This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution license (http://creativecommons.org/licenses/by/3.0/).

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

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 [

Several extensions of Hartman-Knobloch results of perturbations of the ordinary vector

Although Hartman’s

The

The main idea behind the method allowing us to study (3) can be traced back to [

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

After the Introduction, the paper is organized as follows. In

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,

For a subgroup,

For a

Consider two subgroups,

Let

Suppose that

Let

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, [

Let

A multivalued map,

Assume now that

Let

there exists

for all

In such a case,

Take

(

Lemma 2.1 allows us to extend the

Using Lemma 2.1(ii), one can easily verify that

(

For the equivariant topology/representation theory background, we refer the reader to [

To formulate a result on (symmetric) multivalued BVPs, recall some standard notions and facts.

For any Banach space,

Let

A multivalued map,

for every

for every

The following result is well-known (see [

Put

for any

there exist

there is a function,

In addition, we will assume that problem (11) is asymptotically linear at the origin and the linearization at the origin is non-degenerate,

the linear system:

Finally, we assume that

We make the following assumptions with respect to

the multivalued map,

It follows immediately from condition (H6) that the multivalued map,

for every non-zero,

The simple observation, following below, will be essentially used in the sequel.

Assume for contradiction that

Then,

Since

Take the Sobolev space,

Observe that both

Furthermore, define the multivalued map,

Since the operator

(i) The

(ii) The multivalued map,

(iii) The map,

(iv) By conditions (H4) and (H5), there exists

Put:

Using conditions (H0)–(H3) and following the standard argument (see, for example, [

To establish Statement (b), it is enough to combine Statement (a) with assumption (H7) and Lemma 3.4. ☐

In this section, we will apply Theorem 3.6 to study problem (3) in the symmetric setting. Below, we formulate assumptions on

As is very well-known, the set of fixed points of a non-expansive map is

Next, three conditions present the adaptation of the Hartman-Nagumo conditions for the implicit BVP. Namely, we assume that there exists

for any

There exist constants,

There is a function,

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:

For any

the

Finally, as in

the function,

the function

A careful analysis of the proof of Lemma 4.1 shows that under the assumption that

The Lemma, following below, plays an important role in our considerations.

In light of [

Suppose for contradiction that (38) is not true. Then, there exists

Then, since, by (29),

Combining Lemma 4.3 with Theorem 3.6, one obtains the following:

Theorem 4.4 reduces studying symmetric multiple solutions of (3) to the computation of

Assume

Since

Since

Formula (43) requires effective computations of the negative spectrum of

In this section, we describe a class of examples illustrating Theorem 4.4. Throughout this section,

We start with describing a class of functions,

Let

Let

For two vectors,

Let

Let

there exist real constants,

The proof of the statement following below is straightforward.

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

Let

Define:

Clearly,

Observe that the (symmetric) spectral properties of the linearization

In

Orbit types in

The matrix,

We make the following assumptions regarding

(a1)

(a2)

Then, formula (43) reads as follows:

We refer to [

Z. Balanov was supported by a grant No. PAPIIT IN-117511-3 from Universidad Nacional Autonoma de Mexico, 01000 Mexico D.F., Mexico.

W. Krawcewicz was supported by Chutian Scholars Program at China Three Gorges University, 8 University avenue, Hubei 443002, China.

The authors declare no conflict of interest.

Let

The collection determined by (57) is served as a domain of the

Denote by

One can define an operation of

By using the partial order on

In this subsection, we will present a practical “definition” of the

Combining the standard (equivariant) finite-dimensional approximations with the suspension property (

Any degree (including the equivariant one) applied to a concrete (nonlinear) problem can be often computed using the so-called linearization techniques based on local or global linear approximations.

Let

for each

for any irreducible representation,

We have the following effective computational formula for

For the detailed exposition of the equivariant degree theory, one can use [

Represent the dihedral group,

To describe (up to conjugacy) subgroups of

(i) the subgroups

(ii) the cyclic subgroups,

For the complete list of irreducible

(a) Clearly, there is the one-dimensional trivial representation,

(b) For every integer number,

(c) For

Lattices of orbit types for irreducible

Assume that