Periodic solutions of delay differential systems via Hamiltonian systems

Nonlinear Analysis 102 (2014) 159–167

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Periodic solutions of delay differential systems via Hamiltonian systems Qi Wang a , Chungen Liu b,∗ a

Institute of Contemporary Mathematics, School of Mathematics and Information Science, Henan University, Kaifeng 475000, PR China

b

School of Mathematics and LPMC, Nankai University, Tianjin 300071, PR China

article

info

Article history: Received 30 December 2013 Accepted 13 February 2014 Communicated by S. Carl

abstract With a new definition of the (J , M )-index theory developed in Liu (2006), we give another application of this index theory in delay differential systems via Hamiltonian systems, where we do not need the potential function to be C 2 continuous. © 2014 Elsevier Ltd. All rights reserved.

MSC: 58F05 58E05 34C25 58F10 Keywords: Hamiltonian system Boundary value problem Delay differential system

1. Introduction and main results In this paper, we will consider the following delay differential systems



x˙ (t ) = ∇x V (t , x(t − τ )), x(t ) = x(t + 4τ ),

(DDS.1)

where the function V ∈ C 1 (R × RN , R) is τ -periodic in variable t and is even in variables x. This is the follow-up work of [1], where the second author of this paper developed a new method to solve the delay differential systems via Hamiltonian systems and the (J , M )-index theory developed in [2]. The purpose of this paper has two aspects. Firstly, we will give another definition of the (J , M )-index and establish the relationship between this index and the Ekeland type of index developed by Y. Dong in [3]. Secondly, we will give some applications of this (J , M )-index theory as the supplement of [1]. For details, in the remaining part of this section, we will give a brief introduction of differential delay systems (DDS), the M-boundary problem of a Hamiltonian systems (HS) and the (J , M )-index theory. We will display the relationship between systems (DDS) and (HS). With the (J , M )-index we will state our main results Theorems 1.2 and 1.3, where we do not need the function V to be C 2 continuous. In Section 2, after some preliminaries of variational setting, we will give another equivalent definition of the (J , M )-index by Fredholm operator theory and the relative Morse index. In Section 3, we will prove our main results, where we need the concept of spectral flow to display the relationship between (J , M )-index and

∗

Corresponding author. Tel.: +86 22 23501233. E-mail addresses: [email protected] (Q. Wang), [email protected] (C. Liu).

http://dx.doi.org/10.1016/j.na.2014.02.008 0362-546X/© 2014 Elsevier Ltd. All rights reserved.

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the index theory developed in [3]. Lastly, in Section 4, we claim that the (J , M )-index theory can also be used to consider other delay differential systems, such as the examples in [1] and other delay differential systems. One is the following first order delay differential systems:



x˙ (t ) = ∇y V (t , x(t ), x(t − τ )), x(t ) = x(t + 4τ ),

(DDS.2)

where V = V (t , x, y) ∈ C 1 (R × RN × RN , R) satisfying τ -periodic in t and V (t , x, y) = V (t , y, −x). The other is the following second order delay differential systems:



x¨ (t ) = ∇x V (t , x(t − τ )), x(t ) = x(t + 2τ ),

(DDS.3)

where V = V (t , x) ∈ C 1 (R × RN ), satisfying τ -periodic in t. 1.1. Background and related work Delay models usually appear in some biological modeling. They have been used to describe several aspects of infectious disease dynamics: primary infection [4], drug therapy [5] and immune response [6], to name a few. Delays have also appeared in the study of chemostat models [7], circadian rhythms [8], epidemiology [9], the respiratory system [10], tumor growth [11] and neural networks [12]. Delay effects even appeared in the population dynamics of many species [13,14]. In 1974, Kaplan and Yorke in [15] studied the autonomous delay differential equation as (DDS.1) and introduced a new technique for establishing the existence of periodic solutions. More precisely, the authors of [15] considered the periodic solutions of the following kinds of delay differential equations x˙ (t ) = f (x(t − 1)), and x˙ (t ) = f (x(t − 1)) + f (x(t − 2)), with odd function f . They turned their problems into the problems of periodic solution of autonomous Hamiltonian system and under some twisted condition on the origin and infinity for the function f , it was proved that there exists an energy surface of the Hamiltonian function containing at least one periodic solution. Since then many papers (see [16–20] and the references therein) used Kaplan and Yorke’s original idea to search for periodic solutions of more general differential delay equations of the following form x˙ (t ) = f (x(t − 1)) + f (x(t − 2)) + · · · + f (x(t − m + 1)). The existence of periodic solutions of above delay differential equation has been investigated by Nussbaum in [21] using different techniques. Recently, many results on delay differential systems were obtained, the readers may refer to the Refs. [22–29] and the references therein. The readers can also refer [30] for systematic introduction on delay differential equations. 1.2. M -boundary problem of a Hamiltonian system and (J , M )-index theory For a skew-symmetric non-degenerate 2N × 2N matrix J = (aij ), it can define a symplectic structure on R2N by

ω(v, w) = v T · J −1 · w  i ,j −1 or ω = 21 = (a)ij . A 2N × 2N matrix M is called J -symplectic if there holds MT · J −1 · M = J −1 . i,j a dxi ∧ dxj with J We denote the set of all J -symplectic matrices by SpJ (2N ). The usual symplectic group Sp(2N ) is the special case of   0 −IN J = JN = I , i.e., Sp(2N ) = SpJN (2N ). Here IN is the N × N identity matrix. We will write J for JN if the dimension 0 N 2N is clear from the text. For a J -symplectic matrix M with M k = I2N and a function H ∈ C 2 (R × R2N ) with H (t + τ , M z ) = H (t , z ), we consider kτ -periodic solution of the following M -boundary value problem z˙ (t ) = J ∇ H (t , z (t )), z (τ ) = M z (0).



(HS)

The linearized system along a solution z (t ) of the nonlinear Hamiltonian system in (HS) is the following linear Hamiltonian system y˙ (t ) = J H ′′ (t , z (t ))y(t ). Its fundamental solution γz (t ) with γz (0) = I2N should satisfy

γ˙z (t ) = J H ′′ (t , z (t ))γz (t ). The following result is well known and the proof is omitted.

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161

Lemma 1.1. γz is a J -symplectic path, i.e., γz (t )T J −1 γz (t ) = J −1 for all t ∈ R. For simplicity we take τ = 1. For a symmetric continuous matrix function B(t ) satisfying M T B(t + 1)M = B(t ), suppose γB (t ) is the fundamental solution of the linear Hamiltonian system y˙ (t ) = J B(t )y(t ).

(1.1)

For the standard case of J = JN and a matrix P ∈ SpJN (2N ), the (J , P )-nullity and Maslov-type index of a symmetric matrix function B was first defined in [2] by algebra method. For the general linear Hamiltonian systems (1.1), the index J J pair (iM (B), νM (B)) ∈ Z × {0, 1, . . . , 2N } was defined by the second author of this paper in [1] for all symmetric continuous J

J

matrix function B(t ) satisfying M T B(t + 1)M = B(t ). In the next section, we will give another definition of (iM (B), νM (B)) by the relative Morse index. 1.3. Delay differential systems For simplicity, as in [15] we first consider the 4τ -periodic solutions of the following delay differential systems



x˙ (t ) = ∇ V (t , x(t − τ )), x(t ) = x(t + 4τ ),

(DDS.1)

where the function V ∈ C 1 (R × RN , R) is τ -periodic in variable t and is even in variables x. To find 4τ -periodic solution x(t ), we only need to find solution with x(t + 2τ ) = −x(t ). If x(t ) is such a solution, let x1 (t ) = x(t ), x2 (t ) = x(t − τ ) and z (t ) = (x1 (t ), x2 (t ))T , then there holds



x′1 (t ) = ∇ V (t , x2 (t )), x′2 (t ) = −∇ V (t , x1 (t )).

(1.2)

Set H (t , x1 , x2 ) = V (t , x1 ) + V (t , x2 ), then we can rewrite (1.2) as z˙ (t ) = JN ∇ H (t , z (t )), where JN =



0 −IN

IN 0



(1.3)

as defined above. Moreover, if z (t ) = (x1 (t ), x2 (t )) is a 4τ -periodic solution of (1.3) with z (t ) = T

σ z (t ) for the 4-periodic action σ z (t ) = JN−1 z (t − τ ),

(1.4)

then x(t ) = x1 (t ) is a solution of (DDS.1) with x(t + 2τ ) = −x(t ). Condition (1.4) is equivalent to z (τ ) = JN−1 z (0).

(1.5) −1

So the problem (DDS.1) can be transformed to the problem (HS) with J = JN and M = JN . 1.4. Our main results For simplicity, let τ = 1. Consider the delay differential systems (DDS.1). Let V ∈ C 1 (R × RN , R) be 1-periodic in variable t and even in variables x. Assume there exist 1-periodic symmetric continuous matrix functions B1 and B2 satisfying the following conditions: (1) B1 (t ) − B2 (t ) is semi-positive for all t ∈ R. (2) 21 (B1 (t )x, x) − V (t , x) is convex for (t , x) ∈ R × RN , and for some constant c ∈ R V (t , x) ≥ Denote by Bˆ 1 =



1 2

(B2 (t )x, x) + c ,

B1

0

0

B1



and Bˆ 2 =

∀(t , x) ∈ R × RN .



B2

0

0

B2



. Then we have the following result.

Theorem 1.2. Suppose V (t , x) satisfies conditions (1), (2) and J

J

iM (Bˆ 1 ) = iM (Bˆ 2 ). Then (DDS.1) has a 4-periodic solution x with x(t + 2) = −x(t ). Moreover, if we further assume V satisfying the following conditions: (3) Vx′ (t , 0) ≡ 0 ∀t ∈ R. There exists 1-periodic symmetric continuous matrix function B0 satisfying

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(4) B1 (t ) − B0 (t ) is semi-positive ∀t ∈ R. (5) V (t , x) ≤ 21 (B0 (t )x, x), ∀t ∈ R with ∥x∥ small enough.



Denote by Bˆ 0 =

B0

0

0

B0



, then we have the following result.

Theorem 1.3. Let V (t , x) satisfy the conditions in Theorem 1.2 and above conditions (3)–(5) together with J

J

J

iM (Bˆ 0 ) + νM (Bˆ 0 ) < iM (Bˆ 1 ). Then (DDS.1) has at least one nontrivial 4-periodic solution x with x(t + 2) = −x(t ). 2. Variational setting and a new definition of (J , M )-index J

J

In this section, we will give another definition of (iM (B), νM (B)) via the concept of relative Morse index. For this purpose, the notion of spectral flow will be used. 2.1. Variational setting From the discussion in Section 1.3, in order to solve the delay differential systems (DDS.1), we only need to consider the corresponding Hamiltonian systems (HS) with the so called M -boundary conditions. In generally, for a J -symplectic matrix M with Mk = I2N and a function H ∈ C 1 (R × R2N ) with H (t + τ , M z ) = H (t , z ), we consider kτ -periodic solution of the following M -boundary value problem



z˙ (t ) = J ∇ H (t , z (t )), z (τ ) = M z (0).

(HS)

The corresponding functional ϕ is defined on E = W 1/2,2 (S 1 , R2N ) with S 1 = R/(kτ Z) by

ϕ(z ) =

1

kτ



2

(J −1 z˙ (t ), z (t ))dt −

kτ



0

H (t , z (t ))dt ,

∀z ∈ E .

(2.1)

0

The critical point of ϕ in E is a kτ -periodic solution of the nonlinear Hamiltonian system in (HS). In order to solve the problem (HS), we define a group action σ on E by

σ z (t ) = Mz (t − τ ). It is clear that σ k = id and ϕ is σ -invariant, i.e., there holds

ϕ(σ z ) = ϕ(z ). Setting Eσ = Wσ1/2,2 (S 1 , R2N ) , {z ∈ E |σ z = z } = fix(σ ),

(2.2)

by the well known Palais symmetric principal (see [31]), a critical point of ϕ in Eσ is a solution of the boundary problem (HS). For z ∈ Eσ , there holds

ϕ(z ) =

τ



k 2

(J −1 z˙ (t ), z (t ))dt − k

0

τ



H (t , z (t ))dt .

0

Denote the bounded self-adjoint operator A on Eσ by

(Az , z )E = (J −1 z˙ , z )L2 .

(2.3)

Then A is Fredholm operator on Eσ . Denote the functional Φ : Eσ → R by

Φ (z ) = −

τ



H (t , z (t ))dt .

(2.4)

0

Then ϕ can be written as

ϕ(z ) = k



1 2

 (Az , z )L2 + Φ (z ) ,

∀z ∈ Eσ .

The Hamiltonian systems (HS) with the M -boundary conditions are equivalent to the following operator equation on Eσ , Az + Φ ′ (z ) = 0,

z ∈ Eσ .

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163

2.2. Relative Morse index and a new definition of (J , M )-index In general, let E be a separable Hilbert space, for any self-adjoint operator A on E, there is a unique A-invariant orthogonal splitting

E = E+ (A) ⊕ E− (A) ⊕ E0 (A),

(2.5)

where E0 (A) is the null space of A, A is positive definite on E+ (A) and negative definite on E− (A). We denote by PA the orthogonal projection from E to E− (A). For any bounded self-adjoint Fredholm operator F and a compact self-adjoint operator T on E, PF − PF −T is compact (see Lemma 2.7 of [32]), where PF : E → E− (F ) and PF −T : E → E− (F − T ) are the respective projections. Then by Fredholm operator theory, PF |E− (F −T ) : E− (F − T ) → E− (F ) is a Fredholm operator. Here and in the sequel, we denote by ind(·) the Fredholm index of a Fredholm operator. Definition 2.1. For any bounded self-adjoint Fredholm operator F and a compact self-adjoint operator T on E, the relative Morse index pair (µF (T ), υF (T )) is defined by

µF (T ) = ind(PF |E− (F −T ) )

(2.6)

υF (T ) = dim E0 (F − T ).

(2.7)

and

On the other hand, let {Fθ |θ ∈ [0, 1]} be a continuous path of self-adjoint Fredholm operators on the Hilbert space E. Now, we need the concept of spectral flow. It is well known that the concept of spectral flow Sf (Fθ ) was first introduced by Atiyah, Patodi and Singer in [33], and then extensively studied in [34–37,32]. The following proposition displays the relationship between spectral flow and the relative Morse index defined above. Proposition 2.2 (See [38, Proposition 3]). Suppose that, for each θ ∈ [0, 1], Fθ − F0 is a compact operator on E, then ind(PF0 |E− (F1 ) ) = −Sf (Fθ , 0 ≤ θ ≤ 1). Thus, from Definition 2.1,

µF0 (T ) = −Sf (Fθ , 0 ≤ θ ≤ 1), where Fθ = F − θ T , T is a compact operator. More over, if σ (T ) ⊂ [0, ∞) and 0 ̸∈ σP (T ), from the definition of Spectral flow, we have

µF0 (T ) = −Sf (Fθ , 0 ≤ θ ≤ 1)  = υF (θ T ) θ∈[0,1)

=



dim E0 (F − θ T ).

(2.8)

θ∈[0,1)

Now, we can define the (J , M )-index. For any symmetric matrix function B ∈ C (R, L(R2N )) satisfying M T B(t + 1)M = B(t ), define the self-adjoint operator BM : Eσ → Eσ by

⟨BM x, y⟩ =

τ



(B(t )x(t ), y(t ))dt ,

∀x, y ∈ Eσ .

(2.9)

0

Since the embedding Eσ ↩→ L2 is compact, then BM is a compact operator on Eσ . Now, let E = Eσ and the self-adjoint J J operator A defined by (2.3). Thus by Definition 2.1, we can define our (J , M )-index pair (iM (B), νM (B)) by

J

iM (B) = µA (BM ), J νM (B) = υA (BM ). J

From Definition 2.3 in [1], Proposition 2.2 and (2.8), it is easy to see the index iM (B) defined here coincides with the definition in [2] up to a constant.

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3. Relations with the Ekeland type index and the proofs of main results In order to prove our main results, we need to give the relationship between the (J , M )-index and the Ekeland type index developed by Dong in [3], the key theorem is Theorem 3.1.7 in [3]. As a matter of fact, the so called Ekeland type index is a kind of dual Morse index. The dual action principal in Hamiltonian framework was first established by Clarke in [39–41] and Ekeland [42], and had been adapted by many mathematicians to the study of various variational problems. The dual Morse index theory has been studied deeply for Hamiltonian systems. For examples, the dual Morse index theory for periodic boundary condition was studied by Girardi and Matzeu for the cases of super-quadratic Hamiltonian systems in [43], and by the second author for the sub-quadratic and asymptotical Hamiltonian systems in [44,45] respectively. Now, we give a brief introduction of the Ekeland type index developed by Dong. Let X be an infinite-dimensional separable Hilbert space with inner product (·, ·), and norm ∥ · ∥. Let Y ⊂ X be a Banach space with norm ∥ · ∥Y , and the embedding Y ↩→ X is compact. Let A : Y → X be continuous, self-adjoint, i.e. (Ax, y) = (x, Ay) for any x, y ∈ Y with the inner product of X , Im(A) is a closed subspace of X and Im(A) ⊕ ker(A) = X . Let B ∈ Ls (X ) (the set of bounded self-adjoint operator). Definition 3.1 ([3, Definition 3.1.1]). For any B ∈ Ls (X ), we define

νA (B) = dimker(A + B). It has been proved in [3] that the nullity νA (B) is finite. Definition 3.2 ([3]). For any B1 , B2 ∈ Ls (X ) with B1 < B2 , we define IA (B1 , B2 ) =



νA ((1 − λ)B1 + λB2 );

λ∈[0,1)

and for any B1 , B2 ∈ Ls (X ) we define IA (B1 , B2 ) = IA (B1 , K · Id) − IA (B2 , K · Id), where Id : X → X is the identity map and K · Id > B1 , K · Id > B2 for some real number K > 0. Let 0 ∈ Ls (X ) be the zero operator. We give the following definition for related index. Definition 3.3. For any B ∈ Ls (X ) we define iA (B) = IA (0, B). We call iA (B) index of B related to A. By Proposition 3.1.5 in [3], we have the following result. Proposition 3.4. The following statements hold. (1) (2) (3) (4)

For any B1 , B2 ∈ Ls (X ), IA (B1 , B2 ) and iA (B) are well-defined and finite; For any B1 , B2 , B3 ∈ Ls (X ), IA (B1 , B2 ) + IA (B2 , B3 ) = IA (B1 , B3 ); For any B1 , B2 ∈ Ls (X ), IA (B1 , B2 ) = iA (B2 ) − iA (B1 ); For any B1 , B2 ∈ Ls (X ) with B1 < B2 , νA (B1 ) + iA (B1 ) ≤ iA (B2 ). With this index, Dong considered the following operator equation Ax + Φ ′ (x) = 0,

x ∈ X,

(3.1)

where Φ : X → R is differential. Let Ls (X ; B0 ) denote the subset of Ls (X ), which consists of such elements B with B − B0 positive definite. One of his main results is +

Theorem 3.5 (Theorem 3.1.7 [3]). Assume that 1 (1) there exists B1 ∈ Ls (X ) and B2 ∈ L+ s (X ; B1 ), iA (B1 ) + νA (B1 ) = iA (B2 ), νA (B2 ) = 0 such that Φ (x) − 2 (B1 x, x) is convex and 1 Φ (x) ≤ (B2 x, x) + c , ∀x ∈ X . 2 Then (3.1) has a solution. Moreover, if we further assume that (2) Φ ′ (θ ) = θ , Φ (θ ) = 0 and there exists B0 ∈ L+ s (X ; B1 ) satisfying

1 (B0 x, x), ∀x ∈ X , with ∥x∥ small, 2 iA (B0 ) > iA (B1 ) + νA (B1 ).

Φ (x) ≥

Then (3.1) has at least one nontrivial solution.

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165

Now, let the Hilbert space X be L2σ , where L2σ = {z ∈ L2 (S 1 , R2N )|σ z = z }, and the action σ was defined in Section 2. Let the operator A = J −1 dtd and the Banach space Y = Wσ1,2 , where Wσ1,2 = W 1,2 (S 1 , R2N )



L2σ .

Then we have (A, Y , X ) satisfying the properties stated before Definition 3.1, without confusion, we also denote A the 1/2,2 bounded self-adjoint Fredholm operator on D(|A|1/2 ) = Wσ which is generated by

(Az , z )W 1/2,2 = (J −1 z˙ , z )L2 , σ  1/2,2 where Wσ = W 1/2,2 Wσ1,2 .

∀z ∈ Wσ1/2,2 ,

Lemma 3.6. There exist a constant n0 ∈ Z, such that for any symmetric matrix function B ∈ C (R, L(R2N )) with M T B(t + 1)M = B(t ), we have J

νM (B) = νA (−B), and J

iM (B) + iA (−B) + νA (−B) = n0 . J

J

Proof. From Definitions 2.1, 3.1 and the definition of νM (B), it is easy to see νM (B) = νA (−B). In order to prove the second equality, we only need to prove that for any B1 , B2 ∈ C (R, L(R2N )) with M T Bj (t + 1)M = Bj (t ), i = 1, 2, satisfying J

J

iM (B1 ) + iA (−B1 ) + νA (−B1 ) = iM (B2 ) + iA (−B2 ) + νA (−B2 ).

(3.2)

Let k ∈ R large enough, such that k · id − Bj is positive definite for j = 1, 2, where id is the identity map on X . From the J

definition of iM (B) and the catenation property of spectral flow, we have J

J

iM (k · id) − iM (Bj ) = −Sf (A − θ k · id, 0 ≤ θ ≤ 1) + Sf (A − θ Bj , 0 ≤ θ ≤ 1)

= −Sf (A − Bj − θ (k · id − Bj ), 0 ≤ θ ≤ 1)  J J = νM (Bj ) + νM (Bj + λ(k · id − Bj )), j = 1, 2.

(3.3)

λ∈(0,1)

On the other hand, we have k · id − Bj is positive definite on X . From Definition 3.2 and Proposition 3.4, we have iA (−Bj ) − iA (−k · id) = νA (−k · id) +



νA (Bj − λ(k · id − Bj )),

j = 1, 2

(3.4)

λ∈(0,1)

and from (3.2)–(3.4) we have J

J

iM (Bj ) + iA (−Bj ) + νA (−Bj ) = iM (k · id) + iA (−k · id) + νA (−k · id), The proof is completed.

j = 1, 2.



Lemma 3.7. If V (t , x) satisfies condition (2), then we have

∥∇ V (t , x)∥ ≤ c (∥x∥ + 1),

∀(t , x) ∈ R × RN ,

for some constant c ∈ R. Proof. We prove it indirectly. Assume there are {xn } ⊂ RN , satisfying ∥∇ V (t , xn )∥ ≥ n(∥xn ∥ + 1), n → ∞. Since V is C 1 continuous, we have

∥xn ∥ → ∞,

n → ∞.

(3.5)

Denote by yn = ∇ V (t , xn )/∥∇ V (t , xn )∥, and fn,t (s) =

1 2

(B1 (t )(xn + syn ), (xn + syn )) − V (t , xn + syn ).

Since 12 (B1 (t )x, x) − V (t , x) is convex, we have fn,t (s) is convex, and fn,t (s) ≥ fn,t (0) + fn′,t (0)s

≥ (B1 (t )xn , xn ) − V (t , 0) − (∇ V (t , 0), xn ) + s(B1 (t )xn , yn ) − s(∇ V (t , xn ), yn ).

(3.6)

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Let s = −∥xn ∥, we have fn,t (∥xn ∥) ≥ (B1 (t )xn , xn ) − V (t , 0) − (∇ V (t , 0), xn ) − ∥xn ∥(B1 (t )xn , yn ) + ∥xn ∥∥∇ V (t , xn )∥

≥ (B1 (t )xn , xn ) − V (t , 0) − (∇ V (t , 0), xn ) − ∥xn ∥(B1 (t )xn , yn ) + n∥xn ∥2 . On the other hand, since V (t , x) ≥ 1

fn,t (∥xn ∥) ≤

2

1 2

(3.7)

(B2 (t )x, x) + c, we have

((B1 (t ) − B2 (t ))(xn + ∥xn ∥yn ), (xn + ∥xn ∥yn )) − c .

Thus, from (3.7) and (3.8), we have {xn } is bounded which is contradict to (3.5).

(3.8) 

Now, we can prove our main results Theorems 1.2 and 1.3. As claimed in Section 1.3, let x1 (t ) = x(t ), x2 (t ) = x(t − 1) and z (t ) = (x1 (t ), x2 (t ))T , if x(t ) is a 4-periodic solution satisfying x(t + 2) = −x(t ), then z (t ) is a solution of Hamiltonian systems (HS), with J = JN , M = JN−1 and H (t , z ) = V (t , x1 ) + V (t , x2 ). From Lemma 3.7, if V is C 1 continuous and satisfying condition (2), then H (t , z ) is also C 1 continuous and satisfying

∥∇z H (t , z )∥ ≤ c (∥z ∥ + 1),

∀(t , z ) ∈ R × R2N .

Now, let the functional Φ in (3.1) be defined in (2.4), B0 = −Bˆ 0 , B1 = −Bˆ 1 and B2 = −Bˆ 2 , where Bˆ 0 , Bˆ 1 , Bˆ are defined in Section 1.4. Then by the conditions in Theorems 1.2 and 1.3, we have B2 − B1 and B0 − B1 are positive definite, and by the catenation property of spectral flow displayed in Lemma 3.6, we have iA (B1 ) + νA (B1 ) = iA (B2 ), νA (B2 ) = 0 and iA (B1 ) + νA (B1 ) < iA (B0 ). It is easy to verify other conditions in Theorem 3.5, thus we complete the proofs of our main results. 4. Further results In addition to systems (DDS.1) considered above, we can also deal with other delay differential systems, such as the examples considered in [1]. Additionally, we can consider the following delay differential systems.



x˙ (t ) = Vy (t , x(t ), x(t − τ )), x(t ) = x(t + 4τ ),

(DDS.2)

where V = V (t , x, y) ∈ C 1 (R × RN × RN , R) satisfying τ -periodic in t and V (t , x, y) = V (t , y, −x). To find 4τ -periodic solution x(t ) with x(t + 2τ ) = −x(t ), let z (t ) = (x(t ), x(t − τ ))T , then the solutions of (DDS.2) is corresponding to the solutions of the following Hamiltonian systems



z˙ (t ) = JN ∇ V (t , z ), z (t ) = JN−1 z (t + τ ),

(4.1)

with JN defined above. Thus the problem (DDS.2) can also be turned into systems (HS) with J = JN and M = JN−1 . With the (J , M)-index, we can also consider the existence and multiplicity of the periodic solutions of systems (DDS.2). For examples, we can also get the similar results as Theorems 1.2, 1.3 stated above and Theorems 4.1, 4.2 stated in [1]. In [1], the author also consider the second order delay Hamiltonian systems which can be turned into systems (HS). Here we can deal with another type of second order delay differential systems:



x¨ (t ) = ∇ V (t , x(t − τ )), x(t ) = x(t + 2τ ),

(DDS.3)

where V = V (t , x) ∈ C 1 (R × RN ), satisfying τ -periodic in t. Let E = W 1/2,2 (S 1 , RN ), S 1 = R/(2τ Z). Define unbounded self-adjoint operator A on L2 (S 1 , RN ) by Ax(t ) = x¨ (t + τ ),

(4.2)

then A generates a bounded self-adjoint Fredholm operator on E, without confusion, we also denote it by A, that is

(Ax, y)E , (¨x(t + τ ), y(t ))L2 ,

∀x, y ∈ E .

(4.3)

∀x ∈ E .

(4.4)

The functional Φ on E defined by

Φ (x) = −



2τ

V (t , x(t ))dt ,

0

Thus systems (DDS.3) can be turned into the operator equation Ax + Φ ′ (x) = 0,

x ∈ E.

(4.5)

Then we can used the index theory defined above and consider the existence and multiplicity of the periodic solutions of systems (DDS.3).

Q. Wang, C. Liu / Nonlinear Analysis 102 (2014) 159–167

167

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