Existence of Periodic Solutions for a Class of Asymptotically 𝑝-Linear Discrete Systems Involving 𝑝-Laplacian

Abstract

By applying Mountain Pass Theorem in critical point theory, two existence results are obtained for the following asymptotically 𝑝-linear 𝑝-Laplacian discrete system Δ(|Δ𝑢(𝑡−1)|^𝑝−2Δ𝑢(𝑡−1))+∇[−𝐾(𝑡,𝑢(𝑡))+𝑊(𝑡,𝑢(𝑡))]=0. The results obtained generalize some known works.

1. Introduction

Consider the periodic solutions of the following ordinary 𝑝-Laplacian discrete system Δ||Δ𝑢(𝑡−1)||^𝑝−2Δ𝑢(𝑡−1)+∇𝐹(𝑡,𝑢(𝑡))=0,(1.1) where Δ is the forward difference operator defined by Δ𝑢(𝑡)=𝑢(𝑡+1)−𝑢(𝑡), Δ²𝑢(𝑡)=Δ(Δ𝑢(𝑡)), 𝑝∈(1,+∞),𝑡∈ℤ,𝑢∈ℝ^𝑁, 𝐹∶ℤ×ℝ^𝑁→ℝ,𝐹(𝑡,𝑥) is continuously differentiable in 𝑥 for every 𝑡∈ℤ and 𝑇-periodic in 𝑡 for all 𝑥∈ℝ^𝑁, and 𝐹 satisfies (𝐹) 𝐹(𝑡,𝑥)=−𝐾(𝑡,𝑥)+𝑊(𝑡,𝑥), and 𝐾 and 𝑊 are 𝑇-periodic in 𝑡 for all 𝑥∈ℝ^𝑁 with 𝑇>1.

Difference equations provide a natural description of many discrete models in real world. Discrete models exist in various fields of science and technology such as statistics, computer science, electrical circuit analysis, biology, neural network, and optimal control; so it is very important to study the solutions of difference equations. For more details about difference equations, we refer the readers to the books [1–3].

In some recent papers [4–17], the authors investigated the existence of periodic solutions and subharmonic solutions of difference equations by applying critical point theory. These papers imply that the critical point theory is a useful method to the study of periodic solutions for difference equations. Motivated by the above papers and the paper [18], we will generalize the results of [18] to 𝑝-Laplacian systems (1.1). Here are our main results.

Theorem 1.1. Assume that 𝐹 satisfies (𝐹) and 𝐾 and 𝑊 satisfy the following conditions:
(A1) there exist constants 𝑎₁>0 and 𝛾∈(𝑝−1,𝑝] such that 𝐾(𝑡,0)=0,𝐾(𝑡,𝑥)≥𝑎₁|𝑥|^𝛾,∀(𝑡,𝑥)∈ℤ[1,𝑇]×ℝ^𝑁;(1.2)
(A2)  (∇𝐾(𝑡,𝑥),𝑥)≤𝑝𝐾(𝑡,𝑥) for all (𝑡,𝑥)∈ℤ[1,𝑇]×ℝ^𝑁, where ℤ[𝑎,𝑏]∶=ℤ∩[𝑎,𝑏] for every 𝑎,𝑏∈ℤ with 𝑎≤𝑏;
(A3)  𝑊(𝑡,0)=0, limsup_|𝑥|→0(𝑊(𝑡,𝑥)/|𝑥|^𝑝)<𝑎₁ uniformly for 𝑡∈ℤ[1,𝑇];
(A4) there exists a function 𝑔∈𝑙¹(ℤ[1,𝑇],ℝ) such that (∇𝑊(𝑡,𝑥),𝑥)−𝑝𝑊(𝑡,𝑥)≥𝑔(𝑡)for(𝑡,𝑥)∈ℤ[1,𝑇]×ℝ^𝑁,(1.3)lim_|𝑥|→∞[(∇𝑊(𝑡,𝑥),𝑥)−𝑝𝑊(𝑡,𝑥)]=+∞∀𝑡∈ℤ[1,𝑇];(1.4)
(A5) there exist constants 𝑎₂>0 and 𝑑>0 such that 𝑊(𝑡,𝑥)≤𝑎₂|𝑥|^𝑝+𝑑,∀(𝑡,𝑥)∈ℤ[1,𝑇]×ℝ^𝑁;(1.5)
(A6) there exists 𝑥₀∈ℝ^𝑁 such that ^𝑇_𝑡=1𝐾𝑡,𝑥⁰−𝑊𝑡,𝑥₀−𝑔𝑝(𝑡)<0.(1.6)
Then problem (1.1) possesses one nontrivial periodic solution.

Corollary 1.2. Assuming that 𝐹 satisfies (𝐹) and that 𝐾 and 𝑊 satisfy (A1)–(A4) and
(A7) there exists a function 𝑉_∞∈𝑙^∞(ℤ[1,𝑇],ℝ) such that lim_|𝑥|→∞||∇𝑊(𝑡,𝑥)−𝑉_∞(𝑡)|𝑥|^𝑝−2𝑥|||𝑥|^𝑝−1=0uniformlyfor𝑡∈ℤ[1,𝑇],(1.7)^𝑇_𝑡=1max_|𝑥|=1𝐾(𝑡,𝑥)−𝑉^∞(𝑡)𝑝<0.(1.8)
Then problem (1.1) possesses one nontrivial periodic solution.

Remark 1.3. As far as we know, similar results of discrete system (1.1) which satisfies (𝐹) and is asymptotically 𝑝-linear at infinity cannot be found in the literature. From this point, our results are new.

2. Preliminaries

Let 𝐸_𝑇 be the Sobolev space defined by𝐸_𝑇=𝑢∶ℤ⟶ℝ^𝑁∣𝑢(𝑡+𝑇)=𝑢(𝑡),𝑡∈ℤ,(2.1) with the norm‖𝑢‖=^𝑇_𝑡=1||Δ𝑢(𝑡)||^𝑝+||𝑢(𝑡)||^𝑝^1/𝑝,𝑢∈𝐸_𝑇,(2.2) where |⋅| denote the usual norm in ℝ^𝑁. It is easy to see that (𝐸_𝑇,‖⋅‖) is a finite dimensional Banach space and linear homeomorphic to ℝ^𝑁𝑇. As usual, let ‖𝑢‖_∞=sup||𝑢(𝑡)||∶𝑡∈ℤ[1,𝑇],∀𝑢∈𝑙^∞ℤ[1,𝑇],ℝ^𝑁.(2.3) Since 𝐸_𝑇 is finite dimensional Banach space, there exists a positive constant 𝐶₀ such that‖𝑢‖_∞≤𝐶₀‖𝑢‖.(2.4)

For any 𝑢∈𝐸_𝑇, let 𝜑(𝑢)=1𝑝^𝑇_𝑡=1||Δ𝑢(𝑡)||^𝑝+^𝑇_𝑡=1[𝐾(𝑡,𝑢(𝑡))−𝑊(𝑡,𝑢(𝑡))].(2.5) We can compute the Fréchet derivative of (2.5) as 𝜕𝜑(𝑢)𝜕𝑢=Δ||Δ𝑢(𝑡−1)||^𝑝−2Δ𝑢(𝑡−1)+∇[𝐾(𝑡,𝑢(𝑡))−𝑊(𝑡,𝑢(𝑡))],𝑡∈ℤ[1,𝑇].(2.6) Hence, 𝑢 is a critical point of 𝜑 on 𝐸_𝑇 if and only ifΔ||Δ𝑢(𝑡−1)||^𝑝−2Δ𝑢(𝑡−1)+∇[𝐾(𝑡,𝑢(𝑡))−𝑊(𝑡,𝑢(𝑡))]=0,𝑡∈ℤ[1,𝑇],𝑢∈ℝ^𝑁.(2.7) So, the critical points of 𝜑 are classical solutions of (1.1). We will use the following lemma to prove our main results.

Lemma 2.1 (see [19]). Let 𝐸 be a real Banach space and 𝜑∈𝐶¹(𝐸,ℝ) satisfying the (PS) condition. Suppose 𝜑(0)=0 and (a)there exist constants 𝜌,𝛼>0 such that 𝜑|_{𝜕𝐵𝜌(0)}≥𝛼;(b)there exists an 𝑒∈𝐸⧵𝐵_𝜌(0) such that 𝜑(𝑒)≤0.Then 𝜑 possesses a critical value 𝑐≥𝛼 which can be characterized as 𝑐=inf_ℎ∈Γmax_𝑠∈[0,1]𝜑(ℎ(𝑠)), where Γ={ℎ∈𝐶([0,1],𝐸)∣ℎ(0)=0,ℎ(1)=𝑒} and 𝐵_𝜌(0) is an open ball in 𝐸 of radius 𝜌 centered at 0.
It is well known that a deformation lemma can be proved with the weaker condition (C) replacing the usual (PS) condition. So Lemma 2.1 holds true under condition (C).

3. Proofs of Main Results

Proof of Theorem 1.1. The proof is divided into three steps.
Step 1. The functional 𝜑 satisfies condition (C). Let {𝑢_𝑛}⊂𝐸_𝑇 satisfying (1+‖𝑢_𝑛‖)‖𝜑^′(𝑢_𝑛)‖→0 as 𝑛→∞ and 𝜑(𝑢_𝑛) is bounded. Hence, there exists a positive constant 𝐶₁ such that ||𝜑𝑢_𝑛||≤𝐶₁,1+‖‖𝑢_𝑛‖‖‖‖𝜑^𝑢_𝑛‖‖≤𝐶₁.(3.1) We prove {𝑢_𝑛} is bounded by contradiction. If {𝑢_𝑛} is unbounded, without loss of generality, we can assume that ‖𝑢_𝑛‖→∞ as 𝑛→∞. Let 𝑧_𝑛=𝑢_𝑛/‖𝑢_𝑛‖, then we have ‖𝑧_𝑛‖=1. Going to a subsequence if necessary, we may assume that 𝑧_𝑛⇀𝑧 weakly in 𝐸_𝑇 and so 𝑧_𝑛→𝑧 strongly in 𝑙¹(ℤ[1,𝑇],ℝ). It follows from (3.1) and (A2) that (𝑝+1)𝐶₁≥=𝑝_𝑇𝜑𝑢_𝑛−𝜑^𝑢_𝑛,𝑢_𝑛≥_𝑡=1∇𝑊𝑡,𝑢_𝑛,𝑢_𝑛−𝑝𝑊𝑡,𝑢_𝑛+^𝑇_𝑡=1𝑝𝐾𝑡,𝑢_𝑛−∇𝐾𝑡,𝑢_𝑛,𝑢_𝑛_𝑇_𝑡=1∇𝑊𝑡,𝑢_𝑛,𝑢_𝑛−𝑝𝑊𝑡,𝑢_𝑛.(3.2) From (A1) and (A5), we obtain 𝜑𝑢_𝑛=1𝑝^𝑇_𝑡=1||Δ𝑢_𝑛(𝑡)||^𝑝+^𝑇_𝑡=1𝐾𝑡,𝑢_𝑛≥1(𝑡)−𝑊𝑡,𝑢_𝑛(𝑡)𝑝^𝑇_𝑡=1||Δ𝑢_𝑛(𝑡)||^𝑝−𝑎_2𝑇_𝑡=1=1||𝑢_𝑛(𝑡)||^𝑝−𝑑𝑇𝑝‖‖𝑢^𝑛‖‖−1𝑝+𝑎²^𝑇_𝑡=1||𝑢_𝑛(𝑡)||^𝑝−𝑑𝑇.(3.3) Hence, we have 𝜑𝑢_𝑛‖‖𝑢_𝑛‖‖^𝑝≥1𝑝−1𝑝+𝑎²^𝑇_𝑡=1||𝑧_𝑛(𝑡)||^𝑝−𝑑𝑇‖‖𝑢_𝑛‖‖^𝑝.(3.4) Passing to the limit in the above inequality, by using the fact that 𝜑(𝑢_𝑛) is bounded and {𝑧_𝑛(𝑡)} converges uniformly to 𝑧(𝑡) on ℤ[1,𝑇], we obtain 1𝑝+𝑎²^𝑇_𝑡=1||𝑧(𝑡)||^𝑝≥1𝑝,(3.5) which implies that 𝑧≠0. Let Ω⊂ℤ[1,𝑇] be the set on which 𝑧≠0, then the measure of Ω is positive. Moreover, |𝑢_𝑛(𝑡)|→∞ as 𝑛→∞ for 𝑡∈Ω. Thus, from (A4), we get ^𝑇_𝑡=1∇𝑊𝑡,𝑢_𝑛,𝑢_𝑛−𝑝𝑊𝑡,𝑢_𝑛=+_Ω∇𝑊𝑡,𝑢_𝑛,𝑢_𝑛−𝑝𝑊𝑡,𝑢_𝑛_{ℤ[1,𝑇]⧵Ω}≥∇𝑊𝑡,𝑢_𝑛,𝑢_𝑛−𝑝𝑊𝑡,𝑢_𝑛_Ω∇𝑊𝑡,𝑢_𝑛,𝑢_𝑛−𝑝𝑊𝑡,𝑢_𝑛+_{ℤ[]1,𝑇⧵Ω}𝑔(𝑡).(3.6) It follows from Fatou’s lemma and (A4) that lim_{𝑇𝑛→∞}_𝑡=1∇𝑊𝑡,𝑢_𝑛,𝑢_𝑛−𝑝𝑊𝑡,𝑢_𝑛=+∞,(3.7) which contradicts with (3.2). Therefore, {𝑢_𝑛} is bounded in 𝐸_𝑇. Hence, there exists a subsequence, still denoted by {𝑢_𝑛}, such that 𝑢_𝑛⇀𝑢₀weaklyin𝐸_𝑇.(3.8) Since 𝐸_𝑇 is finite dimensional space, we have 𝑢_𝑛→𝑢₀ in 𝐸_𝑇. Therefore, the functional 𝜑 satisfies condition (C).Step 2. From (A3) and (A5), there exist constants 0<𝜀<1/𝑝,𝑞>𝑝 and 𝐶₂>1/𝑇𝐶^𝑝0 such that 𝑊(𝑡,𝑢)≤𝑎₁−𝜀|𝑢|^𝑝+𝐶₂|𝑢|^𝑞for𝑢∈ℝ^𝑁,𝑡∈ℤ[1,𝑇].(3.9) Let 𝛿=𝑝𝜀𝑞𝑇𝐶₂𝐶^𝑝0^{1/(𝑞−𝑝)}.(3.10) Then 0<𝛿<1. If 𝑢∈𝐸_𝑇 and ‖𝑢‖=𝛿/𝐶₀∶=𝜌, then it follows from (2.4) that |𝑢(𝑡)|≤𝛿 for 𝑡∈ℤ[1,𝑇]. Set 𝛼=(𝑞−𝑝)𝜀𝑞𝜌^𝑝.(3.11) Then from (A1), (2.4), (3.9), (3.10), and (3.11), we have𝜑(𝑢)=1𝑝^𝑇_𝑡=1||Δ𝑢(𝑡)||^𝑝+^𝑇_𝑡=1𝐾(𝑡,𝑢(𝑡))−^𝑇_𝑡=1≥1𝑊(𝑡,𝑢(𝑡))𝑝^𝑇_𝑡=1||Δ𝑢(𝑡)||^𝑝+𝑎_1𝑇_𝑡=1||𝑢(𝑡)||^𝛾−𝑎₁−𝜀^𝑇_𝑡=1||𝑢(𝑡)||^𝑝−𝐶_2𝑇_𝑡=1||𝑢(𝑡)||^𝑞≥1𝑝^𝑇_𝑡=1||Δ𝑢(𝑡)||^𝑝+𝜀^𝑇_𝑡=1||𝑢(𝑡)||^𝑝−𝐶_2𝑇_𝑡=1||𝑢(𝑡)||^𝑞≥𝜀^𝑇_𝑡=1||Δ𝑢(𝑡)||^𝑝+^𝑇_𝑡=1||𝑢(𝑡)||^𝑝−𝐶²𝑇‖𝑢‖^𝑞∞≥𝜀‖𝑢‖^𝑝−𝐶₂𝑇𝐶^𝑞0‖𝑢‖^𝑞=(𝑞−𝑝)𝜀𝑞𝜌^𝑝=𝛼∀𝑢∈𝐸_𝑇with‖𝑢‖=𝜌.(3.12)Step 3. Set 𝑓(𝑠)=𝑠^−𝑝𝑊(𝑡,𝑠𝑥₀) for 𝑠>0. Then it follows from (A4) that 𝑓^(𝑠)=𝑠^−𝑝−1−𝑝𝑊𝑡,𝑠𝑥₀+∇𝑊𝑡,𝑠𝑥₀,𝑠𝑥₀≥𝑠^−𝑝−1𝑔(𝑡)∀𝑡∈ℤ[1,𝑇],𝑠>0.(3.13) Integrating the above inequality from 1 to 𝜉>1, we have 𝑊𝑡,𝜉𝑥₀≥𝜉^𝑝𝑊𝑡,𝑥₀+𝑔(𝑡)𝑝(𝜉^𝑝−1)∀𝑡∈ℤ[1,𝑇],𝜉>1.(3.14) From (A2), it is easy to see that 𝐾𝑡,𝜉𝑥₀≤𝜉^𝑝𝐾𝑡,𝑥₀∀𝑡∈ℤ[1,𝑇],𝜉>1.(3.15) From (3.14), (3.15), and (A6), we have 𝜑𝜉𝑥₀=^𝑇_𝑡=1𝐾𝑡,𝜉𝑥₀−𝑊𝑡,𝜉𝑥₀≤𝜉^𝑝𝑇_𝑡=1𝐾𝑡,𝑥⁰−𝑊𝑡,𝑥₀−𝑔𝑝(𝑡)+1𝑝^𝑇_𝑡=1≤0𝑔(𝑡)forlargeenough𝜉>1.(3.16) Choose 𝜉₁>1 such that 𝑇^1/𝑝|𝜉₁𝑥₀|>𝜌 and 𝜑(𝜉₁𝑥₀)≤0. Let 𝑒=𝜉₁𝑥₀, then ‖𝑒‖=𝑇^1/𝑝|𝜉₁𝑥₀|>𝜌 and 𝜑(𝑒)≤0. It is easy to see that 𝜑(0)=0. Hence, by Lemma 2.1, there exists 𝑢∈𝐸_𝑇 such that 𝜑(𝑢)=𝑐,𝜑^′(𝑢)=0.(3.17) Then the function 𝑢 is a desired nontrivial 𝑇-periodic solution of (1.1). The proof is complete.

Proof of Corollary 1.2. Let 𝐶₃=−(1/3𝑇)∑^𝑇𝑡=1[max_|𝑥|=1𝐾(𝑡,𝑥)−(𝑉_∞(𝑡)/𝑝)]. Then it follows from (A7) that 𝐶₃>0 and there exists a positive constant 𝐶₄>0 such that ||∇𝑊(𝑡,𝑥)−𝑉_∞(𝑡)|𝑥|^𝑝−2𝑥||≤𝐶₃|𝑥|^𝑝−1for𝑡∈ℤ[1,𝑇],|𝑥|≥𝐶₄.(3.18) For any 𝑥∈ℝ^𝑁⧵{0}, let 𝑥_∗=𝐶₄𝑥/|𝑥|. Then it follows from (3.18) that for all 𝑡∈ℤ[1,𝑇] and 𝑥∈ℝ^𝑁 with |𝑥|>𝐶₄𝑊(𝑡,𝑥)−𝑉^∞(𝑡)𝑝|𝑥|^𝑝=𝑊𝑡,𝑥_∗−1𝑝𝑉^∞(𝑡)||𝑥_∗||^𝑝≤+𝑊¹⁰𝑡,∇𝑥𝑊_∗𝑡,𝑥_∗+𝑠𝑥−𝑥_∗−𝑉_∞(𝑡)𝑥_∗+𝑠𝑥−𝑥_∗,𝑥−𝑥_∗𝑑𝑠≤𝑊𝑡,𝑥_∗−𝐶𝑝^𝑝4𝑉^∞(𝑡)+𝐶₃¹⁰||𝑥_∗+𝑠𝑥−𝑥_∗||||𝑥−𝑥_∗||𝑑𝑠−𝐶𝑝^𝑝4𝑉^∞(𝑡)+2𝐶₃|𝑥|^𝑝,(3.19) which implies that 𝑊(𝑡,𝑥)≤𝑉^∞(𝑡)𝑝+2𝐶³|𝑥|^𝑝+𝑊𝑡,𝑥_∗−𝐶^𝑝4𝑝𝑉^∞(𝑡)for𝑡∈ℤ[1,𝑇],|𝑥|>𝐶₄,(3.20) which together with (A3) shows that (A5) holds. Similarly, we have 𝑊(𝑡,𝑥)≥𝑉^∞(𝑡)𝑝−2𝐶³|𝑥|^𝑝+𝑊𝑡,𝑥_∗−𝐶^𝑝4𝑝𝑉^∞(𝑡)for𝑡∈ℤ[1,𝑇],|𝑥|>𝐶₄.(3.21) From (A2), it is easy to show that 𝐾(𝑡,𝑥)≤𝐾𝑡,𝑥|𝑥||𝑥|^𝑝for𝑡∈ℤ[1,𝑇],|𝑥|>1.(3.22) Choose 𝑥₀∈ℝ^𝑁 such that |𝑥₀|>𝐶₄+1 and 𝐶₃𝑇||𝑥₀||^𝑝+^𝑇_𝑡=1min^|𝑥|=𝐶₄𝑊(𝑡,𝑥)+𝑔(𝑡)𝑝−𝐶𝑝^𝑝4𝑉^∞(𝑡)>0.(3.23) It follows from (3.21), (3.22), and (3.23) that ^𝑇_𝑡=1≤𝐾||𝑥₀𝑡,𝑥⁰−𝑊𝑡,𝑥₀−𝑔𝑝(𝑡)||^𝑝𝑇_𝑡=1𝐾𝑡,||𝑥𝑥⁰₀||−𝑉^∞(𝑡)𝑝+2𝐶³−^𝑇_𝑡=1𝑊𝑡,𝑥^∗+𝑔𝑝(𝑡)−𝐶𝑝^𝑝4𝑉^∞≤||𝑥(𝑡)₀||^𝑝𝑇_𝑡=1max_|𝑥|=1𝐾(𝑡,𝑥)−𝑉^∞(𝑡)𝑝+2𝐶³−^𝑇_𝑡=1min^|𝑥|=𝐶₄𝑊(𝑡,𝑥)+𝑔(𝑡)𝑝−𝐶𝑝^𝑝4𝑉^∞≤−𝐶₃(𝑡)𝑇||𝑥₀||^𝑝−^𝑇_𝑡=1min^|𝑥|=𝐶₄𝑊(𝑡,𝑥)+𝑔(𝑡)𝑝−𝐶𝑝^𝑝4𝑉^∞<0.(𝑡)(3.24) This implies that (A6) holds. By Theorem 1.1, the conclusion of Corollary 1.2 holds true. The proof is complete.

4. An Example

In this section, we give an example to illustrate our results.

Example 4.1. In problem (1.1), let 𝑝=4/3 and 𝑊(𝑡,𝑥)=𝑎(𝑡)|𝑥|^4/31−ln(𝑒1+|𝑥|),𝐾(𝑡,𝑥)=𝑏|𝑥|^𝜃+𝑐(𝑡)|𝑥|^𝜎,(4.1) where 𝑏>0,𝑎,𝑐∈𝑙¹(ℤ,[0,+∞)),1<𝜃<𝜎≤4/3,𝑎(𝑡+𝑇)=𝑎(𝑡),𝑐(𝑡+𝑇)=𝑐(𝑡). It is easy to check that (𝐹), (A1)–(A3), and (A5) hold. On the one hand, we have (∇𝑊(𝑡,𝑥),𝑥)−43𝑊(𝑡,𝑥)=(𝑒+(4|𝑥/3|))(𝑎ln(𝑡()𝑒|𝑥+||^7/3𝑥|))².(4.2) Then, it is easy to check that condition (A4) holds. On the other hand, we have ^𝑇_𝑡=1𝐾(𝑡,𝑥)−𝑊(𝑡,𝑥)−𝑔𝑝(𝑡)=^𝑇_𝑡=1𝑏|𝑥|^𝜃+𝑐(𝑡)|𝑥|^𝜎−𝑎(𝑡)|𝑥|^4/3=𝑏𝑇|𝑥|_𝜃1−ln(𝑒1+|𝑥|)−𝑔𝑝(𝑡)+|𝑥|^𝜎𝑇_𝑡=1𝑐(𝑡)−‖𝑔‖^𝑙1𝑝−|𝑥|^4/31−ln(𝑒1+|𝑥|)^𝑇_𝑡=1𝑎(𝑡),(4.3) which implies that there exists 𝑥₀∈ℝ^𝑁 such that (A6) holds if ^𝑇_𝑡=1𝑎(𝑡)>^𝑇_𝑡=1𝑐(𝑡).(4.4) Hence, from Theorem 1.1, problem (1.1) with 𝑊 and 𝐾 as in (4.1) has one nontrivial 𝑇-periodic solution if (4.4) holds.

Acknowledgment

This work is partially supported by Scientific Research Foundation of Guilin University of Technology and Scientific Research Foundation of Guangxi Education Office of China (200911MS270).