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 [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 ), ..., 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 [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 ) = ∫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