# Existence and Uniqueness of Periodic Solutions for Second Order Differential Equations

Authors: Yuanhong Wei
Journal: Journal of Function Spaces
 (2014)

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## Abstract

We study some second order ordinary differential equations. We establish the existence and uniqueness in some appropriate function space. By using Schauder’s fixed-point theorem, new results on the existence and uniqueness of periodic solutions are obtained.


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## Body

## 1. Introduction

In this paper, we are concerned with the existence and uniqueness of periodic solution for the nonlinear equation(1)x′′=f(t,x,x′),
where f:R×R×R→R is continuous, periodic in t with period T, and C1 with respect to (x,x′).Because of wide interests in physics and engineering, periodic solutions of second order differential equations have been investigated by many authors. We refer the reader to [1–7] and the references cited therein.The purpose of this paper is to study the existence and uniqueness of periodic solution in some appropriate function spaceCT1. To be precise, we first derive a result of existence and uniqueness, when the nonlinearity is a C1 function with respect to (x,x′). Then, a similar result concerning the existence of periodic solutions is obtained, when the nonlinearity is not a C1 function.Throughout this paper we use the following assumptions.(A1)
There exist two continuous functionsa(t) and b(t) such that, for all t∈[0,T],
(2)0≨a(t)⩽fx(t,x,x′)⩽b(t),where0≨a(t) means 0⩽a(t) and 0≡a(t).(A2)
There exists a positive constantM such that, for all t∈[0,T],
(3)|fx′(t,x,x′)|⩽M.We now present our main results of this paper.Theorem 1.
Let assumptions (A1) and (A2) hold. Then (1) has a unique T-periodic solution.Remark 2.
We point out that the condition0≨a(t)⩽fx in (A1) is necessary. Let us take the following equation, for example:
(4)x′′=C,
where C is a constant and C≠0. The equation has no periodic solution.Remark 3.
Consider the following example:(5)x′′=sin⁡t·sin⁡x·sin⁡x′+x.
We can easily check that assumptions (A1) and (A2) hold. By Theorem 1, the equation has a unique 2π-periodic solution.We also consider the case when the right-hand side of the equation is only continuous. In this case, we establish the existence of periodic solutions for the differential equation(6)x′′=f(t,x,x′)x+g(t,x,x′),
where f, g:R×R×R→R are continuous and periodic in t with period T. We assume the following.(A3)
There exist continuous functionsa(t) and b(t) and a positive constant M such that
(7)0≨a(t)⩽f(t,x,x′)⩽b(t),|g(t,x,x′)|⩽M,for allt∈[0,T].Then, we have the following result.Theorem 4.
Let (A3) hold. Then (6) has at least one T-periodic solution.Remark 5.
Consider the following example:(8)x′′=2x+xsin⁡x′+cos⁡⁡t·cos⁡⁡x.
By Theorem 4, the equation has at least one 2π-periodic solution.To establish the main results, we introduce some appropriate function space. By using Schauder’s fixed-point theorem, the existence and uniqueness of periodic solutions are obtained. We know from the anonymous referee that the proof of this paper can be simplified, by using the results in [8]. The proof of this paper can be seen as an application of Schauder’s fixed-point theorem.


## 2. Preliminary

In this section, We first introduce the function space, in which we will obtain the periodic solution for the problem. Then, some preliminary lemma is introduced, which is valuable for the proof of our main results.LetCT1 be the space of continuously differentiable T-periodic functions, with norm ∥·∥ given by
(9)∥x∥=max⁡t∈[0,T]⁡|x(t)|+max⁡t∈[0,T]⁡|x′(t)|.
It is well known that CT1 is a Banach space.We now introduce the following lemma.Lemma 6.
u
(
t
) and v(t) are T-periodic functions. If 0≨v(t), for all t∈R, then the equation
(10)x′′=u(t)x′+v(t)x
has a unique T-periodic solution x(t)≡0.Proof.
Assume there exists aT-periodic solution x(t), x(t)≡0. Since 0≨v(t), x(t) is not a constant. Otherwise we will get a contradiction by substituting x=C≠0 into (10). Then we claim that there exist t0 and t1, t0<t1, such that
(11)x(t)>0,fort∈(t0,t1);x′(t0)>0,x′(t1)=0.
Now we prove it.
Case
1. x(t) has zero points. Assume that x(θ)=0; then, we have x′(θ)≠0 because the initial value problem of (10) has a unique solution. If x′(θ)>0, then we let t0=θ; if x′(θ)<0, by the periodic condition, there exists t0, t0>θ, such that
(12)x(t)<0,t∈(θ,t0);x(t0)=0.
So x′(t0)>0. Also by the periodic condition there exists η, η>t0, such that
(13)x(t)>0,t∈(t0,η);x(η)=0.
So we can find a t1∈(t0,η) such that x′(t1)=0. Thus, under Case 1 we verify (11) holds.
Case
2. x(t) has no zero point. Then x(t) has a constant sign. Suppose x(t)>0 (if not we can consider the T-periodic solution x¯(t)=-x(t), x¯(t)>0). Since x(t) is periodic and is not a constant, there exists t0 such that x′(t0)>0, x(t0)=x(t0+T). So we can find a t1, t1>t0, x′(t1)=0. Then we prove that (11) holds.
Multiplying both sides of (10) by exp⁡{-∫t0t1‍u(s)ds} and integrating between t0 and t1 we get
(14)0>-x′(t0)=∫t0t1‍v(t)x(t)exp⁡{-∫t0t1‍u(s)ds}dt⩾0,
which leads to a contradiction. This completes the proof of Lemma 6.Remark 7.
We can also prove the following. Ifu(0)=u(T), v(0)=v(T), 0≨v(t), a.e. t∈[0,T], then the following periodic boundary value problem,
(15)x′′=u(t)x′+v(t)x,t∈[0,T],x(0)=x(T),x′(0)=x′(T),
also has a unique solution x(t)≡0.


## 3. Proof of the Main Results

Proof of Theorem1.
First prove the uniqueness. Assume thatx1(t) and x2(t) are two T-periodic solutions of (1). Setting
(16)x(t)=x1(t)-x2(t),
we get
(17)x(t+T)=x(t),x′′=f(t,x1,x1′)-f(t,x2,x2′)=fx(t,x1+θ1(x1-x2),x1′)x+fx′(t,x2,x2+θ2(x1-x2))x′.
From assumption (A1) we know
(18)0≨a(t)⩽fx(t,x1+θ1(x1-x2),x1′).
Hence, by Lemma 6, x(t)≡0.
We next prove the existence. Rewrite (1) in the following form:
(19)x′′=f(t,x,x′)=(f(t,x,x′)-f(t,x,0))+(f(t,x,0)-f(t,0,0))+f(t,0,0)=∫01‍fx′(t,x,θ1x′)dθ1x′+∫01‍fx(t,θ2x,0)dθ2x+f(t,0,0).
From Lemma6, for each x∈CT1, the equation
(20)y′′=∫01‍fx′(t,x,θ1x′)dθ1y′+∫01‍fx(t,θ2x,0)dθ2y+f(t,0,0)
has a unique T-periodic solution because the corresponding homogeneous equation only has trivial T-periodic solution x(t)≡0.
We define operator(21)P:CT1⟶CT1,
for given x∈CT1,  y(t)=P[x](t) is the unique T-periodic solution of (20). Then the existence of the T-periodic solution is equivalent to the existence of fixed point of P in the space CT1. We will prove that P is continuous and compact, and P(CT1) is a bounded subset of CT1.
Proof of Continuity. For any convergent sequence {xk}⊂CT1, satisfying xk→x0(k→∞), let yk=Pxk. Then
(22)yk′′=∫01‍fx′(t,xk,θ1xk′)dθ1yk′+∫01‍fx(t,θ2xk,0)dθ2yk+f(t,0,0).
We claim that {yk} is a bounded sequence in CT1. If not, we can find a subsequence of {yk} (for convenience we also use the same notations) such that ∥yk∥→∞(k→∞). Let ωk=yk/∥yk∥. Then {ωk}⊂CT1, ∥ωk∥=1,
(23)ωk′′=∫01‍fx′(t,xk,θ1xk′)dθ1ωk′+∫01‍fx(t,θ2xk,0)dθ2ωk+f(t,0,0)∥yk∥.
So ∥ωk′′∥⩽M+max⁡t∈[0,T]⁡b(t)+1<∞(k→∞). Since {ωk′′} is bounded and
(24)ωk′(t)=ωk′(0)+∫0t‍ωk′′(s)ds,{ωk′} is bounded and equicontinuous sequence of functions. Furthermore, by
(25)ωk(t)=ωk(0)+∫0t‍ωk′(s)ds,{ωk} is bounded and equicontinuous sequence of functions. By Ascoli-Arzelà Theorem, {ωk′} and {ωk} contain a uniformly convergent subsequence, respectively, (for convenience we also use the same notations) such that
(26)ωk⟶1ω0,ωk′⟶1v0,
where the notation “→1” means uniform convergence. Obviously, ω0, v0∈CT1. From (23) and (24), we obtain
(27)ωk′(t)=ωk′(0)+∫0t‍(f(s,0,0)∥yk∥∫01‍fx′(s,xk,θ1xk′)dθ1ωk′mm+∫01‍fx(s,θ2xk,0)dθ2ωk+f(s,0,0)∥yk∥)ds.
Let k→∞. From (25) and (27), we get
(28)ω0(t)=ω0(0)+∫0t‍v0(s)ds,v0(t)=v0(0)+∫0t‍(∫01‍fx′(s,xk,θ1xk′)dθ1v0(s)mmmmmm+∫01‍fx(s,θ2xk,0)dθ2ω0(s))ds.
Hence
(29)ω0′′=∫01‍fx′(t,x0,θ1x0′)dθ1ω0′+∫01‍fx(t,θ2x0,0)dθ2ω0.
By Lemma 6, we have ω0≡0, which is in contradiction with ∥ω0∥=1, so {yk} is a bounded sequence. Then, by (22), we know {yk′′} is bounded, so {yk′} and {yk} are bounded and equicontinuous sequences of functions. By Ascoli-Arzelà Theorem, {yk′} and {yk} contain a uniformly convergent subsequence, respectively, (for convenience we also use the same notations) such that
(30)yk⟶1y0,yk′⟶1v¯0.
We know
(31)yk′(t)=yk′(0)+∫0t‍yk′′(s)ds=yk′(0)+∫0t‍(∫01‍fx′(s,xk,θ1xk′)dθ1yk′mmmmmmmm+∫01‍fx(s,θ2xk,0)dθ2ykmmmmmmmm+f(s,0,0)∫01‍)ds,yk(t)=yk(0)+∫0t‍yk′(s)ds.
When k→∞, from (31), we obtain
(32)y0′′=∫01‍fx′(t,x0,θ1x0′)dθ1y0′+∫01‍fx(t,θ2x0,0)dθ2y0+f(t,0,0).
By the uniqueness we know y0=Px0;  thus, operator P is continuous.
Proof of Compactness. For each bounded set S⊂CT1, we claim that P(S) is bounded in CT1. If not, by an analogous manner as above we will reach a contradiction. For every x∈S, y=Px is defined by (20). Since ∥y′∥, ∥y∥, ∥fx∥, and ∥fx′∥ are all bounded, then ∥y′′∥<∞. Proceeding as proof of continuity we conclude that {y′} and {y} are bounded and equicontinuous sequences of functions. By the Ascoli-Arzelà Theorem, P is a compact operator.
We claim thatP(CT1) is bounded in CT1. If not, there exist {xk}, k=1,2,…, such that ∥Pxk∥→∞(k→∞). Let yk=Pxk. Then (22) holds. Take ωk=yk/∥yk∥. Then {ωk}⊂CT1, ∥ωk∥=1, and (23), (24), (25), and (27) hold. By the above proof we know {ωk′} and {ωk} are bounded and equicontinuous sequences of functions and contain a uniformly convergent subsequence, respectively, (also use the same notations) such that
(33)ωk⟶1ω0,ωk′⟶1v0,∥ω0∥=1.
Sequences {∫01‍fx′(t,xk,θ1xk′)dθ1}k=1∞ and {∫01‍fx(t,θ2xk,0)dθ2}k=1∞ are bounded sequences in L2[0,T]. By the weakly sequential compactness of L2 space, both of them have a weakly convergent subsequence (also use the same notations) such that
(34)∫01‍fx′(t,xk,θ1xk′)dθ1⟶wf1(t),∫01‍fx(t,θ2xk,0)dθ2⟶wf2(t),
in L2[0,T], where “→w” means weak convergence. Obviously,
(35)|f1(t)|⩽M,0≨a(t)⩽f2(t)⩽b(t),mmmmmmmmmma.e.t∈[0,T].
Whenk→∞, from (25) and (27), for a.e. t∈[0,T], we have
(36)v0′(t)=f1(t)v0(t)+f2(t)ω0(t),ω0′(t)=v0(t).
Then
(37)ω0′′(t)=f1(t)ω0′(t)+f2(t)ω0(t),ω0(0)=ω0(T),ω0′(0)=ω0′(T).
By Remark3, we obtain ω0≡0, which is in contradiction with ∥ω0∥=1. Then there exists a constant K>0 such that ∥Px∥⩽K, x∈CT1.
AssumeD={x∈CT1,∥x∥⩽K+1}. By Schauder’s fixed-point theorem, P:D→D has at least one fixed point. This completes the proof of Theorem 1.Proof of Theorem4.
For eachx∈CT1, from Lemma 6, we know the equation
(38)y′′=f(t,x,x′)y+g(t,x,x′)
has a unique T-periodic solution because the corresponding homogeneous equation only has trivial T-periodic solution x(t)≡0.
We defineP¯:CT1→CT1, for each given x∈CT1  y(t)=P¯[x](t) is the unique T-periodic solution of (38). Hence the existence of the periodic solutions is equivalent to the existence of fixed points of P¯ in the Banach space CT1.
Proceeding as the proof of Theorem1 we can prove that P¯ is a compact continuous operator and P¯(CT1) is a bounded subset of CT1.
Then there exists a constantK1>0 such that ∥P¯x∥⩽K1, for all x∈CT1. Let D¯={x∈CT1,∥x∥⩽K1+1}. By Schauder’s fixed-point theorem, P¯:D¯→D¯ has at least one fixed point. This completes the proof of Theorem 4.


## 4. Another Simple Proof

Actually, by the anonymous referee, we know that the proof of the theorem can be much simplified if we use the theorem in [8]. In fact, for equation
(39)u(n)=f(t,u,…,u(n-1)),
where f:Rn+1→R is continuous and T-periodic in t, Theorem 2.2 of [8] implies the following propositions.Lemma 8.
Let there exist continuous andT-periodic in the first argument functions fk:Rn+1→R(k=1,2,…,n) such that
(40)mmm|f(t,x1,…,xn)-∑k=1n‍fk(t,x1,…,xn)xk|⩽r,p1k(t)⩽fk(t,x1,…,xn)⩽p2k(t)(k=1,2,…,n)
are satisfied on Rn+1, where r=const>0 and pik:R→R(i=1,2;k=1,2,…,n) are continuous T-periodic functions. Let, moreover, for any continuous T-periodic functions pk:R→R(k=1,2,…,n), satisfying
(41)p1k(t)⩽pk(t)⩽p2k(t)(k=1,2,…,n),
the equation
(42)u(n)=∑k=1n‍pk(t)u(k-1)
have no nontrivial T-periodic solution. Then equation has at least one T-periodic solution.Lemma 9.
Let the functionf in the last n arguments have continuous partial derivatives, satisfying
(43)p1k(t)⩽∂fk(t,x1,…,xn)∂xk⩽p2k(t)(k=1,2,…,n),
where pik:R→R(i=1,2;k=1,2,…,n) are continuous T-periodic functions. Let, moreover, for any T-periodic pk:R→R(k=1,2,…,n), satisfying (41), (42) has no nontrivial T-periodic solution. Then, the equation has a unique T-periodic solution.On the basis of these theorems, we can prove the theorem by checking the conditions in the previous lemma. By Lemma6, the conditions above can be easily proved.



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*Source: 246258-2014-08-27.xml*