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
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.
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 (Nu) , 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 . 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, (Nu) admits a finite number of solutions.
We will study (Nu) 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 , 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 , 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.  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  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 ) 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.
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  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 (Nu) .
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 (Lu,?) .
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 (Nu) , depending on the function . We recall that, for every , a solution of (Nu) is a function such that for every We observe that, by the results of [6, (Section 1.4)], whenever is a solution of (Nu) 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 (Nu) . 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.
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.
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 (Nu) . 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.
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 (Lu,?) , depending on the function and on the real parameter . We recall that, for every and , a solution of (Lu,?) 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 (Lu,?) 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.
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 (Lu,?) . 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 © 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.