The Existence of Multiple Periodic Solutions of Nonautonomous Delay Differential Equations

Abstract

We study the multiplicity of periodic solutions of nonautonomous delay differential equations which are asymptotically linear both at zero and at infinity. By making use of a theorem of Benci, some sufficient conditions are obtained to guarantee the existence of multiple periodic solutions.

1. Introduction

The existence and multiplicity of periodic solutions of delay differential equations have received a great deal of attention. In 1962, Jones [1] firstly investigated the existence of periodic solutions to the following scalar equation:𝑢^′(𝑡)=−𝑎𝑢(𝑡−1)[1+𝑢(𝑡)]. (1.1) By making use of Browder fixed point theorem, the author showed that there exist periodic solutions of (1.1) for each 𝑎>𝜋/2. Since then, various fixed point theorems have been used to study the existence of periodic solutions of delay differential equations(cf. [2]). As pointed out in [3], by making change of variable 1+𝑢=𝑒^𝑥, (1.1) turns into𝑥^′(𝑡)=−𝑓(𝑥(𝑡−1)). (1.2) In 1974, Kaplan and Yorke [4] studied the following more general form of  (1.2)𝑥^′(𝑡)=−𝑓(𝑥(𝑡−1))−𝑓(𝑥(𝑡−2))−⋯−𝑓(𝑥(𝑡−𝑛)). (1.3) They introduced a technique which translates the problem of the existence of periodic solutions of a scalar delay differential equation to that of the existence of critical points of an associated ordinary differential system. Using this method, they proved that (1.3) has a periodic solution with minimal period 4 (resp., 6) when (1.3) has one delay (resp., two delays). In this direction, Fei, Li and He did some excellent work and got some signification results (cf. [5–8]).

Many other approaches, such as coincidence degree theory, the Hopf bifurcation theorem, and the Poincaré-Bendixson theorem, have also been used to study the existence of periodic solutions of delay differential equations (cf. [9, 10]). However, most of those results are concerned with scalar delay equations. In 2005, Guo and Yu [3] studied vector delay differential system (1.2). They built a variational structure for (1.2) on certain suitable spaces. Then they reduced the existence of periodic solutions of (1.2) to that of critical points of an associated variational functional. By making use of pseudoindex theory, they obtained some sufficient conditions to guarantee the existence of multiple periodic solutions.

In spite of so many papers on periodic solutions of delay differential equations, there are a quite few researches on nonautonomous case (see for example [11]). The main goal of this paper is to investigate the following nonautonomous system:𝑥^′(𝑡)𝜋=−𝑓𝑡,𝑥𝑡−2. (1.4)

We assume that (𝐟₁) there exists 𝐹∈𝐶¹([0,𝜋/2]×ℝ^𝑛,ℝ) such that 𝑓 is the gradient of 𝐹 with respect to 𝑥, and𝐹()()𝜋𝑡,𝑥=𝐹𝑡,−𝑥,𝐹𝑡+2,𝑥=𝐹(𝑡,𝑥),∀(𝑡,𝑥)∈ℝ×ℝ^𝑛, (1.5)(𝐟₂)𝑓(𝑡,𝑥)=𝐵₀(𝑡)𝑥+𝑜(|𝑥|) as |𝑥|→0 uniformly for 𝑡∈[0,𝜋/2],(𝐟₃)𝑓(𝑡,𝑥)=𝐵_∞(𝑡)𝑥+𝑜(|𝑥|) as |𝑥|→∞ uniformly for 𝑡∈[0,𝜋/2],

where 𝐵₀,𝐵_∞ are 𝑛×𝑛 symmetric continuous 𝜋/2-periodic matrix functions.

Hypothesis (𝐟₃) is known as asymptotically linear condition at infinity. Hypothesis (𝐟₂) is an asymptotically linear condition at zero, which implies that 0 is a trivial solution of (1.4). We are interested in nontrivial periodic solutions of (1.4). Similar to [3], we build a variational structure for (1.4) and convert the existence of periodic solutions to that of critical points of variational functional. Since the asymptotically linear hypothesis at infinity is given by a periodic loop of symmetric matrix, it will be more difficult to deal with more than a constant matrix. However, we can prove the existence of multiple periodic solutions by making use of a multiple critical points theorem of Benci (cf. [12]).

The rest of this paper is organized as follows: in Section 2, we build the variational functional and state some useful lemmas; in Section 3, the main results will be proved.

2. Variational Tools

Denote 𝑆¹=ℝ/(2𝜋ℤ). The space 𝐻=𝐻^1/2(𝑆¹,ℝ^𝑛) has been introduced in [3]. The space 𝐻 can be equipped with inner product as follows: 𝑥,𝑦=𝑎₀,𝑐₀+^∞_𝑗=11+𝑗𝑎_𝑗,𝑐_𝑗+𝑏_𝑗,𝑑_𝑗, (2.1) where 𝑥=𝑎₀/√2𝜋+1/√𝜋∑^∞_𝑗=1(𝑎_𝑗cos𝑗𝑡+𝑏_𝑗sin𝑗𝑡), 𝑦=𝑐₀/√2𝜋+1/√𝜋∑^∞_𝑗=1(𝑐_𝑗cos𝑗𝑡+𝑑_𝑗sin𝑗𝑡), 𝑎₀,𝑐₀∈ℝ^𝑛, 𝑎_𝑗,𝑐_𝑗,𝑏_𝑗,𝑑_𝑗∈ℝ^𝑛, 𝑗∈ℕ.

Set 𝐸={𝑥∈𝐻∣𝑥(𝑡+𝜋)=−𝑥(𝑡),∀𝑡∈ℝ}. (2.2) Then 𝐸 is a closed subspace of 𝐻. If 𝑥∈𝐸, it has Fourier expansion 𝑥(𝑡)=1√𝜋^∞_𝑗=1𝑎_𝑗cos2𝑗−1𝑡+𝑏_𝑗sin2𝑗−1𝑡. (2.3)

Let 𝑥∈𝐿²(𝑆¹,ℝ^𝑛). If for every 𝑧∈𝐶^∞(𝑆¹,ℝ^𝑛)₀^2𝜋𝑥(𝑡),𝑧^′(𝑡)𝑑𝑡=−₀^2𝜋𝑦(𝑡),𝑧(𝑡)𝑑𝑡, (2.4) then 𝑦 is called a weak derivative of 𝑥, denoted by 𝑥̇.

The variational functional defined on 𝐻, corresponding to (1.4), is𝐽(𝑥)=₀^2𝜋12𝑥𝜋𝑡+2,𝑥̇(𝑡)−𝐹(𝑡,𝑥(𝑡))𝑑𝑡. (2.5) Define a linear bounded operator 𝐴∶𝐻→𝐻 by setting𝐴𝑥,𝑦=₀^2𝜋𝑥𝜋𝑡+2,𝑦̇(𝑡)𝑑𝑡. (2.6) It is easy to prove that 𝐸 is an invariant subspace of 𝐻 with respect to 𝐴 and 𝐴 is self-adjoint if it is restricted to 𝐸.

Lemma 2.1 (see [3]). The essential spectrum of the operator 𝐴 restricted to 𝐸 is just {2,−2}.

Define 𝜑(𝑥)=−₀^2𝜋𝐹(𝑡,𝑥(𝑡))𝑑𝑡,∀𝑥∈𝐻. (2.7) Then 𝐽 can be rewritten as𝐽(𝑥)=12⟨𝐴𝑥,𝑥⟩+𝜑(𝑥),∀𝑥∈𝐻. (2.8)

Similar to the argument as in [3], we can prove the following two basic lemmas.

Lemma 2.2. Assume that 𝑓 satisfies (𝐟₁)–(𝐟₃). Then 𝐽 is continuous differentiable on 𝐻 and 𝐽^′(𝑥),ℎ=₀^2𝜋12𝑥̇𝜋𝑡−2−𝑥̇𝜋𝑡+2,ℎ(𝑡)−𝑓(𝑡,𝑥(𝑡)),ℎ(𝑡)𝑑𝑡,∀ℎ∈𝐻. (2.9) Moreover, 𝜑^′∶𝐻→𝐻^∗ is a compact mapping defined as follows: 𝜑^′(𝑥),ℎ=−₀^2𝜋𝑓(𝑡,𝑥(𝑡)),ℎ(𝑡)𝑑𝑡,∀𝑥,ℎ∈𝐻. (2.10)

Lemma 2.3. The existence of 2𝜋-periodic solutions of (1.4) belonging to 𝐸 is equivalent to the existence of critical points of functional 𝐽 restricted to 𝐸.

Lemma 2.3 implies that we can restrict our discussion on space 𝐸. At the end of this section, we recall a useful embedding theorem.

Lemma 2.4 (see [13]). For every 𝑝∈[1,+∞), 𝐻 is compactly embedded into the Banach space 𝐿^𝑝(𝑆¹,ℝ^𝑛). In particular, there is an 𝛼_𝑝 such that ‖𝑥‖_𝐿^𝑝≤𝛼_𝑝‖𝑥‖,∀𝑥∈𝐻. (2.11)

Remark 2.5. Here and hereafter, 𝛼_𝑝(𝑝∈[1,∞)) denotes the real number satisfying (2.11).

3. Main Results

Let 𝐵(𝑡) be an 𝑛×𝑛 symmetric continuous 𝜋/2-periodic matrix function. We define a bounded self-adjoint linear operator 𝐵∈𝐿(𝐸) by extending the bilinear forms𝐵𝑥,𝑦=₀^2𝜋𝐵(𝑡)𝑥(𝑡),𝑦(𝑡)𝑑𝑡,∀𝑥,𝑦∈𝐸. (3.1) It is well known that 𝐵 is compact (cf. [14]).

Denote by 𝐵₀,𝐵_∞ the operators defined by (3.1), corresponding to 𝐵₀(𝑡),𝐵_∞(𝑡), respectively. Set 𝑛₀=dimKer𝐴−𝐵₀,𝑛_∞=dimKer𝐴−𝐵_∞,𝐺_𝑖()()−1𝑡,𝑥=𝐹𝑡,𝑥2𝐵_𝑖(𝑡)𝑥,𝑥,𝜓_𝑖(𝑥)=₀^2𝜋𝐺_𝑖(𝑡,𝑥)𝑑𝑡,𝑖=0,∞. (3.2) Then the functional 𝐽 defined by (2.5) can be rewritten as𝐽(𝑥)=12𝐴−𝐵_𝑖𝑥,𝑥−𝜓_𝑖(𝑥),∀𝑥∈𝐸,𝑖=0,∞. (3.3)

Lemma 3.1. Suppose that 𝑓 satisfies (𝐟₁)–(𝐟₃). Then lim_{‖𝑥‖→0}𝜓^′₀(𝑥)‖𝑥‖=0,lim_{‖𝑥‖→+∞}𝜓^′_∞(𝑥)‖𝑥‖=0. (3.4)

The proof uses the same arguments of [5].

In order to prove our results, we need an abstract theorem by Benci [12].

Proposition 3.2. Let 𝜒∈𝐶¹(𝐸,ℝ) satisfy the following:(J1)𝜒(𝑥)=1/2⟨𝐿𝑥,𝑥⟩+𝜔(𝑥), where 𝐿 is a bounded linear self-adjoint operator and 𝜔^′ is compact, where 𝜔^′ denotes the Frechét derivative of 𝜔;(J2)every sequence {𝑥_𝑗} such that 𝜒(𝑥_𝑗)→𝑐<𝜑(0) and ‖𝜒^′(𝑥_𝑗)‖→0 as 𝑗→+∞ has a convergent subsequence;(J3)𝜔(𝑥)=𝜔(−𝑥), 𝑥∈𝐸;(J4)there are two closed subspaces of 𝐸, 𝐸⁺, and 𝐸⁻, and some constant 𝑐₀,𝑐_∞,𝜌 with 𝑐₀<𝑐_∞<𝜔(0) and 𝜌>0 such that(a)𝜒(𝑥)>𝑐₀ for 𝑥∈𝐸⁺,(b)𝜒(𝑥)<𝑐_∞<𝜔(0) for 𝑢∈𝐸⁻∩𝑆_𝜌(𝑆_𝜌={𝑢∈𝐸‖𝑥‖=𝜌}).Then the number of pairs of nontrivial critical points of 𝜒 is greater than or equal to dim(𝐸⁺∩𝐸⁻)−codim(𝐸⁻+𝐸⁺). Moreover, the corresponding critical values belong to [𝑐₀,𝑐_∞].

Definition 3.3. Let 𝐵₁(𝑡) and 𝐵₂(𝑡) be symmetric matrices function in ℝ^𝑛, continuous and 𝜋/2-periodic in 𝑡. A index 𝐼 of 𝐵₁(𝑡) and 𝐵₂(𝑡) is defined as follows: 𝐼𝐵₁(𝑡),𝐵₂(𝑡)=dim𝑀⁺𝐴−𝐵₁𝑀⁻𝐴−𝐵₂−dim𝑀⁻𝐴−𝐵₁⊕𝑀⁰𝐴−𝐵₁𝑀⁺𝐴−𝐵₂⊕𝑀⁰𝐴−𝐵₂, (3.5) where 𝐵_𝑖(𝑖=1,2) are the operators, defined by (3.1), corresponding to 𝐵_𝑖(𝑡)(𝑖=1,2) and 𝑀⁺(𝐴−𝐵_𝑖) (resp., 𝑀⁻(𝐴−𝐵_𝑖(𝑡)), 𝑀⁰(𝐴−𝐵_𝑖)) denotes the subspace of 𝐸 on which 𝐴−𝐵_𝑖 is positive definite (resp., negative definite, null).

Lemma 3.4. If 𝑓 satisfies (𝐟₁)–(𝐟₃), then 𝐽, defined by (2.8), satisfies (J1), (J3) and (J4).

Proof. Hypothesis (𝐟₁), (2.8), and Lemma 2.2 imply both (J1) and (J3). By definition of 𝜓₀ and Lemma 3.1, we have 𝜓₀(𝑥)=−𝜑(0)+𝑜‖𝑥‖²,for‖𝑥‖⟶0. (3.6) Since 𝐵₀ and 𝐵_∞ are compact operators from 𝐸 to 𝐸, it follows from Lemma 2.1 and a well-known theorem (cf. [15]) that the essential spectrum of 𝐴−𝐵₀ and 𝐴−𝐵_∞ is {2,−2}. Thus 0 is either an isolated eigenvalue of finite multiplicity or it belongs to the resolvent. Hence, we decompose 𝐸 as follows: 𝐸=𝑀⁺𝐴−𝐵₀⊕𝑀⁰𝐴−𝐵₀⊕𝑀⁻𝐴−𝐵₀=𝑀⁺𝐴−𝐵_∞⊕𝑀⁰𝐴−𝐵_∞⊕𝑀⁻𝐴−𝐵_∞. (3.7) Setting 𝐸⁺=𝑀⁺(𝐴−𝐵_∞), 𝐸⁻=𝑀⁻(𝐴−𝐵₀), there exists positive constant 𝛼,𝛽 such that 𝐴−𝐵₀𝑥,𝑥≤−𝛼‖𝑥‖²,∀𝑥∈𝐸⁻,𝐴−𝐵_∞𝑥,𝑥≥𝛽‖𝑥‖²,∀𝑥∈𝐸⁺. (3.8) It follows that, for any 𝑥∈𝐸⁻, it is 𝐽(𝑥)𝛼≤−2‖𝑥‖²+𝜑(0)−𝑜‖𝑥‖²,as‖𝑥‖⟶0. (3.9) Then there exist constants 𝜌>0 and 𝛾>0 such that 𝐽(𝑥)<−𝛾+𝜑(0),∀𝑥∈𝐸⁻∩𝑆_𝜌. (3.10) Setting 𝑐_∞=−𝛾/2+𝜑(0), (J4)(b) is satisfied.
By (𝐟₃), there exists 𝑅₀>0 such that 𝐺_∞()≤𝛽𝑡,𝑥4𝛼²₂|𝑥|²,∀|𝑥|>𝑅₀. (3.11) Since 𝐺_∞(𝑡,𝑥) is continuous with respect to (𝑡,𝑥), denote by 𝑀=max_{0≤𝑡≤𝜋/2,|𝑥|=𝑅0}{𝐺_∞(𝑡,𝑥)}. Then 𝑀 is finite. Thus𝜓_∞(𝑥)≤₀^2𝜋𝐺_∞(𝑡,𝑥)𝑑𝑡≤₀^2𝜋𝛽4𝛼²₂|𝑥|²𝛽+𝑀𝑑𝑡≤4‖𝑥‖²+2𝜋𝑀. (3.12) Then, for every 𝑥∈𝐸⁺, 𝐽(𝑥)=12𝐴−𝐵_∞𝑥,𝑥−𝜓_∞(𝑥)≥𝛽2‖𝑥‖²−𝜓_∞(𝑥≥𝛽)4‖𝑥‖²−2𝜋𝑀. (3.13) Thus 𝐽 is bounded from below on 𝐸⁺. Setting 𝑐₀=inf_𝑥∈𝐸+𝐽(𝑥)−𝜔 (3.14) with 𝜔>0 such that 𝑐₀<𝑐_∞, then (J4)(a) is satisfied.

Remark 3.5. Supposing that 0∉𝜎_𝑒(𝐴−𝐵_∞), any bounded sequence has a convergent subsequence (cf. [12]).

Theorem 3.6. Suppose that 𝑓 satisfies (𝐟₁)–(𝐟₃), and 𝑛₀=𝑛_∞=0, then (1.4) has at least |𝐼(𝐵_∞,𝐵₀)| pairs of nonconstant 2𝜋−periodic solutions if |𝐼(𝐵_∞,𝐵₀)|>0.

Proof. Since 𝑛_∞=0, dim𝑀⁰(𝐴−𝐵_∞)=0. By Proposition 3.2 and Lemma 3.4, we only need to check (J2). Let {𝑥_𝑗} be a sequence such that 𝐽^′𝑥_𝑗→0,𝐽𝑥_𝑗⟶𝑐, (3.15) where 𝑐∈ℝ, 𝑐<𝜑(0). Suppose to the contrary that we can choose ‖𝑥_𝑗‖→+∞ as 𝑗→+∞. Clearly, 𝑥_𝑗 can be written as 𝑥_𝑗=𝑥⁺_𝑗+𝑥⁻_𝑗∈𝑀⁺(𝐴−𝐵_∞)⊕𝑀⁻(𝐴−𝐵_∞). On one hand, 𝐽^′𝑥_𝑗,𝑥⁺_𝑗−𝑥⁻_𝑗𝑥_𝑗,𝑥_𝑗≤𝐽^′𝑥_𝑗𝑥_𝑗𝑥_𝑗², (3.16) then we have 0≤limsup_𝑗→+∞𝐽^′𝑥_𝑗,𝑥⁺_𝑗−𝑥⁻_𝑗𝑥_𝑗,𝑥_𝑗≤limsup_𝑗→+∞𝐽^′𝑥_𝑗𝑥_𝑗𝑥_𝑗²=0. (3.17) Thus limsup_𝑗→+∞𝐽^′𝑥_𝑗,𝑥⁺_𝑗−𝑥⁻_𝑗𝑥_𝑗,𝑥_𝑗=0. (3.18)
On the other hand, 𝐽^′𝑥_𝑗,𝑥⁺_𝑗−𝑥⁻_𝑗=𝐴−𝐵_∞𝑥_𝑗,𝑥⁺_𝑗−𝑥⁻_𝑗−𝜓^′_∞𝑡,𝑥_𝑗,𝑥⁺_𝑗−𝑥⁻_𝑗. (3.19) Since 𝜓^′_∞𝑥_𝑗,𝑥⁺_𝑗−𝑥⁻_𝑗𝑥_𝑗,𝑥_𝑗≤𝜓^′_∞𝑥_𝑗𝑥_𝑗𝑥_𝑗²=𝜓^′_∞𝑥_𝑗𝑥_𝑗, (3.20) it follows by Lemma 3.1 that lim_𝑗→+∞𝜓^′_∞𝑥_𝑗,𝑥⁺_𝑗−𝑥⁻_𝑗𝑥_𝑗,𝑥_𝑗=0. (3.21) Using a similar discussion as (3.8), there exists 𝛽₀>0 such that ⟨(𝐴−𝐵_∞)𝑥,𝑥⟩≤−𝛽₀‖𝑥‖² for all 𝑥∈𝑀⁻(𝐴−𝐵_∞). Choosing 𝛽^′=min(𝛽,𝛽₀)>0, we have 𝐴−𝐵_∞𝑥_𝑗,𝑥⁺_𝑗−𝑥⁻_𝑗=𝐴−𝐵_∞𝑥⁺_𝑗,𝑥⁺_𝑗−𝐴−𝐵_∞𝑥⁻_𝑗,𝑥⁻_𝑗≥𝛽^′‖𝑥‖². (3.22) Thus, liminf_𝑗→+∞𝐽^′𝑥_𝑗,𝑥⁺_𝑗−𝑥⁻_𝑗𝑥_𝑗,𝑥_𝑗=liminf_𝑗→+∞𝐴−𝐵_∞𝑥_𝑗,𝑥⁺_𝑗−𝑥⁻_𝑗−𝑔^′_∞𝑥_𝑗,𝑥⁺_𝑗−𝑥⁻_𝑗𝑥_𝑗,𝑥_𝑗=liminf_𝑗→+∞𝐴−𝐵_∞𝑥_𝑗,𝑥⁺_𝑗−𝑥⁻_𝑗𝑥_𝑗,𝑥_𝑗≥𝛽^′>0 (3.23) which contradicts (3.18). This proves (J2). By Lemma 3.1, (1.4) has at least dim(𝐸⁺∩𝐸⁻)−codim(𝐸⁻+𝐸⁺)=𝐼(𝐵_∞,𝐵₀) pairs of nontrivial solutions if 𝐼(𝐵_∞,𝐵₀)>0. Since the Sobolev space 𝐸 does not contain ℝ^𝑛 as its subspace, all nontrivial periodic solutions are nonconstant periodic solutions.
If 𝐼(𝐵_∞,𝐵₀)<0, then 𝐼(𝐵₀,𝐵_∞)=−𝐼(𝐵_∞,𝐵₀)>0. In this case, we replace 𝐽 by −𝐽 and let 𝐸⁺=𝑀⁻(𝐴−𝐵₀) and 𝐸⁻=𝑀⁺(𝐴−𝐵_∞). It is easy to see that (J1)–(J4) are satisfied. Similarly, we can show that (1.4) has at least 𝐼(𝐵₀,𝐵_∞) pairs of nonconstant solutions.

Remark 3.7. When Theorem 3.6 is applied to autonomous delay differential equations, we obtain the same number of periodic solutions as that in [3].

Theorem 3.8. Suppose 𝑓 satisfies (𝐟₁)–(𝐟₃) and (𝐟₄)𝐺^′_∞(𝑡,𝑥) is bounded, where 𝐺^′_∞ denotes the derivative of 𝐺_∞ with respect to 𝑥,(𝐟₅)^±𝐺_∞(𝑡,𝑥)→±∞ as |𝑥|→+∞, uniformly for 𝑡∈[0,𝜋/2].
Then (1.4) has at least 𝐼(𝐵_∞,𝐵₀) pairs of nonconstant 2𝜋-periodic solutions provided 𝐼(𝐵_∞,𝐵₀)>0.

Proof. By Proposition 3.2 and Lemma 3.4, it suffices to check condition (J2). Let {𝑥_𝑗} be a sequence satisfying (3.15). Suppose to the contrary that {𝑥_𝑗} is unbounded. Clearly, 𝑥_𝑗 can be written as 𝑥_𝑗=𝑥⁺_𝑗+𝑥⁰_𝑗+𝑥⁻_𝑗∈𝑀⁺(𝐴−𝐵_∞)⊕𝑀⁰(𝐴−𝐵_∞)⊕𝑀⁻(𝐴−𝐵_∞). Since 𝐽^′(𝑥_𝑗)→0, for 𝑗 large enough, we get 𝐴−𝐵_∞𝑥_𝑗,𝑥⁺_𝑗−₀^2𝜋𝐺^′_∞𝑡,𝑥_𝑗,𝑥⁺_𝑗𝑑𝑡≤𝑥⁺_𝑗. (3.24) By (𝐟₄), there exists 𝑐₁>0 such that |𝐺^′_∞(𝑡,𝑥)|≤𝑐₁. Then the above inequality and (3.8) imply 𝛽𝑥⁺_𝑗²≤𝐴−𝐵_∞𝑥_𝑗,𝑥⁺_𝑗≤𝑥⁺_𝑗+𝑐₁𝛼₂√2𝜋𝑥⁺_𝑗. (3.25) This gives a uniform bound for {𝑥⁺_𝑗}. In the same manner, one gets a uniform bound for {𝑥⁻_𝑗}. Since {𝐽(𝑥_𝑗)} is convergent, it is bounded and there exist positive constants c₂,𝑐₃,𝑐₄ such that 𝑐₂≤𝐽𝑥_𝑗≤−𝜓_∞𝑥_𝑗+12𝐴−𝐵_∞𝑥_𝑗,𝑥_𝑗≤−𝜓_∞𝑥⁰_𝑗+𝜓_∞𝑥⁰_𝑗−𝜓_∞𝑥_𝑗+𝑐₃≤−𝜓_∞𝑥⁰_𝑗+𝑐₁₀^2𝜋𝑥⁰_𝑗−𝑥_𝑗𝑑𝑡+𝑐₃≤−𝜓_∞𝑥⁰_𝑗+𝑐₄. (3.26) Therefore, 𝜓_∞(𝑥⁰_𝑗) is bounded from above. (𝐟₅)⁺ implies that ‖𝑥⁰_𝑗‖ is bounded. Otherwise, since the kernel of 𝐴−𝐵_∞ is a finite dimensional space, thus 𝜓_∞(𝑥⁰_𝑗)=∫₀^2𝜋𝐺_∞(𝑡,𝑥⁰_𝑗)𝑑𝑡→∞ as 𝑗→∞, which contradicts to (3.26).
If (𝐟₅)⁻ holds, we replace (3.26) by 𝑐₂≥𝐽𝑥_𝑗≥−𝜓_∞𝑥_𝑗−𝐴−𝐵_∞𝑥_𝑗,𝑥_𝑗. (3.27) Arguing as above, we can get a contradiction and complete our proof.

Theorem 3.9. Suppose that 𝑓 satisfies (𝐟₁)–(𝐟₃) and (𝐟₆)there exist constants 𝑟>0, 𝑝∈(1,2), 𝑎₁>0, and 𝑎₂>0 such that 𝑝𝐺_∞(𝑡,𝑥)≥𝑥,𝐺^′_∞()|𝑥|𝜋𝑡,𝑥>0for≥𝑟,𝑡∈0,2;𝐺_∞(𝑡,𝑥)≥𝑎₁|𝑥|^𝑝−𝑎₂,∀𝑥∈ℝ^𝑛𝜋,𝑡∈0,2. (3.28) Then (1.4) has at least 𝐼(𝐵_∞,𝐵₀) pairs of nonconstant 2𝜋-periodic solutions provided 𝐼(𝐵_∞,𝐵₀)>0.

Proof. Let {𝑥_𝑗} be a sequence satisfying (3.15). We want to show that {𝑥_𝑗} is a bounded sequence in 𝐸. Decompose 𝑥_𝑗 as 𝑥_𝑗=𝑥⁺_𝑗+𝑥⁰_𝑗+𝑥⁻_𝑗∈𝑀⁺(𝐴−𝐵_∞)⊕𝑀⁰(𝐴−𝐵_∞)⊕𝑀⁻(𝐴−𝐵_∞). Then 𝐽^′𝑥_𝑗,𝑥⁺_𝑗=𝐴−𝐵_∞𝑥⁺_𝑗,𝑥⁺_𝑗−𝜓^′_∞𝑥_𝑗,𝑥⁺_𝑗≥𝛽𝑥⁺_𝑗²−𝜓^′_∞𝑥_𝑗⋅𝑥⁺_𝑗. (3.29) Combining the above inequality with (3.15) and Lemma 3.1, we have 𝑥⁺_𝑗𝑥_𝑗→0,as𝑗→∞. (3.30) Similarly, we have 𝑥⁻_𝑗𝑥_𝑗⟶0,as𝑗⟶∞. (3.31)
Then by (3.30) and (3.31), there exists a positive integer 𝑗₀ such that for 𝑗≥𝑗₀𝑥⁰_𝑗≥𝑥⁺_𝑗+𝑥⁻_𝑗. (3.32) It follows that 𝑥_𝑗=𝑥⁺_𝑗+𝑥⁻_𝑗+𝑥⁰_𝑗≤𝑥⁺_𝑗+𝑥⁻_𝑗+𝑥⁰_𝑗≤2𝑥⁰_𝑗. (3.33)
By (𝐟₆), there exist positive constants 𝑀₁,𝑀₂,𝑀₃,𝑀₄ such that for 𝑗 large 𝑀₁+12𝑥_𝑗≥12𝐽^′𝑥_𝑗,𝑥_𝑗−𝐽𝑥_𝑗=₀^2𝜋𝐺_∞𝑡,𝑥_𝑗−12𝐺^′_∞𝑡,𝑥_𝑗,𝑥_𝑗≥𝑝𝑑𝑡1−2₀^2𝜋𝐺_∞𝑡,𝑥_𝑗𝑑𝑡−𝑀₂≥𝑀₃𝑥_𝑗^𝑝_𝐿𝑝−𝑀₄. (3.34) Let 𝑞 be such that 𝑝⁻¹+𝑞⁻¹=1. Since 𝐸⊂𝐿^𝑞(𝑆¹,ℝ^𝑛), the embedding being continuous, the dual space 𝐸^∗ of 𝐸, contains 𝐿^𝑝(𝑆¹,ℝ^𝑛) with continuous embedding. Therefore, by (3.34) 𝑀₅1+𝑥_𝑗≥𝑥_𝑗^𝑝_𝐸∗. (3.35) Since ‖𝑥_𝑗‖_𝐸∗=sup_{‖𝑤‖𝐸≤1}(𝑥_𝑗,𝑤)_𝐿2=sup_{‖𝑤‖𝐸≤1}[(𝑥⁰_𝑗,𝑤⁰)_𝐿2+(𝑥⁻_𝑗,𝑤⁻)_𝐿2+(𝑥⁺_𝑗,𝑤⁺)_𝐿2], taking 𝑤=𝑥⁰_𝑗/‖𝑥⁰_𝑗‖_𝐸, it follows that 𝑥_𝑗𝐸∗≥1𝑥⁰_𝑗𝐸𝑥⁰_𝑗²_𝐿2. (3.36) Owing to the fact that 𝑀⁰(𝐴−𝐵_∞) is a finite dimensional subspace of 𝐸, there exist two positive constants 𝑐₁ and 𝑐₂ such that 𝑐₁‖𝑥⁰_𝑗‖_𝐸≤‖𝑥⁰_𝑗‖_𝐿²≤𝑐₂‖𝑥⁰_𝑗‖_𝐸. (3.37)
Therefore by (3.35), (3.36), and (3.37), 𝑀₆1+𝑥_𝑗≥𝑥⁰_𝑗^𝑝. (3.38) Both (3.33) and (3.38) imply that there exists 𝑀₈>0 such that 𝑀₇1+𝑥⁰_𝑗≥𝑥⁰_𝑗^𝑝 (3.39) which yields a bound for ‖𝑥⁰_𝑗‖ and hence 𝑥_𝑗 via (3.33). Thus (J2) holds.

Theorem 3.10. Suppose that 𝑓 satisfies (𝐟₁)–(𝐟₃) and
(𝐟₇)^± there exist positive constants 𝑐₁,𝑐₂>0 such that ±2𝐺_∞(𝑡,𝑥)−𝐺^′_∞(𝑡,𝑥),𝑥≥𝑐₁|𝑥|−𝑐₂∀𝑥∈ℝ^𝑛𝜋,𝑡∈0,2. (3.40) Then (1.4) has at least 𝐼(𝐵_∞,𝐵₀) pairs of nonconstant 2𝜋-periodic solutions provided 𝐼(𝐵_∞,𝐵₀)>0.

Proof. Let {𝑥_𝑗} be a sequence satisfying (3.15). We want to prove that {𝑥_𝑗} is bounded in 𝐸. Suppose, to the contrary, {𝑥_𝑗} is unbounded in 𝐸. Decompose 𝑥_𝑗 as 𝑥_𝑗=𝑥⁺_𝑗+𝑥⁰_𝑗+𝑥⁻_𝑗∈𝑀⁺(𝐴−𝐵_∞)⊕𝑀⁰(𝐴−𝐵_∞)⊕𝑀⁻(𝐴−𝐵_∞). Clearly, (3.30)–(3.33) still hold.
Assume that (𝐟₇)⁺ holds. Since 𝑀⁰(𝐴−𝐵_∞) is a finite dimensional subspace of 𝐸, we have 𝐽^′𝑥_𝑗,𝑥_𝑗−2𝐽𝑥_𝑗=₀^2𝜋2𝐺_∞𝑡,𝑥_𝑗−𝐺^′_∞𝑡,𝑥_𝑗,𝑥_𝑗𝑑𝑡≥𝑐₁₀^2𝜋𝑥_𝑗𝑑𝑡−2𝜋𝑐₂≥𝑐₁₀^2𝜋𝑥⁰_𝑗𝑑𝑡−𝑐₁₀^2𝜋𝑥⁺_𝑗+𝑥⁻_𝑗𝑑𝑡−2𝜋𝑐₂≥𝑐₃𝑥⁰_𝑗−𝑐₄𝑥⁺_𝑗+𝑥⁻_𝑗+1. (3.41) Combining the above inequality with (3.30), (3.31), we have 𝑥⁰_𝑗𝑥_𝑗⟶0as𝑗⟶∞. (3.42) But this implies the following contradiction: 𝑥1=^𝑗𝑥_𝑗=𝑥⁰_𝑗+𝑥⁻_𝑗+𝑥⁺_𝑗𝑥_𝑗⟶0as𝑗⟶+∞, (3.43) therefore, {𝑥_𝑗} must be a bounded sequence.
If (𝐟₇)⁻ holds, using a similar argument, we can get a contradiction which completes our proof.

Theorem 3.11. Suppose that 𝑓 satisfies (𝐟₁)–(𝐟₃) and (f₈)^±there exist constants 1≤𝛾<2, 0<𝛿<𝛾/2, and 𝑏₁,𝑏₂,𝐿>0 such that𝐺^′_∞(𝑡,𝑥)≤𝑏₁|𝑥|^𝛿,±𝐺_∞(𝑡,𝑥)≥𝑏₂|𝑥|^𝛾|𝑥|𝜋,∀≥𝐿,𝑡∈0,2. (3.44) Then (1.4) has at least 𝐼(𝐵_∞,𝐵₀) pairs of nonconstant 2𝜋-periodic solutions provided 𝐼(𝐵_∞,𝐵₀)>0.

Proof. Let {𝑥_𝑗} be a sequence satisfying (3.15). Suppose, to the contrary, ‖𝑥_𝑗‖→+∞ as 𝑗→+∞. Decompose 𝑥_𝑗 as 𝑥_𝑗=𝑥⁺_𝑗+𝑥⁰_𝑗+𝑥⁻_𝑗∈𝑀⁺(𝐴−𝐵_∞)⊕𝑀⁰(𝐴−𝐵_∞)⊕𝑀⁻(𝐴−𝐵_∞). First, we show that for 𝑗 large enough 𝑥⁺_𝑗+𝑥⁻_𝑗≤𝑏₃𝑥⁰_𝑗^𝛿+𝜂, (3.45) where 𝑏₃>0 and 𝜂>0 are constants independent of 𝑗. Since |𝑥_𝑗|≥𝐿 for sufficiently large 𝑗, therefore, |𝐺^′_∞(𝑡,𝑥_𝑗)|≤𝑏₁|𝑥_𝑗|^𝛿, and we have 𝐺^′_∞𝑡,𝑥_𝑗²≤𝑏²₁𝑥_𝑗^2𝛿+𝑏₄;𝜓^′_∞𝑥_𝑗,𝑦≤₀^2𝜋𝐺^′_∞𝑡,𝑥_𝑗𝑦𝑑𝑡≤₀^2𝜋𝐺^′_∞𝑡,𝑥_𝑗²𝑑𝑡^1/2𝑦_𝐿2≤𝛼₂𝑏²₁(2𝜋)^1−𝛿𝛼₂^2𝛿𝑥_𝑗^2𝛿+2𝜋𝑏₄^1/2𝑦,forany𝑦∈𝐸. (3.46) This implies that for 𝑗 large enough 𝜓^′_∞𝑥_𝑗𝑥_𝑗^𝛿≤𝑏₅. (3.47) By (3.15), (3.22), (3.33) and (3.47), for 𝑗 large enough, we have 𝐽^′𝑥_𝑗,𝑥⁺_𝑗−𝑥⁻_𝑗=𝐴−𝐵_∞𝑥_𝑗,𝑥⁺_𝑗−𝑥⁻_𝑗−𝜓^′_∞𝑥_𝑗,𝑥⁺_𝑗−𝑥⁻_𝑗≥𝛽^′𝑥⁺_𝑗+𝑥⁻_𝑗²−𝑏₅𝑥_𝑗^𝛿𝑥⁺_𝑗−𝑥⁻_𝑗≥𝛽^′𝑥⁺_𝑗+𝑥⁻_𝑗²−𝑏₅2^𝛿𝑥⁰_𝑗^𝛿𝑥⁺_𝑗−𝑥⁻_𝑗. (3.48) Therefore, for sufficiently large 𝑗, 𝐽^′𝑥_𝑗≥𝛽^′𝑥⁺_𝑗+𝑥⁻_𝑗−𝑏₅2^𝛿𝑥⁰_𝑗^𝛿. (3.49) This implies that (3.45) holds, where 𝑏₃=𝑏₅2^𝛿/𝛽^′.
By (3.15) and (3.45), for 𝑗 large enough, there exist positive constants 𝑏₆,𝑏₇,𝑏^′₇,𝑏₈ such that 𝜓_∞𝑥_𝑗=12𝐴−𝐵_∞𝑥⁺_𝑗+𝑥⁻_𝑗,𝑥⁺_𝑗+𝑥⁻_𝑗−𝐽𝑥_𝑗≤𝑏₆𝑥⁺_𝑗+𝑥⁻_𝑗²+𝑏^′₇≤𝑏₈𝑥⁰_𝑗^2𝛿+𝑏₇. (3.50) Now, we claim that there exists 𝑏₉>0 such that, for 𝑗 large enough, ₀^2𝜋𝑥_𝑗^𝛾𝑑𝑡≥𝑏₉𝑥⁰_𝑗^𝛾, (3.51) In fact, for 𝛾>1, by (3.45) and the fact that 𝛿<1, we have ₀^2𝜋𝑥_𝑗,𝑥⁰_𝑗𝑑𝑡≤₀^2𝜋𝑥_𝑗^𝛾𝑑𝑡^1/𝛾₀^2𝜋𝑥⁰_𝑗^{𝛾/𝛾−1}𝑑𝑡^{𝛾−1/𝛾}≤𝑏₁₀₀^2𝜋𝑥_𝑗^𝛾𝑑𝑡^1/𝛾𝑥⁰_𝑗;_02𝜋𝑥_𝑗,𝑥⁰_𝑗𝑑𝑡=₀^2𝜋𝑥⁰_𝑗,𝑥⁰_𝑗𝑑𝑡+₀^2𝜋𝑥⁺_𝑗+𝑥⁻_𝑗,𝑥⁰_𝑗≥𝑑𝑡_02𝜋𝑥⁰_𝑗²𝑑𝑡−𝑥⁺_𝑗+𝑥⁻_𝑗𝐿2𝑥⁰_𝑗𝐿2≥𝑏₁₁𝑥⁰_𝑗²−𝑏₃𝛼²₂𝑥⁰_𝑗^1+𝛿−𝛼²₂𝜂𝑥⁰_𝑗≥𝑏₁₂𝑥⁰_𝑗², (3.52) for 𝑗 large enough. This implies (3.51) for 𝛾>1.
For 𝛾=1, since 𝑀⁰(𝐴−𝐵_∞) is a finite dimensional subspace of 𝐸, we know that for any 𝑗, 𝑏₁₃𝑥⁰_𝑗≤𝑥⁰_𝑗∞≤𝑏₁₄𝑥⁰_𝑗. (3.53) where 𝑏₁₃,𝑏₁₄>0 are constants independent of 𝑗. Now we have ₀^2𝜋𝑥_𝑗,𝑥⁰_𝑗𝑑𝑡≤₀^2𝜋𝑥_𝑗𝑥⁰_𝑗𝑑𝑡≤₀^2𝜋𝑥_𝑗𝑑𝑡𝑥⁰_𝑗∞≤𝑏₁₅𝑥⁰_𝑗0^2𝜋𝑥_𝑗𝑑𝑡. (3.54) Combining (3.52) with (3.54), we get (3.51) for 𝛾=1.
On the other hand, by (𝐟₈)⁺𝜓_∞𝑥_𝑗=₀^2𝜋𝐺_∞𝑡,𝑥_𝑗𝑑𝑡≥₀^2𝜋𝑏₂𝑥_𝑗^𝛾𝑑𝑡−2𝜋𝑏₁₆≥𝑏₁₇𝑥⁰_𝑗^𝛾−2𝜋𝑏₁₆. (3.55) Since that 𝛾>2𝛿, we get a contradiction from (3.50) and (3.55). Therefore, {𝑥_𝑗} is bounded.
If (𝐟₈)⁻ holds, using a similar argument as above, we get a contradiction and completes our proof.

Theorem 3.12. Suppose 𝑓 satisfies (𝐟₁)–(𝐟₃) and (f₉)^±there exist positive constants 1≤𝛾<2, 0<𝛿<𝛾/2, and 𝑏₁,𝑏₂,𝐿 such that𝐺^′_∞(𝑡,𝑥)≤𝑏₁|𝑥|^𝛿,±𝐺^′_∞(𝑡,𝑥),𝑥≥𝑏₂|𝑥|^𝛾|𝑥|𝜋,∀≥𝐿,𝑡∈0,2. (3.56) Then (1.4) has at least 𝐼(𝐵_∞,𝐵₀) pairs of nonconstant 2𝜋-periodic solutions provided 𝐼(𝐵_∞,𝐵₀)>0.

Proof. If (f₉)⁺ holds, for 𝑗 large enough, by (3.45) and (3.51), we have ₀^2𝜋𝐺^′_∞𝑡,𝑥_𝑗,𝑥_𝑗𝑑𝑡≤−𝐽^′𝑥_𝑗,𝑥_𝑗+𝐴−𝐵_∞𝑥⁺_𝑗+𝑥⁻_𝑗,𝑥⁺_𝑗+𝑥⁻_𝑗≤𝑥_𝑗+𝑀₁𝑥⁺_𝑗+𝑥⁻_𝑗²≤𝑥⁰_𝑗+𝑀₂𝑥⁰_𝑗^𝛿+𝑀₃𝑥⁰_𝑗^2𝛿+𝑀₄;_02𝜋𝐺^′_∞𝑡,𝑥_𝑗,𝑥_𝑗𝑑𝑡≥𝑏₂₀^2𝜋𝑥_𝑗^𝛾𝑑𝑡−𝑀₅≥𝑀₆𝑥⁰_𝑗^𝛾−𝑀₅. (3.57) Since 𝛾>2𝛿, {𝑥⁰_𝑗} is bounded, so is {𝑥_𝑗}. Therefore, 𝐽 satisfies (J2).
In the case that (f₉)⁻ holds, using a similar argument, we can verify (J2). This completes the proof.

Example 3.13. Consider the following nonautonomous delay differential equation 𝑥^′(𝑡)𝜋=−𝑀𝑥𝑡−2𝑎+𝑏(𝑡)|𝑥(𝑡−(𝜋/2))|^5/2+𝑐|𝑥(𝑡−(𝜋/2))|⁴1+|𝑥(𝑡−(𝜋/2))|⁴, (3.58) where 𝑀 is a 4×4 matrix, 𝑎,𝑐 are constants, 𝑏∈𝐶([0,𝜋/2],ℝ⁺).
Case 1. Let 𝐴=diag(0.3,2.7,7.3,9.3), 𝑎=1, 𝑐=2, and 𝑏 arbitrary. Computing directly, we have 𝐼(𝐵₀(𝑡),𝐵_∞(𝑡))=10. Applying Theorem 3.6, equation (3.58) has at least 10 pairs of 2𝜋-periodic solutions.
Case 2. Let 𝐴=diag(0.3,2.7,5,10.5), 𝑎=2, 𝑐=1, and 𝑏 arbitrary. Then by Theorem 3.8, (3.58) has at least 8 pairs of nonconstant 2𝜋-periodic solutions.

Acknowledgments

This paper is supported by Xinmiao Program of Guangzhou University and National Natural Science Foundation of China (no. 11026059), SRF of Guangzhou Education Bureau (no. 10A012), and Foundation for Distinguished Young Talents in Higher Education of Guangdong (no. LYM09105).