# The Existence and Uniqueness of Periodic Solutions for Some Nonlinearnth-Order Differential Equations

Authors: Qiyuan Zhou; Shuhua Gong
Journal: Abstract and Applied Analysis
 (2012)

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## Abstract

The nonlinearnth-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.


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## Body

## 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    (1.1)x(n)+∑j=1n-1ajx(j)+g(t,x)=e(t),
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 ai(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(H̅)
g(t,x) is continuous, and there are positive constants m0 and M0 such that(1.2)m0≤|gx(t,x)|≤M0,∀(t,x)∈R×R
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 (H̅) 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 ofT-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 (H̅) 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 useΦ to denote the empty set. For n∈N, we denote by CTn the Banach space(2.1)CTn={u∈Cn(R,R):u(t)=u(t+T),∀t∈R},
endowed with the norm(2.2)‖u‖(n)=∑k=0n|u(k)|∞,(u∈CTn),
where, for a function v∈CT0, we have that(2.3)|v|∞=max⁡[0,T]⁡|v|.
For x∈CT0, we will denote(2.4)|x|p=(∫0T|x(t)|pdt)1/p,(p>0).
Now, let f̃:Rn+1→R be a continuous function, T-periodic with respect to the first variable, and consider the nth-order differential equation(2.5)u(n)=f̃(t,u,u′,u′′,…,u(n-1)).Lemma 2.1 (see [14]).
Assume that the following conditions hold.(i)
There existsρ>0 such that, for each λ∈(0,1], one has that any possible T-periodic solution u of the problem
(2.6)u(n)=λf̃(t,u,u′,u′′,…,u(n-1))
satisfies the priori estimation ∥u∥(n-1)<ρ.(ii)
The continuous functionF:R→R defined by
(2.7)F(x)=∫0Tf̃(t,x,0,0,…,0)dt,(x∈R)
satisfies F(-ρ)F(ρ)<0.Then, (2.5) has at least one T-periodic solution u such that ∥u∥(n-1)<ρ.From Lemma2.2 in [15] and the proof of inequality (10) in [7, pp 3402], one obtains the following.Lemma 2.2.
Letx(t)∈CT1. Suppose that there exists a constant D≥0 such that
(2.8)|x(τ0)|≤D,τ0∈[0,T],
then
(2.9)|x|2≤Tπ|x′|2+TD,|x|∞≤D+12T(∫0T|x′(t)|2dt)1/2.Lemma 2.3.
For anyu∈CT2, one has that
(2.10)∫0T|u′(t)|2dt≤(T2π)2∫0T|u′′(t)|2dt.Proof.
Lemma2.3 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 anyu∈CTn, one has that
(2.11)|u(j)|∞≤T(n-1)-j(12)(n-1)-(j-1)∫0T|u(n)(t)|dt,(j=1,2,…,n-1).Lemma 2.5.
Letk be an even number, n=2k, and(2.12)Λ={j:j∈{1,2,…,k-1},(-1)ja2j<0}.
Assume that one of the following conditions is satisfied:  (H1)
fort,x1,x2∈R,x1≠x2,(2.13)1+∑j∈Λ(T2π)2(k-j)(-1)ja2j>0,(g(t,x1)-g(t,x2))(x1-x2)>0,(H2)
there exists a nonnegative constantB such that(2.14)1>B(Tπ)2(T2π)2(k-1),1+∑j∈Λ(T2π)2(k-j)(-1)ja2j>B(Tπ)2(T2π)2(k-1),B(x1-x2)2≥-(g(t,x1)-g(t,x2))(x1-x2)>0,
wheret,x1,x2∈R,x1≠x2, then (1.1) has at most one T-periodic solution.Proof.
Suppose thatu1(t)and u2(t)are two T-periodic solutions of (1.1). Set Z(t)=u1(t)-u2(t). Then, we obtain
(2.15)(u1(t)-u2(t))(n)+∑j=1n-1aj(u1(t)-u2(t))(j)+(g(t,u1(t))-g(t,u2(t)))=0.
Integrating (2.15) from 0 to T, it results that
(2.16)∫0T(g(t,u1(t))-g(t,u2(t)))dt=0.
Therefore, in view of integral mean value theorem, it follows that there exists a constant γ∈[0,T] such that
(2.17)g(γ,u1(γ))-g(γ,u2(γ))=0.
Since g(t,x) is a strictly monotone function in x, (2.17) implies that
(2.18)Z(γ)=u1(γ)-u2(γ)=0.
Then, from (2.9), we have
(2.19)|Z|2≤Tπ|Z′|2.
Multiplying (2.15) by Z(t) and then integrating it from 0 to T, it follows that
(2.20)∫0T|Z(k)(t)|2dt+∑j=1k-1(-1)ja2j∫0T|Z(j)(t)|2dt=-∫0TZ(t)(g(t,u1(t))-g(t,u2(t)))dt=-∫0T(u1(t)-u2(t))(g(t,u1(t))-g(t,u2(t)))dt.
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
(2.21)0≤(1+∑j∈Λ(T2π)2(k-j)(-1)ja2j)∫0T|Z(k)(t)|2dt≤∫0T|Z(k)(t)|2dt+∑j=1k-1(-1)ja2j∫0T|Z(j)(t)|2dt=-∫0T(u1(t)-u2(t))(g(t,u1(t))-g(t,u2(t)))dt≤0,whereΛ≠∅,0≤∫0T|Z(k)(t)|2dt≤∫0T|Z(k)(t)|2dt+∑j=1k-1(-1)ja2j∫0T|Z(j)(t)|2dt=-∫0T(u1(t)-u2(t))(g(t,u1(t))-g(t,u2(t)))dt≤0,whereΛ=∅,
which, together with (2.18), implies that
(2.22)Z(t)≡Z′(t)≡⋯≡Z(k)(t)≡0,∀t∈R.
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
(2.23)(1+∑j∈Λ(T2π)2(k-j)(-1)ja2j)∫0T|Z(k)(t)|2dt≤∫0T|Z(k)(t)|2dt+∑j=1k-1(-1)ja2j∫0T|Z(j)(t)|2dt=-∫0TZ(t)(g(t,u1(t))-g(t,u2(t)))dt≤B∫0T|Z(t)|2dt≤B(Tπ)2∫0T|Z′(t)|2dt≤B(Tπ)2(T2π)2(k-1)∫0T|Z(k)(t)|2dt,whereΛ≠∅,∫0T|Z(k)(t)|2dt≤∫0T|Z(k)(t)|2dt+∑j=1k-1(-1)ja2j∫0T|Z(j)(t)|2dt≤B(Tπ)2(T2π)2(k-1)∫0T|Z(k)(t)|2dt,whereΛ=∅.
From (2.18) and (H2), (2.23) yield that
(2.24)Z(t)≡Z′(t)≡⋯≡Z(k)(t)≡0,∀t∈R.
Therefore, (1.1) has at most one T-periodic solution. The proof of Lemma 2.5 is now complete.Similar to the proof of Lemma2.5, one can prove the following result.Lemma 2.6.
Letk be an odd number, n=2k, and
(2.25)Λ̅={j:j∈{1,2,…,k-1},(-1)j+1a2j<0}.
Assume that one of the following conditions is satisfied: (H1̅)
fort,x1,x2∈R,x1≠x2,(2.26)1+∑j∈Λ̅(T2π)2(k-j)(-1)j+1a2j>0,(g(t,x1)-g(t,x2))(x1-x2)<0,(H2̅)
there exists a nonnegative constantB such that(2.27)1>B(Tπ)2(T2π)2(k-1),1+∑j∈Λ¯(T2π)2(k-j)(-1)j+1a2j>B(Tπ)2(T2π)2(k-1),B(x1-x2)2≥(g(t,x1)-g(t,x2))(x1-x2)>0,
where t,x1,x2∈R,x1≠x2, then (1.1) has at most one T-periodic solution.Lemma 2.7.
Letk be an even number, n=2k+1, and
(2.28)Λ*={j:j∈{1,2,…,k-1},(-1)ja2j<0}.
Assume that one of the following conditions is satisfied:(H3)
fort,x1,x2∈R,x1≠x2,(2.29)a2k+∑j∈Λ*(T2π)2(k-j)(-1)ja2j>0,(g(t,x1)-g(t,x2))(x1-x2)>0,(H4)
there exists a nonnegative constantB such that(2.30)a2k>B(Tπ)2(T2π)2(k-1),1+∑j∈Λ*(T2π)2(k-j)(-1)ja2j>B(Tπ)2(T2π)2(k-1),B(x1-x2)2≥-(g(t,x1)-g(t,x2))(x1-x2)>0,
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
(2.31)a2k∫0T|Z(k)(t)|2dt+∑j=1k-1(-1)ja2j∫0T|Z(j)(t)|2dt=-∫0TZ(t)(g(t,u1(t))-g(t,u2(t)))dt=-∫0T(u1(t)-u2(t))×(g(t,u1(t))-g(t,u2(t)))dt.
Now the proof proceeds in the same way as in Lemma 2.5.Similar to the proof of Lemma2.7, we can prove the following results.Lemma 2.8.
Letk be an odd number, n=2k+1, and
(2.32)Λ*̅={j:j∈{1,2,…,k-1},(-1)j+1a2j<0}.
Assume that one of the following conditions is satisfied:(H3̅)
fort,x1,x2∈R,x1≠x2,(2.33)a2k+∑j∈Λ*̅(T2π)2(k-j)(-1)j+1a2j>0,(g(t,x1)-g(t,x2))(x1-x2)<0,(H4¯)
there exists a nonnegative constantB such that(2.34)a2k>B(Tπ)2(T2π)2(k-1),a2k+∑j∈Λ*̅(T2π)2(k-j)(-1)j+1a2j>B(Tπ)2(T2π)2(k-1),B(x1-x2)2≥(g(t,x1)-g(t,x2))(x1-x2)>0,
where t,x1,x2∈R,x1≠x2, then (1.1) has at most one T-periodic solution.


## 3. Main Results

Theorem 3.1.
Letk 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 that(3.1)(g(t,u)-e(t))u>0,∀t∈R,|u|≥d0,(H2)*
there exist nonnegative constantsd0 and B such that (H2) holds,(3.2)(g(t,u)-e(t))u<0,∀t∈R,|u|≥d0,
then (1.1) has a unique T-periodic solution.Proof.
From Lemma2.5, 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 f̃:Rn+1→R given by
(3.3)f̃(t,u,u′,u′′,…,u(n-1))=e(t)-∑j=1n-1aju(j)-g(t,u).
Forλ∈(0,1], we consider the nth-order differential equation
(3.4)u(n)(t)+λ∑j=1n-1aju(j)+λg(t,u(t))=λe(t).
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
(3.5)‖u‖(n-1)<ρ.
Let λ∈(0,1] and let u be a possible T-periodic solution of (3.4). In what follows, Cj denotes a fixed constant independent of λ and u. Integrating (3.4) from 0 to T, it results that
(3.6)∫0T[g(t,u(t))-e(t)]dt=0,
which together with (H1)* (or (H2)*) implies that
(3.7)∃ξ∈[0,T]:|u(ξ)|<d0.
Hence, from (2.9), we have that
(3.8)|u|∞≤d0+12T(∫0T|u′(t)|2dt)1/2.
In view of (2.10), (3.8) implies that
(3.9)|u|∞≤d0+12T(T2π)k-1(∫0T|u(k)(t)|2dt)1/2.
It follows that
(3.10)|∫0Te(t)u(t)dt|≤Td0|e|∞+12TT(T2π)k-1|e|∞(∫0T|u(k)(t)|2dt)1/2,|∫0Tg(t,0)u(t)dt|≤Td0|g(t,0)|∞+12TT(T2π)k-1|g(t,0)|∞(∫0T|u(k)(t)|2dt)1/2.
On the other hand, multiplying (3.4) by u and integrating from 0 to T, it follows that
(3.11)∫0T|u(k)(t)|2dt+λ∑j=1k-1(-1)ja2j∫0T|u(j)(t)|2dt=-λ∫0Tu(t)g(t,u(t))dt+λ∫0Te(t)u(t)dt.
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
(3.12)(1+∑j∈Λ(T2π)2(k-j)(-1)ja2j)∫0T|u(k)(t)|2dt≤∫0T|u(k)(t)|2dt+λ∑j=1k-1(-1)ja2j∫0T|u(j)(t)|2dt≤-λ∫0T(u(t)-0)(g(t,u(t))-g(t,0))dt-λ∫0Tu(t)g(t,0)dt+Td0|e|∞+12TT(T2π)k-1|e|∞(∫0T|u(k)(t)|2dt)1/2≤Td0(|g(t,0)|∞+|e|∞)+12TT(T2π)k-1(|g(t,0)|∞+|e|∞)(∫0T|u(k)(t)|2dt)1/2,whereΛ≠∅,∫0T|u(k)(t)|2dt≤∫0T|u(k)(t)|2dt+λ∑j=1k-1(-1)ja2j∫0T|u(j)(t)|2dt≤-λ∫0T(u(t)-0)(g(t,u(t))-g(t,0))dt-λ∫0Tu(t)g(t,0)dt+Td0|e|∞+12TT(T2π)k-1|e|∞(∫0T|u(k)(t)|2dt)1/2≤Td0(|g(t,0)|∞+|e|∞)+12TT(T2π)k-1(|g(t,0)|∞+|e|∞)(∫0T|u(k)(t)|2dt)1/2,whereΛ=∅,
which imply that there exists a positive constant C1 satisfying
(3.13)∫0T|u(k)(t)|2dt≤C1,∫0T|u(k)(t)|dt≤TC1.Case 2.
If(H2)* holds, using (2.9), (2.10), (3.10), and (3.11), we obtain
(3.14)(1+∑j∈Λ(T2π)2(k-j)(-1)ja2j)∫0T|u(k)(t)|2dt≤∫0T|u(k)(t)|2dt+λ∑j=1k-1(-1)ja2j∫0T|u(j)(t)|2dt≤-λ∫0T(u(t)-0)(g(t,u(t))-g(t,0))dt-λ∫0Tu(t)g(t,0)dt+Td0|e|∞+12TT(T2π)k-1|e|∞(∫0T|u(k)(t)|2dt)1/2≤B|u|22+Td0(|g(t,0)|∞+|e|∞)+12TT(T2π)k-1(|g(t,0)|∞+|e|∞)(∫0T|u(k)(t)|2dt)1/2≤B(Tπ|u′|2+Td0)2+Td0(|g(t,0)|∞+|e|∞)+12TT(T2π)k-1(|g(t,0)|∞+|e|∞)(∫0T|u(k)(t)|2dt)1/2≤B(Tπ(T2π)k-1|u(k)|2+Td0)2+Td0(|g(t,0)|∞+|e|∞)+12TT(T2π)k-1(|g(t,0)|∞+|e|∞)(∫0T|u(k)(t)|2dt)1/2,whereΛ≠∅,∫0T|u(k)(t)|2dt≤∫0T|u(k)(t)|2dt+λ∑j=1k-1(-1)ja2j∫0T|u(j)(t)|2dt≤-λ∫0T(u(t)-0)(g(t,u(t))-g(t,0))dt-λ∫0Tu(t)g(t,0)dt+Td0|e|∞+12TT(T2π)k-1|e|∞(∫0T|u(k)(t)|2dt)1/2≤B(Tπ(T2π)k-1|u(k)|2+Td0)2+Td0(|g(t,0)|∞+|e|∞)+12TT(T2π)k-1(|g(t,0)|∞+|e|∞)(∫0T|u(k)(t)|2dt)1/2,whereΛ=∅,
which together with (H2) yield that (3.13) holds.
Using (3.9) and (3.13), it follows that there exists C2 such that
(3.15)|u|∞≤C2.
Now, we shall estimatex(j)(j=1,2,…,2k-1), multiplying (3.4) by x(2k) and integrating from 0 to T, we have
(3.16)∫0T|u(2k)(t)|2dt+λ∑j=1k-1(-1)ja2j∫0T|u(k+j)(t)|2dt=-λ∫0Tu(2k)(t)g(t,u(t))dt+λ∫0Te(t)u(2k)(t)dt
Using (2.10), (3.15), and (3.16), we have
(3.17)(1+∑j∈Λ(T2π)2(k-j)(-1)ja2j)∫0T|u(2k)(t)|2dt≤∫0T|u(2k)(t)|2dt+λ∑j=1k-1(-1)ja2j∫0T|u(k+j)(t)|2dt=-λ∫0Tu(2k)(t)g(t,u(t))dt+λ∫0Te(t)u(2k)(t)dt≤(Tsup⁡|u|≤C2,t∈R|g(t,u)|+T|e|∞)(∫0T|u(2k)(t)|2dt)1/2,whereΛ≠∅,∫0T|u(2k)(t)|2dt≤∫0T|u(2k)(t)|2dt+λ∑j=1k-1(-1)ja2j∫0T|u(k+j)(t)|2dt≤(Tsup⁡|u|≤C2,t∈R|g(t,u)|+T|e|∞)(∫0T|u(2k)(t)|2dt)1/2,whereΛ=∅,
which imply that there exists a positive constant C3 satisfying
(3.18)∫0T|u(2k)(t)|2dt≤C3,∫0T|u(2k)(t)|dt≤TC3.
This implies the existence of a constant ρ>d0 such that (3.5) holds.
Now, to show that (ii) in Lemma2.1 is satisfied, it suffices to remark that
(3.19)F(x)=∫0T[-g(t,x)+e(t)]dt,(x∈R).
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 Theorem3.1, from Lemma 2.6, one can prove the following results.Theorem 3.2.
Letk 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 that(3.20)(g(t,u)-e(t))u<0,∀t∈R,|u|≥d0,(H2¯)*
there exist nonnegative constantsd0 and B such that (H2¯) holds,
(3.21)(g(t,u)-e(t))u>0,∀t∈R,|u|≥d0,
then (1.1) has a unique T-periodic solution.Theorem 3.3.
Letk 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 that(3.22)(g(t,u)-e(t))u>0,∀t∈R,|u|≥d0,(H4)*
there exist nonnegative constantsd0 and B such that (H4) holds,
(3.23)(g(t,u)-e(t))u<0,∀t∈R,|u|≥d0,
then (1.1) has a unique T-periodic solution.Proof.
From Lemma2.7, 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
(3.24)λa2k∫0T|u(k)(t)|2dt+λ∑j=1k-1(-1)ja2j∫0T|u(j)(t)|2dt=-λ∫0Tu(t)g(t,u(t))dt+λ∫0Te(t)u(t)dt,λa2k∫0T|u(2k)(t)|2dt+λ∑j=1k-1(-1)ja2j∫0T|u(k+j)(t)|2dt=-λ∫0Tu(2k)(t)g(t,u(t))dt+λ∫0Te(t)u(2k)(t)dt.
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
(3.25)|u|∞≤C5,∫0T|u(2k)(t)|dt≤C5,(3.26)∫0T|u(j)(t)|2dt≤(T2π)2k-j∫0T|u(2k)(t)|2dt≤C5,j=1,2,…,2k-1.
In view of (3.4), (3.25) and (3.26) yield that
(3.27)∫0T|u(2k+1)(t)|dt=∫0T|-λ∑j=12kaju(j)-λg(t,u(t))+λe(t)|dt≤∑j=12k|aj|T(∫0T|u(j)|2dt)1/2+T(sup⁡|u|≤C5,t∈R|g(t,u)|+|e|∞)≤∑j=12k|aj|TC5+T(sup⁡|u|≤C5,t∈R|g(t,u)|+|e|∞),
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 Theorem3.1.Similar to the proof of Theorem3.3, from Lemma 2.8, we obtain the following.Theorem 3.4.
Letk 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 that(3.28)(g(t,u)-e(t))u<0,∀t∈R,|u|≥d0,(H4¯)*
there exist nonnegative constantsd0 and B such that (H4¯) holds,
(3.29)(g(t,u)-e(t))u>0,∀t∈R,|u|≥d0,
then (1.1) has a unique T-periodic solution.Remark 3.5.
If∫0Te(t)dt=0 and g(t,u) satisfies the following condition:(H5)
there existd>0 and ϵ∈{-1,1} such that, for any continuous T-periodic function u, we have(3.30)ϵ∫0Tg(t,u(t))dt<0,ifmin⁡R⁡u≥d,ϵ∫0Tg(t,u(t))dt>0,ifmin⁡R⁡u≤-d.
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.
Leta,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
(4.1)u′′′′(t)+400u′′′(t)+(1.6πT)2u′′(t)+20u′(t)+a(t)u(t)+b(t)u3(t)+c(t)u5(t)=e(t)
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
(4.2)(g(t,u)-e(t))u>0,∀t∈R,|u|≥d0,
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.
Letβ: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
(4.3)u′′′′′(t)+1000003u′′′′(t)+δu′′′(t)-20u′′(t)+τu′(t)+β(t)u3(t)=e(t)
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=1000003,a3=δ,a2=-20, and a1=τ satisfies (H3)*. Hence, the result follows from Theorem 3.3.Remark 4.3.
Sinceg(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.



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