Nonlinear regularity models
Math. Program., Ser. B (2013) 139:223–242 DOI 10.1007/s10107-013-0670-z FULL LENGTH PAPER
Nonlinear regularity models Alexander D. Ioffe
Received: 9 February 2011 / Accepted: 7 September 2011 / Published online: 22 March 2013 © Springer-Verlag Berlin Heidelberg and Mathematical Optimization Society 2013
Abstract The paper studies regularity properties of set-valued mappings between metric spaces. In the context of metric regularity, nonlinear models correspond to nonlinear dependencies of estimates of error bounds in terms of residuals. Among the questions addressed in the paper are equivalence of the corresponding concepts of openness and “pseudo-Hölder” behavior, general and local regularity criteria with special emphasis on “regularity of order k”, for local settings, and variational methods to extimate regularity moduli in case of length range spaces. The majority of the results presented in the paper are new. Keywords Metric regularity · Regularity criterion · Regularity of order k · Error bound · Length space Mathematics Subject Classification (2000)
47H04 · 49J53 · 90C31
1 Introduction Regularity is one of the basic concepts of modern analysis. Among its many faces, the most popular in the variational analysis community is “metric regularity”, the term coined by Borwein in 1986 [4] for estimates of distances to preimages of set-valued mappings in terms of residuals. The first estimates of that sort starting with the famous Hoffman’s estimate for systems of linear inequalities [15] and later results of Ioffe– Tikhomirov and Robinson [16,21,25,26], as well as the estimate in the definition
The paper is dedicated to Jon Borwein’s sixtieth anniversary. A. D. Ioffe (B) Department of Mathematics, Technion, 32000 Haifa, Israel e-mail: [email protected]
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in [4], were linear in the sense that the bound for the first distance was equal to the second one times a certain positive number. We consider regularity models which, in the context of metric regularity, correspond to nonlinear dependence of the bound on the residual. Such models are in particular useful in the analysis of sensitivity and controllability in optimization and control (see e.g. [3,13]). The study of nonlinear regularity models was initiated in late 80s by Borwein–Zuang [6], Frankowska [10–12] and Penot [23]. The main attention in [6,23] was focused on equivalent characterizations of nonlinear regularity with arbitrary nonlinearities. We in particular wish to stress the mention of approximate openness among the equivalences in [6]. On the other hand, [10,11] and subsequent studies by Frankowska [13] and Frankowska and Quincampoix [14] were devoted to analysis, mainly to variational approach to regularity (for mappings into Banach spaces) with power nonlinearities. All mentioned papers deal with local regularity properties in an arbitrary small neighborhood of a given point of the graph of the (generally setvalued) mapping. In our survey [17] we briefly discussed nonlinear regularity, mainly to emphasize similarities with the linear theory and the role of the slope-based metric infinitesimal mechanisms. Here we try to present a systematic study of nonlinear regularity theory in metric spaces. The latter means that (apart from a few corollaries) we consider set-valued mappings between metric spaces. In the main part of the paper we consider general nonlinearities associated with gauge functions and then consider in more details higher order regularity properties associated with power nonlinearities. We begin by proving fairly general equivalence theorem saying that (also in the non-local case) nonlinear versions of openness, metric regularity and pseudo-Hölder properties of the inverse mapping are equivalent. We then prove a sufficient regularity criterion which is a generalization of a nonlinear criterion stated without proof in [17], on the one hand, and of the criterion for linear regularity for mappings between metric spaces in [20], on the other. The criterion is then applied to get a nonlinear density theorem (in the spirit of Dmitruk’s theorem in [9] for the standard linear global regularity model1 ) which contains quantitative measures of density of the images of balls in the domain space in corresponding balls in the range space that guarantee that the latter actually fully lie within the image. Note that the equivalence of the approximate openness and openness proved in [6] follows from the density theorem. The mentioned results are most general in the sense that they are applied not just for arbitrary metric spaces but also for arbitrary gauge functions. The subsequent results are all dealing with power gauge functions μ(t) = r t k with k ≥ 1. The immediate advantage is the possibility to define numerical measures of regularity which we call moduli of order k. Our main tool of analysis is a higher order extensions of the strong slope of DeGiorgi–Marino–Tosques [8]. As in the linear case these slopes allow to get estimates for higher order moduli of regularity and higher order error bounds for sublevel sets of lower semicontinuous functions. In the concluding section we consider a variational approach to computing regularity moduli of higher order. The core of this approach is the study of the behavior of a 1 It is appropriate to mention here a very general result of Khan [22] which contains as particular case the results of Pták [24] and Dmitruk.
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set-valued mapping along geodesic curves in the range space viewed as variations of their initial points. For the range space being Rn a result in this direction was recently obtained in [14]. It turns out that the key property of Rn (in addition to compactness of the unit ball) is that the unit sphere S n−1 is a homogeneous space with respect to the group of orthogonal transformations of Rn . We show that similar formulas can be obtained for set-valued mappings into arbitrary homogeneous locally compact length spaces. Notation In what follows X , Y etc. are metric spaces. We use the same symbol d(·, ·) to denote the distance in all of them and hope this will not be a cause of any ◦
confusion. By B(x, ρ) and B(x, ρ) we denote the closed and open balls of radius ρ ◦
around x. Likewise, if Q ⊂ X , then B(Q, ρ) and B(Q, ρ) are the closed and open ◦
ρ-neighborhoods of Q, that is, say B(Q, ρ) = {x ∈ X : d(x, Q) < ρ}. Given two sets P and Q in X , we denote by ex(P, Q) = inf{ε > 0 : P ⊂ B(Q, ε)} the excess of P over Q. By S(x, ρ) we denote the sphere of radius ρ around x. If F : X ⇒ Y is a set-valued mapping from X into Y , then Graph F = {(x, y) ∈ X × Y : y ∈ F(x)} is the graph of F and dom F = {X : F(x) = ∅} is the domain of F. In the product space X × Y we shall usually consider the ξ -metric (with some ξ > 0) dξ (x, y), (u, v) = max{d(x, u), ξ d(y, v)}. For a Q ⊂ X we denote by i Q the indicator of Q which is the function equal to zero on Q and infinity outside of Q. 2 Definitions and equivalences We shall make an extensive use of gauge functions (or just gauges) by which we mean a continuous strictly increasing functions t → μ(t) on [0, ∞) equal to zero at 0 and going to infinity when t → ∞. Clearly, the inverse of a gauge is also a gauge. Definition 1 Given an F : X ⇒ Y , where X and Y are metric spaces. Let U ⊂ X and V ⊂ Y be open sets, let γ be a function on X which is positive on U , and let δ be a function on Y which is positive on V . Assume finally that we are given three gauge functions μ, ν and η. (a) F is γ -open on U ×V with functional modulus (not smaller than) μ if the inclusion ◦
B(F(x), μ(t)) ∩ V ⊂ F(B(x, t)) holds whenever x ∈ U and 0 < t < γ (x). (b) F is γ -metrically regular on U × V with functional modulus (not greater than) ν if the inequality d(x, F −1 (y)) ≤ ν(d(y, F(x))) holds whenever x ∈ U, y ∈ V and 0 < ν(d(y, F(x))) < γ (x).
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(c) F is δ-Hölder on U × V with functional modulus (not greater than) η if the inequality d(y, F(x)) ≤ η(d(x, u)), holds provided x ∈ U, y ∈ V ∩ F(u) and 0 < η(d(x, u)) < δ(y). Note that the definition does not mean to introduce “functional moduli” as specific objects, although it could be possible in principle. We do not need them and shall use this expression only in the same context as in the definition. This should be kept in mind because occasionally, when it does not lead to a confusion, we shall for simplicity omit expressions “not smaller than” etc. and shall speak about openness with functional modulus μ and so on. If we compare e.g. the openness part of this definition with available definition of the standard “openness at a linear rate”, we notice two major differences. The first, and the obvious, is that r t has been replaced by μ(t). The second difference is the appearance of the “reach function” γ which determines the set of (x, v) ∈ Graph F which are actually involved in implementation or verification of the properties. The following simple example shows that this is an essential part of the definition and changing γ we may change the parameters of regularity or even to kill regularity at all. Example 1 Set X = Y = R, U = V = (0, ∞), F(x) = {x 2 }, μ(t) = t 3 . Then ◦
for any positive x we have B(F(x), μ(t)) = (x 2 − t 3 , x 2 + t 3 ) and F(B(x, t)) = [((x − t)+ )2 , (x + t)2 ]. By definition F is γ -open on U × V with functional modulus not smaller than t 3 if and only if (x 2 − t 3 )+ ≥ ((x − t)+ )2 and x 2 + t 3 ≤ (x + t)2 for all 0 ≤ t < γ (x). The second relation means that t (t − 1) ≤ 2x for all t < γ (x) which is the same as γ (x)(γ (x) − 1) ≤ 2x. To analyze the first relation we consider separately the cases γ 3 (x) ≤ x 2 and γ 3 (x) > x 2 . In the first case the first relation reduces to γ (x)(γ (x) + 1) ≤ 2x in which case the second relation is automatically satisfied. In the second case the first relation cannot be satisfied if x > 1. Indeed in this case for t < x sufficiently close to x we would have t 3 > x 2 . On the other hand if x ≤ 1, then the first relation holds for all t < γ (x) (indeed, (x − t)+ = 0 if t ∈ [x, γ (x)) and t (t + 1) ≤ 2x if t ≤ x). Thus F is γ -open with functional modulus t 3 if and only if either γ 3 (x) ≤ x 2 and γ (x)(γ (x) + 1) ≤ 2x or γ 3 (x) > x 2 , x ≤ 1 and γ (x)(γ (x) − 1) ≤ 2x. Moreover if we take γ satisfying 8γ 3 (x) ≤ x 2 for x > 1 and γ (x)(2γ (x)+1) ≤ x, then F is γ -open on U × V with functional modulus not smaller than (2t)3 . On the contrary, local properties that hold in small neighborhoods of a point of the graph are insensitive to variations of γ , so that the latter turns out to be needless. Local versions of the definitions, near a certain (x, ¯ y¯ ) ∈ Graph F, correspond to U and V being arbitrarily small neighborhoods of x and y respectively. Before stating them in a formal manner we shall prove a simple proposition which implies the mentioned insensitivity of local regularity properties to variations of γ .
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Proposition 1 Let F : X ⇒ Y , let (x, ¯ y¯ ) ∈ Graph F, and let r > 0. Then the following properties are equivalent: (a) there is an ε > 0 such that ◦
◦
B(F(x), μ(t)) ∩ B(y, ε) ⊂ F(B(x, t)), if d(x, x) < ε, (b) there is an ε > 0 such that ◦
◦
B(F(x), μ(t)) ∩ B(y, ε) ⊂ F(B(x, t)), if d(x, x) < ε, 0 < t < ε, (c) there is an ε > 0 such that ◦
◦
B(F(x) ∩ B(y, ε), μ(t)) ⊂ F(B(x, t)), if d(x, x) < ε, 0 < t < ε, (d) there is an ε > 0 such that ◦
◦
◦
B(F(x) ∩ B(y, ε), μ(t)) ∩ B(y, ε) ⊂ F(B(x, t)), if d(x, x) < ε, 0 ≤ t < ε. Proof It is clear that (a) ⇒ (b) ⇒ (d) and (c) ⇒ (d). Thus we have to show that (d) implies (a) and (c). So let (d) hold with some ε > 0. Take a τ > 0 and α > 0 such that α + μ(α) < τ < ε, μ(τ ) + μ(α) < ε
(1)
and set δ = min{α, μ(α)}. Clearly δ < ε We shall first prove that (a) holds with ε replaced by δ. The inclusion y ∈ ◦
◦
B(F(x), μ(t)) ∩ B(y, δ) implies that d(y, y) < δ and d(y, v) < μ(t) for some v ∈ F(x). So if t < τ and d(x, x) < δ, then it follows from (1) that d(v, y) < μ(t) + δ ≤ μ(τ ) + μ(α) < ε and therefore by (d) ◦
◦
◦
◦
◦
B(F(x), μ(t)) ∩ B(y, δ) ⊂ B(F(x) ∩ B(y, ε), μ(t)) ∩ B(y, ε) ⊂ F(B(x, t)). On the other hand, again by (d) ◦
◦
◦
B(y, δ) ⊂ B(F(x) ∩ B(y, ε), μ(α)) ∩ B(y, ε) ⊂ F(B(x, α)), so for t ≥ τ , we have ◦
◦
B(F(x), μ(t)) ∩ B(y, δ) ⊂ F(B(x, α)) ⊂ F(B(x, t)) because by (1) α + d(x, x) ≤ α + μ(α) ≤ τ ≤ t, and (a) follows.
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◦
Let us prove (c). If y ∈ B(F(x) ∩ B(y, δ), μ(t)) for some t < δ, then there is a v ∈ F(x) such that d(v, y) < δ and d(y, v) < μ(t). As α < τ < ε, this means that d(y, y) < δ + μ(t) ≤ μ(α) + μ(τ ) < ε, and consequently by (d) ◦
◦
◦
◦
◦
B(F(x) ∩ B(y, δ), μ(t)) ⊂ B(F(x) ∩ B(y, ε), μ(t)) ∩ B(y, ε) ⊂ F(B(x, t))
and (c) follows.
The above proposition says that speaking about local regularity properties, we can either omit any mention of γ (which is tantamount to taking γ (x) ≡ ∞) or take it arbitrarily small. Each of the four properties (a)–(d) can be equally used as a basis for the definition of the local version of “nonlinear” openness. Definition 2 Let F : X ⇒ Y , let (x, ¯ y¯ ) ∈ Graph F and let μ be a gauge function. We say that F is open at (x, ¯ y¯ ) with modulus germ not smaller than μ if the equivalent properties of Proposition 1 hold. We leave to the reader the simple task of reformulating the local analogues of the other two properties of Definition 1. The equivalence of the local versions of the three properties was basically established by Penot in [23]. The theorem below extends this result to the general situation of Definition 1. Theorem 1 Let F : X ⇒ Y , let U ⊂ X , V ⊂ Y be open sets, and let γ be a function on X which is positive on U . The following properties are equivalent, given a gauge function μ. (a) F is γ -open on U × V with functional modulus not smaller than μ; (b) F is γ -metrically regular on U × V with functional modulus not greater than μ−1 ; (c) F −1 is γ -Hölder on V × U with functional modulus not greater than μ−1 . Proof The implication (b) ⇒ (c) is trivial. To prove that (c) ⇒ (a), let t < γ (x), and ◦
◦
let y ∈ V and y ∈ B(F(x), μ(t)), that is y ∈ B(v, μ(t)) for some v ∈ F(x).Then μ−1 (d(y, v)) < t < γ (x) and y ∈ V , so by (c) d(x, F −1 (y)) ≤ μ−1 d(y, v) < t. This means that there is a u such that y ∈ F(u) and d(x, u) < t, that is y ∈ F(B(x, t)). ◦
As y is an arbitrary element of B(F(x), μ(t)) ∩ V , (a) follows. (a) ⇒ (b). Take an x ∈ U and y ∈ V with μ−1 (d(y, F(x))) < γ (x). Let further K > 1 be such that t = μ−1 (K d(y, F(x)) < γ (x). Such a K definitely exists as μ−1 is a gauge, hence continuous. We have K −1 μ(t) = d(y, F(x)). Take an r < 1 such ◦
that r K > 1. Then d(y, F(x)) < r μ(t), that is y ∈ B(F(x), r μ(t)). Define t and K by μ(t ) = r μ(t) and t = μ−1 (K d(y, v)). Then t < γ (x) (as r < 1), so by (a) y ∈ F(B(x, t )) which means that there is a u with d(x, u) ≤ t such that y ∈ F(u). In other words, d(x, F −1 (y)) ≤ d(x, u) ≤ t = μ−1 (K d(y, F(x))). It is clear that K and r can be chosen to make K arbitrarily close to 1, and (b) follows. The theorem justifies the following definition. Definition 3 We shall say that F is γ -regular on U × V with functional modulus μ if the three equivalent properties of Theorem 1 hold.
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3 Criterion of nonlinear regularity and some applications Let again U ⊂ X and V ⊂ Y be open sets, and let γ be a function on X with Lipschitz constant 1. We set Uγ =
◦
B(u, γ (u)).
u∈U
Let further μ be a gauge function. Lemma 1 Let F : X ⇒ Y be a set-valued mapping whose graph is complete in the product metric, let U and V be open sets in X and Y respectively, let μ be a gauge function, and let γ be a Lipschitz function on X with Lipschitz constant 1 which is positive on U . Suppose there are y ∈ Y and a ξ > 0 such that for any x ∈ Uγ , v ∈ F(x) with 0 < d(y, v) < μ(γ (x)) there is a pair (u, w) ∈ Graph F, (u, w) = (x, v) such that μ−1 d(y, w) ≤ μ−1 d(y, v) − dξ ((x, v), (u, w)).
(2)
Then for any (x, v) ∈ Graph F with x ∈ Uγ , 0 < d(y, v) < μ(γ (x)) there is a ˆ and uˆ ∈ Uγ such that y ∈ F(u) ˆ y)) ≤ μ−1 (d(y, v)). dξ ((x, v), (u, Proof Set ψ y (x, v) = μ−1 (d(y, v)) + i Graph F (x, v). Let x ∈ U , v ∈ F(x) and d(v, y) < μ(γ (x)). Set ε = ψ y (x, v) and find, using Ekeland’s principle (see e.g. [5]), a pair (u, ˆ w) ˆ such that ˆ w)) ˆ ≤ ε; dξ ((x, v), (u, ψ y (u, ˆ w) ˆ ≤ ψ y (x, v) − dξ ((x, v), (u, ˆ w)); ˆ
(3)
ψ y (u, w) + dξ ((u, w), (u, ˆ w)) ˆ > ψ y (u, ˆ w), ˆ if (u, w) = (u, ˆ w). ˆ We claim that ψ y (u, ˆ w) ˆ = 0, that is wˆ = y. Assuming the contrary we shall get a contradiction. Indeed, note first that d(x, u) ˆ < γ (x). This is a consequence of the first inequality in (3) which implies by the assumptions that d(x, u) ˆ ≤ μ−1 (d(y, v)) < ˆ < γ (u). Indeed, as follows from γ (x). Thus uˆ ∈ Uγ . Next we observe that d(y, w) the assumption on γ and the second inequality in (3), ˆ w) ˆ < γ (x) − d(x, u) ˆ ≤ γ (u). ψ y (u, Therefore by (2) there is a (u, w) = (u, ˆ w) ˆ in Graph F such that μ−1 d(y, w) ≤ μ−1 d(y, w) ˆ w), ˆ (u, w)). ˆ − dξ ((u, The latter contradicts to the last relation in (3). Thus uˆ ∈ F −1 (y) = ∅. The final estimate of the lemma follows from the first inequality in (3).
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Theorem 2 Let F : X ⇒ Y be a set-valued mapping whose graph is complete in the product metric, let γ be a Lipschitz function on X with Lipschitz constant 1, let μ be a gauge function, and let U and V be open subsets of X and Y respectively. Then F is γ -open on U × V with functional modulus μ if the following holds: there is a ξ > 0 such that for any x ∈ Uγ , y ∈ V, v ∈ F(x) with 0 < d(y, v) < μ(γ (x)) there is a pair (u, w) ∈ Graph F, (u, w) = (x, v) such that (2) holds. Conversely, assume that F is γ -open on U × V with functional modulus μ and there is a ξ > 0 such that ξ t ≤ μ−1 (t) for all t < supU γ (x). Then for any x ∈ U , y ∈ V and v ∈ F(x) with 0 < d(y, v) < μ(γ (x)) there is a u such that y ∈ F(u) and (2) holds. Proof The first statement is an immediate consequence of the lemma. Indeed, given x ∈ U , y ∈ V , t < γ (x) and v ∈ F(x) with 0 < d(y, v) < μ(t), we can find by the lemma a pair (u, ˆ w) ˆ ∈ Graph F such that y ∈ F(u), ˆ y = v and d(x, u) ˆ ≤ t. It ◦
follows that B(v, μ(t)) ⊂ F(B(x, t)). To prove the second statement suppose that F is γ -open with functional modulus not smaller than μ. Given x, y, v as above, we find a u such that y ∈ F(u) and d(x, u) ≤ μ−1 (d(y, v)). Let further ξ > 0 satisfies the conditions specified in the statement. Then, dξ ((x, v), (u, y)) = max{d(x, u), ξ d(v, y)} ≤ μ−1 (d(y, v)), and setting w = y, we get 0 = μ−1 (d(y, w)) ≤ μ−1 (d(y, v)) − dξ ((x, v), (u, w))
as claimed.
Note that in case supU γ (x) = ∞, μ−1 (t) ≥ ξ t for all t, hence μ is majorized by a linear function. If on the other hand, the supremum is finite, the conditions mean that μ is majorized by a linear function in an arbitrarily small neighborhood of zero. The criterion can be simplified and strengthened if F is upper semicontinuous in the sense that for any y ∈ Y the function x → d(y, F(x)) is lower semicontinuous. Theorem 3 (criterion for an u.s.c. mapping) Suppose in addition to the assumptions of Theorem 2 that F is upper semicontinuous as defined above. Suppose further that for any x ∈ Uγ , any y ∈ V such that 0 < d(y, F(x)) < μ(γ (x)) there is a u = x such that μ−1 d(y, F(u)) ≤ μ−1 d(y, F(x)) − d(x, u).
(4)
Then F −1 (y) = ∅ for all y ∈ V and d(x, F −1 (y)) ≤ μ−1 (d(y, F(x)) if x ∈ U , y ∈ V . In particular F is γ -regular on U × V with functional modulus μ. Conversely, if F is γ -regular on U × V with functional modulus μ, then for any x ∈ U, y ∈ V there is a u such that (4) holds.
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Proof To prove the first statement, we only need to repeat the proof of the lemma with ψ y (x, v) replaced by f y (x) = d(y, F(x)). The second statement follows from the trivial observation: if y ∈ F(u) and d(x, u) ≤ μ−1 (d(y, v)), then μ−1 (d(y, y)) = 0 ≤ μ−1 (d(y, v)) − d(x, u). As an application of the just proved regularity criteria, we give below a nonlinear extension of one of the main theorems of [9] due to Dmitruk. Theorem 4 (nonlinear density theorem) Let U ⊂ X and V ⊂ Y be open sets, let F : X ⇒ Y be a set-valued mapping with complete graph, and let γ be a Lipschitz function on X with Lipschitz constant 1, positive on U and equal to zero on bdU . We assume that there are a gauge function μ, a λ ∈ (0, 1) and an η > 0 such that μ(ητ ) ≤ τ for all τ < supU μ(γ (x)), and for any x ∈ U and t < γ (x) the set F(B(x, t)) is a λμ(t)-net in B(F(x), μ(t)) ∩ V .2 Let ν be another gauge function satisfying ν −1 (t) − ν −1 (λt) ≥ μ−1 (t), ∀ t ∈ (0, sup γ (x)).
(5)
x∈U
Then F is γ -open on U × V with functional modulus not smaller than ν. Proof Let x ∈ U, y ∈ V, v ∈ F(x), d(y, v) < μ(γ (x)). Set t = μ−1 (d(y, v)). Then t < γ (x). By the assumption there is a (u, w) ∈ Graph F such that d(x, u) ≤ t and d(y, w) ≤ λμ(t) = λd(y, v).
(6)
We have by (5) ν −1 (d(y, w)) ≤ ν −1 (λd(y, v))
= ν −1 (d(y, v)) − ν −1 (d(y, v)) − ν −1 (λd(y, v)) ≤ν
−1
−1
(d(y, v)) − μ
(7)
(d(y, v)).
Take ξ > 0 such that ξ η−1 (1+λ) ≤ 1. By Theorem 2, to complete the proof, we have to verify that μ−1 (d(y, v)) ≥ dξ ((x, v), (u, w)). We have μ−1 (d(y, v)) = t ≥ d(u, x). On the other hand, ηd(y, v) ≤ μ−1 (d(y, v)) as d(y, v) < μ(γ (x)). In view of (6) this means that ξ d(v, w) ≤ ξ(d(y, v) + d(y, w)) ≤ ξ η−1 (1 + λ)(η(d(y, v))) ≤ μ−1 (d(y, v)) and the desired estimate for dξ ((x, v), (u, w)) follows.
Dmitruk’s theorem (Theorem 1.5 in [9]) corresponds to μ(t) = r t and γ (x) = d(x, X \U ). For ν(t) of the form ν(t) = ρμ(t) for some ρ < 1 (in which case ν −1 (t) = μ−1 (t/ρ)) (5) reduces to μ−1 (ρ −1 t) − μ−1 (ρ −1 λt) ≥ μ−1 (t). 2 That is for any y ∈ B(F(x), μ(t)) ∩ V there is a v ∈ F(B(x, t)) such that d(y, v) < λμ(t).
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In the most interesting case of μ(t) = r t k we get Corollary 1 If under the assumptions of Theorem 4 μ(t) = r t k and γ is bounded on U then F is γ -open on U × V with the functional modulus not smaller than r (1 − λ1/k )k t k . In particular for k = 1, we get the density theorem of [20]. An obvious consequence of the theorem for local regularity is Corollary 2 (cf. [27]) Let F : X ⇒ Y be a set-valued mapping whose graph is ¯ y¯ ) ∈ Graph F. Assume that there are an ε > 0 and a gauge locally complete3 at (x, ◦
function μ such that for any x ∈ B(x, ε) and t ∈ (0, ε) the set F(B(x, t)) is dense ◦
in B(F(x), μ(t)) ∩ B(y, ε). Then F is open at (x, ¯ y¯ ) with modulus germ not smaller than μ. 4 Regularity of order k Since this moment we shall focus on a particular but the most important class of gauge functions μ(t) = r t k . Fix as before a set-valued mapping F : X ⇒ Y with complete graph, open U ⊂ X and V ⊂ Y and two functions , γ on X and δ on Y , which are positive on U and V respectively. Let also a certain k ≥ 1 be fixed. Definition 4 (a) by sur (k) γ F(U |V ) we denote the upper bound of r > 0 such that ◦
B(F(x), r t k ) ∩ V ⊂ F(B(x, t)), if x ∈ U, t < γ (x). (k)
(8)
(k)
If no such r exists, we set sur γ F(U |V ) = 0. We shall call sur γ F(U |V ) the γ -surjection modulus of order k of F on U × V . We can further define the order k surjection modulus of F at (x, ¯ y¯ ) ∈ Graph F, sur (k) F(x|y), as the upper bound of r > 0 such that for some ε > 0 (8) holds with ◦
◦
U = B(x, ε), V = B(y, ε) and γ (x) ≡ ε. (b) by reg(k) γ F(U |V ) we denote the lower bound of K > 0 such that 1 1 d(x, F −1 (y)) ≤ K d(y, F(x)) k if x ∈U, y ∈ V, K (d(y, F(x))) k < γ (x). (9) (k)
(k)
If no such K exists, we set regγ F(U |V ) = ∞. We shall call regγ F(U |V ) the γ -metric regularity modulus of order k of F on U × V . We shall also define the order k metric regularity modulus of F at (x, ¯ y¯ ) ∈ Graph F, reg(k) F(x|y), as the lower bound of K > 0 such that for some ε > 0 (9) holds with ◦
◦
U = B(x, ε), V = B(y, ε) and γ (x) ≡ ∞. 3 A set is locally complete at a point if its intersection with a closed neighborhood of the point is a complete
space in the induced metric.
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(k)
(c) by holδ F(U |V ) we denote the lower bound of K > 0 such that 1 1 d(y, F(x)) ≤ K d(x, u) k if x ∈ U, y ∈ V, y ∈ F(u), K (d(x, u)) k < δ(y). (10) (k)
(k)
If no such K exists, we set holδ F(U |V ) = ∞. We shall call holδ F(U |V ) the δ-Hölder modulus of order k of F on U × V . Finally, we define the order k Hölder modulus of F at (x, y) ∈ Graph F, hol(k) F(x|y), as the lower bound of K > 0 such that for some ε > 0 (10) holds ◦
◦
with U = B(x, ε), V = B(y, ε) and δ(x) ≡ ∞. If we adopt the convention that 0 · ∞ = 1 (standard in the quantitative regularity theory of variational analysis), then applying Proposition 1 and Theorem 1 for μ(t) = r t k we get Proposition 2 For any set-valued mapping F : X ⇒ Y and any (x, ¯ y¯ ) ∈ Graph F 1 reg(k) F(x|y) · sur (k) F(x|y) k = 1; reg(k) F(x|y) = hol(k) F −1 (y|x). (k)
(k)
Definition 5 If sur γ F(U |V ) > 0 (or equivalently, regγ F(U |V ) < ∞), we say that F is γ -regular of order k on U × V . Likewise, if sur (k) F(x|y) > 0 etc. we say that F is regular of order k at (x, ¯ y¯ ). Our immediate goal is to produce an infinitesimal criterion for regularity of order k. The criterion will be based on an order k extension of the concept of slope of DeGiorgiMarino and Tosques (see e.g. [8]). Recall that, given a function f on a metric space X which is finite at x, the slope of f at x is defined by |∇ f |(x) = lim sup u→x u =x
( f (x) − f (u))+ d(x, u)
Given a ρ ∈ R, α > 0, we set [ρ]α = |ρ|α signρ. We further define the function [ f ]α by [ f ]α (x) = [ f (x)]α . It is clear that [ f ]α is continuous (lower semicontinuous) at x if so is f . Using this function we further define the slope of order k of f at x as the slope of [ f ]1/k at the point: 1
|∇ (k) f |(x) = |∇[ f ] k |(x). For subsequent discussions we also need the concept of a locally coherent space (see [19]): X is locally coherent if for any x ∈ X lim |∇d(u, ·)|(w) = 1.
u,w→x u =w
It can be shown that any smooth manifold in a Banach space with the induced metric is a locally coherent space. Also any length space (see Sect. 6) is locally coherent.
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We are now ready to prove the slope criterion for regularity of order k. Given a set-valued mapping F : X ⇒ Y , we set for any y ∈ Y ϕ y (x, v) =
d(y, v), if v ∈ F(x), . ∞, otherwise.
(k)
Let |∇ξ ϕ y | stand for the k-slope of ϕ y with respect to the ξ -metric in X × Y . Theorem 5 (slope regularity criterion) Let F : X ⇒ Y be a set-valued mapping whose graph is complete in the product metric, let U and V be open subsets of X and Y respectively, and let γ be a function on X with Lipschitz constant 1 which is positive on U . Let finally k ≥ 1. Suppose there are ξ > 0, r > 0 such that 1 (k) ∇ξ ϕ y (x, v) > r k
(11)
whenever y ∈ V , x ∈ Uγ , v ∈ F(x) and 0 < d(y, v) < r (γ (x))k . Then sur (k) γ F(U |V ) > r > 0. In particular if for some (x, ¯ y¯ ) ∈ Graph F there is an ε > 0 such that (11) holds whenever d(x, x) < ε, d(y, y) < ε, v ∈ F(x), d(y, v) < ε,
(12)
then sur (k) F(x|y) ≥ r. Proof This is immediate from the general criterion of Theorem 2. Indeed, if (11) holds, then for any (x, y, v) satisfying the conditions of the theorem there is a (u, w) ∈ Graph F arbitrarily close to (x, v) and such that 1
1
ϕ yk (x, v) − ϕ yk (u, w) , r < dξ ((x, v), (u, w)) 1 k
so that (d(y, w))1/k ≤ (d(y, v))1/k − r 1/k dξ ((x, v), (u, w)).
If k = 1, the converse is valid (for regularity at a point) if Y is a locally coherent space. Namely if sur F(x|y) > r , then for every δ > 0 there is an ε > 0 such that |∇ξ ϕ y |(x, v) > (1 − δ)r for (x, y, v) satisfying (12)—see [17,18]. It is not clear whether this is true if k > 1, probably not. We can prove only a weaker relation, namely that for any δ > 0 there is a sequence (xn , vn ) ⊂ Graph F converging to (x, v) such that lim sup n→∞
123
d(y, v) − d(y, vn ) d(y, v) − d(y, vn ) ≥ r (1 − δ). ≥r (dξ ((x, v), (xn , vn )))k max{1, r ξ }d(v, vn )
(13)
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Indeed, since Y is locally coherent at (x, ¯ y¯ ), for any δ > 0 there is an ε > 0 such that |∇d(y, ·)|(v) > 1 − δ whenever both y and v belong to the ε-ball around y. Let us fix such an ε for every δ > 0. Since sur (k) F(x|y) > r , the inclusion ◦
B(v, r t k ) ⊂ F(B(x, t))
(14)
holds for (x, v) ∈ Graph F sufficiently close to (x, ¯ y¯ ) and sufficiently small t. We may assume that ε is so small that the inclusion holds in particular for x, v, t satisfying v ∈ F(x), d(x, x) < ε, d(v, y) < ε, 0 ≤ t < ε. Now fix a y with d(y, y) < ε, and let (x, v) ∈ Graph F be such that d(x, x) < ε, y = v and d(v, y) < ε, so that (14) holds for the given x and v. As both y and v belong to the open ε-ball around y, and Y is assumed to be locally coherent, there is a sequence (vn ) converging to v and and such that d(y, v) − d(y, vn ) → |∇d(y, ·)|(v) > 1 − δ. d(vn , v)
(15)
By (14) for sufficiently large n there are xn such that vn ∈ F(xn ) and d(xn , x) ≤ (d(vn , v)/r )1/k which, together with (15), implies (13). The same arguments as in the proof of Theorem 5 with reference to Theorem 3 (rather than Theorem 2) prove a criterion for upper semicontinuous mappings. Set ψ y (x) = d(y, F(x)). Theorem 6 Let as before F : X ⇒ Y have complete graph, and let U ⊂ X, V ⊂ Y be open sets. Suppose that F is upper semicontinuous and there are k ≥ 1, r > 0 and a function γ on X satisfying the Lipschitz condition with constant not greater than 1 which is positive on U and such that 1
|∇ (k) ψ y |(x) > r k
whenever x ∈ Uγ , y ∈ V and 0 < d(y, F(x)) < r (γ (x))k . Then F −1 (y) = ∅ for every y ∈ V and d(x, F −1 (y)) ≤ [r −1 d(y, F(x))]1/k if d(y, F(x)) < r (γ (x))k . In (k) particular F is γ -regular of order k on U × V with sur γ F(U |V ) ≥ r . 5 Error bounds of order k In this short section we shall apply the last criterion to extended-real-valued functions on X in order to get estimates for error bounds associated with level sets. So let f be such a function. For an α ∈ R denote by [ f ≤ α] = {x : f (x) ≤ α} the α-sublevel set of f . The meaning of [ f > 0] is clear. We shall be interested in estimating the distance to [ f ≤ 0] from points of [ f > 0] in terms of the values of f at points of the set.
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Theorem 7 Let f be a lower semicontinuous function on X , let U ⊂ X be an open set, and let γ be a Lipschitz function on X with Lipschitz constant not greater than one which is positive on U . Assume that k ≥ 1 and |∇ (k) f |(x) > r > 0 for all x ∈ [ f > 0] ∩ Uγ . Then [ f ≤ 0] = ∅ and ( f + (x))1/k d x, [ f ≤ 0] ≤ , r for all x ∈ U satisfying f (x) < (r γ (x))k . Proof As f is lower semicontinuous, so is the function f + , and the set-valued mapping E f + (x) = {α ∈ R : α ≥ f + (x)} viewed as a mapping from X into R+ , the nonnegative half line, is upper semicontinuous. In particular for any α ∈ R+ the function x → ϕα (x) = d(α, E f (x)) = ( f (x) − α)+ is also lower semicontinuous. We apply Theorem 6 to F = E f + with Y = R+ , V = Y . If α ≥ 0 does not belong to E f + (x), that is α < f (x), then ϕα (x) = d(α, E f + (x)) = f (x) − α. We notice further that (by concavity of t 1/k ) for any α > 0 and any x, u such that both f (x) and f (u) are greater than α we have 1
1
1
1
( f (x) − α) k − ( f (u) − α) k ≥ ( f (x)) k − ( f (u)) k
and therefore for any x ∈ [ f > 0] and any α ∈ [0, f (x)) we have |∇ (k) ϕα |(x) ≥ |∇ (k) f |(x) > r . Thus the conditions of Theorem 6 are satisfied and we conclude that the set (E f + )−1 (0) = [ f ≤ 0] is nonempty and for any x ∈ U such that f + (x) < (r γ (x))k 1
1 1 ( f + (x)) k d 0, [ f ≤ 0] ≤ (ϕ0 (x)) k = r r
as claimed.
The point to be mentioned in connection with the last theorem is that it gives a global estimate based on purely infinitesimal information. 6 A variational viewpoint In this section we shall discuss the variational approach to local regularity of order k ≥ 1. The core of the approach is the study of the behavior of a set-valued mapping with respect to certain curves in the range space representing variations of their initial points. Clearly, some additional restrictions on the class of possible range spaces is needed to make such an approach implementable. First we recall the definition of length for a curve in a metric space (we refer to [1,7] for the details concerning geometry of metric spaces). Let η be a continuous curve in X parametrized by a continuous mapping m(·) from a line segment [0, 1] into X . The curve is called absolutely continuous (or rectifiable) if for almost any t ∈ (0, 1) the limit
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d(m(t + ε), m(t)) ε→+0 ε
|m |(t) = lim
exists and the function |m | is summable. The integral 1 (m(·)) =
|m |(t)dt
0
is called the length of the curve. The length does not depend on the choice of a parametrization. Every rectifiable curve can be parametrized by a mapping m with |m (t)| ≡ 1. We shall call such a parametrization (which is unique, provided the initial point is specified) natural. X is called a length space if the distance between any two points coincides with the lower bound of lengths of rectifiable curves joining the points. The metric defined by the lengths of curves in a metric space is called intrinsic. It is quite clear that X is a length space only if d(y, v) = t + d(y, B(v, t)) for any y = v and any t ≤ d(y, v). The converse is true if the space is complete. It can be shown (see [2]) that a complete metric space is a length space if and only if |∇d(u, ·)|(x) = 1 for any x and any u = x. A metric in X is called strictly intrinsic if X is a length space and any two points y, v can be connected by a shortest path, that is a rectifiable curve joining y and v whose length is equal to d(y, v). In particular the metric of a locally compact length space is strictly intrinsic. A metric space with strictly intrinsic metric is called a geodesic space. Clearly a normed space is a geodesic space as well as a huge variety of (even nonsmooth) manifolds. Such is for instance the boundary of the unit square in R2 with the intrinsic metric or a product of such spaces (a sort of “square torus”). We shall start however with a general result valid for arbitrary range spaces. Theorem 8 Let X and Y be metric spaces, and let F : X ⇒ Y be a set valued mapping whose graph is complete in the product metric. Let (x, ¯ y¯ ) ∈ Graph F, and let k ≥ 1. Then sur (k) F(x|y) = sup r ≥ 0 :
ex(B(v, r t k ), F(B(x, t))) = 0 . tk (x,v,t)→(x,y,+0) lim sup v∈F(x)
(16) Proof It is clear that sur (k) F(x|y) is not greater than the quantity in the right-hand side. To prove the opposite inequality, take a small λ > 0 and choose δ > 0 such that ex(B(v, r t k ), F(B(x, t)) ≤ λt k if d(x, x) < δ, d(v, y) < δ, 0 < t < δ, v ∈ F(x). This means that F(B(x, t)) is a λt k -net in B(v, r t k ) for such x, v, t. By Corollary 1 F ◦
◦
is regular on B(x, δ)× B(y, δ) with functional modulus not smaller than (1−λ1/k )k r t k which implies that sur (k) F(x|y) ≥ (1−λ1/k )k r . But λ can be chosen arbitrarily small, whence sur (k) F(x|y) ≥ r . If Y is a length space, then (16) can be expressed in variational terms. The following is a possible general pattern for that. Let Z be a topological space, and let Φ be
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a continuous mapping from Y × Z × [0, 1] into Y such that Φ(v, z, 0) = v for all (v, z). In other words, Φ(v, ·, t) is a retraction of Φ(v, Z , 1) to v and for any z ∈ Z the mapping t → Φ(v, z, t) is a parametrization of a certain curve starting at v. We can view such a curve as a variation of v and, accordingly, interpret Z as a space of variations of elements of Y . In fact, we can consider even “set-valued” retractions. We can now state the following theorem. Theorem 9 Assume as always that F : X ⇒ Y is a set-valued mapping with complete graph and (x, ¯ y¯ ) ∈ Graph F. Suppose Z is a compact topological space and Φ : Y × Z × [0, 1] ⇒ Y is a set-valued mapping with closed graph. We assume that there ◦
are ε > 0 and c > 0 such that for all v ∈ B(y, ε), z ∈ Z and t ∈ [0, 1] Φ(v, z, 0) = {v}, B(v, ct) ⊂ Φ v, Z , [0, t]
(17)
and for any ξ > 0 there is a finite set Q ξ ⊂ Z such that Φ(v, Q ξ , [0, t]) is a tξ -net in B(v, ct). If under these conditions there are r > 0 and k ≥ 1 such that ex Φ(v, z, r t k ), F(B(x, t)) =0 lim (x,v,t)→(x,y,+0) tk
(18)
v∈F(x)
for any z ∈ Z , then sur (k) F(x|y) ≥ cr . Proof Let an r > 0 be such that (18) holds. Take a ξ > 0. By (18) for any z ∈ Q ξ there is a δ = δ(z) such that ex Φ(v, z, r t k ), F(B(x, t)) < ξ t k
(19)
d(x, x) < δ, d(v, y) < δ, v ∈ F(x), 0 ≤ t < δ.
(20)
if
Take a δ > 0 smaller then all δ(z) corresponding to z ∈ Q ξ , and let Wξ (t) stand for Q ξ × [0, r t k ]. Then (19) means that ex(Φ(v, Wξ (t)), F(B(x, t))) ≤ ξ t k
(21)
if x, v, t satisfy (20). By the assumption F(v, Wξ (t)) is a (r t k )ξ -net in B(v, cr t k ). Therefore (21) implies that ex B(v, cr t k ), F(B(x, t)) lim sup ≤ (1 + r )ξ. tk (x,v,t)→(x,y,+0) v∈F(x)
As ξ can be chosen arbitrarily small, we conclude that the condition of Theorem 8 is satisfied and therefore sur (k) F(x|y) ≥ cr as claimed.
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Remark 1 The theorem does not give an exact formula for the surjection modulus of order k, just a lower bound for it. However if we assume in (17) that actually Φ(v, Z , [0, t]) = B(v, ct) for all v, z and t, then the fact that sur (k) F(x|y) cannot exceed the upper bound of cr such that (18) holds is immediate from Definition 5. The simplest case when the conditions of the theorem hold occurs when Y is Rn , Z = S n−1 , the unit sphere in Rn , Φ(v, z, t) = v + t z and (18) is valid. Indeed in this case B(v, t) = Φ(v, Z , [0, t]) which immediately implies (17) and the ξ -net assumption, so the assumptions of the theorem follow with c = 1. It is not a difficult matter to see that, in this case, the theorem implies the following characterization of higher order regularity moduli of mappings into Rn established in a recent paper by Frankowska and Quincampoix [14]: ¯ y¯ ) ∈ Corollary 3 ([14]) Let F be a set-valued mapping from X into Rn , and let (x, Graph F. Set A=
lim inf
(x,v,t)→(x,y,+0) v∈F(x)
F(B(x, t)) − v . tk
(22)
Then sur (k) F(x|y) = sup{r ≥ 0 : r B ∈ A}, where B is the unit ball in Rn . Proof Indeed, both (18) with Φ(v, z, t) = v +t z and (22) amount to the statement that for any z ∈ S n−1 and ξ > 0 there is a δ > 0 such that d(v + r t k z, F(B(x, t))) < ξ t k if x, v, and t satisfy (20). Thus Theorem 9 implies that sur (k) F(x|y) ≥ sup{r ≥ 0 : r B ∈ A}. To see that the opposite inclusion holds we refer to Remark 1. The curves Φ(v, z, ·) in the last corollary are of course line segments joining v and v + z. It may seem that the ball in Rn in the statement can be replaced by a ball in a locally compact length space in which case instead of the segments we should consider the shortest paths connecting points of the corresponding sphere with the center of the ball. However such a replacement requires caution because it may happen that the shortest paths joining the center of the ball with points of the sphere do not cover the entire ball. Example 2 Let Y = X be the boundary of the first orthant in R3 : X = Y = {x = (x 1 , x 2 , x 3 ) ∈ R3 : x i ≥ 0, x 1 x 2 x 3 = 0}, and let Y be endowed with the (strictly) intrinsic metric induced by the Euclidean metric in R3 . (That is to say, the distance between two points in Y is the minimal Euclidean length of curves lying in Y and joining the points. Clearly, any curve of minimal length is either a linear segment or a pair of linear segments lying in different coordinate planes and having a common point on the axis common for the planes.) 3 Take √ for instance v = (1, 0, 0) and consider in Y (not in R !) the sphere1 of 2radius 2 around v. The configuration of the pieces of√ the sphere lying in the (x , x ) and (x 1 , x 3 ) planes is obvious: circular arcs of radius 2 around v. The piece of the sphere
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lying in the (x 2 , x 3 )-plane in turn consists of two parts symmetric with respect to the ray x 2 = x 3 and defined by the parametric relation √ 1 + 2t 2 − t max{x , x } = r (t), min{x , x } = tr (t), 0 ≤ t ≤ 1, r (t) = . 1 + t2 2
3
2
3
For a point u = (0, x 2 , x 3 ) belonging to this piece of the sphere, say with x 2 ≥ x 3 , the shortest path joining it with v consists of two line segments, one in the (x 2 , x 3 )-plane joining u with
√ 1 + 2t 2 − t w = 0, √ ,0 t 1 + 2t 2 + 1
(where x 3 = t x 2 )
and the other segment in the (x 1 , x 2 )-plane joining w and v. We see therefore that there is no shortest path connecting points of the sphere with v and passing through points in the (x 2 , x 3 )-plane belonging to the interior of the diamond bounded by the segments connecting the points (0, ξ, 0) and (0, 0, ξ ) with (0, η, η) and the origin, where √ √ 3−1 3−1 η= . ; ξ=√ 2 3+1 The extension mentioned before the example is however possible if Y itself is a homogeneous space. The latter means that for any two finite isometric collections of points in Y there is an isometry of Y that moves one of the collections into the other. We actually need less: for any three points yˆ , y and z such that d(z, yˆ ) = d(y, yˆ ) there is an isometry that keeps yˆ invariant and moves z into y. Such are for instance spheres in Euclidean spaces or tori. Theorem 10 Let X be a metric space, let Y be a locally compact homogeneous length space, and let F : X ⇒ Y be a set-valued mapping with a complete graph. Fix a yˆ ∈ Y , and let Γv be an isometry of Y such that Γv ( yˆ ) = v and the mapping (v, y) → Γv (y) is continuous. For any y ∈ S( yˆ , 1), let M y be the collection of shortest paths joining y and yˆ , and let M y (t) be the collection of points of the paths of M y that are at the distance t from yˆ . Let finally (x, ¯ y¯ ) ∈ Graph F. Then sur (k) F(x|y) is the upper bound of r ≥ 0 such that for all y ∈ S( yˆ , 1) ex(Γv (M y (r t k )), F(B(x, t))) = 0. (x,v,t)→(x,y,+0) tk lim
(23)
v∈F(x)
Proof We note first that for any t ∈ (0, 1) the union of M y (t) over y ∈ S( yˆ , 1) coincides with S( yˆ , t). Indeed let d(y, yˆ ) = t (t < 1), and let η be a shortest path connecting yˆ with some point of the 1-sphere around yˆ . Let z ∈ η ∩ S( yˆ , t), and let Γ be an isometry than keeps yˆ invariant and moves z into y. Then y ∈ Γ (η) and the
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latter is the shortest path whose other end is in S( yˆ , 1). Let Z = S( yˆ , 1). Now set Φ(v, z, t) = Γv (Mz (t)). Then as we have just seen, Φ(v, Z , [0, 1]) = B(v, 1) and it is not a difficult matter to verify that the conditions of Theorem 9 are satisfied. Acknowledgment I wish to express my thanks to the reviewers for the detailed analysis of the text and many good suggestions.
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26. Robinson, S.M.: Regularity and stability for convex multivalued functions. Math. Oper. Res. 1, 130–143 (1976) 27. Tziskaridze, KSh: Extremal problems in Banach spaces. In: Nekotorye Voprosy Matematicheskoy Theorii Optimalnogo Upravleniya (Some Problems of the Mathematical Theory of Optimal Control). Tbilisi State Univ, Inst. Appl. Math. (1975) (in Russian)
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Nonlinear regularity models Alexander D. Ioffe
Received: 9 February 2011 / Accepted: 7 September 2011 / Published online: 22 March 2013 © Springer-Verlag Berlin Heidelberg and Mathematical Optimization Society 2013
Abstract The paper studies regularity properties of set-valued mappings between metric spaces. In the context of metric regularity, nonlinear models correspond to nonlinear dependencies of estimates of error bounds in terms of residuals. Among the questions addressed in the paper are equivalence of the corresponding concepts of openness and “pseudo-Hölder” behavior, general and local regularity criteria with special emphasis on “regularity of order k”, for local settings, and variational methods to extimate regularity moduli in case of length range spaces. The majority of the results presented in the paper are new. Keywords Metric regularity · Regularity criterion · Regularity of order k · Error bound · Length space Mathematics Subject Classification (2000)
47H04 · 49J53 · 90C31
1 Introduction Regularity is one of the basic concepts of modern analysis. Among its many faces, the most popular in the variational analysis community is “metric regularity”, the term coined by Borwein in 1986 [4] for estimates of distances to preimages of set-valued mappings in terms of residuals. The first estimates of that sort starting with the famous Hoffman’s estimate for systems of linear inequalities [15] and later results of Ioffe– Tikhomirov and Robinson [16,21,25,26], as well as the estimate in the definition
The paper is dedicated to Jon Borwein’s sixtieth anniversary. A. D. Ioffe (B) Department of Mathematics, Technion, 32000 Haifa, Israel e-mail: [email protected]
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in [4], were linear in the sense that the bound for the first distance was equal to the second one times a certain positive number. We consider regularity models which, in the context of metric regularity, correspond to nonlinear dependence of the bound on the residual. Such models are in particular useful in the analysis of sensitivity and controllability in optimization and control (see e.g. [3,13]). The study of nonlinear regularity models was initiated in late 80s by Borwein–Zuang [6], Frankowska [10–12] and Penot [23]. The main attention in [6,23] was focused on equivalent characterizations of nonlinear regularity with arbitrary nonlinearities. We in particular wish to stress the mention of approximate openness among the equivalences in [6]. On the other hand, [10,11] and subsequent studies by Frankowska [13] and Frankowska and Quincampoix [14] were devoted to analysis, mainly to variational approach to regularity (for mappings into Banach spaces) with power nonlinearities. All mentioned papers deal with local regularity properties in an arbitrary small neighborhood of a given point of the graph of the (generally setvalued) mapping. In our survey [17] we briefly discussed nonlinear regularity, mainly to emphasize similarities with the linear theory and the role of the slope-based metric infinitesimal mechanisms. Here we try to present a systematic study of nonlinear regularity theory in metric spaces. The latter means that (apart from a few corollaries) we consider set-valued mappings between metric spaces. In the main part of the paper we consider general nonlinearities associated with gauge functions and then consider in more details higher order regularity properties associated with power nonlinearities. We begin by proving fairly general equivalence theorem saying that (also in the non-local case) nonlinear versions of openness, metric regularity and pseudo-Hölder properties of the inverse mapping are equivalent. We then prove a sufficient regularity criterion which is a generalization of a nonlinear criterion stated without proof in [17], on the one hand, and of the criterion for linear regularity for mappings between metric spaces in [20], on the other. The criterion is then applied to get a nonlinear density theorem (in the spirit of Dmitruk’s theorem in [9] for the standard linear global regularity model1 ) which contains quantitative measures of density of the images of balls in the domain space in corresponding balls in the range space that guarantee that the latter actually fully lie within the image. Note that the equivalence of the approximate openness and openness proved in [6] follows from the density theorem. The mentioned results are most general in the sense that they are applied not just for arbitrary metric spaces but also for arbitrary gauge functions. The subsequent results are all dealing with power gauge functions μ(t) = r t k with k ≥ 1. The immediate advantage is the possibility to define numerical measures of regularity which we call moduli of order k. Our main tool of analysis is a higher order extensions of the strong slope of DeGiorgi–Marino–Tosques [8]. As in the linear case these slopes allow to get estimates for higher order moduli of regularity and higher order error bounds for sublevel sets of lower semicontinuous functions. In the concluding section we consider a variational approach to computing regularity moduli of higher order. The core of this approach is the study of the behavior of a 1 It is appropriate to mention here a very general result of Khan [22] which contains as particular case the results of Pták [24] and Dmitruk.
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set-valued mapping along geodesic curves in the range space viewed as variations of their initial points. For the range space being Rn a result in this direction was recently obtained in [14]. It turns out that the key property of Rn (in addition to compactness of the unit ball) is that the unit sphere S n−1 is a homogeneous space with respect to the group of orthogonal transformations of Rn . We show that similar formulas can be obtained for set-valued mappings into arbitrary homogeneous locally compact length spaces. Notation In what follows X , Y etc. are metric spaces. We use the same symbol d(·, ·) to denote the distance in all of them and hope this will not be a cause of any ◦
confusion. By B(x, ρ) and B(x, ρ) we denote the closed and open balls of radius ρ ◦
around x. Likewise, if Q ⊂ X , then B(Q, ρ) and B(Q, ρ) are the closed and open ◦
ρ-neighborhoods of Q, that is, say B(Q, ρ) = {x ∈ X : d(x, Q) < ρ}. Given two sets P and Q in X , we denote by ex(P, Q) = inf{ε > 0 : P ⊂ B(Q, ε)} the excess of P over Q. By S(x, ρ) we denote the sphere of radius ρ around x. If F : X ⇒ Y is a set-valued mapping from X into Y , then Graph F = {(x, y) ∈ X × Y : y ∈ F(x)} is the graph of F and dom F = {X : F(x) = ∅} is the domain of F. In the product space X × Y we shall usually consider the ξ -metric (with some ξ > 0) dξ (x, y), (u, v) = max{d(x, u), ξ d(y, v)}. For a Q ⊂ X we denote by i Q the indicator of Q which is the function equal to zero on Q and infinity outside of Q. 2 Definitions and equivalences We shall make an extensive use of gauge functions (or just gauges) by which we mean a continuous strictly increasing functions t → μ(t) on [0, ∞) equal to zero at 0 and going to infinity when t → ∞. Clearly, the inverse of a gauge is also a gauge. Definition 1 Given an F : X ⇒ Y , where X and Y are metric spaces. Let U ⊂ X and V ⊂ Y be open sets, let γ be a function on X which is positive on U , and let δ be a function on Y which is positive on V . Assume finally that we are given three gauge functions μ, ν and η. (a) F is γ -open on U ×V with functional modulus (not smaller than) μ if the inclusion ◦
B(F(x), μ(t)) ∩ V ⊂ F(B(x, t)) holds whenever x ∈ U and 0 < t < γ (x). (b) F is γ -metrically regular on U × V with functional modulus (not greater than) ν if the inequality d(x, F −1 (y)) ≤ ν(d(y, F(x))) holds whenever x ∈ U, y ∈ V and 0 < ν(d(y, F(x))) < γ (x).
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(c) F is δ-Hölder on U × V with functional modulus (not greater than) η if the inequality d(y, F(x)) ≤ η(d(x, u)), holds provided x ∈ U, y ∈ V ∩ F(u) and 0 < η(d(x, u)) < δ(y). Note that the definition does not mean to introduce “functional moduli” as specific objects, although it could be possible in principle. We do not need them and shall use this expression only in the same context as in the definition. This should be kept in mind because occasionally, when it does not lead to a confusion, we shall for simplicity omit expressions “not smaller than” etc. and shall speak about openness with functional modulus μ and so on. If we compare e.g. the openness part of this definition with available definition of the standard “openness at a linear rate”, we notice two major differences. The first, and the obvious, is that r t has been replaced by μ(t). The second difference is the appearance of the “reach function” γ which determines the set of (x, v) ∈ Graph F which are actually involved in implementation or verification of the properties. The following simple example shows that this is an essential part of the definition and changing γ we may change the parameters of regularity or even to kill regularity at all. Example 1 Set X = Y = R, U = V = (0, ∞), F(x) = {x 2 }, μ(t) = t 3 . Then ◦
for any positive x we have B(F(x), μ(t)) = (x 2 − t 3 , x 2 + t 3 ) and F(B(x, t)) = [((x − t)+ )2 , (x + t)2 ]. By definition F is γ -open on U × V with functional modulus not smaller than t 3 if and only if (x 2 − t 3 )+ ≥ ((x − t)+ )2 and x 2 + t 3 ≤ (x + t)2 for all 0 ≤ t < γ (x). The second relation means that t (t − 1) ≤ 2x for all t < γ (x) which is the same as γ (x)(γ (x) − 1) ≤ 2x. To analyze the first relation we consider separately the cases γ 3 (x) ≤ x 2 and γ 3 (x) > x 2 . In the first case the first relation reduces to γ (x)(γ (x) + 1) ≤ 2x in which case the second relation is automatically satisfied. In the second case the first relation cannot be satisfied if x > 1. Indeed in this case for t < x sufficiently close to x we would have t 3 > x 2 . On the other hand if x ≤ 1, then the first relation holds for all t < γ (x) (indeed, (x − t)+ = 0 if t ∈ [x, γ (x)) and t (t + 1) ≤ 2x if t ≤ x). Thus F is γ -open with functional modulus t 3 if and only if either γ 3 (x) ≤ x 2 and γ (x)(γ (x) + 1) ≤ 2x or γ 3 (x) > x 2 , x ≤ 1 and γ (x)(γ (x) − 1) ≤ 2x. Moreover if we take γ satisfying 8γ 3 (x) ≤ x 2 for x > 1 and γ (x)(2γ (x)+1) ≤ x, then F is γ -open on U × V with functional modulus not smaller than (2t)3 . On the contrary, local properties that hold in small neighborhoods of a point of the graph are insensitive to variations of γ , so that the latter turns out to be needless. Local versions of the definitions, near a certain (x, ¯ y¯ ) ∈ Graph F, correspond to U and V being arbitrarily small neighborhoods of x and y respectively. Before stating them in a formal manner we shall prove a simple proposition which implies the mentioned insensitivity of local regularity properties to variations of γ .
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Proposition 1 Let F : X ⇒ Y , let (x, ¯ y¯ ) ∈ Graph F, and let r > 0. Then the following properties are equivalent: (a) there is an ε > 0 such that ◦
◦
B(F(x), μ(t)) ∩ B(y, ε) ⊂ F(B(x, t)), if d(x, x) < ε, (b) there is an ε > 0 such that ◦
◦
B(F(x), μ(t)) ∩ B(y, ε) ⊂ F(B(x, t)), if d(x, x) < ε, 0 < t < ε, (c) there is an ε > 0 such that ◦
◦
B(F(x) ∩ B(y, ε), μ(t)) ⊂ F(B(x, t)), if d(x, x) < ε, 0 < t < ε, (d) there is an ε > 0 such that ◦
◦
◦
B(F(x) ∩ B(y, ε), μ(t)) ∩ B(y, ε) ⊂ F(B(x, t)), if d(x, x) < ε, 0 ≤ t < ε. Proof It is clear that (a) ⇒ (b) ⇒ (d) and (c) ⇒ (d). Thus we have to show that (d) implies (a) and (c). So let (d) hold with some ε > 0. Take a τ > 0 and α > 0 such that α + μ(α) < τ < ε, μ(τ ) + μ(α) < ε
(1)
and set δ = min{α, μ(α)}. Clearly δ < ε We shall first prove that (a) holds with ε replaced by δ. The inclusion y ∈ ◦
◦
B(F(x), μ(t)) ∩ B(y, δ) implies that d(y, y) < δ and d(y, v) < μ(t) for some v ∈ F(x). So if t < τ and d(x, x) < δ, then it follows from (1) that d(v, y) < μ(t) + δ ≤ μ(τ ) + μ(α) < ε and therefore by (d) ◦
◦
◦
◦
◦
B(F(x), μ(t)) ∩ B(y, δ) ⊂ B(F(x) ∩ B(y, ε), μ(t)) ∩ B(y, ε) ⊂ F(B(x, t)). On the other hand, again by (d) ◦
◦
◦
B(y, δ) ⊂ B(F(x) ∩ B(y, ε), μ(α)) ∩ B(y, ε) ⊂ F(B(x, α)), so for t ≥ τ , we have ◦
◦
B(F(x), μ(t)) ∩ B(y, δ) ⊂ F(B(x, α)) ⊂ F(B(x, t)) because by (1) α + d(x, x) ≤ α + μ(α) ≤ τ ≤ t, and (a) follows.
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◦
Let us prove (c). If y ∈ B(F(x) ∩ B(y, δ), μ(t)) for some t < δ, then there is a v ∈ F(x) such that d(v, y) < δ and d(y, v) < μ(t). As α < τ < ε, this means that d(y, y) < δ + μ(t) ≤ μ(α) + μ(τ ) < ε, and consequently by (d) ◦
◦
◦
◦
◦
B(F(x) ∩ B(y, δ), μ(t)) ⊂ B(F(x) ∩ B(y, ε), μ(t)) ∩ B(y, ε) ⊂ F(B(x, t))
and (c) follows.
The above proposition says that speaking about local regularity properties, we can either omit any mention of γ (which is tantamount to taking γ (x) ≡ ∞) or take it arbitrarily small. Each of the four properties (a)–(d) can be equally used as a basis for the definition of the local version of “nonlinear” openness. Definition 2 Let F : X ⇒ Y , let (x, ¯ y¯ ) ∈ Graph F and let μ be a gauge function. We say that F is open at (x, ¯ y¯ ) with modulus germ not smaller than μ if the equivalent properties of Proposition 1 hold. We leave to the reader the simple task of reformulating the local analogues of the other two properties of Definition 1. The equivalence of the local versions of the three properties was basically established by Penot in [23]. The theorem below extends this result to the general situation of Definition 1. Theorem 1 Let F : X ⇒ Y , let U ⊂ X , V ⊂ Y be open sets, and let γ be a function on X which is positive on U . The following properties are equivalent, given a gauge function μ. (a) F is γ -open on U × V with functional modulus not smaller than μ; (b) F is γ -metrically regular on U × V with functional modulus not greater than μ−1 ; (c) F −1 is γ -Hölder on V × U with functional modulus not greater than μ−1 . Proof The implication (b) ⇒ (c) is trivial. To prove that (c) ⇒ (a), let t < γ (x), and ◦
◦
let y ∈ V and y ∈ B(F(x), μ(t)), that is y ∈ B(v, μ(t)) for some v ∈ F(x).Then μ−1 (d(y, v)) < t < γ (x) and y ∈ V , so by (c) d(x, F −1 (y)) ≤ μ−1 d(y, v) < t. This means that there is a u such that y ∈ F(u) and d(x, u) < t, that is y ∈ F(B(x, t)). ◦
As y is an arbitrary element of B(F(x), μ(t)) ∩ V , (a) follows. (a) ⇒ (b). Take an x ∈ U and y ∈ V with μ−1 (d(y, F(x))) < γ (x). Let further K > 1 be such that t = μ−1 (K d(y, F(x)) < γ (x). Such a K definitely exists as μ−1 is a gauge, hence continuous. We have K −1 μ(t) = d(y, F(x)). Take an r < 1 such ◦
that r K > 1. Then d(y, F(x)) < r μ(t), that is y ∈ B(F(x), r μ(t)). Define t and K by μ(t ) = r μ(t) and t = μ−1 (K d(y, v)). Then t < γ (x) (as r < 1), so by (a) y ∈ F(B(x, t )) which means that there is a u with d(x, u) ≤ t such that y ∈ F(u). In other words, d(x, F −1 (y)) ≤ d(x, u) ≤ t = μ−1 (K d(y, F(x))). It is clear that K and r can be chosen to make K arbitrarily close to 1, and (b) follows. The theorem justifies the following definition. Definition 3 We shall say that F is γ -regular on U × V with functional modulus μ if the three equivalent properties of Theorem 1 hold.
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3 Criterion of nonlinear regularity and some applications Let again U ⊂ X and V ⊂ Y be open sets, and let γ be a function on X with Lipschitz constant 1. We set Uγ =
◦
B(u, γ (u)).
u∈U
Let further μ be a gauge function. Lemma 1 Let F : X ⇒ Y be a set-valued mapping whose graph is complete in the product metric, let U and V be open sets in X and Y respectively, let μ be a gauge function, and let γ be a Lipschitz function on X with Lipschitz constant 1 which is positive on U . Suppose there are y ∈ Y and a ξ > 0 such that for any x ∈ Uγ , v ∈ F(x) with 0 < d(y, v) < μ(γ (x)) there is a pair (u, w) ∈ Graph F, (u, w) = (x, v) such that μ−1 d(y, w) ≤ μ−1 d(y, v) − dξ ((x, v), (u, w)).
(2)
Then for any (x, v) ∈ Graph F with x ∈ Uγ , 0 < d(y, v) < μ(γ (x)) there is a ˆ and uˆ ∈ Uγ such that y ∈ F(u) ˆ y)) ≤ μ−1 (d(y, v)). dξ ((x, v), (u, Proof Set ψ y (x, v) = μ−1 (d(y, v)) + i Graph F (x, v). Let x ∈ U , v ∈ F(x) and d(v, y) < μ(γ (x)). Set ε = ψ y (x, v) and find, using Ekeland’s principle (see e.g. [5]), a pair (u, ˆ w) ˆ such that ˆ w)) ˆ ≤ ε; dξ ((x, v), (u, ψ y (u, ˆ w) ˆ ≤ ψ y (x, v) − dξ ((x, v), (u, ˆ w)); ˆ
(3)
ψ y (u, w) + dξ ((u, w), (u, ˆ w)) ˆ > ψ y (u, ˆ w), ˆ if (u, w) = (u, ˆ w). ˆ We claim that ψ y (u, ˆ w) ˆ = 0, that is wˆ = y. Assuming the contrary we shall get a contradiction. Indeed, note first that d(x, u) ˆ < γ (x). This is a consequence of the first inequality in (3) which implies by the assumptions that d(x, u) ˆ ≤ μ−1 (d(y, v)) < ˆ < γ (u). Indeed, as follows from γ (x). Thus uˆ ∈ Uγ . Next we observe that d(y, w) the assumption on γ and the second inequality in (3), ˆ w) ˆ < γ (x) − d(x, u) ˆ ≤ γ (u). ψ y (u, Therefore by (2) there is a (u, w) = (u, ˆ w) ˆ in Graph F such that μ−1 d(y, w) ≤ μ−1 d(y, w) ˆ w), ˆ (u, w)). ˆ − dξ ((u, The latter contradicts to the last relation in (3). Thus uˆ ∈ F −1 (y) = ∅. The final estimate of the lemma follows from the first inequality in (3).
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Theorem 2 Let F : X ⇒ Y be a set-valued mapping whose graph is complete in the product metric, let γ be a Lipschitz function on X with Lipschitz constant 1, let μ be a gauge function, and let U and V be open subsets of X and Y respectively. Then F is γ -open on U × V with functional modulus μ if the following holds: there is a ξ > 0 such that for any x ∈ Uγ , y ∈ V, v ∈ F(x) with 0 < d(y, v) < μ(γ (x)) there is a pair (u, w) ∈ Graph F, (u, w) = (x, v) such that (2) holds. Conversely, assume that F is γ -open on U × V with functional modulus μ and there is a ξ > 0 such that ξ t ≤ μ−1 (t) for all t < supU γ (x). Then for any x ∈ U , y ∈ V and v ∈ F(x) with 0 < d(y, v) < μ(γ (x)) there is a u such that y ∈ F(u) and (2) holds. Proof The first statement is an immediate consequence of the lemma. Indeed, given x ∈ U , y ∈ V , t < γ (x) and v ∈ F(x) with 0 < d(y, v) < μ(t), we can find by the lemma a pair (u, ˆ w) ˆ ∈ Graph F such that y ∈ F(u), ˆ y = v and d(x, u) ˆ ≤ t. It ◦
follows that B(v, μ(t)) ⊂ F(B(x, t)). To prove the second statement suppose that F is γ -open with functional modulus not smaller than μ. Given x, y, v as above, we find a u such that y ∈ F(u) and d(x, u) ≤ μ−1 (d(y, v)). Let further ξ > 0 satisfies the conditions specified in the statement. Then, dξ ((x, v), (u, y)) = max{d(x, u), ξ d(v, y)} ≤ μ−1 (d(y, v)), and setting w = y, we get 0 = μ−1 (d(y, w)) ≤ μ−1 (d(y, v)) − dξ ((x, v), (u, w))
as claimed.
Note that in case supU γ (x) = ∞, μ−1 (t) ≥ ξ t for all t, hence μ is majorized by a linear function. If on the other hand, the supremum is finite, the conditions mean that μ is majorized by a linear function in an arbitrarily small neighborhood of zero. The criterion can be simplified and strengthened if F is upper semicontinuous in the sense that for any y ∈ Y the function x → d(y, F(x)) is lower semicontinuous. Theorem 3 (criterion for an u.s.c. mapping) Suppose in addition to the assumptions of Theorem 2 that F is upper semicontinuous as defined above. Suppose further that for any x ∈ Uγ , any y ∈ V such that 0 < d(y, F(x)) < μ(γ (x)) there is a u = x such that μ−1 d(y, F(u)) ≤ μ−1 d(y, F(x)) − d(x, u).
(4)
Then F −1 (y) = ∅ for all y ∈ V and d(x, F −1 (y)) ≤ μ−1 (d(y, F(x)) if x ∈ U , y ∈ V . In particular F is γ -regular on U × V with functional modulus μ. Conversely, if F is γ -regular on U × V with functional modulus μ, then for any x ∈ U, y ∈ V there is a u such that (4) holds.
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Proof To prove the first statement, we only need to repeat the proof of the lemma with ψ y (x, v) replaced by f y (x) = d(y, F(x)). The second statement follows from the trivial observation: if y ∈ F(u) and d(x, u) ≤ μ−1 (d(y, v)), then μ−1 (d(y, y)) = 0 ≤ μ−1 (d(y, v)) − d(x, u). As an application of the just proved regularity criteria, we give below a nonlinear extension of one of the main theorems of [9] due to Dmitruk. Theorem 4 (nonlinear density theorem) Let U ⊂ X and V ⊂ Y be open sets, let F : X ⇒ Y be a set-valued mapping with complete graph, and let γ be a Lipschitz function on X with Lipschitz constant 1, positive on U and equal to zero on bdU . We assume that there are a gauge function μ, a λ ∈ (0, 1) and an η > 0 such that μ(ητ ) ≤ τ for all τ < supU μ(γ (x)), and for any x ∈ U and t < γ (x) the set F(B(x, t)) is a λμ(t)-net in B(F(x), μ(t)) ∩ V .2 Let ν be another gauge function satisfying ν −1 (t) − ν −1 (λt) ≥ μ−1 (t), ∀ t ∈ (0, sup γ (x)).
(5)
x∈U
Then F is γ -open on U × V with functional modulus not smaller than ν. Proof Let x ∈ U, y ∈ V, v ∈ F(x), d(y, v) < μ(γ (x)). Set t = μ−1 (d(y, v)). Then t < γ (x). By the assumption there is a (u, w) ∈ Graph F such that d(x, u) ≤ t and d(y, w) ≤ λμ(t) = λd(y, v).
(6)
We have by (5) ν −1 (d(y, w)) ≤ ν −1 (λd(y, v))
= ν −1 (d(y, v)) − ν −1 (d(y, v)) − ν −1 (λd(y, v)) ≤ν
−1
−1
(d(y, v)) − μ
(7)
(d(y, v)).
Take ξ > 0 such that ξ η−1 (1+λ) ≤ 1. By Theorem 2, to complete the proof, we have to verify that μ−1 (d(y, v)) ≥ dξ ((x, v), (u, w)). We have μ−1 (d(y, v)) = t ≥ d(u, x). On the other hand, ηd(y, v) ≤ μ−1 (d(y, v)) as d(y, v) < μ(γ (x)). In view of (6) this means that ξ d(v, w) ≤ ξ(d(y, v) + d(y, w)) ≤ ξ η−1 (1 + λ)(η(d(y, v))) ≤ μ−1 (d(y, v)) and the desired estimate for dξ ((x, v), (u, w)) follows.
Dmitruk’s theorem (Theorem 1.5 in [9]) corresponds to μ(t) = r t and γ (x) = d(x, X \U ). For ν(t) of the form ν(t) = ρμ(t) for some ρ < 1 (in which case ν −1 (t) = μ−1 (t/ρ)) (5) reduces to μ−1 (ρ −1 t) − μ−1 (ρ −1 λt) ≥ μ−1 (t). 2 That is for any y ∈ B(F(x), μ(t)) ∩ V there is a v ∈ F(B(x, t)) such that d(y, v) < λμ(t).
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In the most interesting case of μ(t) = r t k we get Corollary 1 If under the assumptions of Theorem 4 μ(t) = r t k and γ is bounded on U then F is γ -open on U × V with the functional modulus not smaller than r (1 − λ1/k )k t k . In particular for k = 1, we get the density theorem of [20]. An obvious consequence of the theorem for local regularity is Corollary 2 (cf. [27]) Let F : X ⇒ Y be a set-valued mapping whose graph is ¯ y¯ ) ∈ Graph F. Assume that there are an ε > 0 and a gauge locally complete3 at (x, ◦
function μ such that for any x ∈ B(x, ε) and t ∈ (0, ε) the set F(B(x, t)) is dense ◦
in B(F(x), μ(t)) ∩ B(y, ε). Then F is open at (x, ¯ y¯ ) with modulus germ not smaller than μ. 4 Regularity of order k Since this moment we shall focus on a particular but the most important class of gauge functions μ(t) = r t k . Fix as before a set-valued mapping F : X ⇒ Y with complete graph, open U ⊂ X and V ⊂ Y and two functions , γ on X and δ on Y , which are positive on U and V respectively. Let also a certain k ≥ 1 be fixed. Definition 4 (a) by sur (k) γ F(U |V ) we denote the upper bound of r > 0 such that ◦
B(F(x), r t k ) ∩ V ⊂ F(B(x, t)), if x ∈ U, t < γ (x). (k)
(8)
(k)
If no such r exists, we set sur γ F(U |V ) = 0. We shall call sur γ F(U |V ) the γ -surjection modulus of order k of F on U × V . We can further define the order k surjection modulus of F at (x, ¯ y¯ ) ∈ Graph F, sur (k) F(x|y), as the upper bound of r > 0 such that for some ε > 0 (8) holds with ◦
◦
U = B(x, ε), V = B(y, ε) and γ (x) ≡ ε. (b) by reg(k) γ F(U |V ) we denote the lower bound of K > 0 such that 1 1 d(x, F −1 (y)) ≤ K d(y, F(x)) k if x ∈U, y ∈ V, K (d(y, F(x))) k < γ (x). (9) (k)
(k)
If no such K exists, we set regγ F(U |V ) = ∞. We shall call regγ F(U |V ) the γ -metric regularity modulus of order k of F on U × V . We shall also define the order k metric regularity modulus of F at (x, ¯ y¯ ) ∈ Graph F, reg(k) F(x|y), as the lower bound of K > 0 such that for some ε > 0 (9) holds with ◦
◦
U = B(x, ε), V = B(y, ε) and γ (x) ≡ ∞. 3 A set is locally complete at a point if its intersection with a closed neighborhood of the point is a complete
space in the induced metric.
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(k)
(c) by holδ F(U |V ) we denote the lower bound of K > 0 such that 1 1 d(y, F(x)) ≤ K d(x, u) k if x ∈ U, y ∈ V, y ∈ F(u), K (d(x, u)) k < δ(y). (10) (k)
(k)
If no such K exists, we set holδ F(U |V ) = ∞. We shall call holδ F(U |V ) the δ-Hölder modulus of order k of F on U × V . Finally, we define the order k Hölder modulus of F at (x, y) ∈ Graph F, hol(k) F(x|y), as the lower bound of K > 0 such that for some ε > 0 (10) holds ◦
◦
with U = B(x, ε), V = B(y, ε) and δ(x) ≡ ∞. If we adopt the convention that 0 · ∞ = 1 (standard in the quantitative regularity theory of variational analysis), then applying Proposition 1 and Theorem 1 for μ(t) = r t k we get Proposition 2 For any set-valued mapping F : X ⇒ Y and any (x, ¯ y¯ ) ∈ Graph F 1 reg(k) F(x|y) · sur (k) F(x|y) k = 1; reg(k) F(x|y) = hol(k) F −1 (y|x). (k)
(k)
Definition 5 If sur γ F(U |V ) > 0 (or equivalently, regγ F(U |V ) < ∞), we say that F is γ -regular of order k on U × V . Likewise, if sur (k) F(x|y) > 0 etc. we say that F is regular of order k at (x, ¯ y¯ ). Our immediate goal is to produce an infinitesimal criterion for regularity of order k. The criterion will be based on an order k extension of the concept of slope of DeGiorgiMarino and Tosques (see e.g. [8]). Recall that, given a function f on a metric space X which is finite at x, the slope of f at x is defined by |∇ f |(x) = lim sup u→x u =x
( f (x) − f (u))+ d(x, u)
Given a ρ ∈ R, α > 0, we set [ρ]α = |ρ|α signρ. We further define the function [ f ]α by [ f ]α (x) = [ f (x)]α . It is clear that [ f ]α is continuous (lower semicontinuous) at x if so is f . Using this function we further define the slope of order k of f at x as the slope of [ f ]1/k at the point: 1
|∇ (k) f |(x) = |∇[ f ] k |(x). For subsequent discussions we also need the concept of a locally coherent space (see [19]): X is locally coherent if for any x ∈ X lim |∇d(u, ·)|(w) = 1.
u,w→x u =w
It can be shown that any smooth manifold in a Banach space with the induced metric is a locally coherent space. Also any length space (see Sect. 6) is locally coherent.
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We are now ready to prove the slope criterion for regularity of order k. Given a set-valued mapping F : X ⇒ Y , we set for any y ∈ Y ϕ y (x, v) =
d(y, v), if v ∈ F(x), . ∞, otherwise.
(k)
Let |∇ξ ϕ y | stand for the k-slope of ϕ y with respect to the ξ -metric in X × Y . Theorem 5 (slope regularity criterion) Let F : X ⇒ Y be a set-valued mapping whose graph is complete in the product metric, let U and V be open subsets of X and Y respectively, and let γ be a function on X with Lipschitz constant 1 which is positive on U . Let finally k ≥ 1. Suppose there are ξ > 0, r > 0 such that 1 (k) ∇ξ ϕ y (x, v) > r k
(11)
whenever y ∈ V , x ∈ Uγ , v ∈ F(x) and 0 < d(y, v) < r (γ (x))k . Then sur (k) γ F(U |V ) > r > 0. In particular if for some (x, ¯ y¯ ) ∈ Graph F there is an ε > 0 such that (11) holds whenever d(x, x) < ε, d(y, y) < ε, v ∈ F(x), d(y, v) < ε,
(12)
then sur (k) F(x|y) ≥ r. Proof This is immediate from the general criterion of Theorem 2. Indeed, if (11) holds, then for any (x, y, v) satisfying the conditions of the theorem there is a (u, w) ∈ Graph F arbitrarily close to (x, v) and such that 1
1
ϕ yk (x, v) − ϕ yk (u, w) , r < dξ ((x, v), (u, w)) 1 k
so that (d(y, w))1/k ≤ (d(y, v))1/k − r 1/k dξ ((x, v), (u, w)).
If k = 1, the converse is valid (for regularity at a point) if Y is a locally coherent space. Namely if sur F(x|y) > r , then for every δ > 0 there is an ε > 0 such that |∇ξ ϕ y |(x, v) > (1 − δ)r for (x, y, v) satisfying (12)—see [17,18]. It is not clear whether this is true if k > 1, probably not. We can prove only a weaker relation, namely that for any δ > 0 there is a sequence (xn , vn ) ⊂ Graph F converging to (x, v) such that lim sup n→∞
123
d(y, v) − d(y, vn ) d(y, v) − d(y, vn ) ≥ r (1 − δ). ≥r (dξ ((x, v), (xn , vn )))k max{1, r ξ }d(v, vn )
(13)
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Indeed, since Y is locally coherent at (x, ¯ y¯ ), for any δ > 0 there is an ε > 0 such that |∇d(y, ·)|(v) > 1 − δ whenever both y and v belong to the ε-ball around y. Let us fix such an ε for every δ > 0. Since sur (k) F(x|y) > r , the inclusion ◦
B(v, r t k ) ⊂ F(B(x, t))
(14)
holds for (x, v) ∈ Graph F sufficiently close to (x, ¯ y¯ ) and sufficiently small t. We may assume that ε is so small that the inclusion holds in particular for x, v, t satisfying v ∈ F(x), d(x, x) < ε, d(v, y) < ε, 0 ≤ t < ε. Now fix a y with d(y, y) < ε, and let (x, v) ∈ Graph F be such that d(x, x) < ε, y = v and d(v, y) < ε, so that (14) holds for the given x and v. As both y and v belong to the open ε-ball around y, and Y is assumed to be locally coherent, there is a sequence (vn ) converging to v and and such that d(y, v) − d(y, vn ) → |∇d(y, ·)|(v) > 1 − δ. d(vn , v)
(15)
By (14) for sufficiently large n there are xn such that vn ∈ F(xn ) and d(xn , x) ≤ (d(vn , v)/r )1/k which, together with (15), implies (13). The same arguments as in the proof of Theorem 5 with reference to Theorem 3 (rather than Theorem 2) prove a criterion for upper semicontinuous mappings. Set ψ y (x) = d(y, F(x)). Theorem 6 Let as before F : X ⇒ Y have complete graph, and let U ⊂ X, V ⊂ Y be open sets. Suppose that F is upper semicontinuous and there are k ≥ 1, r > 0 and a function γ on X satisfying the Lipschitz condition with constant not greater than 1 which is positive on U and such that 1
|∇ (k) ψ y |(x) > r k
whenever x ∈ Uγ , y ∈ V and 0 < d(y, F(x)) < r (γ (x))k . Then F −1 (y) = ∅ for every y ∈ V and d(x, F −1 (y)) ≤ [r −1 d(y, F(x))]1/k if d(y, F(x)) < r (γ (x))k . In (k) particular F is γ -regular of order k on U × V with sur γ F(U |V ) ≥ r . 5 Error bounds of order k In this short section we shall apply the last criterion to extended-real-valued functions on X in order to get estimates for error bounds associated with level sets. So let f be such a function. For an α ∈ R denote by [ f ≤ α] = {x : f (x) ≤ α} the α-sublevel set of f . The meaning of [ f > 0] is clear. We shall be interested in estimating the distance to [ f ≤ 0] from points of [ f > 0] in terms of the values of f at points of the set.
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Theorem 7 Let f be a lower semicontinuous function on X , let U ⊂ X be an open set, and let γ be a Lipschitz function on X with Lipschitz constant not greater than one which is positive on U . Assume that k ≥ 1 and |∇ (k) f |(x) > r > 0 for all x ∈ [ f > 0] ∩ Uγ . Then [ f ≤ 0] = ∅ and ( f + (x))1/k d x, [ f ≤ 0] ≤ , r for all x ∈ U satisfying f (x) < (r γ (x))k . Proof As f is lower semicontinuous, so is the function f + , and the set-valued mapping E f + (x) = {α ∈ R : α ≥ f + (x)} viewed as a mapping from X into R+ , the nonnegative half line, is upper semicontinuous. In particular for any α ∈ R+ the function x → ϕα (x) = d(α, E f (x)) = ( f (x) − α)+ is also lower semicontinuous. We apply Theorem 6 to F = E f + with Y = R+ , V = Y . If α ≥ 0 does not belong to E f + (x), that is α < f (x), then ϕα (x) = d(α, E f + (x)) = f (x) − α. We notice further that (by concavity of t 1/k ) for any α > 0 and any x, u such that both f (x) and f (u) are greater than α we have 1
1
1
1
( f (x) − α) k − ( f (u) − α) k ≥ ( f (x)) k − ( f (u)) k
and therefore for any x ∈ [ f > 0] and any α ∈ [0, f (x)) we have |∇ (k) ϕα |(x) ≥ |∇ (k) f |(x) > r . Thus the conditions of Theorem 6 are satisfied and we conclude that the set (E f + )−1 (0) = [ f ≤ 0] is nonempty and for any x ∈ U such that f + (x) < (r γ (x))k 1
1 1 ( f + (x)) k d 0, [ f ≤ 0] ≤ (ϕ0 (x)) k = r r
as claimed.
The point to be mentioned in connection with the last theorem is that it gives a global estimate based on purely infinitesimal information. 6 A variational viewpoint In this section we shall discuss the variational approach to local regularity of order k ≥ 1. The core of the approach is the study of the behavior of a set-valued mapping with respect to certain curves in the range space representing variations of their initial points. Clearly, some additional restrictions on the class of possible range spaces is needed to make such an approach implementable. First we recall the definition of length for a curve in a metric space (we refer to [1,7] for the details concerning geometry of metric spaces). Let η be a continuous curve in X parametrized by a continuous mapping m(·) from a line segment [0, 1] into X . The curve is called absolutely continuous (or rectifiable) if for almost any t ∈ (0, 1) the limit
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d(m(t + ε), m(t)) ε→+0 ε
|m |(t) = lim
exists and the function |m | is summable. The integral 1 (m(·)) =
|m |(t)dt
0
is called the length of the curve. The length does not depend on the choice of a parametrization. Every rectifiable curve can be parametrized by a mapping m with |m (t)| ≡ 1. We shall call such a parametrization (which is unique, provided the initial point is specified) natural. X is called a length space if the distance between any two points coincides with the lower bound of lengths of rectifiable curves joining the points. The metric defined by the lengths of curves in a metric space is called intrinsic. It is quite clear that X is a length space only if d(y, v) = t + d(y, B(v, t)) for any y = v and any t ≤ d(y, v). The converse is true if the space is complete. It can be shown (see [2]) that a complete metric space is a length space if and only if |∇d(u, ·)|(x) = 1 for any x and any u = x. A metric in X is called strictly intrinsic if X is a length space and any two points y, v can be connected by a shortest path, that is a rectifiable curve joining y and v whose length is equal to d(y, v). In particular the metric of a locally compact length space is strictly intrinsic. A metric space with strictly intrinsic metric is called a geodesic space. Clearly a normed space is a geodesic space as well as a huge variety of (even nonsmooth) manifolds. Such is for instance the boundary of the unit square in R2 with the intrinsic metric or a product of such spaces (a sort of “square torus”). We shall start however with a general result valid for arbitrary range spaces. Theorem 8 Let X and Y be metric spaces, and let F : X ⇒ Y be a set valued mapping whose graph is complete in the product metric. Let (x, ¯ y¯ ) ∈ Graph F, and let k ≥ 1. Then sur (k) F(x|y) = sup r ≥ 0 :
ex(B(v, r t k ), F(B(x, t))) = 0 . tk (x,v,t)→(x,y,+0) lim sup v∈F(x)
(16) Proof It is clear that sur (k) F(x|y) is not greater than the quantity in the right-hand side. To prove the opposite inequality, take a small λ > 0 and choose δ > 0 such that ex(B(v, r t k ), F(B(x, t)) ≤ λt k if d(x, x) < δ, d(v, y) < δ, 0 < t < δ, v ∈ F(x). This means that F(B(x, t)) is a λt k -net in B(v, r t k ) for such x, v, t. By Corollary 1 F ◦
◦
is regular on B(x, δ)× B(y, δ) with functional modulus not smaller than (1−λ1/k )k r t k which implies that sur (k) F(x|y) ≥ (1−λ1/k )k r . But λ can be chosen arbitrarily small, whence sur (k) F(x|y) ≥ r . If Y is a length space, then (16) can be expressed in variational terms. The following is a possible general pattern for that. Let Z be a topological space, and let Φ be
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a continuous mapping from Y × Z × [0, 1] into Y such that Φ(v, z, 0) = v for all (v, z). In other words, Φ(v, ·, t) is a retraction of Φ(v, Z , 1) to v and for any z ∈ Z the mapping t → Φ(v, z, t) is a parametrization of a certain curve starting at v. We can view such a curve as a variation of v and, accordingly, interpret Z as a space of variations of elements of Y . In fact, we can consider even “set-valued” retractions. We can now state the following theorem. Theorem 9 Assume as always that F : X ⇒ Y is a set-valued mapping with complete graph and (x, ¯ y¯ ) ∈ Graph F. Suppose Z is a compact topological space and Φ : Y × Z × [0, 1] ⇒ Y is a set-valued mapping with closed graph. We assume that there ◦
are ε > 0 and c > 0 such that for all v ∈ B(y, ε), z ∈ Z and t ∈ [0, 1] Φ(v, z, 0) = {v}, B(v, ct) ⊂ Φ v, Z , [0, t]
(17)
and for any ξ > 0 there is a finite set Q ξ ⊂ Z such that Φ(v, Q ξ , [0, t]) is a tξ -net in B(v, ct). If under these conditions there are r > 0 and k ≥ 1 such that ex Φ(v, z, r t k ), F(B(x, t)) =0 lim (x,v,t)→(x,y,+0) tk
(18)
v∈F(x)
for any z ∈ Z , then sur (k) F(x|y) ≥ cr . Proof Let an r > 0 be such that (18) holds. Take a ξ > 0. By (18) for any z ∈ Q ξ there is a δ = δ(z) such that ex Φ(v, z, r t k ), F(B(x, t)) < ξ t k
(19)
d(x, x) < δ, d(v, y) < δ, v ∈ F(x), 0 ≤ t < δ.
(20)
if
Take a δ > 0 smaller then all δ(z) corresponding to z ∈ Q ξ , and let Wξ (t) stand for Q ξ × [0, r t k ]. Then (19) means that ex(Φ(v, Wξ (t)), F(B(x, t))) ≤ ξ t k
(21)
if x, v, t satisfy (20). By the assumption F(v, Wξ (t)) is a (r t k )ξ -net in B(v, cr t k ). Therefore (21) implies that ex B(v, cr t k ), F(B(x, t)) lim sup ≤ (1 + r )ξ. tk (x,v,t)→(x,y,+0) v∈F(x)
As ξ can be chosen arbitrarily small, we conclude that the condition of Theorem 8 is satisfied and therefore sur (k) F(x|y) ≥ cr as claimed.
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Remark 1 The theorem does not give an exact formula for the surjection modulus of order k, just a lower bound for it. However if we assume in (17) that actually Φ(v, Z , [0, t]) = B(v, ct) for all v, z and t, then the fact that sur (k) F(x|y) cannot exceed the upper bound of cr such that (18) holds is immediate from Definition 5. The simplest case when the conditions of the theorem hold occurs when Y is Rn , Z = S n−1 , the unit sphere in Rn , Φ(v, z, t) = v + t z and (18) is valid. Indeed in this case B(v, t) = Φ(v, Z , [0, t]) which immediately implies (17) and the ξ -net assumption, so the assumptions of the theorem follow with c = 1. It is not a difficult matter to see that, in this case, the theorem implies the following characterization of higher order regularity moduli of mappings into Rn established in a recent paper by Frankowska and Quincampoix [14]: ¯ y¯ ) ∈ Corollary 3 ([14]) Let F be a set-valued mapping from X into Rn , and let (x, Graph F. Set A=
lim inf
(x,v,t)→(x,y,+0) v∈F(x)
F(B(x, t)) − v . tk
(22)
Then sur (k) F(x|y) = sup{r ≥ 0 : r B ∈ A}, where B is the unit ball in Rn . Proof Indeed, both (18) with Φ(v, z, t) = v +t z and (22) amount to the statement that for any z ∈ S n−1 and ξ > 0 there is a δ > 0 such that d(v + r t k z, F(B(x, t))) < ξ t k if x, v, and t satisfy (20). Thus Theorem 9 implies that sur (k) F(x|y) ≥ sup{r ≥ 0 : r B ∈ A}. To see that the opposite inclusion holds we refer to Remark 1. The curves Φ(v, z, ·) in the last corollary are of course line segments joining v and v + z. It may seem that the ball in Rn in the statement can be replaced by a ball in a locally compact length space in which case instead of the segments we should consider the shortest paths connecting points of the corresponding sphere with the center of the ball. However such a replacement requires caution because it may happen that the shortest paths joining the center of the ball with points of the sphere do not cover the entire ball. Example 2 Let Y = X be the boundary of the first orthant in R3 : X = Y = {x = (x 1 , x 2 , x 3 ) ∈ R3 : x i ≥ 0, x 1 x 2 x 3 = 0}, and let Y be endowed with the (strictly) intrinsic metric induced by the Euclidean metric in R3 . (That is to say, the distance between two points in Y is the minimal Euclidean length of curves lying in Y and joining the points. Clearly, any curve of minimal length is either a linear segment or a pair of linear segments lying in different coordinate planes and having a common point on the axis common for the planes.) 3 Take √ for instance v = (1, 0, 0) and consider in Y (not in R !) the sphere1 of 2radius 2 around v. The configuration of the pieces of√ the sphere lying in the (x , x ) and (x 1 , x 3 ) planes is obvious: circular arcs of radius 2 around v. The piece of the sphere
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lying in the (x 2 , x 3 )-plane in turn consists of two parts symmetric with respect to the ray x 2 = x 3 and defined by the parametric relation √ 1 + 2t 2 − t max{x , x } = r (t), min{x , x } = tr (t), 0 ≤ t ≤ 1, r (t) = . 1 + t2 2
3
2
3
For a point u = (0, x 2 , x 3 ) belonging to this piece of the sphere, say with x 2 ≥ x 3 , the shortest path joining it with v consists of two line segments, one in the (x 2 , x 3 )-plane joining u with
√ 1 + 2t 2 − t w = 0, √ ,0 t 1 + 2t 2 + 1
(where x 3 = t x 2 )
and the other segment in the (x 1 , x 2 )-plane joining w and v. We see therefore that there is no shortest path connecting points of the sphere with v and passing through points in the (x 2 , x 3 )-plane belonging to the interior of the diamond bounded by the segments connecting the points (0, ξ, 0) and (0, 0, ξ ) with (0, η, η) and the origin, where √ √ 3−1 3−1 η= . ; ξ=√ 2 3+1 The extension mentioned before the example is however possible if Y itself is a homogeneous space. The latter means that for any two finite isometric collections of points in Y there is an isometry of Y that moves one of the collections into the other. We actually need less: for any three points yˆ , y and z such that d(z, yˆ ) = d(y, yˆ ) there is an isometry that keeps yˆ invariant and moves z into y. Such are for instance spheres in Euclidean spaces or tori. Theorem 10 Let X be a metric space, let Y be a locally compact homogeneous length space, and let F : X ⇒ Y be a set-valued mapping with a complete graph. Fix a yˆ ∈ Y , and let Γv be an isometry of Y such that Γv ( yˆ ) = v and the mapping (v, y) → Γv (y) is continuous. For any y ∈ S( yˆ , 1), let M y be the collection of shortest paths joining y and yˆ , and let M y (t) be the collection of points of the paths of M y that are at the distance t from yˆ . Let finally (x, ¯ y¯ ) ∈ Graph F. Then sur (k) F(x|y) is the upper bound of r ≥ 0 such that for all y ∈ S( yˆ , 1) ex(Γv (M y (r t k )), F(B(x, t))) = 0. (x,v,t)→(x,y,+0) tk lim
(23)
v∈F(x)
Proof We note first that for any t ∈ (0, 1) the union of M y (t) over y ∈ S( yˆ , 1) coincides with S( yˆ , t). Indeed let d(y, yˆ ) = t (t < 1), and let η be a shortest path connecting yˆ with some point of the 1-sphere around yˆ . Let z ∈ η ∩ S( yˆ , t), and let Γ be an isometry than keeps yˆ invariant and moves z into y. Then y ∈ Γ (η) and the
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latter is the shortest path whose other end is in S( yˆ , 1). Let Z = S( yˆ , 1). Now set Φ(v, z, t) = Γv (Mz (t)). Then as we have just seen, Φ(v, Z , [0, 1]) = B(v, 1) and it is not a difficult matter to verify that the conditions of Theorem 9 are satisfied. Acknowledgment I wish to express my thanks to the reviewers for the detailed analysis of the text and many good suggestions.
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