Variational Approaches for the Existence of Multiple Periodic Solutions of Differential Delay Equations

Abstract

The existence of multiple periodic solutions of the following differential delay equation 𝑥^′(𝑡)=−𝑓(𝑥(𝑡−𝑟)) is established by applying variational approaches directly, where 𝑥∈ℝ, 𝑓∈𝐶(ℝ,ℝ) and 𝑟>0 is a given constant. This means that we do not need to use Kaplan and Yorke's reduction technique to reduce the existence problem of the above equation to an existence problem for a related coupled system. Such a reduction method introduced first by Kaplan and Yorke in (1974) is often employed in previous papers to study the existence of periodic solutions for the above equation and its similar ones by variational approaches.

1. Introduction

We are concerned in this paper with the search for 4𝑟-periodic solutions (𝑟>0) of a class of differential delay equations with the following form

𝑥^′(𝑡)=−𝑓(𝑥(𝑡−𝑟)), (1.1) where 𝑥∈ℝ, 𝑓∈𝐶(ℝ,ℝ), and 𝑟>0 is a given constant. Equation (1.1) occurs in variety of applications and describes many interesting types of phenomena. Taking 𝑓(𝑥)=−𝛿+sin𝑥, then (1.1) with the form

𝑥^′(𝑡)=𝛿−sin(𝑥(𝑡−𝑟)),𝑡≥0,𝛿≥0,𝑟≥0 (1.2) arises in the study of phase-locked loops which are widely used in communication systems [1]. Furthermore, many equations occurring in other fields of applications can be changed to the form of (1.1) by changing variables. For example, letting 𝑥(𝑡)=𝑒^𝑢(𝑡)−1, then the equation

𝑥^′(𝑡)=𝛼𝑥(𝑡−1)(1+𝑥(𝑡)) (1.3) can be changed to the form of (1.1). Equation (1.3) was first proposed by Cunningham [2] as a nonlinear population growth model. Later Wright in [3] mentioned it as arising in the application of probability methods to the theory of asymptotic prime number density. For applications of (1.3) and its similar ones on ecology, one may see [4].

Because of extensive applications, (1.1) and (1.3) have been studied by many authors through various methods [1–3, 5–23]. In 1962, Jones in his paper of [13] considered (1.3) and obtained the existence of periodic solutions of (1.3) by applying fixed point theory. Nussbaum [14] also used fixed point theory [24] for the truncated cones of Krasnosel’skii to give an existence result on periodic solutions.

Besides various fixed point theorems, the global Hopf bifurcation theorem for differential delay equations introduced by Chow and Mallet-Paret [25], qualitative theory of ordinary differential equations, some well-known results on the existence of closed orbits for Hamiltonian vector fields, coincidence degree theory introduced by Mawhin [26, 27], and the Poincaré-Bendixson theorem are proved to be very useful tools in searching for periodic solutions of differential delay (1.1) and (1.3).

A different approach for establishing the existence of periodic solutions for the differential delay (1.1) was introduced by Kaplan and Yorke in their paper in [7]. Employing the method, they could reduce the search for periodic solutions of (1.1) to the problem of seeking periodic solutions for a related system of ordinary differential equations, which is called the coupled system to (1.1). Following the reduction idea of Kaplan and Yorke, Li and He [17, 18] were able to translate (1.1) with more than one delay to a coupled Hamiltonian system. Then they used variational approaches to study the coupled Hamiltonian system and obtained some existence results of multiple periodic solutions of the equations. This proves that variational approaches [28, 29] also are very powerful tools to study periodic solutions of (1.1), (1.3), and their similar ones.

Recently, Guo and Yu [19] do not use Kaplan and Yorke’s reduction technique and apply variational methods directly to study the existence of multiple periodic solutions of (1.1) with 𝑥 and 𝑓 being vectors in ℝ^𝑛. That is to say they do not reduce the existence problem of (1.1) to an existence problem of a related coupled Hamiltonian system. By applying the pseudo-index introduced by Benci in [30], they obtained a sufficient condition on the existence and multiplicity of periodic solutions for (1.1). To the author’s knowledge, this is the first time in which the existence of periodic solutions of (1.1) is studied by variational methods directly.

Let us say some words about the two methods. The advantage of direct variational method is that the function 𝑓 could be a vector in ℝ^𝑛 while 𝑓 is only being a scalar in Kaplan and Yorke’s reduction method. But Kaplan and Yorke’s reduction method can deal with (1.1) with more than two delays while direct variational method used by Guo and Yu only admits one delay in (1.1).

Motivated by the work of Guo and Yu, in this paper we will also use variational approaches directly to study the existence of periodic solutions of (1.1). But our arguments are quite different from theirs. Throughout this paper, we make the following assumptions.

(𝐻1)𝑓(𝑥)∈𝐶(ℝ,ℝ), and for any 𝑥∈ℝ, 𝑓(𝑥)+𝑓(−𝑥)=0, that is, 𝑓(𝑥) is odd.(𝐻2)𝑓(𝑥) satisfies

lim_𝑥→0𝑓(𝑥)𝑥=−𝛼₀∈ℝ,lim_𝑥→∞𝑓(𝑥)𝑥=−𝛼_∞∈ℝ. (1.4) In order to state our main result, we need the following definition.

Definition 1.1. For each 𝛼∈ℝ, define 𝜇(𝛼)=^∞_𝑚=1{𝜎⁻[(−1)^𝑚𝜋2𝑟(2𝑚−1)−𝛼]−𝜎⁻((−1)^𝑚)},𝜈(𝛼)=^∞_𝑚=1𝜎⁰[(−1)^𝑚𝜋2𝑟(2𝑚−1)−𝛼], (1.5) where the two functions 𝜎⁻(𝑡) and 𝜎⁰(𝑡) are given by 𝜎⁻(𝑡)=𝜎1,𝑡<0,0,𝑡≥0,⁰(𝑡)=1,𝑡=0,0,𝑡≠0. (1.6)

It is easy to see that 𝜇(𝛼) and 𝜈(𝛼) are well defined. Let 𝜇₀=𝜇(𝛼₀), 𝜇_∞=𝜇(𝛼_∞), 𝜈₀=𝜈(𝛼₀), and 𝜈_∞=𝜈(𝛼_∞). Then our main result states as follows.

Theorem 1.2. Suppose that 𝑓 satisfies (𝐻1) and (𝐻2). Then the following conclusions are true.(i)If 𝛼_∞∉(𝜋/2𝑟)(2ℤ−1), then (1.1) possesses at least max{𝜇₀−𝜇_∞,𝜇_∞−𝜇₀−𝜈₀} nontrivial geometrically different 4𝑟-periodic solutions.(ii)If 𝛼_∞∈(𝜋/2𝑟)(2ℤ−1), |𝑓(𝑥)+𝛼_∞𝑥| is bounded, and ∫^𝑥₀𝑓(𝑦)𝑑𝑦+(1/2)𝛼_∞𝑥² converges to −∞ as |𝑥|→∞, then the conclusion of (i) also holds.

Remark 1.3. We say that two solutions of (1.1) are geometrically different if one cannot be obtained by time rescaling of the other. We will use [31, Theorem 2.4] to prove the main result.

2. Variational Functional on Hilbert Space

In this section, we will construct a variational functional of (1.1) defined on a suitable Hilbert space such that finding 4𝑟-periodic solutions of (1.1) is equivalent to seeking critical points of the functional.

We work in the Hilbert space 𝐻=𝐻^1/2,2(𝑆¹,ℝ) which consists of those functions 𝑥∈𝐿²(𝑆¹,ℝ) having weak derivative 𝑥^′∈𝐿²(𝑆¹,ℝ). The simplest way to introduce this space seems as follows. Each function 𝑥∈𝐿²(𝑆¹,ℝ) has the following Fourier expansion:

𝑥(𝑡)=𝑎₀+^+∞_𝑚=1(𝑎_𝑚𝜋cos2𝑟𝑚𝑡+𝑏_𝑚𝜋sin2𝑟𝑚𝑡), (2.1) where 𝑎_𝑚,𝑏_𝑚∈ℝ. 𝐻 is the set of those functions satisfying

‖𝑥‖²=4𝑟𝑎²₀+2𝑟^+∞_𝑚=1𝑚(𝑎²_𝑚+𝑏²_𝑚)<+∞. (2.2) With this norm ‖⋅‖, 𝐻 is a Hilbert space with the following inner product:

⟨𝑥,𝑦⟩=4𝑟𝑎₀𝑎^′₀+2𝑟^∞_𝑚=1𝑚(𝑎_𝑚𝑎^′_𝑚+𝑏_𝑚𝑏^′_𝑚), (2.3) where 𝑦=𝑎^′₀+∑^+∞_𝑚=1(𝑎^′_𝑚cos(𝜋/2𝑟)𝑚𝑡+𝑏^′_𝑚sin(𝜋/2𝑟)𝑚𝑡).

For each 𝑥∈𝐻, we define a functional 𝐼∶𝐻→ℝ by

𝐼(𝑥)=₀^4𝑟12𝑥^′(𝑡+𝑟)𝑥(𝑡)𝑑𝑡+₀^4𝑟𝐹(𝑥(𝑡))𝑑𝑡, (2.4) where 𝐹(𝑥)=∫^𝑥₀𝑓(𝑦)𝑑𝑦.

By Riesz representation theorem, 𝐻 identifies its dual space 𝐻^∗. Then we define an operator 𝐽∶𝐻→𝐻^∗=𝐻 by extending the bilinear form

⟨𝐽𝑥,𝑦⟩=₀^4𝑟𝑥^′(𝑡+𝑟)𝑦(𝑡)𝑑𝑡,∀𝑥,𝑦∈𝐻. (2.5) In fact, define a map 𝐿∶𝐻×𝐻→𝐻 by 𝐿(𝑥,𝑦)=⟨𝐽𝑥,𝑦⟩. Then 𝐿 is linear in 𝑥 and 𝑦, respectively, and there exists a positive number 𝑀>0 such that

‖𝐿(𝑥,𝑦)‖≤𝑀‖𝑥‖‖𝑦‖. (2.6) Thus 𝐿 is a bilinear form from 𝐻×𝐻→𝐻. So 𝐽 is a bounded linear operator on 𝐻 and ker𝐽=ℝ.

For any 𝑥∈𝐻, define a mapping 𝐾∶𝐻→ℝ as

𝐾(𝑥)=₀^4𝑟𝐹(𝑥(𝑡))𝑑𝑡. (2.7) Then the functional 𝐼 can be rewritten as

1𝐼(𝑥)=2⟨𝐽𝑥,𝑥⟩+𝐾(𝑥),∀𝑥∈𝐻. (2.8) According to a standard argument in [30], one has, for any 𝑥,𝑦∈𝐻,

⟨𝐼^′(𝑥),𝑦⟩=₀^4𝑟12(𝑥^′(𝑡+𝑟)−𝑥^′(𝑡−𝑟))𝑦(𝑡)𝑑𝑡+₀^4𝑟𝑓(𝑥(𝑡))𝑦(𝑡)𝑑𝑡. (2.9) Moreover 𝐾^′∶𝐻→𝐻 is a compact operator defined by

⟨𝐾^′(𝑥),𝑦⟩=₀^4𝑟𝑓(𝑥(𝑡))𝑦(𝑡)𝑑𝑡. (2.10) Our aim is to reduce the existence of periodic solutions of (1.1) to the existence of critical points of 𝐼. For this we introduce a shift operator Γ∶𝐻→𝐻 defined by

Γ𝑥(𝑡)=𝑥(𝑡+𝑟). (2.11) It is easy to compute that Γ is bounded and linear. Moreover Γ is isometric, that is, ‖Γ𝑥‖=‖𝑥‖ and Γ⁴=𝑖𝑑, where 𝑖𝑑 denotes the identity mapping on 𝐻.

Write

𝐸={𝑥∈𝐻∶Γ²𝑥(𝑡)=−𝑥(𝑡)}. (2.12) By a direct computation, solutions of the Euler equation of 𝐼 in 𝐸 are exactly solutions of (1.1).

Lemma 2.1. As a closed subspace of 𝐻, 𝐸 has the following property: 𝐸={𝑥(𝑡)=^+∞_𝑚=1(𝑎_2𝑚−1𝜋cos2𝑟(2𝑚−1)𝑡+𝑏_2𝑚−1𝜋sin2𝑟(2𝑚−1)𝑡)}. (2.13)

Proof. It is easy to see that 𝐸 is a closed subspace of 𝐻. For any 𝑥(𝑡)=𝑎₀+∑^+∞_𝑚=1(𝑎_𝑚cos(𝜋/2𝑟)𝑚𝑡+𝑏_𝑚sin(𝜋/2𝑟)𝑚𝑡)∈𝐸, Γ²𝑥=−𝑥 implies that 𝑥(𝑡+2𝑟)=−𝑥(𝑡). Then one has 𝑎₀=−𝑎₀,𝑎_𝑚=(−1)^𝑚+1𝑎_𝑚,𝑏_𝑚=(−1)^𝑚+1𝑏_𝑚, (2.14) that is, 𝑎_𝑚=𝑏_𝑚=0foreven𝑚. (2.15) Thus, for any 𝑥(𝑡)∈𝐸, 𝑥(𝑡)=^+∞_𝑚=1(𝑎_2𝑚−1𝜋cos2𝑟(2𝑚−1)𝑡+𝑏_2𝑚−1𝜋sin2𝑟(2𝑚−1)𝑡). (2.16) Hence we get the conclusion.

If we restrict 𝐽 on 𝐸, a direct check shows that 𝐽 over 𝐸 is self-adjoint. For any 𝑥∈𝐸, let 𝐽𝑥(𝑡)=∑^+∞_𝑚=1(𝑎^𝐽_2𝑚−1cos(𝜋/2𝑟)(2𝑚−1)𝑡+𝑏^𝐽_2𝑚−1sin(𝜋/2𝑟)(2𝑚−1)𝑡). Let 𝑦=∑^+∞_𝑚=1(𝑎^′_2𝑚cos(𝜋/2𝑟)(2𝑚−1)𝑡+𝑏^′_2𝑚−1sin(𝜋/2𝑟)(2𝑚−1)𝑡). Then

𝑥^′(𝑡+𝑟)=^+∞_𝑚=1[(−1)^𝑚𝜋2𝑟(2𝑚−1)(𝑎_2𝑚−1𝜋cos2𝑟(2𝑚−1)𝑡+𝑏_2𝑚−1𝜋sin2𝑟(2𝑚−1)𝑡)],_04𝑟𝑥^′(𝑡+𝑟)𝑦(𝑡)𝑑𝑡=(−1)^𝑚[𝜋(2𝑚−1)𝑎_2𝑚−1𝑎^′_2𝑚−1+𝜋(2𝑚−1)𝑏_2𝑚−1𝑏^′_2𝑚−1]. (2.17) According to the definition of the inner product in 𝐸, we have

𝑎^𝐽_2𝑚−1=(−1)^𝑚𝜋𝑎2𝑟^2𝑚−1,𝑏^𝐽_2𝑚−1=(−1)^𝑚𝜋𝑏2𝑟^2𝑚−1, (2.18) that is,

𝐽𝑥(𝑡)=^+∞_𝑚=1[(−1)^𝑚𝜋2𝑟(𝑎_2𝑚−1𝜋cos2𝑟(2𝑚−1)𝑡+𝑏_2𝑚−1𝜋sin2𝑟(2𝑚−1)𝑡)]. (2.19)

Let 𝐼|_𝐸 denote the restriction of 𝐼 on 𝐸. Then we have the following lemma.

Lemma 2.2. Critical points of 𝐼|_𝐸 over 𝐸 are critical points of 𝐼 on 𝐻.

Proof. Note that any 𝑥∈𝐸 is 4𝑟periodic and 𝑓 is odd. It is enough for us to prove that ⟨𝐼^′(𝑥),𝑦⟩=0 for any 𝑦∈𝐻 and 𝑥 being a critical point of 𝐼 in 𝐸.
For any 𝑦∈𝐻, we have
⟨Γ²𝐼^′(𝑥),𝑦⟩=⟨Γ²𝐽𝑥,𝑦⟩+⟨Γ²𝐾^′(𝑥),𝑦⟩=⟨𝐽𝑥,Γ⁻²𝑦⟩+⟨𝐾^′(𝑥),Γ⁻²=𝑦⟩_04𝑟𝑥^′(𝑡+𝑟)𝑦(𝑡−2𝑟)𝑑𝑡+₀^4𝑟𝑓(𝑥(𝑡))𝑦(𝑡−2𝑟)𝑑𝑡=−₀^4𝑟𝑥^′(𝑡+𝑟)𝑦(𝑡)𝑑𝑡−₀^4𝑟𝑓(𝑥(𝑡))𝑦(𝑡)𝑑𝑡=⟨−𝐼^′(𝑥),𝑦⟩. (2.20) This yields Γ²𝐼^′(𝑥)=−𝐼^′(𝑥), that is, 𝐼^′(𝑥)∈𝐸.
Suppose that 𝑥(𝑡) is a critical point of 𝐼 in 𝐸. We only need to show that ⟨𝐼^′(𝑥),𝑦⟩=0 for any 𝑦∈𝐻. Writing 𝑦=𝑦₁⊕𝑦₂ with 𝑦₁∈𝐸,𝑦₂∈𝐸^⟂ and noting that 𝐼^′(𝑥)∈𝐸, one has
⟨𝐼^′(𝑥),𝑦⟩=⟨𝐼^′(𝑥),𝑦₁⟩+⟨𝐼^′(𝑥),𝑦₂⟩=0+0=0. (2.21) The proof is complete.

Remark 2.3. By Lemma 2.2, we only need to consider 𝐼|_𝐸. Therefore in the following 𝐼 will be assumed on 𝐸.

For any 𝛼∈ℝ, we define an operator 𝐿_𝛼∶𝐻→𝐻 by

⟨𝐿_𝛼𝑥,𝑦⟩=₀^4𝑟𝛼𝑥(𝑡)𝑦(𝑡)𝑑𝑡. (2.22)

It is not difficult to see that 𝐿 is a bounded self-adjoint linear mapping on 𝐸. Furthermore, for any 𝑥(𝑡)∈𝐸, let 𝐿_𝛼𝑥(𝑡)=∑^+∞_𝑚=1(𝑎^𝐿𝛼_2𝑚−1cos(𝜋/2𝑟)(2𝑚−1)𝑡+𝑏^𝐿𝛼_2𝑚−1sin(𝜋/2𝑟)(2𝑚−1)𝑡). Let 𝑦=∑^+∞_𝑚=1(𝑎^′_2𝑚−1cos(𝜋/2𝑟)(2𝑚−1)𝑡+𝑏^′_2𝑚−1sin(𝜋/2𝑟)(2𝑚−1)𝑡). Then

₀^4𝑟𝛼𝑥(𝑡)𝑦(𝑡)𝑑𝑡=2𝑟𝛼𝑎_2𝑚−1𝑎^′_2𝑚−1+2𝑟𝛼𝑏_2𝑚−1𝑏^′_2𝑚−1. (2.23) Similar to (2.19), we have

𝑎^𝐿𝛼_2𝑚−1=𝛼𝑎2𝑚−1^2𝑚−1,𝑏^𝐿𝛼_2𝑚−1=𝛼𝑏2𝑚−1^2𝑚−1, (2.24) that is,

𝐿_𝛼𝑥(𝑡)=^+∞_𝑚=1𝛼2𝑚−1[𝑎_2𝑚−1𝜋cos2𝑟(2𝑚−1)𝑡+𝑏_2𝑚−1𝜋sin2𝑟(2𝑚−1)𝑡]. (2.25)

Set

𝐸(𝑚)={𝑥∈𝐸∶𝑥(𝑡)=𝑎_2𝑚−1𝜋cos2𝑟(2𝑚−1)𝑡+𝑏_2𝑚−1𝜋sin𝐸2𝑟(2𝑚−1)𝑡},_𝑛=𝐸(1)⊕𝐸(2)⊕⋯⊕𝐸(𝑛). (2.26) Let 𝑃_𝑛 be the orthogonal projection from 𝐸 to 𝐸_𝑛, then one has

𝑃_𝑛𝑥(𝑡)=^𝑛_𝑚=1(𝑎_2𝑚−1𝜋cos2𝑟(2𝑚−1)𝑡+𝑏_2𝑚−1𝜋sin2𝑟(2𝑚−1)𝑡). (2.27) Denote by 𝑀⁺(𝐽−𝐿_𝛼),𝑀⁻(𝐽−𝐿_𝛼), and 𝑀⁰(𝐽−𝐿_𝛼) the positive definite, the negative definite, and the null subspace of the self-adjoint operator 𝐽−𝐿_𝛼, respectively. Then we have the following lemma.

Lemma 2.4. For 𝑚≥1, let 𝜆_𝑚=(−1)^𝑚(𝜋/2𝑟)(2𝑚−1)−𝛼. Then the following conclusions hold: 𝑀⁺(𝐽−𝐿_𝛼)=^∞_{𝑚=1,𝜆𝑚>0}𝑀𝐸(𝑚),₋(𝐽−𝐿_𝛼)=^∞_{𝑚=1,𝜆𝑚<0}𝑀𝐸(𝑚),₀(𝐽−𝐿_𝛼)=^∞_{𝑚=1,𝜆𝑚=0}𝐸(𝑚). (2.28) Moreover for 𝑚 large enough, dim𝑀⁻(𝑃_𝑚(𝐽−𝐿_𝛼)𝑃_𝑚)=2𝜇(𝛼)+2^𝑚_𝑗=1𝜎⁻((−1)^𝑗),dim𝑀⁰(𝑃_𝑚(𝐽−𝐿_𝛼)𝑃_𝑚)=2𝜈(𝛼). (2.29)

Proof. For 𝑥_𝑚=𝑎_2𝑚−1cos(𝜋/2𝑟)(2𝑚−1)𝑡+𝑏_2𝑚−1sin(𝜋/2𝑟)(2𝑚−1)𝑡, consider the following eigenvalue problem: (𝐽−𝐿_𝛼)𝑥_𝑚=𝜆𝑥_𝑚. (2.30) From (2.19) and (2.25), one has [(−1)^𝑚𝜋−𝛼2𝑟2𝑚−1]𝑎_2𝑚−1=𝜆𝑎_2𝑚−1,[(−1)^𝑚𝜋−𝛼2𝑟2𝑚−1]𝑏_2𝑚−1=𝜆𝑏_2𝑚−1. (2.31) The above two equalities show that 𝐽−𝐿_𝛼 is positive definite, negative definite, and null on 𝐸(𝑚) if and only if 𝜆_𝑚=(−1)^𝑚(𝜋/2𝑟)(2𝑚−1)−𝛼 is positive, negative, and zero, respectively.
By Definition 1.1 and for 𝑚 large enough,
dim𝑀⁻(𝑃_𝑚(𝐽−𝐿_𝛼)𝑃_𝑚)=2^𝑚_𝑗=1𝜎⁻[(−1)^𝑚𝜋2𝑟(2𝑚−1)−𝛼]=2^𝑚_𝑗=1𝜎⁻[(−1)^𝑚𝜋2𝑟(2𝑚−1)−𝛼]−2^𝑚_𝑗=1𝜎⁻((−1)^𝑗)+2^𝑚_𝑗=1𝜎⁻((−1)^𝑗)=2𝜇(𝛼)+2^𝑚_𝑗=1𝜎⁻((−1)^𝑗). (2.32) The last equality of Lemma 2.4 is obvious. The proof is complete.

3. Proof of the Main Result

In this section, we will use [31, Theorem 2.4] to prove our main result. To state the theorem, we need the following notation and definition. For 𝜃∈𝑆¹ and 𝑥∈𝐸, define an action 𝑇 on 𝐸 by

𝑇(𝜃)𝑥(𝑡)=𝑥(𝑡+𝜃). (3.1) A direct computation shows that

𝐹𝑖𝑥(𝑆¹)={𝑥∈𝐸∶𝑇(𝜃)(𝑥)=𝑥,∀𝜃∈𝑆¹}={0}. (3.2)

Definition 3.1. 𝐼 satisfies (𝑃𝑆)_𝑐 condition on 𝐸 if every sequence {𝑥_𝑚}⊂𝐸 with 𝐼(𝑥_𝑚)→𝑐 and 𝐼^′(𝑥_𝑚)→0 possesses a convergent subsequence.

Write 𝐿₀=𝐿_𝛼0 and 𝐿_∞=𝐿_𝛼∞. Then [31, Theorem 2.4] applied to 𝐼 on 𝐸 can be stated as follows.

Lemma 3.2. Suppose that there exist two closed 𝑆¹-invariant linear subspace of 𝐸, 𝑉and, 𝑊 and a positive number 𝑠>0 such that(1)(𝑉+𝑊) is closed and of finite codimension in 𝐸,(2)𝐿(𝑊)⊆𝑊 with 𝐴=𝐽−𝐿₀ or 𝐴=𝐽−𝐿_∞,(3) there exists 𝛽₀∈ℝ such that inf_𝑥∈𝑉𝐼(𝑥)≥𝛽₀, (3.3)(4) there exists 𝛽_∞∈ℝ such that 𝐼(𝑥)≤𝛽_∞<𝐼(0),∀𝑥∈𝑊∩𝐵_𝑠={𝑥∈𝑊∶‖𝑥‖=𝑠}, (3.4)(5)𝐼 satisfies (𝑃𝑆)_𝑐 condition for 𝛽₀≤𝑐≤𝛽_∞.
Then 𝐼 possesses at least (1/2)[dim(𝑉∩𝑊)−codim(𝑉+𝑊)] geometrically distinct critical orbits in 𝐼⁻¹([𝛽₀,𝛽_∞]).

Lemma 3.3. Assume that (𝐻1), (𝐻2), and the assumptions of (i) or (ii) of Theorem 1.2 hold. Then 𝐼 satisfies (𝑃𝑆)_𝑐 condition on 𝐸.

Proof. We first show that {𝑥_𝑚}⊂𝐸 is bounded. Assume that {𝑥_𝑚} is not bounded. By passing to a subsequence, if necessary, we may assume that ‖𝑥_𝑚‖→∞ as 𝑚→∞.
Case (i). Suppose that 𝛼_∞∉(𝜋/2𝑟)(2ℤ−1). Then by Definition 1.1, 𝜈_∞=0, which yields 𝑀⁰(𝐽−𝐿_∞)=0. Thus 𝐽−𝐿_∞ has bounded inverse, that is, ∃𝜅>0 such that
‖(𝐽−𝐿_∞)⁻¹1‖≤𝜅. (3.5)
For any 𝑥∈𝐸, define
𝐼₀1(𝑥)=𝐾(𝑥)+2⟨𝐿₀𝑥,𝑥⟩,𝐼_∞1(𝑥)=𝐾(𝑥)+2⟨𝐿_∞𝑥,𝑥⟩. (3.6) Then 𝐼(𝑥) can be written as 1𝐼(𝑥)=2⟨(𝐽−𝐿₀)𝑥,𝑥⟩+𝐼₀1(𝑥),or𝐼(𝑥)=2⟨(𝐽−𝐿_∞)𝑥,𝑥⟩+𝐼_∞(𝑥). (3.7) By condition (𝐻2), we have ‖𝐼^′_∞(𝑥)‖‖𝑥‖→0,as‖𝑥‖→∞. (3.8) This means that for any 0<𝛾<𝜅 there exists 𝛿>0 such that ‖𝐼^′_∞(𝑥)‖≤𝛾‖𝑥‖,as‖𝑥‖≥𝛿. (3.9)
Note that (𝐽−𝐿_∞)𝑥_𝑚−𝐼^′(𝑥_𝑚)=−𝐼^′_∞(𝑥_𝑚). We have
‖𝑥_𝑚‖−‖(𝐽−𝐿_∞)⁻¹𝐼^′(𝑥_𝑚)‖≤‖𝑥_𝑚−(𝐽−𝐿_∞)⁻¹𝐼^′(𝑥_𝑚)‖=‖(𝐽−𝐿_∞)⁻¹((𝐽−𝐿_∞)𝑥_𝑚−𝐼^′(𝑥_𝑚))‖≤‖(𝐽−𝐿_∞)⁻¹‖‖𝐼^′_∞(𝑥_𝑚≤1)‖𝜅𝛾‖𝑥_𝑚‖. (3.10) Then by the above estimates, ‖𝑥_𝑚𝜅‖≤𝜅−𝛾‖(𝐽−𝐿_∞)⁻¹‖‖𝐼^′(𝑥_𝑚𝜅)‖≤1𝜅−𝛾𝜅‖𝐼^′(𝑥_𝑚)‖→0,as𝑚→∞. (3.11) This contradicts ‖𝑥_𝑚‖→∞ as 𝑚→∞. Hence ‖𝑥_𝑚‖ is bounded.
Now we show that ‖𝑥_𝑚‖ has a convergent subsequence. Notice that 𝑃₀∶𝐸→ker𝐽=ℝ is of finite rank and 𝐾^′∶𝐻→𝐻 is compact. Therefore we may suppose that
𝐾^′(𝑥_𝑚)→𝑦,𝑃₀𝑥_𝑚→𝑧in𝐸as𝑚→∞. (3.12) Since 𝐽+𝑃₀ has continuous inverse (𝐽+𝑃₀)⁻¹, it follows from (𝐽+𝑃₀)𝑥_𝑚=𝐼^′(𝑥_𝑚)−𝐾^′(𝑥_𝑚)+𝑃₀𝑥_𝑚 (3.13) that 𝑥_𝑚=(𝐽+𝑃₀)⁻¹(𝐼^′(𝑥_𝑚)−𝐾^′(𝑥_𝑚)+𝑃₀𝑥_𝑚)→(𝐽+𝑃₀)⁻¹(𝑧−𝑦)as𝑚→∞. (3.14) Thus {𝑥_𝑚} has a convergent subsequence.
Case (ii). Assume that {𝑥_𝑚} is not bounded. Since |𝑓(𝑥)+𝛼_∞𝑥| is bounded, there exists a constant 𝐶>0 such that
‖𝐼^′_∞(𝑥)‖<𝐶,∀𝑥∈𝐸. (3.15) Observe that (𝐽−𝐿_∞)𝑥=𝐼^′(𝑥)−𝐼^′_∞(𝑥) and 𝐼^′(𝑥_𝑚)→0 as 𝑚→∞, ‖(𝐽−𝐿_∞)𝑥_𝑚‖≤‖𝐼^′(𝑥_𝑚)‖+‖𝐼^′_∞(𝑥_𝑚)‖≤𝐶+1as𝑚→∞. (3.16) Since ‖𝑥_𝑚‖→∞ and 𝐹(𝑥_𝑚)+(1/2)𝛼_∞𝑥²_𝑚 converges to −∞ as 𝑚→∞, 𝐼_∞(𝑥_𝑚)=𝐼(𝑥_𝑚1)−2⟨(𝐽−𝐿_∞)𝑥_𝑚,𝑥_𝑚⟩>𝐼(𝑥_𝑚)−𝐶+12‖𝑥_𝑚‖. (3.17) Let 𝑚→∞, then (3.17) implies −∞>−∞. This is a contradiction. Therefore {𝑥_𝑚} is bounded. With the same discussion as that of Case (i), we can prove that {𝑥_𝑚} has a convergent subsequence. The proof is complete.

Now we are ready to prove our main result.

Proof of Theorem 1.2. Since 𝑥(𝑡)∈𝐸 is periodic and 𝐹(𝑥(𝑡)) is independent of 𝑡, 𝐼 is 𝑆¹-invariant and 𝐼^′ is 𝑆¹-equivariant according to the action 𝑇 on 𝑆¹.
Case (i). Assume first 𝜇₀−𝜇_∞>0. Take two subspaces 𝑉 and 𝑊 as
𝑉=𝑀⁺(𝐽−𝐿_∞)⊕𝑀⁰(𝐽−𝐿_∞),𝑊=𝑀⁻(𝐽−𝐿₀). (3.18) Since inner product is continuous and codim(𝑉+𝑊)=0, the assumption (1) of Lemma 3.2 holds. Let 𝐴=𝐽−𝐿₀. We can check easily that 𝐴(𝑊)⊆𝑊. By Lemma 3.3, 𝐼 satisfies (𝑃𝑆)_𝑐 condition. Thus the assumptions (2) and (5) of Lemma 3.2 hold.
By the proof of Lemma 3.3, 𝑀⁰(𝐽−𝐿_∞)=0. For 𝑥∈𝑉=𝑀⁺(𝐽−𝐿_∞), there is a suitable constant 𝑐₁>0 such that
⟨(𝐽−𝐿_∞)𝑥,𝑥⟩≥𝑐₁‖𝑥‖². (3.19) Then by (3.8), for any 𝑥∈𝑉, 1𝐼(𝑥)=2⟨(𝐽−𝐿_∞)𝑥,𝑥⟩+𝐼_∞1(𝑥)≥2𝑐₁‖𝑥‖²−𝑐₂‖𝑥‖, (3.20) where 𝐼_∞(𝑥) is defined in (3.6). Hence there exists 𝛽₀<0 such that the assumption (3) of Lemma 3.2 holds.
By (𝐻2), we have
𝐼₀(0)=0,‖𝐼^′₀(𝑥)‖‖𝑥‖→0as‖𝑥‖→0, (3.21) where 𝐼₀(𝑥) is defined in (3.6). For any 𝑥∈𝑊, we can choose a suitable constant 𝑐₃>0 such that ⟨(𝐽−𝐿₀)𝑥,𝑥⟩≤−𝑐₃‖𝑥‖². (3.22) Then for any 𝑥∈𝑊∩𝐵_𝑠, 1𝐼(𝑥)=2⟨(𝐽−𝐿₀)𝑥,𝑥⟩+𝐼₀1(𝑥)≤−2𝑐₃‖𝑥‖²+𝑐₄1‖𝑥‖=−2𝑐₃𝑟²+𝑐₄𝑟. (3.23) Choose 𝑟 such that −(1/2)𝑐₃𝑟+𝑐₄<0 and take 𝛽_∞=max{𝛽₀,−(1/2)𝑐₁𝑟²+𝑐₂𝑟}. Then the assumption (4) of Lemma 3.2 holds.
Therefore by Lemma 3.2, 𝐼 possesses at least
1♯=2[dim(𝑉∩𝑊)−codim(𝑉+𝑊)] (3.24) geometrically different critical orbits in 𝐼⁻¹([𝛽₀,𝛽_∞]).
From Lemma 2.4 and Definition 1.1, we can show that for 𝑚 large enough
1♯=2[dim𝑀⁻(𝑃_𝑚(𝐽−𝐿₀)𝑃_𝑚)−dim𝑀⁻(𝑃_𝑚(𝐽−𝐿_∞)𝑃_𝑚=1)]2[2𝜇(𝛼₀)−2𝜇(𝛼_∞)]=𝜇₀−𝜇_∞. (3.25)
Secondly, if 𝜇_∞−𝜇₀−𝜈₀>0, then we replace 𝐼 by −𝐼 and set
𝑉=𝑀⁻(𝐽−𝐿_∞),𝑊=𝑀⁺(𝐽−𝐿₀). (3.26) With a similar argument to the case 𝜇₀−𝜇_∞>0, we can show that −𝐼 satisfies the assumptions (1)–(5) of Lemma 3.2. Then −𝐼 has at least ♯=(1/2)[dim(𝑉∩𝑊)−codim(𝑉+𝑊)] geometrically different critical orbits in 𝐼⁻¹([𝛽₀,𝛽_∞]). By Lemma 2.4, Definition 1.1, and for 𝑚 large enough, 1♯=2[dim𝑀⁻(𝑃_𝑚(𝐽−𝐿_∞)𝑃_𝑚)−dim(𝑀⁻(𝑃_𝑚(𝐽−𝐿₀)𝑃_𝑚)⊕𝑀⁰(𝑃_𝑚(𝐽−𝐿₀)𝑃_𝑚=1))]2[2𝜇(𝛼_∞)−(2𝜇(𝛼₀)+2𝜈(𝛼₀))]=𝜇_∞−𝜇₀−𝜈₀. (3.27)
Case (ii). Assume that 𝜇₀−𝜇_∞>0. We take the same subspaces 𝑉 and 𝑊 as in Case (i). By a similar discussion, we can prove that the assumptions of (1), (2), and (4) of Lemma 3.2 hold. By Lemma 3.3, 𝐼 satisfies (𝑃𝑆)_𝑐 condition on 𝐸.
By condition (ii) of Theorem 1.2, 𝐼^′_∞(𝑥) is bounded, that is, ∃𝑐₅>0 such that
‖𝐼^′_∞(𝑥)‖≤𝑐₅,∀𝑥∈𝐸. (3.28) Moreover 𝐼_∞(𝑥₀)→−∞ as 𝑥₀∈𝑀⁰(𝐽−𝐿_∞) with ‖𝑥₀‖→+∞.
Write 𝑥=𝑥₊+𝑥₀∈𝑉, where 𝑥₊∈𝑀⁺(𝐽−𝐿_∞). We have
1𝐼(𝑥)=2⟨(𝐽−𝐿_∞)𝑥₊,𝑥₊⟩+𝐼_∞(𝑥₊+𝑥₀)>12𝑐₆‖𝑥₊‖²+𝑐₅‖𝑥‖+𝐼_∞(𝑥₀). (3.29) Thus the assumption (3) of Lemma 3.2 holds. Then by Lemma 3.2 and the proof of Case (i), 𝐼 possesses at least 𝜇₀−𝜇_∞ geometrically different critical orbits in 𝐼⁻¹([𝛽₀,𝛽_∞]). The subsequent proof is similar to that of Case (i). We omit the details. The proof is complete.