Multiple periodic solutions of differential delay equations via Hamiltonian systems (II)

In this paper, we continue our study for periodic solutions of the following differential delay equation{A formula is presented}where f is odd and n {greater than or slanted equal to} 2 is an integer. When n = 2 N with N {greater than or slanted equal to} 1 being an integer, the author proved in [9] that Kaplan and Yorke's original idea [11] can indeed be used to search for periodic solutions of the differential delay Eq. (1.1). More precisely, if the 2 n-periodic solution X ( t ) = ( x₁ ( t ), x₂ ( t ), ..., x_n ( t ) ) of the following system:{A formula is presented}satisfies the symmetric structure{A formula is presented}then x ( t ) = x₁ ( t ) is a 2 n-periodic solution of (1.1) and x ( t - n ) = - x ( t ). Here A_n is a n × n skew symmetric matrix, and Ψ ( X ) = ( f ( x₁ ), f ( x₂ ), ..., f ( x_n ) )^T. The method used in [9] is variational. Periodic solutions of (1.2) are obtained as critical points of a function φ{symbol} over a Hilbert space E. Since f is odd, the system (1.2) possesses a natural symmetry. Therefore the function φ{symbol} is invariant and φ{symbol}^′ is equivariant under a compact group action related to (1.3). This allows the author to find critical points of φ{symbol} on a subspace of E which is invariant under the group action. The pseudo-index theory [7] is applied directly to obtain critical points of φ{symbol} in this subspace, which surely have the required symmetric structure (1.3) and give solutions to (1.1). The goal of this paper is to handle the case when n is odd, i.e. n = 2 N + 1 with N {greater than or slanted equal to} 1 being an integer. In this case, the function φ{symbol} is still invariant about a similar compact group action related to (1.3). But φ{symbol}^′ is not equivariant about this compact group action anymore. Therefore we cannot apply pseudo-index theory [7] directly on the invariant subspace as in the case when n is even. Special treatment is needed. First we characterize the structure of the mentioned invariant subspace and study the behavior of φ{symbol} over this subspace. Then we use Galerkin approximation to construct equivariant pseudo-gradient vector fields and prove a new deformation theorem. Finally, the S¹-index theory [14] is used to obtain critical points of φ{symbol} with the required symmetric structure. As in [9], we have the following conditions on f: ( f 1 ) f ∈ C ( R, R ) is odd and there exist {A formula is presented} ( f 2^± ) | f ( x ) - α_∞ x | is bounded and G_∞ ( x ) → ± ∞ as | x | → ∞, ( f 3^± ) ± G₀ ( x ) > 0 for | x | > 0 being small,where F ( x ) = ∫₀^x f ( y ) d y, and{A formula is presented}. Denote m^- ( · ) and m⁰ ( · ) the functions given by m^- ( t ) = 1 if t < 0, m^- ( t ) = 0, otherwise; m⁰ ( t ) = 1 if t = 0, m⁰ ( t ) = 0, otherwise. {A formulation is presented}. Let M = ( t_ij ) be the 2 N × 2 N matrix with t_ij = ( - 1 )^{i + j} for i ≠ j and t_ij = 2 for i = j. For α_∞ ∈ R and m {greater than or slanted equal to} 1, denote {A formula is presented}Define{A formula is presented}It is easy to see that n_∞ is well defined. Note that if x ( t ) is a ( 4 N + 2 )-periodic solution of (1.1), so is x ( t + θ ) for each θ ∈ S¹ = R / ( ( 4 N + 2 ) Z ). We say that two ( 4 N + 2 )-periodic solutions x ( t ) and y ( t ) are geometrically different, if there is no θ ∈ S¹ such that x ( t + θ ) = y ( t ), ∀ t ∈ R. For convenience, denote # ( 1.1 ) = the number of geometrically different nonconstant periodic solutions of (1.1) which satisfy x ( t - ( 2 N + 1 ) ) = - x ( t ) for all t ∈ R, and{A formula is presented}. Our main result reads as{A formulation is presented}{A formulation is presented}.