Math. Program., Ser. A DOI 10.1007/s10107-015-0877-2 FULL LENGTH PAPER

A nonsmooth Robinson’s inverse function theorem in Banach spaces R. Cibulka · A. L. Dontchev

Received: 24 January 2014 / Accepted: 17 February 2015 © Springer-Verlag Berlin Heidelberg and Mathematical Optimization Society 2015

Abstract In a recent paper, Izmailov (Math Program Ser A 147:581–590, 2014) derived an extension of Robinson’s implicit function theorem for nonsmooth generalized equations in finite dimensions, which reduces to Clarke’s inverse function theorem when the generalized equation is just an equation. Páles (J Math Anal Appl 209:202–220, 1997) gave earlier a generalization of Clarke’s inverse function theorem to Banach spaces by employing Ioffe’s strict pre-derivative. In this paper we generalize both theorems of Izmailov and Páles to nonsmooth generalized equations in Banach spaces. Keywords Robinson’s inverse function theorem · Clarke’s inverse function theorem · Strict pre-derivative · Generalized equation · Strong metric regularity Mathematics Subject Classification

49J53 · 49J52 · 49K40 · 90C31

A. L. Dontchev: Supported by the National Science Foundation Grant DMS 1008341. R. Cibulka NTIS - New Technologies for the Information Society and Department of Mathematics, Faculty of Applied Sciences, University of West Bohemia, Univerzitní 22, 306 14 Plzeˇn, Czech Republic e-mail: [email protected] A. L. Dontchev (B) Mathematical Reviews and Department of Mathematics, The University of Michigan, Ann Arbor, MI, USA e-mail: [email protected]

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1 Introduction The classical inverse function theorem says that if a function f acting from a Banach space X to itself is continuously Fréchet differentiable around a point x¯ ∈ int dom f ¯ for x¯ then the inverse f −1 has a single-valued graphical localization around f (x) which is continuously differentiable around f (x) ¯ if and only if the Fréchet derivative D f (x) ¯ is invertible, that is, D f (x) ¯ −1 is a linear function. Here and further we use the terminology from [3] according to which, for a set-valued mapping F : X ⇒ Y and y¯ ∈ F(x), ¯ a graphical localization of F at x¯ for y¯ is a mapping F˜ such that ˜ gph F = (U × V ) ∩ gph F for some neighborhoods U of x¯ and V of y¯ ; clearly, dom F˜ depends on the choice of U and V . If a graphical localization at x¯ for y¯ of F is a function whose domain is a neighborhood of x, ¯ we say that this localization is around x¯ for y¯ . By employing his generalized Jacobian, Clarke [1] proved the following nonsmooth inverse function theorem in finite dimensions: if a function f : Rn → Rn is Lipschitz continuous around x, ¯ then the inverse f −1 has a Lipschitz continuous single-valued graphical localization around f (x) ¯ for x¯ provided that every element of the generalized Jacobian, denoted in this paper by ∂¯ f (x), ¯ is a nonsingular matrix. A generalization of Clarke’s theorem was given by Kummer [8] who showed that, if a ¯ then f −1 has a Lipschitz continuous function f : Rn → Rn is continuous around x, single-valued graphical localization around f (x) ¯ for x¯ if and only if the so-called strict graphical derivative of f at x¯ is injective. A different kind of nonsmooth inverse function theorem was derived by Robinson [13] in which the approximation of the underlying function is single-valued and has Lipschitz inverse. Throughout X and Y are Banach spaces. The notation f : X → Y means that f is a function while F : X ⇒ Y denotes a mapping whichmay be set-valued. The   y ∈ F(x) , the domain graph of a mapping F is the set gph F = (x, y) ∈ X × Y    of F is the set dom F =  x ∈ X  F(x) = ∅ , and the inverse of F is the mapping   y → F −1 (y) = x ∈ X  y ∈ F(x) . The closed and open ball centered at x with o

radius r are denoted by Br (x) and Br (x), respectively. A set C is said to be locally ¯ is closed. Whenever closed at x¯ ∈ C when there exists r > 0 such that C ∩ Br (x) a set is singleton we identify it with its only element, e.g., for the Minkowski sum of a singleton {a} and a set C we write a + C instead of {a} + C. The space of all linear bounded mappings acting from X to Y is equipped with the standard operator norm and denoted by L(X, Y ). Given a subset A of L(X, Y ), recall that the measure of non-compactness of A, denoted by χ (A), is defined by      Br (A)  A ∈ B for some finite set B ⊂ A . χ (A) = inf r > 0  A ⊂ Clearly, if A is compact then χ (A) = 0; conversely, if χ (A) = 0 then the closure of A is compact. In a seminal paper Robinson [12] extended the classical implicit function theorem to generalized equations defined as inclusions of the form 0 ∈ f (x) + F(x) where f is a function and F is a set-valued mapping. In the same paper Robinson coined the concept of “strong regularity.” In the terminology of [3], given a set-valued mapping

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F : X ⇒ Y with y¯ ∈ F(x), ¯ if F −1 has a Lipschitz continuous single-valued graphical localization around y¯ for x, ¯ i.e., a graphical localization which is a Lipschitz continuous function in a neighborhood of y¯ , then F is said to be strongly metrically regular at x¯ for y¯ . Recall that the Lipschitz modulus of a function f : X → Y at x, ¯ denoted by lip( f ; x), ¯ is the infimum of all l > 0 for which there exists a neighborhood U of x¯ such that f is Lipschitz continuous on U with a Lipschitz constant l. Clearly, if a mapping F is strongly metrically regular at x¯ for y¯ , then every graphical localization of its inverse F −1 around y¯ for x¯ has the same Lipschitz modulus. Robinson stated his theorem [12, Theorem 2.1] as an implicit function theorem in linear normed spaces. The following symmetric inverse function version of Robinson’s theorem fits our purposes: Theorem 1 (Robinson’s inverse function theorem) Let f : X → Y be a function which is strictly Fréchet differentiable at x¯ with strict Fréchet derivative D f (x) ¯ and let F : X ⇒ Y be any set-valued mapping. Then the following are equivalent: (a) the mapping f (x) ¯ + D f (x)(· ¯ − x) ¯ + F(·) is strongly metrically regular at x¯ for y¯ ; (b) the mapping f + F is strongly metrically regular at x¯ for y¯ . In either case, if h and g denote single-valued localizations of the inverses of the mapping appearing in (a) and (b) respectively, then lip(h; y¯ ) = lip(g; y¯ ). Various versions of Robinson’s theorem, including Theorem 1 above, as well as related results centered around metric regularity together with historical remarks, are broadly discussed in the book [3]. In a recent paper A. F. Izmailov published the following result in [6, Theorem 1.3] which extends both Clarke’s inverse function theorem and a finite-dimensional version of Robinson’s theorem (the implication (a) ⇒ (b) in Theorem 1). Theorem 2 ([6, Theorem 1.3]) Let f : Rn → Rn be a function which is Lipschitz continuous around x, ¯ let y¯ ∈ Rn , and let F : Rn ⇒ Rn be a set-valued mapping. Suppose that for every matrix J in ∂¯ f (x), ¯ the Clarke generalized Jacobian of f at x, ¯ ¯ + J (x − x) ¯ + F(x) is strongly metrically regular at x¯ the mapping G J : x → f (x) for y¯ . Then the mapping f + F is strongly metrically regular at x¯ for y¯ . When F is the zero mapping, then the strong regularity assumption in Theorem 2 reduces to the requirement every matrix in ∂¯ f (x) ¯ be nonsingular, and then Theorem 2 becomes Clarke’s inverse function theorem. In this paper we present the following extension of Izmailov’s theorem to Banach spaces: Theorem 3 (Nonsmooth Robinson’s inverse function theorem in Banach spaces) Let (x, ¯ y¯ ) ∈ X × Y , let f : X → Y and F : X ⇒ Y be such that y¯ ∈ f (x) ¯ + F(x). ¯ Suppose that there exist a convex subset A of L(X, Y ) and a constant c > 0 such that

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(A) there exists r > 0 such that for each u and v in Br (x) ¯ one can find A ∈ A such that

f (v) − f (u) − A(v − u) ≤ c v − u ; (B) for every A ∈ A the mapping ¯ + A(x − x) ¯ + F(x) G A : x → f (x) is strongly metrically regular at x¯ for y¯ ; moreover, if s A is any single-valued ¯ then graphical localization of G −1 A around y¯ for x, (c + χ (A)) sup lip(s A ; y¯ ) < 1. A∈A

Then the mapping f + F is strongly metrically regular at x¯ for y¯ . A proof of Theorem 3 will be given in the following section. Here we first discuss the assumptions (A) and (B) in its statement. Ioffe [4] introduced the strict pre-derivative of f at x¯ which is a homogeneous set-valued mapping T : X ⇒ Y such that for each ε > 0 there exists r > 0 such that ¯ f (x1 ) ∈ f (x2 ) + T (x1 − x2 ) + ε x1 − x2 B, whenever x1 , x2 ∈ Br (x). When X = Y = Rn , as shown in [4, Corollary 9.11], the Clarke generalized Jacobian ∂¯ f (x) ¯ of a locally Lipschitz-continuous function f is a strict pre-derivative of f at x¯ and hence, it satisfies assumption (A) with any positive c. In this case assumption (B) holds exactly when for each J ∈ ∂¯ f (x) ¯ the mapping G J in Theorem 2 is strongly metrically regular at x¯ for y¯ , that is, we obtain Izmailov’s theorem [6, Theorem 1.3]. The strict pre-derivative of f at x¯ is single-valued if and only if f is strictly differentiable at x; ¯ hence in this case we obtain Theorem 1. For a statement of Robinson’s theorem in general metric spaces, see [3, Theorem 5F.1]. In a different direction, Páles extended in [10, Theorem 5] Clarke’s inverse function theorem to functions acting in Banach spaces. Specifically, based on his preceding paper [9], Páles assumed in [10] that the function f has a strict pre-derivative at x¯ generated by a convex set A whose measure of non-compactness is strictly less than the infimum of the surjection moduli of the elements of A. This condition becomes equivalent to our condition (B) when F is the zero mapping. Thus, our Theorem 3 extends [10, Theorem 5] to generalized equations. Theorem 3 can be easily extended to an equivalent implicit function theorem for the generalized equation 0 ∈ f ( p, x) + F(x), where P, X and Y are Banach spaces and f : P × X → Y is Lipschitz continuous around a point ( p, ¯ x) ¯ such that 0 ∈ ¯ + F(x). ¯ For such a generalization one may adapt to infinite dimensions the f ( p, ¯ x) way Clarke’s implicit function theorem is derived in the book of Clarke [2] from its inverse function counterpart. In that case one needs to use a set A in L(P × X, Y ) whose projection on the space L(X, Y ) has properties analogous to those displayed in assumptions (A), (B).

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The authors are aware of the fact that assumptions (A) and (B) in Theorem 3 above may be difficult to check in the general case considered. However, we expect that in special cases, e.g., for optimal control problems, these assumptions can be reduced to numerically tractable conditions. This is a subject of our continuing research on this topic. 2 Proof of Theorem 3 We start with the following simple lemma: Lemma 1 (Local graph closedness from strong metric regularity) Let f : X → Y be a function which is Lipschitz continuous around x, ¯ let F : X ⇒ Y be a set-valued mapping, and suppose that f + F is strongly metrically regular at x¯ for y¯ . Then gph F is locally closed at (x, ¯ y¯ − f (x)). ¯ Proof The assumption that f + F is strongly metrically regular at x¯ for y¯ means that there exist positive α and β such that the mapping Bβ ( y¯ ) y → s(y) := ¯ is a Lipschitz continuous function. Let a > 0 and b > 0 be ( f + F)−1 (y) ∩ Bα (x) ¯ with constant l, and moreover b +la ≤ β such that f is Lipschitz continuous in Ba (x) ¯ × Bb ( y¯ − f (x))) ¯ and a ≤ α. Then dom s ⊃ Bb+la ( y¯ ). Let (xn , z n ) ∈ gph F ∩ (Ba (x) converge to (x, z) as n → ∞. For sufficiently large n we have ¯ + f (xn ) − f (x)

¯ ≤ b + la.

z n + f (xn ) − y¯ ≤ z n − ( y¯ − f (x))

Then z n + f (xn ) ∈ Bb+la ( y¯ ) for large n and also z n + f (xn ) ∈ ( f + F)(xn ); hence, xn = s(z n + f (xn )). Since both f and s are continuous, passing to the limit ¯ that is, with n → ∞ we get x = s(z + f (x)) = ( f + F)−1 (z + f (x)) ∩ Ba (x), z + f (x) ∈ f (x) + F(x), hence (x, z) ∈ gph F. Thus, gph F is locally closed at (x, ¯ y¯ − f (x)). ¯   Without loss of generality, let y¯ = 0. The proof of Theorem 3 will include several steps. Step 1 Set m := sup A∈A lip(s A ; 0) with any choice of single-valued graphical localizations s A of G −1 A ; by assumption (B) this is a finite number. Choose ε and  such that ε > c + χ (A),  > m and ε < 1. (1) We will show that there is a compact convex subset B of a finite dimensional subspace of L(X, Y ) such that ¯ there is A ∈ B such that (a) for each two distinct u and v in Br (x)

f (v) − f (u) − A(v − u) < ε v − u ; (b) sup A∈B lip(s A ; 0) <  with any choice of single-valued graphical localizations ¯ s A of G −1 A around 0 for x.

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To prove this, pick positive γ such that γ m < 1, c + χ (A) + γ < ε, and m/(1 − γ m) < .

(2)

Since χ (A) < ∞, there is a finite set A F := {A1 , . . . , An } ⊂ A, with cardinality n, say, such that A ⊂ A F + Bχ (A)+γ (0). Denote by B the closed convex hull of A F . Clearly, B is a convex compact subset of the affine hull of AF (being a finite ¯ dimensional subspace of L(X, Y )). Fix any two distinct elements u and v in Br (x). From the assumption (A) there exists A0 ∈ A such that

f (v) − f (u) − A0 (v − u) ≤ c v − u . Pick A ∈ A F ⊂ B such that

(A − A0 )h ≤ (χ (A) + γ ) h for each h ∈ X. Hence, the triangle inequality and the fact that u = v reveal that

f (v) − f (u) − A(v − u) ≤ (c + χ (A) + γ ) v − u < ε v − u , which is (a). n

nFix an arbitrary A ∈ B. Hence, there is A := i=1 λi Ai for some λi ≥ 0 with



x). ¯

+(A− A)(·− i=1 λi = 1 such that A ∈ A and A− A < γ . Note that G A = G A −1 ¯ Then by Let s A be any single-valued graphical localization of G A around 0 for x. assumption (B), lip(s A ; 0) ≤ m. Since γ m < 1 by (2), applying [3, Theorem 5F.1], there is a single-valued graphical localization of G −1 A around 0 for x¯ with lip(s A ; 0) ≤

m < . 1 − γm

Since every graphical localization of this kind has the same Lipschitz modulus, (b) is proved. The next step shows that the strong metric regularity of G A is uniform in A ∈ B in the sense that the neighborhoods and the Lipschitz constant associated with the localization of G −1 A are the same for all A ∈ B. Step 2 There exists β > 0 such that for every A ∈ B, the mapping Bβ (0) y → ¯ is a Lipschitz continuous function with Lipschitz constant . G −1 A (y) ∩ Bβ ( x) Fix positive κ and γ such that 2κγ < 1 and

sup lip(s A ; 0) < κ < (1 − κγ ).

A∈B

Choose any A ∈ B. The mapping G −1 A has a single-valued graphical localization s A around 0 for x¯ which is Lipschitz continuous around 0 with the constant κ and ¯ and Bb (0) for some b > 0. neighborhoods Bκb (x)

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Choose any A ∈ B such that A − A < γ . Now consider the mapping G A associated with A as in (B). For every y ∈ Bb/2 (0) and every x ∈ Bκb (x) ¯ we have ¯ < b/2 + γ κb < b/2 + b/2 = b.

y + (A − A )(x − x)

¯ Therefore, for every y ∈ Bb/2 (0) and every x ∈ Bκb (x),  ¯ x ∈ G −1 A (y) ⇐⇒ x ∈ ξ y (x) := s A (y + (A − A )(x − x)).

Let y ∈ Bb/2 (0). Then ¯ − x

¯ = s A (y) − s A (0) ≤ κ y ≤ κb/2.

ξ y (x) ¯ we obtain Moreover, for any x, x  ∈ Bκb (x)

ξ y (x) − ξ y (x  ) ≤ κ (A − A )(x − x  ) ≤ κγ x − x  . Since κγ < 1/2, the above estimate with x  := x¯ and the previous one imply that ¯ ≤ ξ y (x) − ξ y (x)

¯ + ξ y (x) ¯ − x

¯ < κb/2 + κb/2 = κb

ξ y (x) − x

whenever x ∈ Bκb (x). ¯ ¯ into itself. By Banach’s contraction mapping Therefore ξ y is a contraction from Bκb (x) principle, there a unique fixed point x(y) of ξ y in Bκb (x). ¯ Since x(y) = G −1 A (y) ∩ ¯ for every y ∈ Bb/2 (0), we conclude that the mapping Bb/2 (0) y → x(y) = Bκb (x) G −1 ¯ is single-valued. Furthermore, for any y, y  ∈ Bb/2 (0) we have A (y) ∩ Bκb ( x)

x(y) − x(y  ) = ξ y (x(y))−ξ y  (x(y  )) ≤ κ y − y  +κ (A− A )(x(y)−x(y  ))

≤ κ y − y  + κγ x(y) − x(y  ) , hence x(·) is Lipschitz continuous on Bb/2 (0) with Lipschitz constant κ/(1 − κγ ). o

Thus, for any A ∈ B with A ∈ Bγ (A) the mapping G −1 A has a Lipschitz continuous single-valued localization around 0 for x¯ with a Lipschitz constant κ/(1 − κγ ) ¯ and Bb/2 (0). In other words, for any operator A which and neighborhoods Bκb (x) is sufficiently close to A the sizes of the neighborhoods and the Lipschitz constant associated with the localization of G −1 A remain the same. Observe that κ and γ are independent of a particular A ∈ B which is not the case o

for b. Pick any A ∈ B and the corresponding b to obtain that for every A ∈ Bγ (A) the mapping Bb/2 (0) y → G −1 ¯ is a Lipschitz continuous function with A (y) ∩ Bκb ( x) o

the constant κ/(1 − κγ ). Since B is compact, from the open covering ∪ A∈B Bγ (A) of o

B we can choose a finite subcovering with open balls Bγ (Ai ); let bi be the constants associated with the Lipschitz localizations for G −1 Ai . Taking β = min{1/2, κ/} mini bi and observing that κ/(1 − κγ ) <  Step 2 is complete.

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Step 3 Let μ := sup A∈B A . Clearly, μ < ∞ since B is compact. Note that f is ¯ with a Lipschitz constant μ + ε. Indeed, for any two Lipschitz continuous on Br (x) ¯ condition (a) implies that, choosing an appropriate distinct elements u and v of Br (x) A ∈ B, we get that

f (v) − f (u) ≤ f (v) − f (u) − A(v − u) + A(v − u) < (ε + μ) v − u . The combination of Lemma 1 and Step 2 then yields that there is a positive constant β such that the following statements hold:

¯ × Bβ (− f (x)) ¯ is closed; • the set gph F ∩ Bβ (x) ¯ with the constant μ + ε; • f is Lipschitz continuous on Bβ (x) • for each A ∈ B, the mapping ¯ B2β (0) z → s A (z) := G −1 A (z) ∩ B2β ( x) is a Lipschitz continuous function with Lipschitz constant . Recall that  satisfies (1). Define the mapping B × Bβ (0) (A, z) → ϕ(A, z) := s A (z). Note that dom ϕ = B × Bβ (0) thanks to the Lipschitz continuity of s A with Lipschitz constant . Moreover, ϕ has the following properties: (v) For each A ∈ B the mapping ϕ(A, ·) = s A is Lipschitz continuous on Bβ (0) with Lipschitz constant ; (vi) For each A ∈ B one has ϕ(A, 0) = x¯ = s A (0); (vii) ϕ is continuous. One needs to prove (vii) only. Let {An } be a sequence of operators in B which is convergent to A¯ and {z n } be a sequence of points from Bβ (0) convergent to z¯ ; then ¯ z¯ ) and u n = ϕ(An , z n ) for each n ∈ N. For all n A¯ ∈ B and z¯ ∈ Bβ (0). Set u¯ = ϕ( A, we have ¯ + An (u n − x) ¯ + F(u n ), z n ∈ f (x) that is, ¯ n − x) ¯ + F(u n ) z n + ( A¯ − An )(u n − x). ¯ f (x) ¯ + A(u ¯ and An → A¯ as n → ∞, we obtain that Since each u n ∈ Bβ (x) ¯ ∈ B2β (0) for all n sufficiently large. z n + ( A¯ − An )(u n − x) ¯ property (v) reveals that Then, using the definitions of u n and u,

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u n − u

¯ = s A¯ (z n + ( A¯ − An )(u n − x)) ¯ − s A¯ (¯z )

2 ≤  z n − z¯ +  β A¯ − An → 0 as n → ∞. Thus (vii) is established. Step 4 Our next step is to show that the mapping ( f + F)−1 has a nonempty-valued graphical localization around 0 for x. ¯

(3) In preparation for that, apply condition (a) to find δ such that for each two distinct ¯ there exists A ∈ B with elements u and v of B3δ (x)

f (v) − f (u) − A(v − u) < ε v − u .

(4)

Adjust δ if necessary to satisfy 0 < 6δ < Clearly, δ < β. From (1),

β . (1/ + μ)

b := (1 − ε)δ < δ.

(5)

(6)

For any y ∈ Bεb (0), w ∈ Bδ (x), ¯ u˜ ∈ Bδ (x) ¯ and A ∈ B the relations (5) and (6) yield the estimate

y − f (w)+ f (x)+ ¯ A(w − u)

˜ ≤ y +(ε+μ) w − x +μ w ¯ − x +μ

¯ u˜ − x

¯ ≤ εb + (ε + μ)δ + 2μδ < δ(2ε + 3μ) < 3δ(1/ + μ) < β/2. Hence, y − f (w) + f (x) ¯ + A(w − u) ˜ ∈ Bβ/2 (0) whenever (y, w, u, ˜ A) ∈ Bεb (0) × Bδ (x) ¯ × Bδ (x) ¯ × B.

(7)

¯ this will Let y ∈ Bεb (0) be fixed. We will now find x ∈ ( f + F)−1 (y) ∩ Bδ (x); prove (3). Denote K = B2εδ (0)\{0}. ¯ and define the function Fix u ∈ Bδ (x)

¯ + A(u − x) ¯ − u. B A → u (A) = ϕ A, y − f (u) + f (x)

(8)

By (7) with u˜ = x¯ and w = u, each value of the argument of ϕ in (8) is in dom ϕ, hence ¯ dom u = B. Also note that, according to property (vii) above, for any u ∈ Bδ (x),

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the function u is continuous in its domain. If there exist A ∈ B and u ∈ Bδ (x) ¯ such that u (A) = 0, then u ∈ ( f + F)−1 (y). From now on, assume that this is not the case, that is, ¯ (9) u (A) = 0 for all A ∈ B and all u ∈ Bδ (x). In further lines we will construct a sequence of points convergent to an x ∈ ( f + F)−1 (y). To this end, we make use of the following two lemmas: ¯ suppose that there exist v ∈ Bδ (x)\{u} ¯ along with Lemma 2 Given u ∈ Bδ (x), A˜ ∈ B satisfying ˜ − u) ≤ ε v − u and

f (v) − f (u) − A(v

˜ − v) + F(u) y. (10) f (v) + A(u

Then u maps B into K . More precisely, we have 0 < u (A) ≤ ε u − v whenever A ∈ B.

(11)

Proof We show that (10) implies (11). Pick any A ∈ B. The first inequality follows ˜ − v), the inclusion (7) from (9). Then for z := y − f (v) + f (x) ¯ + A(u − x) ¯ − A(u ˜ with A := A, w := v and u˜ = u combined with (5) implies that ˜ − u) + A(u − x)

z ≤ y − f (v)+ f (x)+ ¯ A(v ¯ ≤ β/2+μδ < β/2+β/6 < β. Rearranging the inclusion in (10) gives us ˜ − v) = z. f (x) ¯ + A(u − x) ¯ + F(u) y + f (x) ¯ + A(u − x) ¯ − f (v) − A(u Therefore, u = ϕ(A, z) and we conclude that

u (A) = ϕ A, y − f (u) + f (x) ¯ + A(u − x) ¯ − ϕ(A, z)

˜ − u) ≤ ε u − v ≤ 2εδ. ≤  f (v) − f (u) − A(v Hence (11) is satisfied. The last estimate also shows that u maps B into K .

 

¯ fixed define the following set-valued mapping acting from K Keeping u ∈ Bδ (x) into the subsets of B:    K h → u (h) = A ∈ B  f (u + h) − f (u) − Ah ≤ ε h .

(12)

Lemma 3 Given u ∈ Bδ (x), ¯ suppose that u maps B into K . Then there exists a continuous selection ψu of the mapping u such that the function defined as the composition ψu ◦ u and acting from B into itself has a fixed point. Proof Note that the values of u are closed convex sets. Fix any h ∈ K . Since ε < 1/, we get that u + h − x

¯ ≤ u − x

¯ + h ≤ δ + 2εδ < 3δ. Hence u and u + h ¯ Then (4) with v = u + h implies that u (h) = ∅. are distinct elements of B3δ (x). Therefore dom u = K . We show next that u is inner semi-continuous. Fix any

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h ∈ K , and let  be an open set in B which has a non-empty intersection with u (h). Pick any A from u (h) ∩ . According to (4), there is A˜ ∈ B such that ˜ < ε h .

f (u + h) − f (u) − Ah

Since  is open and A ∈ , there is λ ∈ (0, 1) such that Aλ := (1 − λ)A + λ A˜ ∈ . Put    V = τ ∈ K  f (u + τ ) − f (u) − Aλ τ < ε τ . The estimate

f (u + h) − f (u) − Aλ h ≤ (1 − λ) f (u + h) − f (u) − Ah

˜ + λ f (u + h) − f (u) − Ah

< (1 − λ)ε h + λε h = ε h , tells us that h ∈ V . Employing the continuity of f and the fact that Aλ is a continuous linear mapping, we have that any τ sufficiently close to h belongs to V ; thus V is a neighborhood of h in K . From the definitions of u and V we get that Aλ ∈ u (w) for w ∈ V, therefore u (w) intersects  for each w ∈ V . This proves that u is inner semi-continuous. Michael’s selection theorem yields the existence of a continuous selection ψu for u , that is, a continuous function acting from K into B with the property that if w ∈ K and A := ψu (w) then

f (u + w) − f (u) − Aw ≤ ε w .

(13)

Since B is a compact convex set, u is a continuous function, and by assumption u maps B into K , by the Brouwer’s fixed point theorem the composite mapping ψu ◦ u acting from B into itself has a fixed point.   We will now employ an iterative procedure which generates sequences {xn } in X and {An } in B whose entries have the following properties for each n ∈ N0 : (i) (ii) (iii) (iv)

¯ < δ;

xn − x

0 < xn+1 − xn ≤ (lε)n x1 − x0 ;

f (xn+1 ) − f (xn ) − An (xn+1 − xn ) ≤ ε xn+1 − xn ; f (xn ) + An (xn+1 − xn ) + F(xn+1 ) y.

Clearly, x0 := x¯ verifies (i) for n = 0. For any A ∈ B, we have x0 (A) = ϕ(A, y)−x0 . Using the properties (vi) and (vii) along with (6) and (9), we get 0 < x0 (A) = ϕ(A, y)−ϕ(A, 0) ≤  y ≤ εb < εδ < δ whenever A ∈ B. Hence x0 maps B into K . According to Lemma 3, there exists a continuous selection ψx0 of the mapping x0 such that the composite function ψx0 ◦ x0 has a fixed point. Denote this fixed point by A0 ; that is, A0 = ψx0 (x0 (A0 )) ∈ B. Then x1 := x0 + x0 (A0 ) = ϕ(A0 , y)

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satisfies (i) with n = 1, as well as (ii) and (iv) with n = 0. Since A0 = ψx0 (x1 − x0 ), condition (iii) holds as well thanks to (13). Further, we proceed by induction. Suppose that for a natural N > 0 we have found xn+1 and An that satisfy conditions (i)–(iv) for all natural n < N . Set v := x N −1 and A˜ = A N −1 . Conditions (ii)–(iv) with n = N − 1 imply that the mapping x N satisfies the assumption (10) of Lemma 2. Hence, by using Lemma 2 and then Lemma 3, there exists A N ∈ B such that A N = ψx N (x N (A N )). Put

x N +1 = x N + x N (A N ) = ϕ A N , y − f (x N ) + f (x) ¯ + A N (x N − x) ¯ .

(14)

Then f (x) ¯ + A N (x N +1 − x) ¯ + F(x N +1 ) y − f (x N ) + f (x) ¯ + A N (x N − x). ¯ This is (iv) for n = N . Since A N = ψx N (x N +1 −x N ), (iii) holds for n = N . Combining (11), (14), and (ii) for n = N − 1, gives us (ii) for n = N . Furthermore, since

x1 − x0 = x1 − x

¯ = x0 (A0 ) ≤ εb, using (ii) and (6), we conclude that

x N +1 − x

¯ ≤

N  n=0

xn+1 − xn <

εb ¯

x1 − x

≤ = εδ < δ. 1 − ε 1 − ε

We arrive at (i) for n = N + 1. The induction step is complete. By (ii), the sequence {xn } is a Cauchy sequence, hence it converges to some x ∈ ¯ Fix any n ∈ N. In view of (i), both xn and xn+1 are in Bδ (x) ¯ ⊂ Bβ (x). ¯ Bδ (x). Moreover, (7) for u˜ := xn+1 , w := xn and A := An , along with (ii), implies that

y − f (xn ) + f (x) ¯ − An (xn+1 − xn ) < β/2 < β. Using (iv) we obtain



xn+1 , y − f (xn ) − An (xn+1 − xn ) ∈ gph F ∩ Bβ (x) ¯ × Bβ (− f (x)) ¯ .

Passing to the limit and remembering the first of the key properties stated in the beginning of Step 3, we conclude that f (x) + F(x) y, that is, x ∈ ( f + F)−1 (y) ∩ ¯ Since y ∈ Bεb (0) was chosen arbitrarily, Bδ (x). Bεb (0) y → σ (y) := ( f + F)−1 (y) ∩ Bδ (x) ¯ is a nonempty-valued localization of ( f + F)−1 ; that is, (3) is established.

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Step 5 It remains to show that σ is a Lipschitz continuous function. Choose any y  , y  ∈ Bεb (0). Pick any x  ∈ σ (y  ) and x  ∈ σ (y  ). Then there exists A ∈ B such that

f (x  ) − f (x  ) − A(x  − x  ) ≤ ε x  − x  . Then (7) reveals that both y  − f (x  ) + f (x) ¯ + A(x  − x) ¯ and y  − f (x  ) + f (x) ¯ +  ¯ lie in Bβ (0). Moreover, we have A(x − x) x  = s A (y  − f (x  ) + f (x) ¯ + A(x  − x)) ¯ 





and



x = s A (y − f (x ) + f (x) ¯ + A(x − x)). ¯ Taking the difference, we obtain

x  − x  ≤  y  − y  +  f (x  ) − f (x  ) − A(x  − x  )

≤  y  − y  + ε x  − x  . This gives us

x  − x  ≤



y  − y  . 1 − ε

Thus σ is both single-valued and Lipschitz continuous. The proof of Theorem 3 is complete.   3 Final remarks We will first comment on the proofs given by Izmailov [6] and Páles [10] and compare them to our proof of Theorem 3. The proof in [6] strongly relies on the fact that the spaces are finite dimensional. The main step in that proof is an application of Brouwer’s fixed point theorem to a ¯ (using the notation in [6]) whose continuity is claimed function χ y on a ball Bδ (x) but never proved.1 Páles [10] uses in his proof Banach’s open mapping theorem, Michael’s selection theorem, Ekeland’s variational principle and Kakutani’s fixed point theorem. We also use Banach’s and Michael’s theorems, along with the general formulation in [3, Theorem 5F.1] of Robinson’s implicit function theorem but the whole idea of the proof is different from [10]; the main step of the proof is an iterative procedure which resembles Newton’s method. We would like to also point out that our motivation to consider infinite dimensional spaces is not based only on our desire to develop an alternative proof of Izmailov’s 1 Shortly before this paper was accepted for publication we received a letter by A. Izmailov where he confirmed that indeed his proof is not complete and could be fixed by using some of the arguments in our proof. He also noted that his proof heavily relies on the finite dimensions.

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theorem also covering the infinite-dimensional case. In a subsequent paper [11] Páles gave an application of his theorem to optimal control; we do believe that our more general result has the potential for further applications in that field. Furthermore, there have been a number of developments in the last decade regarding Newton-type methods applied to nonsmooth equations in infinite-dimensional spaces, e.g. in PDEconstrained optimization; some of them are broadly covered in the recent books by Ito and Kunisch [5] and Ulbrich [14]. Strong regularity plays a prominent role in these developments, as it has been since Josephy, a student of Robinson, applied it for showing convergence of Newton’s method for variational inequalities. In particular, Klatte and Kummer considered in [7, Section 10.1] nonsmooth equations where strong regularity appears in the form of certain uniform invertibility of the elements of a set of linear bounded operators which is closely related to but is different from the set A in our paper. Note that Ulbrich [14] and Ito and Kunisch [5] consider variational inequalities representing optimality conditions that are reduced to nonsmooth equations by using e.g. the Fischer–Burmeister function. In our work we consider nonsmooth generalized equations which occur typically in models of optimal control and in C 1,1 optimization. We expect that applying our theorem and the analysis around it to specific nonsmooth Newton-type methods in infinite dimensions will provide valuable contributions to the area. References 1. Clarke, F.H.: On the inverse function theorem. Pac. J. Math. 64, 97–102 (1976) 2. Clarke, F.H.: Optimization and Nonsmooth Analysis. Wiley, New York (1983) 3. Dontchev, A.L., Rockafellar, R.T.: Implicit Functions and Solution Mappings, 2nd edn. Springer, Dordrecht (2014) 4. Ioffe, A.D.: Nonsmooth analysis: differential calculus of nondifferentiable mappings. Trans. Am. Math. Soc. 266, 1–56 (1981) 5. Ito, K., Kunisch, K.: Lagrange Multiplier Approach to Variational Problems and Applications. SIAM, Philadelphia (2008) 6. Izmailov, A.F.: Strongly regular nonsmooth generalized equations. Math. Program. Ser. A 147, 581– 590 (2014) 7. Klatte, D., Kummer, B.: Nonsmooth equations in optimization. Regularity, calculus, methods and applications. In: Nonconvex Optimization and Its Applications, vol. 60. Kluwer, Dordrecht (2002) 8. Kummer, B.: An implicit-function theorem for C 0,1 -equations and parametric C 1,1 -optimization. J. Math. Anal. Appl. 158, 35–46 (1991) 9. Páles, Z.: Linear selections for set-valued functions and extensions of bilinear forms. Arch. Math. 62, 427–432 (1994) 10. Páles, Z.: Inverse and implicit function theorems for nonsmooth maps in Banach spaces. J. Math. Anal. Appl. 209, 202–220 (1997) 11. Páles, Z.: On abstract control problems with non-smooth data. In: Recent Advances in Optimization, pp. 205–216. Springer, Berlin (2006) 12. Robinson, S.M.: Strongly regular generalized equations. Math. Oper. Res. 5, 43–62 (1980) 13. Robinson, S.M.: An implicit-function theorem for a class of nonsmooth functions. Math. Oper. Res. 16, 292–309 (1991) 14. Ulbrich, M.: Semismooth Newton Methods for Variational Inequalities and Constrained Optimization Problems in Function Spaces. SIAM, Philadelphia (2011)

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