Existence and Uniqueness of Homoclinic Solution for a Class of Nonlinear Second-Order Differential Equations

Abstract

The authors study the existence and uniqueness of a set with 2𝑘𝑇-periodic solutions for a class of second-order differential equations by using Mawhin's continuation theorem and some analysis methods, and then a unique homoclinic orbit is obtained as a limit point of the above set of 2𝑘𝑇-periodic solutions.

1. Introduction

In this paper, we study the existence and uniqueness of homoclinic solutions for the following nonlinear second-order differential equations: 𝑢^′′(𝑡)+𝑔𝑢^′(𝑡)+ℎ(𝑢(𝑡))=𝑓(𝑡), (1.1) where 𝑢(𝑡)∈𝑅,  𝑔,  ℎ and 𝑓 are all in 𝐶(𝑅,𝑅).

As usual we say that a nonzero solution 𝑢(𝑡) of (1.1) is homoclinic (to 0) if 𝑢(𝑡)→0 and 𝑢^′(𝑡)→0 as |𝑡|→+∞.

Equation (1.1) is important in the applied sciences such as nonlinear vibration of masses, see [1–3] and the references therein. But most of the authors in those papers are interested in the study of problems of periodic solutions. Recently, the existence of homoclinic solutions for some second-order ordinary differential equation (system) has been extensively studied by using critical point theory, see [4–13] and the references therein. For example, in [9], by using the Mountain Pass theorem, Lv et al. discussed the existence of homoclinic solutions for the following second-order Hamiltonian systems: 𝑢^′′(𝑡)−𝐿(𝑡)𝑢(𝑡)+∇𝑤(𝑡,𝑢(𝑡))=0, (1.2) and in [13], the authors by means of variational method studied the problem of homoclinic solutions for the forced pendulum equation without the first derivative term. But, as far as we know, there were few papers studying the existence of homoclinic solution for the equation such as (1.1). This is due to the fact that (1.1) contains the first derivative term 𝑔(𝑢^′(𝑡)). This implies that the differential equation is not the Euler Lagrange equation associated with some functional 𝐼∶𝑊^1,𝑝_2𝑘𝑇→𝑅. So the method of critical point theory (or variational method) in [4–13] cannot be applied directly. Although paper [13] discussed the existence of homoclinic solutions for the following equation containing the first derivative term: 𝑥^′′(𝑡)+𝑓(𝑡)𝑥^′(𝑡)+𝛽(𝑡)𝑥(𝑡)+𝑔(𝑡,𝑥(𝑡))=0, (1.3) the term containing the first derivative is only linear with respect to 𝑥^′(𝑡).

In order to investigate the homoclinic solutions to (1.1), firstly, we study the existence of 2𝑘𝑇-periodic solutions to the following equation for each 𝑘∈𝐍: 𝑢^′′(𝑡)+𝑔𝑢^′(𝑡)+ℎ(𝑢(𝑡))=𝑓_𝑘(𝑡), (1.4) where 𝑓_𝑘∶𝑅→𝑅 is a 2𝑘𝑇-periodic function such that 𝑓_𝑘(𝑡)=𝑓(𝑡),𝑡∈−𝑘𝑇,𝑘𝑇−𝜀₀𝑓𝑘𝑇−𝜀₀+𝑓(−𝑘𝑇)−𝑓𝑘𝑇−𝜀₀𝜀₀𝑡−𝑘𝑇+𝜀₀,𝑡∈𝑘𝑇−𝜀₀,𝑘𝑇, (1.5)𝑇>0 is a given constant, and 𝜀₀∈(0,𝑇) is a constant independent of 𝑘. Then a homoclinic solution to (1.1) is obtained as a limit point of the set {𝑢_𝑘(𝑡)}, where 𝑢_𝑘(𝑡) is an arbitrary 2𝑘𝑇-periodic solution to (1.4) for each 𝑘∈𝐍.

The significance of present paper is that we not only investigate the existence of homoclinic solution to (1.1), but also study the uniqueness of the homoclinic solution and, the existence of 2𝑘𝑇-periodic solutions to (1.4) is obtained by using Mawhin’s continuation theorem [14], not by using the methods of critical point theory, which is quite different from the approaches of [4–13, 15]. Furthermore, the method to obtain the homoclinic solution to (1.1) is also different from the corresponding ones of [15].

2. Main Lemmas

For each 𝑘∈𝐍, let 𝐶_2𝑘𝑇={𝑥∣𝑥∈𝐶(𝑅,𝑅),𝑥(𝑡+2𝑘𝑇)≡𝑥(𝑡)}, 𝐶¹_2𝑘𝑇={𝑥∣𝑥∈𝐶¹(𝑅,𝑅),𝑥(𝑡+2𝑘𝑇)≡𝑥(𝑡)}, the norms of 𝐶_2𝑘𝑇 and 𝐶¹_2𝑘𝑇 are defined by ‖⋅‖_∞=max_{𝑡∈[−𝑘𝑇,𝑘𝑇]}|𝑥(𝑡)| and ‖𝑥‖_𝐶¹_2𝑘𝑇=max{‖𝑥‖_∞,‖𝑥^′‖_∞}, respectively, then 𝐶_2𝑘𝑇 and 𝐶¹_2𝑘𝑇 are all Banach spaces. Furthermore for 𝑥∈𝐶_2𝑘𝑇, ‖𝑥‖_𝑟=(∫^𝑘𝑇_−𝑘𝑇|𝑥(𝑡)|^𝑟𝑑𝑡)^1/𝑟, where 𝑟∈(1,+∞).

Lemma 2.1 (see [12]). Let 𝑎>0 and 𝑞∈𝑊^1,𝑝(𝑅,𝑅), then for every 𝑡∈𝑅, the following inequality holds: 𝑞(𝑡)≤(2𝑎)^{−1/𝜇𝑡+𝑎}_𝑡−𝑎𝑞(𝑠)^𝜇𝑑𝑠^1/𝜇+𝑎(2𝑎)^{−1/𝑝𝑡+𝑎}_𝑡−𝑎𝑞^′(𝑠)^𝑝𝑑𝑠^1/𝑝, (2.1) where 𝜇,𝑝∈(1,+∞) are constants.

Lemma 2.2 (see [12]). Let 𝑞∈𝑊^1,𝑝_2𝑘𝑇(𝑅,𝑅^𝑛), then the following inequality holds: 𝑞_∞≤𝑇^{−1/𝜈𝑘𝑇}_−𝑘𝑇𝑞(𝑠)^𝜈𝑑𝑠^1/𝜈+𝑇^{(𝑝−1)/𝑝𝑘𝑇}_−𝑘𝑇𝑞^′(𝑠)^𝑝𝑑𝑠^1/𝑝, (2.2) where 𝜈 and 𝑝 are constants with 𝜈>1 and 𝑝>1.

In order to use Mawhin’s continuation theorem for investigating the existence of 2𝑘𝑇-periodic solutions to (1.4), we give some definitions associated with Mawhin’s continuation theorem.

Definition 2.3 (see [14]). Let 𝑋 and 𝑌 be two Banach spaces with norms ‖𝑥‖_𝑋 and ‖𝑥‖_𝑌, respectively. A linear operator 𝐿∶𝐷(𝐿)⊂𝑋→𝑌 (2.3) is said to be a Fredholm opeartor with index zero provided that (1)Im𝐿 is a closed subset of 𝑌; (2)dimKer𝐿=codimIm𝐿<∞.

If 𝐿∶𝐷(𝐿)⊂𝑋→𝑌 is a Fredholm operator with index zero, then 𝑋=ker𝐿⊕𝑋₁ and 𝑌=Im𝐿⊕𝑌₁. Let 𝑃∶𝑋→ker𝐿 and 𝑄∶𝑌→𝑌₁ be the continuous projectors. Clearly, ker𝐿∩(𝐷(𝐿)∩𝑋₁)={0}, thus the restriction 𝐿_𝑃∶=𝐿|_{𝐷(𝐿)∩𝑋1} is invertible. Denote by 𝐾_𝑃 the inverse of 𝐿_𝑃.

Definition 2.4 (see [14]). Let 𝑋 and 𝑌 be two Banach spaces with norms ‖𝑥‖_𝑋 and ‖𝑥‖_𝑌, respectively, and the operator 𝐿∶𝐷(𝐿)⊂𝑋→𝑌 (2.4) is a Fredholm operator with index zero, Ω⊂𝑋 is an open bounded set with 𝐷(𝐿)∩Ω≠𝜙. A continuous operator 𝑁∶Ω⊂𝑋→𝑌 is said to be 𝐿-compact in Ω, provided that (1)𝐾_𝑃(𝐼−𝑄)𝑁(Ω) is a relative compact set of 𝑋; (2)𝑄𝑁(Ω) is a bounded set of 𝑌.

Lemma 2.5 (see [14]). Suppose that 𝑋 and 𝑌 are two Banach spaces, and 𝐿∶𝐷(𝐿)⊂𝑋→𝑌 is a Fredholm operator with index zero. Furthermore, Ω⊂𝑋 is an open bounded subset and 𝑁∶Ω→𝑌 is 𝐿-compact on Ω. If all the following conditions hold:(1)𝐿𝑥≠𝜆𝑁𝑥, for all 𝑥∈𝜕Ω∩𝐷(𝐿),𝜆∈(0,1);(2)𝑁𝑥∉Im𝐿, for all 𝑥∈𝜕Ω∩Ker𝐿;(3)deg{𝐽𝑄𝑁,Ω∩Ker𝐿,0}≠0,

where 𝐽∶Im𝑄→Ker𝐿 is an isomorphism. Then equation 𝐿𝑥=𝑁𝑥 has a solution on Ω∩𝐷(𝐿).

Lemma 2.6. Assume that there are positive constants 𝑚,  𝑚₁,  𝑛,  𝑙₀, and 𝑙₁ with 𝑙₀≥𝑙₁, such that the following conditions hold.(A1)sup_𝑡∈𝑅|𝑓(𝑡)|<+∞, ∫_𝑅|𝑓(𝑡)|^{(𝑙0+1)/𝑙0}𝑑𝑡<+∞,∫_𝑅|𝑓(𝑡)|^{(𝑙1+1)/𝑙1}𝑑𝑡<+∞ and ∫_𝑅|𝑓(𝑡)|²𝑑𝑡<+∞.(A2)−𝑚₁|𝑥|^𝑙0+1≤𝑥𝑔(𝑥)≤−𝑚|𝑥|^𝑙0+1(𝑥)|𝑥|,∀𝑥∈𝑅,𝑥ℎ≤−𝑛^𝑙1+1,∀𝑥∈𝑅. (2.5)(A3)ℎ∈𝐶¹(𝑅,𝑅) with ℎ^′(𝑥)≤0 for all 𝑥∈𝑅.

Then for every 𝑘∈𝐍, (1.4) possesses a 2𝑘𝑇-periodic solution.

Remark 2.7. From (1.5), we see 𝑓_𝑘∞≤sup_𝑡∈𝑅𝑓(𝑡𝑓)<+∞,∀𝑘∈𝐍,_{𝑘(𝑙0+1)/𝑙0}=^𝑘𝑇_−𝑘𝑇𝑓_𝑘(𝑠)^{(𝑙0+1)/𝑙0}𝑑𝑠^{𝑙0/(𝑙0+1)}≤^{𝑘𝑇−𝜀0}_−𝑘𝑇𝑓_𝑘(𝑠)^{(𝑙0+1)/𝑙0}𝑑𝑠^{𝑙0/(𝑙0+1)}+^𝑘𝑇_{𝑘𝑇−𝜀0}𝑓_𝑘(𝑠)^{(𝑙0+1)/𝑙0}𝑑𝑠^{𝑙0/(𝑙0+1)}≤_𝑅𝑓(𝑠)^{(𝑙0+1)/𝑙0}𝑑𝑠^{𝑙0/(𝑙0+1)}+𝜀^𝑙0/(𝑙0₀⁺¹⁾sup_𝑡∈𝑅𝑓(𝑡),∀𝑘∈𝐍, (2.6) which together with assumption (A1) yields that ‖𝑓_𝑘‖_∞ and ‖𝑓_𝑘‖_{(𝑙0+1)/𝑙0} are two constants independent of 𝑘∈𝐍.

Similarly, we have that ‖𝑓_𝑘‖_{(𝑙1+1)/𝑙1}<+∞ and ‖𝑓_𝑘‖₂<+∞ are two constants independent of 𝑘∈𝐍.

Proof. Set 𝑋=𝐶¹_2𝑘𝑇,  𝑌=𝐶_2𝑘𝑇,  𝐿∶𝐷(𝐿)⊂𝑋→𝑌,  𝐿𝑢=𝑢^′′,   where 𝐷(𝐿)={𝑢∣𝑢∈𝐶²_2𝑘𝑇}, and 𝑁∶𝐶¹_2𝑘𝑇→𝐶_2𝑘𝑇,[𝑁𝑢](𝑡)=−𝑔𝑢^′(𝑡)−ℎ(𝑢(𝑡))+𝑓_𝑘(𝑡). (2.7) Clearly, Ker𝐿=𝑅, Im𝐿={𝑦∈𝑌∶∫^𝑘𝑇_−𝑘𝑇𝑦(𝑠)𝑑𝑠=0}, which implies that Im𝐿 is a closed subset of 𝑋, and dimKer𝐿=codimIm𝐿=1<+∞. So 𝐿 is a Fredholm operator with index zero. Let 𝑃∶𝑋→Ker𝐿,𝑄∶𝑌→𝑌/Im𝐿 (2.8) be defined respectively by 𝑃𝑥=(1/2𝑘𝑇)∫₀^2𝑘𝑇𝑥(𝑠)𝑑𝑠,𝑄𝑥=(1/2𝑘𝑇)∫₀^2𝑘𝑇𝑥(𝑠)𝑑𝑠 and let 𝐿_𝑃=𝐿|_{𝑋∩Ker𝑃}∶𝑋∩Ker𝑃→Im𝐿. (2.9) Then 𝐿_𝑃 has a unique continuous pseudo-inverse 𝐿_𝑃⁻¹ on Im𝐿 defined by (𝐿_𝑃⁻¹𝑦)(𝑡)=[𝐹𝑦](𝑡), where 𝐹𝑦(𝑡)=₀^2𝑘𝑇𝐺()𝑦(𝑠)𝐺()=𝑠()𝑡,𝑠𝑑𝑠,𝑡,𝑠2𝑘𝑇−𝑡𝑡()2𝑘𝑇,0≤𝑠<𝑡,2𝑘𝑇−𝑠2𝑘𝑇,𝑡≤𝑠≤2𝑘𝑇. (2.10) For each open bounded set Ω⊂𝐶_2𝑘𝑇, from the above formula, it is easy to see that the mapper 𝑁 is 𝐿-compact on Ω.
Step  1. For each 𝑘∈𝐍, let Ω₁={𝑥∈𝐶¹_2𝑘𝑇∶𝐿𝑥=𝜆𝑁𝑥,𝜆∈(0,1)}, that is, Ω₁=𝑥∈𝐶¹_2𝑘𝑇∶𝑥^′′(𝑡)+𝜆𝑔𝑥^′(𝑡)+𝜆ℎ(𝑥(𝑡))=𝜆𝑓_𝑘(𝑡),𝜆∈(0,1). (2.11) We will show that Ω₁ is bounded in 𝐶¹_2𝑘𝑇. Suppose that 𝑢∈Ω₁, then 𝑢^′′(𝑡)+𝜆𝑔𝑢^′(𝑡)+𝜆ℎ(𝑢(𝑡))=𝜆𝑓_𝑘(𝑡),𝜆∈(0,1). (2.12) Multiplying both sides of (2.12) by 𝑢^′(𝑡) and integrating on the interval [−𝑘𝑇,𝑘𝑇], we have from assumption (A2) that 𝑚^𝑘𝑇_−𝑘𝑇𝑢^′(𝑡)^𝑙0+1𝑑𝑡≤−^𝑘𝑇_−𝑘𝑇𝑢^′(𝑡)𝑔𝑢^′(𝑡)𝑑𝑡=−^𝑘𝑇_−𝑘𝑇𝑓_𝑘(𝑡)𝑢^′(𝑡)𝑑𝑡. (2.13) By using hólder inequality, we get 𝑢^′𝑙0_𝑙⁺¹₀₊₁≤1𝑚𝑓_{𝑘(𝑙0+1)/𝑙0}⋅𝑢^′_𝑙0+1, (2.14) which together with the conclusion of Remark 2.7 shows 𝑢^′_𝑙0+1≤1𝑚𝑓_{𝑘(𝑙0+1)/𝑙0}^1/𝑙0∶=𝛽₁. (2.15) Clearly, 𝛽₁ is a constant independent of 𝑘 and 𝜆.
Multiplying both sides of (2.12) by 𝑢^′′(𝑡) and integrating on the interval [−𝑘𝑇,𝑘𝑇], we have 𝑢^′′2₂−𝜆^𝑘𝑇_−𝑘𝑇𝑢^′(𝑡)²ℎ^′(𝑢(𝑡))𝑑𝑡=𝜆^𝑘𝑇_−𝑘𝑇𝑓_𝑘(𝑡)𝑢^′′(𝑡)𝑑𝑡. (2.16) It follows from (2.16) and assumption (A3) that 𝑢^′′2₂≤^𝑘𝑇_−𝑘𝑇𝑓_𝑘(𝑡)𝑢^′′(𝑡)𝑑𝑡≤𝑓_𝑘2⋅𝑢^′′₂, (2.17) which implies 𝑢^′′₂≤𝑓_𝑘2∶=𝛽₂. (2.18) By using Lemma 2.2, we have 𝑢^′_∞≤𝑇^{−1/(𝑙0+1)}𝑢^′_𝑙0+1+𝑇^1/2𝑢^′′₂≤𝑇^{−1/(𝑙0+1)}𝛽₁+𝑇^1/2𝛽₂∶=𝛽. (2.19) Clearly, 𝛽 is a constant independent of 𝑘 and 𝜆.
On the other hand, multiplying both sides of (2.12) by 𝑢(𝑡) and integrating on the interval [−𝑘𝑇,𝑘𝑇], we have −^𝑘𝑇_−𝑘𝑇𝑢^′(𝑡)²𝑑𝑡+𝜆^𝑘𝑇_−𝑘𝑇ℎ(𝑢(𝑡))𝑢(𝑡)𝑑𝑡=−𝜆^𝑘𝑇_−𝑘𝑇𝑔𝑢^′(𝑡)𝑢(𝑡)𝑑𝑡+𝜆^𝑘𝑇_−𝑘𝑇𝑓_𝑘(𝑡)𝑢(𝑡)𝑑𝑡. (2.20) It follows from assumption (A2) that 𝜆𝑛^𝑘𝑇_−𝑘𝑇(𝑢(𝑡))^𝑙1+1𝑑𝑡+^𝑘𝑇_−𝑘𝑇𝑢^′(𝑡)²𝑑𝑡≤𝜆𝑚₁^𝑘𝑇_−𝑘𝑇𝑢^′(𝑡)^𝑙0|𝑢(𝑡)|𝑑𝑡+𝜆^𝑘𝑇_−𝑘𝑇𝑓_𝑘(𝑡)|𝑢(𝑡)|𝑑𝑡, (2.21) which together with (2.19) and 𝑙₀≥𝑙₁ results in 𝑛^𝑘𝑇_−𝑘𝑇|𝑢(𝑡)|^𝑙1+1𝑑𝑡≤𝑚₁^𝑘𝑇_−𝑘𝑇𝑢^′(𝑡)^𝑙0⋅|𝑢(𝑡)|𝑑𝑡+^𝑘𝑇_−𝑘𝑇𝑓_𝑘(𝑡)𝑢(𝑡)𝑑𝑡≤𝑚₁^𝑘𝑇_−𝑘𝑇𝑢^′(𝑡)^{𝑙0(𝑙1+1)/𝑙1}𝑑𝑡^{𝑙1/(𝑙1+1)}⋅^𝑘𝑇_−𝑘𝑇|𝑢(𝑡)|^𝑙1+1𝑑𝑡^1/(𝑙1+1)+^𝑘𝑇_−𝑘𝑇𝑓_𝑘(𝑡)^{(𝑙1+1)/𝑙1}𝑑𝑡^{𝑙1/(𝑙1+1)}⋅^𝑘𝑇_−𝑘𝑇|𝑢(𝑡)|^𝑙1+1𝑑𝑡^1/(𝑙1+1)=𝑚₁^𝑘𝑇_−𝑘𝑇𝑢^′(𝑡)^𝑙0+1𝑢^′(𝑡)^{(𝑙0−𝑙1)/𝑙1}𝑑𝑡^{𝑙1/(𝑙1+1)}⋅‖𝑢‖_𝑙1+1+𝑓_{𝑘(𝑙1+1)/𝑙1}‖𝑢‖_𝑙1+1≤𝑚₁𝛽^{(𝑙0−1)/(𝑙1+1)𝑘𝑇}_−𝑘𝑇𝑢^′(𝑡)^𝑙0+1𝑑𝑡^{𝑙1/(𝑙1+1)}‖𝑢‖_𝑙1+1+𝑓_{𝑘(𝑙1+1)/𝑙1}‖𝑢‖_𝑙1+1=𝑚₁𝛽^{(𝑙0−𝑙1)/(𝑙1+1)}𝑢^′_𝑙0+1^{𝑙1(𝑙0+1)/(𝑙1+1)}‖𝑢‖_𝑙1+1+𝑓_{𝑘(𝑙1+1)/𝑙1}‖𝑢‖_𝑙1+1≤𝑚₁𝛽^{(𝑙0−1)/(𝑙1+1)}𝛽^{𝑙1(𝑙0+1)/(𝑙1}₁⁺¹⁾‖𝑢‖_𝑙1+1+𝑓_{𝑘(𝑙1+1)/𝑙1}‖𝑢‖_𝑙1+1 (2.22) that is, ‖𝑢‖^𝑙1_𝑙⁺¹₁₊₁≤1𝑛𝑚₁𝛽^{(𝑙0−𝑙1)/(𝑙1+1)}𝛽^{𝑙1(𝑙0+1)/(𝑙1}₁⁺¹⁾+𝑓_{𝑘(𝑙1+1)/𝑙1}‖𝑢‖_𝑙1+1. (2.23) Therefore ‖𝑢‖_𝑙1+1≤1𝑛^1/𝑙1𝑚₁𝛽^{(𝑙0−𝑙1)/(𝑙1+1)}𝛽^{𝑙1(𝑙0+1)/(𝑙1}₁⁺¹⁾+𝑓_{𝑘(𝑙1+1)/𝑙1}^1/𝑙1∶=𝛼₁, (2.24) where 𝛼₁ is a constant independent of 𝑘 and 𝜆. By using Lemma 2.2 again, we get ‖𝑢‖_∞≤𝑇^{−1/(𝑙1+1)}‖𝑢‖_𝑙1+1+𝑇^{𝑙0/(𝑙0+1)}𝑢^′_𝑙0+1≤𝑇^{−1/(𝑙1+1)}𝛼₁+𝑇^{𝑙0/(𝑙0+1)}𝛽₁∶=𝛼. (2.25) Obviously, 𝛼 is a constant independent of 𝑘 and 𝜆. Therefore, if 𝑢∈Ω₁, then by (2.19) we see that ‖𝑢‖_𝐶¹_2𝑘𝑇=max‖𝑢‖_∞,𝑢^′_∞≤max𝛼,𝛽+sup_𝑡∈𝑅𝑓(𝑡)∶=𝑀. (2.26) Clearly, 𝑀>0 is a constant independent of 𝑘 and 𝜆; that is, Ω₁ is uniformly bounded for all 𝑘∈𝐍 and 𝜆∈(0,1).
Step  2. From assumptions (A2) and (A3), we see that there must be a constant 𝑀₁>0 such that −ℎ(𝑀₁)+𝑓_𝑘>0 and −ℎ(−𝑀₁)+𝑓_𝑘<0. Set Ω₂={𝑢(𝑡)∈𝐶¹_2𝑘𝑇∶‖𝑢‖_𝐶¹_2𝑘𝑇<𝑀}, where 𝑀=max{𝑀₁,𝑀}. We will show that 𝑁𝑢∉Im𝐿, for all 𝑢∈𝜕Ω₂∩Ker𝐿.
In fact, by assumption (A2), we see that 𝑔(0)=0, and if 𝑢∈𝜕Ω₂∩Ker𝐿, then 𝑢(𝑡)≡𝑀 or 𝑢≡−𝑀. So (𝑢)1𝑄𝑁=−2𝑘𝑇^𝑘𝑇_−𝑘𝑇ℎ(𝑀)−𝑓_𝑘(𝑡)𝑑𝑡=−ℎ(𝑀)+𝑓_𝑘≥−ℎ𝑀₁+𝑓_𝑘(𝑡)(𝑢)1>0,∀𝑢≡𝑀,𝑄𝑁=−2𝑘𝑇^𝑘𝑇_−𝑘𝑇ℎ(−𝑀)−𝑓_𝑘(𝑡)𝑑𝑡=−ℎ(−𝑀)+𝑓_𝑘≤−ℎ−𝑀₁+𝑓_𝑘<0,∀𝑢(𝑡)≡−𝑀, (2.27) where 𝑓_𝑘=(1/2𝑘𝑇)∫^𝑘𝑇_−𝑘𝑇𝑓_𝑘(𝑡)𝑑𝑡. This implies that 𝑁𝑢∉Im𝐿, for all 𝑢∈𝜕Ω₂∩Ker𝐿.
Step  3. Set 𝐽∶Im𝑄→Ker𝐿, 𝐽𝑥=𝑥, we will show deg{𝐽𝑄𝑁,Ω₂∩Ker𝐿,0}≠0.
Let 𝐻(𝑥,𝜇)=𝜇𝑥+(1−𝜇)𝐽𝑄𝑁𝑥, for all 𝑥∈Ω₂∩Ker𝐿, when 𝑥∈𝜕(Ω₂∩Ker𝐿), we have 𝑥=±𝑀 and 𝐻(𝑀)(𝑀)𝐻()()𝑀,𝜇=𝜇𝑀+1−𝜇𝐽𝑄𝑁=𝜇𝑀+1−𝜇𝑄𝑁>0,−𝑀,𝜇=−𝜇𝑀+1−𝜇𝐽𝑄𝑁−𝑀=−𝜇𝑀+1−𝜇𝑄𝑁−𝑀<0. (2.28) So for all 𝜇∈[0,1],  𝐻(𝜕(Ω₂∩Ker𝐿),𝜇)≠0, and then deg𝐽𝑄𝑁,Ω₂∩Ker𝐿,0=deg𝐻(⋅,0),Ω₂𝐻()∩Ker𝐿,0=deg⋅,1,Ω₂∩Ker𝐿,0=deg𝐼,Ω₂∩Ker𝐿,0=1. (2.29) Therefore, by Lemma 2.5, (1.4) has a 2𝑘𝑇-periodic solution 𝑢_𝑘∈Ω₂.

Remark 2.8. Suppose that all the conditions in Lemma 2.6 hold. We see that for each 𝑘∈𝐍, (1.4) has a 2𝑘𝑇-periodic solution 𝑢_𝑘∈Ω₂. This implies that 𝑢_𝑘∞≤𝑀,𝑢^′_𝑘∞≤𝑀. (2.30) Furthermore, as same as the proof of step  1 in Lemma 2.6 with replacing 𝑢(𝑡) by 𝑢_𝑘(𝑡), we have 𝑢_𝑘𝑙1+1≤𝛼₁,𝑢^′_𝑘𝑙0+1≤𝛽₁, (2.31) where 𝛼₁ and 𝛽₁ are two positive constants independent of 𝑘∈𝐍.

Lemma 2.9 (see [12]). Let 𝑢_𝑘∈𝐶¹_2𝑘𝑇 be the 2𝑘𝑇-periodic solution for (1.4) and satisfies (2.30) and (2.31) for all 𝑘∈𝐍. Then there exists a function 𝑢₀∈𝐶¹(𝑅,𝑅) such that for each interval [𝑐,𝑑]⊂𝑅, there is a subsequence {𝑢_𝑘𝑗} of {𝑢_𝑘}_𝑘∈𝐍 with 𝑢^′_𝑘𝑗(𝑡)→𝑢^′₀(𝑡) uniformly on [𝑐,𝑑].

3. Main Results

Theorem 3.1. Suppose that assumptions (A1), (A2), and (A3) in Lemma 2.6 hold. Then (1.1) has a unique homoclinic solution.

Proof. Since assumptions (A1), (A2), onsisting of Kuratowski operations we used following principles and (A3) in Lemma 2.6 hold, by using Lemma 2.6, we see that (1.4) has a 2𝑘𝑇-periodic solution 𝑢_𝑘(𝑡) satisfying (2.30) and (2.31) for each 𝑘∈𝐍. It follows from Lemma 2.9 that there exists a 𝑢₀∈𝐶¹(𝑅,𝑅) such that for each interval [𝑐,𝑑]⊂𝑅, there is a subsequence {𝑢_𝑘𝑗} of {𝑢_𝑘}_𝑘∈𝐍 satisfying 𝑢^′_𝑘𝑗(𝑡)→𝑢^′₀(𝑡) uniformly on [𝑐,𝑑]. Below, we will show that 𝑢₀(𝑡) is just a unique homoclinic solution to (1.1).
Step  1. We show that 𝑢₀ is a solution of (1.1).
In view of 𝑢_𝑘𝑗(𝑡) being a 2𝑘_𝑗𝑇-periodic solution to (1.4), we have 𝑢_𝑘^′′_𝑗(𝑡)+𝑔𝑢^′_𝑘𝑗(𝑡)+ℎ𝑢_𝑘𝑗(𝑡)=𝑓_𝑘𝑗(𝑡),for𝑡∈−𝑘_𝑗𝑇,𝑘_𝑗𝑇,𝑗∈𝐍. (3.1) Take 𝑎,𝑏∈𝑅 such that 𝑎<𝑏, there exists 𝑗₀∈𝐍 such that for all 𝑗>𝑗₀𝑢_𝑘^′′_𝑗(𝑡)+𝑔𝑢^′_𝑘𝑗(𝑡)+ℎ𝑢_𝑘𝑗(𝑡)=𝑓(𝑡),for𝑡∈[𝑎,𝑏]. (3.2) Integrating (3.2) from 𝑎 to 𝑡∈[𝑎,𝑏], we have 𝑢^′_𝑘𝑗(𝑡)−𝑢^′_𝑘𝑗(𝑎)=^𝑡_𝑎−𝑔𝑢^′_𝑘𝑗(𝑠)−ℎ𝑢_𝑘𝑗(𝑠)+𝑓(𝑠)𝑑𝑠,for𝑡∈[𝑎,𝑏]. (3.3) Since Lemma 2.9 shows that 𝑢_𝑘𝑗→𝑢₀ uniformly on [𝑎,𝑏] and 𝑢^′_𝑘𝑗→𝑢^′₀ uniformly on [𝑎,𝑏] as 𝑗→∞, let 𝑗→∞ in (3.3), we get 𝑢^′₀(𝑡)−𝑢^′₀(𝑎)=^𝑡_𝑎−𝑔𝑢^′₀(𝑠)−ℎ𝑢₀(𝑠)+𝑓(𝑠)𝑑𝑠,for𝑡∈[𝑎,𝑏]. (3.4) In view of 𝑎 and 𝑏 are arbitrary, (3.4) shows that 𝑢₀(𝑡) is a solution of (1.1).
Step  2. We prove that 𝑢₀(𝑡)→0, as 𝑡→±∞.
Obviously, for every 𝑖∈𝐍, there exists 𝑗_𝑖∈𝐍 such that for all 𝑗>𝑗_𝑖, we have ^𝑖𝑇_−𝑖𝑇𝑢_𝑘𝑗(𝑡)^𝑙1+1+𝑢^′_𝑘𝑗(𝑡)^𝑙0+1𝑑𝑡≤^𝑘𝑗𝑇_{−𝑘𝑗𝑇}𝑢_𝑘𝑗(𝑡)^𝑙1+1+𝑢^′_𝑘𝑗(𝑡)^𝑙0+1𝑑𝑡≤𝛼^𝑙1₁⁺¹+𝛽^𝑙0₁⁺¹∶=𝑀₂. (3.5) It follows that ^+∞_−∞𝑢₀(𝑡)^𝑙1+1+𝑢^′₀(𝑡)^𝑙0+1𝑑𝑡=lim_𝑖→+∞^𝑖𝑇_−𝑖𝑇𝑢₀(𝑡)^𝑙1+1+𝑢^′₀(𝑡)^𝑙0+1𝑑𝑡=lim_𝑖→+∞lim_𝑗→+∞^𝑖𝑇_−𝑖𝑇𝑢_𝑘𝑗(𝑡)^𝑙1+1+𝑢^′_𝑘𝑗(𝑡)^𝑙0+1𝑑𝑡≤𝑀₂, (3.6) and then _{|𝑡|≥𝑟}𝑢₀(𝑡)^𝑙1+1+𝑢^′₀(𝑡)^𝑙0+1𝑑𝑡→0,as𝑟→+∞, (3.7) which yields _{|𝑡|≥𝑟}𝑢₀(𝑡)^𝑙1+1𝑑𝑡→0,_{|𝑡|≥𝑟}𝑢^′₀(𝑡)^𝑙0+1𝑑𝑡→0,as𝑟→+∞. (3.8) By using Lemma 2.1, as 𝑡→±∞, 𝑢₀(𝑡)≤(2𝑎)^{−1/(𝑙1+1)𝑡+𝑎}_𝑡−𝑎𝑢₀(𝑠)^𝑙1+1𝑑𝑠^1/(𝑙1+1)+𝑎⋅(2𝑎)^{−1/(𝑙0+1)𝑡+𝑎}_𝑡−𝑎𝑢^′₀(𝑠)^𝑙0+1𝑑𝑠^1/(𝑙0+1)→0. (3.9) So we have 𝑢₀(𝑡)→0, as 𝑡→±∞.
Step  3. We will show that 𝑢^′₀(𝑡)→0,as𝑡→±∞. (3.10) From the Remark 2.8 and Lemma 2.9, we have 𝑢₀(𝑡)≤𝑀,𝑢^′₀(𝑡)≤𝑀,forall𝑡∈𝑅, (3.11) which together with (1.1) implies that 𝑢₀^′′_∞≤𝑔_𝑀+ℎ_𝑀+sup_𝑡∈𝑅𝑓(𝑡), (3.12) where 𝑔_𝑀=max_{|𝑥|≤𝑀}|𝑔(𝑥)| and ℎ_𝑀=max_{|𝑥|≤𝑀}|ℎ(𝑥)|. If 𝑢^′₀(𝑡)↛0, as 𝑡→±∞, then there exist a 𝜀₀∈(0,1/2) and a sequence {𝑡_𝑘} such that 𝑡₁<𝑡₂<𝑡₃<⋯,𝑡_𝑘+1<𝑡_𝑘+1𝑢,𝑘∈𝐍,_′0𝑡_𝑘≥2𝜀₀,𝑘∈𝐍. (3.13) From this, we have for 𝑡∈[𝑡_𝑘,𝑡_𝑘+𝜀₀/(1+𝑀₁)]𝑢^′₀(𝑡)=𝑢^′₀𝑡_𝑘+^𝑡_𝑡𝑘𝑢₀^′′(𝑠)𝑑𝑠≥𝑢^′₀𝑡_𝑘−^𝑡_𝑡𝑘𝑢₀^′′(𝑠)𝑑𝑠≥𝜀₀. (3.14) It follows that ^+∞_−∞𝑢^′₀(𝑡)^𝑙0+1𝑑𝑡≥^∞_𝑘=1^{𝑡𝑘+𝜀0/(1+𝑀1)}_𝑡𝑘𝑢^′₀(𝑡)^𝑙0+1𝑑𝑡=∞, (3.15) which contradicts (3.6), and so (3.10) holds.
Step  4. Finally, we will prove that (1.1) possesses a unique homoclinic solution. In order to do it, let 𝑢(𝑡)=𝑢₁(𝑡)−𝑢₂(𝑡), where 𝑢₁(𝑡) and 𝑢₂(𝑡) are two arbitrary homoclinic solutions of (1.1). Then 𝑢(𝑡)→0as𝑡→±∞. (3.16) We will show that 𝑢(𝑡)≡0. (3.17) If (3.17) does not hold, then there must be a 𝑡^∗∈𝑅 such that 𝑢𝑡^∗>0 (3.18) or 𝑢𝑡^∗<0. (3.19)
If 𝑢(𝑡^∗)>0, then from (3.16), we see that there is a constant 𝑋>0 such that 𝑡^∗∈(−𝑋,𝑋) and 𝑢(𝑡)<𝑢(𝑡^∗)/2 for 𝑡∈(−∞,𝑋)∪(𝑋,+∞). Let 𝑡^∗∗∈[−𝑋,𝑋] such that 𝑢(𝑡^∗∗)=max_{𝑡∈[−𝑋,𝑋]}𝑢(𝑡), then 𝑢𝑡^∗∗≥𝑢𝑡^∗𝑢𝑡>0,_∗∗≥𝑢𝑡^∗>sup_{𝑡∈(−∞,𝑋)∪(𝑋,+∞)}𝑢(𝑡), (3.20)(3.21) that is, 𝑢𝑡^∗∗=max_𝑡∈𝑅𝑢(𝑡). (3.22) So 𝑢^′(𝑡^∗∗)=0 and 𝑢^′′(𝑡^∗∗)≤0, and then from (1.1), we see −ℎ𝑢₁𝑡^∗∗−ℎ𝑢₂𝑡^∗∗=𝑢₁^′′𝑡^∗∗−𝑢₂^′′𝑡^∗∗=𝑢^′′𝑡^∗∗≤0. (3.23) By using the condition (A3), we have that 𝑢𝑡^∗∗=𝑢₁𝑡^∗∗−𝑢₂𝑡^∗∗≤0, (3.24) which contradicts to (3.20). This contradiction implies that (3.18) does not hold. Similarly, we can prove that (3.19) does not hold, either. So 𝑢(𝑡)≡0.
As an application, we consider the following example: 𝑢^′′(𝑡)−𝑚𝑢^′(𝑡)³(𝑢(𝑡=𝑒−𝑛))^𝑡/2𝑒^−𝑡+𝑒^𝑡, (3.25) where 𝑚,𝑛>0 are constants and, 𝑓(𝑡)=𝑒^𝑡/2/(𝑒^−𝑡+𝑒^𝑡). Corresponding to (1.1), we can chose 𝑙₀=3 and 𝑙₁=1 such that assumptions (A2) and (A3) hold. Furthermore, by the direct calculation, we can easily obtain that _𝑅𝑓(𝑡)^{(𝑙1+1)/𝑙1}𝑑𝑡=^+∞_−∞𝑓(𝑡)²𝜋𝑑𝑡=4<∞,_𝑅𝑓(𝑡)^{(𝑙0+1)/𝑙0}𝑑𝑡=^+∞_−∞𝑓(𝑡)^4/33𝑑𝑡=2<∞. (3.26) This implies that assumption (A1) also holds. So by applying Theorem 3.1, we know that (3.25) possesses a unique homoclinic solution.

Acknowledgments

The authors are very grateful to the referee for her/his careful reading of the original paper and for her/his valuable suggestions for improving this paper. This work was sponsored by the key NNSF of China (no. 11271197) and Science Foundation of NUIST (no. 20090202; 2012r101).