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:

$\dot{x}=\varphi (x)+\epsilon \psi (t,x,\epsilon ),$

(1)

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 *T*-periodic with respect to time and $\epsilon \ge 0$ is a small parameter. Precisely, one seeks for the existence of a family of *T*-periodic 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 so-called Malkin bifurcation function:

${f}_{0}(\theta )={\int}_{0}^{T}\u3008{z}_{0}(\tau ),\psi (\tau -\theta ,{x}_{0}(\tau ),0)\u3009\phantom{\rule{0.2em}{0ex}}d\tau ,$

where

${z}_{0}$ is a

*T*-periodic solution of

$\dot{z}=-{\left({\varphi}^{\prime}({x}_{0}(t))\right)}^{\ast}z$

the adjoint system of the linearized system

$\dot{y}={\varphi}^{\prime}({x}_{0}(t))y.$

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 *T*-periodic 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:

$P(x)+\epsilon Q(x,\epsilon )=0,$

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 *T*-periodic 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 Leray-Schauder 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 *T*-periodic 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:

$\dot{x}=Ax+\varphi (x)+\epsilon \psi (t,x,\epsilon ),\phantom{\rule{1em}{0ex}}t\ge 0,\epsilon >0,$

(2)

where *A* is the infinitesimal generator of a strongly continuous semigroup ${e}^{At}$, $t\ge 0$, acting in the Banach space *E*, satisfying the Radon-Nikodym 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 *T*-periodic 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 *T*-periodic 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

*T*-periodic 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

*T*-periodic solutions to (2) and that we rewrite in the following form:

$\tilde{P}(x)+\epsilon \tilde{Q}(x,\epsilon )=0,$

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

$M(\theta )={\int}_{0}^{T}\u3008\tilde{Q}({x}_{0}(\theta ),0)(t),{z}_{0}(t+\theta )\u3009\phantom{\rule{0.2em}{0ex}}dt,\phantom{\rule{1em}{0ex}}\theta \in [0,T],$

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 *T*-periodic 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 *T*-periodic 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.