A bifurcation problem for a class of periodically perturbed autonomous parabolic equations

  • Mikhail Kamenskii1,

    Affiliated with

    • Boris Mikhaylenko1 and

      Affiliated with

      • Paolo Nistri2Email author

        Affiliated with

        Boundary Value Problems20132013:101

        DOI: 10.1186/1687-2770-2013-101

        Received: 24 December 2012

        Accepted: 6 April 2013

        Published: 23 April 2013

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        Abstract

        The paper deals with the problem of the existence of a branch of T-periodic solutions originating from the isolated limit cycle of an autonomous parabolic equation in a Banach space when it is perturbed by a nonlinear T-periodic term of small amplitude.

        We solve this problem by first introducing a novel integral operator, whose fixed points are T-periodic 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 well-known assumption of the existence of a simple zero of the Malkin bifurcation function guarantees the existence of the branch.

        MSC:35K58, 35B10, 35B20, 35B32.

        Keywords

        autonomous parabolic equations periodic perturbations limit cycle bifurcation periodic solutions

        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 [15] 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:
        x ˙ = ϕ ( x ) + ε ψ ( t , x , ε ) , http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-101/MediaObjects/13661_2012_Article_347_Equ1_HTML.gif
        (1)

        where ϕ C 2 ( R n , R n ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-101/MediaObjects/13661_2012_Article_347_IEq1_HTML.gif, ψ C 1 ( R × R n × [ 0 , 1 ] , R n ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-101/MediaObjects/13661_2012_Article_347_IEq2_HTML.gif, ψ is T-periodic with respect to time and ε 0 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-101/MediaObjects/13661_2012_Article_347_IEq3_HTML.gif is a small parameter. Precisely, one seeks for the existence of a family of T-periodic solutions originating from a limit cycle x 0 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-101/MediaObjects/13661_2012_Article_347_IEq4_HTML.gif 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 ( θ ) = 0 T z 0 ( τ ) , ψ ( τ θ , x 0 ( τ ) , 0 ) d τ , http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-101/MediaObjects/13661_2012_Article_347_Equa_HTML.gif
        where z 0 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-101/MediaObjects/13661_2012_Article_347_IEq5_HTML.gif is a T-periodic solution of
        z ˙ = ( ϕ ( x 0 ( t ) ) ) z http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-101/MediaObjects/13661_2012_Article_347_Equb_HTML.gif
        the adjoint system of the linearized system
        y ˙ = ϕ ( x 0 ( t ) ) y . http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-101/MediaObjects/13661_2012_Article_347_Equc_HTML.gif

        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 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-101/MediaObjects/13661_2012_Article_347_IEq4_HTML.gif is not isolated was considered. By means of suitably defined bifurcation functions f m , n http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-101/MediaObjects/13661_2012_Article_347_IEq6_HTML.gif, m , n N http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-101/MediaObjects/13661_2012_Article_347_IEq7_HTML.gif, called Melnikov subharmonic functions, the existence of periodic solutions near to x 0 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-101/MediaObjects/13661_2012_Article_347_IEq4_HTML.gif was proved. The periods of the solutions are in m : n http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-101/MediaObjects/13661_2012_Article_347_IEq8_HTML.gif 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 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-101/MediaObjects/13661_2012_Article_347_IEq4_HTML.gif 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 [1620]. Furthermore, the use of a Melnikov function permits to detect chaotic behavior of a suitable iterate of the Poincaré map Π ε http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-101/MediaObjects/13661_2012_Article_347_IEq9_HTML.gif 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 Π ε http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-101/MediaObjects/13661_2012_Article_347_IEq9_HTML.gif, ε 0 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-101/MediaObjects/13661_2012_Article_347_IEq3_HTML.gif, associated to (1) into the problem of the existence of zeros of an equation of the form:
        P ( x ) + ε Q ( x , ε ) = 0 , http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-101/MediaObjects/13661_2012_Article_347_Equd_HTML.gif

        where P : R n R n http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-101/MediaObjects/13661_2012_Article_347_IEq10_HTML.gif and Q : R n × [ 0 , 1 ] R n http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-101/MediaObjects/13661_2012_Article_347_IEq11_HTML.gif are given by P ( x ) = Π 0 ( x ) x http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-101/MediaObjects/13661_2012_Article_347_IEq12_HTML.gif and Q ( x , ε ) = Π ε ( x ) Π 0 ( x ) ε http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-101/MediaObjects/13661_2012_Article_347_IEq13_HTML.gif with singular P ( x 0 ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-101/MediaObjects/13661_2012_Article_347_IEq14_HTML.gif. Then, by a convenient scaling of the variable x, we introduce an equivalent equation Ψ ( w , ε ) = 0 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-101/MediaObjects/13661_2012_Article_347_IEq15_HTML.gif. 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 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-101/MediaObjects/13661_2012_Article_347_IEq4_HTML.gif.

        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 ( ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-101/MediaObjects/13661_2012_Article_347_IEq16_HTML.gif of the limit cycle. Conditions to ensure the existence of several branches of T-periodic solution emanating from x 0 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-101/MediaObjects/13661_2012_Article_347_IEq4_HTML.gif are provided by means of suitably defined Malkin bifurcation functions.

        In all the papers cited before, the existence of periodic solutions for ε 0 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-101/MediaObjects/13661_2012_Article_347_IEq3_HTML.gif small is a consequence of the application of a convenient version of the implicit function theorem. This requires, as assumed for system (1), that ϕ C 2 ( R n , R n ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-101/MediaObjects/13661_2012_Article_347_IEq1_HTML.gif and ψ C 1 ( R × R n × [ 0 , 1 ] , R n ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-101/MediaObjects/13661_2012_Article_347_IEq2_HTML.gif. Under less restrictive regularity conditions, by using topological tools such as the coincidence degree [25], the Leray-Schauder degree and the related continuation principles [2628], existence results for ε 0 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-101/MediaObjects/13661_2012_Article_347_IEq3_HTML.gif 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 ε 0 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-101/MediaObjects/13661_2012_Article_347_IEq3_HTML.gif small in the case when the unperturbed system is nonautonomous and the perturbation consists of two nonlinear periodic terms with multiplicative different powers of ε 0 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-101/MediaObjects/13661_2012_Article_347_IEq3_HTML.gif. 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:
        x ˙ = A x + ϕ ( x ) + ε ψ ( t , x , ε ) , t 0 , ε > 0 , http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-101/MediaObjects/13661_2012_Article_347_Equ2_HTML.gif
        (2)

        where A is the infinitesimal generator of a strongly continuous semigroup e A t http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-101/MediaObjects/13661_2012_Article_347_IEq17_HTML.gif, t 0 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-101/MediaObjects/13661_2012_Article_347_IEq18_HTML.gif, acting in the Banach space E, satisfying the Radon-Nikodym property, ϕ : E E http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-101/MediaObjects/13661_2012_Article_347_IEq19_HTML.gif is twice continuously Frechét differentiable and ψ : R × E × [ 0 , 1 ] E http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-101/MediaObjects/13661_2012_Article_347_IEq20_HTML.gif 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 ε = 0 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-101/MediaObjects/13661_2012_Article_347_IEq21_HTML.gif has a continuous T-periodic isolated solution x 0 : R E http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-101/MediaObjects/13661_2012_Article_347_IEq22_HTML.gif, i.e., x 0 C T ( E ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-101/MediaObjects/13661_2012_Article_347_IEq23_HTML.gif.

        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 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-101/MediaObjects/13661_2012_Article_347_IEq4_HTML.gif. 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:
        P ˜ ( x ) + ε Q ˜ ( x , ε ) = 0 , http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-101/MediaObjects/13661_2012_Article_347_Eque_HTML.gif
        where P ˜ : C T ( E ) C T ( E ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-101/MediaObjects/13661_2012_Article_347_IEq24_HTML.gif and Q ˜ : C T ( E ) × [ 0 , 1 ] C T ( E ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-101/MediaObjects/13661_2012_Article_347_IEq25_HTML.gif. This equation has a branch of solutions originating from x 0 ( θ 0 ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-101/MediaObjects/13661_2012_Article_347_IEq26_HTML.gif if the Malkin bifurcation function given by
        M ( θ ) = 0 T Q ˜ ( x 0 ( θ ) , 0 ) ( t ) , z 0 ( t + θ ) d t , θ [ 0 , T ] , http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-101/MediaObjects/13661_2012_Article_347_Equf_HTML.gif

        has θ 0 [ 0 , T ] http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-101/MediaObjects/13661_2012_Article_347_IEq27_HTML.gif as simple zero. Here, for any θ [ 0 , T ] http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-101/MediaObjects/13661_2012_Article_347_IEq28_HTML.gif, x 0 ( θ ) C T ( E ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-101/MediaObjects/13661_2012_Article_347_IEq29_HTML.gif is given by x 0 ( θ ) ( ) : = x θ ( ) = x ( + θ ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-101/MediaObjects/13661_2012_Article_347_IEq30_HTML.gif, z 0 ( θ ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-101/MediaObjects/13661_2012_Article_347_IEq31_HTML.gif is the eigenvector corresponding to the simple eigenvalue 0 of ( P ˜ ( x 0 ( θ ) ) ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-101/MediaObjects/13661_2012_Article_347_IEq32_HTML.gif and , http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-101/MediaObjects/13661_2012_Article_347_IEq33_HTML.gif denotes the duality pairing of E with its dual E http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-101/MediaObjects/13661_2012_Article_347_IEq34_HTML.gif. The main difficulty to verify the conditions of [[36], Theorem 2] consists in proving that the zero eigenvalue of P ˜ ( x 0 ( θ ) ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-101/MediaObjects/13661_2012_Article_347_IEq35_HTML.gif is simple. In fact, in this case, the assumption that the linearized equation, around the limit cycle x 0 ( θ ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-101/MediaObjects/13661_2012_Article_347_IEq36_HTML.gif, θ [ 0 , T ] http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-101/MediaObjects/13661_2012_Article_347_IEq28_HTML.gif, of the autonomous system at ε = 0 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-101/MediaObjects/13661_2012_Article_347_IEq21_HTML.gif, does not have neither T-periodic solutions linearly independent with x 0 ( θ ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-101/MediaObjects/13661_2012_Article_347_IEq37_HTML.gif nor Floquet adjoint solutions to x 0 ( θ ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-101/MediaObjects/13661_2012_Article_347_IEq37_HTML.gif does not guarantee that the zero eigenvalue of P ˜ ( x 0 ( θ ) ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-101/MediaObjects/13661_2012_Article_347_IEq35_HTML.gif 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 ) + ε Q ( x , ε ) = 0 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-101/MediaObjects/13661_2012_Article_347_IEq38_HTML.gif, the operator P ( x 0 ( θ ) ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-101/MediaObjects/13661_2012_Article_347_IEq39_HTML.gif 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 ε 0 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-101/MediaObjects/13661_2012_Article_347_IEq3_HTML.gif 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 T-periodic solutions for the T-periodically perturbed autonomous equation of the form
        x ˙ = A x + ϕ ( x ) + ε ψ ( t , x , ε ) , t 0 , ε 0 , http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-101/MediaObjects/13661_2012_Article_347_Equ3_HTML.gif
        (3)

        from a T-periodic limit cycle x 0 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-101/MediaObjects/13661_2012_Article_347_IEq4_HTML.gif of the unperturbed system corresponding to ε = 0 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-101/MediaObjects/13661_2012_Article_347_IEq21_HTML.gif. Here, A is the infinitesimal generator of a strongly continuous semigroup e A t http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-101/MediaObjects/13661_2012_Article_347_IEq17_HTML.gif, t 0 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-101/MediaObjects/13661_2012_Article_347_IEq40_HTML.gif, acting in the Banach space E, which satisfies the Radon-Nikodym property; see [[37], Theorem 23, p.276]; ϕ : E E http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-101/MediaObjects/13661_2012_Article_347_IEq41_HTML.gif, ψ : R × E × [ 0 , 1 ] E http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-101/MediaObjects/13661_2012_Article_347_IEq42_HTML.gif is T-periodic and x C T ( E ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-101/MediaObjects/13661_2012_Article_347_IEq43_HTML.gif, the space of T-periodic continuous functions x : R E http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-101/MediaObjects/13661_2012_Article_347_IEq44_HTML.gif.

        Throughout the paper, we assume the following conditions on A, ϕ and ψ.

        • (H1) ( e A t ) = e A t http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-101/MediaObjects/13661_2012_Article_347_IEq45_HTML.gif and there exists α 0 > 0 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-101/MediaObjects/13661_2012_Article_347_IEq46_HTML.gif such that
          e A t E e α 0 t , t 0 . http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-101/MediaObjects/13661_2012_Article_347_Equg_HTML.gif
        • (H2) ϕ is twice continuously Fréchet differentiable, ψ is continuously Fréchet differentiable with respect to the pair ( x , ε ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-101/MediaObjects/13661_2012_Article_347_IEq47_HTML.gif. Moreover, for any nonempty, bounded set Ω E http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-101/MediaObjects/13661_2012_Article_347_IEq48_HTML.gif we have
          χ ( ϕ ( Ω ) ) k χ ( Ω ) ; χ ( ψ ( [ 0 , T ] × Ω × [ 0 , 1 ] ) ) l χ ( Ω ) , http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-101/MediaObjects/13661_2012_Article_347_Equh_HTML.gif

          where 0 < k / α 0 < 1 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-101/MediaObjects/13661_2012_Article_347_IEq49_HTML.gif, l > 0 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-101/MediaObjects/13661_2012_Article_347_IEq50_HTML.gif and χ ( ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-101/MediaObjects/13661_2012_Article_347_IEq51_HTML.gif is the Hausdorff measure of noncompactness [38].

        • (H3) The unperturbed equation
          x ˙ = A x + ϕ ( x ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-101/MediaObjects/13661_2012_Article_347_Equ4_HTML.gif
          (4)
          has a T-periodic isolated solution x 0 C T ( E ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-101/MediaObjects/13661_2012_Article_347_IEq23_HTML.gif, hence the set of shifts x θ ( ) = x 0 ( + θ ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-101/MediaObjects/13661_2012_Article_347_IEq52_HTML.gif, for any θ [ 0 , T ] http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-101/MediaObjects/13661_2012_Article_347_IEq53_HTML.gif, represents a family of T-periodic solutions to (4). Moreover, for θ [ 0 , T ] http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-101/MediaObjects/13661_2012_Article_347_IEq28_HTML.gif, we have that y θ ( t ) : = x θ ( t ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-101/MediaObjects/13661_2012_Article_347_IEq54_HTML.gif is a T-periodic solution to the linearized equation
          y ˙ = A y + ϕ ( x 0 ( θ ) ) y , http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-101/MediaObjects/13661_2012_Article_347_Equ5_HTML.gif
          (5)
          where x 0 ( θ ) : = x θ http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-101/MediaObjects/13661_2012_Article_347_IEq55_HTML.gif. We assume that x 0 ( θ ) 0 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-101/MediaObjects/13661_2012_Article_347_IEq56_HTML.gif for any θ [ 0 , T ] http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-101/MediaObjects/13661_2012_Article_347_IEq28_HTML.gif and that (5) does not possess neither T-periodic solution linearly independent with y θ ( t ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-101/MediaObjects/13661_2012_Article_347_IEq57_HTML.gif nor Floquet adjoint solution to y θ ( t ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-101/MediaObjects/13661_2012_Article_347_IEq57_HTML.gif, whenever θ [ 0 , T ] http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-101/MediaObjects/13661_2012_Article_347_IEq28_HTML.gif, i.e., (5) does not have solutions of the form
          y ( t ) = v ( t ) + t T y θ ( t ) , http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-101/MediaObjects/13661_2012_Article_347_Equi_HTML.gif

          where v ( t ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-101/MediaObjects/13661_2012_Article_347_IEq58_HTML.gif is a T-periodic function.

        We pose the following.

        Problem To find conditions to ensure the existence of a branch of T-periodic solutions to (3) parameterized by ε 0 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-101/MediaObjects/13661_2012_Article_347_IEq3_HTML.gif, originating, for some θ 0 [ 0 , T ] http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-101/MediaObjects/13661_2012_Article_347_IEq27_HTML.gif, from the family of T-periodic solutions x 0 ( θ ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-101/MediaObjects/13661_2012_Article_347_IEq36_HTML.gif.

        To solve this problem, we first reduce the existence of T-periodic 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 ) C T ( E ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-101/MediaObjects/13661_2012_Article_347_IEq59_HTML.gif as follows:
        ( J y ) ( t ) : = e A t ( I e A T ) 1 0 T e A ( T s ) y ( s ) d s + 0 t e A ( t s ) y ( s ) d s . http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-101/MediaObjects/13661_2012_Article_347_Equj_HTML.gif
        Therefore, if we let
        ( J r ε ( y ) ) ( t ) : = J ( ϕ ( y ( t ) ) + ε ψ ( t , y ( t ) , ε ) ) , http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-101/MediaObjects/13661_2012_Article_347_Equk_HTML.gif
        then a function y C T ( E ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-101/MediaObjects/13661_2012_Article_347_IEq60_HTML.gif satisfying
        J ( r ε ( y ) ) = y http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-101/MediaObjects/13661_2012_Article_347_Equ6_HTML.gif
        (6)
        is a solution to (3) and vice versa. Moreover, it is easy to verify that the equation
        ( J ( r 0 ( x θ ) ) ) y = y http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-101/MediaObjects/13661_2012_Article_347_Equl_HTML.gif
        is equivalent to the linearized unperturbed equation
        y ˙ = A y + a θ ( t ) y , http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-101/MediaObjects/13661_2012_Article_347_Equm_HTML.gif
        where a θ ( t ) : = ϕ ( x 0 ( t + θ ) ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-101/MediaObjects/13661_2012_Article_347_IEq61_HTML.gif. Hence, we can rewrite (6) in the following form:
        P ˜ ( y ) + ε Q ˜ ( y , ε ) = 0 , http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-101/MediaObjects/13661_2012_Article_347_Equ7_HTML.gif
        (7)
        where P ˜ : C T ( E ) C T ( E ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-101/MediaObjects/13661_2012_Article_347_IEq24_HTML.gif and Q ˜ : C T ( E ) × [ 0 , 1 ] C T ( E ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-101/MediaObjects/13661_2012_Article_347_IEq25_HTML.gif are defined as follows:
        P ˜ ( y ) ( t ) : = y ( t ) e A t ( I e A T ) 1 0 T e A ( T s ) ϕ ( y ( s ) ) d s 0 t e A ( t s ) ϕ ( y ( s ) ) d s , http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-101/MediaObjects/13661_2012_Article_347_Equn_HTML.gif
        and
        Q ˜ ( y , ε ) ( t ) : = e A t ( I e A T ) 1 0 T e A ( T s ) ψ ( s , y ( s ) , ε ) d s 0 t e A ( t s ) ψ ( s , y ( s ) , ε ) d s . http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-101/MediaObjects/13661_2012_Article_347_Equo_HTML.gif

        In conclusion, our problem will be solved if we show that for ε 0 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-101/MediaObjects/13661_2012_Article_347_IEq3_HTML.gif 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 B http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-101/MediaObjects/13661_2012_Article_347_IEq62_HTML.gif be a twice continuously Fréchet differentiable map and Q : B × [ 0 , 1 ] B http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-101/MediaObjects/13661_2012_Article_347_IEq63_HTML.gif continuously Fréchet differentiable with respect to both the variables.

        Assume that the equation P ( x ) = 0 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-101/MediaObjects/13661_2012_Article_347_IEq64_HTML.gif has one-dimensional set of solutions x 0 ( θ ) B http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-101/MediaObjects/13661_2012_Article_347_IEq65_HTML.gif, parameterized by θ [ 0 , T ] http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-101/MediaObjects/13661_2012_Article_347_IEq28_HTML.gif, such that there exists x 0 ( θ ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-101/MediaObjects/13661_2012_Article_347_IEq66_HTML.gif for any θ [ 0 , T ] http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-101/MediaObjects/13661_2012_Article_347_IEq28_HTML.gif and x 0 ( θ ) 0 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-101/MediaObjects/13661_2012_Article_347_IEq67_HTML.gif for any θ [ 0 , T ] http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-101/MediaObjects/13661_2012_Article_347_IEq28_HTML.gif. Assume that the eigenvalue 0 σ ( P ( x 0 ( θ ) ) ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-101/MediaObjects/13661_2012_Article_347_IEq68_HTML.gif is simple and the operator I P ( x 0 ( θ ) ) : B B http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-101/MediaObjects/13661_2012_Article_347_IEq69_HTML.gif is compact, whenever θ [ 0 , T ] http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-101/MediaObjects/13661_2012_Article_347_IEq28_HTML.gif. Consider the function defined by
        M ( θ ) : = 0 T Q ( x 0 ( θ ) , 0 ) ( t ) , z 0 ( t + θ ) d t , http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-101/MediaObjects/13661_2012_Article_347_Equp_HTML.gif

        where z 0 ( θ ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-101/MediaObjects/13661_2012_Article_347_IEq31_HTML.gif is the eigenvector corresponding to the simple eigenvalue 0 σ ( P ( x 0 ( θ ) ) ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-101/MediaObjects/13661_2012_Article_347_IEq70_HTML.gif. Here, ∗ denotes the adjoint operator.

        Then, for each θ 0 [ 0 , T ] http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-101/MediaObjects/13661_2012_Article_347_IEq27_HTML.gif such that M ( θ 0 ) = 0 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-101/MediaObjects/13661_2012_Article_347_IEq71_HTML.gif and M ( θ 0 ) 0 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-101/MediaObjects/13661_2012_Article_347_IEq72_HTML.gif the equation P ( x ) + ε Q ( x , ε ) = 0 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-101/MediaObjects/13661_2012_Article_347_IEq38_HTML.gif is solvable, for ε 0 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-101/MediaObjects/13661_2012_Article_347_IEq3_HTML.gif sufficiently small, in a neighborhood of the point x 0 ( θ 0 ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-101/MediaObjects/13661_2012_Article_347_IEq73_HTML.gif and the solution has the form
        x ( ε ) = x 0 ( θ 0 ) + ε w + o ( ε ) , http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-101/MediaObjects/13661_2012_Article_347_Equq_HTML.gif

        where w B http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-101/MediaObjects/13661_2012_Article_347_IEq74_HTML.gif 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 ( x 0 ( θ ) ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-101/MediaObjects/13661_2012_Article_347_IEq75_HTML.gif can be replaced by the condensivity of I P ( x 0 ( θ ) ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-101/MediaObjects/13661_2012_Article_347_IEq75_HTML.gif with respect to the Hausdorff measure of noncompactness. Indeed, as it is shown in [40], under assumptions (H1)-(H2), the operator I P http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-101/MediaObjects/13661_2012_Article_347_IEq76_HTML.gif and thus I P ( x 0 ( θ ) ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-101/MediaObjects/13661_2012_Article_347_IEq75_HTML.gif, see [[38], Theorem 1.5.4], are condensing with constant k / α 0 < 1 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-101/MediaObjects/13661_2012_Article_347_IEq77_HTML.gif. Furthermore, [[38], Theorem 2.6.11] ensures that zero is an eigenvalue of P ( x 0 ( θ ) ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-101/MediaObjects/13661_2012_Article_347_IEq39_HTML.gif 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 P ˜ ( x 0 ( θ ) ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-101/MediaObjects/13661_2012_Article_347_IEq35_HTML.gif, θ [ 0 , T ] http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-101/MediaObjects/13661_2012_Article_347_IEq28_HTML.gif. In fact, the assumption that (5) does not possess neither T-periodic solutions linearly independent with y θ ( t ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-101/MediaObjects/13661_2012_Article_347_IEq57_HTML.gif, nor Floquet adjoint solutions to y θ ( t ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-101/MediaObjects/13661_2012_Article_347_IEq57_HTML.gif, whenever θ [ 0 , T ] http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-101/MediaObjects/13661_2012_Article_347_IEq28_HTML.gif, does not imply that the zero eigenvalue of P ˜ ( x 0 ( θ ) ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-101/MediaObjects/13661_2012_Article_347_IEq35_HTML.gif is simple, despite the existence of a bijection between the T-periodic solutions to (5) and the T-periodic solutions to P ˜ ( x 0 ( θ ) ) = 0 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-101/MediaObjects/13661_2012_Article_347_IEq78_HTML.gif. Moreover, as it shown in [41] the simplicity of the zero eigenvalue of P ˜ ( x 0 ( θ ) ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-101/MediaObjects/13661_2012_Article_347_IEq35_HTML.gif does not imply that the T-periodic solution y θ ( t ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-101/MediaObjects/13661_2012_Article_347_IEq57_HTML.gif 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 T-periodic solutions to (3). Moreover, we will show that the zero eigenvalue of the corresponding operator P ( x 0 ( θ ) ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-101/MediaObjects/13661_2012_Article_347_IEq39_HTML.gif is simple. This is the aim of the next section.

        3 A novel equivalent integral operator

        Let F ε ( y ) : = P ˜ ( y ) + ε Q ˜ ( y , ε ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-101/MediaObjects/13661_2012_Article_347_IEq79_HTML.gif, hence equation (7) reads as F ε ( y ) = 0 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-101/MediaObjects/13661_2012_Article_347_IEq80_HTML.gif. Consider the integral equation
        F ε ( y ) ( t ) ξ ( t ) 0 τ F ε ( y ) ( s ) d s = 0 , t [ 0 , T ] , http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-101/MediaObjects/13661_2012_Article_347_Equ8_HTML.gif
        (8)

        where ξ : E C T ( E ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-101/MediaObjects/13661_2012_Article_347_IEq81_HTML.gif is defined in the next lemma and τ [ 0 , T ] http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-101/MediaObjects/13661_2012_Article_347_IEq82_HTML.gif is a given point.

        For any fixed θ [ 0 , T ] http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-101/MediaObjects/13661_2012_Article_347_IEq28_HTML.gif, let e 0 : = x 0 ( θ ) 0 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-101/MediaObjects/13661_2012_Article_347_IEq83_HTML.gif, γ 0 : = 0 τ e 0 ( s ) d s http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-101/MediaObjects/13661_2012_Article_347_IEq84_HTML.gif and β : = f , 0 τ ( J e 0 ) ( s ) d s http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-101/MediaObjects/13661_2012_Article_347_IEq85_HTML.gif where f E http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-101/MediaObjects/13661_2012_Article_347_IEq86_HTML.gif.

        We can now formulate the following result.

        Lemma 1 Assume (H1)-(H3) and that τ [ 0 , T ] http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-101/MediaObjects/13661_2012_Article_347_IEq87_HTML.gif and f E http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-101/MediaObjects/13661_2012_Article_347_IEq86_HTML.gif satisfy the conditions:

        • (H4) γ 0 0 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-101/MediaObjects/13661_2012_Article_347_IEq88_HTML.gif.

        • (H5) β 0 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-101/MediaObjects/13661_2012_Article_347_IEq89_HTML.gif.

        • (H6) f , γ 0 = 1 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-101/MediaObjects/13661_2012_Article_347_IEq90_HTML.gif.

        Define ξ : E C T ( E ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-101/MediaObjects/13661_2012_Article_347_IEq91_HTML.gif as follows:
        x E , ξ ( t ) x : = f , x g ( t ) , t [ 0 , T ] , http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-101/MediaObjects/13661_2012_Article_347_Equr_HTML.gif
        where g : = e 0 + 1 β ( J e 0 ) C T ( E ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-101/MediaObjects/13661_2012_Article_347_IEq92_HTML.gif. Then (8) is equivalent to (6). Moreover, the zero eigenvalue of P ( x 0 ( θ ) ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-101/MediaObjects/13661_2012_Article_347_IEq39_HTML.gif is simple, where P : C T ( E ) C T ( E ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-101/MediaObjects/13661_2012_Article_347_IEq93_HTML.gif is given by
        P ( y ) ( t ) : = P ˜ ( y ) ( t ) ξ ( t ) 0 τ P ˜ ( y ) ( s ) d s , t [ 0 , T ] . http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-101/MediaObjects/13661_2012_Article_347_Equs_HTML.gif
        Proof First of all observe that, under our assumptions, we have that
        1 σ ( 0 τ ξ ( s ) d s ) . http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-101/MediaObjects/13661_2012_Article_347_Equt_HTML.gif
        Indeed, arguing by contradiction assume that y ˆ E http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-101/MediaObjects/13661_2012_Article_347_IEq94_HTML.gif is such that
        ( 0 τ ξ ( s ) d s ) y ˆ = y ˆ , http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-101/MediaObjects/13661_2012_Article_347_Equu_HTML.gif
        then, by the definition of ξ, we obtain
        ( 0 τ ξ ( s ) d s ) y ˆ = f , y ˆ 0 τ g ( s ) d s , http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-101/MediaObjects/13661_2012_Article_347_Equv_HTML.gif
        hence
        y ˆ = f , y ˆ 0 τ g ( s ) d s , http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-101/MediaObjects/13661_2012_Article_347_Equw_HTML.gif
        thus y ˆ http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-101/MediaObjects/13661_2012_Article_347_IEq95_HTML.gif and 0 τ g ( s ) d s http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-101/MediaObjects/13661_2012_Article_347_IEq96_HTML.gif are linearly dependent, i.e., 0 τ g ( s ) d s http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-101/MediaObjects/13661_2012_Article_347_IEq96_HTML.gif is also an eigenvector of 0 τ ξ ( s ) d s http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-101/MediaObjects/13661_2012_Article_347_IEq97_HTML.gif, that is,
        0 τ ξ ( s ) d s 0 τ g ( s ) d s = 0 τ g ( s ) d s . http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-101/MediaObjects/13661_2012_Article_347_Equx_HTML.gif
        In conclusion, we should have
        f , 0 τ g ( s ) d s = 1 . http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-101/MediaObjects/13661_2012_Article_347_Equy_HTML.gif
        On the other hand, as it is easy to verify our conditions imply that
        f , 0 τ g ( s ) d s = 2 . http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-101/MediaObjects/13661_2012_Article_347_Equ9_HTML.gif
        (9)
        We now prove the equivalence between (6) and (8). Clearly, if F ε ( y ) = 0 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-101/MediaObjects/13661_2012_Article_347_IEq80_HTML.gif for some y C T ( E ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-101/MediaObjects/13661_2012_Article_347_IEq60_HTML.gif then (8) is satisfied. Conversely, assume that y C T ( E ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-101/MediaObjects/13661_2012_Article_347_IEq60_HTML.gif is a solution to (8), hence
        F ε ( y ) ( t ) ξ ( t ) 0 τ F ε ( y ) ( s ) d s = 0 , t [ 0 , T ] . http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-101/MediaObjects/13661_2012_Article_347_Equz_HTML.gif
        Integrating on the interval [ 0 , τ ] http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-101/MediaObjects/13661_2012_Article_347_IEq98_HTML.gif, we obtain
        0 τ F ε ( y ) ( s ) d s 0 τ ξ ( s ) d s 0 τ F ε ( y ) ( s ) d s = 0 , http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-101/MediaObjects/13661_2012_Article_347_Equaa_HTML.gif
        or equivalently,
        ( I 0 τ ξ ( s ) d s ) 0 τ F ε ( y ) ( s ) d s = 0 . http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-101/MediaObjects/13661_2012_Article_347_Equ10_HTML.gif
        (10)

        Since 1 σ ( 0 τ ξ ( s ) d s ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-101/MediaObjects/13661_2012_Article_347_IEq99_HTML.gif from (10), it follows that 0 τ F ε ( y ) ( s ) d s = 0 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-101/MediaObjects/13661_2012_Article_347_IEq100_HTML.gif and so from (8) we get F ε ( y ) = 0 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-101/MediaObjects/13661_2012_Article_347_IEq80_HTML.gif.

        It remains to prove the second part of the lemma. For this, given θ [ 0 , T ] http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-101/MediaObjects/13661_2012_Article_347_IEq28_HTML.gif, let a θ ( t ) : = ϕ ( x 0 ( t + θ ) ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-101/MediaObjects/13661_2012_Article_347_IEq61_HTML.gif, t [ 0 , T ] http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-101/MediaObjects/13661_2012_Article_347_IEq101_HTML.gif, and let ( α θ y ) ( t ) : = a θ ( t ) y ( t ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-101/MediaObjects/13661_2012_Article_347_IEq102_HTML.gif. To simplify the notation in the sequel, we omit the subscript θ. Observe that
        F 0 ( x 0 ( θ ) ) y = P ˜ ( x 0 ( θ ) ) y = y J ( α y ) . http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-101/MediaObjects/13661_2012_Article_347_Equab_HTML.gif
        Then the equation
        ( F 0 ( x 0 ( θ ) ) y ) ( t ) ξ ( t ) 0 τ ( F 0 ( x 0 ( θ ) ) y ) ( s ) d s = 0 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-101/MediaObjects/13661_2012_Article_347_Equac_HTML.gif
        can be rewritten as follows:
        y ( t ) ( J ( α y ) ) ( t ) ξ ( t ) 0 τ ( y ( s ) ( J ( α y ) ) ( s ) ) d s = 0 . http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-101/MediaObjects/13661_2012_Article_347_Equ11_HTML.gif
        (11)
        For t [ 0 , T ] http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-101/MediaObjects/13661_2012_Article_347_IEq101_HTML.gif, define
        P ( y ) ( t ) : = P ˜ ( y ) ( t ) ξ ( t ) 0 τ P ˜ ( y ) ( s ) d s http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-101/MediaObjects/13661_2012_Article_347_Equad_HTML.gif
        and
        Q ( y , ε ) ( t ) : = Q ˜ ( y , ε ) ( t ) ξ ( t ) 0 τ Q ˜ ( y , ε ) ( s ) d s , http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-101/MediaObjects/13661_2012_Article_347_Equae_HTML.gif
        then the equation (11) takes the form
        P ( x 0 ( θ ) ) y = 0 . http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-101/MediaObjects/13661_2012_Article_347_Equaf_HTML.gif
        Clearly, e 0 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-101/MediaObjects/13661_2012_Article_347_IEq103_HTML.gif is an eigenvector of P ( x 0 ( θ ) ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-101/MediaObjects/13661_2012_Article_347_IEq39_HTML.gif corresponding to the zero eigenvalue, i.e., P ( x 0 ( θ ) ) e 0 = 0 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-101/MediaObjects/13661_2012_Article_347_IEq104_HTML.gif. Assume now that there exists an adjoint vector e 1 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-101/MediaObjects/13661_2012_Article_347_IEq105_HTML.gif to e 0 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-101/MediaObjects/13661_2012_Article_347_IEq103_HTML.gif, namely
        P ( x 0 ( θ ) ) e 1 = e 0 , http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-101/MediaObjects/13661_2012_Article_347_Equag_HTML.gif
        or
        e 1 ( t ) ( J ( α e 1 ) ) ( t ) ξ ( t ) 0 τ ( e 1 ( s ) ( J ( α e 1 ) ) ( s ) ) d s = e 0 ( t ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-101/MediaObjects/13661_2012_Article_347_Equ12_HTML.gif
        (12)
        for any t [ 0 , T ] http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-101/MediaObjects/13661_2012_Article_347_IEq101_HTML.gif. By assumption, there are no adjoint Floquet solutions to (5), thus
        e ˙ 1 ( t ) = A e 1 ( t ) + a ( t ) e 1 ( t ) e 0 ( t ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-101/MediaObjects/13661_2012_Article_347_Equ13_HTML.gif
        (13)
        does not possess T-periodic solution e 1 ( t ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-101/MediaObjects/13661_2012_Article_347_IEq106_HTML.gif. The integral form of (13) is given by
        e 1 ( t ) = ( J ( α e 1 ) ) ( t ) ( J e 0 ) ( t ) . http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-101/MediaObjects/13661_2012_Article_347_Equ14_HTML.gif
        (14)
        Therefore, it remains to show that (12) and (14) coincide, namely
        ξ ( t ) 0 τ ( e 1 ( s ) ( J ( α e 1 ) ) ( s ) ) d s + e 0 ( t ) = ( J e 0 ) ( t ) . http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-101/MediaObjects/13661_2012_Article_347_Equ15_HTML.gif
        (15)
        For this, integrating (12) on the interval [ 0 , τ ] http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-101/MediaObjects/13661_2012_Article_347_IEq98_HTML.gif, we obtain
        0 τ ( e 1 ( s ) ( J ( α e 1 ) ) ( s ) ) d s 0 τ ξ ( s ) d s 0 τ ( e 1 ( s ) ( J ( α e 1 ) ) ( s ) ) d s = 0 τ e 0 ( s ) d s , http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-101/MediaObjects/13661_2012_Article_347_Equah_HTML.gif
        that is
        ( I 0 τ ξ ( s ) d s ) 0 τ ( e 1 ( s ) ( J ( α e 1 ) ) ( s ) ) d s = 0 τ e 0 ( s ) d s . http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-101/MediaObjects/13661_2012_Article_347_Equai_HTML.gif
        On the other hand 1 σ ( 0 τ ξ ( s ) d s ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-101/MediaObjects/13661_2012_Article_347_IEq99_HTML.gif, thus
        0 τ ( e 1 ( s ) ( J ( α e 1 ) ) ( s ) ) d s = ( I 0 τ ξ ( s ) d s ) 1 0 τ e 0 ( s ) d s . http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-101/MediaObjects/13661_2012_Article_347_Equ16_HTML.gif
        (16)
        From (16), we get the following form for (15)
        ξ ( t ) ( I 0 τ ξ ( s ) d s ) 1 0 τ e 0 ( s ) d s = e 0 ( t ) ( J e 0 ) ( t ) . http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-101/MediaObjects/13661_2012_Article_347_Equ17_HTML.gif
        (17)
        Then
        ( I 0 τ ξ ( s ) d s I ) ( I 0 τ ξ ( s ) d s ) 1 0 τ e 0 ( s ) d s = 0 τ ( e 0 ( s ) + ( J e 0 ) ( s ) ) d s , http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-101/MediaObjects/13661_2012_Article_347_Equaj_HTML.gif
        hence
        0 τ e 0 ( s ) d s ( I 0 τ ξ ( s ) d s ) 1 0 τ e 0 ( s ) d s = 0 τ ( e 0 ( s ) + ( J e 0 ) ( s ) ) d s , http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-101/MediaObjects/13661_2012_Article_347_Equak_HTML.gif
        and
        ( I 0 τ ξ ( s ) d s ) 1 0 τ e 0 ( s ) d s = 0 τ ( J e 0 ) ( s ) d s . http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-101/MediaObjects/13661_2012_Article_347_Equ18_HTML.gif
        (18)
        Finally substituting (18) into (17), we obtain
        ξ ( t ) 0 τ ( J e 0 ) ( s ) d s = e 0 ( t ) + ( J e 0 ) ( t ) . http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-101/MediaObjects/13661_2012_Article_347_Equ19_HTML.gif
        (19)

        By our definition ξ ( t ) x = f , x g ( t ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-101/MediaObjects/13661_2012_Article_347_IEq107_HTML.gif, g ( t ) = 1 β ( e 0 ( t ) + ( J e 0 ) ( t ) ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-101/MediaObjects/13661_2012_Article_347_IEq108_HTML.gif, with f , 0 τ e 0 ( s ) d s = 1 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-101/MediaObjects/13661_2012_Article_347_IEq109_HTML.gif and f , 0 τ ( J e 0 ) ( s ) d s = β http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-101/MediaObjects/13661_2012_Article_347_IEq110_HTML.gif. Therefore, (19) is satisfied and this concludes the proof. □

        Remark 1 Observe that a little though convinces of the existence of τ [ 0 , T ] http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-101/MediaObjects/13661_2012_Article_347_IEq87_HTML.gif and f E http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-101/MediaObjects/13661_2012_Article_347_IEq86_HTML.gif satisfying the conditions (H4)-(H6) of Lemma 1.

        Furthermore, recall that assumptions (H1)-(H2) ensure that the operator I P http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-101/MediaObjects/13661_2012_Article_347_IEq76_HTML.gif is condensing with constant k / α 0 < 1 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-101/MediaObjects/13661_2012_Article_347_IEq111_HTML.gif (see [40]); moreover, [[38], Theorem 1.5.4] guarantees that I P ( x 0 ( θ ) ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-101/MediaObjects/13661_2012_Article_347_IEq75_HTML.gif is also condensing with the same constant. Finally, by [[38], Theorem 2.6.11], zero turns out to be an eigenvalue of P ( x 0 ( θ ) ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-101/MediaObjects/13661_2012_Article_347_IEq39_HTML.gif 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 ξ ( θ ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-101/MediaObjects/13661_2012_Article_347_IEq112_HTML.gif associated to (8)
        M ξ ( θ ) : = 0 T Q ˜ ( x 0 ( θ ) , 0 ) ( t ) , z 0 ( t + θ ) d t 0 T ξ ( t ) 0 τ Q ˜ ( x 0 ( θ ) , 0 ) ( s ) d s , z 0 ( t + θ ) d t , http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-101/MediaObjects/13661_2012_Article_347_Equal_HTML.gif
        where z 0 ( θ ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-101/MediaObjects/13661_2012_Article_347_IEq31_HTML.gif is an eigenvector of ( P ( x 0 ( θ ) ) ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-101/MediaObjects/13661_2012_Article_347_IEq113_HTML.gif, i.e.,
        ( P ( x 0 ( θ ) ) ) z 0 ( θ ) = 0 . http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-101/MediaObjects/13661_2012_Article_347_Equam_HTML.gif

        For notational convenience, we simply denote z 0 ( θ ) C T ( E ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-101/MediaObjects/13661_2012_Article_347_IEq114_HTML.gif by z 0 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-101/MediaObjects/13661_2012_Article_347_IEq5_HTML.gif. In order to compute M ξ ( θ ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-101/MediaObjects/13661_2012_Article_347_IEq112_HTML.gif, it is necessary to determine z 0 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-101/MediaObjects/13661_2012_Article_347_IEq5_HTML.gif in explicit form. The following result solves the problem.

        Lemma 2 Assume (H1)-(H6), we have that
        z 0 ( t ) = γ ( t ) 1 [ 0 , τ ] ( t ) 0 T g ( t ) , γ ( t ) f d t , http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-101/MediaObjects/13661_2012_Article_347_Equ59_HTML.gif

        where γ is an eigenvector of ( J α ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-101/MediaObjects/13661_2012_Article_347_IEq115_HTML.gif corresponding to the eigenvalue 1 and 1 [ 0 , τ ] http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-101/MediaObjects/13661_2012_Article_347_IEq194_HTML.gif is the characteristic function of the interval [ 0 , τ ] http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-101/MediaObjects/13661_2012_Article_347_IEq98_HTML.gif.

        Proof By assumption, E has the Radon-Nikodym property, then the eigenvalue of ( P ( x 0 ( θ ) ) ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-101/MediaObjects/13661_2012_Article_347_IEq113_HTML.gif can be determined, without loss of generality in the Hilbert space H T 1 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-101/MediaObjects/13661_2012_Article_347_IEq117_HTML.gif. Hence, for any x , y C T ( E ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-101/MediaObjects/13661_2012_Article_347_IEq118_HTML.gif we have
        0 T P ( x 0 ( θ ) ) x ( t ) , y ( t ) d t = 0 T x ( t ) ( J ( α x ) ) ( t ) ξ ( t ) 0 τ ( x ( s ) ( J ( α x ) ) ( s ) ) d s , y ( t ) d t . http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-101/MediaObjects/13661_2012_Article_347_Equao_HTML.gif
        First, by using assumption (H1), we calculate
        0 T ( J ( α x ) ) ( t ) , y ( t ) d t = 0 T e A t ( I e A T ) 1 0 T e A ( T s ) a ( s ) x ( s ) d s + 0 t e A ( t s ) a ( s ) x ( s ) d s , y ( t ) d t = 0 T 0 T e A ( T s ) a ( s ) x ( s ) d s , ( I e A T ) 1 e A t y ( t ) d t + 0 T 0 t e A ( t s ) a ( s ) x ( s ) d s , y ( t ) d t = 0 T 0 T e A ( T s ) a ( s ) x ( s ) , ( I e A T ) 1 e A t y ( t ) d s d t + 0 T 0 t e A ( t s ) a ( s ) x ( s ) , y ( t ) d s d t = 0 T 0 T e A s a ( s ) x ( s ) , e A T ( I e A T ) 1 e A t y ( t ) d t d s + 0 T s T e A s a ( s ) x ( s ) , e A t y ( t ) d t d s = 0 T x ( s ) , 0 T a ( s ) e A ( T s ) ( I e A T ) 1 e A t y ( t ) d t d s + 0 T x ( s ) , s T a ( s ) e A ( t s ) y ( t ) d t d s . http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-101/MediaObjects/13661_2012_Article_347_Equap_HTML.gif
        Therefore, we have
        ( ( J α ) y ) ( t ) = 0 T a ( t ) e A ( T t ) ( I e A T ) 1 e A s y ( s ) d s + t T a ( t ) e A ( s t ) y ( s ) d s . http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-101/MediaObjects/13661_2012_Article_347_Equ20_HTML.gif
        (20)
        We now calculate the adjoint operator for ξ 0 τ ( x ( s ) ( J ( α x ) ) ( s ) ) d s http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-101/MediaObjects/13661_2012_Article_347_IEq119_HTML.gif. For this, consider
        0 T ξ ( t ) 0 τ ( x ( s ) ( J ( α x ) ) ( s ) ) d s , y ( t ) d t = 0 T 0 τ ( x ( s ) ( J ( α x ) ) ( s ) ) d s , ξ ( t ) y ( t ) d t = 0 T 0 T 1 [ 0 , τ ] ( s ) ( x ( s ) ( J ( α x ) ) ( s ) ) d s , ξ ( t ) y ( t ) d t = 0 T 1 [ 0 , τ ] ( s ) ( x ( s ) ( J ( α x ) ) ( s ) ) , 0 T ξ ( t ) y ( t ) d t d s = 0 T x ( s ) ( J ( α x ) ) ( s ) , 1 [ 0 , τ ] ( s ) 0 T ξ ( t ) y ( t ) d t d s = 0 T x ( s ) , 1 [ 0 , τ ] ( s ) 0 T ξ ( t ) y ( t ) d t d s 0 T x ( s ) , ( J α ) 1 [ 0 , τ ] ( s ) 0 T ξ ( t ) y ( t ) d t d s = 0 T x ( s ) , 1 [ 0 , τ ] ( s ) 0 T ξ ( t ) y ( t ) d t ( J α ) 1 [ 0 , τ ] ( s ) 0 T ξ ( t ) y ( t ) d t d s . http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-101/MediaObjects/13661_2012_Article_347_Equ63_HTML.gif
        Thus,
        ( ξ ( t ) 0 τ ( y ( s ) ( ( J α ) y ) ( s ) ) d s ) = 1 [ 0 , τ ] ( t ) 0 T ξ ( t ) y ( t ) d t ( J α ) 1 [ 0 , τ ] ( t ) 0 T ξ ( t ) y ( t ) d t . http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-101/MediaObjects/13661_2012_Article_347_Equ64_HTML.gif
        Finally, we calculate ξ ( t ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-101/MediaObjects/13661_2012_Article_347_IEq120_HTML.gif, for this consider
        ξ ( t ) x , y = f , x g ( t ) , y = f , x g ( t ) , y = x , g ( t ) , y f , http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-101/MediaObjects/13661_2012_Article_347_Equas_HTML.gif
        i.e.
        ξ ( t ) y = g ( t ) , y f . http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-101/MediaObjects/13661_2012_Article_347_Equat_HTML.gif
        Now, we are in the position to determine the eigenvector z 0 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-101/MediaObjects/13661_2012_Article_347_IEq5_HTML.gif of ( P ( x 0 ( θ ) ) ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-101/MediaObjects/13661_2012_Article_347_IEq113_HTML.gif. We have that
        z 0 ( t ) = ( J α ) z 0 ( t ) + 1 [ 0 , τ ] ( t ) 0 T g ( s ) , z 0 ( s ) f d s ( J α ) 1 [ 0 , τ ] ( t ) 0 T g ( s ) , z 0 ( s ) f d s . http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-101/MediaObjects/13661_2012_Article_347_Equ67_HTML.gif
        Then
        z 0 ( t ) 1 [ 0 , τ ] ( t ) 0 T g ( s ) , z 0 ( s ) f d s = ( J α ) ( z 0 ( t ) 1 [ 0 , τ ] ( t ) 0 T g ( s ) , z 0 ( s ) f d s ) . http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-101/MediaObjects/13661_2012_Article_347_Equ68_HTML.gif
        Let γ ( t ) : = z 0 ( t ) 1 [ 0 , τ ] ( t ) 0 T g ( s ) , z 0 ( s ) f d s http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-101/MediaObjects/13661_2012_Article_347_IEq203_HTML.gif, then the previous equation takes the form
        γ = ( J α ) γ , http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-101/MediaObjects/13661_2012_Article_347_Equ21_HTML.gif
        (21)
        namely γ is an eigenvector of the linear operator ( J α ) : C T ( E ) C T ( E ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-101/MediaObjects/13661_2012_Article_347_IEq122_HTML.gif corresponding to the simple eigenvalue 1. Therefore, by replacing y with γ in (20), we obtain
        γ ( t ) = 0 T a ( t ) e A ( T t ) ( I e A T ) 1 e A s γ ( s ) d s + t T a ( t ) e A ( s t ) γ ( s ) d s . http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-101/MediaObjects/13661_2012_Article_347_Equaw_HTML.gif
        The aim now is to find γ, for this consider the adjoint equation to (5)
        v ˙ ( t ) = A v ( t ) a ( t ) v ( t ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-101/MediaObjects/13661_2012_Article_347_Equ22_HTML.gif
        (22)
        and the solution of (22), defined for t [ 0 , T ] http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-101/MediaObjects/13661_2012_Article_347_IEq101_HTML.gif, given by
        v ( t ) = e A ( t T ) v ( T ) + T t e A ( t s ) ( a ( s ) ) v ( s ) d s . http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-101/MediaObjects/13661_2012_Article_347_Equ23_HTML.gif
        (23)
        For t = 0 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-101/MediaObjects/13661_2012_Article_347_IEq123_HTML.gif, we have
        v ( 0 ) = e A T v ( T ) + 0 T e A s a ( s ) v ( s ) d s . http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-101/MediaObjects/13661_2012_Article_347_Equax_HTML.gif
        Since v is T-periodic, we obtain
        v ( T ) = ( I e A T ) 1 0 T e A s a ( s ) v ( s ) d s . http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-101/MediaObjects/13661_2012_Article_347_Equ24_HTML.gif
        (24)
        By using (24) into (23), we get
        v ( t ) = e A ( T t ) ( I e A T ) 1 0 T e A s a ( s ) v ( s ) d s + t T e A ( t s ) a ( s ) v ( s ) d s . http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-101/MediaObjects/13661_2012_Article_347_Equ25_HTML.gif
        (25)
        Put ω ( t ) : = v ˙ ( t ) A v ( t ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-101/MediaObjects/13661_2012_Article_347_IEq124_HTML.gif, then ω ( t ) = a ( t ) v ( t ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-101/MediaObjects/13661_2012_Article_347_IEq125_HTML.gif and (25) becomes
        v ( t ) = e A ( T t ) ( I e A T ) 1 0 T e A s ω ( s ) d s + t T e A ( s t ) ω ( s ) d s . http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-101/MediaObjects/13661_2012_Article_347_Equay_HTML.gif
        Therefore,
        ω ( t ) = 0 T a ( t ) e A ( T t ) ( I e A T ) 1 e A s ω ( s ) d s + t T a ( t ) e A ( s t ) ω ( s ) d s , http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-101/MediaObjects/13661_2012_Article_347_Equaz_HTML.gif
        i.e., ω ( t ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-101/MediaObjects/13661_2012_Article_347_IEq126_HTML.gif is a solution to (21). Hence,
        γ ( t ) = v ˙ ( t ) A v ( t ) , http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-101/MediaObjects/13661_2012_Article_347_Equba_HTML.gif
        where v ( t ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-101/MediaObjects/13661_2012_Article_347_IEq58_HTML.gif is the T-periodic solution to the adjoint equation (22). Finally, from
        z 0 ( t ) 1 [ 0 , τ ] ( t ) 0 T g ( s ) , z 0 ( s ) f d s = γ ( t ) , http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-101/MediaObjects/13661_2012_Article_347_Equ79_HTML.gif
        we obtain
        0 T g ( t ) , z 0 ( t ) d t 0 T 1 [ 0 , τ ] ( t ) 0 T g ( s ) , z 0 ( s ) d s g ( t ) , f d t = 0 T g ( t ) , γ ( t ) d t , http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-101/MediaObjects/13661_2012_Article_347_Equ80_HTML.gif
        and
        0 T g ( t ) , z 0 ( t ) d t 0 T g ( s ) , z 0 ( s ) d s 0 τ g ( t ) d t , f = 0 T g ( t ) , γ ( t ) d t . http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-101/MediaObjects/13661_2012_Article_347_Equbd_HTML.gif
        By (9), we get
        0 T g ( t ) , z 0 ( t ) d t = 0 T g ( t ) , γ ( t ) d t . http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-101/MediaObjects/13661_2012_Article_347_Eqube_HTML.gif
        In conclusion,
        z 0 ( t ) = γ ( t ) 1 [ 0 , τ ] ( t ) 0 T g ( t ) , γ ( t ) f d t . http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-101/MediaObjects/13661_2012_Article_347_Equ83_HTML.gif

         □

        Lemmas 1 and 2, together with the fact that (H1)-(H2) ensure the condensivity of I P ( x 0 ( θ ) ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-101/MediaObjects/13661_2012_Article_347_IEq75_HTML.gif, θ [ 0 , T ] http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-101/MediaObjects/13661_2012_Article_347_IEq28_HTML.gif, of constant 0 < k / α 0 < 1 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-101/MediaObjects/13661_2012_Article_347_IEq127_HTML.gif (see [40]) allow to apply Theorem 1 to state the following.

        Theorem 2 Assume (H1)-(H6). If there exists θ 0 [ 0 , T ] http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-101/MediaObjects/13661_2012_Article_347_IEq128_HTML.gif such that M ξ ( θ 0 ) = 0 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-101/MediaObjects/13661_2012_Article_347_IEq129_HTML.gif and M ξ ( θ 0 ) 0 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-101/MediaObjects/13661_2012_Article_347_IEq130_HTML.gif. Then there exists a branch of T-periodic solutions to (3) of the form
        x ( ε ) = x 0 ( θ 0 ) + ε w + o ( ε ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-101/MediaObjects/13661_2012_Article_347_Equbg_HTML.gif

        for ε 0 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-101/MediaObjects/13661_2012_Article_347_IEq3_HTML.gif sufficiently small and w C T ( E ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-101/MediaObjects/13661_2012_Article_347_IEq131_HTML.gif.

        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 θ [ 0 , T ] http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-101/MediaObjects/13661_2012_Article_347_IEq28_HTML.gif, assume that 0 σ ( P ˜ ( x 0 ( θ ) ) ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-101/MediaObjects/13661_2012_Article_347_IEq132_HTML.gif is simple. Then the Malkin bifurcation function M ( θ ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-101/MediaObjects/13661_2012_Article_347_IEq133_HTML.gif associated to system (6) coincide with the Malkin bifurcation function M ξ ( θ ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-101/MediaObjects/13661_2012_Article_347_IEq112_HTML.gif associated to system (8).

        Proof Consider
        M ξ ( θ ) = 0 T Q ˜ ( x 0 ( θ ) , 0 ) ( t ) , z 0 ( t ) d t 0 T ξ ( t ) 0 τ Q ˜ ( x 0 ( θ ) , 0 ) ( s ) d s , z 0 ( t ) d t = 0 T Q ˜ ( x 0 ( θ ) , 0 ) ( t ) , γ ( t ) d t 0 T Q ˜ ( x 0 ( θ ) , 0 ) ( t ) , 1 [ 0 , τ ] ( t ) 0 T g ( s ) , γ ( s ) d s f d t 0 T ξ ( t ) 0 τ Q ˜ ( x 0 ( θ ) , 0 ) ( s ) d s , γ ( t ) 1 [ 0 , τ ] ( t ) 0 T g ( s ) , γ ( s ) d s f d t . http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-101/MediaObjects/13661_2012_Article_347_Equ85_HTML.gif
        Let μ : = 0 T g ( s ) , γ ( s ) d s http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-101/MediaObjects/13661_2012_Article_347_IEq134_HTML.gif, then by (9) we have
        M ξ ( θ ) = 0 T Q ˜ ( x 0 ( θ ) , 0 ) ( t ) , γ ( t ) d t 0 T Q ˜ ( x 0 ( θ ) , 0 ) ( t ) , 1 [ 0 , τ ] ( t ) μ f d t 0 T f , 0 τ Q ˜ ( x 0 ( θ ) , 0 ) ( s ) d s g ( t ) , γ ( t ) 1 [ 0 , τ ] ( t ) μ f d t = 0 T Q ˜ ( x 0 ( θ ) , 0 ) ( t ) , γ ( t ) d t 0 T Q ˜ ( x 0 ( θ ) , 0 ) ( t ) , 1 [ 0 , τ ] ( t ) μ f d t 0 T f , 0 τ Q ˜ ( x 0 ( θ ) , 0 ) ( s ) d s g ( t ) , γ ( t ) d t + 0 T f , 0 τ Q ˜ ( x 0 ( θ ) , 0 ) ( s ) d s g ( t ) , 1 [ 0 , τ ] ( t ) μ f d t = 0 T Q ˜ ( x 0 ( θ ) , 0 ) ( t ) , γ ( t ) d t 2 0 T Q ˜ ( x 0 ( θ ) , 0 ) ( t ) , 1 [ 0 , τ ] ( t ) μ f d t + 0 T f , 0 τ Q ˜ ( x 0 ( θ ) , 0 ) ( s ) d s g ( t ) , 1 [ 0 , τ ] ( t ) μ f d t = 0 T Q ˜ ( x 0 ( θ ) , 0 ) ( t ) , γ ( t ) d t 2 0 τ Q ˜ ( x 0 ( θ ) , 0 ) ( t ) , μ f d t + 2 μ f , 0 τ Q ˜ ( x 0 ( θ ) , 0 ) ( s ) d s = 0 T Q ˜ ( x 0 ( θ ) , 0 ) ( t ) , γ ( t ) d t = M ( θ ) . http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-101/MediaObjects/13661_2012_Article_347_Equ86_HTML.gif

         □

        Remark 3 M ( θ ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-101/MediaObjects/13661_2012_Article_347_IEq133_HTML.gif can be rewritten in the classical form of the Malkin bifurcation function f 0 ( θ ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-101/MediaObjects/13661_2012_Article_347_IEq135_HTML.gif for ordinary differential equations as (1) of the Introduction. In fact, consider
        M ( θ ) = 0 T Q ˜ ( x 0 ( θ ) , 0 ) ( t ) , γ ( t ) d t = 0 T ( J Ψ ( x 0 ( θ ) ) ) ( t ) , v ˙ ( t ) A v ( t ) d t = 0 T ( J Ψ ( x 0 ( θ ) ) ) ( s ) , v ˙ ( s ) d s + 0 T ( J Ψ ( x 0 ( θ ) ) ) ( s ) , A v ( s ) d s = ( J Ψ ( x 0 ( θ ) ) ) ( t ) , v ( t ) | 0 T 0 T A ( J Ψ ( x 0 ( θ ) ) ) ( t ) + Ψ ( x 0 ( θ ) ) ( t ) , v ( t ) d t + 0 T A ( J Ψ ( x 0 ( θ ) ) ) ( t ) , v ( t ) d t . http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-101/MediaObjects/13661_2012_Article_347_Equbj_HTML.gif
        Since ( J Ψ ( x 0 ( θ ) ) ) ( 0 ) = ( J Ψ ( x 0 ( θ ) ) ) ( T ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-101/MediaObjects/13661_2012_Article_347_IEq136_HTML.gif, v ( 0 ) = v ( T ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-101/MediaObjects/13661_2012_Article_347_IEq137_HTML.gif, Ψ is the superposition operator generated by ψ and v solves (22) we have
        M ( θ ) = 0 T ψ ( t , x 0 ( t + θ ) , 0 ) , v ( t + θ ) d t . http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-101/MediaObjects/13661_2012_Article_347_Equbk_HTML.gif

        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 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-101/MediaObjects/13661_2012_Article_347_IEq138_HTML.gif-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
        2 u 1 t 2 = Δ x u 1 + 1 λ 2 2 u 1 y 2 β 1 u 1 t α 1 u 1 + ϕ 1 ( x , λ y , u 1 , u 2 ) , 2 u 2 t 2 = Δ x u 2 + 1 λ 2 2 u 2 y 2 β 2 u 2 t α 2 u 2 + ϕ 2 ( x , λ y , u 1 , u 2 ) , u 1 ν = u 2 ν = 0 , http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-101/MediaObjects/13661_2012_Article_347_Equ26_HTML.gif
        (26)

        where ( x , y ) Q : = Ω × ( 0 , 1 ) R n + 1 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-101/MediaObjects/13661_2012_Article_347_IEq139_HTML.gif, n 2 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-101/MediaObjects/13661_2012_Article_347_IEq140_HTML.gif, Ω is a C 2 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-101/MediaObjects/13661_2012_Article_347_IEq141_HTML.gif-smooth bounded domain in R n http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-101/MediaObjects/13661_2012_Article_347_IEq142_HTML.gif, 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, α 1 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-101/MediaObjects/13661_2012_Article_347_IEq143_HTML.gif, α 2 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-101/MediaObjects/13661_2012_Article_347_IEq144_HTML.gif, β 1 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-101/MediaObjects/13661_2012_Article_347_IEq145_HTML.gif, β 2 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-101/MediaObjects/13661_2012_Article_347_IEq146_HTML.gif are positive constants and the functions ϕ 1 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-101/MediaObjects/13661_2012_Article_347_IEq147_HTML.gif, ϕ 2 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-101/MediaObjects/13661_2012_Article_347_IEq148_HTML.gif are of class C 1 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-101/MediaObjects/13661_2012_Article_347_IEq149_HTML.gif jointly in their arguments. The linear part of system (26) generates an exponentially stable C 0 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-101/MediaObjects/13661_2012_Article_347_IEq138_HTML.gif-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 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-101/MediaObjects/13661_2012_Article_347_IEq150_HTML.gif-periodic solution u 0 = ( u 1 0 , u 2 0 ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-101/MediaObjects/13661_2012_Article_347_IEq151_HTML.gif of the limit problem, obtained as λ 0 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-101/MediaObjects/13661_2012_Article_347_IEq152_HTML.gif, and suitable conditions on the growth of the derivatives of ϕ 1 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-101/MediaObjects/13661_2012_Article_347_IEq147_HTML.gif, ϕ 2 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-101/MediaObjects/13661_2012_Article_347_IEq148_HTML.gif with respect to their arguments, it is shown in [44] that [[43], Theorem 1] applies. This result guarantees the existence of λ 0 > 0 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-101/MediaObjects/13661_2012_Article_347_IEq153_HTML.gif such that, for fixed λ ˆ ( 0 , λ 0 ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-101/MediaObjects/13661_2012_Article_347_IEq154_HTML.gif, system (26) has an isolated T λ ˆ http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-101/MediaObjects/13661_2012_Article_347_IEq155_HTML.gif-periodic solution u λ ˆ = ( u 1 λ ˆ , u 2 λ ˆ ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-101/MediaObjects/13661_2012_Article_347_IEq156_HTML.gif. The crucial assumption of [[43], Theorem 1] is that the zero eigenvalue of the linearized system around u 0 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-101/MediaObjects/13661_2012_Article_347_IEq157_HTML.gif 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 λ ˆ http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-101/MediaObjects/13661_2012_Article_347_IEq155_HTML.gif-periodic perturbation of (26) of small amplitude ε > 0 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-101/MediaObjects/13661_2012_Article_347_IEq158_HTML.gif
        2 u 1 t 2 = Δ x u 1 + 1 λ ˆ 2 2 u 1 y 2 β 1 u 1 t α 1 u 1 + ϕ 1 ( x , λ ˆ y , u 1 , u 2 ) 2 u 1 t 2 = + ε ψ 1 ( ε , t , x , λ ˆ y , u 1 , u 2 ) , 2 u 2 t 2 = Δ x u 2 + 1 λ ˆ 2 2 u 2 y 2 β 2 u 2 t α 2 u 2 + ϕ 2 ( x , λ ˆ y , u 1 , u 2 ) 2 u 2 t 2 = + ε ψ 2 ( ε , t , x , λ ˆ y , u 1 , u 2 ) , u 1 ν = u 2 ν = 0 . http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-101/MediaObjects/13661_2012_Article_347_Equ27_HTML.gif
        (27)

        If we assume that the superposition operators generated by the functions ϕ 1 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-101/MediaObjects/13661_2012_Article_347_IEq147_HTML.gif, ϕ 2 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-101/MediaObjects/13661_2012_Article_347_IEq148_HTML.gif, ψ 1 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-101/MediaObjects/13661_2012_Article_347_IEq159_HTML.gif, ψ 2 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-101/MediaObjects/13661_2012_Article_347_IEq160_HTML.gif satisfy assumption (H2) of this paper, then (H1) and [[47], Theorem 4.3.1] ensure that the C 0 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-101/MediaObjects/13661_2012_Article_347_IEq138_HTML.gif-semigroup generated by the linearization around u λ ˆ http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-101/MediaObjects/13661_2012_Article_347_IEq161_HTML.gif of the unperturbed system, corresponding to ε = 0 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-101/MediaObjects/13661_2012_Article_347_IEq21_HTML.gif in (27), is strongly contractive with respect to the Hausdorff measure of noncompactness χ ( ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-101/MediaObjects/13661_2012_Article_347_IEq51_HTML.gif, i.e., χ-strongly contractive. Therefore, our abstract bifurcation result Theorem 2 applies to system (27).

        Declarations

        Acknowledgements

        Dedicated to professor Jean Mawhin on the occasion of his seventieth birthday.

        The first two authors acknowledge the support by RFBR Grants 10-01-93112-a and 12-01-0392-a. 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.

        Authors’ Affiliations

        (1)
        Department of Mathematics, Voronezh State University
        (2)
        Dipartimento di Ingegneria dell’Informazione e Scienze Matematiche, Università di Siena

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