Existence of periodic solution for generalized neutral Rayleigh equation with variable parameter

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R E S E A R C H

Open Access

Existence of periodic solution for

generalized neutral Rayleigh equation with

variable parameter

Yun Xin

*

_{and Shan Zhao}

*_{Correspondence:}

[email protected]

College of Computer Science and Technology, Henan Polytechnic University, Jiaozuo, 454000, China

Abstract

In this paper, we consider a generalized neutral Rayleigh equation with variable parameter

(x(t) –c(t)x(t–

δ(t))

+f

(t,

x(t))+g(t,x(t–

τ

(t)))=e(t),

where|c(t)| = 1,c,

δ

∈C1₍_R_,_R_{) and}_c,

_δ

_are

_{ω-periodic functions for some}

_ω

_{> 0.}

By applications of coincidence degree theory and some analysis skills, suﬃcient conditions for the existence of periodic solution is established.

Keywords: neutral operator; periodic solution; Rayleigh equation; variable parameter

1 Introduction

In the paper, we consider the generalized neutral Rayleigh diﬀerential equation with vari-able parameter

x(t) –c(t)xt–δ(t)+ft,x(t)+gt,xt–τ(t)=e(t), (.) where|c(t)| = ,c,δ∈C(R,R) andc, δareω-periodic functions for someω> ,τ,e∈ C[,ω] and_ωe(t)dt= ;f andgare continuous functions deﬁned onR_{and periodic in} twithf(t,·) =f(t+ω,·),g(t,·) =g(t+ω,·), andf(t, ) = .

Neutral differential equations manifest themselves in many fields including biology, me-chanics and economics [–]. For example, in population dynamics, since a growing popu-lation consumes more (or less) food than a matured one, depending on individual species, this leads to neutral equations []. These equations also arise in classical ‘cobweb’ models in economics where current demand depends on price, but supply depends on the previ-ous periodic []. The study on neutral differential equations is more intricate than that on ordinary delay differential equations. In recent years, there is a good amount of work on periodic solutions for neutral differential equations (see [–] and the references cited therein). For example, in [], Luet al.considered the following neutral differential equa-tion with deviating arguments:

x(t) –cx(t–δ)+fx(t)+fxt–τ(t)=p(t).

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By using the continuation theorem and some analysis techniques, some new results on the existence of periodic solutions are obtained. Afterwards, Duet al.[] investigated the second order neutral equation

x(t) –c(t)x(t–δ))+fx(t)x(t) +gxt–γ(t)=e(t), (.)

by using Mawhin’s continuous theorem, the authors obtained the existence of periodic solution to (.). Recently, Renet al.[] studied the neutral equation with variable delay

x(t) –cxt–δ(t)= –a(t)x(t) +λb(t)fxt–τ(t), (.)

by an application of the ﬁxed-point index theorem, the authors obtained suﬃcient con-ditions for the existence, multiplicity and nonexistence of positive periodic solutions to (.).

Motivated by [, , ], in this paper, we consider the generalized neutral Rayleigh equa-tion (.). Notice that here the neutral operatorAis a natural generalization of the familiar operatorA=x(t) –cx(t–δ),A=x(t) –c(t)x(t–δ),A=x(t) –cx(t–δ(t)). ButApossesses a more complicated nonlinearity thanAi,i= , , . For example, the neutral operatorA

is homogeneous in the following sense (Ax)(t) = (Ax)(t), whereas the neutral operator

Ain general is inhomogeneous. As a consequence, many of the new results for diﬀerential equations with the neutral operatorAwill not be a direct extension of known theorems for neutral diﬀerential equations.

The paper is organized as follows. In Section , we ﬁrst analyze qualitative properties of the generalized neutral operatorAwhich will be helpful for further studies of diﬀerential equations with this neutral operator; in Section , by Mawhin’s continuation theorem, we obtain the existence of periodic solutions for the generalized neutral Rayleigh equation with variable parameter. We will give an example to illustrate our results, and an example is also given in this section. Our results improve and extend the results in [, , –].

2 Analysis of the generalized neutral operator with variable parameter

Let

c∞= max

t∈[,ω]

c(t), c= min t∈[,ω]

c(t).

LetX={x∈C(R,R) :x(t+ω) =x(t),t∈R}with normx=max_t_∈[,ω]|x(t)|. Then (X,

· ) is a Banach space. Moreover, deﬁne operatorsA,B:Cω→Cωby

(Ax)(t) =x(t) –c(t)xt–δ(t), (Bx)(t) =c(t)xt–δ(t).

Lemma . If|c(t)| = ,then the operator A has a continuous inverse A–_{on C}

ω,satisfying

()

A–f(t) =

⎧ ⎪ ⎨ ⎪ ⎩

f(t) +∞_j=_i=j c(Di)x(t–

j

i=δ(Di)) for|c(t)|< ,∀f ∈Cω,

–f(t+_c(t+_δδ(t))_(t))–∞_j=f(t+δ(t)+

j

i=δ(Di))

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≤

By deﬁnition of the linear operatorB, we have

hereDiis deﬁned as in Case . Summing overjyields

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On the other hand, from (Ax)(t) =x(t) –c(t)x(t–δ(t)), we have

(Ax)(t) =x(t) –c(t)xt–δ(t)= –c(t)

xt–δ(t)– 

c(t)x(t)

,

i.e.,

(Ax)(t) = –c(t)(Ex)t–δ(t).

Letf ∈Cωbe arbitrary. We are looking forxsuch that

(Ax)(t) =f(t),

i.e.,

–c(t)(Ex)t–δ(t)=f(t).

Therefore

(Ex)(t) = –f(t+δ(t))

c(t+δ(t))=:f(t), and hence

x(t) =E–f

(t) =f(t) +

∞

j=

Bj_f

(t) = –f(t+δ(t))

c(t+δ(t))–

∞

j=

Bj_f(t+δ(t)) c(t+δ(t)),

proving thatA–_{exists and satisﬁes}

A–f(t) = –f(t+δ(t))

c(t+δ(t))–

∞

j=

Bj_f(t+δ(t)) c(t+δ(t))

= –f(t+δ(t))

c(t+δ(t))–

∞

j=

f(t+δ(t) +j_i=δ(D_i))

c(t+δ(t))j_i=c(D_i) and

A–f(t)=

–

f(t+δ(t))

c(t+δ(t))–

∞

j=

f(t+δ(t) +j_i=δ(D_i))

c(t+δ(t))j_i=c(D_i)

≤

f c– .

Statements () and () are proved. From the above proof, () can easily be deduced.

3 Periodic solution for (1.1)

We ﬁrst recall Mawhin’s continuation theorem which our study is based upon. LetXand

Y be real Banach spaces andL:D(L)⊂X→Y be a Fredholm operator with index zero, hereD(L) denotes the domain ofL. This means thatImLis closed inY anddim KerL=

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natural projections. Clearly,KerL∩(D(L)∩X) ={}, thus the restrictionLP:=L|D(L)∩X

is invertible. LetL–_P_denote the inverse ofLP.

Letbe an open bounded subset ofXwithD(L)∩=∅. A mapN:→Yis said to be

L-compact inifQN() is bounded and the operatorL–

P(I–Q)N:→Xis compact.

Lemma .(Gaines and Mawhin []) Suppose that X and Y are two Banach spaces,and L:D(L)⊂X→Y is a Fredholm operator with index zero.Furthermore,⊂X is an open bounded set and N:→Y is L-compact on.Assume that the following conditions hold:

() Lx=λNx,∀x∈∂∩D(L),λ∈(, ); () Nx∈/ImL,∀x∈∂∩KerL;

() deg{JQN,∩KerL, } = ,whereJ:ImQ→KerLis an isomorphism. Then the equation Lx=Nx has a solution in∩D(L).

In order to use Mawhin’s continuation theorem to study the existence ofω-periodic solutions for (.), we rewrite (.) in the following form:

(Ax)(t) =x(t),

x_(t) = –f(t,x_(t)) –g(t,x(t–τ(t))) +e(t).

(.)

Clearly, ifx(t) = (x(t),x(t))is anω-periodic solution to (.), thenx(t) must be anω -periodic solution to (.). Thus, the problem of ﬁnding anω-periodic solution for (.) reduces to ﬁnding one for (.).

Recall thatCω={φ∈C(R,R) :φ(t+ω)≡φ(t)}with normφ=maxt∈[,ω]|φ(t)|. Deﬁne X=Y =Cω×Cω={x= (x(·),x(·))∈C(R,R) :x(t) =x(t+ω),t∈R}with normx= max{x,x}. Clearly,XandYare Banach spaces. Moreover, deﬁne

L:D(L) =x∈CR,R:x(t+ω) =x(t),t∈R⊂X→Y

by

(Lx)(t) =

(Ax)(t)

x_(t)

andN:X→Yby

(Nx)(t) =

x(t)

–f(t,x_(t)) –g(t,x(t–τ(t))) +e(t)

. (.)

Then (.) can be converted to the abstract equationLx=Nx. From the deﬁnition ofL, one can easily see that

KerL∼=R, ImL=

y∈Y:

ω



y



(

s

)

y



(

s

)

ds=



.

SoLis a Fredholm operator with index zero. LetP:X→KerLandQ:Y→ImQ⊂R

be deﬁned by

Px=

(Ax)()

x()

; Qy= 

ω



y(s)

y(s)

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thenImP=KerL,KerQ=ImL. SetLP=L|D(L)∩KerPandL–P:ImL→D(L) denotes the

inverse ofLP, then

L–_P_y(t) =

(A–Fy)(t) (Fy)(t)

,

[Fy](t) =

t



y(s)ds, [Fy](t) =

t



y(s)ds. (.)

From (.) and (.), it is clear thatQNandL–_P_(I–Q)Nare continuous, andQN() is bounded, and thenL–_P_(I–Q)N() is compact for any open bounded⊂X, which meansNisL-compact on¯.

For the sake of convenience, we list the following assumptions which will be used re-peatedly in the sequel:

(H) there exists a positive constantKsuch that|f(t,u)| ≤Kfor(t,u)∈R×R;

(H) there exists a positive constantDsuch thatx·g(t,x) > and|g(x,x)|>Kfor|x|>D; (H) there exists a positive constantMandM>esuch thatg(t,x)≥–Mforx≤–Dand

t∈R;

(H) there exists a positive constantMandM>esuch thatg(t,x)≤Mforx≥Dand t∈R;

Now we give our main results on periodic solutions for (.).

Theorem . Assume that conditions(H)-(H)hold.Suppose that one of the following conditions is satisﬁed:

(i) Ifc∞< and –c∞–δc∞–cω> ; (ii) Ifc> andc–  –δc∞–cω> ; whereδ=maxt∈[,ω]|δ(t)|,c=maxt∈[,ω]|c(t)|.

Then(.)has at least one solution with periodω.

Proof By construction, (.) has anω-periodic solution if and only if the following operator equation

Lx=Nx

has anω-periodic solution. From (.), we see thatNisL-compact on¯, whereis any open, bounded subset ofCω. Forλ∈(, ], deﬁne

={x∈Cω:Lx=λNx}.

Thenx= (x,x)∈satisﬁes

⎧ ⎨ ⎩

(Ax)(t) =λx(t),

x_(t) = –λf(t,x_(t)) –λg(t,x(t–τ(t))) +λe(t). (.) Substitutingx(t) = _λ(Ax)(t) into the second equation of (.) yields



λ(Ax)(t)

= –λft,x_(t)–λgt,x

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i.e.,

(Ax)(t)= –λft,x_(t)–λgt,xt–τ(t)+λe(t). (.) We ﬁrst claim that there is a constantξ∈Rsuch that

x(ξ)≤D. (.)

Integrating both sides of (.) over [,ω], we have

ω



ft,x_(t)+gt,xt–τ(t)dt= , (.)

which yields that there at least exists a pointtsuch that

ft,x_(t)+gt,xt–τ(t)= , then by (H) we have

gt,xt–τ(t)=–ft,x_(t)≤K,

and in view of (H) we get that|x(t–τ(t))| ≤D. Sincex(t) is periodic with periodicω.

Sot–τ(t) =nω+ξ,ξ∈[,ω], wherenis some integer, then|x(ξ)| ≤D. Equation (.) is proved.

Then we have

_x₍_t₎₌_x₍ξ) +

t

ξ

x(s)ds≤D+

t

ξ

_x₍_s₎_ds_, _t_∈_[_ξ_,_ξ₊_ω_],

and

x(t)=x(t–ω)=x(ξ) –

ξ

t–ω

x(s)ds≤D+

ξ

t–ω

x(s)ds, t∈[ξ,ξ+ω].

Combining the above two inequalities, we obtain

|x|= max t∈[,ω]

_x₍_t₎₌ max t∈[ξ,ξ+ω]

_x₍_t₎_≤ max t∈[ξ,ξ+ω]

D+ 

t

ξ

_x₍_s₎_ds₊ ξ

t–ω

_x₍_s₎_ds

≤D+ 

ω



_x₍_s₎_ds_. _(.)

On the other hand, multiplying both sides of (.) by (Ax)(t) and integrating over [,ω], we get

ω



(Ax)(t)Ax(t)dt= –

ω



(Ax)(t)dt= –λ

ω



ft,x_(t)(Ax)(t)dt

–λ

ω



gt,xt–τ(t)(Ax)(t)dt

+λ

ω



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Using (H), we have

Besides, we can assert that there exists some positive constantNsuch that

ω With these sets we get

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That is,

ω



gt,xt–τ(t)dt=

E

gt,xt–τ(t)dt+

E

gxt–τ(t)dt

≤

E gt,x

t–τ(t)dt+Kω

≤ωmax

M, sup t∈[,ω],|x(t–τ(t))|<D

g(t,x)+Kω

= ωN+Kω, whereN=max{M,sup_t_∈_[,_ω_],_|_x

(t–τ(t))|<D|g(t,x)|}, proving (.).

Substituting (.) into (.) and recalling (.), we get

ω



(Ax)(t)dt≤( +c∞)xKω+ ωN+ωe

≤( +c∞)Kω+ ωN+ωeD+



ω



x_(t)dt

= ( +c∞)ND+ ( +c∞)N



ω



x_(t)dt, (.)

whereN= Kω+ ωN+ωe. Since (Ax)(t) =x(t) –c(t)x(t–δ(t)), we have

(Ax)(t) =x(t) –c(t)xt–δ(t)

=x_(t) –c(t)xt–δ(t)–c(t)x_t–δ(t) –δ(t)

=x_(t) –c(t)xt–δ(t)–c(t)x_t–δ(t)+c(t)x_t–δ(t)δ(t) =Ax_(t) –c(t)xt–δ(t)+c(t)x_t–δ(t)δ(t),

and

Ax_(t) = (Ax)(t) +c(t)xt–δ(t)–c(t)x_t–δ(t)δ(t). Case (i): Ifc∞< , by applying Lemma ., we have

ω



x_(t)dt=

ω



A–Ax_(t)dt

≤ ω

 |(Ax)(t)|dt

 –c∞

=

ω

 |(Ax)(t) +c(t)x(t–δ(t)) –c(t)x(t–δ(t))δ(t)|dt

 –c∞ ≤

ω

 |(Ax)(t)|dt+

ω

 |c(t)x(t–δ(t))|dt+

ω

 |c(t)x(t–δ(t))δ(t)|dt

 –c∞ ≤

ω

 |(Ax)(t)|dt+cωx+c∞δ

ω

 |x(t)|dt

 –c∞ ≤

ω

 |(Ax)(t)|dt+cωD+ (_cω+c∞δ)

ω

 |x(t)|dt

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wherec=maxt∈[,ω]|c(t)|,δ=maxt∈[,ω]|δ(t)|. Since  –c∞–cω–c∞δ> , so we get

It is easy to see that there exists a constantM>  (independent ofλ) such that

ω



x_(t)dt≤M.

It follows from (.) that

x ≤D+ equation of (.) we obtain

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So, from (H) and (.), we have

x ≤

ω



ft,x_(t)dt+

ω



gt,xt–τ(t)dt+

ω



e(t)dt

≤Kω+ ωN+ωe:=M.

LetM=

M

+M+ ,={x= (x,x):x<M,x<M}, then∀x∈∂∩KerL

QNx= 

ω



x(t)

–f(t,x_(t)) –g(t,x(t–τ(t))) +e(t)

dt.

IfQNx= , thenx(t) = ,x=Mor –M. But ifx(t) =M, we know

 =

ω



g(t,M)dt,

there exists a pointtsuch thatg(t,M) = . From assumption (H), we knowM≤D,

which yields a contradiction. Similarly if x = –M. We also haveQNx= ,i.e., ∀x∈

∂∩KerL,x∈/ ImL, so conditions () and () of Lemma . are both satisﬁed. Deﬁne the isomorphismJ:ImQ→KerLas follows:

J(x,x)= (–x,x).

LetH(μ,x) =μx+ ( –μ)JQNx, (μ,x)∈[, ]×, then∀(μ,x)∈(, )×(∂∩KerL),

H(μ,x) =

μx(t) +–_ωμ_ω[f(t,x_(t)) +g(t,x(t–τ(t))) –e(t)]dt

(μ+ ( –μ))x(t)

.

We have_ωe(t)dt= . So, we can get

H(μ,x) =

μx(t) +–_ωμ_ω[f(t,x_(t)) +g(t,x(t–τ(t)))]dt

(μ+ ( –μ))x(t)

,

∀(μ,x)∈(, )×(∂∩KerL).

From (H), it is obvious thatxH(μ,x) > ,∀(μ,x)∈(, )×(∂∩KerL). Hence

deg{JQN,∩KerL, }=degH(,x),∩KerL,  =degH(,x),∩KerL,  =deg{I,∩KerL, } = .

So condition () of Lemma . is satisﬁed. By applying Lemma ., we conclude that equa-tionLx=Nxhas a solutionx= (x,x)on¯ ∩D(L),i.e., (.) has anω-periodic solution

x(t).

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Theorem . Assume that conditions(H), (H), (H)hold.Suppose one of the following conditions is satisﬁed:

(i) Ifc∞< and –c∞–δc∞–cω> ; (ii) Ifc> andc–  –δc∞–cω> . Then(.)has at least one solution with periodω.

Remark . If_ωe(t)dt=  andf(t, )= , the problem of existence ofω-periodic solu-tions to (.) can be converted to the existence ofω-periodic solutions to the equation

x(t) –c(t)xt–δ(t)+f

t,x(t)+g

t,xt–τ(t)=e(t), (.)

where f(t,x) = f(t,x) –f(t, ),g(t,x) =g(t,x) + _ω_ωe(t)dt+f(t, ) and e(t) =e(t) –



ω

 e(t)dt. Clearly,

ω

 e(t)dt=  andf(t, ) = , (.) can be discussed by using

Theo-rem . (or TheoTheo-rem .).

Example . Consider the following equation:

x(t) – 

sin(t)x

t–  sint

+costsinx(t)

+arctan

x(t–sint)  +cos_(_t₎

=cost. (.)

Comparing (.) to (.), we have ω= π_, f(t,u) =costsinu, g(t,x) =arctan x +cos_(t),

c(t) = _ sint, δ(t) = _ sint, τ(t) =sint, e(t) =cost and δ =maxt∈[,π_]| × cost|=_,c∞=maxt∈[,π_]| sint|=



< ,c=maxt∈[,π_]| cost|= 

. We can

easily chooseK= ,D>π_ andM=π_ such that (H)-(H) hold. And

 –c∞–δc∞–

cω=  –  –

 ×

 –

 ×

 ×

π

 > . Hence, by Theorem ., (.) has at least one π_-periodic solution.

Competing interests

The authors declare that they have no competing interests.

Authors’ contributions

YX and SZ worked together in the derivation of the mathematical results. Both authors read and approved the ﬁnal manuscript.

Acknowledgements

YX and SZ would like to thank the referee for invaluable comments and insightful suggestions. This work was supported by NSFC project (No. 11326124) and the Fundamental Research Funds for the Universities of Henan Province

(NSFRF140142).

Received: 13 November 2014 Accepted: 29 May 2015

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