Multiple Solutions to Implicit Symmetric Boundary Value Problems for Second Order Ordinary Differential Equations ( ODEs ) : Equivariant Degree Approach

In this paper, we develop a general framework for studying Dirichlet Boundary Value Problems (BVP) for second order symmetric implicit differential systems satisfying the Hartman-Nagumo conditions, as well as a certain non-expandability condition. The main result, obtained by means of the equivariant degree theory, establishes the existence of multiple solutions together with a complete description of their symmetric properties. The abstract result is supported by a concrete example of an implicit system respecting D4-symmetries.


Subject and Goal
Boundary value/periodic problems for second order nonlinear Ordinary Differential Equations (ODEs) have been within the focus of the nonlinear analysis community for a long time (see, for example [1][2][3]). In [2,4], P. Hartman established the existence result for the boundary value problem:   ÿ = f (t, y,ẏ) y(0) = y(1) = 0 (1) where the function, f : [0, 1] × R n × R n → R n , satisfies the so-called Hartman-Nagumo conditions, which, informally, means that f is a reasonable function having a sub-quadratic growth onẏ. Later on, H.W. Knobloch [5,6] observed that a similar result is true for the corresponding periodic problem:   ÿ = f (t, y,ẏ) Several extensions of Hartman-Knobloch results of perturbations of the ordinary vector p-Laplacian operator were suggested by J. Mawhin et al. (see, for example, [7][8][9] and the references therein).
Although Hartman's existence result was extended to more general settings by many authors, to the best of our knowledge, the problems of estimating a minimal number of solutions to (1), as well as classifying their symmetric properties have not been carefully studied. To some extent, our recent paper [10] opened a door to a systematic usage of the equivariant degree theory for analysis of multiple solutions to symmetric (1) and its generalizations. The starting point for our discussion was Example 6.1 from [11] in which a particular case of BVP (1) in the presence of D 4 -symmetries was considered (see also [12] in which a "multivalued perturbation of this example" was discussed).
The goal of this paper is to study multiple solutions to boundary value problems for implicit symmetric second order differential systems using the equivariant degree-based method. To simplify our exposition, we will restrict ourselves to the Dirichlet boundary conditions. More specifically, we are interested in the BVPs of the form:   ÿ = h(t, y,ẏ,ÿ), a.e. t ∈ [0, 1] where V := R n is an orthogonal G-representation and h : [0, 1] × V × V × V → V is a G-equivariant map satisfying the so-called Carathéodory condition. For some motivating examples from mechanics (including, in particular, the ones modeled by the so-called generalized Liénard equation), we refer the reader to [13] and the references therein.

Method
The main idea behind the method allowing us to study (3) can be traced back to [14,15]. Namely, assume that h satisfies the Hartman-Nagumo conditions with respect to (y,ẏ), and, in addition, it is non-expansive with respect toÿ. Since the set of fixed points of a non-expansive map of an Euclidean space is convex, one can canonically associate with problem (3) the "explicit" differential inclusion of the form: withF : [0, 1] × V × V → K c (V ) (here, K c (V ) stands for the set of all non-empty convex and compact subsets of V ). By the (equivariant) homotopy argument, the later problem can be reduced to the "explicit" single-valued symmetric BVP of the form of Equation (1), and the equivariant degree-based method developed in [10] can be applied.
Recall that the equivariant degree is a topological tool allowing "counting" orbits of solutions to (symmetric) equations in the same way as the usual Brouwer degree does, but according to their symmetric properties. This method is an alternative and/or complement to the equivariant singularity theory developed by M. Golubitsky et al. (see, for example, [16]), as well as to a variety of methods rooted in Morse Theory, Lusternik-Schnirelmann Theory and Morse-Floer complex techniques (see, for example, [17][18][19][20]) used for the treatment of variational problems with symmetries. These standard methods, although being quite effective in the settings in which they are usually applied, encounter technical difficulties when: (i) the group of symmetries is large; (ii) multiplicities of eigenvalues of linearizations are large; (iii) phase spaces are of a high dimension; and (iv) the operators involved exhibit a lack of smoothness. Furthermore, one would expect to use computer routines for complex computations, while it is not clear if these approaches are "open enough" to be computerized. On the other hand, the equivariant degree theory has all the attributes allowing its application in settings related to (i)-(iv), and in many cases, it allows computerization. For instance, in the case of the dihedral group, the tools required for the symbolic computations of the equivariant degree can be found at [21]. For a detailed exposition of the equivariant degree theory, we refer the reader to [11,[22][23][24][25].

Overview
After the Introduction, the paper is organized as follows. In Section 2, we collect the standard equivariant background together with basic properties of the equivariant degree (without free parameters) for compact equivariant multivalued fields. This theory is applied in Section 3 for studying "explicit" second order equivariant inclusions (see (11)). We reformulate (11) as a fixed-point problem with a compact equivariant multivalued vector field defined on the Sobolev space, H 2 ([0, 1]; R n ), and associate to this field an invariant, ω(C, F ), expressed in terms of the equivariant degree (see (22)). Using ω(C, F ), we formulate our result for (11) (see Theorem 3.6). In Section 4, we combine Theorem 3.6 with the equivariant version of the well-known result from [15] (cf. Lemma 4.1) to obtain our main abstract result for the "explicit" BVP (3) (in fact, this result (see Theorem 4.4) is expressed in terms of ω(C, F ) provided by Lemma 4.1). In Section 5, we describe a wide class of BVPs (3) symmetric with respect to the dihedral group representations for which Theorem 4.4 can be applied to obtain a complete symmetric classification of solutions (see Proposition 5.1). We also give a concrete D 4 -symmetric example supporting Proposition 5.1 (see Theorem 5.2). To make our exposition self-contained, we conclude with two Appendices (Appendix 1 is related to the equivariant degree theory for single-valued maps (in particular, the concept of a Burnside ring, and a computational formula for the equivariant degree of a linear equivariant isomorphism is given); in Appendix 2, we collected all the facts frequently used in this paper that are related to D 4 -representations and D 4 -equivariant degree).

G-Actions and Equivariant Degree without Parameters for Multivalued Fields
In this section, we briefly recall the standard "equivariant jargon" and present basic facts related to the equivariant degree without free parameters for equivariant multivalued fields. In what follows, G stands for a finite group and V for an orthogonal G-representation. For a G-space, X and x ∈ X, denote by G x := {g ∈ G : gx = x} the isotropy of x and by G(x) := {gx : g ∈ G} G/G x the orbit of x. Given an isotropy, G x , call (G x ) the orbit type in Let X and Y be two G-spaces. A continuous map, f : X → Y , is said to be G-equivariant (or, simply, a G-map) if f (gx) = gf (x) for all x ∈ X and g ∈ G.

G-Actions
Convention: For a (finite) group, G, we denote by V 0 , V 1 , V 2 , . . . , V r the complete list of all irreducible orthogonal (real) G-representations.
Suppose that V is an orthogonal G-representation (in general, reducible). Then, it is possible to represent V as the following direct sum: called the G-isotypical decomposition of V , where the isotypical components V k are modeled on the irreducible G-representations, V k . In other words, the component, V k , is the minimal subrepresentation of V containing all the irreducible subrepresentations of V that are equivalent to V k . Notice that if Let S([a, b]; V ) be a Banach space of reasonable (e.g., continuous, differentiable, Sobolev differentiable, etc.) functions, [a, b] → V , where V is an orthogonal G-representation. Then, S([a, b]; V ) can be equipped with the structure of a Banach G-representation by letting: Combining Equations (5) and (6) yields the isotypical decomposition:

Equivariant Degree for Multivalued Vector Fields
In order to treat implicit symmetric BVPs, we will use an extension of the equivariant degree without free parameters to multivalued compact equivariant vector fields with compact convex images. Up to several standard steps, such an extension is very simple (see, for example, [12]). Therefore, below, we will only outline the key steps of the construction (we refer the reader to Appendix 1 of the present paper, where the axiomatic definition for single-valued fields is presented).
Let E be a Banach space. Denote by C (E) (respectively K c (E)) the family of all non-empty convex subsets of E (respectively all non-empty convex and compact subsets of E). Let X be a subset of a Banach space, Y. A map, F : X → C (E) (respectively F : X → K c (E)), is called a multivalued map with convex values from X to E (respectively multivalued map with convex and compact values from X to E).
A multivalued map, F : X → K c (E), is said to be upper semi-continuous (in short, u.s.c.) if for every open set, U ⊂ E, the set, {x ∈ X : F (x) ⊂ U }, is open in X. A u.s.c multivalued map, F : X → K c (E), is called compact if, for any bounded set, S ⊂ X, the closure of In what follows, we will write F ∈ M K to indicate that F is a u.s.c. compact multivalued map with non-empty compact convex values.
Assume now that E and Y are isometric Banach G-representations, X ⊂ Y is a G-invariant set and Clearly, F f is u.s.c. (as a multivalued map) and, therefore, Similarly to the single-valued case, a multivalued map, F : Ω → K c (E), is called an Ω-admissible compact G-equivariant field if the following two conditions are satisfied: x ∈ F (x) (by the same token, F has no fixed-points in ∂Ω).
In such a case, (F, Ω) is called an admissible G-pair in E. Denote by A M G K (E) the set of all such admissible G-pairs in E and put: (here, the union is taken over all isometric Banach G-representations). [12,27]).
Then, there exists a single-valued equivariant Ω-admissible homotopy joining f 0 and f 1 .

Lemma 2.1 allows us to extend the G-equivariant degree defined for single-valued admissible G-pairs to the fields from
Using Lemma 2.1(ii), one can easily verify that G-Deg(F, Ω), defined by (9), is independent of a choice of a single-valued representative, f . Moreover, by applying the standard argument, one can show that G-Deg satisfies the standard properties. More precisely: [12]).
For the equivariant topology/representation theory background, we refer the reader to [26,[28][29][30]. For all the "multivalued" backgrounds frequently used here, we refer the reader to [27,31]. The detailed exposition of the equivariant degree theory can be found in [22,25].

Basic Definitions and Facts
To formulate a result on (symmetric) multivalued BVPs, recall some standard notions and facts. For any Banach space, E, the set, K c (E), of all nonempty compact convex sets in E can be equipped with the so-called Hausdorff metric, D(·, ·). To be more specific, if A, B ∈ K c (E), put: (here, B r (0) stands for the ball of radius r centered at the origin). One can easily verify that the function D is indeed a metric on K c (E).
is called a Carathéodory if it satisfies the following two conditions: The following result is well-known (see [32,33]) and plays an important role in our considerations.
be a Carathéodory multivalued map satisfying the following condition: (A) For any bounded set, B ⊂ R m , there exists ϕ B ∈ L 1 ([0, 1]; R), such that: Then, the formula:

Hypotheses
Put V := R n . We are interested in studying the BVP for second order differential inclusion of the type: where F : and C : V → V is a linear operator. As usual, the differentiation is understood in the sense of Sobolev derivatives. We need the following adaptation of the Hartman-Nagumo conditions (cf. [2,4]) for the multivalued map, C + F .
(H0) F is a Carathéodory map satisfying condition (A), and there exists a constant R > 0, such that: In addition, we will assume that problem (11) is asymptotically linear at the origin and the linearization at the origin is non-degenerate, i.e.: has only the trivial solution, y ≡ 0, i.e., σ(C) ∩ {−π 2 n 2 : n ∈ N} = ∅, where σ(C) stands for the spectrum of C.
It follows immediately from condition (H6) that the multivalued map, is well-defined and u.s.c. We assume additionally: (H7) for every non-zero, v 0 ∈ V G , there is δ = δ(v 0 ) > 0, such that: The simple observation, following below, will be essentially used in the sequel.
(ii) The multivalued map, F : E → K c (E), given by is a compact G-equivariant multivalued field (cf. condition (H6) and compactness of the operator, j).
(b) If, in addition, (H j ) is a maximal orbit type in V \ V G , then (G u ) = (H j ).

General Result
In this section, we will apply Theorem 3.6 to study problem (3) in the symmetric setting. Below, we formulate assumptions on h. The following condition essentially allows a passage from the "implicit" problem to the single-valued "explicit" one via multivalued equivariant homotopy techniques. and: and for all u, v ∈ V , w 1 , w 2 ∈ B r := {w ∈ V : w < r}, where r := r(t, u, v) = α(t,u,v)

1−c
As is very well-known, the set of fixed points of a non-expansive map is convex. The following statement was proven in [15]: where r(t, u, v) is given in condition (A0), is well-defined and satisfies the Carathéodory condition.
In addition, we will assume that problem (3) is asymptotically linear at the origin and that the linearization at the origin is non-degenerate. More precisely: Finally, as in Subsection 3.2, we assume that V is a coordinate permutation G-representation given by (12), and condition (13) is satisfied. Furthermore, assume that: h(t, gu, gv, gw) = gh(t, u, v, w), for all t ∈ [0, 1] and u, v, w ∈ V (A7) the function h o = h| [0,1]×V G ×V G ×V G satisfies the condition: for any u o ∈ V , there is δ = δ(u o ) > 0, such that: where r := r(t, u, v) is given in (A0).

Remark 4.2.
A careful analysis of the proof of Lemma 4.1 shows that under the assumption that h is G-equivariant, one can construct F to be G-equivariant, as well.
The Lemma, following below, plays an important role in our considerations. (b) any solution, u ∈ E, to (11) is also a solution to (3).

Proof:
In light of [15] (see also (12) and (13), conditions (A6) and (A7)), we need to check only condition (H4), i.e.: uniformly with respect to t ∈ [0, 1]. Suppose for contradiction that (38) is not true. Then, there exists ε > 0 and a sequence, {(t n , u n , v n , w n )} ∞ n=1 , such that t n → t o and (u n , v n ) → (0, 0), where w n ∈ F (t, u n , v n ) and: Therefore, by the definition of F : w n + Cu n = g(t n , u n , v n , w n + Cu n ), n ∈ N Therefore: w n + Cu n = Au n + B(w n + Cu n ) + r(t n , u n , v n ; u n + Cw n ) where r(t, u, v, w)/ (u, v, w) → 0 as (u, v, w) → 0 (uniformly with respect to t), which leads to: (Id − B)w n = r(t n , u n , v n ; u n + Cw n ) Then, since, by (29), w n ≤ α(t n , u n , v n ) → 0 as n → ∞, it follows that: lim n→∞ (Id − B)w n (u n , v n ) = lim n→∞ r(t n , u n , v n ; u n + Cw n ) (u n , v n ) = lim n→∞ r(t n , u n , v n ; u n + Cw n ) (u n , v n , u n + Cw n ) · lim n→∞ u n + Cw n = 0 · 0 = 0 and we obtain a contradiction with (39).

General Formula for ω(C, F )
Theorem 4.4 reduces studying symmetric multiple solutions of (3) to the computation of ω(C, F ) (or that is the same (cf. Remark 3.5, (23) and (24)) as the computation of the equivariant degree, G-Deg(A , Ω δo ) ∈ A(G)). Below, we give a general formula for it.
Assume V admits the isotypical decomposition (5). Then, E := H 2 ([0, 1]; V ) has the following G-isotypical decomposition: where: A (E k ) = E k , k = 0, 1, 2, . . . , r Since A is a compact linear field, all points of the spectrum of A are of finite multiplicity, and one can be the only accumulation point of the spectrum of A . Hence, the negative spectrum, σ − (A ), is composed of a finite number of eigenvalues (of finite multiplicity). For each λ ∈ σ − (A ), denote by E(λ) the generalized eigenspace of λ and put (cf. (64)) to denote the V k -multiplicity of the eigenvalue, λ. Consequently, one has (see Theorem 5.5) the following formula: Formula (43) requires effective computations of the negative spectrum of A . In the next section, we will show that very often, it is a feasible task.

Examples of Implicit D n -Symmetric BVPs with Multiple Solutions
In this section, we describe a class of examples illustrating Theorem 4.4. Throughout this section, V stands for a D n -representation given by (72) and (73) and admitting the isotypical decomposition (76).
Let g : V × V → V be a function satisfying the following conditions: (g1) g is D n -equivariant (in particular, continuous); (g2) there exist real constants, α > 0 and 1 > c ≥ 0, such that g(v, w) ≤ α + c w for all The proof of the statement following below is straightforward.

Example
One can easily construct a wide class of illustrative examples of implicit BVPs for differential systems symmetric with respect to various classical finite groups (including, in particular, arbitrary dihedral groups D n , a tetrahedral group A 4 , an octahedral group S 4 , an icosahedral group A 5 , etc. (see [25])). However, being motivated by simplicity and the transparency of our exposition, we restrict ourselves to one of the simplest non-abelian symmetry groups, namely D 4 .
We refer to [10] as an appropriate source of examples of explicit D n -symmetric BVPs that can be converted to implicit D n -symmetric BVPs admitting an arbitrary large number of symmetric solutions.
where B(V k ) is the unit ball in V k and T k := T | V k : V k → V k . Denote by σ − (T ) the set of all negative real eigenvalues of the operator, T . Choose λ ∈ σ − (T ), and let: denote the generalized eigenspace of T corresponding to λ. Then, define: (i) for each k = 0, 1, . . . , r, put: and call the number, m k (λ), the V k -multiplicity of the eigenvalue, λ, of T ; (ii) for any irreducible representation, V k , put: and call deg V k the basic G-degree corresponding to the representation, V k .
Theorem 5.5. Let V be an orthogonal G-representation with isotypical decomposition (5) and T ∈ GL G (V ). Then: where B(V ) stands for the unit ball in V , σ − (T ) denotes the set of negative real eigenvalues of T and the product is taken in the Burnside ring, A(G).

A2.2. Irreducible D n -Representations and Basic Degrees
For the complete list of irreducible D n -representations and the corresponding basic degrees, we refer the reader, for instance, to [25], p. 174. Here, we restrict ourselves with the data important for the present paper.
(a) Clearly, there is the one-dimensional trivial representation, V 0 . In this case: For every integer number, 1 ≤ j < n 2 , there is a D n -representation, V j , on C given by: γz := γ j · z, for γ ∈ Z n and z ∈ C κz := z where γ j · z denotes the usual complex multiplication. Put: h := gcd(j, n) and q := n/h For the lattices of orbit types related to this case, we refer to Figure 2i (the case, q, is odd) and Figure 2ii (the case, q, is even). Furthermore, we have the following degrees of the basic maps: (c) For n being even, there is an irreducible representation, V n 2 , given by d : D n → Z 2 = O(1), such that ker d = D n/2 . In this case, the lattice of orbit types is given in Figure 2iii and the degree of the corresponding basic map is: Figure 2. Lattices of orbit types for irreducible D n -representations.