Research Article | Open Access

Francesca Faraci, Antonio Iannizzotto, "Bifurcation for Second-Order Hamiltonian Systems with Periodic Boundary Conditions", *Abstract and Applied Analysis*, vol. 2008, Article ID 756934, 13 pages, 2008. https://doi.org/10.1155/2008/756934

# Bifurcation for Second-Order Hamiltonian Systems with Periodic Boundary Conditions

**Academic Editor:**Jean Mawhin

#### Abstract

Through variational methods, we study nonautonomous systems of second-order ordinary differential equations with periodic boundary conditions. First, we deal with a nonlinear system, depending on a function , and prove that the set of bifurcation points for the solutions of the system is not -compact. Then, we deal with a linear system depending on a real parameter and on a function , and prove that there exists such that the set of the functions , such that the system admits nontrivial solutions, contains an accumulation point.

#### 1. Introduction

In the present paper, we will deal with nonautonomous
Hamiltonian systems with periodic boundary conditions, depending on a function , of the following type: where is a real
interval, is an symmetric,
positive definite matrix; and the function , referred to as the *potential*, is measurable
with respect to the scalar variable and continuously differentiable with
respect to the vector variable (note that by we will always
mean the gradient of with respect to
the vector variable).

The function is regarded as
a parameter, and we are interested in studying the structure of the set of the
solutions of (N_{u})
, as varies in a
suitable function space . In particular, we will focus on those functions which are *bifurcation
points* for the problem (see Definition 3.4 below), according to the very
general definition given by Chow and Hale [1]. Actually, our result ensures
that, under convenient assumptions on the potential , the set of such bifurcation points is
“large,” that is, it is not -compact.
Moreover, whenever is not a
bifurcation point, (N_{u})
admits a finite
number of solutions.

We will study (N_{u})
in the
equivalent form of an equation in , involving a nonlinear operator from to itself: as
the bifurcation points are exactly the *singular values* of , we will be able to apply a result established by
Ricceri [2], assuring that the set of the singular values of is not -compact. The
equivalence between bifurcation points and singular values was already employed
by Durikovic and Durikovicová [3],
where a single second-order nonlinear differential equation is studied.

Our fundamental assumptions are that, for all , the function is positively homogeneous of degree and not quasiconvex (see condition (F3) below for a more precise statement). Homogeneity assumptions have already been used in the study of Hamiltonian system, for instance, Ben-Naoum et al. [4] proved the existence of a nonconstant solution, provided that satisfies certain sign assumptions and is positively homogeneous with degree , . Our assumption that places in the class of subquadratic potentials: in a similar framework, Tang and Wu proved in [5] the existence of a solution for a Hamiltonian system involving a matrix which is not necessarily positive definite.

We will also present a result (based on another theorem from [2]) in the framework of eigenvalue problems for linear second-order systems with periodic boundary conditions, depending on the function and on the real parameter of the following type: where denotes the Hessian matrix of some potential , which is assumed to be twice differentiable. In this case, we replace the homogeneity condition by assuming a subquadratic growth of the potential, and obtain the existence of a real such that the set of the functions , such that the system admits nontrivial solutions, contains an accumulation point.

#### 2. Preliminaries

In this section, we introduce the common hypotheses and notation of the nonlinear and the linear cases, and recall some results which will be useful in the sequel.

Let (), , ; let be an real symmetric matrix, whose entries are functions , and assume that there exists such that, for a.e. and every , Following the monograph [6] by Mawhin and Willem, we denote by the space of the functions with weak derivative and satisfying . In particular, every is an absolutely continuous function, hence it admits a classical derivative, equal to , a.e. in . Due to the above assumptions on the matrix , is endowed with a scalar product defined by putting for every , the induced norm is defined for every by It is well known that is an infinite-dimensional real Hilbert space, compactly embedded in ; in particular, there exists such that (C) for every . In the proofs of our results, we will employ some basic facts about nonlinear operators in Hilbert spaces, which we recall for the reader's convenience.

Let , be Banach
spaces, an operator. We
recall that is *proper* if, for any compact subset of , the set is compact. We
also recall the following result by Sadyrkhanov.

Theorem 2.1 (see [7, Theorem 1.1]). *
Let
be an
infinite-dimensional Hilbert space,
a continuous,
closed, nonconstant operator. Then, is proper. *

We also present the following statement of the domain invariance theorem.

Theorem 2.2 (see [8, Theorem 16.C]). *Let
be a Banach
space, let
be an open
subset of , let
be a
continuous, compact operator, and let be defined by for every . Assume that
is injective.
Then,
is an open mapping. *

We recall that is Gâteaux differentiable in if there exists a mapping (by we mean the space of bounded linear operators mapping into ) such that, for all , if is continuous, we will write .

We denote by the set of the *singular
points* of , that is, the set of all the points such that is not a local
homeomorphism at . A point is said to be a *singular value* of if it is the
correspondent of some singular point, that is, if . If, in addition, , we denote by the set of all
the points such that is not
surjective. Finally, we recall that a set is called -*compact* if it is the union of an at most countable family of compact sets.

Let be a Hilbert space, with scalar product : we denote by the topological dual of and recall that, by the Riesz theorem, there exists a surjective linear isometry , satisfying the following equality for all : Let be a Gâteaux differentiable functional, then its derivative is defined as a mapping , and we define an operator by putting for all , note that, if , the mapping is continuous.

The following result, due to Ricceri, will play a
fundamental role in the study of problem (N_{u})
.

Theorem 2.3 (see [2, Theorem 1]). *Let
be an
infinite-dimensional Hilbert space, . Assume that
is sequentially
weakly lower semicontinuous, not quasiconvex, and positively homogeneous of
degree ; moreover, suppose that
is a closed
mapping. Then, both sets and are not -compact. *

Now, let be a twice-differentiable functional, then its second derivative (i.e., the derivative of ) is defined as a mapping ; for all we define by putting for all , We observe that is a Gâteaux differentiable mapping, whose derivative is a mapping expressed for all by (this follows from the composite map formula, see Ambrosetti and Prodi [9, Chapter 1, Proposition 1.4]); in particular, if (i.e., ), then .

We will employ the following result, due to Ricceri as well, for the study of
the problem (L_{u,?})
.

Theorem 2.4 (see [2, Theorem 3]). * Let
be an
infinite-dimensional Hilbert space, let
be a
non-quasiconvex functional. Assume that is compact and
that ** Moreover, suppose that ** Then, there exists such that the
set contains at
least an accumulation point.*

#### 3. The Nonlinear Case

Let be a function satisfying the following conditions:(F1) is measurable for every , for a.e. ;(F2) there exist and a nonnegative such that for every and a.e. , (F3) there exist , , , and a closed interval such that for a.e. , (F4) there exists such that for every , , and a.e. ,

In (F2). we can assume increasing, without loss of generality. Note that from (F4) it follows that a.e. in .

In this section, we deal with the nonlinear problem (N_{u})
, depending on the function . We recall that, for every , a *solution* of (N_{u})
is a function such that for
every We observe that, by the results of [6, (Section 1.4)], whenever is a solution
of (N_{u})
in the above
sense, actually with derivative . Thus, clearly ; moreover, is absolutely
continuous, hence the second derivative exists a.e. in and satisfies Now, we are going to introduce a suitable variational
setting for the problem (N_{u})
. Firstly, we put, for every , The following lemma describes the properties of the
functional .

Lemma 3.1. *
Let , , , and
be satisfied.
Then, the functional
with compact
derivative . Moreover, is not
quasiconvex. *

*Proof. *By
standard arguments, it is proved that , its derivative is an operator , expressed by for every . We are now going to prove that the map is compact: let be a bounded
sequence in , then there exist a subsequence, still denoted
by , and some such that ; hence for all , where is as in (C).
Conditions (F1) and (F2) ensure that the right-hand side tends to zero as , by an application of the Lebesgue theorem.

Let us prove now that the functional is not
quasiconvex, that is, it has a nonconvex sublevel set. Define . Since , there exists such that for
every measurable set with (where denotes the
Lebesgue measure of ), we have Without loss of generality, we can choose such that so that , and functions for , such that for every and for every , satisfying the condition Since due to (F4),
for , we have Moreover, if , then an analogous argument leads to Thus, it is proved that the set is not
convex.

Lemma 3.2. *Let , , and
be satisfied.
Then, and are positively
homogeneous with exponents , , respectively (in particular, they both vanish at
0). Moreover, *

*Proof. *
The
homogeneity properties of and are easily
obtained from (F4). Condition (3.13) is quickly deduced as well, as we recall that .

We define the operator by putting, for every , (where is as in Section 2). The next lemma yields some properties of the operator .

Lemma 3.3. *
Let , , and
be satisfied.
Then,
satisfies the
condition ** and it is
closed and proper.*

*Proof. *
Since is an isometry,
for all , we get so from condition (3.13) we deduce (3.15).

Now we prove that is a closed
mapping. Let be a closed
subset of , we need to prove that is closed as
well. Assume that is a sequence
in such that converges to
some . Then, by (3.15), is bounded;
since is compact
(Lemma 3.1) and is surjective,
there exists a subsequence such that converges to for some . We get hence, . Since is continuous
(Lemma 3.1), , so which implies .

Finally, we note that is continuous
and is not constant, then by Theorem 2.1, is proper.

The operator provides the
desired variational setting for (N_{u})
. Indeed, let us define the set and for every , By the definition of , it is clear that, for every , Indeed, for all , we have if and only if for all , which is equivalent to .

Next, we give the definition of bifurcation point for , equivalent to the one of [1, page 2].

*Definition 3.4. *A bifurcation point for is a function
such that there
exist and sequences , , and in such that , for every , and As pointed out in the introduction, we are interested
in the “size” of the set of the bifurcation points for . Our main result, which is based on Theorem 2.3,
ensures that such set is not -compact and
that whenever is not a
bifurcation point, the set is nonempty and
finite. The precise statement is the following.

Theorem 3.5. *Let , , , and
be satisfied.
Then, the following assertions hold: *(I)*the set of the
bifurcation points for is closed and
not -compact;*(II)* for every which is not a
bifurcation point for , the set is nonempty and
finite.*

*Proof. *In
order to prove (I), we are going to apply Theorem 2.3, all of its hypotheses are
fulfilled due to Lemmas 3.1, 3.2, and 3.3 (in particular, we point out that, since is compact, turns out to be
sequentially weakly continuous). Thus, the sets and are not -compact.
Moreover, from the definition of a singular point, it is immediately deduced
that is closed;
since is a closed
operator, is closed too.

All that remains to prove is that is the set of
the bifurcation points for .

Indeed, choose , then by (3.21), there is a such that . Moreover, for every , denote the open ball
in centered in with radius : since , is not a
homeomorphism.

Now we prove that is not
injective, arguing by contradiction. Assume that is injective,
we already know that is a compact
operator, so clearly is compact as
well; thus, by Theorem 2.2, would be open,
hence a homeomorphism, which is a contradiction.

Thus, there are with such that . Clearly, so, denoting for , Definition 3.4 is fulfilled and is a
bifurcation point for .

On the other hand, choose , then by (3.21), it clearly follows that Then, is not a
bifurcation point for ; indeed, for every we have , that is, is a local
homeomorphism in ; in particular, there exists an open neighborhood of , such that the restriction of to is injective,
hence cannot comply
with Definition 3.4.

Now we prove (II). Choose again . Let us define an energy functional by putting for
every , it is easily seen that and its
derivative satisfies, for all , the following equality: so, by (3.21), is the set of
the critical points of .

We prove now that is coercive;
with this aim in mind, we note that , as a functional with
compact derivative, is sequentially weakly continuous, hence its restriction to
the closed unit ball admits minimum ; for big enough,
from (F4), we easily get and the latter
goes to as (since ).

We observe, also, that is sequentially
weakly l.s.c. Thus, admits a global
minimum, that is, .

Finally, we prove that has a finite
number of elements: first, recalling that is proper
(Lemma 3.3), we observe that is compact due
to (3.21). Besides, is a discrete
set. Indeed, for every , we have already observed that admits an open
neighborhood such that the
restriction of to is injective,
in particular Being compact and discrete, is finite,
which concludes the proof.

Before concluding this section, we give an example of application of Theorem 3.5 to a system of two equations.

*Example 3.6. *Let , be a matrix as in
Section 2, let
be a real
number, and consider the following problem, depending on the function : We are led to the study of the potential defined by which satisfies all the assumptions of Theorem 3.5.
Thus, the set of bifurcation points related to the system is not -compact, and
for every which is not a
bifurcation point, the set of solutions of the problem is nonempty and
finite.

#### 4. The Linear Case

Let be a function satisfying the following conditions:(F5) is measurable for every , , and for a.e. ;(F6)there exist and a nonnegative such that for every and a.e. , (F7)

As above, in (F6) we can assume increasing. Besides, in this section we will also assume that condition (F3), stated as in Section 3, is fulfilled. In the sequel, for every , we denote by the Hessian matrix of in .

In this section, we deal with the linear problem (L_{u,?})
, depending on the function and on the real
parameter . We recall that, for every and , a *solution* of (L_{u,?})
is a function such that for
every , (the meaning of such definition being the same as in
Section 3). For every , we denote by the set of all such that (L_{u,?})
admits at least
a nonzero solution.

We define the functional over as in Section 3, and collect its properties in the following lemma.

Lemma 4.1. *
Let , , , and
be satisfied.
Then, , its first derivative is a compact
mapping and its second derivative is such that
is a compact
linear operator for all . Moreover,
is not
quasiconvex and satisfies (3.13) and *

*Proof. *Clearly, (F5) and (F6) imply (F1) and (F2), respectively, so as in Lemma 3.1, with a compact
derivative (note that, in
Lemma 3.1, the homogeneity assumption (F4) was employed only to deduce , which here is explicitly assumed in (F5)). Also,
from (F5) and (F6), it is easily deduced that the operator is continuously
differentiable and its derivative is a mapping expressed by for all .

Next we prove that, for all , is a compact linear
operator. Let be a sequence
in with for all (); we wish to
prove that admits a
subsequence which converges to some element of . To this end, we fix and observe
that there exists such that for all with . We choose now and consider
the sequence , which is bounded; by the compactness of , we can assume that (up to a subsequence) converges in , hence in particular, it is a Cauchy sequence. So
there is such that for all . It is easily seen that for all , so since is a complete
metric space, is convergent.

From (F7), through an application of the mean value
theorem, we obtain The asymptotic behaviors of and are easily
deduced from those of , , so (3.13) and (4.3) hold.

Finally, the existence of a nonconvex sublevel set of is proved as in
Lemma 3.1.

For every , we define the mapping by putting for all , and show its properties in the next lemma.

Lemma 4.2. *
Let , , and
be satisfied.
Then, for every ,
and the
following condition holds: *

*Proof. * Fix . Since , from what is observed in Section 2, we deduce that and its
derivative is a mapping defined by for all . In order to achieve (4.10), we proceed as in the proof
of Lemma 3.3, using (3.13).

The next theorem describes the structure of the set for a convenient .

Theorem 4.3. *
Let , , , and
be satisfied.
Then, there exists such that
contains at
least one accumulation point.*

*Proof. *
The
assumptions of Theorem 2.4 are satisfied due to Lemmas 4.1 and 4.2, hence there
exists such that the
set , consisting of the points such that is not
surjective, has an accumulation point .

We note that is a closed
set. To prove this assertion, we observe that the set of all
surjective bounded linear operators from into itself is
open in (see Dieudonné
[10, Théorème 1]).

Besides, is a continuous
mapping, so the set is open. Thus, .

To conclude the proof, it remains to show that for all With this aim
in mind, we fix and note that
the linear operator satisfies the hypotheses of the Fredholm alternative
theorem ([9, Theorem 0.1]). Indeed, by Lemma 4.1, is compact
(recall that is a linear
isometry). Thus, is injective
iff it is surjective. Hence, belongs to iff is not
injective, that is, iff there exists satisfying . Resuming, lies in iff there
exists such that for
all
that is, is a solution
of (L_{u,?})
. Thus, (4.13) is proved and we may conclude that the set contains an
accumulation point.

We conclude by presenting the following example.

*Example 4.4. *
Let , be an
matrix as in
Section 2, and consider the following problem, depending on the function
and on the real
parameter : (here is the
Kronecker symbol). We are led to the study of the potential defined by which satisfies all the assumptions of Theorem 4.3.
Thus, there exists such that the
set contains at
least one accumulation point.

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#### Copyright

Copyright © 2008 Francesca Faraci and Antonio Iannizzotto. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.