Generalized Gradient Equivariant Multivalued Maps, Approximation and Degree

Consider the Euclidean space Rn with the orthogonal action of a compact Lie group G. We prove that a locally Lipschitz G-invariant mapping f from Rn to R can be uniformly approximated by G-invariant smooth mappings g in such a way that the gradient of g is a graph approximation of Clarke’s generalized gradient of f . This result enables a proper development of equivariant gradient degree theory for a class of set-valued gradient mappings.


Introduction
It is well known that various versions of degree theory are very useful in nonlinear analysis; see the books [1][2][3] and their extensive references. A powerful special degree for maps commuting with an action of a Lie group gives many multiplicity results; see the book [4]. In order to extend the degrees to set-valued maps, a graph approximation method in the spirit of [5] appeared to be very successful. In [6], a series of selection and graph approximation results for convex-valued mappings was obtained in the presence of symmetries given by a compact group action as extensions of many classical results. Some of them can be basic tools in the construction of equivariant degree theory. An equivariant version of the Cellina approximation theorem for convex-valued upper semicontinuous mappings was used in [7] to define the equivariant degree, which extended the one from [4,8]. The degree theory was applied in [7] to obtain nontrivial solutions to multivalued boundary value problems. See also [9,10] for further applications of that degree to obtain multiple solutions to some implicit functional differential equations.
On the other hand, in many applications, the considered maps are additionally gradients of smooth functionals, and special invariants for this class are involved like the gradient equivariant degree and Conley index; see [11,12]. There are plenty of papers on various applications, and we mention only a few of them concerning second order ODEs [13], symmetry-breaking [14], bifurcations of the Neumann problem [15] and symmetric Newtonian systems [16]. In the non-smooth case, one can use Clarke's generalized gradient notion [17] for locally Lipschitz functions, which is a convex-valued u.s.c. map in this case (see Definition 3 or [18] for definition of u.s.c. for multifunction). It appears that an approximation result is valid for such functions defined in R n (see [19]), which may be treated as an analogue of the Whitney approximation theorem.
The main purpose of this paper is to prove the equivariant version of the approximation theorem ofĆwiszewski and Kryszewski. Then, we use this to provide an equivariant gradient degree theory with all the usual properties including the Hopf classification theorem.
It is worth pointing out that the proof is strictly finite-dimensional, and it is a challenging problem to extend the approximation theorem to Hilbert spaces. Nevertheless, the degree can be defined for compact vector fields, i.e., maps of the form Id − ∂ f , by standard Leray-Schauder-type techniques.

Definition 2. Let X and Y be G-sets. A set-valued map
) for all g ∈ G and x ∈ X.
Note that ϕ is G-equivariant if and only if ϕ(gx) ⊂ gϕ(x) for all g ∈ G and x ∈ X (or gϕ(x) ⊂ ϕ(gx) for all g ∈ G and x ∈ X). Moreover, it is easy to see that ϕ is G-equivariant if and only if its graph Gr (F) is a G-invariant subset of X × Y with a natural action g(x, y) := (gx, gy), x ∈ X, y ∈ Y. Observe that if ϕ : X Y, where Y is a topological space, is G-equivariant, then so is its closure, i.e., the map ϕ : X Y given by ϕ( Let us collect simple examples of G-equivariant set-valued maps. x) for 0 ≤ t ≤ T and x ∈ U. Consider a differential inclusion (under suitable assumptions assuring the existence of solutions): , then gx is also a solution to this problem with the initial condition gx 0 . Therefore, the solution map P : U C([0, T], E) that assigns to each initial value x 0 ∈ U the set of all solutions is G-equivariant, whenever well defined.
(2) Let f : E → R ∪ {∞}, where E is a real Banach representation of G, be a convex function. For each which gives the assertion. In a similar manner, one shows that the Clarke generalized gradient ∂ f : U → E * , where U is a G-invariant open in E and f : U → R is a G-invariant locally Lipschitz function, is G-equivariant.

Gradient approximations
In R n , we shall use various analytic characterizations of graph-approximations. For A ∈ R n , we recall the notion of a support function σ A : R n → R ∪ {∞} given by: One easily checks that σ A is finite and continuous if A is bounded.
We also define: |||A||| := sup 2) Let U ⊂ R n , ϕ : U R n with closed convex values, g : U → R n , and ε : U → (0, +∞). The following conditions are equivalent: Let U ⊂ R n be an open subset, and consider a locally Lipschitz function f : U → R. A generalized directional Clarke derivative: It is well known that the map ∂ f : U R n is an u.s.c. mapping with compact convex values (see [17] for this and more of the properties of the generalized gradient). One observes also that Recall only the local u.s.c. property: The following approximation result was proved in [19]: Theorem 1. Let f : U → R be locally Lipschitz and ε : U → (0, ∞) a continuous function. Then, there exists a C ∞ -map g : U → R such that: In other words, g is a uniform approximation of f , and ∇g is an ε-approximation of ∂ f . We shall call such pairs (g, ∇g) Whitney approximations.
Let us assume that V is an n-dimensional orthogonal representation of a compact Lie group G and U ⊂ V is open and G-invariant. Assume that a locally Lipschitz function f : It is easy to verify that ∂ f is then equivariant, i.e., ∂(gx) = g(∂(x)) as sets.
We now modify the proof from [19] in order to obtain an equivariant version of Theorem 1. Let us start from the remark that since G acts orthogonally, there exists an invariant C ∞ -mollifier, i.e., a nonnegative invariant function ω : R n → R with the support supp ω ⊂ B 1 (0) and such that: We can take, e.g., a function given by: and then, we normalize it: Let f : R n → R be an invariant L-Lipschitz map with a compact support, and let λ > 0. We define the λ-regularization of f by the formula: The following is a straightforward consequence of the definition. Proposition 1. The regularization g λ satisfies the following properties: The following technical lemma was proven in detail in [19], Lemma 3.4.
Lemma 2. Let f 1 , ..., f k : R n → R be Lipschitz functions of common rank L and with compact supports. For any ε > 0, there exists sufficiently small µ > 0 with the property that for every x ∈ R n , there exists x ∈ B ε (x) such that: A local approximation result is a consequence of the above.
Proposition 2. Let f i (i = 1, ..., k) be G-invariant functions as in Lemma 2. Then, for any ε > 0, there exists λ 0 > 0 such that: (i) for any λ ≤ λ 0 , the regularization g i λ of f i is a uniform ε-approximation of f i , for each i = 1, ..., k; (ii) for every x ∈ R n , there exists x ∈ B ε (x) such that for all λ ≤ λ 0 and i = 1, ..., k: Proof. We put λ 0 := min{µ, ε L } > 0, and let λ ≤ λ 0 . Then, Property (i) follows from Proposition 1. We prove (ii) by means of Property (iii) from Lemma 1. For this, we fix a point x and u ∈ B 1 (0). There exists a δ = δ(λ, x, u) < µ such that, for all i, we have: On the other hand, we have: and by the use of Lemma 2, we find x such that: Therefore, we have the inequality ∇g i Theorem 2. Let f : U → R be a locally G-invariant map as in Theorem 1. Then, for every ε > 0, there exists a C ∞ -smooth G-invariant map g : U → R such that (g, ∇g) is an ε-WT-approximation of ( f , ∂ f ).
Proof. Since U is G-invariant, there exists an increasing family of G-invariant compact sets {K k } ∞ k=0 such that: For each k = 1, the set P k := int K k+1 \ K k−1 is also G-invariant. Therefore, the family {P k } is an open G-invariant covering of U. Let {ϕ k } ∞ k=1 be a C ∞ -smooth and G-invariant partition of unity subordinated to {P k }.
Thus, the functions ϕ k · f (k ≥ 1) are globally Lipschitz, G-invariant, and they can be considered as defined on R n .
We define a sequence of positive numbers: Now, we can apply Proposition 2 for each k ≥ 1. We find λ k such that for any 0 < λ ≤ λ k invariant λ-regularizations: g k λ of ϕ k · f and g k+1 for all 0 < λ ≤ λ n . We can assume that supp g k λ ⊂ P k , and the sequence {λ k } ∞ k=1 is nonincreasing. The desired map g : U → R is defined by the formula: It is obviously well defined, since at most two terms are different from zero at given x. By definition, it is C ∞ -smooth and G-invariant. We have to check the approximation conditions. For a fixed x ∈ U, there is a unique k ≥ 0 such that x ∈ K k+1 \ K k . If k = 0, then g(y) = g 1 λ 1 in some neighbourhood of x, and the claim is clear from the choice of ε 1 .
If k ≥ 1, then in some neighbourhood of x contained in B ε k (x), we have g(y) = g k λ k (y) + g k+1 λ k+1 (y) and ϕ k (y) + ϕ k+1 (y) = 1. Therefore: By the properties of the generalized gradient, we obtain: Because of Lemma 1, the last condition means that ∇g is a graph ε-approximation of ∂ f . This ends the proof. Theorem 3. Let X be a compact G-invariant subset of V. Suppose that f : X → R is locally Lipschitz and G-invariant. Then, for all ε > 0, there exists a δ > 0 with δ < ε and such that, for every two WTδ 2 -approximations (g 1 , ∇g 1 ), (g 2 , ∇g 2 ) of ( f , ∂ f ), the pair (g t , ∇g t ) is an WT ε -approximation of ( f , ∂ f ), where: is a linear homotopy.

Proof.
For all x ∈ X, ∂ f (·) is upper semicontinuous (1) at x, so we can apply Definition 3 with a given ε and obtain an appropriate, small δ(x).
Then, from (2) and (3), we get: From (2) and (3), we have: We can use the upper semicontinuity property of ∂ f in x i : for pointsx,x, which are close enough to x i : However, B ε (∂ f (x i )) is a convex set, and therefore: Finally, from (2) and (4), we have: This ends the proof.

Equivariant Gradient Degree
In this section, we outline an axiomatic approach to the equivariant gradient degree. To this end, some preliminaries are in order.
Recall that the Euler ring U(G) of a compact Lie group G as an abelian group is equal to Z(Φ(G)), where Φ(G) is the set of all conjugacy classes of normal subgroups of G. The multiplicative structure of U(G) is quite involved (see [22] or [4] for details on the ring structure).
Let V be an orthogonal G-representation. Denote by C 2 In an obvious way, one can define a G-gradient Ω-admissible homotopy between two G-gradient Ω-admissible maps.
The notion of equivariant gradient degree we consider here was defined originally by K. Gȩba [12]. We use an axiomatic approach here adapted from [8].
Since the equivariant gradient degree satisfies the Hopf property with respect to gradient G-homotopies, as well as the ordinary equivariant degree with respect to equivariant homotopies, the gradient degree is more delicate homotopy invariant.
The ordinary equivariant degree for multivalued convex-valued u.s.c. maps was defined in [7] by means of the equivariant version of the Cellina approximation theorem (see [6]). The approximation results from the previous section enable us to develop the equivariant degree theory in quite an analogous way.
Let V be a finite dimensional orthogonal representation of a compact Lie group G, and let Ω be an open, bounded G-invariant subset of V. Let f : V → R be a locally Lipschitz G-invariant map such that its Clarke's generalized gradient ∂ f : V → V is Ω-admissible, i.e., 0 ∈ ∂ f (x) for x ∈ bd Ω. Since the map ∂ f is u.s.c., the image of the boundary ∂ f (bd Ω) is compact, and its distance from zero is positive. The same is true for a sufficiently fine neighbourhood of bd Ω. Thus, there exists an ε 0 > 0 such that for every 0 < ε < ε 0 , each ε-WT-approximation of ( f , ∂ f ) is Ω-admissible. Moreover, Theorem 3 assures that there exists a smaller δ > 0 such that all δ-approximations are Ω-admissible homotopic by a linear homotopy. Take such a small δ.

Definition 5.
We define the equivariant degree of Ω-admissible pair ( f , ∂ f ) as follows: The properties of the degree are consequences of Theorem 4. The first three of them are the most important for applications. Properties 1 and 2 are quite straightforward. The homotopy property needs some comment. Recall that the generalized gradient of a locally Lipschitz map is a multivalued u.s.c. map with convex compact values. Considering homotopies as continuous families of maps, we need here a stronger regularity than the u.s.c. property, which may admit quite a rapid change of values.

Definition 6.
A gradient G-homotopy is a pair ( f t , ∂ f t ), where f t : V → R are locally Lipschitz G-invariant maps and the family of maps ∂ f t : V → V is continuous with respect to t ∈ [0, 1], as well as the family f t .
Observe that in the most popular case of linear homotopy f t (x) = t f 1 (x) + (1 − t) f 2 (x), the Clarke gradient with respect to the x variable satisfies the condition: In some cases, an equality holds, e.g., if at least one of the f i is of class C 1 , or if both of them are convex functions.

Theorem 5. Let the linear homotopy of locally Lipschitz maps f t
be Ω-admissible, and let the condition ∂ f t (x) = t∂ f 1 (x) + (1 − t)∂ f 2 (x) be satisfied for all x and t ∈ [0, 1]. Then: Proof. The proof is similar to the proof of Theorem 3. For all x ∈ X, ∂ f i (·) (i = 1, 2) are upper semicontinuous (1) at x, so we can apply Definition 3 with a given ε and obtain an appropriate δ(x). Let δ(x) < |x| and δ(0) = δ 0 > 0. Then, For all x ∈ X, we have that x ∈ B(x, δ(x)) implies ∂ f i (x ) ⊂ B ε (∂ f i (x)) and {B(x, δ(x))} x∈X is an open covering of X. We can choose a finite subcovering: Finally, from (5) and (7), we have: This ends the proof.
The above theorem is satisfactory in many applications. In particular, the Hopf property follows from this weak version of the homotopy property. Theorem 6. Assume that Ω = B(V) is the unit ball of an orthogonal representation V and ( f 1 , ∂ f 1 ), ( f 2 , ∂ f 2 )) are Ω-admissible pairs such that: Then, the pairs are Ω-admissible homotopic.
Proof. It is enough to observe that if we choose a δ-WT-approximation (g, ∇g) of ( f , ∂ f ) defining the equivariant degree, then the linear formula: defines an Ω-admissible homotopy satisfying the assumptions of Theorem 5. Therefore, we apply this to both pairs. Now, we can apply the Hopf property for smooth maps ∇g i , i = 1, 2. The transitivity property of the homotopy relation for single-valued maps ends the proof.
We are ready to prove the full homotopy property now. 1] be an Ω-admissible gradient G-homotopy. Then: Proof. First, we observe that because of the continuity of the homotopy with respect to t ∈ [0, 1], for each ε > 0 and every fixed t, there exist a δ(t) > 0 and a Whitney approximation (g t , ∇g t ) of ( f t , ∂ f t ), which is also an ε-Whitney approximation of ( f τ , ∂ f τ ) for τ ∈ [t − δ(t), t + δ(t)]. Therefore, (g t , ∇g t ) is linearly homotopic to ( f t−δ , ∂ f t−δ ), as well as to ( f t+δ , ∂ f t+δ ). Thus, by Theorem 5, the gradient degree is constant in the interval [t − δ(t), t + δ(t)]. Now, we consider the open covering of [0, 1] by the intervals (t − δ(t), t + δ(t)) (for t = 0, 1, we put half-open intervals). We find a finite subcovering and apply the transitivity of the homotopy property to finish the proof.

Discussion and Conclusions
The main technical result of this paper is contained in Theorem 2. It says that for a symmetric (G-invariant, where G acts orthogonally on R n ) and locally Lipschitz function f : R n → R, there exists an arbitrarily fine smooth and G-invariant uniform approximation g of f such that the gradient ∇g is a graph approximation of Clarke's generalized gradient ∂ f . This is the equivariant version of the result from [19]. It is known that a version of the Whitney smooth approximation theorem is valid not only in finite-dimensional spaces, but also in Hilbert spaces. However, the presented proof does not work in infinite dimensions because it uses the compactness of closed balls. Moreover, the generalized Clarke gradient is only a weakly upper semicontinuous map with bounded closed values in general. Thus, it as an open question if a similar approximation theorem is true in general (it is also an open problem in the non-equivariant case).
As an application, we developed a gradient equivariant degree theory for the class of set-valued equivariant maps, which are generated by locally Lipschitz potentials together with the Hopf-type classification theorem. It should be mentioned that thanks to our approximation theorem, the classes of homotopy equivalence of such maps can be represented by smooth equivariant gradient maps, and these are different from the usual equivariant homotopy classes.
Further applications into nonlinear analysis are expected, for example some multiplicity results for Hamiltonian systems as those considered for C 2 -maps in [8,16] and some references therein. By means of an infinite-dimensional extension of our degree theory by Leray-Schauder-type finite-dimensional approximations, we can expect similar results under a bit weaker smoothness assumptions. In search of periodic solutions for differential equations (inclusions), a natural action of the group S 1 is considered besides some additional spatial symmetries as one can see in the book [4] and the references therein. Compact multivalued perturbations of the identity operator in a Hilbert space will be considered, as well as perturbations of some unbounded self-adjoint operators (comp. [20]). We treat our paper as the preliminary step towards this direction.