Nontrivial Periodic Solutions for Nonlinear Second-Order Difference Equations

Abstract

This paper is concerned with the existence of nontrivial periodic solutions and positive periodic solutions to a nonlinear second-order difference equation. Under some conditions concerning the first positive eigenvalue of the linear equation corresponding to the nonlinear second-order equation, we establish the existence results by using the topological degree and fixed point index theories.

1. Introduction

Let 𝐑, 𝐙, 𝐍 be the sets of real numbers, integers, and natural numbers, respectively. For 𝑎,𝑏∈𝐙, define 𝐙[𝑎,𝑏]={𝑎,𝑎+1,…,𝑏} when 𝑎≤𝑏.

In this paper, we deal with the existence of nontrivial periodic solutions and positive periodic solutions for a nonlinear second-order difference equationΔ²𝑢(𝑡−1)+𝑞(𝑡)𝑢(𝑡)=𝑓(𝑡,𝑢(𝑡)),𝑢(𝑡+𝑇)=𝑢(𝑡),𝑡∈𝐙,(1.1) where 𝑇 is a positive integer, 𝑞∶𝐙→𝐑 and 𝑞(𝑡+𝑇)=𝑞(𝑡) for any 𝑡∈𝐙, 𝑓∶𝐙×𝐑→𝐑 is continuous in the second variable and 𝑓(𝑡+𝑇,𝑥)=𝑓(𝑡,𝑥) for any (𝑡,𝑥)∈𝐙×𝐑, and Δ𝑢(𝑡)=𝑢(𝑡+1)−𝑢(𝑡), Δ²𝑢(𝑡)=Δ(Δ𝑢(𝑡)).

From the 𝑇-periodicity of 𝑞 and 𝑓, it is easy to verify that the 𝑇-periodic solution to (1.1) is equivalent to the solution to the following periodic boundary value problem (PBVP for short):Δ²𝑢(𝑡−𝑢1()0+)=𝑞(𝑢𝑡)(𝑇𝑢)(𝑡,)=𝑓Δ(𝑢𝑡,(𝑢0)(𝑡=))Δ,𝑢𝑡(𝑇∈)𝐙.[1,𝑇],(1.2)

The theory of nonlinear difference equations has been widely used to study discrete models appearing in many fields such as computer science, economics, neural network, ecology, and cybernetics. In recent years, there are many papers to study the existence of periodic solutions for second-order difference equations. By using various methods and techniques, for example, fixed point theorems, the method of upper and lower solutions, coincidence degree theory, critical point theory, a series of existence results of periodic solutions have been obtained. We refer the reader to [1–16] and references therein.

In [2], by using the method of upper and lower solutions, Atici and Cabada investigated the existence and uniqueness of periodic solutions for PBVP (1.2) provided that 𝑞(𝑡)≤0, 𝑞(𝑡)≢0. Of course the natural question is what would happen if 𝑞(𝑡)≥0. In this paper, we will assume that 0≤𝑞(𝑡)<4sin²𝜋2𝑇,𝑞(𝑡)≢0.(H)And we will use the topological degree and fixed point index theories to establish the existence of nontrivial periodic solutions and positive periodic solutions for (1.1). We note that some ideas of this paper are from [17–19].

This paper is organized as follows. In Section 2, we give Green’s function associated with PBVP (1.2) and then present some preliminary lemmas that will be used to prove our main results. In Sections 4 and 5, by computing the topological degree and fixed point index, we establish some existence results of nontrivial periodic solutions and positive periodic solutions to (1.1). The final section of the paper contains some examples to illustrate our results, and we also remark that the results obtained in previous papers and ours are mutually independent.

2. Preliminaries

In this section, we are going to construct Green’s function associated with PBVP (1.2) and then present some preliminary lemmas. Consider 𝑇-dimensional Banach space 𝐸=𝑢={𝑢(𝑡)}^𝑇1∶𝑢(𝑡)∈𝐑,𝑡∈𝐙[1,𝑇](2.1) equipped with the norm ‖𝑢‖=max{|𝑢(𝑡)|,𝑡∈𝐙[1,𝑇]} for all 𝑢∈𝐸 and the cone 𝑃={𝑢∈𝐸∶𝑢(𝑡)≥0,𝑡∈𝐙[1,𝑇]}. Then the cone 𝑃 is normal and has nonempty interiors int𝑃. It is clear that 𝑃 is also a total cone of 𝐸, that is, 𝐸=𝑃−𝑃, which means the set 𝑃−𝑃={𝑢−𝑣∶𝑢,𝑣∈𝑃} is dense in 𝐸. For each 𝑢,𝑣∈𝐸, we write 𝑢≤𝑣 if 𝑣−𝑢∈𝑃. For 𝑟>0, let 𝐵_𝑟={𝑢∈𝐸∶‖𝑢‖<𝑟} and 𝜕𝐵_𝑟={𝑢∈𝐸∶‖𝑢‖=𝑟}. Put 𝑄=max_{𝑡∈𝐙[1,𝑇]}𝑞(𝑡).

Lemma 2.1. If   0<𝑄<4sin²(𝜋/2𝑇), then, for each 𝑣∈𝐸, the problem Δ²𝑢𝑢(𝑡(0−)1=)+𝑢(𝑄𝑇𝑢),(𝑡)=Δ𝑣𝑢(𝑡()0,)=𝑡Δ∈𝑢𝐙(𝑇[1),.𝑇],(2.2) has a unique solution 𝑢(𝑡)=^𝑇_𝑘=1𝐺(𝑡,𝑘)𝑣(𝑘),𝑡∈𝐙[0,𝑇+1],(2.3) where 𝐺(𝑡,𝑘) is given by 𝐺(𝑡,𝑘)=𝜌𝜌((𝑡𝑇−+𝑘𝑡)−,𝑘),00≤≤𝑘𝑡≤≤𝑘𝑡≤≤𝑇𝑇++11,(2.4) with 𝜌(𝑡)=(1/(2sin𝜑sin(𝜑𝑇/2)))cos𝜑((𝑇/2)−𝑡) and 𝜑∶=arctan(√𝑄(4−𝑄)/(2−𝑄)).

Proof. (i) Taking into account that 𝑄∈(0,4sin²(𝜋/2𝑇)), an easy computation ensures that 𝜑∶=arctan(√𝑄(4−𝑄)/(2−𝑄))∈(0,(𝜋/𝑇)). Hence, 𝜌(𝑡)>0,𝑡∈𝐙[0,𝑇]. It is easy to verify that Δ²𝜌(𝑡−1)+𝑄𝜌(𝑡)=0,𝜌(0)=𝜌(𝑇),𝜌(1)=𝜌(𝑇+1)+1.(2.5)
Let 𝑢(𝑡)=∑^𝑇𝑘=1𝐺(𝑡,𝑘)𝑣(𝑘),𝑡∈𝐙[0,𝑇+1]. Then 𝑢(𝑡)=^𝑡_𝑘=1𝜌(𝑡−𝑘)𝑣(𝑘)+^𝑇_𝑘_=𝑡+1𝜌(𝑇+𝑡−𝑘)𝑣(𝑘),𝑡∈𝐙[1,𝑇],(2.6) where, and in what follows, we denote ∑^{𝑙𝑘=𝑠}𝑥(𝑘)=0 when 𝑙<𝑠. We have Δ𝑢(𝑡)=^𝑡_𝑘=1Δ𝜌(𝑡−𝑘)𝑣(𝑘)+𝜌(0)𝑣(𝑡+1)+^𝑇_𝑘_=𝑡+1Δ𝜌(𝑇+𝑡−𝑘)𝑣(𝑘)−𝜌(𝑇)𝑣(𝑡+1).(2.7) Then, Δ²𝑢(𝑡−1)+𝑄𝑢(𝑡)=^𝑡−1_𝑘=1Δ²+𝜌(𝑡−𝑘−1)+𝑄𝜌(𝑡−𝑘)𝑣(𝑘)+𝑄𝜌(0)𝑣(𝑡)+Δ𝜌(0)𝑣(𝑡)_𝑇_𝑘=𝑡Δ²=𝑣(𝑡),𝑡∈𝜌(𝐙𝑇+[1,𝑡𝑇−]𝑘.−1)+𝑄𝜌(𝑇+𝑡−𝑘)𝑣(𝑘)−𝑄𝜌(𝑇)𝑣(𝑡)−Δ𝜌(𝑇)𝑣(𝑡)(2.8) From the definitions of 𝑢 and 𝐺, it is easy to see that 𝑢(0)=𝑢(𝑇) and Δ𝑢(0)=Δ𝑢(𝑇). This completes the proof of the lemma.

From the expression of 𝐺, we see that 𝐺(𝑡,𝑘)>0 and 𝐺(𝑡,𝑘)=𝐺(𝑘,𝑡) for all 𝑡,𝑘∈𝐙[1,𝑇]. Define operators 𝐾,𝐿∶𝐸→𝐸, respectively, by(𝐾𝑢)(𝑡)=^𝑇_𝑘=1(𝐿𝑢)(𝑡)=(𝑄𝐺−(𝑞𝑡,(𝑘𝑡)))𝑢𝑢((𝑘𝑡)),,𝑢𝑢∈∈𝐸𝐸,,𝑡𝑡∈∈𝐙𝐙[[11,,𝑇𝑇]].,(2.9) Obviously, 𝐾(𝑃)⊆𝑃 and 𝐿(𝑃)⊆𝑃. It is clear that 𝐾 is strongly positive, that is, 𝐾(𝑢)∈int𝑃 for 𝑢∈𝑃⧵{𝜃}.

Lemma 2.2. Assume that (H) holds. Then, 𝐾𝐿∶𝐸→𝐸 is a linear completely continuous operator with ‖𝐾𝐿‖<1, and (I−𝐾𝐿)⁻¹, the inverse mapping of 𝐼−𝐾𝐿, exists and is bounded.

Proof. It is obvious that 𝐾𝐿∶𝐸→𝐸 is a linear completely continuous operator. Since 𝑢₀(𝑡)≡1/𝑄 is a solution of PBVP (2.2) with 𝑣₀(𝑡)≡1, we have ^𝑇_𝑘=1𝐺(𝑡,𝑘)=1𝑄,𝑡∈𝐙[1,𝑇].(2.10) Then by (H) and the fact that 𝐾 is strongly positive, one has ||(𝐾𝐿)𝑢(𝑡)||=^𝑇_𝑘=1𝐺(𝑡,𝑘)(𝑄−𝑞(𝑘))||𝑢(𝑘)||≤‖𝑢‖^𝑇_𝑘=1𝐺(𝑡,𝑘)(𝑄−𝑞(𝑘))=‖𝑢‖(1−(𝐾𝑞)(𝑡))<‖𝑢‖,(2.11) where 𝑢∈𝐸, 𝑡∈𝐙[1,𝑇]. Hence ‖𝐾𝐿‖<1, and (𝐼−𝐾𝐿)⁻¹, the inverse mapping of 𝐼−𝐾𝐿, exists and is bounded. The proof of Lemma 2.2 is completed.

Let 𝑆∶=(𝐼−𝐾𝐿)⁻¹𝐾=(𝐼+𝐾𝐿+⋯+(𝐾𝐿)^𝑛+⋯)𝐾=𝐾+(𝐾𝐿)𝐾+⋯+(𝐾𝐿)^𝑛𝐾+⋯.(2.12) The complete continuity of 𝐾 together with the continuity of (𝐼−𝐾𝐿)⁻¹ implies that the operator 𝑆∶𝐸→𝐸 is completely continuous.

Lemma 2.3. Assume that (H) holds. Then, for each 𝑣∈𝐸, the following linear periodic boundary value problem Δ²𝑢(𝑡𝑢−(01))=+𝑢𝑞((𝑇𝑡))𝑢,(𝑡)=Δ𝑣𝑢((𝑡0)),=𝑡Δ∈𝑢(𝐙𝑇[)1,𝑇],(2.13) has a unique solution {𝑢(𝑡)}^𝑇+1𝑡=0,  where 𝑢(𝑡)=(𝑆𝑣)(𝑡),𝑡∈𝐙[1,𝑇], and 𝑢(0)=𝑢(𝑇),𝑢(1)=𝑢(𝑇+1).

Proof. It is easy to see that PBVP (2.13) is equivalent to the operator equation 𝑢=𝐾𝐿𝑢+𝐾𝑣. Therefore, PBVP (2.13) has a unique solution {𝑢(𝑡)}^𝑇+1𝑡=0, where 𝑢(𝑡)=(𝑆𝑣)(𝑡)=((𝐼−𝐾𝐿)⁻¹𝐾𝑣)(𝑡),𝑡∈𝐙[1,𝑇], and 𝑢(0)=𝑢(𝑇),𝑢(𝑇+1)=𝑢(1).

Lemma 2.4. Assume that (H) holds. Then, for the operator 𝑆 defined by (2.12), the spectral radius 𝑟(𝑆)>0 and there exists 𝜉∈𝐸 with 𝜉>0 on 𝐙[1,𝑇] such that 𝑆𝜉=𝑟(𝑆)𝜉 and ∑^𝑇𝑡=1𝜉(𝑡)=1/𝑟(𝑆). Moreover, 𝜆₁=1/𝑟(𝑆) is the first positive eigenvalue of the linear PBVP corresponding to PBVP (1.2) and ^𝑇_𝑡=1(𝑆𝑢)(𝑡)𝜉(𝑡)=1𝜆_1𝑇𝑡=1𝑢(𝑡)𝜉(𝑡),∀𝑢∈𝐸.(2.14)

Proof. An obvious modification of the proof of [13, Lemma 2.3] yields this result. We omit the details here.

Lemma 2.5. Assume that (H) holds. Then, 𝑆(𝑃)⊆𝑃₁, where 𝑃₁=𝑢∈𝑃∶^𝑇_𝑡=1𝑢(𝑡)𝜉(𝑡)≥𝛿‖𝑢‖,𝛿=2sin𝜑sin(𝜑𝑇/2)min^{𝑡∈𝐙[1,𝑇]}𝜉(𝑡)𝜆₁‖‖(𝐼−𝐾𝐿)⁻¹‖‖,(2.15) and 𝜆₁,𝜉 are given in Lemma 2.4, 𝜑 is given in Lemma 2.1.

Proof. By (2.14), we have, for any 𝑢∈𝑃, ^𝑇_𝑡=1(𝑆𝑢)(𝑡)𝜉(𝑡)=1𝜆_1𝑇𝑡=1𝑢(𝑡)𝜉(𝑡)≥min^{𝑡∈𝐙[1,𝑇]}𝜉(𝑡)𝜆_1𝑇𝑡=1𝑢(𝑡).(2.16) On the other hand, one has ‖𝑆𝑢‖=‖‖(𝐼−𝐾𝐿)⁻¹𝐾𝑢‖‖≤‖‖(𝐼−𝐾𝐿)⁻¹‖‖‖𝐾𝑢‖=‖‖(𝐼−𝐾𝐿)⁻¹‖‖max_{𝑇𝑡∈𝐙[1,𝑇]}_𝑘=1≤‖‖(𝐼−𝐾𝐿)₋₁𝐺(𝑡,𝑘)𝑢(𝑘)‖‖max_{𝑡,𝑘∈𝐙[1,𝑇]}𝐺(𝑡,𝑘)^𝑇_𝑘=1𝑢(𝑘)=‖‖(𝐼−𝐾𝐿)⁻¹‖‖2sin𝜑sin(𝜑𝑇/2)^𝑇_𝑘=1𝑢(𝑘).(2.17) Then, ^𝑇_𝑡=1(𝑆𝑢)(𝑡)𝜉(𝑡)≥2sin𝜑sin(𝜑𝑇/2)min^{𝑡∈𝐙[1,𝑇]}𝜉(𝑡)𝜆₁‖‖(𝐼−𝐾𝐿)⁻¹‖‖‖𝑆𝑢‖=𝛿‖𝑆𝑢‖.(2.18) Hence, 𝑆(𝑃)⊆𝑃₁. The proof is complete.

Define operators f,𝐴∶𝐸→𝐸, respectively, by(f𝑢)(𝑡)=𝑓(𝑡,𝑢(𝑡)𝐴),=𝑆𝑢f∈.𝐸,𝑡∈𝐙[1,𝑇],(2.19) It follows from the continuity of 𝑓 together with the complete continuity of 𝑆 that 𝐴∶𝐸→𝐸 is completely continuous.

Remark 2.6. By Lemma 2.3, it is easy to see that 𝑢={𝑢(𝑡)}^𝑇𝑡=1∈𝐸 is a fixed point of the operator 𝐴 if and only if 𝑢={𝑢(𝑡)}^𝑇+1𝑡=0 is a solution of PBVP (1.2), where 𝑢(0)=𝑢(𝑇),𝑢(1)=𝑢(𝑇+1).

The proofs of the main theorems of this paper are based on the topological degree and fixed point index theories. The following four well-known lemmas in [20–22] are needed in our argument.

Lemma 2.7. Let Ω be a bounded open set in a real Banach space 𝐸 with 𝜃∈Ω, let and 𝐴∶Ω→𝐸 be completely continuous. If there exists 𝑥₀∈𝐸⧵{𝜃} such that 𝑥−𝐴𝑥≠𝜇𝑥₀ for all 𝑥∈𝜕Ω and 𝜇≥0, then the topological degree deg(I−A,Ω,𝜃)=0.

Lemma 2.8. Let Ω be a bounded open set in a real Banach space 𝐸 with 𝜃∈Ω, and let 𝐴∶Ω→𝐸 be completely continuous. If 𝐴𝑥≠𝜇𝑥 for all 𝑥∈𝜕Ω and 𝜇≥1, then the topological degree deg(𝐼−𝐴,Ω,𝜃)=1.

Lemma 2.9. Let 𝐸 be a Banach space and 𝑋⊂𝐸 a cone in 𝐸. Assume that Ω is a bounded open subset of 𝐸. Suppose that 𝐴∶𝑋∩Ω→𝑋 is a completely continuous operator. If there exists 𝑥₀∈𝑋⧵{𝜃} such that 𝑥−𝐴𝑥≠𝜇𝑥₀ for all 𝑥∈𝑋∩𝜕Ω and 𝜇≥0, then the fixed point index 𝑖(𝐴,𝑋∩Ω,𝑋)=0.

Lemma 2.10. Let 𝐸 be a Banach space and 𝑋⊂𝐸 a cone in 𝐸. Assume that Ω is a bounded open subset of 𝐸 with 𝜃∈Ω. Suppose that 𝐴∶𝑋∩Ω→𝑋 is a completely continuous operator. If 𝐴𝑥≠𝜇𝑥 for all 𝑥∈𝑋∩𝜕Ω and 𝜇≥1, then the fixed point index 𝑖(𝐴,𝑋∩Ω,𝑋)=1.

3. Existence of Nontrivial Periodic Solutions

Theorem 3.1. Assume that (H) holds. If the following conditions are satisfied limsup_𝑥→∞max_{𝑡∈𝐙[1,𝑇]}|||𝑓(𝑡,𝑥)𝑥|||<𝜆¹,(3.1)liminf_𝑥→0+min_{[]𝑡∈𝐙1,𝑇}𝑓(𝑡,𝑥)𝑥>𝜆¹,(3.2)limsup_𝑥→0−max_{[]𝑡∈𝐙1,𝑇}𝑓(𝑡,𝑥)𝑥<𝜆¹,(3.3) where 𝜆₁ is the first positive eigenvalue of the linear operator 𝑆 given in Lemma 2.4, then (1.1) has at least one nontrivial periodic solution.

Proof. In view of Remark 2.6, it suffices to prove that the operator 𝐴 has at least fixed point in 𝐸⧵{𝜃}. It follows from (3.2) and (3.3) that there exist 𝑟>0 and 𝜎∈(0,1) such that 𝑓𝑓(𝑡(,𝑡,𝑥𝑥))≥≥𝜆𝜆₁₁(1+𝜎)𝑥≥𝜆₁(1−𝜎)𝑥,∀𝑥∈[0,𝑟],𝑡∈𝐙[1,𝑇],(1−𝜎)𝑥≥𝜆₁(1+𝜎)𝑥,∀𝑥∈[−𝑟,0],𝑡∈𝐙[1,𝑇].(3.4) By the above two inequalities, we have 𝑓𝑓((𝑡𝑡,,𝑥𝑥))≥≥𝜆𝜆₁₁(1+𝜎)𝑥,∀|𝑥|≤𝑟,𝑡∈𝐙[1,𝑇],(3.5)(1−𝜎)𝑥,∀|𝑥|≤𝑟,𝑡∈𝐙[1,𝑇].(3.6) We may suppose that 𝐴 has no fixed point on 𝜕𝐵_𝑟. Otherwise, the proof is finished. Now we will prove 𝑢≠𝐴𝑢+𝜇𝜉,∀𝑢∈𝜕𝐵_𝑟,𝜇≥0,(3.7) where 𝜉 is given in Lemma 2.4. Suppose the contrary; then there exist 𝑢₀∈𝜕𝐵_𝑟 and 𝜇₀≥0 such that 𝑢₀=𝐴𝑢₀+𝜇₀𝜉. Then 𝜇₀>0. Multiplying the equality 𝑢₀=𝐴𝑢₀+𝜇₀𝜉 by 𝜉 on its both sides, summing from 1 to 𝑇, and using (2.14) and (3.5), it follows that ^𝑇_𝑡=1𝑢₀(𝑡)𝜉(𝑡)=^𝑇_𝑡=1𝐴𝑢₀(𝑡)𝜉(𝑡)+𝜇^0𝑇_𝑡=1𝜉²(𝑡)=1𝜆_1𝑇𝑡=1𝑓𝑡,𝑢₀(𝑡)𝜉(𝑡)+𝜇_0𝑇_𝑡=1≥(1+𝜎)_𝑇𝜉²(𝑡)_𝑡=1𝑢₀(𝑡)𝜉(𝑡)+𝜇_0𝑇_𝑡=1𝜉²(𝑡).(3.8) Similarly, by (3.6), we know also that ^𝑇_𝑡=1𝑢₀(𝑡)𝜉(𝑡)≥(1−𝜎)^𝑇_𝑡=1𝑢₀(𝑡)𝜉(𝑡)+𝜇_0𝑇_𝑡=1𝜉²(𝑡).(3.9) If ∑^𝑇𝑡=1𝑢₀(𝑡)𝜉(𝑡)≥0, then (3.8) implies that ∑^𝑇𝑡=1𝜉²(𝑡)≤0, which contradicts 𝜉>0 on 𝐙[1,𝑇]. If ∑^𝑇𝑡=1𝑢₀(𝑡)𝜉(𝑡)<0, then (3.9) also implies that ∑^𝑇𝑡=1𝜉²(𝑡)<0, which is a contradiction. Thus, (3.7) holds. On the basis of Lemma 2.7, we have deg𝐼−𝐴,𝐵_𝑟,𝜃=0.(3.10)
From (3.1) it follows that there exist 𝐺>0 and 𝜀∈(0,1) such that |𝑓(𝑡,𝑥)|<𝜆₁(1−𝜀)|𝑥| for |𝑥|>𝐺,𝑡∈𝐙[1,𝑇]. Let 𝐶=sup_{𝑡∈𝐙[1,𝑇],|𝑥|≤𝐺}𝑓(𝑡,𝑥). Obviously, ||𝑓(𝑡,𝑥)||≤𝜆₁(1−𝜀)|𝑥|+𝐶,∀𝑥∈𝐑,𝑡∈𝐙[1,𝑇].(3.11) Choose 𝑅 such that 𝑅>max{𝑟,(𝑎𝜀)⁻¹𝐶}, where 𝑎=min_{𝑡∈𝐙[1,𝑇]}𝜉(𝑡). We next show 𝐴𝑢≠𝜇𝑢,forall𝑢∈𝜕𝐵_𝑅,𝜇≥1. In fact, if there exist 𝑢₁∈𝜕𝐵_𝑅 and 𝜇₁≥1 such that 𝐴𝑢₁=𝜇₁𝑢₁, then, by the definition of 𝐴 and (3.11), we obtain ||𝑢₁(𝑡)||≤||𝐴𝑢₁(𝑡)||≤(𝐼−𝐾𝐿)⁻¹^𝑇_𝑘=1≤𝜆₁𝐺(𝑡,𝑘)||𝑓𝑘,𝑢₁(𝑘)||(1−𝜀)(𝐼−𝐾𝐿)⁻¹^𝑇_𝑘=1𝐺(𝑡,𝑘)||𝑢₁(𝑘)||+𝐶(𝐼−𝐾𝐿)^𝑇−1_𝑘=1𝐺(𝑡,𝑘).(3.12) Set 𝑢₂(𝑡)=|𝑢₁(𝑡)|. Then 𝑢₂∈𝑃⧵{𝜃}, and, for any 𝑡∈𝐙[1,𝑇],𝜆₁(1−𝜀)(𝑆𝑢₂)(𝑡)+𝐶(𝑆𝑣₀)(𝑡)≥𝑢₂(𝑡), where 𝑣₀(𝑡)≡1. Then, by (2.14), we have ^𝑇_𝑡=1(1−𝜀)𝑢²(𝑡)+𝐶𝜆₁𝑣₀(𝑡)𝜉(𝑡)=𝜆₁(1−𝜀)^𝑇_𝑡=1𝑆𝑢₂(𝑡)𝜉(𝑡)+𝐶^𝑇_𝑡=1𝑆𝑣₀(𝑡)𝜉(𝑡)≥^𝑇_𝑡=1𝑢₂(𝑡)𝜉(𝑡).(3.13) Using the above inequality and noticing that ∑^𝑇𝑡=1𝜉(𝑡)=𝜆₁ (see Lemma 2.4), we have that 𝜀⁻¹𝐶≥∑^𝑇𝑡=1𝑢₂(𝑡)𝜉(𝑡)≥𝑎∑^𝑇𝑡=1𝑢₂(𝑡). This implies that 𝑅=‖𝑢₂‖≤∑^𝑇𝑡=1𝑢₂(𝑡)≤(𝑎𝜀)⁻¹𝐶, which contradicts the choice of 𝑅. It follows from Lemma 2.8 that deg𝐼−𝐴,𝐵_𝑅,𝜃=1.(3.14) According to the additivity of Leray-Schauder degree, by (3.14) and (3.10), we get deg𝐼−𝐴,𝐵_𝑅⧵𝐵_𝑟,𝜃=deg𝐼−𝐴,𝐵_𝑅,𝜃−deg𝐼−𝐴,𝐵_𝑟,𝜃=1,(3.15) which implies that the nonlinear operator 𝐴 has at least one fixed point in 𝐵_𝑅⧵𝐵_𝑟. Thus, (1.1) has at least one nontrivial periodic solution. The proof is complete.

Theorem 3.2. Assume that (H) holds. If the following conditions are satisfied limsup_𝑥→0max_{𝑡∈𝐙[1,𝑇]}|||𝑓(𝑡,𝑥)𝑥|||<𝜆¹,(3.16)liminf_𝑥→+∞min_{𝑡∈𝐙[1,𝑇]}𝑓(𝑡,𝑥)𝑥>𝜆¹,(3.17)limsup_{𝑥→−∞}max_{𝑡∈𝐙[1,𝑇]}𝑓(𝑡,𝑥)𝑥<𝜆¹,(3.18) where 𝜆₁ is the first positive eigenvalue of the linear operator 𝑆 given in Lemma 2.4, then (1.1) has at least one nontrivial periodic solution.

Proof. It suffices to prove that the operator 𝐴 has at least fixed point in 𝐸⧵{𝜃}. From (3.16), we find that there exist 𝜀∈(0,1) and 𝑟>0 such that ||𝑓(𝑡,𝑥)||≤𝜆₁(1−𝜀)|𝑥|,∀|𝑥|≤𝑟,𝑡∈𝐙[1,𝑇],(3.19) Now we prove 𝐴𝑢≠𝜇𝑢,∀𝑢∈𝜕𝐵_𝑟,𝜇≥1.(3.20) If (3.20) does hold, there exist 𝜇₀≥1 and 𝑢₀∈𝜕𝐵_𝑟 such that 𝐴𝑢₀=𝜇₀𝑢₀. Then, by (3.19), we have ||𝑢₀(𝑡)||≤||𝐴𝑢₀(𝑡)||≤(𝐼−𝐾𝐿)⁻¹^𝑇_𝑘=1≤𝜆₁𝐺(𝑡,𝑘)||𝑓𝑘,𝑢₀(𝑘)||(1−𝜀)(𝐼−𝐾𝐿)⁻¹^𝑇_𝑘=1𝐺(𝑡,𝑘)||𝑢₀(𝑘)||,𝑡∈𝐙[1,𝑇].(3.21) Set 𝑢₁(𝑡)=|𝑢₀(𝑡)|. Then 𝑢₁∈𝑃⧵{𝜃} and 𝜆₁(1−𝜀)𝑆𝑢₁≥𝑢₁. Multiplying this inequality by 𝜉 and summing from 1 to 𝑇, it follows from (2.14) that (1−𝜀)^𝑇_𝑡=1𝑢₁(𝑡)𝜉(𝑡)=𝜆₁(1−𝜀)^𝑇_𝑡=1𝑆𝑢₁(𝑡)𝜉(𝑡)≥^𝑇_𝑡=1𝑢₁(𝑡)𝜉(𝑡).(3.22) This together with ∑^𝑇𝑡=1𝑢₁(𝑡)𝜉(𝑡)>0 implies that 1−𝜀≥1, which contradicts the choice of 𝜀, and so (3.20) holds. It follows from Lemma 2.8 that deg𝐼−𝐴,𝐵_𝑟,𝜃=1.(3.23)
By (3.17), (3.18), and the continuity of 𝑓(𝑡,𝑥) with respect to 𝑥, we know that there exist 𝜎∈(0,1) and 𝐶>0 such that 𝑓𝑓((𝑡𝑡,,𝑥𝑥))≥≥𝜆𝜆₁₁(1+𝜎)𝑥−𝐶,∀𝑥≥0,𝑡∈𝐙[1,𝑇],(1−𝜎)𝑥−𝐶,∀𝑥≤0,𝑡∈𝐙[1,𝑇].(3.24) Then, 𝑓𝑓((𝑡𝑡,,𝑥𝑥))≥≥𝜆𝜆₁₁(1+𝜎)𝑥−𝐶≥𝜆₁(1−𝜎)𝑥−𝐶,∀𝑥≥0,𝑡∈𝐙[1,𝑇],(1−𝜎)𝑥−𝐶≥𝜆₁(1+𝜎)𝑥−𝐶,∀𝑥≤0,𝑡∈𝐙[1,𝑇].(3.25) By the above two inequalities, we have 𝑓(𝑡,𝑥)≥𝜆₁(1+𝜎)𝑥−𝐶,∀𝑥∈𝐑,𝑡∈𝐙[1,𝑇],(3.26)𝑓(𝑡,𝑥)≥𝜆₁(1−𝜎)𝑥−𝐶,∀𝑥∈𝐑,𝑡∈𝐙[1,𝑇].(3.27) Set Ω={𝑢∈𝐸∶𝑢=𝐴𝑢+𝜏𝜉forsome𝜏≥0},(3.28) where 𝜉 is given in Lemma 2.4. We claim that Ω is bounded in 𝐸. In fact, for any 𝑢∈Ω, there exists 𝜏≥0 such that 𝑢=𝐴𝑢+𝜏𝜉≥𝐴𝑢. Then, by (3.26), we have 𝑢(𝑡)≥𝜆₁(1+𝜎)(𝑆𝑢)(𝑡)−𝐶𝑆𝑣₀(𝑡),𝑡∈𝐙[1,𝑇],(3.29) where 𝑣₀(𝑡)≡1. Multiplying the above inequality by 𝜉(𝑡) on both sides and summing from 1 to 𝑇, it follows from (2.14) that ^𝑇_𝑡=1𝑢(𝑡)𝜉(𝑡)≥𝜆₁(1+𝜎)^𝑇_𝑡=1(𝑆𝑢)(𝑡)𝜉(𝑡)−𝐶^𝑇_𝑡=1𝑆𝑣₀(𝑡)𝜉(𝑡)=(1+𝜎)^𝑇_𝑡=1𝑢(𝑡)𝜉(𝑡)−𝐶𝜆_1𝑇𝑡=1𝜉(𝑡).(3.30) Then, noticing that ∑^𝑇𝑡=1𝜉(𝑡)=𝜆₁, we have 𝜎^𝑇_𝑡=1𝑢(𝑡)𝜉(𝑡)≤𝐶.(3.31) On the other hand, bearing in mind that 𝜉=𝜆₁𝑆𝜉, we obtain that, for 𝑢∈Ω, 𝑢−𝜆₁(1−𝜎)𝑆𝑢+𝐶𝑆𝑣₀=𝑆f𝑢−𝜆₁(1−𝜎)𝑆𝑢+𝐶𝑆𝑣₀+𝜏𝜉=𝑆f𝑢−𝜆₁(1−𝜎)𝑢+𝐶𝑣₀+𝜏𝜆₁𝜉.(3.32) By (3.27), we obtain that f𝑢−𝜆₁(1−𝜎)𝑢+𝐶𝑣₀+𝜏𝜆₁𝜉∈𝑃. Lemma 2.5 yields that 𝑢−𝜆₁(1−𝜎)𝑆𝑢+𝐶𝑆𝑣₀∈𝑃₁. Then by (2.14) and (3.31), we obtain that ‖‖𝑢−𝜆₁(1−𝜎)𝑆𝑢+𝐶𝑆𝑣₀‖‖≤1𝛿^𝑇_𝑡=1𝑢(𝑡)−𝜆₁=1(1−𝜎)(𝑆𝑢)(𝑡)+𝐶𝑆𝑣₀(𝑡)𝜉(𝑡)𝛿^𝑇_𝑡=1𝑢(𝑡)𝜉(𝑡)−(1−𝜎)𝑢(𝑡)𝜉(𝑡)+𝜆𝐶₁𝜉(𝑡)≤2𝐶𝛿.(3.33) This gives ‖‖𝑢−𝜆₁(1−𝜎)𝑆𝑢‖‖≤2𝐶𝛿+𝐶‖‖𝑆𝑣⁰‖‖,∀𝑢∈Ω.(3.34) Hence, (𝐼−𝜆₁(1−𝜎)𝑆)(Ω)⊂𝐵_𝑅1, where 𝑅₁=2𝐶/𝛿+𝐶‖𝑆𝑣₀‖>0. It follows from 𝜆₁(1−𝜎)𝑟(𝑆)<1 that 𝐼−𝜆₁(1−𝜎)𝑆 has a linear bounded inverse (𝐼−𝜆₁(1−𝜎)𝑆)⁻¹. Therefore, there exists 𝑅₂>0 such that Ω⊂𝐼−𝜆₁(1−𝜎)𝑆⁻¹𝐵_𝑅1⊂𝐵_𝑅2.(3.35) Then, we can conclude that Ω is bounded in 𝐸, proving our claim. Thus, there exists 𝑅>max{𝑟,𝑅₂} such that 𝑢≠𝐴𝑢+𝜏𝜉,∀𝑢∈𝜕𝐵_𝑅,𝜏≥0.(3.36) This and Lemma 2.7 give deg(𝐼−𝐴,𝐵_𝑅,𝜃)=0. Taking (3.23) into account, we have deg(𝐼−𝐴,𝐵_𝑅⧵𝐵_𝑟,𝜃)=−1. Then, 𝐴 has at least one fixed point in 𝐵_𝑅⧵𝐵_𝑟, which means that (1.1) has at least one nontrivial periodic solution. The proof is completed.

4. Existence of Positive Periodic Solutions

Theorem 4.1. Assume that (H) holds. If the following conditions are satisfied 𝑥𝑓(𝑡,𝑥)≥0,∀𝑥∈𝐑,𝑡∈𝐙[1,𝑇],(4.1)limsup_𝑥→∞max_{𝑡∈𝐙[1,𝑇]}𝑓(𝑡,𝑥)𝑥<𝜆¹,(4.2)liminf_𝑥→0min_{𝑡∈𝐙[1,𝑇]}𝑓(𝑡,𝑥)𝑥>𝜆¹,(4.3) where 𝜆₁ is the first positive eigenvalue of the linear operator 𝑆 given in Lemma 2.4, then (1.1) has at least one positive periodic solution and one negative periodic solution.

Proof. From (4.1), we know that 𝐴(𝑃)⊂𝑃. Similar to the proof of Theorem 3.1, it follows from (4.1)–(4.3) and Lemmas 2.9 and 2.10 that there exist 0<𝑟<𝑅 such that 𝑖𝐴,𝐵_𝑟∩𝑃,𝑃=0,𝑖𝐴,𝐵_𝑅∩𝑃,𝑃=1.(4.4) Hence, by the additivity of the fixed point index, we have 𝑖𝐴,𝐵_𝑅⧵𝐵_𝑟∩𝑃,𝑃=𝑖𝐴,𝐵^𝑅∩𝑃,𝑃−𝑖𝐴,𝐵_𝑟∩𝑃,𝑃=1.(4.5) Then, the nonlinear operator 𝐴 has at least one fixed point on (𝐵_𝑅⧵𝐵_𝑟)∩𝑃. So (1.1) has at least one positive periodic solution.
Put 𝑓₁(𝑡,𝑥)=−𝑓(𝑡,−𝑥),forall(𝑡,𝑥)∈𝐙[1,𝑇]×𝐑. Define operators f₁,𝐴₁∶𝐸→𝐸, respectively, by f₁𝑢(𝑡)=𝑓₁(𝑡,𝑢𝐴(𝑡)),𝑢∈𝐸,𝑡∈𝐙[1,𝑇₁],=𝑆f₁.(4.6) Obviously,  𝐴₁(𝑃)⊂𝑃. Following almost the same procedure as above, from limsup_𝑥→∞max_{𝑡∈𝐙[1,𝑇]}(𝑓₁(𝑡,𝑥)/𝑥)<𝜆₁ and liminf_𝑥→0min_{𝑡∈𝐙[1,𝑇]}(𝑓₁(𝑡,𝑥)/𝑥)>𝜆₁, we know also that the nonlinear operator 𝐴₁ has at least one fixed point 𝜁∈𝑃⧵{𝜃}. Then 𝐴₁𝜁=𝜁. This means that 𝐴(−𝜁)=𝑆(f(−𝜁))=𝑆(−f₁(𝜁))=−𝐴₁(𝜁)=−𝜁. Hence, (1.1) has at least one negative periodic solution −𝜁, and the conclusion is achieved.

Theorem 4.2. Assume that (H) and (4.1) hold. If the following conditions are satisfied limsup_𝑥→0max_{𝑡∈𝐙[1,𝑇]}𝑓(𝑡,𝑥)𝑥<𝜆¹liminf_𝑥→∞,min_{𝑡∈𝐙[1,𝑇]}𝑓(𝑡,𝑥)𝑥>𝜆¹,(4.7) where 𝜆₁ is the first positive eigenvalue of the linear operator 𝑆 given in Lemma 2.4, then (1.1) has at least one positive periodic solution and one negative periodic solution.

The proof is similar to that of Theorem 4.1 and so we omit it here.

5. Examples

Example 5.1. Let 𝑓(𝑡,𝑥)=√|𝑥|. It is easy to see that limsup_𝑥→∞max_{𝑡∈𝐙[1,𝑇]}|𝑓(𝑡,𝑥)/𝑥|=0<𝜆₁,liminf_𝑥→0⁺min_{𝑡∈𝐙[1,𝑇]}(𝑓(𝑡,𝑥)/𝑥)=+∞>𝜆₁, and limsup_𝑥→0⁻max_{𝑡∈𝐙[1,𝑇]}(𝑓(𝑡,𝑥)/𝑥)=−∞<𝜆₁. Then, it follows from Theorem 3.1 that (1.1) has at least one nontrivial periodic solution.

Example 5.2. Let 𝑓(𝑡,𝑥)=2𝑥⁴+𝑥³. It is not difficult to see that limsup_𝑥→0max_{𝑡∈𝐙[1,𝑇]}|𝑓(𝑡,𝑥)/𝑥|=0<𝜆₁,liminf_𝑥→+∞min_{𝑡∈𝐙[1,𝑇]}(𝑓(𝑡,𝑥)/𝑥)=+∞>𝜆₁, and limsup_{𝑥→−∞}max_{𝑡∈𝐙[1,𝑇]}(𝑓(𝑡,𝑥)/𝑥)=−∞<𝜆₁. Then, it follows from Theorem 3.2 that (1.1) has at least one nontrivial periodic solution.

Example 5.3. Let 𝑓(𝑡,𝑥)=3𝑥⁵𝑒^𝑥6. Obviously, 𝑥𝑓(𝑡,𝑥)≥0 for all  𝑥∈𝐑 and 𝑡∈𝐙[1,𝑇]. Moreover, limsup_𝑥→0max_{𝑡∈𝐙[1,𝑇]}(𝑓(𝑡,𝑥)/𝑥)=0<𝜆₁ and liminf_𝑥→∞min_{𝑡∈𝐙[1,𝑇]}(𝑓(𝑡,𝑥)/𝑥)=+∞>𝜆₁. Then it follows from Theorem 4.2 that (1.1) has at least one positive periodic solution and one negative periodic solution.

Remark 5.4. It is easy to see that the existence of nontrivial periodic solutions in Examples 5.1–5.3 could not be obtained by any theorems in [1–16, 19].

Ackowledments

This project is supported by the Natural Science Foundation of Guangdong Province (no. S2011010001900) and by the Scientific Research Plan Item of Fujian Provincial Department of Education (no. JA06035).