The Existence and Uniqueness of Periodic Solutions for Some Nonlinear 𝑛th-Order Differential Equations

Abstract

The nonlinear 𝑛th-order differential equations are considered. By using inequality techniques and coincidence degree theory, some criteria are obtained to guarantee the existence and uniqueness of 𝑇-periodic solutions for the equations. The obtained results are also valid and new for the problem discussed in the previous literature. Moreover, two illustrative examples are provided to illustrate the effectiveness of our results.

1. Introduction

In applied science, some practical problems are associated with the periodic solutions for nonlinear high-order differential equations, such as nonlinear oscillations [1, 2], electronic theory [3], biological model, and other models [4–6]. In particular, during the past thirty years, there has been a great amount of work on the existence and uniqueness of periodic solutions for the 𝑛th-order nonlinear differential equation 𝑥^(𝑛)+^𝑛−1_𝑗=1𝑎_𝑗𝑥^(𝑗)+𝑔(𝑡,𝑥)=𝑒(𝑡), (1.1) where 𝑒∶𝑅→𝑅 and 𝑔∶𝑅×𝑅→𝑅 are continuous functions, 𝑒(𝑡) is 2𝜋-periodic with respect to 𝑡, 𝑔 is 2𝜋-periodic in the first variable, and 𝑎_𝑖(𝑖=1,2,…,𝑛−1) are constants. Many of these results can be found in [7–11] and the references cited therein. Among the known results, we find that the assumption(𝐻)𝑔(𝑡,𝑥) is continuous, and there are positive constants 𝑚₀ and 𝑀₀ such that𝑚₀≤𝑔_𝑥(𝑡,𝑥)≤𝑀₀,∀(𝑡,𝑥)∈𝑅×𝑅 (1.2) is employed, and it plays an important role in the proofs of these known results (see, e.g., [9–11]). Recently, under some spectral conditions of linear differential operator, Li [12, 13] discussed the existence and uniqueness of 𝑇-periodic solutions of nonlinear differential equations.

However, to the best of our knowledge, there exist few results for the existence and uniqueness of periodic solutions of (1.1) without (𝐻) and the spectral conditions of linear differential operator. Thus, in this case, it is worth to study the problem of existence and uniqueness of periodic solutions of 𝑛th-order nonlinear differential equation (1.1).

The purpose of this paper is to investigate the existence and uniqueness of 𝑇-periodic solutions of (1.1). By using some inequality techniques and Mawhin’s continuation theorem, we establish some sufficient conditions for the existence and uniqueness of 𝑇-periodic solutions of (1.1) when (𝐻) and the spectral conditions are avoided. Moreover, two illustrative examples are given in Section 4.

2. Preliminary Results

Let us introduce some notations. We will use Φ to denote the empty set. For 𝑛∈𝐍, we denote by 𝐶^𝑛_𝑇 the Banach space𝐶^𝑛_𝑇=𝑢∈𝐶^𝑛(𝑅,𝑅)∶𝑢(𝑡)=𝑢(𝑡+𝑇),∀𝑡∈𝑅, (2.1) endowed with the norm‖𝑢‖_(𝑛)=∑^𝑛_𝑘=0𝑢^(𝑘)_∞,𝑢∈𝐶^𝑛_𝑇, (2.2) where, for a function 𝑣∈𝐶⁰_𝑇, we have that|𝑣|_∞=max_[]0,𝑇|𝑣|. (2.3) For 𝑥∈𝐶⁰_𝑇, we will denote|𝑥|_𝑝=^𝑇₀|𝑥(𝑡)|^𝑝𝑑𝑡^1/𝑝,𝑝>0. (2.4) Now, let 𝑓∶𝑅^𝑛+1→𝑅 be a continuous function, 𝑇-periodic with respect to the first variable, and consider the 𝑛th-order differential equation𝑢^(𝑛)=𝑓𝑡,𝑢,𝑢^′,𝑢^′′,…,𝑢^(𝑛−1). (2.5)

Lemma 2.1 (see [14]). Assume that the following conditions hold.(i)There exists 𝜌>0 such that, for each 𝜆∈(0,1], one has that any possible 𝑇-periodic solution 𝑢 of the problem 𝑢^(𝑛)=𝜆𝑓𝑡,𝑢,𝑢^′,𝑢^′′,…,𝑢^(𝑛−1) (2.6) satisfies the priori estimation ‖𝑢‖_(𝑛−1)<𝜌.(ii)The continuous function 𝐹∶𝐑→𝐑 defined by 𝐹(𝑥)=^𝑇₀𝑓(𝑡,𝑥,0,0,…,0)𝑑𝑡,(𝑥∈𝐑) (2.7) satisfies 𝐹(−𝜌)𝐹(𝜌)<0.Then, (2.5) has at least one 𝑇-periodic solution 𝑢 such that ‖𝑢‖_(𝑛−1)<𝜌.

From Lemma 2.2 in [15] and the proof of inequality (10) in [7, pp 3402], one obtains the following.

Lemma 2.2. Let 𝑥(𝑡)∈𝐶¹_𝑇. Suppose that there exists a constant 𝐷≥0 such that 𝑥𝜏₀≤𝐷,𝜏₀∈[0,𝑇], (2.8) then |𝑥|₂≤𝑇𝜋𝑥^′₂+√𝑇𝐷,|𝑥|_∞1≤𝐷+2√𝑇^𝑇₀𝑥^′(𝑡)²𝑑𝑡^1/2. (2.9)

Lemma 2.3. For any 𝑢∈𝐶²_𝑇, one has that ^𝑇₀𝑢^′(𝑡)²𝑇𝑑𝑡≤2𝜋^2𝑇₀𝑢^′′(𝑡)²𝑑𝑡. (2.10)

Proof. Lemma 2.3 is a direct consequence of the Wirtinger inequality, and see [16, 17] for its proof.

By the same approach used in the proof of Lemma  3 of [7], we have the following.

Lemma 2.4. For any 𝑢∈𝐶^𝑛_𝑇, one has that 𝑢^(𝑗)_∞≤𝑇^{(𝑛−1)−𝑗}12^{(𝑛−1)−(𝑗−1)𝑇}₀𝑢^(𝑛)(𝑡)𝑑𝑡,𝑗=1,2,…,𝑛−1. (2.11)

Lemma 2.5. Let 𝑘 be an even number, 𝑛=2𝑘, andΛ=𝑗∶𝑗∈{1,2,…,𝑘−1},(−1)^𝑗𝑎_2𝑗<0. (2.12) Assume that one of the following conditions is satisfied: (𝐻₁) for𝑡,𝑥₁,𝑥₂∈𝑅,𝑥₁≠𝑥₂,1+_𝑗∈Λ𝑇2𝜋^{2(𝑘−𝑗)}(−1)^𝑗𝑎_2𝑗>0,𝑔𝑡,𝑥₁−𝑔𝑡,𝑥₂𝑥₁−𝑥₂>0, (2.13)(𝐻₂) there exists a nonnegative constant 𝐵 such that𝑇1>𝐵𝜋²𝑇2𝜋^2(𝑘−1),1+_𝑗∈Λ𝑇2𝜋^{2(𝑘−𝑗)}(−1)^𝑗𝑎_2𝑗𝑇>𝐵𝜋²𝑇2𝜋^2(𝑘−1),𝐵𝑥₁−𝑥₂²≥−𝑔𝑡,𝑥₁−𝑔𝑡,𝑥₂𝑥₁−𝑥₂>0, (2.14) where𝑡,𝑥₁,𝑥₂∈𝑅,𝑥₁≠𝑥₂, then (1.1) has at most one 𝑇-periodic solution.

Proof. Suppose that 𝑢₁(𝑡)and 𝑢₂(𝑡)are two 𝑇-periodic solutions of (1.1). Set 𝑍(𝑡)=𝑢₁(𝑡)−𝑢₂(𝑡). Then, we obtain 𝑢₁(𝑡)−𝑢₂(𝑡)^(𝑛)+^𝑛−1_𝑗=1𝑎_𝑗𝑢₁(𝑡)−𝑢₂(𝑡)^(𝑗)+𝑔𝑡,𝑢₁(𝑡)−𝑔𝑡,𝑢₂(𝑡)=0. (2.15) Integrating (2.15) from 0 to 𝑇, it results that ^𝑇₀𝑔𝑡,𝑢₁(𝑡)−𝑔𝑡,𝑢₂(𝑡)𝑑𝑡=0. (2.16) Therefore, in view of integral mean value theorem, it follows that there exists a constant 𝛾∈[0,𝑇] such that 𝑔𝛾,𝑢₁𝛾−𝑔𝛾,𝑢₂𝛾=0. (2.17) Since 𝑔(𝑡,𝑥) is a strictly monotone function in 𝑥, (2.17) implies that 𝑍𝛾=𝑢₁𝛾−𝑢₂𝛾=0. (2.18) Then, from (2.9), we have |𝑍|₂≤𝑇𝜋𝑍^′₂. (2.19) Multiplying (2.15) by 𝑍(𝑡) and then integrating it from 0 to 𝑇, it follows that ^𝑇₀𝑍^(𝑘)(𝑡)²𝑑𝑡+^𝑘−1_𝑗=1(−1)^𝑗𝑎_2𝑗^𝑇₀𝑍^(𝑗)(𝑡)²𝑑𝑡=−^𝑇₀𝑍(𝑡)𝑔𝑡,𝑢₁(𝑡)−𝑔𝑡,𝑢₂(𝑡)𝑑𝑡=−^𝑇₀𝑢₁(𝑡)−𝑢₂(𝑡)𝑔𝑡,𝑢₁(𝑡)−𝑔𝑡,𝑢₂(𝑡)𝑑𝑡. (2.20) Now suppose that (𝐻₁) (or (𝐻₂)) holds, and we will consider two cases as follows.
Case  i. If (𝐻₁) holds, (2.10) and (2.20) yield that 0≤1+_𝑗∈Λ𝑇2𝜋^{2(𝑘−𝑗)}(−1)^𝑗𝑎_2𝑗^𝑇₀𝑍^(𝑘)(𝑡)²≤𝑑𝑡_𝑇0𝑍^(𝑘)(𝑡)²𝑑𝑡+^𝑘−1_𝑗=1(−1)^𝑗𝑎_2𝑗^𝑇₀𝑍^(𝑗)(𝑡)²𝑑𝑡=−^𝑇₀𝑢₁(𝑡)−𝑢₂(𝑡)𝑔𝑡,𝑢₁(𝑡)−𝑔𝑡,𝑢₂(𝑡)𝑑𝑡≤0,whereΛ≠∅,0≤^𝑇₀𝑍^(𝑘)(𝑡)²≤𝑑𝑡_𝑇0𝑍^(𝑘)(𝑡)²𝑑𝑡+^𝑘−1_𝑗=1(−1)^𝑗𝑎_2𝑗^𝑇₀𝑍^(𝑗)(𝑡)²𝑑𝑡=−^𝑇₀𝑢₁(𝑡)−𝑢₂(𝑡)𝑔𝑡,𝑢₁(𝑡)−𝑔𝑡,𝑢₂(𝑡)𝑑𝑡≤0,whereΛ=∅, (2.21) which, together with (2.18), implies that 𝑍(𝑡)≡𝑍^′(𝑡)≡⋯≡𝑍^(𝑘)(𝑡)≡0,∀𝑡∈𝑅. (2.22) Hence, (1.1) has at most one 𝑇-periodic solution.Case  ii. If (𝐻₂) holds, using (2.9), (2.10), (2.19), and (2.20), we obtain that 1+_𝑗∈Λ𝑇2𝜋^{2(𝑘−𝑗)}(−1)^𝑗𝑎_2𝑗^𝑇₀𝑍^(𝑘)(𝑡)²≤𝑑𝑡_𝑇0𝑍^(𝑘)(𝑡)²𝑑𝑡+^𝑘−1_𝑗=1(−1)^𝑗𝑎_2𝑗^𝑇₀𝑍^(𝑗)(𝑡)²𝑑𝑡=−^𝑇₀𝑍(𝑡)𝑔𝑡,𝑢₁(𝑡)−𝑔𝑡,𝑢₂(𝑡)𝑑𝑡≤𝐵^𝑇₀|𝑍(𝑡)|²𝑇𝑑𝑡≤𝐵𝜋^2𝑇₀𝑍^′(𝑡)²𝑇𝑑𝑡≤𝐵𝜋²𝑇2𝜋^{2(𝑘−1)𝑇}₀𝑍^(𝑘)(𝑡)²𝑑𝑡,whereΛ≠∅,_𝑇0𝑍^(𝑘)(𝑡)²𝑑𝑡≤^𝑇₀𝑍^(𝑘)(𝑡)²𝑑𝑡+^𝑘−1_𝑗=1(−1)^𝑗𝑎_2𝑗^𝑇₀𝑍^(𝑗)(𝑡)²𝑇𝑑𝑡≤𝐵𝜋²𝑇2𝜋^{2(𝑘−1)𝑇}₀𝑍^(𝑘)(𝑡)²𝑑𝑡,whereΛ=∅. (2.23) From (2.18) and (𝐻₂), (2.23) yield that 𝑍(𝑡)≡𝑍^′(𝑡)≡⋯≡𝑍^(𝑘)(𝑡)≡0,∀𝑡∈𝑅. (2.24) Therefore, (1.1) has at most one 𝑇-periodic solution. The proof of Lemma 2.5 is now complete.

Similar to the proof of Lemma 2.5, one can prove the following result.

Lemma 2.6. Let 𝑘 be an odd number, 𝑛=2𝑘, and Λ=𝑗∶𝑗∈{1,2,…,𝑘−1},(−1)^𝑗+1𝑎_2𝑗<0. (2.25) Assume that one of the following conditions is satisfied: (𝐻₁)for𝑡,𝑥₁,𝑥₂∈𝑅,𝑥₁≠𝑥₂, 1+_𝑗∈Λ𝑇2𝜋^{2(𝑘−𝑗)}(−1)^𝑗+1𝑎_2𝑗>0,𝑔𝑡,𝑥₁−𝑔𝑡,𝑥₂𝑥₁−𝑥₂<0, (2.26)(𝐻₂) there exists a nonnegative constant 𝐵 such that 𝑇1>𝐵𝜋²𝑇2𝜋^2(𝑘−1),1+_𝑗∈Λ𝑇2𝜋^{2(𝑘−𝑗)}(−1)^𝑗+1𝑎_2𝑗𝑇>𝐵𝜋²𝑇2𝜋^2(𝑘−1),𝐵𝑥₁−𝑥₂²≥𝑔𝑡,𝑥₁−𝑔𝑡,𝑥₂𝑥₁−𝑥₂>0, (2.27) where 𝑡,𝑥₁,𝑥₂∈𝑅,𝑥₁≠𝑥₂, then (1.1) has at most one 𝑇-periodic solution.

Lemma 2.7. Let 𝑘 be an even number, 𝑛=2𝑘+1, and Λ^∗=𝑗∶𝑗∈{1,2,…,𝑘−1},(−1)^𝑗𝑎_2𝑗<0. (2.28) Assume that one of the following conditions is satisfied:(𝐻₃)for𝑡,𝑥₁,𝑥₂∈𝑅,𝑥₁≠𝑥₂,𝑎_2𝑘+_𝑗∈Λ∗𝑇2𝜋^{2(𝑘−𝑗)}(−1)^𝑗𝑎_2𝑗>0,𝑔𝑡,𝑥₁−𝑔𝑡,𝑥₂𝑥₁−𝑥₂>0, (2.29)(𝐻₄)there exists a nonnegative constant 𝐵 such that𝑎_2𝑘𝑇>𝐵𝜋²𝑇2𝜋^2(𝑘−1),1+_𝑗∈Λ∗𝑇2𝜋^{2(𝑘−𝑗)}(−1)^𝑗𝑎_2𝑗𝑇>𝐵𝜋²𝑇2𝜋^2(𝑘−1),𝐵𝑥₁−𝑥₂²≥−𝑔𝑡,𝑥₁−𝑔𝑡,𝑥₂𝑥₁−𝑥₂>0, (2.30) where𝑡,𝑥₁,𝑥₂∈𝑅,𝑥₁≠𝑥₂, then (1.1) has at most one 𝑇-periodic solution.

Proof. Multiplying (2.15) by 𝑍(𝑡) and then integrating it from 0 to 𝑇, yields that 𝑎_2𝑘^𝑇₀𝑍^(𝑘)(𝑡)²𝑑𝑡+^𝑘−1_𝑗=1(−1)^𝑗𝑎_2𝑗^𝑇₀𝑍^(𝑗)(𝑡)²𝑑𝑡=−^𝑇₀𝑍(𝑡)𝑔𝑡,𝑢₁(𝑡)−𝑔𝑡,𝑢₂(𝑡)𝑑𝑡=−^𝑇₀𝑢₁(𝑡)−𝑢₂(𝑡)×𝑔𝑡,𝑢₁(𝑡)−𝑔𝑡,𝑢₂(𝑡)𝑑𝑡. (2.31) Now the proof proceeds in the same way as in Lemma 2.5.

Similar to the proof of Lemma 2.7, we can prove the following results.

Lemma 2.8. Let 𝑘 be an odd number, 𝑛=2𝑘+1, and Λ^∗=𝑗∶𝑗∈{1,2,…,𝑘−1},(−1)^𝑗+1𝑎_2𝑗<0. (2.32) Assume that one of the following conditions is satisfied:(𝐻₃)for𝑡,𝑥₁,𝑥₂∈𝑅,𝑥₁≠𝑥₂, 𝑎_2𝑘+_𝑗∈Λ∗𝑇2𝜋^{2(𝑘−𝑗)}(−1)^𝑗+1𝑎_2𝑗>0,𝑔𝑡,𝑥₁−𝑔𝑡,𝑥₂𝑥₁−𝑥₂<0, (2.33)(𝐻₄)there exists a nonnegative constant 𝐵 such that𝑎_2𝑘𝑇>𝐵𝜋²𝑇2𝜋^2(𝑘−1),𝑎_2𝑘+_𝑗∈Λ∗𝑇2𝜋^{2(𝑘−𝑗)}(−1)^𝑗+1𝑎_2𝑗𝑇>𝐵𝜋²𝑇2𝜋^2(𝑘−1),𝐵𝑥₁−𝑥₂²≥𝑔𝑡,𝑥₁−𝑔𝑡,𝑥₂𝑥₁−𝑥₂>0, (2.34) where 𝑡,𝑥₁,𝑥₂∈𝑅,𝑥₁≠𝑥₂, then (1.1) has at most one 𝑇-periodic solution.

3. Main Results

Theorem 3.1. Let 𝑘 be an even number and 𝑛=2𝑘. Assume that one of the following conditions is satisfied: (𝐻₁)^∗ let (𝐻₁) hold, and there exists a nonnegative constant 𝑑₀ such that𝑔(𝑡,𝑢)−𝑒(𝑡)𝑢>0,∀𝑡∈𝑅,|𝑢|≥𝑑₀, (3.1)(𝐻₂)^∗ there exist nonnegative constants 𝑑₀ and 𝐵 such that (𝐻₂) holds, 𝑔(𝑡,𝑢)−𝑒(𝑡)𝑢<0,∀𝑡∈𝑅,|𝑢|≥𝑑₀, (3.2) then (1.1) has a unique 𝑇-periodic solution.

Proof. From Lemma 2.5, together with (𝐻₁)^∗ (or (𝐻₂)^∗), it is easy to see that (1.1) has at most one 𝑇-periodic solution. Thus, to prove Theorem 3.1, it suffices to show that (1.1) has at least one 𝑇-periodic solution. To do this, we shall use Lemma 2.1 with the nonlinearity 𝑓∶𝐑^𝑛+1→𝐑 given by 𝑓𝑡,𝑢,𝑢^′,𝑢^′′,…,𝑢^(𝑛−1)=𝑒(𝑡)−^𝑛−1_𝑗=1𝑎_𝑗𝑢^(𝑗)−𝑔(𝑡,𝑢). (3.3)
For 𝜆∈(0,1], we consider the 𝑛th-order differential equation 𝑢^(𝑛)(𝑡)+𝜆^𝑛−1_𝑗=1𝑎_𝑗𝑢^(𝑗)+𝜆𝑔(𝑡,𝑢(𝑡))=𝜆𝑒(𝑡). (3.4) Let us show that (i) in Lemma 2.1 is satisfied, which means that there exists 𝜌>0 such that any possible 𝑇-periodic solution 𝑢 of (3.4) is such that ‖𝑢‖_(𝑛−1)<𝜌. (3.5) Let 𝜆∈(0,1] and let 𝑢 be a possible 𝑇-periodic solution of (3.4). In what follows, 𝐶_𝑗 denotes a fixed constant independent of 𝜆 and 𝑢. Integrating (3.4) from 0 to 𝑇, it results that ^𝑇₀𝑔(𝑡,𝑢(𝑡))−𝑒(𝑡)𝑑𝑡=0, (3.6) which together with (𝐻₁)^∗ (or (𝐻₂)^∗) implies that ∃𝜉∈[0,𝑇]∶𝑢(𝜉)<𝑑₀. (3.7) Hence, from (2.9), we have that |𝑢|_∞≤𝑑₀+12√𝑇^𝑇₀𝑢^′(𝑡)²𝑑𝑡^1/2. (3.8) In view of (2.10), (3.8) implies that |𝑢|_∞≤𝑑₀+12√𝑇𝑇2𝜋^𝑘−1𝑇₀𝑢^(𝑘)(𝑡)²𝑑𝑡^1/2. (3.9) It follows that ^𝑇₀𝑒(𝑡)𝑢(𝑡)𝑑𝑡≤𝑇𝑑₀|𝑒|_∞+12𝑇√𝑇𝑇2𝜋^𝑘−1|𝑒|_∞^𝑇₀𝑢^(𝑘)(𝑡)²𝑑𝑡^1/2,_𝑇0𝑔(𝑡,0)𝑢(𝑡)𝑑𝑡≤𝑇𝑑₀𝑔(𝑡,0)_∞+12𝑇√𝑇𝑇2𝜋^𝑘−1𝑔(𝑡,0)_∞^𝑇₀𝑢^(𝑘)(𝑡)²𝑑𝑡^1/2. (3.10)
On the other hand, multiplying (3.4) by 𝑢 and integrating from 0 to 𝑇, it follows that ^𝑇₀𝑢^(𝑘)(𝑡)²𝑑𝑡+𝜆^𝑘−1_𝑗=1(−1)^𝑗𝑎_2𝑗^𝑇₀𝑢^(𝑗)(𝑡)²𝑑𝑡=−𝜆^𝑇₀𝑢(𝑡)𝑔(𝑡,𝑢(𝑡))𝑑𝑡+𝜆^𝑇₀𝑒(𝑡)𝑢(𝑡)𝑑𝑡. (3.11) Now suppose that (𝐻₁)^∗(or (𝐻₂)^∗) holds, and we will consider two cases as follows.
Case 1. If (𝐻₁)^∗ holds, using (2.10), (3.10), and (3.11), we have 1+_𝑗∈Λ𝑇2𝜋^{2(𝑘−𝑗)}(−1)^𝑗𝑎_2𝑗^𝑇₀𝑢^(𝑘)(𝑡)²𝑑t≤^𝑇₀𝑢^(𝑘)(𝑡)²𝑑𝑡+𝜆^𝑘−1_𝑗=1(−1)^𝑗𝑎_2𝑗^𝑇₀𝑢^(𝑗)(𝑡)²𝑑𝑡≤−𝜆^𝑇₀(𝑢(𝑡)−0)𝑔(𝑡,𝑢(𝑡))−𝑔(𝑡,0)𝑑𝑡−𝜆^𝑇₀𝑢(𝑡)𝑔(𝑡,0)𝑑𝑡+𝑇𝑑₀|𝑒|_∞+12𝑇√𝑇𝑇2𝜋^𝑘−1|𝑒|_∞^𝑇₀𝑢^(𝑘)(𝑡)²𝑑𝑡^1/2≤𝑇𝑑₀𝑔(𝑡,0)_∞+|𝑒|_∞+12𝑇√𝑇𝑇2𝜋^𝑘−1𝑔(𝑡,0)_∞+|𝑒|_∞^𝑇₀𝑢^(𝑘)(𝑡)²𝑑𝑡^1/2,whereΛ≠∅,_𝑇0𝑢^(𝑘)(𝑡)²≤𝑑𝑡_𝑇0𝑢^(𝑘)(𝑡)²𝑑𝑡+𝜆^𝑘−1_𝑗=1(−1)^𝑗𝑎_2𝑗^𝑇₀𝑢^(𝑗)(𝑡)²𝑑𝑡≤−𝜆^𝑇₀(𝑢(𝑡)−0)𝑔(𝑡,𝑢(𝑡))−𝑔(𝑡,0)𝑑𝑡−𝜆^𝑇₀𝑢(𝑡)𝑔(𝑡,0)𝑑𝑡+𝑇𝑑₀|𝑒|_∞+12𝑇√𝑇𝑇2𝜋^𝑘−1|𝑒|_∞^𝑇₀𝑢^(𝑘)(𝑡)²𝑑𝑡^1/2≤𝑇𝑑₀𝑔(𝑡,0)_∞+|𝑒|_∞+12𝑇√𝑇𝑇2𝜋^𝑘−1𝑔(𝑡,0)_∞+|𝑒|_∞^𝑇₀𝑢^(𝑘)(𝑡)²𝑑𝑡^1/2,whereΛ=∅, (3.12) which imply that there exists a positive constant 𝐶₁ satisfying ^𝑇₀𝑢^(𝑘)(𝑡)²𝑑𝑡≤𝐶₁,^𝑇₀𝑢^(𝑘)(𝑡)𝑑𝑡≤𝑇𝐶₁. (3.13)Case 2. If (𝐻₂)^∗ holds, using (2.9), (2.10), (3.10), and (3.11), we obtain 1+_𝑗∈Λ𝑇2𝜋^{2(𝑘−𝑗)}(−1)^𝑗𝑎_2𝑗^𝑇₀𝑢^(𝑘)(𝑡)²≤𝑑𝑡_𝑇0𝑢^(𝑘)(𝑡)²𝑑𝑡+𝜆^𝑘−1_𝑗=1(−1)^𝑗𝑎_2𝑗^𝑇₀𝑢^(𝑗)(𝑡)²𝑑𝑡≤−𝜆^𝑇₀(𝑢(𝑡)−0)𝑔(𝑡,𝑢(𝑡))−𝑔(𝑡,0)𝑑𝑡−𝜆^𝑇₀𝑢(𝑡)𝑔(𝑡,0)𝑑𝑡+𝑇𝑑₀|𝑒|_∞+12𝑇√𝑇𝑇2𝜋^𝑘−1|𝑒|_∞^𝑇₀𝑢^(𝑘)(𝑡)²𝑑𝑡^1/2≤𝐵|𝑢|²₂+𝑇𝑑₀𝑔(𝑡,0)_∞+|𝑒|_∞+12𝑇√𝑇𝑇2𝜋^𝑘−1𝑔(𝑡,0)_∞+|𝑒|_∞^𝑇₀𝑢^(𝑘)(𝑡)²𝑑𝑡^1/2𝑇≤𝐵𝜋𝑢^′₂+√𝑇𝑑₀²+𝑇𝑑₀𝑔(𝑡,0)_∞+|𝑒|_∞+12𝑇√𝑇𝑇2𝜋^𝑘−1𝑔(𝑡,0)_∞+|𝑒|_∞^𝑇₀𝑢^(𝑘)(𝑡)²𝑑𝑡^1/2𝑇≤𝐵𝜋𝑇2𝜋^𝑘−1𝑢^(𝑘)₂+√𝑇𝑑₀²+𝑇𝑑₀𝑔(𝑡,0)_∞+|𝑒|_∞+12𝑇√𝑇𝑇2𝜋^𝑘−1𝑔(𝑡,0)_∞+|𝑒|_∞^𝑇₀𝑢^(𝑘)(𝑡)²𝑑𝑡^1/2,whereΛ≠∅,_𝑇0𝑢^(𝑘)(𝑡)²≤𝑑𝑡_𝑇0𝑢^(𝑘)(𝑡)²𝑑𝑡+𝜆^𝑘−1_𝑗=1(−1)^𝑗𝑎_2𝑗^𝑇₀𝑢^(𝑗)(𝑡)²𝑑𝑡≤−𝜆^𝑇₀(𝑢(𝑡)−0)𝑔(𝑡,𝑢(𝑡))−𝑔(𝑡,0)𝑑𝑡−𝜆^𝑇₀𝑢(𝑡)𝑔(𝑡,0)𝑑𝑡+𝑇𝑑₀|𝑒|_∞+12𝑇√𝑇𝑇2𝜋^𝑘−1|𝑒|_∞^𝑇₀𝑢^(𝑘)(𝑡)²𝑑𝑡^1/2𝑇≤𝐵𝜋𝑇2𝜋^𝑘−1𝑢^(𝑘)₂+√𝑇𝑑₀²+𝑇𝑑₀𝑔(𝑡,0)_∞+|𝑒|_∞+12𝑇√𝑇𝑇2𝜋^𝑘−1𝑔(𝑡,0)_∞+|𝑒|_∞^𝑇₀𝑢^(𝑘)(𝑡)²𝑑𝑡^1/2,whereΛ=∅, (3.14) which together with (𝐻₂) yield that (3.13) holds.
Using (3.9) and (3.13), it follows that there exists 𝐶₂ such that |𝑢|_∞≤𝐶₂. (3.15)
Now, we shall estimate 𝑥^(𝑗)(𝑗=1,2,…,2𝑘−1), multiplying (3.4) by 𝑥^(2𝑘) and integrating from 0 to 𝑇, we have ^𝑇₀𝑢^(2𝑘)(𝑡)²𝑑𝑡+𝜆^𝑘−1_𝑗=1(−1)^𝑗𝑎_2𝑗^𝑇₀𝑢^(𝑘+𝑗)(𝑡)²𝑑𝑡=−𝜆^𝑇₀𝑢^(2𝑘)(𝑡)𝑔(𝑡,𝑢(𝑡))𝑑𝑡+𝜆^𝑇₀𝑒(𝑡)𝑢^(2𝑘)(𝑡)𝑑𝑡 (3.16)
Using (2.10), (3.15), and (3.16), we have 1+_𝑗∈Λ𝑇2𝜋^{2(𝑘−𝑗)}(−1)^𝑗𝑎_2𝑗^𝑇₀𝑢^(2𝑘)(𝑡)²≤𝑑𝑡_𝑇0𝑢^(2𝑘)(𝑡)²𝑑𝑡+𝜆^𝑘−1_𝑗=1(−1)^𝑗𝑎_2𝑗^𝑇₀𝑢^(𝑘+𝑗)(𝑡)²𝑑𝑡=−𝜆^𝑇₀𝑢^(2𝑘)(𝑡)𝑔(𝑡,𝑢(𝑡))𝑑𝑡+𝜆^𝑇₀𝑒(𝑡)𝑢^(2𝑘)(𝑡)≤√𝑑𝑡𝑇sup_{|𝑢|≤𝐶2,𝑡∈𝑅}𝑔(𝑡,𝑢)+√𝑇|𝑒|_∞^𝑇₀𝑢^(2𝑘)(𝑡)²𝑑𝑡^1/2,whereΛ≠∅,_𝑇0𝑢^(2𝑘)(𝑡)²≤𝑑𝑡_𝑇0𝑢^(2𝑘)(𝑡)²𝑑𝑡+𝜆^𝑘−1_𝑗=1(−1)^𝑗𝑎_2𝑗^𝑇₀𝑢^(𝑘+𝑗)(𝑡)²≤√𝑑𝑡𝑇sup_{|𝑢|≤𝐶2,𝑡∈𝑅}𝑔(𝑡,𝑢)+√𝑇|𝑒|_∞^𝑇₀𝑢^(2𝑘)(𝑡)²𝑑𝑡^1/2,whereΛ=∅, (3.17) which imply that there exists a positive constant 𝐶₃ satisfying ^𝑇₀𝑢^(2𝑘)(𝑡)²𝑑𝑡≤𝐶₃,^𝑇₀𝑢^(2𝑘)(𝑡)𝑑𝑡≤𝑇𝐶₃. (3.18) This implies the existence of a constant 𝜌>𝑑₀ such that (3.5) holds.
Now, to show that (ii) in Lemma 2.1 is satisfied, it suffices to remark that 𝐹(𝑥)=^𝑇₀−𝑔(𝑡,𝑥)+𝑒(𝑡)𝑑𝑡,(𝑥∈𝐑). (3.19) Hence, from (𝐻₁)^∗ (or (𝐻₂)^∗) and 𝜌>𝑑₀, it results that 𝐹(−𝜌)𝐹(𝜌)<0. Then, using Lemma 2.1, it follows that (3.4) has at least one 𝑇-periodic solution 𝑢 satisfying (3.5). This completes the proof.

Similar to the proof of Theorem 3.1, from Lemma 2.6, one can prove the following results.

Theorem 3.2. Let 𝑘 be an odd number and 𝑛=2𝑘+1. Assume that one of the following conditions is satisfied: (𝐻₁)^∗ let (𝐻₁) hold, and there exists a nonnegative constant 𝑑₀ such that𝑔(𝑡,𝑢)−𝑒(𝑡)𝑢<0,∀𝑡∈𝑅,|𝑢|≥𝑑₀, (3.20)(𝐻₂)^∗there exist nonnegative constants 𝑑₀ and 𝐵 such that (𝐻₂) holds, 𝑔(𝑡,𝑢)−𝑒(𝑡)𝑢>0,∀𝑡∈𝑅,|𝑢|≥𝑑₀, (3.21) then (1.1) has a unique 𝑇-periodic solution.

Theorem 3.3. Let 𝑘 be an even number and 𝑛=2𝑘+1. Assume that one of the following conditions is satisfied:(𝐻₃)^∗ let (𝐻₃) hold, and there exists a nonnegative constant 𝑑₀ such that𝑔(𝑡,𝑢)−𝑒(𝑡)𝑢>0,∀𝑡∈𝑅,|𝑢|≥𝑑₀, (3.22)(𝐻₄)^∗ there exist nonnegative constants 𝑑₀ and 𝐵 such that (𝐻₄) holds, 𝑔(𝑡,𝑢)−𝑒(𝑡)𝑢<0,∀𝑡∈𝑅,|𝑢|≥𝑑₀, (3.23) then (1.1) has a unique 𝑇-periodic solution.

Proof. From Lemma 2.7, together with (𝐻₃)^∗ (or (𝐻₄)^∗), it is easy to see that (1.1) has at most one 𝑇-periodic solution. Thus, to prove Theorem 3.3, it suffices to show that (1.1) has at least one 𝑇-periodic solution.
Multiplying (3.4) by 𝑢^(𝑖)(𝑖=0,2𝑘) and integrating from 0 to 𝑇, it follows that 𝜆𝑎_2𝑘^𝑇₀𝑢^(𝑘)(𝑡)²𝑑𝑡+𝜆^𝑘−1_𝑗=1(−1)^𝑗𝑎_2𝑗^𝑇₀𝑢^(𝑗)(𝑡)²𝑑𝑡=−𝜆^𝑇₀𝑢(𝑡)𝑔(𝑡,𝑢(𝑡))𝑑𝑡+𝜆^𝑇₀𝑒(𝑡)𝑢(𝑡)𝑑𝑡,𝜆𝑎_2𝑘^𝑇₀𝑢^(2𝑘)(𝑡)²𝑑𝑡+𝜆^𝑘−1_𝑗=1(−1)^𝑗𝑎_2𝑗^𝑇₀𝑢^(𝑘+𝑗)(𝑡)²𝑑𝑡=−𝜆^𝑇₀𝑢^(2𝑘)(𝑡)𝑔(𝑡,𝑢(𝑡))𝑑𝑡+𝜆^𝑇₀𝑒(𝑡)𝑢^(2𝑘)(𝑡)𝑑𝑡. (3.24) Then, from (3.24), by using similar arguments in proof of (3.15) and (3.18), we can obtain that there exists a constant 𝐶₅ such that |𝑢|_∞≤𝐶₅,^𝑇₀𝑢^(2𝑘)(𝑡)𝑑𝑡≤𝐶₅,_𝑇0𝑢^(𝑗)(𝑡)²𝑇𝑑𝑡≤2𝜋^{2𝑘−𝑗𝑇}₀𝑢^(2𝑘)(𝑡)²𝑑𝑡≤𝐶₅,𝑗=1,2,…,2𝑘−1. (3.25)(3.26) In view of (3.4), (3.25) and (3.26) yield that ^𝑇₀𝑢^(2𝑘+1)(𝑡)𝑑𝑡=^𝑇₀−𝜆^2𝑘_𝑗=1𝑎_𝑗𝑢^(𝑗)((𝑡(𝑡)≤−𝜆𝑔𝑡,𝑢))+𝜆𝑒𝑑𝑡_2𝑘𝑗=1𝑎_𝑗√𝑇^𝑇₀𝑢^(𝑗)2𝑑𝑡^1/2+𝑇sup_{|𝑢|≤𝐶5,𝑡∈𝑅}𝑔(𝑡,𝑢)+|𝑒|_∞≤^2𝑘_𝑗=1𝑎_𝑗𝑇𝐶₅+𝑇sup_{|𝑢|≤𝐶5,𝑡∈𝑅}𝑔(𝑡,𝑢)+|𝑒|_∞, (3.27) which together with (2.11) and (3.25) implies the existence of a constant 𝜌>𝑑₀ such that (3.5) holds.
Now the proof proceeds in the same way as in Theorem 3.1.

Similar to the proof of Theorem 3.3, from Lemma 2.8, we obtain the following.

Theorem 3.4. Let 𝑘 be an odd number and 𝑛=2𝑘+1. Assume that one of the following conditions is satisfied:(𝐻₃)^∗ let (𝐻₃) hold, and there exists a nonnegative constant 𝑑₀ such that𝑔(𝑡,𝑢)−𝑒(𝑡)𝑢<0,∀𝑡∈𝑅,|𝑢|≥𝑑₀, (3.28)(𝐻₄)^∗ there exist nonnegative constants 𝑑₀ and 𝐵 such that (𝐻₄) holds, 𝑔(𝑡,𝑢)−𝑒(𝑡)𝑢>0,∀𝑡∈𝑅,|𝑢|≥𝑑₀, (3.29) then (1.1) has a unique 𝑇-periodic solution.

Remark 3.5. If ∫^𝑇₀𝑒(𝑡)𝑑𝑡=0 and 𝑔(𝑡,𝑢) satisfies the following condition:(𝐻₅) there exist 𝑑>0 and 𝜖∈{−1,1} such that, for any continuous 𝑇-periodic function 𝑢, we have𝜖^𝑇₀𝑔(𝑡,𝑢(𝑡))𝑑𝑡<0,ifmin_𝑅𝜖𝑢≥𝑑,_𝑇0𝑔(𝑡,𝑢(𝑡))𝑑𝑡>0,ifmin_𝑅𝑢≤−𝑑. (3.30) Moreover, one of conditions (𝐻₁)–(𝐻₄) holds. Then, by using the methods similarly to those used in Theorem 3.1, one may also establish the results similar to those in Theorems 3.1–3.4.

4. Examples and Remarks

Example 4.1. Let 𝑎,𝑏,𝑐∶𝑅→𝑅 be three continuous, strictly positive, and 𝑇-periodic functions, and let 𝑒∶𝑅→𝑅 be continuous, 𝑇-periodic, then the fourth-order differential equation 𝑢^′′′′(𝑡)+400𝑢^′′′(𝑡)+1.6𝜋𝑇²𝑢^′′(𝑡)+20𝑢^′(𝑡)+𝑎(𝑡)𝑢(𝑡)+𝑏(𝑡)𝑢³(𝑡)+𝑐(𝑡)𝑢⁵(𝑡)=𝑒(𝑡) (4.1) has a unique 𝑇-periodic solution. For the proof, it suffices to remark that the function 𝑔(𝑡,𝑢)≡𝑎(𝑡)𝑢+𝑏(𝑡)𝑢³+𝑐(𝑡)𝑢⁵ satisfies 𝑔(𝑡,𝑢)−𝑒(𝑡)𝑢>0,∀𝑡∈𝑅,|𝑢|≥𝑑₀, (4.2) where 𝑑₀∈𝐑 is sufficiently large. Hence, 𝑝=−(1.6𝜋/𝑇)² and 𝑔 satisfy (𝐻₁)^∗, and the result follows from Theorem 3.1.

Example 4.2. Let 𝛽∶𝑅→𝑅 be continuous, strictly positive, and 𝑇-periodic, let 𝛿,𝜏∈𝑅 be constants, and let 𝑒∶𝑅→𝑅 be continuous, 𝑇-periodic, then the five-order differential equations 𝑢^{′′′′′}(𝑡)+100000√3𝑢^′′′′(𝑡)+𝛿𝑢^′′′(𝑡)−20𝑢^′′(𝑡)+𝜏𝑢^′(𝑡)+𝛽(𝑡)𝑢³(𝑡)=𝑒(𝑡) (4.3) have a unique 𝑇-periodic solution. For the proof, it suffices to remark that the function 𝑔(𝑡,𝑢)≡𝛽(𝑡)𝑢³(𝑡) with 𝑎₄=100000√3,𝑎₃=𝛿,𝑎₂=−20, and 𝑎₁=𝜏 satisfies (𝐻₃)^∗. Hence, the result follows from Theorem 3.3.

Remark 4.3. Since 𝑔(𝑡,𝑢) in Examples 4.1 and 4.2 does not satisfy (𝐻), the main results in [9–11] and the references therein cannot be applicable to (4.1)–(4.3) to obtain the existence and uniqueness of 2-periodic solutions. Moreover, all the results in this present paper avoid the spectral conditions in [12, 13]. This implies that the results of this paper are new, and they complement previously known results.

Acknowledgments

The authors would like to express their sincere appreciation to the anonymous referee for the valuable comments which have led to an improvement in the presentation of the paper. This work was supported by the Construct Program of the Key Discipline in Hunan Province (Mechanical Design and Theory), the Key Project of Chinese Ministry of Education (Grant no. 210 151), the Scientific Research Fund of Hunan Provincial Natural Science Foundation of PR China (Grant no. 11JJ6006), the Natural Scientific Research Fund of Hunan Provincial Education Department of PR China (Grants nos. 11C0916, 11C0915, 11C1186), the Natural Scientific Research Fund of Zhejiang Provincial of P.R. China (Grants nos. Y6110436, Y12A010059), and the Natural Scientific Research Fund of Zhejiang Provincial Education Department of P.R. China (Grant no. Z201122436).