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

Consider 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. In 1974, Kaplan and Yorke [11] introduced a new technique which allows them to "reduce the search for periodic solutions of a differential delay equation to the problem of finding periodic solutions for a related system of ordinary differential equations". More precisely, if the solution x ( t ) of (1.1) satisfies x ( t ) = - x ( t - n ), let{A formula is presented}then X ( t ) = ( x₁ ( t ), x₂ ( t ), ..., x_n ( t ) )^T satisfies{A formula is presented}i.e., A_n is a n × n skew symmetric matrix, and Ψ ( X ) = ( f ( x₁ ), f ( x₂ ), ..., f ( x_n ) )^T. In fact, by direct computation, one has the following proposition. {A formulation is presented}. For n = 2, 3, when f ∈ C ( R, R ) is odd, xf ( x ) > 0 for x ≠ 0 and f satisfies some suitable conditions near 0 and ∞, Kaplan and Yorke proved that (1.3) has periodic solutions with the symmetric structure (1.4), which give periodic solutions of (1.1) with period 4 and 6, respectively. They further conjectured that similar result should be true for the general case n {greater than or slanted equal to} 2, i.e. under similar conditions for f, (1.1) has a 2 n-periodic solution x ( t ) satisfying x ( t ) = - x ( t - n ). In 1978, Nussbaum [19] proved the existence of a 2 n-periodic solution for a class of differential delay equations which includes (1.1) as a special case, i.e., Kaplan and Yorke's conjecture was solved. However the method used in [19] is quite different from the one outlined by Kaplan and Yorke in [11]. Recently, in an attempt to reuse Kaplan and Yorke's original idea, Li and He [12-15] were able to translate (1.3) into Hamiltonian systems by changing variables. Then they applied Lyapunov Center Theorem and some known results about convex Hamiltonian systems [18, Theorem 7.2] to obtain 2 n-periodic solutions of (1.3). However, as shown in Proposition 1.1, only solutions of (1.3) with the symmetric structure (1.4) will give solutions to (1.1). Thus more arguments are needed to show that those 2 n-periodic solutions of (1.3) obtained in this way do have the symmetric structure (1.4). In [11], Kaplan and Yorke's proof depends heavily on the condition xf ( x ) > 0 for x ≠ 0 and the fact that the corresponding system (1.3) has solutions with prescribed minimal period. We have to point out that those 2 n-periodic solutions obtained by [18, Theorem 7.2] give no information about the symmetric structure (1.4) or the minimal period. In fact, Example 3.2 shows that (1.3) could have 2 n-periodic solutions without the required symmetric structure (1.4), which certainly will not generate nonconstant solutions of (1.1). See Remark 3.3 for more details. In this paper we shall take a different approach. Since f is odd, system (1.3) possesses a natural symmetry. Periodic solutions of (1.3) are still obtained as critical points of a function φ{symbol} over a Hilbert space E. However, instead of finding critical points of φ{symbol} over E directly (as in [12-15]), we shall work on a subspace of E which has the symmetric structure (1.4). When n is even, the function φ{symbol} is invariant and φ{symbol}^′ is equivariant about a compact group action related to (1.4). This allows us to find critical points of φ{symbol} on a subspace of E which is invariant under the mentioned group action. Then we can apply the pseudo-index theory [7] to obtain periodic solutions in this subspace, which surely have the required symmetric structure (1.4) and give solutions to (1.1). This is carried out in Section 2 of this paper. When n is odd, the function φ{symbol} is still invariant about a similar compact group action related to (1.4). However, φ{symbol}^′ is not equivariant about this compact group action anymore. Therefore we cannot directly apply the same idea as in the case when n is even. In order to overcome the difficulty, we have to construct equivariant pseudo-gradient vector fields and prove a new deformation theorem. Then we can combine Galerkin approximation with the S¹-index theory [18, Chapter 6] to obtain critical points of φ{symbol} with the required symmetric structure. Since the idea used in this part is quite different from the one used in the even case, we shall carry out it in the subsequent paper [9]. From now on in this paper, we always assume that n = 2 N with N {greater than or slanted equal to} 1 being an integer in (1.1). More specifically, 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 ) dy, and{A formula is presented}. Denote m^- ( · ) and m⁰ ( · ) the two 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}. It is easy to see that i ( α, 2 N ) and ν ( α, 2 N ) are well defined. Note that if x ( t ) is a 4 N-periodic solution of (1.1), so is x ( t + θ ) for each θ ∈ S¹ = R / ( 4 N Z ). We say that two 4 N-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 ) = - x ( t ), ∀ t ∈ R, and {A formula is presented}. Our main result reads as: {A formulation is presented}{A formulation is presented}. This paper is organized as follows. In Section 2, we introduce the mentioned group action and characterize the structure of the invariant subspace. Then we use pseudo-index theory to prove Theorem 1.3. Some examples are given in Section 3.