 1 − k k≤ Moreover, if X is finite dimensional, we have f (x) = max( f 0(x; u)): (8) |∇ | u 1 − k k≤ Proof. Assuming that x is a local minimum, then f 0(x; u) 0 for all u X, thus f x f x u u X ≥u f∈x u ( ) sup u 1( 0( ; ). Otherwise, fix 0 with 1. If 0( ; ) 0, |∇ | ≥ k k≤ − ∈ \{ } k k ≤ − ≤ the inequality f 0(x; u) f (x) is obvious. Suppose that f 0(x; u) > 0 and choose sequences t 0− and v ≤u |∇such| that − n ↓ n → f(x) f(x + tnvn) f 0(x; u) := lim − > 0: n t − →∞ n Then f(x) f(x + tnvn) f(x) f(x + tnvn) f 0(x; u) lim − = lim − − ≤ n t u n t v →∞ nk k →∞ nk nk yielding f x f y f x u ( ) ( ) f x ; 0( ; ) lim sup x − y = ( ) − ≤ = |∇ | y 6 x → k − k from which we get (7). Assume now that X is finite dimensional. If x is a local minimum, f x f x x x then ( ) = 0 = 0( ; 0). Otherwise, let ( n)n N be a sequence converging to |∇ | − ∈ 232 D. Az´e/ A Unified Theory for Metric Regularity of Multifunctions f(x) f(xn) with xn = x such that f (x) = lim sup − . Setting tn = xn x and 6 |∇ | n x xn k − k →∞ u t 1 x x k − k u n = n− ( n ), there exists a subsequence, still denoted by ( n)n N converging to some ∈ u X with −u = 1. Thus we get ∈ k k f x t 1 f x t u f x f x; u ; ( ) = lim inf n− ( ( + n n) ( )) 0( ) n −|∇ | →∞ − ≥ hence (8). Estimates from above of the slope are also available in some special cases. These estimates will be useful when aimed at seeking for necessary conditions for metric regularity of multifunctions. Proposition 2.9. Let f : X R + be a closed proper function defined on a Banach space X and let x dom−→f. We∪ { have∞} ∈ a) d (0; @fÝ (x)) f (x), where @Ý denotes the Fr´echetsubdifferential; ∗ ≥ |∇ | b) d(0;@sf(x)) f (x) whenever @s denotes the s-H¨oldersmooth subdifferential. ≥ |∇ | c) d (0;@−f(x)) f (x) whenever X is finite dimensional and @− denotes the Dini ∗ subdifferential.≥ |∇ | Proof. a) We may assume that x is not a local minimum, since in that case the right member of the inequality in a) is equal to 0. Let ‚ @fÝ (x) and let " > 0. We can find r > 0 such that f(y) f(x) ‚; y x " y x ∈for all y B (x), yielding − ≥ h − i − k − k ∈ r f(x) f(y) − ‚ + " for all y Br(x); y x ≤ k k∗ ∈ k − k f x f y ( ) ( ) ‚ ‚ @fÝ x and then lim sup x − y for all ( ), from which we get = ≤ k k∗ ∈ y 6 x → k − k d (0; @fÝ (x)) f (x): ∗ ≥ |∇ | b) follows from a) since @sf(x) @fÝ (x). ⊂ c) From Proposition 2.8, putting … := f (x), we can find a unit vector u X such |∇ | ∈ that f 0(x; u) = …. Thus if ‚ @−f(x), we have … = f 0(x; u) ‚; u , so that ‚ ‚; u −…. ∈ − ≥ h i k k∗ ≥ h − i ≥ Remark 2.10. If f : X R + is a closed proper convex function defined on a Banach space X, then 7−→ ∪ { ∞} f (x) = d (0; @f(x)) for all x dom f: (9) |∇ | ∗ ∈ Indeed, we get from part a) of Proposition 2.9 that d (0; @f(x)) f (x). Now if ∗ 0 < » < d (0; @f(x)), then, x is not a minimum point of the function ≥z |∇f(|z)+» x z ∗ for, otherwise, from standard convex calculus, we have 0 @f(x)+»B¯ ,7→ where B¯ denotesk − k ∈ ∗ ∗ the closed unit ball of X∗, contradicting the choice of ». Let thus z X be such that f(z) + » x z < f(x), so that for every ]0; 1]: ∈ k − k ∈ f x f x z x ( ) ( + ( )) > » ; − x z − k − k showing that f (x) », whence f (x) d (0; @f(x)). |∇ | ≥ |∇ | ≥ ∗ D. Az´e/ A Unified Theory for Metric Regularity of Multifunctions 233 2.1.3. Parametric error bounds When dealing with parametric family of multifunctions, a parametric error bound result may be useful. Let us begin with a result of [10] which is of independant interest. Theorem 2.11. Let X be complete, f : X R + be a lower semicontinuous → ∪ { ∞} function, U be a subset of X, « R, and »; ” > 0. Assume that U [f<« + »”] = and that ∈ ∩ 6 ∅ inf f »: Bρ(U) [α Applying Corollary 2.3, we find x Br(¯x) with g(x) g(¯x) and g (x) < ». Then, x B (U) [« Theorem 2.13. Let X be a complete metric space, let P be a topological space and let f X P x ; p X P : R + and ( 0 0) . Assume that × −→ ∪ { ∞} ∈ × a) f (x ) 0; p0 0 ≤ b) fp is lower semicontinuous for all p near p0; c) f is epi-upper semicontinuous at (x0; p0); d) there exist neighborhoods U0 of x0, N0 of p0 and … > 0 such that inf f (x) …: |∇ p| ≥ (x; p) U0 N0 f x; p∈> × ( ) 0 Then there exist neighborhoods V of x0 and N of p0 such that [fp 0] = for all p N, and for all › 0, ≤ 6 ∅ ∈ ≥ … d(x; [f ›]) (f (x) ›)+ for all (x; p) V N: p≤ ≤ p − ∈ × Proof. Let ” > 0 be such that B4ρ(x0) U0. From assumption c), we can find N (p0) N N ⊂ f x < …” U B x ∈ N such that 0 and supp N infx Bρ(x ) p( ) . Setting = ρ( 0), we get, for all ∈ 0 p N that U⊂ [f <…”] = ∈and ∈ ∩ p 6 ∅ inf fp (x) … for all p N: x Bρ(U) [0 Thus we obtain from Theorem 2.11 that for all p N we have [f 0] = and ∈ p≤ 6 ∅ … d(x; [f 0]) f (x)+ for all x U [f <…”] and p N; p≤ ≤ p ∈ ∩ p ∈ yielding [f 0] B (x ) = for all p N. p≤ ∩ 2ρ 0 6 ∅ ∈ Now we claim that … d(x; [f ›]) (f (x) ›)+ for all x V := B (x ) for all p N and p≤ ≤ p − ∈ ρ 0 ∈ for all › 0. If not, we can find x Bρ(x0), p N and › 0 such that 0 < fp(x) › < … d x; f ≥› r d x; f∈ › ∈ < r < ≥” g x < g− …r ( [ p ]). Setting = ( [ p ]), we get 0 3 and ( ) infB¯r(x) + where g =≤ (f ›)+. From Corollary≤ 2.3, there exists y B (x) such that g (y) < …. As p − ∈ r |∇ | d(y; x) < d(x; [fp ›]), we get fp(y) > ›, so that fp (y) = g (y) < … and y B4ρ(x0), a contradiction. ≤ |∇ | |∇ | ∈ Remark 2.14. Theorem 2.13 fails to be true without epi-upper semicontinuity of f at x ; p f ( 0 0) as shown by the following example. Let : R R R be defined by × −→ ex p f x; p if = 0 ( ) = + 6 (x if p = 0; so that f is lower semicontinuous on R R and f(0; 0) = 0. For all p R and for all × ∈1 x [f >0] ( 1; 1), we derive from (2) that f (x) > r with r = e− . Thus we get ∈ p ∩ − |∇ p| p x f > ; = @Ýεf x from (5) that for all R and for all [ p 0] ( 1 1) we have 0 p( ) for all ε ∈ ∈ ∩ − ∈ " (0; r) where @Ý fp(x) is the "-Fr´echet subdifferential of fp at x. However [fp 0] = for∈ all p = 0 and then f provides a counterexample to [15, Theorem 2]. ≤ ∅ 6 D. Az´e/ A Unified Theory for Metric Regularity of Multifunctions 235 In the previous theorem, we saw that the fact for the slope to be bounded away from 0 on the complement of a level set in a neighborhood of some point yields a local error bound result for all greater level sets. The powerfulness of the strong slope lies in the fact that the converse is true as shown in the following proposition which is [9, Remark 3.2]. Proposition 2.15. Let f : X R + be a function defined on a metric space X, → ∪ { ∞} U be a subset of X, and « R, ‹ R + with « < ‹. Assume that, for some … > 0 we have ∈ ∈ ∪ { ∞} … d(x; [f ›]) (f(x) ›)+ for all › « and for all x U [f<‹]: ≤ ≤ − ≥ ∈ ∩ Then inf f …: U [α 1 … d(x; F − (y)) d(y; F (x)) for all (x; y) W: (11) ≤ ∈ We shall need the following definition 236 D. Az´e/ A Unified Theory for Metric Regularity of Multifunctions z X d X d x d z; x Definition 3.1. For , let z : R be defined by z( ) := ( ). We say that the metric space X is coherent∈ at z X→if ∈ d (x) = 1 for every x = z : |∇ z| 6 If this is true for every z X, we just say that X is coherent. ∈ Clearly, if X is a convex subset of a normed vector space, then X (with the metric associated with the norm) is coherent. In general, since dz is 1-Lipschitzian, so that d 1, the fact that a complete metric space X is coherent at z is equivalent to: |∇ z| ≤ (d(z; x) ›)+ d(x; B¯ (z)) for all x X and for all › 0 ; (12) − ≥ γ ∈ ≥ according to [8, Theorem 2.4]. We now consider two metric spaces X and Y (we shall use the same notation d for both metrics), and, for fi > 0, the product space X Y as endowed with the metric: × d ((x; y); (x0; y0)) := max d(x; x0); fid(y; y0) : δ { } Accordingly, if f : X Y R + is a lower semicontinuous function, we shall let × → ∪ { ∞} δf denote the strong slope of f with respect to the metric dδ. The next theorem is a slight|∇ | variant of [10, Theorem 5.3]. Theorem 3.2. Let X and Y be complete metric spaces, F X Y be a closed multi- ⊂ × function, (x0; y0) F , let Z be a subset of Y , let V and W be neighborhoods of x0 and y , respectively, »∈ > 0, and 0 < fi 1=». 0 ≤ (a) Assume that: f (x; y) » for all (x; y; z) V W Z ; y = z ; |∇δ z| ≥ ∈ × × 6 where fz(x; y) = d(z; y) + iF (x; y). Then, there exists " > 0 such that: 1 d(z; F (x)) »d(x; F − (z)) for all (x; z) B (x ) (B (y ) Z) ; ≥ ∈ ε 0 × ε 0 ∩ (in particular B (y ) Z F (X)). ε 0 ∩ ⊂ (b) Conversely, assume that Y is coherent and that: 1 d(z; F (x)) »d(x; F − (z)) for all (x; z) V W: (13) ≥ ∈ × Then, there exists r > 0 such that f (x; y) » for all (x; y; z) B (x ) B (y ) B (y ) ; y = z : |∇δ z| ≥ ∈ r 0 × r 0 × r 0 6 Proof. (a) Let ” > 0 be such that B (x ; y ) V W , and let z B (y ) Z W . 2ρ 0 0 ⊂ × ∈ σρ 0 ∩ ⊂ Applying Theorem 2.11 to fz with U := Bρ(x0; y0), « := 0, and the given » and ”, since (x0; y0) U [fz<»”] and δfz (x; y) » for every (x; y) [fz>0] Bρ(U), we obtain that, for∈ all z∩ B (y ) Z|∇ W|: ≥ ∈ ∩ ∈ σρ 0 ∩ ⊂ f (x; y) »d ((x; y); [f 0]) for all (x; y) [f <»”] B (x ; y ) : (14) z ≥ δ z≤ ∈ z ∩ ρ 0 0 D. Az´e/ A Unified Theory for Metric Regularity of Multifunctions 237 Let 0 < r ” be such that Br(x0; y0) F [fz<»”] for every z Bσr(y0). Then, taking the definitions≤ of f and of d , the fact∩ that⊂»fi 1, and (10), into∈ account, (14) yields: z δ ≤ 1 d(z; y) »d(x; F − (z)) for all (x; y) B (x ; y ) F; for all z B (y ) Z; (15) ≥ ∈ r 0 0 ∩ ∈ σr 0 ∩ and in particular: 1 d(z; y ) »d(x ;F − (z)) for all z B (y ) Z: (16) 0 ≥ 0 ∈ σr 0 ∩ Assume now that the conclusion does not hold. Then, there exist sequences (xn; yn) F and (z ) Z such that d(x ; x ) 0, d(z ; y ) 0, and: ⊂ n ⊂ n 0 → n 0 → 1 d(zn; yn) < »d(xn;F − (zn)) : (17) Taking (16) into account yields: 1 d(y ; y ) d(z ; y ) < »d(x ; x ) + »d(x ;F − (z )) »d(x ; x ) + d(z ; y ) ; n 0 − n 0 n 0 0 n ≤ n 0 n 0 which shows that d(y ; y ) 0, so that (17) contradicts (15) for large n. n 0 → (b) Let r > 0 be such that Br(x0) V and B3r(y0) W . Let z Br(y0) be fixed, let then › 0 and (x; y) F (B (x )⊂ B (y )) [f >›⊂], so that › <∈ d(z; y) 2r. Then, ≥ ∈ ∩ r 0 × r 0 ∩ z ≤ for every y0 B¯ (z) V , we have: ∈ γ ⊂ 1 d(y0; y) »d(x; F − (y0)) ≥ 1 (in particular, F − (y0) = for every such y0), hence: 6 ∅ 1 d(y0; y) »d ((x; y);F − (y0) y0 ) ; ≥ δ × { } and finally, since y0 is arbitrary in B¯γ(z): d(y; B¯ (z)) »d ((x; y); [f ›]) : γ ≥ δ z≤ According to (12), we have: f (x; y) › »d ((x; y); [f ›]) : z − ≥ δ z≤ Letting U := B (x ) B (y ), we thus have: r 0 × r 0 f (x; y) › inf inf z − »; γ>0 (x,y) U [fz>γ] d ((x; y); [f ›]) ≥ ∈ ∩ δ z≤ and the conclusion follows from Proposition 2.15 applied to f with « := 0 and ‹ := + , z ∞ since z is arbitrary in Br(y0). 4. Metric regularity in Banach spaces In this section we survey a large selection of results on metric regularity and we show how they enter in the general framework described in Section 3. The following result is the well-known Ursescu-Robinson Theorem ([57, 54]). 238 D. Az´e/ A Unified Theory for Metric Regularity of Multifunctions 4.1. The Ursescu-Robinson Theorem Let us recall that the core of a convex set C X is the set core(C) of those x C such that X = (0; + )(C x). ⊂ ∈ ∞ − Theorem 4.1. Let F X Y be a closed convex multifunction where X, Y are Banach spaces. Assume that y ⊂ core× F (X), then F is metrically regular near any (x ; y ) F . 0 ∈ 0 0 ∈ z Y f X Y Proof. Let us consider, for , the closed convex function z : R + ∈ × 7−→ ∪ { ∞} defined by fz(x; y) = z y + iF (x; y) . As y0 core F (X), Baire’s Theorem yields r1, r > B¯ ky − k F B¯ x ∈< » < r =r fi r =r < =» 2 0 such that r1 ( 0) cl ( ( r2 ( 0)). Let 0 1 2, let = 2 1 1 and 1 ⊂ let " = (1 + »)− (r r ») (0; r ). Let (x; y) [f >0] (B (x ) B (y )) F and let 1 − 2 ∈ 1 ∈ z ∩ ε 0 × ε 0 ∩ (‚; ) @fz(x; y) so that (‚; ) = (‚2; 1 + 2) with 1 = 1 and (‚2; 2) NF (x; y). As B¯ ∈y B¯ y v r " B¯k k∗ x ; y ∈ F r1 ε( ) r1 ( 0) we can find, for any ( 1 ) Y , a sequence (( n n))n N such − y ⊂ y v x ∈ −B¯ x ∈ ⊂ X Y that ( n)n N converges to and ( n)n N r2 ( 0). Now we get, endowing ∈ ∈ with the norm (u; v) = max(− u ; fi v ), ⊂ × k kδ k k k k (x xn; y yn) δ (‚; ) ‚2; x xn + 1; y yn + 2; y yn 1; y yn ; k − − k k k∗ ≥ h − i h − i h − i ≥ h − i r " 1 » yielding 1; v (r2 + ") (‚; ) and then (‚; ) − = . Thus we derive from h i ≤ k k∗ k k∗ ≥ r2 + " (9) that, for all z Y , ∈ inf δfz »; [fz>0] (Bε(x0) Bε(x0)) |∇ | ≥ ∩ × and then the conclusion of the theorem follows from Theorem 3.2 (a) applied with Z = Y . 4.2. Normal conditions Given a multifunction F X Y and a subdifferential operator @, the coderivative ⊂ × D∗F (x; y) at a point (x; y) F is the multifunction D∗F (x; y) Y ∗ X∗ defined by ∈ ⊂ × D∗F (x; y) = ( ; ‚) Y ∗ X∗ :(‚; ) NF (x; y) where NF (x; y) = @iF (x; y) is the { ∈ × − ∈ } @ S normal cone associated to the subdifferential operator . We shall further denote by Y ∗ the unit sphere in Y ∗. Given a metrix space Z, a subset S Z and z cl S, the notation ⊂ 0 ∈ z S z will mean z goes to z in S. → 0 0 4.2.1. Characterization using coderivatives The following basic result is a mixing of the main results of [33, Section 6] and [47]. Theorem 4.2. Let X, Y be Banach spaces, let @ be a subdifferential satisfying (P 1) and (P 2) on X Y and let F X Y be a closed multifunction. Assume that × ⊂ × d ;D F x; y S > … > : lim inf (0 ∗ ( )( Y ∗ )) 0 F ∗ (x,y) (x ,y ) → 0 0 Then there exists a neighborhood W of (x0; y0) such that 1 … d(x; F − (y)) d(y; F (x)) for all (x; y) W: (18) ≤ ∈ Conversely, assuming that (18) holds true, then d ;D F x; y S …; lim inf (0 ∗ ( )( Y ∗ )) F ∗ (x,y) (x ,y ) ≥ → 0 0 D. Az´e/ A Unified Theory for Metric Regularity of Multifunctions 239 whenever @ is the Fr´echetsubdifferential or the s-H¨older-smooth subdifferential or the Dini subdifferential where X and Y are finite dimensional. Proof. Let us set as usual fz(x; y) = z y +iF (x; y) = g(x; y)+iF (x; y) and let us endow k − k 1 X Y with the norm (x; y) = max( x ;… − y ) whose dual norm is (‚; ) = ‚ + × k k k k k k k k∗ k k∗ … . We can find an open neighborhood W0 of (x0; y0) such that d (0;D∗F (x; y)(SY )) > k k∗ ∗ ∗ … for all (x; y) W0 F . Let z Y and (x; y) W0 F [fz>0] and let (x1; y1), (x ; y ) W ∈F , ‚ ∩ @g(x ; y ),∈ (‚ ; ) N (x∈; y ).∩ Assume∩ that (x ; y ) is close 2 2 ∈ 0 ∩ 1 ∈ 1 1 2 2 ∈ F 2 2 1 1 enough to (x; y) in order that z y1 > 0. We have ‚1 = (0; 1) with 1 = 1 so that k − k k k∗ (0; 1) + (‚2; 2) = ‚2 + … 1 + 2 ‚2 … 2 + … … k k∗ k k∗ k k∗ ≥ k k∗ − k k∗ ≥ since ‚ D∗F (x ; y )( ). Thus we derive from Proposition 2.6 that f (x; y) … 2 ∈ 2 2 − 2 |∇ z| ≥ for all (x; y) W0 [fz>0] and then the conclusion of the theorem follows from Theorem 3.2. ∈ ∩ Assume now that (18) holds true. We derive from Theorem 3.2 that there exists r > 0 such that δfz (x; y) … for all (x; y; z) Br(x0) Br(y0) Br(y0) such that fz(x; y) > 0 with fi = 1|∇=…. For| any≥ of the three subdifferentials∈ × involved,× we have @g(x; y)+@i (x; y) F ⊂ @f (x; y). Now let (x; y) F (B (x ) B (y )), let S and let ‚ D∗F (x; y)( ), z ∈ ∩ r 0 × r 0 ∈ Y ∗ ∈ so that (‚; ) NF (x; y). Given " > 0, we can find by the Bishop-Phelps Theorem − ∈ ¿ S v S w Y w (see e.g. [19, Theorem 7.2]) Y ∗ and Y = : = 1 such that 1 ∈ ∈ { ∈ k k } ¿ … − " and ¿; v = 1. Let t > 0 small enough in order that z = y tv Br(y0). ∗ kWe− havek ≤ (0;¿) @g(x;h y)i so that (‚;¿ ) @f (x; y) and then, by Proposition− ∈ 2.9, ∈ − ∈ z (‚;¿ ) = ‚ + … ¿ δfz (x; y) …; k − k∗ k k∗ k − k∗ ≥ |∇ | ≥ from which we get ‚ … " and then ‚ … by letting " go to 0. k k∗ ≥ − k k∗ ≥ 4.2.2. The Borwein-Zhu sufficient condition Given a metric space X, a subset A X and a mapping h defined near x A with values ⊂ 0 ∈ in a metric space space Y , we say that (h; A) is metrically regular near x0 whenever there 1 exist … > 0 and neighborhoods V of x0 and W of y0 = h(x0) such that … d(x; h− (y) A) d(y; h(x)) for all (x; y) (V A) W . This amounts to say that the multifunction∩ ≤ (x; h(x)) : x A is metrically∈ ∩ regular× near (x ; y ). { ∈ } 0 0 Lemma 4.3. Let X be a Banach space, let A X be a closed set and let h be a continuous mapping defined near x A with values in a⊂ normed space Y . Then (h; A) is metrically 0 ∈ regular near x0 if and only if there exist neighborhoods V of x0 and W of y0 such that inf fy (x) > 0; (19) (x; y) (V A) W |∇ | ∈ ∩ × h(x) = y 6 where f (x) = h(x) y + i (x). y k − k A 1 Proof. Observing that [f 0] = h− (y) A and that f(x; ) is continuous for all x X, we y≤ ∩ · ∈ derive from Theorem 2.13 that (h; A) is metrically regular near x0. Conversely, assume that (h; A) is metrically regular near x0, so that we can find r > 0, ” > 0 such that 240 D. Az´e/ A Unified Theory for Metric Regularity of Multifunctions 1 … d(x; h− (y) A) h(x) y all (x; y) (B (x ) A) B (y ). Let V B (x ) and ∩ ≤ k − k ∈ r 0 ∩ × 2ρ 0 ⊂ r 0 W B2ρ(y0) be such that h(x) y < ” for all (x; y) V W and let (x; y) (V A) W ⊂ k − k 1∈ × ∈ ∩ × be such that h(x) = y. Setting ›n = fy(x) n− , we get ›n > 0, fy(x) > ›n and 1 6 − … n− > 0 eventually. Thus there exist † (0; 1) and z Y such that z y = › − n ∈ n ∈ k n − k n and zn = y + †n(h(x) y), from which we get zn B2ρ(y0). As h(x) = zn, we can find 1 − ∈ 6 x h− (z ) A such that n ∈ n ∩ 1 (… n− ) x x z h(x) = (1 † ) h(x) y = f (x) f (x ) = f (x) › : − k − nk ≤ k n − k − n k − k y − y n y − n x x It follows that ( n)n N converges to and that ∈ fy(x) fy(xn) … lim sup − fy (x); ≤ n x xn ≤ |∇ | →∞ k − k yielding the conclusion of the lemma. Theorem 4.4. Let X, Y be Banach spaces with ‹-smooth norms, let A X be a closed set, and let h be defined and continuous near x A. Assume that h is strictly⊂ differen- 0 ∈ tiable at x0 and that (h; A) is not metrically regular near x0. Then d ; S N β x lim inf (0 Λ∗( Y ∗ ) + A( )) = 0 (20) A ∗ x x → 0 where Λ = Dh(x0) and where SY = Y ∗ : = 1 is the unit sphere in Y ∗. ∗ { ∈ k k∗ } Proof. Let " > 0, fi > 0 and let 0 < – < fi such that h Λ is "-Lipschitzian on B (x ). From Lemma 4.3, we can find x B (x ) A [f >−0] and y B (y ) such 2η 0 ∈ η 0 ∩ ∩ y ∈ η 0 that fy (x) < ", so that the function fy( ) + " y admits a local minimum at x. Setting|∇g(|z; y) = h(z) y and applying [13,· Theoremk − ·k 2.9], we can find x B (x ), k − k 1 ∈ η 0 x B (x ) A and ‚ Dβg (x ), ‚ N β(x ) such that h(x ) y > 0 and 2 ∈ η 0 ∩ 1 ∈ y 1 2 ∈ A 2 k 1 − k ‚1 + ‚2 ". Let u B¯X so that we can find tu > 0 such that k k∗ ≤ ∈ 1 t− ( h(x + tu) y h(x ) y ) ‚ ; u " ‚ ; u 2" for all t (0; t ): k 1 − k − k 1 − k ≥ h 1 i − ≥ −h 2 i − ∈ u Let us set (u) = β( )(h(x +tu) y) which is well defined for all t > 0 small enough. t ∇ k·k 1 − As t(u) = 1, we have for all t > 0 small enough k k∗ 1 (u); t− (h(x + tu) y (h(x ) y)) Λ(u) "; h t 1 − − 1 − − i ≤ 1 hence t− ( (h(x1 + tu) y ( (h(x1) y ) t(u); Λ(u) + " for all t (0; tu). As k − k − kβ − k ≤ h i ∈ t(u) weakly converges to = ( )(h(x1) y), we get Λ∗( ) + ‚2; u 3" for all u B¯ ‚ ∇ k ·" k − d ; S h N β x i ≥ −" X yielding Λ∗( ) + 2 3 and then (0 Λ∗( Y ∗ ) + A( 2)) 3 , so that ∈ k βk∗ ≤ ∗ ≤ lim inf A d (0; Λ∗(S ) + N (x)) = 0. x x Y ∗ A → 0 ∗ Remark 4.5. In [13, Theorem 4.3] the preceding result is obtained with (20) replaced d ; S N˜ β x N˜ β x LDβd x N β x by lim inf A (0 Λ∗( Y ∗ ) + A( )) = 0 with A( ) = L>0 A( ). As A( ) x x0 ∗ ⊂ N˜ β x → A( ), Theorem 4.4 extends the quoted result. S D. Az´e/ A Unified Theory for Metric Regularity of Multifunctions 241 4.2.3. Pointwise criteria Given a subdifferential operator @ and a multifunction F X Y , we say that F is @- x ; y F ⊂ × x ; y F codirectionally compact at ( 0 0) whenever for any sequence (( n n))n N con- x ; y ∈ ‚ ; X Y ‚ D F∈ x⊂; y verging to ( 0 0) and for any sequence (( n n))n N ∗ ∗ with n ∗ ( n n)( n) ‚ ∈ ⊂ × ∈ such that ( n)n N converges to 0, then ( n)n N converges to 0 whenever it -weakly con- ∈ ∈ verges to 0. It is clear that any closed multifunction is @-codirectionally compact∗ at each (x0; y0) F for any subdifferential operator whenever Y is finite dimensional. The next theorem∈ captures the main results of [46, 48, 53]. Theorem 4.6. Let X, Y be Banach spaces and let F X Y be a closed multifunction which is @Ý-codirectionally compact at (x ; y ) F where⊂ @Ý is× the Fr´echetsubdifferential. 0 0 ∈ a) Assume that X and Y are Asplund and that Ker D∗F (x ; y ) = 0 ; 0 0 { } where D∗F (x0; y0) is the Mordukhovich (limiting Fr´echet)coderivative and Ker D∗F (x ; y ) = Y ∗ : 0 D∗F (x ; y )( ) : 0 0 { ∈ ∈ 0 0 } Then F is metrically regular near (x0; y0). b) Conversely, if X is finite dimensional and if F is metrically regular near (x0; y0), then Ker D∗F (x ; y ) = 0 . 0 0 { } Proof. a) We claim that lim inf F d (0; DÝ ∗F (x; y)(S )) > 0 where DÝ ∗F (x; y) (x,y) (x ,y ) Y ∗ → 0 0 ∗ stands for the Fr´echet coderivative, which will give the conclusion of the theorem by x ; y F Theorem 4.2. Indeed, if not, we can find sequences (( n n))n N converging to x ; y ‚ ; X Y S ‚ DÝ F∈ x⊂; y ( 0 0) and (( n n))n N ∗ ∗ with ( n)n N Y ∗ , n ∗ ( n n)( n) such that ‚ ∈ ⊂ × ∈ ⊂ ∈ ( n)n N converges to 0. Denoting also by ( n)n N a subsequence that -weakly converges ∈ Y D F x ; y ∈ ∗ to some ∗, we get Ker ∗ ( 0 0) thus = 0. It follows that ( n)n N converges ∈ ∈ ∈ to 0 contradicting the fact that n 1. k k∗ ≡ b D F x ; y S ) Conversely, assume that Ker ∗ ( 0 0) = 0 , so that there exists Y ∗ such that ; N x ; y 6 { } x ; y F ∈ x ; y (0 ) F ( 0 0). Thus we can find sequences (( n n))n N converging to ( 0 0) ∈‚ ; X Y ‚ ∈ ⊂ and (( n n))n N ∗ ∗ such that ( n)n N converges to 0 and ( n)n N - weakly con- ∈ ⊂ × ∈ ‚ ; ∈ ∗ verges to . Considering a subsequence still denoted by (( n n))n N such that ( n )n N ∈ k 1 k∗ ∈ converges to some c > 0, we derive that (‚Ýn; Ýn) NÝF (xn; yn) where ‚Ýn = n − ‚n and Ý 1 S ‚Ý ∈ d k ; kDÝ F x; y n = n − n Y ∗ . As ( n) N converges to 0, we get lim inf F (0 ∗ ( ) k k ∈ ∈ (x,y) (x0,y0) ∗ S F x ; y → ( Y ∗ )) = 0 thus is not metrically regular near ( 0 0) by applying again Theorem 4.2. As a consequence, we derive the nice characterization given by Mordukhovich in [46]. Theorem 4.7. Let X, Y be finite dimensional spaces and let F X Y be a closed multifunction. Then F is metrically regular near (x ; y ) F if and⊂ only× if 0 0 ∈ Ker D∗F (x ; y ) = 0 ; 0 0 { } where D∗F (x0; y0) is the Mordukhovich (limiting Fr´echet)coderivative. Proof. Follows immediately from Theorem 4.6 since finite dimensional spaces are As- plund and each closed multifunction F X Y is @-codirectionally compact at each (x ; y ) F for any subdifferential operator.⊂ × 0 0 ∈ 242 D. Az´e/ A Unified Theory for Metric Regularity of Multifunctions 4.3. Tangencial conditions Let us list the definitions of the tangent cones and directional derivatives we shall use in this section. Given a subset C X of a normed space and " > 0, the "-contingent C x C T⊂ε x u X cone to at 0 cl ( ) is the set C ( 0) of those such that there exist sequences u B¯ u; "∈ u t ; ∈ x t u C ( n)n N ( ) and ( n)n N (0 + ) converging to 0 such that 0 + n n ∈ ⊂ k k ∈ ⊂ ∞ ∈ for all n N. We also need a notion of "-directional derivative namely, given f : X ∈ " > x f u X → R + , 0, 0 dom and : ∪ { ∞} ∈ ∈ f x tv f x f x u ( 0 + ) ( 0) : ε0( 0; ) = lim inf inf − t 0 v B¯(u,ε u ) t ↓ ° ∈ k k ± i x i ε Observe that these two notions depend on the norm and that ( C )ε0 ( 0; ) = TC (x0). Then, it is clear that · (1 + ") u f (x ) f 0(x ; u): (21) k k|∇ | 0 ≥ − ε 0 Moreover, it is also clear that f 0(x ; u) g0(x ; u) + K" u + h0 (x ; u) (22) ε 0 ≤ 0 k k ε 0 f g h g X K x whenever = + with : R convex and Lipschitz continuous of rank near 0. → The contingent cone (or Bouligand cone, see [16, p. 32]) to C at x0 cl (C) is the set T x u X u ∈ u t C ( 0) of for which there exist sequences ( n)n N converging to and ( n)n N ; ∈ x t u C ∈ n T x∈ ⊂ (0 + ) converging to 0 such that 0 + n n for all N. In other words C ( 0) = ∞ ∈ ∈ f x tv f x T ε x i x i f x u ( + ) ( ) ε>0 C ( 0) and we have C0 ( 0; ) = TC (x0) where 0( ; ) := lim inf − . t 0 t · v&u T → Given a subset C X of a normed space, the Clarke tangent cone to C at x0 cl (C) is ⊂ T x ∈ the closed convex cone C↑ ( 0) defined by T x u X t 1d x tu; C : C↑ ( 0) = : lim − ( + ) = 0 { ∈ C } x x0 → t 0 ↓ 4.3.1. The Borwein-Aubin-Frankowska Theorem The next theorem which is [3, Theorem 2.3] is also a consequence of [11, Theorem 3.1]. Theorem 4.8. Let X, Y be Banach spaces such that Y is finite dimensional, let F ⊂ X Y be a closed multifunction, and let (x ; y ) F . Assume that T ↑ (x ; y )(X) = Y . × 0 0 ∈ F 0 0 Then F is metrically regular near (x0; y0). Y B¯ v ; ; v N Proof. Let us endow with a norm such that Y = co ( 1 N ) for some N and ··· ∈ some v1; ; vN Y . From our assumption, we can find … > 0 and u1; ; uN X such u ;··· v T ∈x ; y u … 1 v i ; ;N ··· X ∈ Y that ( i i) F↑ ( 0 0) and i − i , = 1 . Let us endow with ∈ k 1k ≤ k k ··· × the norm (x; y) = max( x ;… − y ) and let us set k k k k k k N K u ; v ; ; u ; v u ; v = pos (( 1 1) ( N N )) = R+( i i) ··· i=1 X D. Az´e/ A Unified Theory for Metric Regularity of Multifunctions 243 K T x ; y …B¯ K B¯ so that is a closed convex cone contained in F↑ ( 0 0) and Y ( X ). Given " (0; 1), we can find, by Ascoli’s Theorem and by the definition of the⊂ Clarke tangent ∈ 1 cone, a neighborhood W of (x ; y ) and – > 0 such that d((u; v); t− (F (x; y)) < " for 0 0 − all (u; v) K B¯X Y , for all (x; y) F W and for all t (0;–). For z Y , let us ∈ ∩ × ∈f ∩X Y ∈ ∈ define a lower semicontinuous function z : R + by × −→ ∪ { ∞} f (x; y) = z y + i (x; y); z k − k F z y and let z Y and let (x; y) F W be such that f (x; y) > 0. Setting v = … − , ∈ ∈ ∩ z y z k − k we can find u B¯X such that (u; v) K B¯X Y . Thus there exists, for all t (0;–) a × vector (u ; v ) ∈ X Y such that (x; y∈) + t∩(u ; v ) F and (u ; v ) (u; v) <∈ ", so that t t ∈ × t t ∈ k t t − k f ((x; y) + t(u ; v )) f (x; y) y z + tv y z + t…" t… + t…"; z t t − z ≤ k − k − k − k ≤ − yielding (1 + ") fz (x; y) …(1 "). Thus the conclusion of the theorem follows from Theorem 3.2 applied|∇ | with Z≥= Y .− In fact the Borwein-Aubin-Frankowska Theorem can be deduced from a more general result exposed in the next subsection. 4.3.2. Equi-circatangency The results of this subsection are taken from [6]. Definition 4.9. Let X be a Banach space and let C X be a subset of X. We say that a cone K is equi-circatangent to C at x cl (C) whenever⊂ 0 ∈ C x e K B¯ ; : lim X −t = 0 x C x ∩ 0 ° ± → t 0 ↓ Thus if K is equi-circatangent to C at x we have K T ↑ (x ). 0 ⊂ C 0 Theorem 4.10. Let X, Y be Banach spaces, let F X Y be a closed multifunction, and let (x ; y ) F . Assume that there exists a cone⊂ K × X Y such that K is equi- 0 0 ∈ ⊂ × circatangent to F at (x0; y0) and there exists … > 0 such that … B¯Y K(B¯X ). Then for all » (0;…), there exists a neighborhood W of (x ; y ) such that ⊂ ∈ 0 0 1 » d(x; F − (y)) d(y; F (x)) for all (x; y) W: ≤ ∈ z Y f X Y Proof. For , let us define a lower semicontinuous function z : R + by ∈ × −→ ∪{ ∞} f (x; y) = z y + i (x; y): z k − k F 1 Let us endow X Y with the norm (x; y) = max( x ;… − y ) and let " (0; 1) be 1× " k k k k k k ∈ such that » … − . We can find a neighborhood W of (x ; y ) and – > 0 such that ≤ 1 + " 0 0 1 d((u; v); t− (F (x; y))) < " for all (u; v) K B¯X Y ,(x; y) F W and t (0;–). Now − ∈ ∩ × ∈ ∩ ∈ 1 let z Y and let (x; y) F W be such that f (x; y) > 0. Setting v = … y z − (z y), ∈ ∈ ∩ z k − k − 244 D. Az´e/ A Unified Theory for Metric Regularity of Multifunctions we can find u B¯X such that (u; v) K. As (u; v) B¯X Y , there exists, for all t (0;–) ∈ ∈ ∈ × ∈ a vector (ut; vt) X Y such that (x; y) + t(ut; vt) F and (ut; vt) (u; v) < ", so that ∈ × ∈ k − k f ((x; y) + t(u ; v )) f (x; y) y z + tv y z + t…" t… + t…"; z t t − z ≤ k − k − k − k ≤ − yielding (1 + ") fz (x; y) …(1 ") and then fz (x; y) ». Thus the conclusion of the theorem follows|∇ | from Theorem≥ − 3.2 applied with|∇ Z| = Y .≥ Remark 4.11. a) By the Ursescu-Robinson theorem, the assumptions of Theorem 4.10 are fulfilled whenever the equi-circatangent cone K is closed and convex and satisfies K(X) = Y . b) In [5], Aubin and Frankowska define a closed set C to be uniformly sleek at x A if 0 ∈ lim C e(T (x ) B¯ ;T (x)) = 0. In that case the contingent and the Clarke tangent x x C 0 X C → 0 ∩ cone at x0 do coincide. One easily checks (see [6, Proposition 2.4]) that, assuming C to x T x C be uniformly sleek at 0, the Clarke tangent cone C↑ ( 0) is then equi-circatangent to at x . Nevertheless, there exist (see [6]) closed sets C x whose Clarke tangent cone is 0 3 0 equi-circatangent to C at x0 which are not uniformly sleek at x0. It follows that Theorem 4.10 extends the results of [5] based on uniform sleekness. 4.3.3. "-Contingent cone The next theorem is a slight improvement of [7, Theorem 3.2]. Theorem 4.12. Let X, Y be Banach spaces, let F X Y be a closed multifunction, ⊂ × and let (x0; y0) F . Assume that there exist " > 0, … > 0, a neighborhood W0 of (x0; y0) and › 0 such∈ that … › "(… + ›) > 0 and ≥ − − …B¯ T ε (x; y)(B¯ ) + ›B¯ for all (x; y) W F: (23) Y ⊂ F X Y ∈ 0 ∩ Then there exists a neighborhood W of (x0; y0) such that 1 » d(x; F − (y)) d(y; F (x)) for all (x; y) W; ≤ ∈ … › "(… + ›) where » = − − > 0. " … 1› (1 + )(1 + − ) 1 Proof. Let us endow X Y with the norm (x; y) = max( u ;… − v ), and let us z Y × k f Xk Y k k k k f x; y define, for a lower semicontinuous function z : R + by z( ) = ∈ × −→ ∪ { ∞} z y + iF (x; y). Now let z Y and let (x; y) W0 F such that fz(x; y) > 0. Setting vk −… ky z 1 z y ∈ u; w T ε x;∈ y ∩ u w v › = − ( ), we can find ( ) F ( ) such that 1 and . Relyingk − on (22)k and− observing that (x; y) ∈ z y is …-Lipschitzian,k k ≤ wek get− k ≤ 7−→ k − k f x; y ; u; w y z w …" u; w i x; y ; u; w ( z)ε0 (( ) ( )) 0( ; ) + ( ( ) + ( F )ε0 (( ) ( )) ≤ k · k − k 1k 0(y; v) + › + …"(1 + … − ›); ≤ k · k f x; y ; u; w … › " … › f x; y thus ( z)ε0 (( ) ( )) + + ( + ), from which we get by (21) that z ( ) ». Thus the conclusion of≤ the − theorem follows from Theorem 3.2 applied with|∇Z =| Y . ≥ A very slight modification of the proof of Theorem 4.12 yields [2, Theorem 2]: D. Az´e/ A Unified Theory for Metric Regularity of Multifunctions 245 Theorem 4.13. Let X, Y be Banach spaces, let F X Y be a closed multifunction, ⊂ × and let (x0; y0) F . Assume that there exist … > 0, a neighborhood W0 of (x0; y0) and › [0;…) such that∈ ∈ …B¯ T (x; y)(B¯ ) + ›B¯ for all (x; y) W F: (24) Y ⊂ F X Y ∈ 0 ∩ Then there exists a neighborhood W of (x0; y0) such that … › 1 − d(x; F − (y)) d(y; F (x)) for all (x; y) W: … 1› 1 + − ≤ ∈ … › Proof. For all " 0; − , and for all z Y we have (23) with W independent of ". ∈ … + › ∈ 0 For all (x; y) W ° [f >0], we± know from the proof of Theorem 4.12 that f (x; y) ∈ 0 ∩ z |∇ z| ≥ … › "(… + ›) … › − − . Letting " decrease to 0, we derive that f (x; y) − , hence " … 1› z … 1› (1 + )(1 + − ) |∇ | ≥ 1 + − the conclusion of the theorem follows by applying Theorem 3.2 with Z = Y . Remark 4.14. a) One can have › = 0 in (24). It follows that if, for some … > 0 and some neighborhood W0 of (x0; y0) we have …B¯ T (x; y)(B¯ ) for all (x; y) W F: (25) Y ⊂ F X ∈ 0 ∩ 1 Then there exists a neighborhood W of (x0; y0) such that … d(x; F − (y)) d(y; F (x)) for all (x; y) W . Conversely, assuming that F is metrically regular near (≤x ; y ), then ∈ 0 0 there exists … > 0, – > 0 and neighborhoods U of x0 and V of y0 such that for all (x; y) (U V ) F , v B¯Y and for all t (0;–), we have y + …tv F (x + tut) for some u B¯∈ , from× which∩ we∈ get ∈ ∈ t ∈ X 1 …B¯ t− (F (x + tB¯ ) y) for all (x; y) (U V ) F: (26) Y ⊂ X − ∈ × ∩ t (0,η) ∈\ Then for all (x; y) (U V ) F , v …B¯ and for t > 0 small enough, there exists u B¯ ∈ × ∩ ∈ t ∈ such that (x + tut; y + tv) F . It follows that assuming Y to be finite dimensional, there exists (u; v) T (x; y) with∈ u 1, thus condition (25) is in fact a characterization of ∈ F k k ≤ metric regularity of F near (x0; y0). b) Assume that assumption (25) is replaced by the weaker one: there exists some … > 0 and some neighborhood W0 of (x0; y0) such that …B¯ (co T (x; y))(B¯ ) for all (x; y) W F: (27) Y ⊂ F X ∈ 0 ∩ Then, it is shown in [23] that F is metrically regular near (x0; y0) whenever X and Y are finite dimensional. In fact this conclusion still holds under the weaker condition that X and Y are Asplund spaces. Indeed it is easily shown (see e.g. [53, Lemma 3.1]) that (27) leads to ‚ … for all (‚; ) TF (x; y)− and for all (x; y) W0 where TF (x; y)− k k∗ ≥ k k∗ ∈ ∈ denotes the negative polar cone of TF (x; y). As the Fr´echet normal cone NÝF (x; y) is contained in TF (x; y)−, it then follows from Theorem 4.2 that, assuming X and Y to be Asplund spaces, then the multifunction F is metrically regular near (x0; y0). 246 D. Az´e/ A Unified Theory for Metric Regularity of Multifunctions c) Assume that there exist … > 0 and a neighborhood W of (x0; y0) such that F (x + tB¯X ) y …B¯Y Lim sup − for all (x; y) W F; (28) ⊂ t 0 t ∈ ∩ ↓ S S X Y u; v where Lim supt 0 t = η>0 cl t (0,η) t . Let us endow with the norm ( ) = ↓ ∈ × k k u ;… 1 v z Y f x; y z y i x; y max( − ) andT let us° set,S for any± , z( ) = + F ( ). For any x; y k kf > k kW t∈ ; k − k u ( ) [ z 0] , we can find sequences ( n)n N (0 + ) converging to 0, ( n)n N ∈ ∩ z y∈ ⊂ ∞ ∈ ⊂ B¯ v Y v … x t u ; y t v F X and ( n)n N converging to = − such that ( + n n + n n) for ∈ ⊂ z y ∈ n n k − k t … z y 1 all N. It then follows, for large enough (so that 1 n − 0) ∈ − k − k ≥ f (x + t u ; y + t v ) y + t v z + t v v f (x; y) t … + t v v z n n n n ≤ k n − k nk n − k ≤ z − n nk n − k thus, for n large enough in order that v = 0, n 6 f x; y f x; y t u ; v 1 z( ) z(( ) + n( n n)) max(1;… − v ) − … v v k nk t (u ; v ) ≥ − k n − k nk n n k yielding f (x; y) …. Thus we get from Theorem 3.2 applied with Z = Y that F is |∇ z| ≥ metrically regular near (x0; y0), this is [27, Theorem 6.1]. In fact (26) shows that (28) is a characterization of metric regularity. d) In fact Theorem 4.10 follows from Theorem 4.12. Indeed, we know from [6, Proposition 2.3] that a cone K is equi-circatangent to a set C X, X Banach, at x0 cl C if and " > V ⊂ x K ∈ T ε x only if, for all 0, there exists a neighborhood of 0 such that x V C C ( ). ⊂ ∈ ∩ T 4.3.4. The Ursescu Theorem Theorem 4.15. Let X, Y be Banach spaces and let F X Y be a closed multifunction. Assume that there exists an open neighborhood W of (x⊂; y ×) F and … > 0 such that 0 0 ∈ …B T (x; y)(X) cl T (x; y)(B ) for all (x; y) W F: Y ∩ F ⊂ F X ∈ ∩ Then there exists neighborhoods U of x0 and V of y¡0 such that 1 … d(x; F − (y)) d(y; F (x)) for all (x; y) U (V Z): ≤ ∈ × ∩ where Z = (y + cl (TF (x; y)(X))). (x,y) W F \∈ ∩ 1 Proof. Let us endow X Y with the norm (x; y) = max( x ;… − y ) and let us z Z × f X kY k k k k k f x; y consider, for any , the function z : R + defined by z( ) = ∈ × −→ ∪ { ∞} z y + iF (x; y). Let (x; y) F W and z Z be such that (x; y) [fz>0]. Given k − k ∈ ∩ 1 ∈ ∈ " (0;…) and setting v = (… ") y z − (z y), we can find w0 T (x; y)(X) such that ∈ − k − k − ∈ F w0 v " and u B¯X , w Y such that (u; w) TF (x; y) and w0 w ". Given k − k ≤t ∈ ; ∈ ∈u ; w Xk −Y k ≤ sequences ( n)n N (0 + ) converging to 0 and (( n n))n N converging to ∈ ∈ (u; w) such that (x;⊂ y) + t∞(u ; w ) F , we have ⊂ × n n n ∈ fz(x; y) fz(x + tnun; y + tnwn) y z y + tnv z − k − k − k − k wn w 2" tn ≥ tn − k − k − … 3"; ≥ − D. Az´e/ A Unified Theory for Metric Regularity of Multifunctions 247 yielding (u; w) fz (x; y) … 3" and then, by letting " go to 0, fz (x; y) …. Thus the conclusionk ofk|∇ the| theorem≥ follows− from Theorem 3.2. |∇ | ≥ Remark 4.16. In [58, Theorem 2] Ursescu proves a partial converse to Theorem 4.15. Namely, assuming that X is finite dimensional and that, for some subset Z Y , there ⊂ exist neighborhoods U of x0 and V of y0 such that 1 … d(x; F − (y)) d(y; F (x)) for all (x; y) U (V Z); ≤ ∈ × ∩ then it follows from the quoted theorem that, for any » (0;…), ∈ »B T (y) T (x; y)(B ) for all (x; y) F (U V ): Y ∩ Z ⊂ F X ∈ ∩ × In particular, if X is finite dimensional and if F is metrically regular near (x0; y0) (which corresponds to the case Z = Y ), then »BY TF (x; y)(BX ) for all » (0;…) and for all (x; y) F (U V ). ⊂ ∈ ∈ ∩ × 4.4. Parametric results 4.4.1. Tangencial results Given Banach spaces X, Y and a metric space P , we say that a mapping h( ; ) defined · · in a neighborhood of (x0; p0) X P with values in Y is partially strictly differentiable in x at (x ; p ) whenever there∈ exists× a linear continuous mapping L(X;Y ) such that 0 0 ∈ for all " > 0, there exist neighborhoods V0 of x0 and U0 of p0 such that h(x; p) h(z; p) (x z) " x z for all (x; z; p) V V U : k − − − k ≤ k − k ∈ 0 × 0 × 0 Dh x h h ; p Observe that we then have = p0 ( 0) where p( ) = ( ). The following theorem is [11, Theorem 4.1]. · · Theorem 4.17. Let X, Y be Banach spaces, let P be a topological space, let A X be a closed convex set, and let h be defined and continuous near (x ; p ) A P with⊂ values 0 0 ∈ × in Y . Assume that h is partially strictly differentiable in x at (x0; p0) and that 0 core Dh (x )(A x ) : (29) ∈ p0 0 − 0 ¡ Then there exist … > 0 and neighborhoods U of x0, V of y0 = h(x0; p0) and N of p0 such that 1 … d(x; h− (y) A) h (x) y for all (x; y; p) (A U) V N: p ∩ ≤ k p − k ∈ ∩ × × Proof. From assumption (29), we can find … > 0 and an open neighborhood U˜0 of x0 such that …B¯ cl ((A x) B¯ ) for all x U˜ where = Dh (x ). Given " (0; …=2), Y ⊂ − ∩ X ∈ 0 p0 0 ∈ UÝ x N p we can find open neighborhoods¡ 0 of 0 and 0 of 0 such that h(x; p) h(z; p) (x z) " x z for all (x; z; p) UÝ UÝ N : k − − − k ≤ k − k ∈ 0 × 0 × 0 1 Let us set W = U˜ UÝ and f (x) = h(x; p) y +i (x), so that [f 0] = h− (y) A. 0 0 ∩ 0 p,y k − k A p,y≤ p ∩ y h(x; p) Let (p; y) N Y and x [f >0] A W . Settingw Ý = … − , we can find ∈ 0 × ∈ p,y ∩ ∩ 0 h(x; p) y k − k 248 D. Az´e/ A Unified Theory for Metric Regularity of Multifunctions w Y and v B¯X such that w wÝ ", w = (v) and v (A x) B¯X . We then have,∈ for all t∈ > 0 small enoughk − k ≤ ∈ − ∩ h(x + tv; p) y h(x; p) y + twÝ + 2t" h(x; p) y t… + 2t": k − k ≤ k − k ≤ k − k − Thus we have fp,y (x) … 2" for all (p; y) N0 Y and x [fp,y>0] W0 and then the conclusion of the|∇ theorem| ≥ follows− from Theorem∈ 2.13× applied∈ with the∩ parameter space P Y , observing that the function (p; y) f (x ) is continuous. × 7→ p,y 0 A multifunction F P X between topological spaces is said to be lower semicontinuous p ; x F ⊂ × x F p at ( 0 0) whenever 0 Lim infp p0 ( ). This is equivalent to the fact that the f ∈ X P ∈ → f x; p i p; x i x function : R + defined by ( ) = F ( ) = F (p)( ) is epi-upper × → ∪ { ∞} semicontinuous at (x0; p0). In [9, Corollary 5.5], one can find a parametric metric regularity result involving a condi- tion based on the "-contingent derivative of the multifunctions. Theorem 4.18. Let X, Y be Banach spaces, P a metric space, and let F P (X Y ) ⊂ × × be a closed-valued multifunction which is lower semicontinuous at (p ; (x ; y )) F . As- 0 0 0 ∈ sume that there exist " > 0, … > 0, neighborhoods W0 of (x0; y0), U0 of p0 and › 0 such that … › "(… + ›) > 0 and ≥ − − …B¯ T ε (x; y)(B¯ ) + ›B¯ for all (x; y) W F and for all p U : (30) Y ⊂ Fp X Y ∈ 0 ∩ ∈ 0 1 Then there exists neighborhoods U of x , W of y , N of p such that F − (z) = for all 0 0 0 p 6 ∅ (p; z) N W and such that ∈ × 1 » d(x; F − (z)) d(z; F (x)) for all (x; z; p) U W N; p ≤ p ∈ × × … › "(… + ›) where » = − − . " … 1› (1 + )(1 + − ) p; z P Y f X Y f x; y Proof. For ( ) , let us define p,z : R + by p,z( ) = z y i x; y∈ × ×f −→ ∪ { ∞} + Fp ( ). Observe that the functions p,z are lower semicontinuous, that kf − xk; y f x; y ; p; z f x; y p0,y0 ( 0 0) = 0 and that the function ( ) ( )) = p,z( ) is epi-upper semi- continuous at ((x0; y0); (p0; y0)) due to the lower semicontinuity of F at (p0; (x0; y0)) and to the Lipschitz continuity of the norm. Let us endow X Y with the norm 1 × (u; v) = sup( u ;… − v ) and let (p; z) U0 Y and (x; y) W0 Fp be such that fk (x;k y) > 0. Fromk k thek proofk of Theorem 4.12,∈ we× obtain that ∈ f (∩x; y) ». Apply- p,z |∇ p,z| ≥ ing Theorem 2.13 yields the existence of neighborhoods NÝ of p0, WÝ , VÝ of y0 and UÝ of x0 such that [f 0] = for all (p; z) NÝ WÝ and such that » d((x; y); [f 0]) f (x; y) p,z≤ 6 ∅ ∈ × p,z≤ ≤ p,z for all ((x; y); (p; z)) UÝ VÝ NÝ WÝ from which we get ∈ × × × 1 » d(x; F − (z)) z y for all (p; (x; y)) F (NÝ UÝ VÝ ) and z W:Ý (31) p ≤ k − k ∈ ∩ × × ∈ Assume now that the conclusion of the theorem fails, so that there exists a sequence x ; z ; p x ; y ; p y F x d z ; y < » d x ; (( n n n))n N converging to ( 0 0 0) and n pn ( n) such that ( n n) ( n ∈ ∈ D. Az´e/ A Unified Theory for Metric Regularity of Multifunctions 249 F 1 z p x ; y x ; y F x ; y p− ( n)). Considering a mapping ( p p) such that ( p p) p and limp p0 ( p p) = n 7→ ∈ → (x0; y0), we have 1 d(y ; y ) d(z ; y ) < »d(x ; x ) + » d(x ;F − (z )) »d(x ; x ) + » d(z ; y ); n pn − n pn n pn pn pn n ≤ n pn n pn y y so that ( n)n N converges to 0 contradicting (31). ∈ 4.4.2. Normal conditions The next result is [9, Corollary 5.7]. Theorem 4.19. Let X, Y be Banach spaces, let P be a topological space, let @ be a subdifferential satisfying (P 1) and (P 2) on X Y and let F P (X Y ) be a closed × ⊂ × × valued multifunction which is lower semicontinuous at some (p0; (x0; y0)) F . Assume that ∈ d ;D F x; y S > … > : lim inf (0 ∗ p( )( Y ∗ )) 0 F ∗ (p,x,y) (p ,x ,y ) → 0 0 0 Then there exist neighborhoods W of (x0; y0) and N of p0 such that 1 … d(x; F − (y)) d(y; F (x)) for all ((x; y); p) W N: p ≤ p ∈ × Proof. Let us introduce the closed proper function f :(X Y ) (P Y ) R + f x; y ; p; z f x; y z y i x;× y × g x;× y −→i x;∪{ y ∞} defined by (( ) ( )) = (p,z)( ) = + Fp ( ) = ( ) + Fp ( ) and k − k 1 let us endow X Y with the norm (x; y) = max( x ;… − y ) whose dual norm is × k k k k k k (‚; ) = ‚ + … . We can find open neighborhoods U0 of x0, V0 of y0 and N0 k k∗ k k∗ k k∗ of p0 such that d (0;D∗Fp(x; y)(SY )) > … for all (p; x; y) (N0 U0 V0) F . Let ∗ ∗ ∈ × × ∩ p N0, z Y and (x; y) (U0 V0) [fp,z>0] Fp. Let (p; z) N0 Y and let (x1; y1), (x∈; y ) ∈(U V ) F ,∈‚ ×@g(x ∩; y ), (‚ ; ∩) N (x ; y ).∈ Assume× that (x ; y ) is 2 2 ∈ 0 × 0 ∩ p 1 ∈ 1 1 2 2 ∈ Fp 2 2 1 1 close to (x; y) in order that z y1 > 0. We have ‚1 = (0; 1) with 1 = 1 so that k − k k k∗ (0; 1) + (‚2; 2) = ‚2 + … 1 + 2 ‚2 … 2 + … … k k∗ k k∗ k k∗ ≥ k k∗ − k k ≥ since ‚2 … 2. Thus we derive from Proposition 2.6 that fp,z(x; y) … for all ∗ (p; z) k Nk ≥Y kandk for all (x; y) [f >0] (U V ). As F is lower|∇ semicontinuous≥ at ∈ 0 × ∈ p,z ∩ 0 × 0 (p0; (x0; y0)) and as the norm is Lipschitzian, it follows that f is epi-upper semicontinuous at ((x0; y0); (p0; y0)). Applying Theorem 2.13, there exists neighborhoods UÝ of x0, VÝ , WÝ 1 of y and NÝ of p such that … d(x; F − (z)) z y for all (p; (x; y)) F (NÝ UÝ VÝ ) 0 0 p ≤ k − k ∈ ∩ × × and for all z WÝ from which one easily gets the conclusion of the theorem as in Theorem 4.18. ∈ Remark 4.20. 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