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A bifurcation problem for a class of periodically perturbed autonomous parabolic equations
Boundary Value Problemsvolume 2013, Article number: 101 (2013)
Abstract
The paper deals with the problem of the existence of a branch of Tperiodic solutions originating from the isolated limit cycle of an autonomous parabolic equation in a Banach space when it is perturbed by a nonlinear Tperiodic term of small amplitude.
We solve this problem by first introducing a novel integral operator, whose fixed points are Tperiodic solutions of the considered equation and vice versa. Then we compute the Malkin bifurcation function associated to this integral operator and we provide conditions under which the wellknown assumption of the existence of a simple zero of the Malkin bifurcation function guarantees the existence of the branch.
MSC:35K58, 35B10, 35B20, 35B32.
1 Introduction
In recent years, bifurcation problems for smooth and nonsmooth dynamical systems have received a renewed attention and interest from different fields of engineering, physics and mathematics. We mention here, among others, the monographs [1–5] and the review papers [6, 7]. Of particular interest is the study of the bifurcation of periodic solutions for periodically perturbed autonomous systems of the form:
where $\varphi \in {C}^{2}({\mathbb{R}}^{n},{\mathbb{R}}^{n})$, $\psi \in {C}^{1}(\mathbb{R}\times {\mathbb{R}}^{n}\times [0,1],{\mathbb{R}}^{n})$, ψ is Tperiodic with respect to time and $\epsilon \ge 0$ is a small parameter. Precisely, one seeks for the existence of a family of Tperiodic solutions originating from a limit cycle ${x}_{0}$ of the autonomous unperturbed system.
Existence, uniqueness and asymptotic stability of bifurcating periodic solutions for system (1) are classical problems; see [8, 9]. The main tool employed in these papers is the socalled Malkin bifurcation function:
where ${z}_{0}$ is a Tperiodic solution of
the adjoint system of the linearized system
It is assumed that the linearized system has only one characteristic multiplier with absolute value 1.
Since the pioneering papers [8, 9], a relevant bibliography devoted to this subject has been developed. From this bibliography, we quote in the sequel some of the papers more related to the present paper. In [10], the case when the cycle ${x}_{0}$ is not isolated was considered. By means of suitably defined bifurcation functions ${f}_{m,n}$, $m,n\in \mathbb{N}$, called Melnikov subharmonic functions, the existence of periodic solutions near to ${x}_{0}$ was proved. The periods of the solutions are in $m:n$ ratio with respect to the period of the perturbation term. The case when 1 is not a simple multiplier of the linearized system was treated in [11]. The existence of at least two branches of Tperiodic solutions originating from ${x}_{0}$ is shown in [12, 13] and their stability, in the sense of Lyapunov, follows from the results of [14, 15]. Developments of the Malkin’s and Melnikov’s approaches have permitted to prove several results about the existence of bifurcating solutions in [16–20]. Furthermore, the use of a Melnikov function permits to detect chaotic behavior of a suitable iterate of the Poincaré map ${\mathrm{\Pi}}_{\epsilon}$ associated to the differential equation (1), which is assumed to have a homoclinic orbit. Indeed, the existence of a simple zero of the considered Melnikov function ensures such a chaotic behavior; see [21, 22].
Very recently, in [23], a new method to prove bifurcation of a branch of asymptotically stable periodic solutions to (1) has been proposed. The method consists first in converting the problem of finding fixed points of the singular Poincaré map ${\mathrm{\Pi}}_{\epsilon}$, $\epsilon \ge 0$, associated to (1) into the problem of the existence of zeros of an equation of the form:
where $P:{\mathbb{R}}^{n}\to {\mathbb{R}}^{n}$ and $Q:{\mathbb{R}}^{n}\times [0,1]\to {\mathbb{R}}^{n}$ are given by $P(x)={\mathrm{\Pi}}_{0}(x)x$ and $Q(x,\epsilon )=\frac{{\mathrm{\Pi}}_{\epsilon}(x){\mathrm{\Pi}}_{0}(x)}{\epsilon}$ with singular ${P}^{\prime}({x}_{0})$. Then, by a convenient scaling of the variable x, we introduce an equivalent equation $\mathrm{\Psi}(w,\epsilon )=0$. For this equation, under the usual assumption of the existence of a simple zero of the Malkin bifurcation function associated to (1), the classical implicit function theorem can be applied to prove the existence of a branch of solutions originating from ${x}_{0}$.
The same approach has been employed in [24] for a class of systems for which the resulting operators P and Q satisfy regularity conditions, which permit to apply the implicit function theorem, only along certain directions at the point ${x}_{0}(\cdot )$ of the limit cycle. Conditions to ensure the existence of several branches of Tperiodic solution emanating from ${x}_{0}$ are provided by means of suitably defined Malkin bifurcation functions.
In all the papers cited before, the existence of periodic solutions for $\epsilon \ge 0$ small is a consequence of the application of a convenient version of the implicit function theorem. This requires, as assumed for system (1), that $\varphi \in {C}^{2}({\mathbb{R}}^{n},{\mathbb{R}}^{n})$ and $\psi \in {C}^{1}(\mathbb{R}\times {\mathbb{R}}^{n}\times [0,1],{\mathbb{R}}^{n})$. Under less restrictive regularity conditions, by using topological tools such as the coincidence degree [25], the LeraySchauder degree and the related continuation principles [26–28], existence results for $\epsilon \ge 0$ small have been proved in [29] when the autonomous system is Hamiltonian and in [30, 31] when the limit cycle is isolated. More precisely, in [29, 31] the existence of two branches of Tperiodic solutions was proved. Roughly speaking, in these papers, the bifurcation functions are employed to guarantee that the topological degree of certain operators is different from zero, rather than for the application of an implicit function theorem.
Topological degree arguments have been also employed in [32] to show the existence of periodic solutions for $\epsilon \ge 0$ small in the case when the unperturbed system is nonautonomous and the perturbation consists of two nonlinear periodic terms with multiplicative different powers of $\epsilon \ge 0$. Finally, the behavior of the bifurcating periodic solutions when the perturbation vanishes has been studied in [33] for a nonsmooth system of the form (1) having an isolated limit cycle and in [34] for nonsmooth planar Hamiltonian systems.
A first attempt to extend to infinite dimensional bifurcation problem the approach outlined in [23] has been presented in [35], with the aim of studying the bifurcation of periodic solutions for a functional differential equation of neutral type. In the present paper, we precise and generalize the idea of how to use a suitable abstract Malkin bifurcation function to deal with infinite dimensional bifurcation problems. To this aim, we consider the following autonomous differential equation of parabolic type periodically perturbed by a nonlinear term of small amplitude:
where A is the infinitesimal generator of a strongly continuous semigroup ${e}^{At}$, $t\ge 0$, acting in the Banach space E, satisfying the RadonNikodym property, $\varphi :E\to E$ is twice continuously Frechét differentiable and $\psi :\mathbb{R}\times E\times [0,1]\to E$ is continuously Frechét differentiable with respect to x, ε and Tperiodic with respect to time. The functions ϕ and ψ satisfy suitable condensivity conditions with respect to the Hausdorff measure of noncompactness. The crucial assumption is that the unperturbed equation at $\epsilon =0$ has a continuous Tperiodic isolated solution ${x}_{0}:\mathbb{R}\to E$, i.e., ${x}_{0}\in {C}_{T}(E)$.
The paper is organized as follows. In Section 2, we precise the conditions under which there is at least a branch of Tperiodic solutions to (2) emanating from ${x}_{0}$. This existence result follows from the application of [[36], Theorem 2]; this theorem relies on the method introduced in [23]. Precisely, to solve the bifurcation problem for (2), we introduce an equivalent integral equation whose zeros are the Tperiodic solutions to (2) and that we rewrite in the following form:
where $\tilde{P}:{C}_{T}(E)\to {C}_{T}(E)$ and $\tilde{Q}:{C}_{T}(E)\times [0,1]\to {C}_{T}(E)$. This equation has a branch of solutions originating from ${x}_{0}({\theta}_{0})$ if the Malkin bifurcation function given by
has ${\theta}_{0}\in [0,T]$ as simple zero. Here, for any $\theta \in [0,T]$, ${x}_{0}(\theta )\in {C}_{T}(E)$ is given by ${x}_{0}(\theta )(\cdot ):={x}_{\theta}(\cdot )=x(\cdot +\theta )$, ${z}_{0}(\theta )$ is the eigenvector corresponding to the simple eigenvalue 0 of ${({\tilde{P}}^{\prime}({x}_{0}(\theta )))}^{\ast}$ and $\u3008\cdot ,\cdot \u3009$ denotes the duality pairing of E with its dual ${E}^{\prime}$. The main difficulty to verify the conditions of [[36], Theorem 2] consists in proving that the zero eigenvalue of ${\tilde{P}}^{\prime}({x}_{0}(\theta ))$ is simple. In fact, in this case, the assumption that the linearized equation, around the limit cycle ${x}_{0}(\theta )$, $\theta \in [0,T]$, of the autonomous system at $\epsilon =0$, does not have neither Tperiodic solutions linearly independent with ${x}_{0}^{\prime}(\theta )$ nor Floquet adjoint solutions to ${x}_{0}^{\prime}(\theta )$ does not guarantee that the zero eigenvalue of ${\tilde{P}}^{\prime}({x}_{0}(\theta ))$ is simple. To overcome this difficulty, we define in Section 3 a novel integral operator, equivalent to that associated to (2) with the property that for the resulting equation $P(x)+\epsilon Q(x,\epsilon )=0$, the operator ${P}^{\prime}({x}_{0}(\theta ))$ has 0 as simple eigenvalue.
Furthermore, in Section 4, we calculate the Malkin bifurcation function associated to the integral operator introduced in Section 3, and we formulate in Theorem 2 the result of the existence of a branch of Tperiodic solutions parameterized by $\epsilon \ge 0$ small. Proposition 1 of Section 5 states a somewhat surprising result: the Malkin functions associated to the two integral operators coincide and they have the common form of the classical Malkin function introduced for ordinary differential equations in finite dimensional spaces of the form (1). Finally, in Section 6, we provide a concrete example of a system of partial differential equations to which our abstract bifurcation result applies.
2 Assumptions and statement of the problem
The paper deals with the problem of the existence of bifurcation of Tperiodic solutions for the Tperiodically perturbed autonomous equation of the form
from a Tperiodic limit cycle ${x}_{0}$ of the unperturbed system corresponding to $\epsilon =0$. Here, A is the infinitesimal generator of a strongly continuous semigroup ${e}^{At}$, $t\ge 0$, acting in the Banach space E, which satisfies the RadonNikodym property; see [[37], Theorem 23, p.276]; $\varphi :E\to E$, $\psi :\mathbb{R}\times E\times [0,1]\to E$ is Tperiodic and $x\in {C}_{T}(E)$, the space of Tperiodic continuous functions $x:\mathbb{R}\to E$.
Throughout the paper, we assume the following conditions on A, ϕ and ψ.

(H1) ${({e}^{At})}^{\ast}={e}^{{A}^{\ast}t}$ and there exists ${\alpha}_{0}>0$ such that
$${\parallel {e}^{At}\parallel}_{E}\le {e}^{{\alpha}_{0}t},\phantom{\rule{1em}{0ex}}t\ge 0.$$ 
(H2) ϕ is twice continuously Fréchet differentiable, ψ is continuously Fréchet differentiable with respect to the pair $(x,\epsilon )$. Moreover, for any nonempty, bounded set $\mathrm{\Omega}\subset E$ we have
$$\begin{array}{c}\chi (\varphi (\mathrm{\Omega}))\le k\chi (\mathrm{\Omega});\hfill \\ \chi \left(\psi ([0,T]\times \mathrm{\Omega}\times [0,1])\right)\le l\chi (\mathrm{\Omega}),\hfill \end{array}$$where $0<k/{\alpha}_{0}<1$, $l>0$ and $\chi (\cdot )$ is the Hausdorff measure of noncompactness [38].

(H3) The unperturbed equation
$$\dot{x}=Ax+\varphi (x)$$(4)has a Tperiodic isolated solution ${x}_{0}\in {C}_{T}(E)$, hence the set of shifts ${x}_{\theta}(\cdot )={x}_{0}(\cdot +\theta )$, for any $\theta \in [0,T]$, represents a family of Tperiodic solutions to (4). Moreover, for $\theta \in [0,T]$, we have that ${y}_{\theta}(t):={x}_{\theta}^{\prime}(t)$ is a Tperiodic solution to the linearized equation
$$\dot{y}=Ay+{\varphi}^{\prime}({x}_{0}(\theta ))y,$$(5)where ${x}_{0}(\theta ):={x}_{\theta}$. We assume that ${x}_{0}^{\prime}(\theta )\ne 0$ for any $\theta \in [0,T]$ and that (5) does not possess neither Tperiodic solution linearly independent with ${y}_{\theta}(t)$ nor Floquet adjoint solution to ${y}_{\theta}(t)$, whenever $\theta \in [0,T]$, i.e., (5) does not have solutions of the form
$$y(t)=v(t)+\frac{t}{T}{y}_{\theta}(t),$$where $v(t)$ is a Tperiodic function.
We pose the following.
Problem To find conditions to ensure the existence of a branch of Tperiodic solutions to (3) parameterized by $\epsilon \ge 0$, originating, for some ${\theta}_{0}\in [0,T]$, from the family of Tperiodic solutions ${x}_{0}(\theta )$.
To solve this problem, we first reduce the existence of Tperiodic solutions to (3) to the problem of finding fixed points of an integral equation. For this, we introduce the linear operator $J:{C}_{T}(E)\to {C}_{T}(E)$ as follows:
Therefore, if we let
then a function $y\in {C}_{T}(E)$ satisfying
is a solution to (3) and vice versa. Moreover, it is easy to verify that the equation
is equivalent to the linearized unperturbed equation
where ${a}_{\theta}(t):={\varphi}^{\prime}({x}_{0}(t+\theta ))$. Hence, we can rewrite (6) in the following form:
where $\tilde{P}:{C}_{T}(E)\to {C}_{T}(E)$ and $\tilde{Q}:{C}_{T}(E)\times [0,1]\to {C}_{T}(E)$ are defined as follows:
and
In conclusion, our problem will be solved if we show that for $\epsilon \ge 0$ sufficiently small, equation (7) has a solution. To this end, it would be sufficient to verify the conditions of the following result.
Theorem 1 ([[36], Theorem 2])
Let B be a Banach space, let $P:B\to B$ be a twice continuously Fréchet differentiable map and $Q:B\times [0,1]\to B$ continuously Fréchet differentiable with respect to both the variables.
Assume that the equation $P(x)=0$ has onedimensional set of solutions ${x}_{0}(\theta )\in B$, parameterized by $\theta \in [0,T]$, such that there exists ${x}_{0}^{\u2033}(\theta )$ for any $\theta \in [0,T]$ and ${x}_{0}^{\prime}(\theta )\ne 0$ for any $\theta \in [0,T]$. Assume that the eigenvalue $0\in \sigma ({P}^{\prime}({x}_{0}(\theta )))$ is simple and the operator $I{P}^{\prime}({x}_{0}(\theta )):B\to B$ is compact, whenever $\theta \in [0,T]$. Consider the function defined by
where ${z}_{0}(\theta )$ is the eigenvector corresponding to the simple eigenvalue $0\in \sigma {({P}^{\prime}({x}_{0}(\theta )))}^{\ast}$. Here, ∗ denotes the adjoint operator.
Then, for each ${\theta}_{0}\in [0,T]$ such that $M({\theta}_{0})=0$ and ${M}^{\prime}({\theta}_{0})\ne 0$ the equation $P(x)+\epsilon Q(x,\epsilon )=0$ is solvable, for $\epsilon \ge 0$ sufficiently small, in a neighborhood of the point ${x}_{0}({\theta}_{0})$ and the solution has the form
where $w\in B$ can be determined in explicit form as shown in [[35], Theorem 2.3] and [[36], Lemma 3].
As it has been observed in [39], the compactness of the operator $I{P}^{\prime}({x}_{0}(\theta ))$ can be replaced by the condensivity of $I{P}^{\prime}({x}_{0}(\theta ))$ with respect to the Hausdorff measure of noncompactness. Indeed, as it is shown in [40], under assumptions (H1)(H2), the operator $IP$ and thus $I{P}^{\prime}({x}_{0}(\theta ))$, see [[38], Theorem 1.5.4], are condensing with constant $k/{\alpha}_{0}<1$. Furthermore, [[38], Theorem 2.6.11] ensures that zero is an eigenvalue of ${P}^{\prime}({x}_{0}(\theta ))$ of finite multiplicity. Therefore, under assumptions (H1)(H3), one can easily verify that the conditions of the previous Theorem 1 are satisfied for (7) except the condition of the simplicity of the zero eigenvalue of ${\tilde{P}}^{\prime}({x}_{0}(\theta ))$, $\theta \in [0,T]$. In fact, the assumption that (5) does not possess neither Tperiodic solutions linearly independent with ${y}_{\theta}(t)$, nor Floquet adjoint solutions to ${y}_{\theta}(t)$, whenever $\theta \in [0,T]$, does not imply that the zero eigenvalue of ${\tilde{P}}^{\prime}({x}_{0}(\theta ))$ is simple, despite the existence of a bijection between the Tperiodic solutions to (5) and the Tperiodic solutions to ${\tilde{P}}^{\prime}({x}_{0}(\theta ))=0$. Moreover, as it shown in [41] the simplicity of the zero eigenvalue of ${\tilde{P}}^{\prime}({x}_{0}(\theta ))$ does not imply that the Tperiodic solution ${y}_{\theta}(t)$ to (5) has the property mentioned above.
In conclusion, in order to apply Theorem 1, we will introduce a novel integral operator whose fixed points are also fixed points of (6) and vice versa, and thus Tperiodic solutions to (3). Moreover, we will show that the zero eigenvalue of the corresponding operator ${P}^{\prime}({x}_{0}(\theta ))$ is simple. This is the aim of the next section.
3 A novel equivalent integral operator
Let ${F}_{\epsilon}(y):=\tilde{P}(y)+\epsilon \tilde{Q}(y,\epsilon )$, hence equation (7) reads as ${F}_{\epsilon}(y)=0$. Consider the integral equation
where $\xi :E\to {C}_{T}(E)$ is defined in the next lemma and $\tau \in [0,T]$ is a given point.
For any fixed $\theta \in [0,T]$, let ${e}_{0}:={x}_{0}^{\prime}(\theta )\ne 0$, ${\gamma}_{0}:={\int}_{0}^{\tau}{e}_{0}(s)\phantom{\rule{0.2em}{0ex}}ds$ and $\beta :=\u3008f,{\int}_{0}^{\tau}(J{e}_{0})(s)\phantom{\rule{0.2em}{0ex}}ds\u3009$ where $f\in {E}^{\prime}$.
We can now formulate the following result.
Lemma 1 Assume (H1)(H3) and that $\tau \in [0,T]$ and $f\in {E}^{\prime}$ satisfy the conditions:

(H4) ${\gamma}_{0}\ne 0$.

(H5) $\beta \ne 0$.

(H6) $\u3008f,{\gamma}_{0}\u3009=1$.
Define $\xi :E\to {C}_{T}(E)$ as follows:
where $g:={e}_{0}+\frac{1}{\beta}(J{e}_{0})\in {C}_{T}(E)$. Then (8) is equivalent to (6). Moreover, the zero eigenvalue of ${P}^{\prime}({x}_{0}(\theta ))$ is simple, where $P:{C}_{T}(E)\to {C}_{T}(E)$ is given by
Proof First of all observe that, under our assumptions, we have that
Indeed, arguing by contradiction assume that $\stackrel{\u02c6}{y}\in E$ is such that
then, by the definition of ξ, we obtain
hence
thus $\stackrel{\u02c6}{y}$ and ${\int}_{0}^{\tau}g(s)\phantom{\rule{0.2em}{0ex}}ds$ are linearly dependent, i.e., ${\int}_{0}^{\tau}g(s)\phantom{\rule{0.2em}{0ex}}ds$ is also an eigenvector of ${\int}_{0}^{\tau}\xi (s)\phantom{\rule{0.2em}{0ex}}ds$, that is,
In conclusion, we should have
On the other hand, as it is easy to verify our conditions imply that
We now prove the equivalence between (6) and (8). Clearly, if ${F}_{\epsilon}(y)=0$ for some $y\in {C}_{T}(E)$ then (8) is satisfied. Conversely, assume that $y\in {C}_{T}(E)$ is a solution to (8), hence
Integrating on the interval $[0,\tau ]$, we obtain
or equivalently,
Since $1\notin \sigma ({\int}_{0}^{\tau}\xi (s)\phantom{\rule{0.2em}{0ex}}ds)$ from (10), it follows that ${\int}_{0}^{\tau}{F}_{\epsilon}(y)(s)\phantom{\rule{0.2em}{0ex}}ds=0$ and so from (8) we get ${F}_{\epsilon}(y)=0$.
It remains to prove the second part of the lemma. For this, given $\theta \in [0,T]$, let ${a}_{\theta}(t):={\varphi}^{\prime}({x}_{0}(t+\theta ))$, $t\in [0,T]$, and let $({\alpha}_{\theta}y)(t):={a}_{\theta}(t)y(t)$. To simplify the notation in the sequel, we omit the subscript θ. Observe that
Then the equation
can be rewritten as follows:
For $t\in [0,T]$, define
and
then the equation (11) takes the form
Clearly, ${e}_{0}$ is an eigenvector of ${P}^{\prime}({x}_{0}(\theta ))$ corresponding to the zero eigenvalue, i.e., ${P}^{\prime}({x}_{0}(\theta )){e}_{0}=0$. Assume now that there exists an adjoint vector ${e}_{1}$ to ${e}_{0}$, namely
or
for any $t\in [0,T]$. By assumption, there are no adjoint Floquet solutions to (5), thus
does not possess Tperiodic solution ${e}_{1}(t)$. The integral form of (13) is given by
Therefore, it remains to show that (12) and (14) coincide, namely
For this, integrating (12) on the interval $[0,\tau ]$, we obtain
that is
On the other hand $1\notin \sigma ({\int}_{0}^{\tau}\xi (s)\phantom{\rule{0.2em}{0ex}}ds)$, thus
From (16), we get the following form for (15)
Then
hence
and
Finally substituting (18) into (17), we obtain
By our definition $\xi (t)x=\u3008f,x\u3009g(t)$, $g(t)=\frac{1}{\beta}({e}_{0}(t)+(J{e}_{0})(t))$, with $\u3008f,{\int}_{0}^{\tau}{e}_{0}(s)\phantom{\rule{0.2em}{0ex}}ds\u3009=1$ and $\u3008f,{\int}_{0}^{\tau}(J{e}_{0})(s)\phantom{\rule{0.2em}{0ex}}ds\u3009=\beta $. Therefore, (19) is satisfied and this concludes the proof. □
Remark 1 Observe that a little though convinces of the existence of $\tau \in [0,T]$ and $f\in {E}^{\prime}$ satisfying the conditions (H4)(H6) of Lemma 1.
Furthermore, recall that assumptions (H1)(H2) ensure that the operator $IP$ is condensing with constant $k/{\alpha}_{0}<1$ (see [40]); moreover, [[38], Theorem 1.5.4] guarantees that $I{P}^{\prime}({x}_{0}(\theta ))$ is also condensing with the same constant. Finally, by [[38], Theorem 2.6.11], zero turns out to be an eigenvalue of ${P}^{\prime}({x}_{0}(\theta ))$ of finite multiplicity. The second part of the proof of Lemma 1 shows that it is simple.
4 The Malkin bifurcation function
In the previous section, Lemma 1 states that the operator P, associated to the integral equation (8) satisfies the conditions of Theorem 1. This section is devoted to the computation of the following Malkin bifurcation function ${M}_{\xi}(\theta )$ associated to (8)
where ${z}_{0}(\theta )$ is an eigenvector of ${({P}^{\prime}({x}_{0}(\theta )))}^{\ast}$, i.e.,
For notational convenience, we simply denote ${z}_{0}(\theta )\in {C}_{T}(E)$ by ${z}_{0}$. In order to compute ${M}_{\xi}(\theta )$, it is necessary to determine ${z}_{0}$ in explicit form. The following result solves the problem.
Lemma 2 Assume (H1)(H6), we have that
where γ is an eigenvector of ${(J\alpha )}^{\ast}$ corresponding to the eigenvalue 1 and ${\mathbb{1}}_{[0,\tau ]}$ is the characteristic function of the interval $[0,\tau ]$.
Proof By assumption, E has the RadonNikodym property, then the eigenvalue of ${({P}^{\prime}({x}_{0}(\theta )))}^{\ast}$ can be determined, without loss of generality in the Hilbert space ${H}_{T}^{1}$. Hence, for any $x,y\in {C}_{T}(E)$ we have
First, by using assumption (H1), we calculate
Therefore, we have
We now calculate the adjoint operator for $\xi {\int}_{0}^{\tau}(x(s)(J(\alpha x))(s))\phantom{\rule{0.2em}{0ex}}ds$. For this, consider
Thus,
Finally, we calculate ${\xi}^{\ast}(t)$, for this consider
i.e.
Now, we are in the position to determine the eigenvector ${z}_{0}$ of ${({P}^{\prime}({x}_{0}(\theta )))}^{\ast}$. We have that
Then
Let $\gamma (t):={z}_{0}(t){\mathbb{1}}_{[0,\tau ]}(t){\int}_{0}^{T}\u3008g(s),{z}_{0}(s)\u3009f\phantom{\rule{0.2em}{0ex}}ds$, then the previous equation takes the form
namely γ is an eigenvector of the linear operator ${(J\alpha )}^{\ast}:{C}_{T}(E)\to {C}_{T}(E)$ corresponding to the simple eigenvalue 1. Therefore, by replacing y with γ in (20), we obtain
The aim now is to find γ, for this consider the adjoint equation to (5)
and the solution of (22), defined for $t\in [0,T]$, given by
For $t=0$, we have
Since v is Tperiodic, we obtain
By using (24) into (23), we get
Put $\omega (t):=\dot{v}(t){A}^{\ast}v(t)$, then $\omega (t)={a}^{\ast}(t)v(t)$ and (25) becomes
Therefore,
i.e., $\omega (t)$ is a solution to (21). Hence,
where $v(t)$ is the Tperiodic solution to the adjoint equation (22). Finally, from
we obtain
and
By (9), we get
In conclusion,
□
Lemmas 1 and 2, together with the fact that (H1)(H2) ensure the condensivity of $I{P}^{\prime}({x}_{0}(\theta ))$, $\theta \in [0,T]$, of constant $0<k/{\alpha}_{0}<1$ (see [40]) allow to apply Theorem 1 to state the following.
Theorem 2 Assume (H1)(H6). If there exists ${\theta}_{0}\in [0,T]$ such that ${M}_{\xi}({\theta}_{0})=0$ and ${M}_{\xi}^{\prime}({\theta}_{0})\ne 0$. Then there exists a branch of Tperiodic solutions to (3) of the form
for $\epsilon \ge 0$ sufficiently small and $w\in {C}_{T}(E)$.
Remark 2 The function w can be calculated in an explicit form as shown in [[35], Theorem 2.3] and [[36], Lemma 3].
5 An invariance property of the Malkin bifurcation function
In what follows, we state an interesting property of the Malkin bifurcation functions introduced before. Precisely, we can prove the following result.
Proposition 1 Let $\theta \in [0,T]$, assume that $0\in \sigma ({\tilde{P}}^{\prime}({x}_{0}(\theta )))$ is simple. Then the Malkin bifurcation function $M(\theta )$ associated to system (6) coincide with the Malkin bifurcation function ${M}_{\xi}(\theta )$ associated to system (8).
Proof Consider
Let $\mu :={\int}_{0}^{T}\u3008g(s),\gamma (s)\u3009\phantom{\rule{0.2em}{0ex}}ds$, then by (9) we have
□
Remark 3 $M(\theta )$ can be rewritten in the classical form of the Malkin bifurcation function ${f}_{0}(\theta )$ for ordinary differential equations as (1) of the Introduction. In fact, consider
Since $(J\mathrm{\Psi}({x}_{0}(\theta )))(0)=(J\mathrm{\Psi}({x}_{0}(\theta )))(T)$, $v(0)=v(T)$, Ψ is the superposition operator generated by ψ and v solves (22) we have
6 An example
In order to introduce an example consistent with the general approach of the paper that requires the employ of the theory of condensing operators, we are led to consider partial differential equations of hyperbolic type, whose abstract formulation in Banach spaces gives rise to infinitesimal generators of noncompact ${C}_{0}$semigroups; see, e.g., [42].
Precisely, following [43] and [44], we present a concrete, not academic example concerning the existence of periodic solutions of a system of two autonomous damped wave equations in a thin domain with Neumann boundary conditions. The study of the dynamics of partial differential equations in thin domains has received many attention in the past few years; see [45] and the extensive references therein. The system has the form
where $(x,y)\in Q:=\mathrm{\Omega}\times (0,1)\subset {\mathbb{R}}^{n+1}$, $n\ge 2$, Ω is a ${C}^{2}$smooth bounded domain in ${\mathbb{R}}^{n}$, ν denotes the outward unit normal vector to Q, λ is a small positive parameter representing the thickness of the domain of the variable λy, ${\alpha}_{1}$, ${\alpha}_{2}$, ${\beta}_{1}$, ${\beta}_{2}$ are positive constants and the functions ${\varphi}_{1}$, ${\varphi}_{2}$ are of class ${C}^{1}$ jointly in their arguments. The linear part of system (26) generates an exponentially stable ${C}_{0}$semigroup in a suitable Banach space; see [43] as well as the related references therein. Under the assumption of the existence of a ${T}_{0}$periodic solution ${u}_{0}=({u}_{1}^{0},{u}_{2}^{0})$ of the limit problem, obtained as $\lambda \to 0$, and suitable conditions on the growth of the derivatives of ${\varphi}_{1}$, ${\varphi}_{2}$ with respect to their arguments, it is shown in [44] that [[43], Theorem 1] applies. This result guarantees the existence of ${\lambda}_{0}>0$ such that, for fixed $\stackrel{\u02c6}{\lambda}\in (0,{\lambda}_{0})$, system (26) has an isolated ${T}_{\stackrel{\u02c6}{\lambda}}$periodic solution ${u}_{\stackrel{\u02c6}{\lambda}}=({u}_{1}^{\stackrel{\u02c6}{\lambda}},{u}_{2}^{\stackrel{\u02c6}{\lambda}})$. The crucial assumption of [[43], Theorem 1] is that the zero eigenvalue of the linearized system around ${u}_{0}$ is simple. For a single damped wave equation of system (26) with the nonlinear term ϕ depending periodically on time t, the existence of periodic solutions was studied in [46].
Consider now a ${T}_{\stackrel{\u02c6}{\lambda}}$periodic perturbation of (26) of small amplitude $\epsilon >0$
If we assume that the superposition operators generated by the functions ${\varphi}_{1}$, ${\varphi}_{2}$, ${\psi}_{1}$, ${\psi}_{2}$ satisfy assumption (H2) of this paper, then (H1) and [[47], Theorem 4.3.1] ensure that the ${C}_{0}$semigroup generated by the linearization around ${u}_{\stackrel{\u02c6}{\lambda}}$ of the unperturbed system, corresponding to $\epsilon =0$ in (27), is strongly contractive with respect to the Hausdorff measure of noncompactness $\chi (\cdot )$, i.e., χstrongly contractive. Therefore, our abstract bifurcation result Theorem 2 applies to system (27).
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Acknowledgements
Dedicated to professor Jean Mawhin on the occasion of his seventieth birthday.
The first two authors acknowledge the support by RFBR Grants 100193112a and 12010392a. The third one acknowledges the support by the GNAMPA of the Istituto di Alta Matematica. The authors would like also to thank the referees for their helpful comments and suggestions which improved the presentation of the paper.
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The authors wrote this paper in collaboration and with the same responsibility. All authors read and approved the final version of the manuscript.
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Keywords
 autonomous parabolic equations
 periodic perturbations
 limit cycle
 bifurcation
 periodic solutions