Antiperiodic Solutions for a Generalized High-Order (𝑝,𝑞)-Laplacian Neutral Differential System with Delays in the Critical Case

Abstract

By applying the method of coincidence degree, some criteria are established for the existence of antiperiodic solutions for a generalized high-order (𝑝,𝑞)-Laplacian neutral differential system with delays (𝜑_𝑝((𝑥(𝑡)−𝑐𝑥(𝑡−𝜏))^(𝑘)))^{(𝑚−𝑘)}=𝐹(𝑡,𝑥_𝜃0(𝑡),𝑥^′_𝜃1(𝑡),…,𝑥_𝜃^(𝑘)_𝑘(𝑡),𝑦_𝜗0(𝑡),𝑦^′_𝜗1(𝑡),…,𝑦_𝜗^(𝑙)_𝑙(𝑡)), (𝜑_𝑞((𝑦(𝑡)−𝑑𝑦(𝑡−𝜎))^(𝑙)))^{(𝑛−𝑙)}=𝐺(𝑡,𝑦_𝜇0(𝑡),𝑦^′_𝜇1(𝑡),…,𝑦_𝜇^(𝑙)_𝑙(𝑡),𝑥_𝜈0(𝑡),𝑥^′_𝜈1(𝑡),…,𝑥_𝜈^(𝑘)_𝑘(𝑡)) in the critical case |𝑐|=|𝑑|=1. The results of this paper are completely new. Finally, an example is employed to illustrate our results.

1. Introduction

During the last twenty years, there have been quite a few results on the existence of periodic solutions for delay differential equations and neutral functional differential equations. We can see [1–7]. For example, the authors of [8–11] investigated the existence of periodic solutions for the following types of neutral functional differential equations: (𝑥(𝑡)−𝑐𝑥(𝑡−𝜏))^′=𝑔₁(𝑡,𝑥(𝑡))+𝑔₂(((𝑡),(𝑥(𝑡)(𝑡,𝑥𝑡−𝛿))+𝑒−𝑐𝑥𝑡−𝜏))^′′(𝑥((𝑡),𝜑+𝑔𝑡−𝛿))=𝑒_𝑝(𝑥(𝑡)−𝑐𝑥(𝑡−𝜏))^′′(𝑡(𝑡),𝜑+𝑔𝑡,𝑥𝑡−𝜂)=𝑒_𝑝𝑥^′(𝑡)−𝑐𝑥^′(𝑡−𝜏)^′=𝑓(𝑥(𝑡))𝑥^′(𝑡)+𝜉(𝑡)𝑔𝑥𝑡−𝜂(𝑡)+𝑒(𝑡), (1) respectively. But the condition of constant |𝑐|≠1 is required. For example, under the assumption |𝑐|≠1, they obtain that 𝐴∶𝐶_2𝜋∶={𝑥∶𝑥∈𝐶(ℝ,ℝ),𝑥(𝑡+2𝜋)≡𝑥(𝑡)}→𝐶_2𝜋, (𝐴𝑥)(𝑡)=𝑥(𝑡)−𝑐𝑥(𝑡−𝜏) has a unique inverse 𝐴⁻¹∶𝐶_2𝜋→𝐶_2𝜋 defined by 𝐴⁻¹𝑓(𝑡)=_𝑗∑_≥0𝑐^𝑗𝑓𝑡−𝑗𝜏,|𝑐|<1,−∑_𝑗≥1𝑐^−𝑗𝑓𝑡+𝑗𝜏,|𝑐|>1, (2) and then ₀^2𝜋𝐴⁻¹𝑓(𝑠)d1𝑠≤|1−|𝑐||₀^2𝜋𝑓(𝑠)d𝑠,∀𝑓∈𝐶_2𝜋, (3) which was crucial to obtaining estimation of a priori bounds of periodic solutions in the noncritical case |𝑐|≠1.

Under the critical case |𝑐|=1, the authors of [12–15] studied a first-order neutral differential equation (𝑥(𝑡)−𝑐𝑥(𝑡−𝜏))^′=𝑔(𝑡,𝑥(𝑡−𝛿(𝑡)))+𝑒(𝑡), (4) a Duffing differential equation of neutral type (𝑥(𝑡)−𝑐𝑥(𝑡−𝜏))^′′=𝑔(𝑡,𝑥(𝑡−𝛿(𝑡)))+𝑒(𝑡), (5) a Rayleigh differential equation of neutral type (𝑥(𝑡)−𝑐𝑥(𝑡−𝜏))^′′=𝑓(𝑥(𝑡))𝑥^′(𝑡)+𝑔(𝑡,𝑥(𝑡−𝛿(𝑡)))+𝑒(𝑡), (6) and a 𝑝-Laplacian differential equation of neutral type 𝜑_𝑝(𝑥(𝑡)−𝑐𝑥(𝑡−𝜏))^′′=𝑓(𝑥(𝑡))𝑥^′(𝑡)+𝑔(𝑡,𝑥(𝑡−𝛿(𝑡)))+𝑒(𝑡), (7) respectively.

In the past thirty years, there has been a great deal of work on the problem of the periodic solutions of high-order nonlinear differential equations, especially for the third-order and fourth-order differential equations which have been used to describe nonlinear oscillations [16–20], and fluid mechanical and nonlinear elastic mechanical phenomena [21–27]. In [28], Jin and Lu discussed the existence of periodic solutions of third-order 𝑝-Laplacian equation with a deviating argument 𝜑_𝑝𝑥^′′(𝑡)^′+𝑓𝑡,𝑥^′(𝑡),𝑥^′′(𝑡)+𝑔𝑡,𝑥𝑡−𝜂(𝑡)=𝑒(𝑡). (8)

Before continuing, by applying Mawhin’s continuation theorem of coincidence degree theory, the authors of [29] studied the existence of periodic solutions for a fourth-order 𝑝-Laplacian equation with a deviating argument 𝜑_𝑝𝑥^′′(𝑡)^′′+𝑓𝑥^′′(𝑡)+𝑔𝑥𝑡−𝜂(𝑡)=𝑒(𝑡). (9)

Arising from problems in applied sciences, it is well known that the existence of antiperiodic solutions plays a key role in characterizing the behavior of nonlinear differential equations as a special periodic solution and has been extensively studied by many authors during the past twenty years; see [30–44] and references therein. For example, antiperiodic trigonometric polynomials are important in the study of interpolation problems [45, 46], and antiperiodic wavelets are discussed in [47].

However, to the best of our knowledge, due to the neutral term and 𝑝-Laplace operator term, the existence of antiperiodic solutions for (4)–(7) is very difficult to obtain by applying traditional researching methods. Therefore, to date, there are few papers to investigate the existence of antiperiodic solutions for (4)–(7).

Motivated by above statements, in this paper, we will apply the method of coincidence degree to study the existence of antiperiodic solutions for a generalized high-order (𝑝,𝑞)-Laplacian neutral differential system with delays in the critical case 𝜑_𝑝(𝑥(𝑡)−𝑐𝑥(𝑡−𝜏))^{(𝑘)(𝑚−𝑘)}=𝐹𝑡,𝑥_𝜃0(𝑡),𝑥^′_𝜃1(𝑡),…,𝑥_𝜃^(𝑘)_𝑘(𝑡),𝑦_𝜗0(𝑡),𝑦^′_𝜗1(𝑡),…,𝑦_𝜗^(𝑙)_𝑙(𝑡),𝜑_𝑞𝑦(𝑡)−𝑑𝑦(𝑡−𝜎)^{(𝑙)(𝑛−𝑙)}=𝐺𝑡,𝑦_𝜇0(𝑡),𝑦^′_𝜇1(𝑡),…,𝑦_𝜇^(𝑙)_𝑙(𝑡),𝑥_𝜈0(𝑡),𝑥^′_𝜈1(𝑡),…,𝑥_𝜈^(𝑘)_𝑘(𝑡), (10) where |𝑐|=|𝑑|=1, 𝜑_𝑝(𝑠)=|𝑠|^𝑝−2𝑠, 𝜑_𝑞(𝑠)=|𝑠|^𝑞−2𝑠, and 𝑠∈ℝ, 𝑝,𝑞≥2; 𝜃_𝑖(𝑡)(0≤𝑖≤𝑘), 𝜗_𝑖(𝑡)(0≤𝑖≤𝑙), 𝜇_𝑖(𝑡)(0≤𝑖≤𝑙), and 𝜈_𝑖(𝑡)(0≤𝑖≤𝑘)∈𝐶(ℝ,ℝ) are 𝜋-periodic functions; for any 𝜌(𝑡)∈𝐶(ℝ,ℝ), 𝑥_𝜌(𝑡) is defined by 𝑥_𝜌(𝑡)=𝑥(𝑡−𝜌(𝑡)); 𝐹,𝐺∈𝐶(ℝ^𝑘+𝑙+3,ℝ) are 2𝜋-periodic in their first arguments; 𝜏, 𝜎 are constants; 𝑚, 𝑛, 𝑘, and 𝑙 are nonnegative integers, 𝑘<𝑚, 𝑙<𝑛.

Throughout this paper, we will denote by ℕ the set of nonnegative integers and by ℕ₁ the set of odd positive integers.

Let 𝑝=𝑞, 𝑘=𝑙, 𝑚=𝑛, 𝑥=𝑦, 𝑐=𝑑, 𝜏=𝜎, and 𝐹=𝐺 in system (10), then system (10) is reformulated as 𝜑_𝑝(𝑥(𝑡)−𝑐𝑥(𝑡−𝜏))^{(𝑘)(𝑚−𝑘)}=𝐹𝑡,𝑥_𝜃0(𝑡),𝑥^′_𝜃1(𝑡),…,𝑥_𝜃^(𝑘)_𝑘(𝑡),𝑥_𝜗0(𝑡),𝑥^′_𝜗1(𝑡),…,𝑥_𝜗^(𝑙)_𝑙(𝑡). (11) Furthermore, one can easily obtain the following.(a) If 𝑝=𝑞=2,𝑥=𝑦,|𝑐|=|𝑑|=1,𝑘=𝑙=0,𝑚=𝑛=1,𝜃₀(𝑡)=𝛿(𝑡), 𝐹=𝐹𝑡,𝑥_𝜃0(𝑡)=𝑔(𝑡,𝑥(𝑡−𝛿(𝑡)))+𝑒(𝑡), (12) then system (10) reduces to (4). (b) If 𝑝=𝑞=2,𝑥=𝑦,|𝑐|=|𝑑|=1,𝑘=𝑙=0,𝑚=𝑛=2,𝜃₀(𝑡)=𝛿(𝑡), 𝐹=𝐹𝑡,𝑥_𝜃0(𝑡)=𝑔(𝑡,𝑥(𝑡−𝛿(𝑡)))+𝑒(𝑡), (13) then system (10) reduces to (5). (c) If 𝑝=𝑞=2,𝑥=𝑦,|𝑐|=|𝑑|=1,𝑘=𝑙=0,𝑚=𝑛=2,𝜃₀(𝑡)=𝜃₁(𝑡)≡0,𝜗₀(𝑡)=𝛿(𝑡), 𝐹=𝐹𝑡,𝑥_𝜃0(𝑡),𝑥^′_𝜃1(𝑡),𝑥_𝜗0(𝑡)=𝑓(𝑥(𝑡))𝑥^′(𝑡)+𝑔(𝑡,𝑥(𝑡−𝛿(𝑡)))+𝑒(𝑡), (14) then system (10) reduces to (6). (d) If 𝑝=𝑞,𝑥=𝑦,|𝑐|=|𝑑|=1,𝑘=𝑙=1,𝑚=𝑛=2,𝜃₀(𝑡)=𝜃₁(𝑡)≡0,𝜗₀(𝑡)=𝛿(𝑡), 𝐹=𝐹𝑡,𝑥_𝜃0(𝑡),𝑥^′_𝜃1(𝑡),𝑥_𝜗0(𝑡)=𝑓(𝑥(𝑡))𝑥^′(𝑡)+𝑔(𝑡,𝑥(𝑡−𝛿(𝑡)))+𝑒(𝑡), (15) then system (10) reduces to (7).

The main purpose of this paper is to establish sufficient conditions for the existence of 𝜋-antiperiodic solutions to system (10) by using the method of coincidence degree.

The organization of this paper is as follows. In Section 2, we make some preparations. In Section 3, by using the method of coincidence degree, we establish sufficient conditions for the existence of 𝜋-antiperiodic solutions to system (10). An illustrative example is given in Section 4.

2. Preliminaries

The following continuation theorem of coincidence degree is crucial in the arguments of our main results.

Lemma 1 (see [48]). Let 𝕏,𝕐 be two Banach spaces; let Ω⊂𝕏 be open bounded and symmetric with 0∈Ω. Suppose that 𝐿∶𝐷(𝐿)⊂𝕏→𝕐 is a linear Fredholm operator of index zero with 𝐷(𝐿)∩Ω≠∅ and 𝑁∶Ω→𝕐 is 𝐿-compact. Further, one also assumes that
(H) 𝐿𝑥−𝑁𝑥≠𝜆(−𝐿𝑥−𝑁(−𝑥)), for all 𝑥∈𝐷(𝐿)∩𝜕Ω,𝜆∈(0,1].
Then equation 𝐿𝑥=𝑁𝑥 has at least one solution on 𝐷(𝐿)∩Ω.

Definition 2. Let 𝑢(𝑡)∶ℝ→ℝ be continuous. 𝑢(𝑡) is said to be 𝑇/2-antiperiodic on ℝ, if 𝑢()(𝑡)𝑇𝑡+𝑇=𝑢,𝑢𝑡+2=−𝑢(𝑡),∀𝑡∈ℝ. (16)

We will adopt the following notations: 𝐶^𝑘_2𝜋∶=𝑢∈𝐶(ℝ,ℝ)∶𝑢is2𝜋-periodic,𝑘∈ℕ,|𝑢|_∞=max_{[]𝑡∈0,2𝜋}|𝑢(𝑡)|, (17) where 𝑢 is a 2𝜋-periodic function.

For the sake of convenience, we introduce the following assumptions. (𝐻₁) There exist nonnegative constants 𝛼₁,𝛼₂,…,𝛼_𝑘+𝑙+2,𝛽₁,𝛽₂,…,𝛽_𝑘+𝑙+2, such that 𝐹𝑡,𝑠₁,𝑠₂,…,𝑠_𝑘+𝑙+2−𝐹𝑡,𝑧₁,𝑧₂,…,𝑧_𝑘+𝑙+2≤^𝑘+𝑙+2_𝑖=1𝛼_𝑖𝑠_𝑖−𝑧_𝑖,𝐺𝑡,𝑠₁,𝑠₂,…,𝑠_𝑘+𝑙+2−𝐺𝑡,𝑧₁,𝑧₂,…,𝑧_𝑘+𝑙+2≤^𝑘+𝑙+2_𝑖=1𝛽_𝑖𝑠_𝑖−𝑧_𝑖 (18) for any (𝑡,𝑠₁,𝑠₂,…,𝑠_𝑘+𝑙+2), (𝑡,𝑧₁,𝑧₂,…,𝑧_𝑘+𝑙+2)∈ℝ^𝑘+𝑙+3. (𝐻₂)For all (𝑡,𝑠₁,𝑠₂,…,𝑠_𝑘+𝑙+2)∈ℝ^𝑘+𝑙+3, 𝐹𝑡+𝜋,−𝑠₁,−𝑠₂,…,−𝑠_𝑘+𝑙+2=−𝐹𝑡,𝑠₁,𝑠₂,…,𝑠_𝑘+𝑙+2,𝐺𝑡+𝜋,−𝑠₁,−𝑠₂,…,−𝑠_𝑘+𝑙+2=−𝐺𝑡,𝑠₁,𝑠₂,…,𝑠_𝑘+𝑙+2. (19)

In order to apply Lemma 1 to study the existence of antiperiodic solutions for system (10), we set 𝕏=𝑥=𝑥₁(𝑡),𝑥₂(𝑡),𝑦₁(𝑡),𝑦₂(𝑡)^𝑇∈𝐶^𝑘_2𝜋×𝐶^{𝑚−𝑘−1}_2𝜋×𝐶^𝑙_2𝜋×𝐶^{𝑛−𝑙−1}_2𝜋()(𝑡,𝑥∶𝑥𝑡+𝜋=−𝑥)𝕐=𝑥=₁(𝑡),𝑥₂(𝑡),𝑦₁(𝑡),𝑦₂(𝑡)^𝑇∈𝐶⁰_2𝜋×𝐶⁰_2𝜋×𝐶⁰_2𝜋×𝐶⁰_2𝜋∶𝑥(𝑡+𝜋)=−𝑥(𝑡) (20) are two Banach spaces with the norms ‖𝑥‖_𝕏=^𝑘_𝑖=0𝑥₁^(𝑖)_∞+^{𝑚−𝑘−1}_𝑖=0𝑥₂^(𝑖)_∞+^𝑙_𝑖=0𝑦₁^(𝑖)_∞+^{𝑛−𝑙−1}_𝑖=0𝑦₂^(𝑖)_∞,‖𝑥‖_𝕐=²_𝑗=1𝑥_𝑗∞+𝑦_𝑗∞, (21) respectively. Define 𝔻=𝑥=𝑥₁(𝑡),𝑥₂(𝑡),𝑦₁(𝑡),𝑦₂(𝑡)^𝑇∈𝐶^𝑘_2𝜋×𝐶^𝑚−𝑘_2𝜋×𝐶^𝑙_2𝜋×𝐶^𝑛−𝑙_2𝜋∶𝑥(𝑡+𝜋)=−𝑥(𝑡) (22) and two difference operators 𝐴 and 𝐵 as follows: (𝑡)(𝑡)(),(𝑡)(𝑡)().𝐴∶𝕐⟶𝕐,𝐴𝑥)(=𝑥−𝑐𝑥𝑡−𝜏𝐵∶𝕐⟶𝕐,𝐵𝑦=𝑦−𝑑𝑦𝑡−𝜎 (23) Then system (10) reduces to 𝐴𝑥₁^(𝑘)(𝑡)=𝜑_𝑝′𝑥₂(𝑡),𝑥₂^{(𝑚−𝑘)}(𝑡)=𝐹𝑡,𝑥_1𝜃0(𝑡),𝑥^′_1𝜃1(𝑡),…,𝑥^(𝑘)_{1𝜃𝑘(𝑡)},𝑦_1𝜗0(𝑡),𝑦^′_1𝜗1(𝑡),…,𝑦_𝜗^(𝑙)_1𝑙(𝑡),𝐵𝑦₁^(𝑙)(𝑡)=𝜑_𝑞′𝑦₂(𝑡),𝑦₂^{(𝑛−𝑙)}(𝑡)=𝐺𝑡,𝑦_1𝜇0(𝑡),𝑦^′_1𝜇1(𝑡),…,𝑦^(𝑙)_{1𝜇𝑙(𝑡)},𝑥_1𝜈0(𝑡),𝑥^′_1𝜈1(𝑡),…,𝑥^(𝑘)_{1𝜈𝑘(𝑡)}, (24) where 1/𝑝+1/𝑝^′=1,1/𝑞+1/𝑞^′=1,1<𝑝^′,𝑞^′≤2. Obviously, the existence of antiperiodic solutions to system (10) is equivalent to that of antiperiodic solutions to system (24). Thus, the problem of finding a 𝜋-antiperiodic solution for system (10) reduces to finding one for system (24).

Define a linear operator 𝐿∶𝐷(𝐿)≡𝔻⊂𝕏→𝕐 by setting 𝑥𝐿𝑥=𝐿¹𝑥₂𝑦₁𝑦₂=𝐴𝑥₁^(𝑘)𝑥₂^{(𝑚−𝑘)}𝐵𝑦₁^(𝑙)𝑦₂^{(𝑛−𝑙)},∀𝑥∈𝐷(𝐿) (25) and 𝑁∶𝕏→𝕐 by setting 𝑥𝑁𝑥=𝑁¹𝑥₂𝑦₁𝑦₂=𝜑_𝑝′𝑥₂(𝑡)𝐹𝑡,𝑥_1𝜃0(𝑡),𝑥^′_1𝜃1(𝑡),…,𝑥^(𝑘)_{1𝜃𝑘(𝑡)},𝑦_1𝜗0(𝑡),𝑦^′_1𝜗1(𝑡),…,𝑦^(𝑙)_{1𝜗𝑙(𝑡)}𝜑_𝑞′𝑦₂(𝑡)𝐺𝑡,𝑦_1𝜇0(𝑡),𝑦^′_1𝜇1(𝑡),…,𝑦^(𝑙)_{1𝜇𝑙(𝑡)},𝑥_1𝜈0(𝑡),𝑥^′_1𝜈1(𝑡),…,𝑥^(𝑘)_{1𝜈𝑘(𝑡)}. (26) It is easy to see that Ker𝐿={0},Im𝐿=𝑥∈𝕐∶₀^2𝜋𝑥₁(𝑠)𝑥₂(𝑠)𝑦₁(𝑠)𝑦₂(𝑠)d0000𝑠=≡𝕐. (27) Thus, dimKer𝐿=0=codimIm𝐿, and 𝐿 is a linear Fredholm operator of index zero.

Define the continuous projector 𝑃∶𝕏→Ker𝐿 and the averaging projector 𝑄∶𝕐→𝕐 by 𝑃𝑥₁𝑥₂𝑦₁𝑦₂𝑥=𝑄¹𝑥₂𝑦₁𝑦₂=12𝜋₀^2𝜋𝑥₁(𝑠)𝑥₂(𝑠)𝑦₁(𝑠)𝑦₂(𝑠)d0000𝑠≡. (28) Hence Im𝑃=Ker𝐿 and Ker 𝑄=Im𝐿. Denoting by 𝐿_𝑃⁻¹∶Im𝐿→𝐷(𝐿)∩Ker𝑃 the inverse of 𝐿|_{𝐷(𝐿)∩Ker}𝑃, we have 𝐿_𝑃⁻¹𝑥₁𝑥₂𝑦₁𝑦₂(𝑡)=𝐴⁻¹𝐾𝑥₁(𝑡)𝐾𝑥₂(𝑡)𝐵⁻¹𝐾𝑦₁(𝑡)𝐾𝑦₂𝑡),( (29) where 𝐾𝑥₁(𝑡)=^𝑘−1_𝑖=01𝑖!𝐴ℎ₁^(𝑖)(0)𝑡^𝑖+1(𝑘−1)!^𝑡₀(𝑡−𝑠)^𝑘−1𝑥₁(𝑠)d𝑠,𝐾𝑥₂(𝑡)=^{𝑚−𝑘−1}_𝑖=01ℎ𝑖!^2(𝑖)(0)𝑡^𝑖+1(𝑚−𝑘−1)!^𝑡₀(𝑡−𝑠)^{𝑚−𝑘−1}𝑥₂(𝑠)d𝑠,𝐾𝑦₁(𝑡)=^𝑙−1_𝑖=01𝑖!𝐵ℎ₃^(𝑖)(0)𝑡^𝑖+1(𝑙−1)!^𝑡₀(𝑡−𝑠)^𝑙−1𝑦₁(𝑠)d𝑠,𝐾𝑦₂(𝑡)=^{𝑛−𝑙−1}_𝑖=01ℎ𝑖!^4(𝑖)(0)𝑡^𝑖+1(𝑛−𝑙−1)!^𝑡₀(𝑡−𝑠)^{𝑛−𝑙−1}𝑦₂(𝑠)d𝑠, (30) in which (𝐴ℎ₁)^(𝑖)(0)(𝑖=0,1,…,𝑘−1) are decided by 𝐸₁𝑍₁=𝐵₁, where 𝐸₁=𝑐200⋯00₁𝑐20⋯00₂𝑐₁𝑐2⋯00⋮⋮⋮⋮⋮_𝑘−2𝑐_𝑘−3𝑐_𝑘−4𝑐⋯20_𝑘−1𝑐_𝑘−2𝑐_𝑘−3⋯𝑐₁2_𝑘×𝑘,𝑍₁=𝐴ℎ₁^(𝑘−1)(0)𝐴ℎ₁^(𝑘−2)(0)𝐴ℎ₁^(𝑘−3)(0)⋮𝐴ℎ₁^′(0)𝐴ℎ₁(0)_𝑘×1, (31)𝐵₁=(𝑏₁₁,𝑏₁₂,…,𝑏_1𝑘)^𝑇,𝑏_1𝑖=−(1/𝑖!)∫₀^2𝜋(𝜋−𝑠)^𝑖𝑥₁(𝑠)d𝑠,𝑐_𝑗=(𝜋)^𝑗/𝑗!, and 𝑗=1,2,…,𝑘−1;ℎ₂^(𝑖)(0)(𝑖=0,1,…,𝑚−𝑘−1) are decided by 𝐸₂𝑍₂=𝐵₂, where 𝐸₂=𝑐20⋯00₁𝑐2⋯00₂𝑐₁𝑐⋯00⋮⋮⋮⋮_{𝑚−𝑘−2}𝑐_{𝑚−𝑘−3}𝑐⋯20_{𝑚−𝑘−1}𝑐_{𝑚−𝑘−2}⋯𝑐₁2_{(𝑚−𝑘)×(𝑚−𝑘)},𝑍₂=ℎ₂^{(𝑚−𝑘−1)}(0)ℎ₂^{(𝑚−𝑘−2)}(0)ℎ₂^{(𝑚−𝑘−3)}(0)⋮ℎ^′₂(0)ℎ₂(0)_{(𝑚−𝑘)×1}, (32)𝐵₂=(𝑏₂₁,𝑏₂₂,…,𝑏_{2(𝑚−𝑘)})^𝑇,𝑏_2𝑖=−(1/𝑖!)∫₀^2𝜋(𝜋−𝑠)^𝑖𝑥₂(𝑠)d𝑠,𝑐_𝑗=(𝜋)^𝑗/𝑗!, and 𝑗=1,2,…,𝑚−𝑘−1;(𝐵ℎ₃)^(𝑖)(0)(𝑖=0,1,…,𝑙−1) are decided by 𝐸₃𝑍₃=𝐵₃, where 𝐸₃=𝑐200⋯00₁𝑐20⋯00₂𝑐₁𝑐2⋯00⋮⋮⋮⋮⋮_𝑙−2𝑐_𝑙−3𝑐_𝑙−4𝑐⋯20_𝑙−1𝑐_𝑙−2𝑐_𝑙−3⋯𝑐₁2_𝑙×𝑙,𝑍₃=𝐵ℎ₃^(𝑙−1)(0)𝐵ℎ₃^(𝑙−2)(0)𝐵ℎ₃^(𝑙−3)(0)⋮𝐵ℎ₃^′(0)𝐵ℎ₃(0)_𝑙×1, (33)𝐵₃=(𝑏₃₁,𝑏₃₂,…,𝑏_3𝑙)^𝑇,𝑏_3𝑖=−(1/𝑖!)∫₀^2𝜋(𝜋−𝑠)^𝑖𝑦₁(𝑠)d𝑠,𝑐_𝑗=(𝜋)^𝑗/𝑗!, and 𝑗=1,2,…,𝑙−1;ℎ₄^(𝑖)(0)(𝑖=0,1,…,𝑛−𝑙−1) are decided by 𝐸₄𝑍₄=𝐵₄, where 𝐸₄=𝑐20⋯00₁𝑐2⋯00₂𝑐₁𝑐⋯00⋮⋮⋮⋮_{𝑛−𝑙−2}𝑐_{𝑛−𝑙−3}𝑐⋯20_{𝑛−𝑙−1}𝑐_{𝑛−𝑙−2}⋯𝑐₁2_{(𝑛−𝑙)×(𝑛−𝑙)},𝑍₄=ℎ₄^{(𝑛−𝑙−1)}(0)ℎ₄^{(𝑛−𝑙−2)}(0)ℎ₄^{(𝑛−𝑙−3)}(0)⋮ℎ^′₄(0)ℎ₄(0)_{(𝑛−𝑙)×1}, (34)𝐵₄=(𝑏₄₁,𝑏₄₂,…,𝑏_{4(𝑛−𝑙)})^𝑇,𝑏_4𝑖=−(1/𝑖!)∫₀^2𝜋(𝜋−𝑠)^𝑖𝑦₂(𝑠)d𝑠,𝑐_𝑗=(𝜋)^𝑗/𝑗!, and 𝑗=1,2,…,𝑛−𝑙−1.

Clearly, 𝑄𝑁 and 𝐿_𝑃⁻¹(𝐼−𝑄)𝑁 are continuous. Using the Arzela-Ascoli theorem, it is not difficult to show that 𝑄𝑁(Ω),𝐿_𝑃⁻¹(𝐼−𝑄)𝑁(Ω) are relatively compact for any open bounded set Ω⊂𝕏. Therefore, 𝑁 is 𝐿-compact on Ω for any open bounded set Ω⊂𝕏.

Lemma 3 (see [12]). Let |𝜏|=(𝑏/𝑎)𝜋, where 𝑎 and 𝑏 are coprime positive integers. Then (1)if 𝑐=−1, 𝑏 is odd and 𝑎 is even, then 𝜔₁∶=inf_𝑘∈ℕ11−𝑐𝑒^{−𝑘𝑖𝜏}=inf_𝑘∈ℕ1[2(1+cos𝑘𝜏)]^1/2>0; (35)(2)if 𝑐=1 and 𝑏 is odd, then 𝜔₂∶=inf_𝑘∈ℕ11−𝑐𝑒^{−𝑘𝑖𝜏}=inf_𝑘∈ℕ1[2(1−cos𝑘𝜏)]^1/2>0; (36)(3)if 𝑐=1 and 𝑎=𝑏=1, then 𝜔₃∶=inf_𝑘∈ℕ11−𝑐𝑒^{−𝑘𝑖𝜏}=inf_𝑘∈ℕ1[2(1−cos𝑘𝜏)]^1/2=2>0. (37)

Lemma 4 (see [13]). Suppose 𝐴∶𝕐→𝕐, (𝐴𝑥)(𝑡)=𝑥(𝑡)−𝑐𝑥(𝑡−𝜏) for all 𝑡∈[0,2𝜋]. Then the following propositions are true. (1)If 𝑐=−1, |𝜏|=(𝑏/𝑎)𝜋, where 𝑎 and 𝑏 are coprime positive integers with 𝑏 odd and 𝑎 even, then 𝐴 has a unique inverse 𝐴⁻¹∶𝕐→𝕐 satisfying ‖𝐴⁻¹‖≤(1/𝜔₁). (2)If 𝑐=1, |𝜏|=(𝑏/𝑎)𝜋, where 𝑎 and 𝑏 are coprime positive integers with 𝑏 odd, then 𝐴 has a unique inverse 𝐴⁻¹∶𝕐→𝕐 satisfying ‖𝐴⁻¹‖≤(1/𝜔₂). (3)If 𝑐=1, |𝜏|=𝜋, then 𝐴 has a unique inverse 𝐴⁻¹∶𝕐→𝕐 satisfying ‖𝐴⁻¹‖≤(1/𝜔₃).

Lemma 5 (see [12]). Let |𝜎|=(𝜈/𝜇)𝜋, where 𝜇 and 𝜈 are coprime positive integers. Then (1)if 𝑑=−1, 𝜈 is odd and 𝜇 is even, then 𝜔₁∶=inf_𝑘∈ℕ11−𝑑𝑒^{−𝑘𝑖𝜏}=inf_𝑘∈ℕ1[2(1+cos𝑘𝜏)]^1/2>0; (38)(2)if 𝑑=1 and 𝜈 is odd, then 𝜔₂∶=inf_𝑘∈ℕ11−𝑑𝑒^{−𝑘𝑖𝜏}=inf_𝑘∈ℕ1[2(1−cos𝑘𝜏)]^1/2>0; (39)(3)if 𝑑=1 and 𝜇=𝜈=1, then 𝜔₃∶=inf_𝑘∈ℕ11−𝑑𝑒^{−𝑘𝑖𝜏}=inf_𝑘∈ℕ1[2(1−cos𝑘𝜏)]^1/2=2>0. (40)

Lemma 6 (see [13]). Suppose 𝐵∶𝕐→𝕐,(𝐵𝑥)(𝑡)=𝑥(𝑡)−𝑑𝑥(𝑡−𝜎) for all 𝑡∈[0,2𝜋]. Then the following propositions are true. (1)If 𝑑=−1,|𝜎|=(𝜈/𝜇)𝜋, where 𝜇 and 𝜈 are coprime positive integers with 𝜈 odd and 𝜇 even, then 𝐵 has a unique inverse 𝐵⁻¹∶𝕐→𝕐 satisfying ‖𝐵⁻¹‖≤(1/𝜔₁). (2)If 𝑑=1,|𝜎|=(𝜈/𝜇)𝜋, where 𝜇 and 𝜈 are coprime positive integers with 𝜈 odd, then 𝐵 has a unique inverse 𝐵⁻¹∶𝕐→𝕐 satisfying ‖𝐵⁻¹‖≤(1/𝜔₂). (3)If 𝑑=1,|𝜎|=𝜋, then 𝐵 has a unique inverse 𝐵⁻¹∶𝕐→𝕐 satisfying ‖𝐵⁻¹‖≤(1/𝜔₃).

3. Main Result

In this section, we will study the existence of 𝜋-antiperiodic solutions for system (10) in the critical case |𝑐|=|𝑑|=1.

Theorem 7. Assume that (H₁)-(H₂) hold. Suppose further that 𝑐=−1, |𝜏|=(𝑏/𝑎)𝜋, where 𝑎 and 𝑏 are coprime positive integers with 𝑏 odd and 𝑎 even, then system (10) has at least one 𝜋-antiperiodic solution, if one of the following conditions holds. (1)𝑑=−1,|𝜎|=(𝜈/𝜇)𝜋, where 𝜇 and 𝜈 are coprime positive integers with 𝜈 odd and 𝜇 even, and 𝑊₁+𝑊₂<1, where 𝑊₁=max^𝑘+1_𝑖=1𝜋^{𝑚−𝑖+1}𝜔₁𝛼_𝑖,^𝑘+1_𝑖=1𝜋^{𝑛−𝑙+𝑘−𝑖+1}𝜔₁𝛽_𝑙+𝑖+1,𝑊₂=max^𝑙+1_𝑖=1𝜋^{𝑚−𝑘+𝑙−𝑖+1}𝜔₁𝛼_𝑘+𝑖+1,^𝑙+1_𝑖=1𝜋^{𝑛−𝑖+1}𝜔₁𝛽_𝑖. (41)(2)𝑑=1,|𝜎|=(𝜈/𝜇)𝜋, where 𝜇 and 𝜈 are coprime positive integers with 𝜈 odd, and 𝑊₁+𝑊₃<1, where 𝑊₃=max^𝑙+1_𝑖=1𝜋^{𝑚−𝑘+𝑙−𝑖+1}𝜔₂𝛼_𝑘+𝑖+1,^𝑙+1_𝑖=1𝜋^{𝑛−𝑖+1}𝜔₂𝛽_𝑖. (42)(3)𝑑=1,|𝜎|=𝜋, and 𝑊₁+𝑊₄<1, where 𝑊₄=max^𝑙+1_𝑖=1𝜋^{𝑚−𝑘+𝑙−𝑖+1}𝜔₃𝛼_𝑘+𝑖+1,^𝑙+1_𝑖=1𝜋^{𝑛−𝑖+1}𝜔₃𝛽_𝑖, (43) in which 𝜔₁,𝜔₂, and 𝜔₃ are constants defined by Lemma 3 or Lemma 5.

Proof. As the proof of other cases works almost exactly as the proof of case (1), we will prove case (1) only. Consider the operator equation 𝐿𝑥−𝑁𝑥=𝜆(−𝐿𝑥−𝑁(−𝑥)),𝜆∈(0,1]. (44) Then we have 𝐴𝑥₁^(𝑘)(𝑡)=1𝜑1+𝜆^𝑝′𝑥₂(𝑡)−𝜆𝜑1+𝜆^𝑝′−𝑥₂(𝑡),𝑥₂^{(𝑚−𝑘)}(𝑡)=1𝐹−𝜆1+𝜆𝑡,𝑥,𝑦𝐹,1+𝜆𝑡,−𝑥,−𝑦𝐵𝑦₁^(𝑙)(𝑡)=1𝜑1+𝜆^𝑞′𝑦₂(𝑡)−𝜆𝜑1+𝜆^𝑞′−𝑦₂(𝑡),𝑦₂^{(𝑛−𝑙)}(𝑡)=1𝐺−𝜆1+𝜆𝑡,𝑦,𝑥𝐺,1+𝜆𝑡,−𝑦,−𝑥 (45) where 𝐹𝑡,𝑥₁,𝑦₁=𝐹𝑡,𝑥_1𝜃0(𝑡),𝑥^′_1𝜃1(𝑡),…,𝑥^(𝑘)_{1𝜃𝑘(𝑡)},𝑦_1𝜗0(𝑡),𝑦^′_1𝜗1(𝑡),…,𝑦^(𝑙)_{1𝜗𝑙(𝑡)},𝐺𝑡,𝑦₁,𝑥₁=𝐺𝑡,𝑦_1𝜇0(𝑡),𝑦^′_1𝜇1(𝑡),…,𝑦^(𝑙)_{1𝜇𝑙(𝑡)},𝑥_1𝜈0(𝑡),𝑥^′_1𝜈1(𝑡),…,𝑥^(𝑘)_{1𝜈𝑘(𝑡)},𝐹𝑡,−𝑥₁,−𝑦₁=𝐹𝑡,−𝑥_1𝜃0(𝑡),−𝑥^′_1𝜃1(𝑡),…,−𝑥^(𝑘)_{1𝜃𝑘(𝑡)},−𝑦_1𝜗0(𝑡),−𝑦^′_1𝜗1(𝑡),…,−𝑦^(𝑙)_{1𝜗𝑙(𝑡)},𝐺𝑡,−𝑦₁,−𝑥₁=𝐺𝑡,−𝑦_1𝜇0(𝑡),−𝑦^′_1𝜇1(𝑡),…,−𝑦^(𝑙)_{1𝜇𝑙(𝑡)},−𝑥_1𝜈0(𝑡),−𝑥^′_1𝜈1(𝑡),…,−𝑥^(𝑘)_{1𝜈𝑘(𝑡)}. (46) Suppose that 𝑥(𝑡)=(𝑥₁(𝑡),𝑥₂(𝑡),𝑦₁(𝑡),𝑦₂(𝑡))^𝑇∈𝔻 is an arbitrary 𝜋-antiperiodic solution of system (45). Because 𝑥₁(𝑡)∈𝐶^𝑘_2𝜋 is 𝑇/2-antiperiodic, hence, we have ₀^2𝜋𝑥₁(𝑠)d𝑠=^𝜋₀𝑥₁(𝑠)d𝑠+_𝜋^2𝜋𝑥₁(𝑠)d𝑠=^𝜋₀𝑥₁(𝑠)d𝑠+^𝜋₀𝑥₁(𝑠+𝜋)d𝑠=0. (47) Then there exists a constant 𝜉∈[0,2𝜋] such that 𝑥₁(𝜉)=0. (48) Therefore, we have 𝑥₁(𝑡)=𝑥₁(𝜉)+^𝑡_𝜉𝑥^′₁(𝑠)d𝑠≤^𝑡_𝜉𝑥^′₁(𝑠)d𝑥𝑠,₁(𝑡)=𝑥₁(𝑡−2𝜋)=𝑥₁(𝜉)−^𝜉_𝑡−2𝜋𝑥^′₁(𝑠)d𝑠≤^𝜉_𝑡−2𝜋𝑥^′₁(𝑠)d𝑠 (49) for all 𝑡∈[𝜉,𝜉+2𝜋]. Combining the above two inequalities, we can get 𝑥_1∞=max_{𝑡∈[0,2𝜋]}𝑥₁(𝑡)=max_{𝑡∈[𝜉,𝜉+2𝜋]}𝑥₁(𝑡)≤max_{𝑡∈𝜉,𝜉+2𝜋}12^𝑡_𝜉𝑥^′₁(𝑠)d𝑠+^𝜉_𝑡−2𝜋𝑥^′₁(𝑠)d𝑠≤12₀^2𝜋𝑥^′₁(𝑠)d𝑠. (50) Similar to (50), one can easily get 𝑥^′_1∞≤12₀^2𝜋𝑥₁^′′(𝑠)d𝑥𝑠,_1′′∞≤12₀^2𝜋𝑥₁^′′′(𝑠)d𝑠,…,𝑥₁^(𝑘−1)_∞≤12₀^2𝜋𝑥₁^(𝑘)(𝑠)d𝑠, (51) which yield 𝑥₁^(𝑖)_∞≤𝜋^𝑘−𝑖𝑥₁^(𝑘)_∞for𝑖=0,1,…,𝑘−1. (52) By a parallel argument to (47)–(52), we can also obtain 𝑦₁^(𝑖)_∞≤𝜋^𝑙−𝑖𝑦₁^(𝑙)_∞,for𝑖=0,1,…,𝑙−1,𝑥₂^(𝑗)_∞≤𝜋^{𝑚−𝑘−𝑗−1}𝑥₂^{(𝑚−𝑘−1)}_∞,for𝑗=0,1,…,𝑚−𝑘−2,𝑦₂^(𝑗)_∞≤𝜋^{𝑛−𝑙−𝑗−1}𝑦₂^{(𝑛−𝑙−1)}_∞,for𝑗=0,1,…,𝑛−𝑙−2. (53)(54)(55) Since 𝑥₂(𝑡)∈𝐶^𝑚−𝑘_2𝜋 is 𝜋-antiperiodic, similar to (47), there exists a constant 𝜁 such that 𝑥₂^{(𝑚−𝑘−1)}(𝜁)=0. By a parallel argument to (50), we can obtain from (45), (𝐻₁), (52), and (53) that 2𝑥₂^{(𝑚−𝑘−1)}_∞≤2𝑥₂^{(𝑚−𝑘−1)}(𝜁)+₀^2𝜋𝑥₂^{(𝑚−𝑘)}(𝑠)d𝑠=₀^2𝜋1𝐹1+𝜆𝑠,𝑥₁,𝑦₁−𝜆𝐹1+𝜆𝑠,−𝑥₁,−𝑦₁d𝑠≤11+𝜆₀^2𝜋𝐹𝑠,𝑥₁,𝑦₁d𝑠+𝜆1+𝜆₀^2𝜋𝐹𝑠,−𝑥₁,−𝑦₁d𝑠≤11+𝜆₀^2𝜋𝐹𝑠,𝑥₁,𝑦₁−𝐹(𝑠,0,0)d𝑠+𝜆1+𝜆₀^2𝜋𝐹𝑠,−𝑥₁,−𝑦₁−𝐹(𝑠,0,0)d𝑠+11+𝜆₀^2𝜋|𝐹(𝑠,0,0)|d𝜆𝑠+1+𝜆₀^2𝜋|𝐹(𝑠,0,0)|d𝑠≤11+𝜆₀^2𝜋𝛼₁𝑥₁𝑠−𝜃₀(𝑠)+𝛼₂𝑥^′₁𝑠−𝜃₁(𝑠)+⋯+𝛼_𝑘+1𝑥₁^(𝑘)𝑠−𝜃_𝑘(𝑠)+𝛼_𝑘+2𝑦₁𝑠−𝜗₀(𝑠)+𝛼_𝑘+3𝑦^′₁𝑠−𝜗₁(𝑠)+⋯+𝛼_𝑘+𝑙+2𝑦₁^(𝑙)𝑠−𝜗_𝑙(𝑠)d𝑠+𝜆1+𝜆₀^2𝜋𝛼₁𝑥₁𝑠−𝜃₀(𝑠)+⋯+𝛼_𝑘+1𝑥₁^(𝑘)𝑠−𝜃_𝑘(𝑠)+𝛼_𝑘+2𝑦₁𝑠−𝜗₀(𝑠)+⋯+𝛼_𝑘+𝑙+2𝑦₁^(𝑙)𝑠−𝜗_𝑙(𝑠)d𝑠+₀^2𝜋|𝐹(𝑠,0,0)|d𝑠≤₀^2𝜋𝛼₁𝑥₁𝑠−𝜃₀(𝑠)+𝛼₂𝑥^′₁𝑠−𝜃₁(𝑠)+⋯+𝛼_𝑘+1𝑥₁^(𝑘)𝑠−𝜃_𝑘(𝑠)+𝛼_𝑘+2𝑦₁𝑠−𝜗₀(𝑠)+𝛼_𝑘+3𝑦^′₁𝑠−𝜗₁(𝑠)+⋯+𝛼_𝑘+𝑙+2𝑦₁^(𝑙)𝑠−𝜗_𝑙(𝑠)d𝑠+₀^2𝜋|𝐹(𝑠,0,0)|d𝑠≤₀^2𝜋𝛼₁𝑥_1∞+𝛼₂𝑥^′_1∞+⋯+𝛼_𝑘𝑥₁^(𝑘−1)_∞+𝛼_𝑘+1𝑥₁^(𝑘)_∞+𝛼_𝑘+2𝑦_1∞+𝛼_𝑘+3𝑦^′_1∞+⋯+𝛼_𝑘+𝑙+1𝑦₁^(𝑙−1)_∞+𝛼_𝑘+𝑙+2𝑦₁^(𝑙)_∞d𝑠+₀^2𝜋|𝐹(𝑠,0,0)|d𝑠≤2𝜋𝛼₁𝑥_1∞+2𝜋𝛼₂𝑥^′_1∞+⋯+2𝜋𝛼_𝑘𝑥₁^(𝑘−1)_∞+2𝜋𝛼_𝑘+1𝑥₁^(𝑘)_∞+2𝜋𝛼_𝑘+2𝑦_1∞+2𝜋𝛼_𝑘+3𝑦^′_1∞+⋯+2𝜋𝛼_𝑘+𝑙+1𝑦₁^(𝑙−1)_∞+2𝜋𝛼_𝑘+𝑙+2𝑦₁^(𝑙)_∞+2𝜋max_{[]𝑠∈0,2𝜋}|𝐹(𝑠,0,0)|≤2𝜋𝛼₁(2𝜋)^𝑘2^𝑘𝑥₁^(𝑘)_∞+2𝜋𝛼₂(2𝜋)^𝑘−12^𝑘−1𝑥₁^(𝑘)_∞+⋯+2𝜋𝛼_𝑘2𝜋2𝑥₁^(𝑘)_∞+2𝜋𝛼_𝑘+1𝑥₁^(𝑘)_∞+2𝜋𝛼_𝑘+2(2𝜋)^𝑙2^𝑙𝑦₁^(𝑙)_∞+2𝜋𝛼_𝑘+3(2𝜋)^𝑙−12^𝑙−1𝑦₁^(𝑙)_∞+⋯+2𝜋𝛼_𝑘+𝑙+12𝜋2𝑦₁^(𝑙)_∞+2𝜋𝛼_𝑘+𝑙+2𝑦₁^(𝑙)_∞+2𝜋max_{[]𝑠∈0,2𝜋}|𝐹(𝑠,0,0)|=2^𝑘+1_𝑖=1𝜋^{𝑘−𝑖+2}𝛼_𝑖𝑥₁^(𝑘)_∞+2^𝑙+1_𝑖=1𝜋^{𝑙−𝑖+2}𝛼_𝑘+𝑖+1𝑦₁^(𝑙)_∞+2𝜋max_{𝑠∈[0,2𝜋]}|𝐹(𝑠,0,0)|. (56) Namely, 𝑥₂^{(𝑚−𝑘−1)}_∞≤^𝑘+1_𝑖=1𝜋^{𝑘−𝑖+2}𝛼_𝑖𝑥₁^(𝑘)_∞+^𝑙+1_𝑖=1𝜋^{𝑙−𝑖+2}𝛼_𝑘+𝑖+1𝑦₁^(𝑙)_∞+𝜋max_{𝑠∈[0,2𝜋]}|𝐹(𝑠,0,0)|. (57) Since 𝑦₂(𝑡)∈𝐶^𝑛−𝑙_2𝜋 is 𝜋-antiperiodic, similar to (50), there exists a constant 𝜂 such that 𝑦₂^{(𝑛−𝑙−1)}(𝜂)=0. By a parallel argument to (50), we can obtain from (45), (𝐻₁), (52), and (53) that 2𝑦₂^{(𝑛−𝑙−1)}_∞≤2𝑦₂^{(𝑛−𝑙−1)}𝜂+₀^2𝜋𝑦₂^{(𝑛−𝑙)}(𝑠)d𝑠=₀^2𝜋1𝐺1+𝜆𝑠,𝑦₁,𝑥₁−𝜆𝐺1+𝜆𝑠,−𝑦₁,−𝑥₁d𝑠≤11+𝜆₀^2𝜋𝐺𝑠,𝑦₁,𝑥₁d𝑠+𝜆1+𝜆₀^2𝜋𝐺𝑠,−𝑦₁,−𝑥₁d𝑠≤11+𝜆₀^2𝜋𝐺𝑠,𝑦₁,𝑥₁−𝐺(𝑠,0,0)d𝑠+𝜆1+𝜆₀^2𝜋𝐺𝑠,−𝑦₁,−𝑥₁−𝐺(𝑠,0,0)d𝑠+11+𝜆₀^2𝜋|𝐺(𝑠,0,0)|d𝜆𝑠+1+𝜆₀^2𝜋|𝐺(𝑠,0,0)|d𝑠≤11+𝜆₀^2𝜋𝛽₁𝑦₁𝑠−𝜇₀(𝑠)+𝛽₂𝑦^′₁𝑠−𝜇₁(𝑠)+⋯+𝛽_𝑙+1𝑦₁^(𝑙)𝑠−𝜇_𝑙(𝑠)+𝛽_𝑙+2𝑥₁𝑠−𝜈₀(𝑠)+𝛽_𝑙+3𝑥^′₁𝑠−𝜈₁(𝑠)+⋯+𝛽_𝑙+𝑘+2𝑥₁^(𝑘)𝑠−𝜈_𝑘(𝑠)d𝑠+𝜆1+𝜆₀^2𝜋𝛽₁𝑦₁𝑠−𝜇₀(𝑠)+𝛽₂𝑦^′₁𝑠−𝜇₁(𝑠)+⋯+𝛽_𝑙+1𝑦₁^(𝑙)𝑠−𝜇_𝑙(𝑠)+𝛽_𝑙+2𝑥₁𝑠−𝜈₀(𝑠)+𝛽_𝑙+3𝑥^′₁𝑠−𝜈₁(𝑠)+⋯+𝛽_𝑙+𝑘+2𝑥₁^(𝑘)𝑠−𝜈_𝑘(𝑠)d𝑠+₀^2𝜋|𝐺(𝑠,0,0)|d𝑠≤₀^2𝜋𝛽₁𝑦₁𝑠−𝜇₀(𝑠)+𝛽₂𝑦^′₁𝑠−𝜇₁(𝑠)+⋯+𝛽_𝑙+1𝑦₁^(𝑙)𝑠−𝜇_𝑙(𝑠)+𝛽_𝑙+2𝑥₁𝑠−𝜈₀(𝑠)+𝛽_𝑙+3𝑥^′₁𝑠−𝜈₁(𝑠)+⋯+𝛽_𝑙+𝑘+2𝑥₁^(𝑘)𝑠−𝜈_𝑘(𝑠)d𝑠+₀^2𝜋|𝐺(𝑠,0,0)|d𝑠≤₀^2𝜋𝛽₁𝑦_1∞+𝛽₂𝑦^′_1∞+⋯+𝛽_𝑙𝑦₁^(𝑙−1)_∞+𝛽_𝑙+1𝑦₁^(𝑙)_∞+𝛽_𝑙+2𝑥_1∞+𝛽_𝑙+3𝑥^′_1∞+⋯+𝛽_𝑙+𝑘+1𝑥₁^(𝑘−1)_∞+𝛽_𝑙+𝑘+2𝑥₁^(𝑘)_∞d𝑠+₀^2𝜋|𝐺(𝑠,0,0)|d𝑠≤2𝜋𝛽₁𝑦_1∞+2𝜋𝛽₂𝑦^′_1∞+⋯+2𝜋𝛽_𝑙𝑦₁^(𝑙−1)_∞+2𝜋𝛽_𝑙+1𝑦₁^(𝑙)_∞+2𝜋𝛽_𝑙+2𝑥_1∞+2𝜋𝛽_𝑙+3𝑥^′_1∞+⋯+2𝜋𝛽_𝑙+𝑘+1𝑥₁^(𝑘−1)_∞+2𝜋𝛽_𝑙+𝑘+2𝑥₁^(𝑘)_∞+2𝜋max_{[]𝑠∈0,2𝜋}|𝐺(𝑠,0,0)|≤2𝜋𝛽₁(2𝜋)^𝑙2^𝑙𝑦₁^(𝑙)_∞+2𝜋𝛽₂(2𝜋)^𝑙−12^𝑙−1𝑦₁^(𝑙)_∞+⋯+2𝜋𝛽_𝑙2𝜋2𝑦₁^(𝑙)_∞+2𝜋𝛽_𝑙+1𝑦₁^(𝑙)_∞+2𝜋𝛽_𝑙+2(2𝜋)^𝑘2^𝑘𝑥₁^(𝑘)_∞+2𝜋𝛽_𝑙+3(2𝜋)^𝑘−12^𝑘−1𝑥₁^(𝑘)_∞+⋯+2𝜋𝛽_𝑙+𝑘+12𝜋2𝑥₁^(𝑘)_∞+2𝜋𝛽_𝑙+𝑘+2𝑥₁^(𝑘)_∞+2𝜋max_{[]𝑠∈0,2𝜋}|𝐺(𝑠,0,0)|=2^𝑙+1_𝑖=1𝜋^{𝑙−𝑖+2}𝛽_𝑖𝑦₁^(𝑙)_∞+2^𝑘+1_𝑖=1𝜋^{𝑘−𝑖+2}𝛽_𝑙+𝑖+1𝑥₁^(𝑘)_∞+2𝜋max_{𝑠∈[0,2𝜋]}|𝐺(𝑠,0,0)|. (58) Namely, 𝑦₂^{(𝑛−𝑙−1)}_∞≤^𝑙+1_𝑖=1𝜋^{𝑙−𝑖+2}𝛽_𝑖𝑦₁^(𝑙)_∞+^𝑘+1_𝑖=1𝜋^{𝑘−𝑖+2}𝛽_𝑙+𝑖+1𝑥₁^(𝑘)_∞+𝜋max_{𝑠∈[0,2𝜋]}|𝐺(𝑠,0,0)|. (59) From (1) of Lemmas 4 and 6, one can obtain 𝑥₁^(𝑘)(𝑡)=𝐴⁻¹𝐴𝑥₁^(𝑘)(𝑡≤1)𝜔₁𝐴𝑥₁^(𝑘)(𝑡=1)𝜔₁𝐴𝑥₁^(𝑘)(𝑡≤1)𝜔₁𝜑_𝑝′𝑥_2∞,𝑦₁^(𝑙)(𝑡)=𝐵⁻¹𝐵𝑦₁^(𝑙)(𝑡≤1)𝜔₁𝐵𝑥₁^(𝑙)(𝑡=1)𝜔₁𝐵𝑦₁^(𝑙)(𝑡≤1)𝜔₁𝜑_𝑞′𝑦_2∞. (60) That is, 𝑥₁^(𝑘)_∞≤1𝜔₁𝜑_𝑝′𝑥_2∞,𝑦₁^(𝑙)_∞≤1𝜔₁𝜑_𝑞′𝑦_2∞. (61) From (54) and (55), we can get 𝑥_2∞≤𝜋^{𝑚−𝑘−1}𝑥₂^{(𝑚−𝑘−1)}_∞,𝑦_2∞≤𝜋^{𝑛−𝑙−1}𝑦₂^{(𝑛−𝑙−1)}_∞. (62) With (57)–(62), we have 𝑥_2∞≤^𝑘+1_𝑖=1𝜋^{𝑚−𝑖+1}𝜔₁𝛼_𝑖𝑥_2∞^𝑝′−1+^𝑙+1_𝑖=1𝜋^{𝑚−𝑘+𝑙−𝑖+1}𝜔₁𝛼_𝑘+𝑖+1𝑦₂^𝑞′_∞⁻¹+𝑇₁,𝑦_2∞≤^𝑘+1_𝑖=1𝜋^{𝑛−𝑙+𝑘−𝑖+1}𝜔₁𝛽_𝑙+𝑖+1𝑥₂^𝑝′_∞⁻¹+^𝑙+1_𝑖=1𝜋^{𝑛−𝑖+1}𝜔₁𝛽_𝑖𝑦₂^𝑞′_∞⁻¹+𝑇₂, (63) where 𝑇₁=𝜋^𝑚−𝑘max_{𝑠∈[0,2𝜋]}|𝐹(,𝑇𝑠,0,0)|₂=𝜋^𝑛−𝑙max_{𝑠∈[0,2𝜋]}|𝐺(𝑠,0,0)|. (64) Let 𝑋₀=max{|𝑥₂|_∞,|𝑦₂|_∞}. Then we have from (63) that 𝑋₀≤𝑊₁𝑋^𝑝′₀⁻¹+𝑊₂𝑋^𝑞′₀⁻¹+max𝑇₁,𝑇₂ (65) in which 𝑊₁=max^𝑘+1_𝑖=1𝜋^{𝑚−𝑖+1}𝜔₁𝛼_𝑖,^𝑘+1_𝑖=1𝜋^{𝑛−𝑙+𝑘−𝑖+1}𝜔₁𝛽_𝑙+𝑖+1,𝑊₂=max^𝑙+1_𝑖=1𝜋^{𝑚−𝑘+𝑙−𝑖+1}𝜔₁𝛼_𝑘+𝑖+1,^𝑙+1_𝑖=1𝜋^{𝑛−𝑖+1}𝜔₁𝛽_𝑖. (66) Since 1<𝑝^′,𝑞^′≤2, (65) implies from (1) that there exists a positive constant 𝑀₀ such that 𝑋₀≤𝑀₀; that is, 𝑥_2∞≤𝑀₀,𝑦_2∞≤𝑀₀, (67) which implies from (61) that there exist positive constants 𝑀₁ and 𝑀₂ such that 𝑥₁^(𝑘)_∞≤𝑀₁,𝑦₁^(𝑙)_∞≤𝑀₂, (68) which implies from (57) and (59) that 𝑥₂^{(𝑚−𝑘−1)}_∞≤^𝑘+1_𝑖=1𝜋^{𝑘−𝑖+2}𝛼_𝑖𝑀₁+^𝑙+1_𝑖=1𝜋^{𝑙−𝑖+2}𝛼_𝑘+𝑖+1𝑀₂+𝜋max_{𝑠∈[0,2𝜋]}|𝐹(𝑠,0,0)|≡𝑀₃,𝑦₂^{(𝑛−𝑙−1)}_∞≤^𝑙+1_𝑖=1𝜋^{𝑙−𝑖+2}𝛽_𝑖𝑀₂+^𝑘+1_𝑖=1𝜋^{𝑘−𝑖+2}𝛽_𝑙+𝑖+1𝑀₁+𝜋max_{𝑠∈[0,2𝜋]}|𝐺(𝑠,0,0)|≡𝑀₄. (69) Substituting (68), (69) into (52), (53), (54), and (55), we get 𝑥₁^(𝑖)_∞≤𝑇^𝑘−𝑖2^𝑘−𝑖𝑥₁^(𝑘)_∞≤𝑇^𝑘−𝑖2^𝑘−𝑖𝑀₁≡𝑁_𝑖for𝑦𝑖=0,1,…,𝑘−1,_1(𝑖)∞≤𝑇^𝑙−𝑖2^𝑙−𝑖𝑦₁^(𝑙)_∞≤𝑇^𝑙−𝑖2^𝑙−𝑖𝑀₂≡𝐿_𝑖for𝑥𝑖=0,1,…,𝑙−1,_2(𝑗)∞≤𝑇^{𝑚−𝑘−𝑗−1}2^{𝑚−𝑘−𝑗−1}𝑥₂^{(𝑚−𝑘−1)}_∞≤𝑇^{𝑚−𝑘−𝑗−1}2^{𝑚−𝑘−𝑗−1}𝑀₃≡𝐻_𝑗for𝑦𝑗=0,1,…,𝑚−𝑘−2,_2(𝑗)∞≤𝑇^{𝑛−𝑙−𝑗−1}2^{𝑛−𝑙−𝑗−1}𝑦₂^{(𝑛−𝑙−1)}_∞≤𝑇^{𝑛−𝑙−𝑗−1}2^{𝑛−𝑙−𝑗−1}𝑀₄≡𝐾_𝑗for𝑗=0,1,…,𝑛−𝑙−2. (70) Let 𝑀=^𝑘−1_𝑖=0𝑁_𝑖+^{𝑚−𝑘−2}_𝑖=0𝐻_𝑖+^𝑙−1_𝑖=0𝐿_𝑖+^{𝑛−𝑙−2}_𝑖=0𝐾_𝑖+⁴_𝑖=1𝑀_𝑖+1Clearly,𝑀isindependentof𝜆. (71) Take Ω=𝑥∈𝕏∶‖𝑥‖_𝕏<𝑀. (72) It is clear that Ω satisfies all the requirements in Lemma 1 and condition (𝐻) is satisfied. In view of all the discussions above, we conclude from Lemma 1 that system (10) has at least one 𝜋-antiperiodic solution. This completes the proof.

Similar to Theorem 7, we can easily obtain the following results.

Theorem 8. Assume that (𝐻₁)-(𝐻₂) hold. Suppose further that 𝑐=1,|𝜏|=(𝑏/𝑎)𝜋, where 𝑎 and 𝑏 are coprime positive integers with 𝑏 odd, then system (10) has at least one 𝜋-antiperiodic solution, if one of the following conditions holds. (1)𝑑=−1,|𝜎|=(𝜈/𝜇)𝜋, where 𝜇 and 𝜈 are coprime positive integers with 𝜈 odd and 𝜇 even, and 𝑊₅+𝑊₂<1, where 𝑊₅=max^𝑘+1_𝑖=1𝜋^{𝑚−𝑖+1}𝜔₂𝛼_𝑖,^𝑘+1_𝑖=1𝜋^{𝑛−𝑙+𝑘−𝑖+1}𝜔₂𝛽_𝑙+𝑖+1,𝑊₂=max^𝑙+1_𝑖=1𝜋^{𝑚−𝑘+𝑙−𝑖+1}𝜔₁𝛼_𝑘+𝑖+1,^𝑙+1_𝑖=1𝜋^{𝑛−𝑖+1}𝜔₁𝛽_𝑖. (73)(2)𝑑=1,|𝜎|=(𝜈/𝜇)𝜋, where 𝜇 and 𝜈 are coprime positive integers with 𝜈 odd, and 𝑊₅+𝑊₃<1, where 𝑊₃=max^𝑙+1_𝑖=1𝜋^{𝑚−𝑘+𝑙−𝑖+1}𝜔₂𝛼_𝑘+𝑖+1,^𝑙+1_𝑖=1𝜋^{𝑛−𝑖+1}𝜔₂𝛽_𝑖. (74)(3)𝑑=1,|𝜎|=𝜋, and 𝑊₅+𝑊₄<1, where 𝑊₄=max^𝑙+1_𝑖=1𝜋^{𝑚−𝑘+𝑙−𝑖+1}𝜔₃𝛼_𝑘+𝑖+1,^𝑙+1_𝑖=1𝜋^{𝑛−𝑖+1}𝜔₃𝛽_𝑖, (75) in which 𝜔₁,𝜔₂, and 𝜔₃ are constants defined by Lemma 3 or Lemma 5.

Theorem 9. Assume that (𝐻₁)-(𝐻₂) hold. Suppose further that 𝑐=1, |𝜏|=𝜋, then system (10) has at least one 𝜋-antiperiodic solution, if one of the following conditions holds. (1)𝑑=−1,|𝜎|=(𝜈/𝜇)𝜋, where 𝜇 and 𝜈 are coprime positive integers with 𝜈 odd and 𝜇 even, and 𝑊₆+𝑊₂<1, where 𝑊₆=max^𝑘+1_𝑖=1𝜋^{𝑚−𝑖+1}𝜔₃𝛼_𝑖,^𝑘+1_𝑖=1𝜋^{𝑛−𝑙+𝑘−𝑖+1}𝜔₃𝛽_𝑙+𝑖+1,𝑊₂=max^𝑙+1_𝑖=1𝜋^{𝑚−𝑘+𝑙−𝑖+1}𝜔₁𝛼_𝑘+𝑖+1,^𝑙+1_𝑖=1𝜋^{𝑛−𝑖+1}𝜔₁𝛽_𝑖. (76)(2)𝑑=1,|𝜎|=(𝜈/𝜇)𝜋, where 𝜇 and 𝜈 are coprime positive integers with 𝜈 odd, and 𝑊₆+𝑊₃<1, where 𝑊₃=max^𝑙+1_𝑖=1𝜋^{𝑚−𝑘+𝑙−𝑖+1}𝜔₂𝛼_𝑘+𝑖+1,^𝑙+1_𝑖=1𝜋^{𝑛−𝑖+1}𝜔₂𝛽_𝑖. (77)(3)𝑑=1,|𝜎|=𝜋, and 𝑊₆+𝑊₄<1, where 𝑊₄=max^𝑙+1_𝑖=1𝜋^{𝑚−𝑘+𝑙−𝑖+1}𝜔₃𝛼_𝑘+𝑖+1,^𝑙+1_𝑖=1𝜋^{𝑛−𝑖+1}𝜔₃𝛽_𝑖, (78) in which 𝜔₁,𝜔₂, and 𝜔₃ are constants defined by Lemma 3 or Lemma 5.

4. An Example

Example 1. Let 𝑝, 𝑞≥2. Then the following third-order (𝑝,𝑞)-Laplacian neutral differential system 𝜑_𝑝𝑥(𝑡)𝜋+𝑥𝑡−2^′′′√=sin𝑡+2𝜋⁴𝑥()+√𝑡−12𝜋⁶𝑦()+√𝑡−12𝜋⁶𝑦^′(),𝜑𝑡−1_𝑞𝑦(𝑡)𝜋+𝑦𝑡−2^′′′√=cos𝑡+2𝜋⁶𝑦()+√𝑡−12𝜋⁴𝑥(𝑡−1) (79) has at least one 𝜋-antiperiodic solution.

Proof. By calculation, 𝜔₁=√2,𝛼₁=√2/𝜋⁴,𝛼₃=𝛼₄=√2/𝜋⁶,𝛼₂=𝛼₅=0,𝛽₁=√2/𝜋⁶,𝛽₄=√2/𝜋⁴, and 𝛽₂=𝛽₃=𝛽₅=0. Therefore, 𝑊₁1=max𝜋,1𝜋²=1𝜋<12,𝑊₂1=max𝜋²+1𝜋³,1𝜋³=1𝜋²+1𝜋³<12. (80) Hence, 𝑊₁+𝑊₂<12+12=1, (81) which implies that case (1) in Theorem 7 holds. It is easy to verify that (𝐻₁)-(𝐻₂) hold and the result follows from Theorem 7. This completes the proof.

Acknowledgment

This work is supported by the National Natural Sciences Foundation of People's Republic of China under Grant 10971183.