Volume 2009, Article ID 950797,14pages doi:10.1155/2009/950797

*Research Article*

**Existence and Uniqueness of Periodic Solutions for**

**a Second-Order Nonlinear Differential Equation**

**with Piecewise Constant Argument**

**Gen-qiang Wang**

**1**

_{and Sui Sun Cheng}

_{and Sui Sun Cheng}

**2**

*1 _{Department of Computer Science, Guangdong Polytechnic Normal University,}*

*Guangzhou, Guangdong 510665, China*

*2 _{Department of Mathematics, Tsing Hua University, Hsinchu, Taiwan 30043, China}*

Correspondence should be addressed to Sui Sun Cheng,[email protected]

Received 4 July 2009; Revised 1 October 2009; Accepted 2 November 2009

Recommended by Enrico Obrecht

Based on a continuation theorem of Mawhin, a unique periodic solution is found for a second-order nonlinear diﬀerential equation with piecewise constant argument.

Copyrightq2009 G.-q. Wang and S. S. Cheng. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

**1. Introduction**

Qualitative behaviors of first-order delay diﬀerential equations with piecewise constant arguments are the subject of many investigationssee, e.g.,1–19, while those of higher-order equations are not.

However, there are reasons for studying higher-order equations with piecewise
constant arguments. Indeed, as mentioned in10, a potential application of these equations
is in the stabilization of hybrid control systems with feedback delay, where a hybrid system is
one with a continuous plant and with a discretesampledcontroller. As an example, suppose
that a moving particle with time variable mass *rt* is subjected to a restoring controller

−*φxt*which acts at sampled time*t.*Then Newton’s second law asserts that

*rtxt*−*φxt.* 1.1

Since this equation is “similar” to the harmonic oscillator equation

we expect that the well-known qualitative behavior of the later equation may also be found
in the former equation, provided appropriate conditions on*rt*and*φx*are imposed.

In this paper we study a slightly more general second-order delay diﬀerential equation with piecewise constant argument:

*rtxtft, xt* *pt,* 1.3

where*ft, x*is a real continuous function defined on*R*2 *with positive integer periodω* for

*t*;*rt*and *pt*are continuous function defined on*R* with period*ω, rt* *>* 0 for*t* ∈ *R*and

*ω*

0*ptdt*0*.*

By a solution of1.3we mean a function*xt*which is defined on*R*and which satisfies
the following conditions: i*xt*is continuous on *R,*ii*rtxt*is diﬀerentiable at each
point*t* ∈ *R,*with the possible exception of the points*t* ∈ *R* where one-sided derivatives
exist, andiiisubstitution of*xt*into1.3leads to an identity on each interval*n, n*1⊂*R*

with integral endpoints.

In this note, existence and uniqueness criteria for periodic solutions of 1.3 will be
established. For this purpose, we will make use of a continuation theorem of Mawhin. Let*X*

and*Y* be two Banach spaces and*L*: Dom*L*⊂*X* → *Y* is a linear mapping and*N* :*X* → *Y*

a continuous mapping. The mapping*L*will be called a Fredholm mapping of index zero if
dim Ker*L* codimIm*L <*∞, and Im*L*is closed in*Y.*If*L*is a Fredholm mapping of index
zero, there exist continuous projectors*P* : *X* → *X* and*Q* :*Y* → *Y* such that Im*P* Ker*L*

and Im*L*Ker*Q*Im*I*−*Q.*It follows that*L*|Dom*L*∩Ker*P* :*I*−*PX* → Im*L*has an inverse

which will be denoted by*KP*. IfΩis an open and bounded subset of*X*, the mapping*N*will
be called*L*-compact onΩif*QN*Ωis bounded and*KPI*−*QN*:Ω → *X*is compact. Since

Im*Q*is isomorphic to Ker*L,*there exists an isomorphism*J*: Im*Q* → Ker*L*.

**Theorem A**Mawhin’s continuation theorem18*. LetLbe a Fredholm mapping of index zero,*
*and letNbeL-compact on*Ω*. Suppose that*

i*for eachλ*∈0*,*1*,* *x*∈*∂*Ω*, Lx /λNx*;

ii*for eachx*∈*∂*Ω∩Ker*L, QNx /0 and degJQN,*Ω∩Ker*,*0*/*0*.*
*Then the equationLxNxhas at least one solution in*Ω∩dom*L.*

**2. Existence and Uniqueness Criteria**

Our main results of this paper are as follows.

**Theorem 2.1. Suppose that there exist constants**D >0 andδ0 such that

i*ft, x*sgn*x >0 fort*∈*Rand*|*x*|*> D,*

iilim*x*→ −∞max0≤*t*≤*ωft, x/x*≤*δ(or limx*→∞max0≤*t*≤*ωft, x/x*≤*δ).*
*Ifω*2_{δ}_{}_{max}

0≤*t*≤*ω*1*/rt* *<* 1*,then*1.3*has anω-periodic solution. Furthermore, theω-periodic*
*solution is unique if in addition one has the following.*

iii*ft, x* *is strictly monotonous in* *x* *and there exists nonnegative constant* *b* *<*

4*/ω*2_{}_{min}

0≤*t*≤*ωrtsuch that*

**Theorem 2.2. Suppose that there exist constants**D >0 andδ0 such that

i*ft, x*sgn*x <0 fort*∈*Rand*|*x*|*> D,*

iilim*x*→ −∞max0≤*t*≤*ωft, x/x*−*δ(or limx*→∞max0≤*t*≤*ωft, x/x*−*δ).*

*Ifω*2_{δ}_{}_{max}

0≤*t*≤*ω*1*/rt* *<* 1*,then*1.3*has anω-periodic solution. Furthermore, theω-periodic*
*solution is unique if in addition one has the following.*

iii*ft, x* *is strictly monotonous in* *x* *and there exists nonnegative constant* *b* *<*

4*/ω*2_{}_{min}

0≤*t*≤*ωrtsuch that*2.1*holds.*

We only give the proof ofTheorem 2.1, asTheorem 2.2can be proved similarly.
First we make the simple observation that *xt* is an *ω*-periodic solution of the
following equation:

*rtxt* *r*0*x*0−

*t*

0

*fs, xs*−*psds,* 2.2

if, and only if, *xt* is an *ω*-periodic solution of 1.3. Next, let *Xω* be the Banach space

of all real*ω*-periodic continuously diﬀerentiable functions of the form *x* *xt* which is
defined on *R* and endowed with the usual linear structure as well as the norm *x* _{1}
_{1}

*i*0max0≤*i*≤*ω*|*xit*|. Let*Yω*be the Banach space of all real continuous functions of the form
*y* *αtht*such that*y*0 0*,*where*α*∈ *R*and*ht*∈ *Xω,*and endowed with the usual
linear structure as well as the norm *y* _{2} |*α*| *h* _{1}*.*Let the zero element of*Xω*and*Yω*be
denoted by*θ*1and*θ*2respectively.

Define the mappings*L*:*Xω* → *Yω*and*N*:*Xω* → *Yω,*respectively, by

*Lxt* *rtxt*−*r*0*x*0*,* 2.3

*Nxt* −
*t*

0

*fs, xs*−*psds.* 2.4

Let

*ht* −
*t*

0

*fs, xs*−*psds* *t*
*ω*

*ω*

0

*fs, xsds.* 2.5

Since*h* ∈ *Xω* and*h*0 0,*N*is a well-defined operator from *Xω* to*Yω.*Let us define*P* :

*Xω* → *Xω*and*Q*:*Yω* → *Yω,*respectively, by

*P xt* *x*0*,* *n*∈*Z* 2.6

for*xxt*∈*Xω*and

*Qyt* *αt* 2.7

* Lemma 2.3. Let the mappingLbe defined by*2.3

*. Then*

Ker*LR.* 2.8

*Proof. It su*ﬃces to show that if*xt*is a real*ω*-periodic continuously diﬀerentiable function
which satisfies

*rtxt* *r*0*x*0*,* *t*∈*R,* 2.9

then*xt*is a constant function. To see this, note that for such a function*xxt,*

*xt* *r*0*x*

_{}_{0}_{}

*rt* *,* *t*∈*R.* 2.10

Hence by integrating both sides of the above equality from 0 to*t,*we see that

*xt* *x*0 *r*0*x*0

*t*

0
*ds*

*rs,* *t*∈*R.* 2.11

Since*rt*is positive, continuous, and periodic,

_{∞}

0
*ds*

*rs* ∞*.* 2.12

Since*xt*is bounded, we may infer from2.11that*x*0 0. But then2.9implies*xt* 0
for*t*∈*R.*The proof is complete.

* Lemma 2.4. Let the mappingLbe defined by*2.3

*. Then*

Im*Ly*∈*Xω*|*y*0 0 ⊂*Yω.* 2.13

*Proof. It su*ﬃces to show that for each*y* *yt* ∈ *Xω* that satisfies*y*0 0*,*there is a*x*
*xt*∈*Xω*such that

*yt* *rtxt*−*r*0*x*0*,* *t*≥0*.* 2.14

But this is relatively easy, since we may let

*α* * _{ω}* 1
0

*ds/rs*

*,* _{}2.15

*xt*
_{t}

0
*ys*
*rsds*−*α*

_{ω}

0
*ys*
*rsds*

_{t}

0
*ds*

*rs,* *t*≥0*.* 2.16

* Lemma 2.5. The mappingLdefined by*2.3

*is a Fredholm mapping of index zero.*

Indeed, from Lemmas2.3and2.4and the definition of*Yω,*dim Ker*L*codim Im*L*

1 *<* ∞*.*From 2.13, we see that Im*L*is closed in*Yω.*Hence*L* is a Fredholm mapping of

index zero.

**Lemma 2.6. Let the mapping**L, P,and*Qbe defined by*2.3*,*2.6*, and* 2.7*, respectively. Then*

Im*P*Ker*Land ImL*Ker*Q.*

Indeed, from Lemmas2.3and2.4and defining conditions2.6and2.7, it is easy to
see that Im*P*Ker*L*and Im*L*Ker*Q.*

* Lemma 2.7. LetLandNbe defined by*2.3

*and*2.4

*, respectively. Suppose that*Ω

*is an open and*

*bounded subset ofXω.ThenNisL-compact on*Ω

*.*

*Proof. It is easy to see that for anyx*∈Ω*,*

*QNxt* −*t*
*ω*

*ω*

0

*fs, xsds,* 2.17

so that

*QNx* _{2}1
*ω*

_{ω}

0

*fs, xsds,* 2.18

*I*−*QNxt* −
_{t}

0

*fs, xs*−*psds* *t*
*ω*

_{ω}

0

*fs, xsds,* *t*≥0*.* 2.19

These lead us to

*KPI*−*QNxt* −
*t*

0

1

*rvdv*
*v*

0

*fs, xs*−*psds*

*α*

*ω*

0
*dv*
*rv*

*v*

0

*fs, xs*−*psds*
*t*

0

1

*rvdv*

1

*ω*
_{t}

0
*v*
*rvdv*

_{ω}

0

*fs, xsds*

− *α*

*ω*

*ω*

0
*vdv*
*rv*

_{ω}

0

*fs, xsds*
*t*

0

1

*rvdv,*

2.20

where *α*is defined by2.15. By2.18, we see that*QN*Ω is bounded. Noting that2.7

holds and*N* is a completely continuous mapping, by means of the Arzela-Ascoli theorem

**Lemma 2.8. Suppose that***gtis a real, bounded and continuous function ona, band limx*→*b*−*gt*
*exists. Then there is a pointξ*∈*a, bsuch that*

_{b}

*a*

*gsdsgξb*−*a.* 2.21

The above result is only a slight extension of the integral mean value theorem and is easily proved.

* Lemma 2.9. Suppose that condition (i) inTheorem 2.1holds. Suppose further thatxt*∈

*Xωsatisfies*

*ω*

0

*fs, xsds*0*.* 2.22

*Then there ist*1∈0*, ωsuch that*|*xt*1| ≤*D.*

*Proof. From*2.22andLemma 2.8**, we have***ξi*∈*i*−1*, i*for*i*1*, . . . , ω*such that

*ω*

*i*1

*fξi, xi*−1
*ω*

*i*1
_{i}

*i*−1

*fs, xsds*
_{ω}

0

*fs, xsds*0*.* 2.23

In case *ω* 1*,*from the conditioni inTheorem 2.1and 2.23, we know that|*x*0| ≤ *D.*

Suppose *ω* ≥ 2*.*Our assertion is true if one of *x*0*, x*1*, . . . , xω* −1 has absolute value
less than or equal to *D.* Otherwise, there should be*xη*1 and *xη*2 among*x*0*, x*1*, . . .*

and*xω*−1such that*xη*1 *> D* and*xη*2 *<* −*D.*Since*xt*is continuous, in view of the

intermediate value theorem, there is*xη*3such that−*D* ≤ *xη*3 ≤*D,*here*η*1 *> η*3 *> η*2 or
*η*2 *> η*3 *> η*1. Since*xt*is periodic, there is*t*1 ∈0*, ω*such that|*xt*1| |*xη*3| ≤ *D.*The

proof is complete.

Now, we consider that following equation:

*rtxt*−*r*0*x*0 −*λ*
_{t}

0

*fs, xs*−*psds,* 2.24

where*λ*∈0*,*1*.*

* Lemma 2.10. Suppose that conditions (i) and (ii) ofTheorem 2.1hold. Ifω*2

_{δ}_{}

_{max}

0≤*t*≤*ω*1*/rt<*

1*,then there are positive constantsD*0*andD*1*such that for anyω-periodic solutionxtof* 2.24*,*

*xit*≤*Di,* *t*∈0*, ω*; *i*0*,*1*.* 2.25

*Proof. Let* *xt* be a *ω*-periodic solution of 2.24. By 2.24 and our assumption that

*ω*

0*psds*0*,*we have

_{ω}

0

ByLemma 2.9, there is*t*1∈0*, ω*such that

|*xt*1| ≤*D.* 2.27

Since*xt*and*xt*are with period*ω,*thus for any*t*∈*t*1*, t*1*ω,*we have

*xt* *xt*1
_{t}

*t*1

*xsds,*

*xt* *xt*1*ω*
*t*

*t*1*ω*

*xsdsxt*1
*t*

*t*1*ω*

*xsds.*

2.28

From2.28, we see that for any*t*∈*t*1*, t*1*ω,*

|*xt*| ≤ |*xt*1|

1 2

*t*1*ω*

*t*1

*xsds*|*xt*1|

1 2

*ω*

0

*xsds.* 2.29

It is easy to see from2.27and2.29that for any*t*∈0*, ω*

|*xt*| ≤ |*xt*1|

1 2

_{ω}

0

*xsds*≤*D* 1

2

_{ω}

0

*xsds.* 2.30

In view of the condition*ω*2_{δ}_{}_{max}

0≤*t*≤*ω*1*/rt<*1*,*we know that there is a positive number
*ε*such that

*η*1:*ω*2*δε*

max

0≤*t*≤*ω*

1

*rt*

*<*1*.* 2.31

From conditionii, we see that there is a*ρ > D*such that for*t*∈*R*and*x <*−*ρ,*

*ft, x*

*x* *< δε.* 2.32

Let

*E*1

*t*|*t*∈0*, ω, xt<*−*ρ* *,* 2.33

*E*2

*t*|*t*∈0*, ω,*|*xt*| ≤*ρ* *,* 2.34

*E*3 0*, ω*\*E*1∪*E*2*,* 2.35

*M*0 max

0≤*t*≤*ω,*|*x*|≤*ρft, x.* 2.36

By2.32and2.33, we have

*E*1

*fs, xsds*≤*δε*

*E*1

|*xs*|*ds*≤*δεω*max

From2.34and2.36, we have

*E*2

_{f}_{}_{s, x}_{}_{s}_{}_{ds}_{≤}_{ωM}_{0}_{.}_{}_{2.38}_{}

In view of conditioni,2.26,2.37, and2.38, we get

*E*3

_{f}_{}_{s, x}_{}_{s}_{}_{ds}_{}
*E*3

*fs, xsds*

−

*E*1

*fs, xsds*−

*E*2

*fs, xsds*

≤

*E*1

*fs, xsds*

*E*2

*fs, xsds*

≤*δεω*max

0≤*t*≤*ω*|*xt*|*ωM*0*.*

2.39

It follows from2.37,2.38, and2.39that

*ω*

0

_{f}_{}_{s, x}_{}_{s}_{}_{ds}_{}
*E*1

_{f}_{}_{s, x}_{}_{s}_{}_{ds}_{}
*E*2

_{f}_{}_{s, x}_{}_{s}_{}_{ds}_{}
*E*3

_{f}_{}_{s, x}_{}_{s}_{}_{ds}

≤2*δεω*max

0≤*t*≤*ω*|*xt*|2*ωM*0*.*

2.40

Since*x*0 *xω,*thus there is a*t*1 ∈0*, ω*such that*xt*1 0*.*In view of2.24and the fact

that*xt*1 0, we conclude that for any*t*∈*t*1*, t*1*ω,*

*rtxt*_{}*rt*1*xt*1−*λ*
_{t}

*t*1

*fs, xs*−*psds*_{}

_{}−*λ*
_{t}

*t*1

*fs, xs*−*psds*_{}

≤_{}

_{t}

*t*1

*fs, xs*−*psds*_{}

≤

*t*1*ω*

*t*1

_{f}_{}_{s, x}_{}_{s}_{}_{ds}_{}*t*1*ω*
*t*1

_{p}_{}_{s}_{}_{ds}

≤

_{ω}

0

*fs, xsds*
_{ω}

0

*psds.*

From2.40and2.41, we see that

max

0≤*t*≤*ω*

_{x}_{}_{t}_{}_{≤}_{max}

0≤*t*≤*ω*

1

*rt*

2*δεω*max

0≤*t*≤*ω*|*xt*|2*ωM*00max≤*t*≤*ω*

_{p}_{}_{t}_{}_{.}_{}_{2.42}_{}

It follows from2.30,2.31, and2.42that

max

0≤*t*≤*ω*|*xt*| ≤*D*

1 2

_{ω}

0

*xsds*

≤*ω*2

max

0≤*t*≤*ω*

1

*rt*

*δε*max

0≤*t*≤*ω*|*xt*|*M*1

*η*1max

0≤*t*≤*ω*|*xt*|*M*1*,*

2.43

where

*M*1 *D*

max

0≤*t*≤*ω*

1

*rt*

2*ωM*0max
0≤*t*≤*ω*

*pt*

*.* 2.44

Let*D*0*M*1*/*1−*η*1*,*then from2.43we have

max

0≤*t*≤*ω*|*xt*| ≤*D*0*.* 2.45

From2.42and2.45, for any*t*∈0*, ω,*we have

max

0≤*t*≤*ωx*

_{}_{t}_{}_{≤}_{D}

1*,* 2.46

where

*D*1

max

0≤*t*≤*ω*

1

*rt*

2*δεωD*02*ωM*0max
0≤*t*≤*ω*

_{p}_{}_{t}_{}_{.}_{}_{2.47}_{}

The proof is complete.

**Lemma 2.11. Suppose that condition (iii) of***Theorem 2.1* *is satisfied. Then* 1.3*has at most one*
*ω-periodic solution.*

*Proof. Suppose thatx*1*t*and*x*2*t*are two*ω*-periodic solutions of1.3. Set*zt* *x*1*t*−
*x*2*t*. Then we have

*Case*i. For all*t*∈0*, ω, zt/*0. Without loss of generality, we assume that*zt>*0, that is,

*x*1*t> x*2*t*for*t*∈0*, ω.*Integrating2.48from 0 to*ω*, we have

*ω*

0

*ft, x*1*x*1*t*−*ft, x*2*t*

*dt*0*.* 2.49

Combining conditioniiiand*x*1*t> x*2*t*, either

*ft, x*1*t*−*ft, x*2*t>*0*,* *t*∈0*, ω* 2.50

or

*ft, x*1*t*−*ft, x*2*t<*0*,* *t*∈0*, ω* 2.51

holds. This is contrary to2.49.

*Case*ii. There exist*ξ*∈0*, ω*such that*zξ* 0. As in the proof of2.30inLemma 2.10, we
have

max

0≤*t*≤*ω*|*zt*| ≤ |*zξ*|

1 2

_{ω}

0

*zsds* 1

2

_{ω}

0

*zsds.* 2.52

On the other hand, since*z*0 *zω,*thus there is a*t*1∈0*, ω*such that*zt*1 0. In view of

2.48, we conclude that for any*t*∈*t*1*, t*1*ω,*

*rtzt* *rt*1*zt*1−
_{t}

*t*1

*fs,x*1*s*−*fs, x*2*s*

*ds,*

*rtzt* *rt*1*ωzt*1*ω*−
*t*

*t*1*ω*

*fs,x*1*s*−*fs, x*2*s*

*ds*

*rt*1*zt*1−
_{t}

*t*1*ω*

*fs,x*1*s*−*fs, x*2*s*

*ds.*

2.53

By2.53and the fact that*zt*1 0, we have for any*t*∈*t*1*, t*1*ω,*

*rtzt* *rt*1*zt*1−1

2

*t*

*t*1

*fs,x*1*s*−*fs, x*2*s*

*ds*

1

2

_{t}_{1}_{}_{ω}

*t*

*fs,x*1*s*−*fs, x*2*s*

*ds.*

−1

2

_{t}

*t*1

*fs,x*1*s*−*fs, x*2*s*

*ds*

1

2

*t*1*ω*

*t*

*fs,x*1*s*−*fs, x*2*s*

*ds.*

It follows that for any*t*∈*t*1*, t*1*ω,*

*rtzt*≤ 1

2

_{t}_{1}_{}_{ω}

*t*1

*fs,x*1*s*−*fs, x*2*sds*

≤ 1

2

*ω*

0

_{f}_{}_{s,}_{}_{x}_{1}_{}_{s}_{}_{−}_{f}_{}_{s, x}_{2}_{}_{s}_{}_{ds}

≤ 1

2*bω*max0≤*t*≤*ω*|*zt*|*.*

2.55

We know that for any*t*∈0*, ω,*

*rtzt*≤ 1

2*bω*max0≤*t*≤*ω*|*zt*|*.* 2.56

From2.56, we have

max

0≤*t*≤*ω*

*zt*≤ *bω*

2

max

0≤*t*≤*ω*

1

*rt*

max

0≤*t*≤*ω*|*zt*|*.* 2.57

By2.52, we get

max

0≤*t*≤*ω*|*zt*| ≤
*ω*

2max0≤*t*≤*ωz*

_{}_{t}_{}_{.}_{}_{2.58}_{}

It is easy to see from2.57and2.58that

max

0≤*t*≤*ω*|*zt*| ≤
*bω*2

4

max

0≤t≤*ω*

1

*rt*

max

0≤*t*≤*ω*|*zt*|*.* 2.59

By condition iiiof Theorem 2.1, we see that*bω*2_{/}_{4}_{}_{max}

0≤*t*≤*ω*1*/rt* *<* 1*.* Thus 2.58

leads us to max0≤*t*≤*ω*|*zt*|0*,*which is contrary to*x*1*/x*2*.*So1.3has at most one*ω*-periodic

solution. The proof is complete.

We now turn to the proof ofTheorem 2.1. Suppose*ω*2_{δ}_{}_{max}

0≤*t*≤*ω*1*/rt* *<* 1*.* Let
*L, N, P,*and*Q*be defined by2.3,2.4,2.6, and2.7, respectively. ByLemma 2.10, there
are positive constants*D*0and*D*1such that for any*ω*-periodic solution*xt*of2.24such that

2.25holds. Set

Ω *x*∈*Xω*| *x* _{1}*< D,* 2.60

where*D*is a fixed number which satisfies*D > DD*0*D*1*.*It is easy to see thatΩis an open

and bounded subset of*Xω.*Furthermore, in view of Lemmas 2.5and 2.7,*L*is a Fredholm

mapping of index zero and*N*is*L*-compact onΩ. Noting that*D > D*0*D*1*,*byLemma 2.10,

constant:*xt*≡*D*or*xt*≡ −*D.*Hence byiand2.17,*xt*≡ −*D.*Hence by conditionsi,

iiiand2.17,

*QNxt* −*t*
*ω*

*ω*

0

*fs, xsds*−*t*
*ω*

*ω*

0

*fs, xds,* 2.61

so*QNx /θ*2*.*The isomorphism*J*: Im*Q* → Ker*L*is defined by*Jtα* *α*for*α*∈*R*and *t*∈*R.*

Then

*JQNx*−1
*ω*

_{ω}

0

*fs, xds*1

*ω/*0*.* 2.62

In particular, we see that if*xD,*then

*JQNx*−1
*ω*

_{ω}

0

*fs, Dds <*0*,* 2.63

and if*x*−*D,*then

*JQNx*−1
*ω*

*ω*

0

*fs,*−*Dds >*0*.* 2.64

Consider the mapping

*Hx, μμx*1−*μJQNx,* 0≤*μ*≤1*.* 2.65

From2.63and2.65, for each*μ*∈0*,*1and*xD,*we have

*Hx, μμD*1−*μ*−1
*ω*

*ω*

0

*fs, Dds <*0*.* 2.66

Similarly, from2.64and2.65, for each*μ*∈0*,*1and*x*−*D,*we have

*Hx, μμD*1−*μ*−1
*ω*

_{ω}

0

*fs,*−*Dds <*0*.* 2.67

By2.66and2.67,*Hx, μ*is a homotopy. This shows that

deg*JQNx,*Ω∩Ker*L, θ*1 deg−*x,*Ω∩Ker*L, θ*1*/*0*.* 2.68

By Theorem A, we see that equation *Lx* *Nx* has at least one solution in Ω∩Dom*L.*

In other words,1.3has an*ω*-periodic solution *xt.*Furthermore, ifiiiis satisfied, from

**3. Example**

Consider the equation

*xt*exp

−2−cos2*πt*
5

3−sin2*πt*
5

arctan*xt* cos2*πt*

5 *,* 3.1

and we can show that it has a nontrivial 5-periodic solution. Indeed, take

*rt* exp

2−cos2*πt*
5

*,* *pt* cos2*πt*
5 *,*

*ft, x* 1

100

3−sin2*πt*
5

arctan*x.*

3.2

We see that min0≤*t*≤5*rt* *e.*Let*D >*0 and*δb*1*/*25*.*Then conditioniofTheorem 2.1is

satisfied:

lim

*x*→ −∞max0≤*t*≤*ω*
*ft, x*

*x*

1

25*.* 3.3

Let*D >* 0 and*δ* *b* 1*/*25*.*Then conditionsi,iiandiii, ofTheorem 2.1are satisfied.
Note further that 52_{δ}_{}_{max}

0≤*t*≤*ω*1*/rt* *e*−1 *<*1*.*Therefore3.1has exactly one 5-periodic

solution. Furthermore, it is easy to see that any solution of3.1must be nontrivial. We have thus shown the existence of a unique nontrivial 5-periodic solution of3.1.

**Acknowledgment**

The first author is supported by the Natural Science Foundation of Guangdong Province of China under Grant no. 9151008002000012.

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