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

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## Abstract

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  that Kaplan and Yorke's original idea  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 ), ..., xn ( 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 An is a n × n skew symmetric matrix, and Ψ ( X ) = ( f ( x1 ), f ( x2 ), ..., f ( xn ) )T. The method used in  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  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  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  is used to obtain critical points of φ{symbol} with the required symmetric structure. As in , 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 ) = ∫0x 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 = ( tij ) be the 2 N × 2 N matrix with tij = ( - 1 )i + j for i ≠ j and tij = 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}.

Original language English (US) 40-58 19 Nonlinear Analysis, Theory, Methods and Applications 65 1 https://doi.org/10.1016/j.na.2005.06.012 Published - Jul 1 2006 Yes