On nonlinear inclusions in non-smooth domains

This paper reviews the old and new landmark extensions of the famous intermediate value theorem (IVT) of Bolzano and Poincaré to a set-valued operator \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\Phi : E \supset X \rightrightarrows E}$$\end{document} defined on a possibly non- convex, non-smooth, or even non-Lipschitzian domain X in a normed space E. Such theorems are most general solvability results for nonlinear inclusions: \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\exists x_{0} \in X}$$\end{document} with \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${0 \in \Phi (x_{0}).}$$\end{document} Naturally, the operator Φ must have continuity properties (essentially upper semi- or hemi-continuity) and its values (assumed to be non-empty closed sets) may be convex or have topological properties that extend convexity. Moreover, as the one-dimensional IVT simplest formulation tells freshmen calculus students, to have a zero, the mapping must also satisfy “direction conditions” on the boundary ∂X which, when \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${X = [a,b] \subset E = \mathbb{R}}$$\end{document}, Φ (x) = f(x) is an ordinary single-valued continuous mapping, consist of the traditional “sign condition” f (a) f (b) ≤ 0. When X is a convex subset of a normed space, this sign condition is expressed in terms of a tangency boundary condition \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\Phi (x) \cap T_{X}(x) \neq \emptyset}$$\end{document}, where TX(x) is the tangent cone of convex analysis to X at \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${x \in \partial X}$$\end{document}. Naturally, in the absence of convexity or smoothness of the domain X, the tangency condition requires the consideration of suitable local approximation concepts of non-smooth analysis, which will be discussed in the paper in relationship to the solvability of general dynamical systems.


Historical background and preliminaries
In 1817, the Bohemian philosopher, theologian and mathematician Bernhard Bolzano published a series of investigations consecrating him as one of the forefathers of modern analysis [19]. This work contained among other things the first modern criterion of convergence and the celebrated Bolzano-Weierstrass theorem, 1 which he used as a lemma to provide the first "purely analytic proof of the theorem that between any two values, which give results of opposite sign, there lies at least one real root of the equation." This is known as the intermediate value theorem (IVT) in dimension one: the first existence theorem in a freshmen calculus course. In today's notation, it reads: if f ∈ c([a, b], R) and f (a) f (b) ≤ 0, then f (x 0 ) = 0 for some x 0 ∈ [a, b].
As part of his program of rigorizing analysis, Augustin Louis Cauchy provided 4 years later (1821) another proof of the IVT in his Cours d'analyse. 2 Cauchy was the first to contribute significantly (in 1831 and 1837) to the study of the planar case ( f 1 (x 1 , x 2 ) = 0, f 2 (x 1 , x 2 ) = 0) with scalar functions f i "defined inside a domain on whose boundary they do not vanish simultaneously." He defined the Cauchy's index, precursor of the notion of degree of a mapping. Jacques Charles François Sturm (1803-1855) and Joseph Liouville (1809-1882) worked to make Cauchy's work more rigorous and precise, laying the bases for the analytic definition of the topological degree of a continuous mapping.
Building on the contributions of Cauchy, Sturm-Liouville and later Leopold Kroenecker generalized in 1869 the Cauchy's index to C 1 vector fields on R n into what became known as the Kronecker index i( f, a), which counts the generalized multiplicity of an isolated root a of a general equation f (x) = 0, f ∈ c 1 (R n , R). (i( f, a) is precisely the Brouwer degree deg( f, B(a, ), 0) for > 0 small enough.) The reader is referred to Dinca-Mahwin [34] for a detailed modern exposition on the Kronecker index and the Brouwer degree.
This paper aims at reviewing, in some detail, the most important extensions of the Bolzano IVT to spaces of arbitrary dimensions and point to set mappings defined on compact domains that may or may not be convex. The remainder of this section is devoted to a survey of landmark generalizations of the IVT to n-dimensional Euclidean spaces and to general normed spaces, followed by continuity concepts for set-valued maps. Tangency conditions for convex and non-smooth domains are discussed in Sect. 2. Section 3 outlines a general method, based on the Browder-Ky Fan fixed point theorem, for the solvability of the inclusion 0 ∈ (x 0 ) where is an upper hemicontinuous point to set map with closed convex values defined on a compact convex subset of a locally convex topological linear space. Section 4 discusses the case of non-smooth domains with or without Lipschitzian behavior and maps with convex or non-convex values.

Extensions of the IVT to arbitrary dimensions
In a qualitative study of nonlinear ordinary differential equations, using the Kroenecker index, Jules Henri Poincaré extended in 1883 the IVT to R n [58]. "Mr. Kronecker, Poincaré writes, has presented to the Berlin Academy, in 1869, a memoir on functions of several variables, including an important theorem from which the following result follows easily: let ξ 1 , ξ 2 , . . . , ξ n be n continuous functions in the n variables x 1 , x 2 , . . . , x n , the variable x i restricted to range among the limits −a i and +a i . Let us suppose that for x i = a i the function ξ i is always positive, and for x i = −a i the function ξ i is always negative. Then, I say that there is a system of values for the x at which all the ξ i vanish. This result can be applied to the three-body problem to prove it has infinitely many special solutions . . ." The IVT in n-dimension was later used in 1910 by Jacques Hadamard to give the first proof of the Brouwer fixed point theorem for arbitrary n. 3 1 It was only 50 years later that Karl Weierstrass proved the theorem again and stressed its importance in the foundations of analysis. 2 The idea was, as one can imagine, older than Bolzano and Cauchy's. Earlier mathematicians considered it intuitively obvious, thus requiring no proof. For instance, Dutch mathematician Simon Stevin (1548-1620) proved the IVT for polynomials, anticipating Cauchy's algorithmic proof. There are views that Cauchy was more than inspired by Bolzano's earlier work (without acknowledgment). Others see in the similarities (e.g., in the definition of continuity, the convergence criterion, and the proof of the IVT) nothing more than common earlier influences, particularly those of Lagrange. We refer to Grabiner [39] for an insightful discussion. 3 The Italian mathematician, Carlo Miranda [53] proved in 1940 that the Brouwer fixed point theorem is equivalent to Poincaré's intermediate value theorem, and thus the appelation, used by some, of the Poincaré-Miranda Theorem.

The sign conditions
Note also that the Brouwer fixed point theorem follows at once from Poincaré's IVT: if f ∈ c(X, X ), then [60]: if X is a closed ball centered at 0 E and ξ is a compact field with (ξ + I d)(∂ X ) ⊂ X, then ξ has a zero in X . Clearly, if X is a closed ball centred at 0 in R n equipped with the maximum norm and if ξ satisfies Rothe's boundary condition, then the IVT sign condition on ∂ X holds. Rothe's theorem remains valid if X is a convex subset of a topological vector space E with 0 ∈ U open ⊂ X, ξ a compact field, and is a (not necessarily continuous) positively homogeneous functional satisfying p −1 (0) = {0} (e.g., p is a seminorm).) • Altman [1]: if X is a closed ball centered at 0 E and ξ is a compact field with ξ(x) for all x ∈ ∂ X, then ξ has a zero in X. In the case where E is a Hilbert space, this theorem was first established by Krasnoselsky [50]. Clearly, if X is a closed ball centred at 0 in R n equipped with the Euclidean norm, then (ξ satisfies Altman's condition on i.e., the IVT sign condition on ∂ X holds. • Yamamuro [66]: let X = cl(G), G being an open subset in E and ξ a compact field verifying: [there exists a ∈ G such that if ξ(x) = λ(a − x) for some x ∈ ∂G, then λ ≥ 0] and ξ has a zero in X . Perhaps, the first most striking generalizations of the Poincaré's IVT to locally convex topological vector spaces are the ones due to Halpern and Halpern-Bergman.
• Halpern [43]: X is a non-empty convex compact subset of a strictly convex normed linear E and f ∈ c(X, E) satisfies the inwardness condition ∀x ∈ ∂ X, Note that here, X need not have a non-empty interior, but also the tangency condition ξ(x) ∈ S X (x) seems to be less general than the Yamamuro boundary condition.
• Halpern and Bergman [46]: if X is a non-empty convex compact subset of a topological vector space E having separating dual and ξ ∈ c(X, E) satisfies the weak inwardness condition ξ(x) ∈ T X (x) = cl(S X (x)), ∀x ∈ ∂ X , then ξ has a zero in X. • The strict convexity assumption was removed first by Halpern and Bergman [46] before the publication of Felix Browder's paper [24]. Before turning our attention to the set-valued case 0 ∈ (x 0 ), being a set-valued map, let us fix a few definitions, conventions and notations and define crucial concepts for what follows.

Continuity concepts and classes of set-valued maps
Throughout this paper, it is assumed that topological (vector) spaces (t.v.s. for short) are Hausdorff (and real).
Set-valued maps are assumed to have non-empty values and are denoted by capital greek letters , , etc., using double arrows ⇒. A zero, 0 ∈ (x 0 ), for a set-valued map is also called an equilibrium for . Single-valued maps are denoted by small letters f, ξ, s, etc. The space of continuous single-valued maps from a topological space X into a topological space Y is denoted by c(X, Y ). 4 Recall that, given a map (with non-empty values) : X ⇒ Y of topological spaces and x 0 ∈ X , the (Kuratowski-Painlevé ) inferior limit lim inf x→x 0 (x) is the set {y ∈ Y : for any net x i → x 0 , there exists a net y i → y with y i ∈ (x i ), for all i}. The (Kuratowski-Painlevé) superior limit lim sup x→x 0 (x) := {y ∈ Y : ∃ x i → x 0 and ∃y i → y with y i ∈ (x i ), for all i}. If Y is metrizable, lim inf x→x 0 (x) = {y : lim sup x→x 0 d(y; (x)) = 0} and lim sup x→x 0 (x) = {y : lim inf x→x 0 d(y; (x)) = 0}. Both limiting sets are closed (possibly empty) and lim inf x→x 0 (x) ⊆ cl( (x)) ⊆ lim sup x→x 0 (x). When equality occurs, the common set is denoted lim x→x 0 (x). For a countable family, lim inf n→∞ (x n ) = (iv) an usco is a u.s.c. map with (non-empty) compact values.
In case X is a topological space and E a t.v.s. with topological dual E , the following regularity properties for a set-valued map : X ⇒ E are more general than upper semicontinuity: This concept has been presented by Fan [35].
is upper hemicontinuous on X (u.h.c.) if for each p ∈ E , the support functional x → σ (x) ( p) = sup y∈ (x) p, y is upper semicontinuous as an extended real-valued function on X, i.e., ∀λ ∈ R ∪ {∞}, the set {x ∈ X : σ (x) ( p) < λ} is open in X. This concept has been presented, as far as we can tell, by Cornet [29].
The interesting fact about u.h.c. maps is that a linear combination of u.h.c. maps is also u.h.c. (one can easily verify that (i) the support functional of a sum of sets σ A 1 +A 2 is the sum of the support functionals σ A 1 + σ A 1 and that the sum of u.s.c. extended real-valued functions is also a u.s.c. function; and (ii) that for a given set A and a real λ, We shall consider in the sequel the classes of convex valued maps:

Tangency and dynamics on non-convex domains
This section discusses boundary conditions extending the IVT's sign condition to convex and non-convex subsets of normed spaces. These boundary conditions are expressed in terms of tangent or normal cones and have an inherent dynamical nature as they appear as necessary and sufficient conditions for the existence of viable solutions to dynamical systems. Although one can conceivably work in the context of topological linear spaces with separating duals, for the sake of simplicity we assume in this section that the underlying space E is a real normed space with topological dual E .

Tangent and normal cones
We start with a brief discussion of some local approximation concepts of tangent cones to a subset X of E near a point x ∈ cl(X ).
As defined earlier, given be the cone pointed at 0 with base X − x. When X is convex, cl(S X (x)) is the tangent cone of convex analysis to X at x, and its negative polar cone is precisely the normal cone of convex analysis: The Bouligand-Severi contingent cone T X (x) is the upper limit in the sense of Painlevé-Kuratowski (i.e., the set of all cluster points) when t ↓ 0, of the family ([23,62]). It is characterized as: where d X (x) = inf{ x − u : u ∈ X }, x ∈ E, is the distance from the point x to the set X. The Clarke (circatangent) cone (see [61]) T C X (x) is the lower limit (i.e., the set of all limit points), when t ↓ 0 and x → X x, of the family x ∈X and is characterized as: Note the following facts: is finite, positively homogeneous, subadditive and Lipschitz continuous on E: The Bipolar theorem (see e.g., [6]) implies that N C , which is itself a closed cone contained in cl(S X (x)).
• The negative polar cone to the Bouligand-Severi contingent cone is the prenormal cone (of regular normals) to X at x given by: Note that the basic normal cone is not convex and that its closed convex hull is the Clarke's normal cone N C X (x) = cl(conv(N X (x))). • The set X is said to be regular at x if the contingent cone T X (x) and the normal cone N X (x) are mutually polar (thus, they are both convex and closed, i.e., is also a convex and closed cone, and x → T C X (x) is also l.s.c.. • Examples of non-sleek sets are:

Proposition 2.1 If X is sleek, then the convex and closed valued map N C
, and hence has closed graph.
Proof Note first that given a sequence of sets {T n } in a normed space, we always have lim inf n→∞ T n ⊆ (σ − lim sup n→∞ T − n ) − (equality occurs, e.g., when T n is a closed convex cone, this is known as the Duality Theorem; see [6]). Indeed, if x ∈ lim inf n→∞ T n , i.e., x = lim n x n , x n ∈ T n , and p ∈ σ − lim sup n→∞ T − n , i.e., p is the weak * −limit σ − lim n k p n k of a subsequence p n k ∈ T − n k , p n k , x n k ≤ 0, then p, i.e., N C X is weak * − u.s.c.. By Proposition 1.3 (5), it has a weakly closed graph, which, being convex, is also strongly closed (Mazur's theorem).
The reader is referred to Mordukhovich [54] for a detailed and lively account of aspects of variational analysis, including tangents and normal cones to non-smooth sets.

Boundary conditions
Assume that : cl(X ) ⇒ E and consider the following boundary conditions: Observe that if : X ⇒ E verifies (x) ∩ X = ∅, ∀x ∈ ∂ X (general case of the Brouwer and Kakutani fixed point theorems), then = − I d verifies (R).
Also, note that all conditions (H, wH, K F and wK F1, 2) are meaningful only on the boundary ∀y ∈ E, and the conditions are trivially satisfied.
Notice that (R) ⇒ (H ). In case X is convex, its tangent cone is the cone of convex analysis cl(S X (x)) which coincides with the circatangent cone of Clarke T C X (x) and, clearly, (H ) ⇒ (w H ) also holds. We have the additional relationships between the boundary conditions above: : p, y ≤ 0} being weakly closed and since p n * p and y n ȳ, it follows that (ȳ,

Tangency and viability of trajectories
The dynamical nature of the tangency condition (w H ) is remarkably captured by the Nagumo viability theorem.

Definition 2.4
A subset X of R n is said to be locally viable with respect to a differential system Here, T X (x) is the Bouligand-Severi contingent cone. The set-valued extension of the Nagumo's theorem is from Bebernes-Schuur [18].
The lack of convexity of values of can be compensated by a stronger regularity of in addition to a stronger tangency condition as shown by Aubin and Cellina (see [4]): if R n ⊇ X is locally closed 7 and : X ⇒ R n is both u.s.c. and l.s.c. with closed values, then Observe that if has closed convex values and is l.s.c., then viability follows from Michael's selection theorem and Nagumo's viability theorem.
The reader is referred to [6,26,33,48] for extensive references on and proofs of viability theorems.

Equilibria and co-equilibria in Hilbert spaces
We clarify the relationship between the existence of equilibria and co-equilibria for a set-valued map. For simplicity, assume in this section that the underlying space is a real Hilbert space (E, ·, · ) identified with its dual. The results below remain valid with a dual pair (E, E ) of a normed space and its topological dual, with the suitable adaptations.
Recall that, given a subset X of E, an element x 0 ∈ cl(X ) is an equilibrium for a set-valued map : is the Clarke's normal cone to X at x 0 as defined in Sect. 2.1. Clearly, an interior co-equilibrium is an equilibrium since, for such a point, N C X = {0}. Observe that x 0 is a co-equilibrium for if and only if the maps and N C X coincide at x 0 , i.e., ( is weakly compact. Indeed, the extended real valued function y → sup v∈T C X (x) y, v is l.s.c. and convex, hence weakly l.s.c.. Therefore, it achieves its infimum on (x 0 ) at some y 0 verifying y 0 , v ≤ 0, ∀v ∈ T C X (x), i.e., y 0 ∈ N C X (x 0 ). By a Hilbert space pair we mean a pair (X, E) with E a real Hilbert space and X a closed subset of E. . By Proposition 2.1 and since X is sleek, the map N C X : X ⇒ E has closed graph. By Proposition 1.3 (3) and since the graph of N C X is also convex and the values N C X (x) ∩ D are closed, convex and bounded, hence weakly compact, it follows that the set-valued map x → N C X (x) ∩ D is u.h.c. with closed convex, and bounded values. As a linear combination of u.h.c. maps, is also u.h.c. Being the sum of a compact convex set and a closed bounded convex set, (x) is closed and convex for each x ∈ X , i.e., ∈ H(X, E). We need to show that verifies (w H ). For any given x ∈ ∂ X, since the cone T C X (x) is closed and convex, the Moreau decomposition theorem [55] implies that any y ∈ (x) can be decomposed as a sum y = y T + y N with y T = Proj T C X (x) (y) and y N = Proj N C X (x) (y) and y N , y T = 0. Hence, 0 = y N , y T = y N , y − y N = y N , y − y N 2 and by the The fact that (X, E) has the equilibrium property for H w H ends the proof. (ii) Halpern (see theorem 6 in [45]) replaced the compactness hypothesis of the domain X by that of the map under the stronger inwardness (x) ∩ S X (x) = ∅, ∀x ∈ ∂ X, and provided the underlying space is complete. More precisely, he showed: if E is a complete locally convex t.v.s., E ⊇ X closed convex, ∈ K(X, E) with cl( (X )) compact, and (x) ∩ S X (x) = ∅, ∀x ∈ ∂ X, then has an equilibrium. (iii) Situation (e) follows from the coincidence Theorem 9 in [37] with g(x) = {0} for all x ∈ ∂ X. (iv) The compactness of the domain X can be replaced by a weaker coercivity condition of "Karamardian type" (see Theorem 3.8 and Remark 3.9(3) below).

A proof based on the Browder-Ky Fan fixed point theorem
We opt to present the proof of the existence of an equilibrium, in the fully convex case, as it opens the door to further generalizations. The Browder-Ky Fan fixed point theorem (Theorem 1 in [24] and Theorem 2 in [36]) asserts that if X is a compact convex subset of a t.v.s. E and : X ⇒ X is a set-valued map with convex values and open fibers, then either has a maximal element x 0 ∈ X, i.e., (x 0 ) = ∅, or has a fixed pointŷ ∈ (ŷ). 8 The analytical expression of this result as an alternative for systems of nonlinear inequalities reads:

Proposition 3.2 ([13]) If X is a convex compact subset of a t.v.s. E and f : X × X −→ R a numerical function satisfying:
Then, for any given λ ∈ R, the following nonlinear alternative holds:
The following result contains all situations (a)-(e) above.

Theorem 3.5 Assume that X is a convex compact subset in a locally convex t.v.s. E and ∈ H(X, E) verifies condition (KF). Then has an equilibrium in X .
is u.s.c.) and concave in ϕ. Condition (KF) opposes (2) of Corollary 3.4. Hence (1) holds: ∃x ∈ X with inf y∈ (x) ϕ, y ≤ 0 for all ϕ ∈ E .
Since convex sets are sleek, the image of a compact set under a compact valued u.h.c. map is compact, Theorems 2.9 and 3.5 imply:

Corollary 3.6 If X is a convex compact subset of a Hilbert space E, any compact-valued map ∈ H(X, E) has a co-equilibrium.
Using the same approach based on the Browder-Ky Fan fixed point theorem, Theorem 3.5 has been extended to surjectivity results for perturbations of H−maps by convex processes by Kryszewski and the author in [15] with the compactness condition on the domain replaced by a coercivity condition (thus extending results of Aubin [2]) as described below.
Let E, F be two normed spaces and let (E, F) be the normed space of all closed convex processes (i.e., set-valued maps whose graphs are closed convex cones in Let E ⊃ X and let L : X −→ (E, F) be a continuous operator. Definition 3.7 A map : X ⇒ F is said to verify the condition (wH) on a pair (X 1 ⊂ X, X 2 ⊂ X ) with respect to the operator L iff:

Theorem 3.8 Assume that X is a convex set in the normed space E,
∈ H(X, F) and L satisfies the boundedness condition: Assume also the existence of a compact subset K of X such that for each finite subset N ⊂ X , there exists a convex compact subset C N ⊂ X such that the map verifies: (i) the condition (wH) on (C N \K , C N ) with respect to L. (ii) the condition (wH) on (K ∩ ∂ X, X ) with respect to L.
(2) In the case where for every x ∈ X, (x) ≡ L is a linear operator from E into F, and C N = C is the same compact convex set for all finite subsets N of K , this result can be found in [10] and (2) guarantees the surjectivity of the perturbation + L onto L(X ). (3) In the case where X is compact and (x) = L(x) is a linear operator, this result is precisely the solvability theorem in Aubin and Frankowska [6]. By putting K = ∅ and C N = X for any N , (i) and (ii) reduce to: satisfying the condition (K F) with respect to L .
As an immediate consequence of Theorem 3.8, we deduce that the existence of viable solutions in a noncompact domain implies the existence of a stationary solution 9 for the inclusion (x (t)) ⊂ (x(t)), t ∈ [0, T ]. More precisely, we have: Corollary 3.10 ( [15]) Let X be a closed convex subset of R n , K a compact subset of X, C a bounded subset of X, : R n ⇒ R m a linear process, 10 and ∈ H(X, R m ).
Assume that the following properties are satisfied: (i) given any x 0 ∈ K , there exist T > 0 and a solution x(.) of the differential inclusion (x (t)) ⊂ (x(t)) on [0, T ) starting at x 0 such that for any T ∈ (0, T ] there exists t ∈ (0, T ] with x(t) ∈ X ; (ii) given any x 0 ∈ X \K , there exists T > 0 and a solution x(.) of the same differential inclusion on [0, T ) starting at x 0 such that for any T ∈ (0, T ] there exists t ∈ (0, T ] such that x(t) ∈ conv(x 0 , C); Then (x (t)) ⊂ (x(t)) has a stationary solution in X. Remark 3.11 (1) Conditions (i)-(ii) state that if a trajectory of the differential inclusion starts in K then it must first enter X, and if it starts in X \K then it is first attracted by C in some weak sense (the trajectory intersects the drop with vertex x 0 and base C). (2) When X is a compact convex viability domain of , R n = R m and = I d R n , then (i) and (ii) are obviously satisfied with K = C = X. This corresponds to the equilibrium theorem of [6].

Equilibria in non-smooth domains
Naturally, the generalization of the IVT that comes immediately to mind, in case of non-convex domains, is the existence of a zero for a tangent vector field defined on a compact smooth manifold with non-trivial Euler characteristic. We shall now discuss extensions of this result to set-valued maps and more general domains.
When considering non-convex domains, it is natural from a topological point of view to look at domains that are homeomorphic to convex sets or contractible (e.g., star shaped), or more generally absolute (neighborhood) retracts of normed spaces. From an optimization perspective, one would consider for example proximally smooth or regular sets defined by smooth or non-smooth inequalities, or subsets of normed spaces that are (locally) copies of epigraphs of Lipschitz continuous functions. Consideration of non-convex domains for extensions of the IVT necessitates adequate boundary conditions described in terms of tangent and normal cones discussed in Sect. 2.
Note that although X is of course locally viable with respect to u = f (u), not all trajectories are viable in X. Indeed, the vector field u(t) = (0, 0, 1 − e −t ) is a trajectory starting at u(0) = (0, 0, 0) but leaving X for t > 0.
The next two subsections of this paper focus on the last two types of domains.
As mentioned earlier, one of the most natural classes of subsets of normed spaces to be considered in extending fixed point and equilibrium theorem to non-convex domains is the class of absolute retracts (ARs) and more generally that of absolute neighborhood retracts (AN Rs) (see e.g., Borsuk's generalizations of the Brouwer fixed point theorem [21]).
Recall that a subset X of a normed space E is said to be a (neighborhood) retract of E if there exists a mapping r : It is well-known that every continuous self-mapping f of a compact AR has a fixed point [21]; while if X is a compact AN R, then f : X −→ X has a fixed point provided its Lefschetz number λ( f ) is non-zero.
Since the Euler characteristic 13 χ(X ) of X is precisely the Lefschetz number of the identity mapping I d X , then assuming that χ(X ) = 0 (which holds true if K is an absolute retract), it follows that a map from X into itself that is homotopic to the identity has a fixed point. It is well established that χ(X ) = 0 for any compact manifold of odd dimension X, and that for such domains there are non-vanishing tangent vector fields.
Thus, χ(X ) = 0 is a standing assumption when considering domains that are not absolute retracts.
In what follows we shall assume that: (A) X is a compact AN R of a normed space E with a given retraction r : U −→ X defined on a neighborhood U of X.
Since X is compact, we may assume with no loss of generality that such a neighborhood U is a uniform neighborhood B(X, β) = {x ∈ E : d(x; X ) < β} of X.
Let us mention that the class of sets satisfying (A) is quite substantial. For instance, if X is compact convex, then X satisfies (A) (a by-product of the Dugundji's Extension Theorem) in addition to having non-trivial Euler characteristic. Furthermore, since the property of being a compact neighborhood retract and the Euler characteristic are both topologically invariant, it follows that any subset Y of E, homeomorphic to X, satisfies (A) and has non-trivial Euler characteristic, provided X has both properties.
For an extensive study of retracts, the reader is referred to the monographs by Borsuk [21] and Hu [47].
there exists an open neighborhood U of X in E and a continuous retraction r : U −→ X and a constant L > 0 such that: . If X is epi-Lipschitzian, then X = cl(I nt (X )), the map x → T X (x) is l.s.c. (thus N C X has closed graph), i.e., X is sleek. 12 X is a proximate retract in R n if there exists a continuous retraction r : R n −→ X such that r (u) − u = d X (u), ∀u ∈ R n . 13 If X is a smooth submanifold of R n then John Milnor defines χ(X ) as the Brouwer degree of the Gauss mapping G X (x) = the unit outward normal vector to X at x ∈ ∂ X. For non-smooth sets, e.g., X is a compact epi-Lipschitzian subset of R n , Cornet [30] defines the Gauss mapping in terms of proximal normal vectors: G X (x) = conv(N C X (x)∩S n−1 ) and χ(X ) = deg(G X , int (X ), 0), the Cellina-Lasota degree. More generally, if X is a compact topological space, the singular cohomology {H q (X )} is a graded linear space of finite type. Denote dim Q (H q (X )) = β q (X ) (qth-Betti number) and define: χ(X ) := q (−1) q β q (X ). It turns out that χ(X ) = λ(I d X ) the Lefschetz number of the identity mapping on X .
Observe that if X is a neighborhood retract of a metric space E with a Lipschitz continuous neighborhood retraction r : U −→ X with modulus k > 0, where U is an open subset of E containing X, then d(r (x), x) ≤ Ld X (x), ∀x ∈ U with L = k + 1. Moreover, if X is compact, it suffices that the neighborhood retraction is locally Lipschitz continuous.
Clearly, closed convex subsets of Hilbert spaces come immediately to mind with the retraction consisting of the metric projectionProj X , which is single valued and locally Lipschitz continuous. These properties of Proj X hold true for more general types of sets such as, e.g., Federer's sets with so-called local positive reach in Euclidean spaces, proximally smooth subsets of Hilbert spaces 14 of Clarke-Stern-Wolenski, sets with the Shapiro property 15 and more generally prox-regular subsets of Hilbert spaces in the sense of Poliquin-Rockafellar-Thibault (see [59] and references there). More precisely (see Sect. 2 above), recalling that the normal cone N X (x) to a closed set X at x ∈ X is the limiting proximal normal cone : λ ≥ 0 and y ∈Proj X (x))} being the proximal normal cone to X at x, we have: Uniform prox-regularity with constant 1/ρ for every 0 < ρ < ρ is equivalent to the Fréchet (or Gâteaux) differentiability of the distance function d X (.) on a tubular neighborhood B(X ; ρ) = {x ∈ H, d X (x) < ρ} of X in H. It is also equivalent to the fact that every non-zero proximal normal to X at any x ∈ X can be realized by a ρ−ball, i.e., ∀x ∈ X,  [59] assert that a necessary and sufficient condition for X to be uniformly prox-regular with constant ρ > 0 is that the projection Proj X be single valued and monotone on the tubular neighborhood B(X ; ρ) and Lipschitz continuous on B(X, ρ ) with modulus ρ ρ−ρ for any 0 < ρ < ρ. A convex set is uniformly prox-regular with constant ρ = +∞. Moreover, since the normal cone N X (x) to a proximally smooth set X at any given point x ∈ X is closed and convex, it follows that N X (x) = N P X (x) for such sets, i.e., every normal is actually a proximal normal. Hence, a proximally smooth closed subset of a Hilbert space is uniformly prox-regular.

Proposition 4.3 Each of the following is an L-retract:
(1) A proximate retract of Plaskacz.
(2) A uniformly prox-regular closed subset X of a Hilbert space H (see also [22]).
(3) A bi-Lipschitz homeomorphic to a convex compact subset of a normed space.
Proof (1) is obvious as a proximate retract is an L-retract with L = 1. To show (2) observe that for any given ρ ∈ (0, ρ) where ρ > 0 is the constant of uniform prox-regularity of X, the projection of H onto X, Proj X = r : B(X, ρ ) −→ X, r (x) = (x) is single valued and continuous with x − r (x) = d X (x) (see Theorem 4.1 and Lemma 4.2 in [59]). Hence X is a neighborhood proximate retract and thus an L-retract (with constant 1).
To show (3), let X be a closed subset of E that is bi-Lipschitz homeomorphic to a closed convex subset Y of a normed space (F, . ) (comp. with [28]), i.e., there exists a Lipschitz homeomorphism h : X −→ Y with Lipschitz inverse g = h −1 . We establish that X is an L-retract. To do this, consider a system {U i , a i ) i∈I for E\X such that: Such a system always exists. Let f : X −→ B be the continuous extension of the map h to the entire space E, defined by the formula: where {λ i } i∈I is a locally finite partition of unity subordinated to {U i } i∈I .
Let r : E −→ X be given by: We show that for each x ∈ E, d(r (x), x) ≤ Ld X (x) where L = 3L g L h + 1, L g and L h the Lipschitz constants of h and g respectively. Indeed, for any given . Take i ∈ I (x), > 0 arbitrary and a ∈ X such that: The property of being an L-retract is intimately linked to a universal extension property for locally Lipschitzian mappings.

ii) NLEP is stable for open subsets. (iii) NLEP is stable for unions of open sets.
(iv) NLEP is hereditary, i.e., it goes from local to global in the presence of compactness; hence a compact metric space that has NLEP locally is an L-retract.
The following definition by [17] extends the concept of an epi-Lipschitz subset of a Euclidean space (due to Rockafellar) to subsets of normed spaces of arbitrary dimension.

Definition 4.6
A subset X of a normed space E is an epi-Lipschitz set if each boundary point y of X has a neighborhood U (in X ) for which there exists a normed space F, an open set C of F, a Lipschitz continuous function g : C −→ R, a point z = (x, g(x)), x ∈ C, a neighborhood V of z, and a bi-Lipschitz homeomorphism h : V ∩ E pigraph(g) −→ U such that h(z) = y. Roughly speaking, such a set is locally the epigraph of a Lipschitz continuous function.

Equilibria in L-retracts
Theorem 4.8 Let X be a compact L-retract in a normed space E with χ(X ) = 0, and let ∈ H(X, E) verifying (x) ∩ T C X (x) = ∅, ∀x ∈ ∂ X. Then has an equilibrium.
The proof of Theorem 4.8 is based on the following approximation under constraint result, a sort of hybrid between the celebrated E. Michael's selection theorem [52] and a crucial theorem of A. Cellina [27] on the graph approximation of u.s.c. maps with convex values. 16

Lemma 4.9 Let (X, d) be a metric space and (E, . ) be a normed space. Let
: X ⇒ E be a lower semicontinuous map with convex values, and ∈ K(X, E) be such that (x) ∩ (x) = ∅ for each x ∈ X. Then for any δ > 0, there is a continuous map f : X −→ E such that for every x ∈ X: For any x ∈ X, choose z x ∈ (x) ∩ (x) and consider the open Let {λ i } i∈I be a locally finite partition of unity subordinated to O. Hence, for each i ∈ I, there are where z i = z x i is clearly continuous. Moreover, for each x ∈ X and each index i in the finite set of essential indices Thus, by convexity of (x), On the other hand, given

Remark 4.10
This lemma guarantees the existence of a δ-approximate selection for which is also a δapproximation of the graph of . As in the Michael's selection theorem [52], assuming that in addition the values of are closed and E is a Banach space, we conclude that there exists a selection of which is also a δ-approximation of the graph of .
Given a compact L-retract X in a normed space (E, . ), let us assume (with (A) in mind): It follows that: Recall the characterization of the Clarke circatangent cone T C X (x), x ∈ X, as being the set T C is u.s.c. on X × E and convex in v (see Sect. 2.1). With these properties at hand, we can present the proof of Theorem 4.8.
Proof of Theorem 4.8 Given > 0 arbitrary, let δ = 2L+1 . Due to the above properties of d 0 X ,the map : X ⇒ E defined by the formula: x ∈ X, has convex values and its graph is open; it is hence lower semicontinuous. By hypothesis, (x) ∩ (x) = ∅ for all x ∈ X. In view of Lemma 33, there is a continuous δ-approximation f of the graph of such that: Since f is continuous, it is bounded on X, say f (x) ≤ M for some M > 0. Choose τ > 0 with Mτ < η where η is given by (A ), and a sequence (t n ) n∈N in (0, τ ], t n ↓ 0 + . For each n ∈ N, the map g n : X −→ X given by: is well defined, since for each x ∈ X, d X (x + t n f (x)) < η.
For each n ∈ N, the homotopy h n : X × [0, 1] −→ X, defined by: joins g n to the identity on X. Therefore, g n has a fixed point x n ∈ X. Observe now that: Since X is compact, a subsequence of (x n ) (again denoted by (x n ) ) converges to some x ∈ X. Hence, Therefore, for any n, Letting n → ∞, we obtain: Taking into account that f is a δ-approximation of , we infer that there exist x ∈ B X (x, δ), y ∈ (x ), such that y − f (x) < δ. Hence, y < (2L + 1)δ = . Since is arbitrary and X is compact, it follows that has a zero.
The following general variational inequality follows at once from Theorems 2.9 and 4.8:

Corollary 4.11
If X is a compact L-retract in a Hilbert space H with χ(X ) = 0, and ∈ H(X, E) is a compact-valued map, then has a co-equilibrium. on ∂ X on int (X ) .
The convexity of the values of the map can be dropped only under a stronger tangency condition: Theorem 4.13 ([16]) Let X be a compact L-retract in a normed space E with χ(X ) = 0, and let ∈ V(X, E) := {usco s with acyclic values} verifying (x) ⊂ T C X (x), ∀x ∈ ∂ X. Then has an equilibrium. For sets defined by locally Lipschitz inequalities and relaxed regularity and compactness conditions, co-equilibria results for H -maps with weak*compact and convex values can be found in Cwiszewski-Kryszewski [32]. A remarkable theorem on the existence of stationary solutions for the Cauchy problem x (t) = f (x(t)), t ≥ 0, x(0) = x ∈ X ⊂ R n , from Kamenskii-Quincampoix [49], is worth bringing to the attention of the reader. Assume that f is Lipschitzian continuous and define X 0 ( f ) as the set of those points x ∈ ∂ X such that the solution of the Cauchy problem starting at x leaves X immediately. Recall from Sect. 2.3 that Nagumo's theorem asserts that for X locally closed: ( f (x) ∈ T X (x), ∀x ∈ X ⇒ X 0 = ∅). ([49]) Let R n ⊃ X be epi-Lipschitz and compact and f : R n −→ R n be a (locally) Lipschitz mapping. Assume that X 0 is closed and χ(X 0 ) is well defined. If χ(X 0 ) = χ(X 0 ), then f has an equilibrium in X.

Theorem 4.15
Related results in the context of Hilbert spaces and for condensing set-valued maps can be found in the more recent paper by Gudovich-Kamenskii-Quincampoix [40].

Non-Lipschitz and non-sleek domains
The last section is devoted to non-Lipschitzian compact neighborhood retracts of normed spaces. The following two examples show that tangency conditions involving the Bouligand-Severi cone T X (x) or the Clarke cone T C X (x) (which are equivalent for tangentially regular sets) are inadequate for the study of equilibria in AN Rs without Lipschitz regularity.
These examples motivate the definition of a notion of retraction normal cone N r X (x) to a compact neighborhood retract X .
Denote by D * the closed unit ball in the dual E of E equipped with the strong topology. Given X ⊂ E, x ∈ X, > 0, we have: Let X ⊂ E be a set satisfying (A). Then, as seen before, the following holds: (A ) ∃η > 0, η < 1 2 β, ∀x ∈ B(X, η), x − r (x) < η.
The class A enjoys a number of stability properties (again, see [11] and references there). One of them, the closedness under composition products, is crucial for the sequel: