Existenceofweightedpseudoanti-periodicsolutionstosomeneutraldifferentialequationswithpiecewiseconstantargument*

,

(1. Department of Mathematics, Guangdong University of Petrochemical Technology, Maoming 525000, China; 2. Department of Mathematics, Huizhou University, Huizhou 516007, China)

Existenceofweightedpseudoanti-periodicsolutionstosomeneutraldifferentialequationswithpiecewiseconstantargument*

LINQuanwen1,ZHUANGRongkun2

(1. Department of Mathematics, Guangdong University of Petrochemical Technology, Maoming 525000, China; 2. Department of Mathematics, Huizhou University, Huizhou 516007, China)

By means of weighted pseudo anti-periodic solutions of relevant difference equations, the existence for weighted pseudo anti-periodic solutions of differential equations with piecewise constant argument is studied. The conditions of existence and uniqueness for the weighted pseudo anti-periodic solutions are presented.

pseudo anti-periodic solutions; pseudo anti-periodic sequences; neutral delay equation; piecewise constant argument

In this paper we consider the following first order neutral delay differential equations with piecewise constant argument of the forms

(1)

(2)

wherep(≠0),a0,a1areconstants, [·]denotesthegreatestintegerfunction.Tostudytheexistenceofweightedpseudoω-anti-periodic solutions to Eqs. (1) and (2), we will assume that the following assumptions hold:

(H1)f: R→R is weighted pseudoω-anti-periodic function.

(H2)g: R×R2→ R is jointly continuous and satisfiesg(t+ω,x,y) =-g(t,x,y) for allt∈ R and (x,y)∈R2. Moreover, the functiongis uniformly Lipschitz with respect tox,yin the following sense: there existsη> 0 such that

(3)

for all (xi,yi)∈ R2,i=1, 2andt∈R.

A functionx: R → R is called a solution of Eq. (1) if the following conditions are satisfied:

(i)xis continuous on R;

(ii) the derivative ofx(t) +px(t-1)existsonR except possibly at the pointst=n,n∈Z, where one-sided derivatives exist;

(iii)xsatisfies Eq. (1) on each interval (n,n+ 1) , with integern∈Z.

The existence of anti-periodic solutions to differential equations is an attractive topic in the qualitative theory of differential equations due to its applications in control theory or engineering and others, see [1-4] and references therein. Motivated by the study of existence of pseudo almost periodic solutions, and weighted pseudo almost solution to differential equations[5-7], Al-Islam[8]et al. introduced the weighted pseudo anti-periodic functions, which is a natural generalization of the classical pseudo almost periodic functions, and has been used in the investigation of a certain non-autonomous second-order abstract differential equation.

Differential equations with piecewise constant arguments are usually referred to as a hybrid system (a combination of continuous and discrete). These equations have the structure of continuous dynamical systems within intervals and the solutions are continuous, and so combine properties of both differential and difference equations. The equations are thus similar in structure to those found in certain sequential-continuous models of disease dynamics as treated by by Busenberg and Cooke[9]. Therefore, there are many papers concerning the differential equations with piecewise constant argument (see [10-19] and the references therein).

We note that there is no results on the weighted pseudo anti-periodic solution for Eq. (1) (or (2) ) still now. The main purpose of this work is to establish an existence and uniqueness result of weighted pseudo anti-periodic solutions of Eqs. (1) and (2).

1 Preliminary definitions and lemmas

For the sake of convenience, we now state some of the preliminary definitions and lemmas. we always denote byBC(R, R) the space of bounded continuous functionsu: R → R,C(R, R) the space of continuous functionsu: R → R, and denote by |·| the Euclidean norm.

Definition 1 A functionf∈C(R,R) is said to beω-anti-periodic function for someω> 0, iff(t+ω)=-f(t) for allt∈R. The least positiveωwith this property is called the anti-period off.DenotebyAPω(R) the set of all such functions.

Proposition 1 Iff(t) is anω-anti-periodic function, thenf(t) is also (2ω+1) -anti-periodic and 2ω-periodic.

LetUbe the collection of functions (weights)ρ: R → (0, +∞), which are locally integrable over R. Ifρ∈U, we set

and

Obviously,UB?U∞?U,withstrictinclusions.

Letρ1,ρ2∈U∞,ρ1issaidtobeequivalenttoρ2,denotingthisasρ1ρ2, ifρ1/ρ2∈UB.ThenisabinaryequivalencerelationonU∞(see [7]). Letρ∈U∞,c∈R, defineρcbyρc(t)=ρ(t+c)fort∈R. We denote

UT{ρ∈U∞:ρρcfor eachc∈R}

It is easy to see thatUTcontainsplentyofweights,say, 1, et, 1 + 1/(1 +t2), 1 + |t|nwithn∈N, etc.

Forρ∈U∞,theweightedergodicspacePAP0(R,ρ)isdefinedby

Lemma 1[14]PAP0(R,ρ)withρ∈UTistranslationinvariant,i.e.φ∈PAP0(R,ρ)ands∈Rimplythatφ(·-s)∈PAP0(R,ρ).

Definition 2[7]Letρ∈U∞. A functionf∈BC(R, R) is called weightedω-anti-periodic function (orρ-pseudoω-anti-periodic function) for someω> 0, iffcanbewrittenasf=fap+fe, wherefap∈APω(R), andfe∈PAP0(R,ρ).fapandfeare called theω-anti-periodic component and the weighted ergodic perturbation, respectively, of the functionf. Denote byPAPω(R,ρ)thesetofallsuchfunctions.

Definition 3 Letρ∈U∞.Afunctiong∈BC(R×R)is called weighted pseudoω-anti-periodic function (orρ-pseudoω-anti-periodic function) intuniformly on R2,ifgcan be written asg=gap+ge, wheregapisω-anti-periodic in t uniformly for R2, and for any compact setW?R2,geiscontinuous,boundedandsatisfies

uniformlyin(x,y)∈W,gapandgearecalledtheω-anti-periodic component and the weighted ergodic perturbation, respectively, of the functiong.DenotebyPAP(R×R,R,ρ)thesetofallsuchfunctions.

Definition 4 A sequencex: Z → R, denoted by {x(n)}, is called aω-anti-periodic sequence ifx(n+ω)=-x(n) for alln∈Z.We denote the set of all such sequences byAPωS(R).

LetUsdenote the collection of sequences (weights)Q:Z→(0,+∞).ForQ∈UsandT∈Z+={n∈ Z:n≥ 0}, set

Denote

and

UsB

LetQ1,Q2∈Us∞,Q1issaidtobeequivalenttoQ2,denotingthisasQ1Q2,if{Q1(n)/Q2(n)}n∈z∈UsB.ThenitiseasytoseethatisabinaryequivalencerelationonUs∞.LetQ∈Us∞,k∈Z,defineQkbyQk(n)=Q(n+k)forn∈Z.Wedenote

ForQ∈Us∞,theweightedergodicsequencesspacePAP0S(R,Q)isdefinedby

PAP0S(R,Q)

Definition 5 LetQ∈Us∞.Asequencex:Z→R, is called a weighted pseudoω-anti-periodicsequence(orQ-pseudoω-anti-periodicsequence)ifxcanbewrittenasx(n)=xap(n)+xe(n),n∈Zwherexap∈APωS(R), andxe∈PAP0S(R,Q).xapandxearecalledtheω-anti-periodiccomponentandtheweightedergodicperturbation,respectively,ofthesequencex.DenotethesetofallsuchsequencesbyPAPωS(R,Q).

Proposition 2 Iff∈APω(R),ω∈Z+,then{f(n)}n∈Z∈APωS(R).

Proof Sincef(t) is anω-anti-periodic function, then for allt∈ R, we havef(t+ω) +f(t) = 0 and

From definition, it follows that {hn}n∈Zisanω-anti-periodic sequence. This completes the proof of Lemma 2.

Lemma 3 Letρ∈UT, and denote

forn∈Z

(4)

ThenQ∈UsT. Moreover, givenc∈R, there exist positive constantsC1,C2such that, for sufficiently largeT,

(5)

Proof Without loss of generality, we assume thatc≥ 0. Sinceρ∈UT,thereexistsM> 0 such thatρc+1(t) ≤Mρ(t) andρ-(c+1)(t) ≤Mρ(t) fort∈ R and

(6)

Notice that

ForT>c+2,i.e.,-T+2c+3

(7)

Similarly,wecanprovethatthereexistsM′> 0 such that, forTlarge enough,

(8)

Thus by (6)-(8), we have

forTlarge enough. This leads to (5), and from which we can get easily thatQ∈UsT. The proof is complete.

Proposition 3PAP0S(R,Q)withQ∈UsTis translation invariant.

This implies that {x(n-k)}n∈Z∈PAP0S(R,Q).Theproofiscomplete.

Itisclearthat|hn|≤‖f‖forn∈Zand

isω-anti-periodic.Let

ForT∈ Z+, we get

ForT∈Z+,s∈ [1, 1], let

Lemma 5[11]Letx:R→R is a continuous function, andw(t)=x(t)+px(t-1).then

t≥t0

Where|p|<1,a=log (1/|p|),b=1/(1-|p|),or

t≤t0

Where|p|>1,b=1/(|p|-1).

2 Main results

Now,wecanformulateourmaintheorems.

Theorem 1 Suppose that

(9)

Then for anyf∈PAPωS(R,ρ), the following results hold:

(i) Ifω=n0∈Z+, Eq. (5) has a uniqueρ-pseudoω-anti-periodic bounded solution.

Theorem 2 Suppose that conditions (H2) and (9) hold. Then there existsη*>0,suchthatifη<η*,thatfollowingresultshold:

(i)Ifω=n0∈ Z+,Eq. (5)hasauniqueρ-pseudoω-anti-periodic bounded solution.

3 Proofs of theorems

Proof of Theorem 1 (i) Letx(t)beasolutionofEq.(1)onR, integrating (1) fromntot, we have that forn≤t

(10)

In view of the continuity of a solution at a point, we obtain that fort→(n+1)-0,

(11)

ThecorrespondinghomogeneousequationofEq. (11)is

(12)

Following[10],weseektheparticularsolutionsasx(n)=λnforhomogeneousdifferenceequation(12),thenwehavethefollowingcharacteristicequationof(12):

(13)

Eq. (13)hastwonontrivialsolutions

Inviewof(9),wehavethat|λ1,2|≠1andλ1≠λ2,then

(14)

isthegeneralsolutionsofEq.(12),wherek1,k2are any constants.

We define a sequence {cn} by

(15)

wherek1,k2will be determined later. We put Eq.(15) into Eq.(11) and compare the coefficients ofhn’s.

For |λ1|<1, |λ2|<1,weobtainalinearsystemink1andk2

(16)

Solving system (16), we have

(17)

is a solution of the difference equation (11).

For other cases we can similarly write out the expression for the solution of Eq.(11).

(ii) Sincef∈PAPωS(R,ρ),itfollowsfromLemma4that{hn}n∈Z∈PAPωS(R,Q),sothat{hn} can be written as a sum

It is easy to see that

Indeed, it is easy to see that forT∈ Z+,

Form∈Z+,let

FromProposition1,weget

(18)

Givenε> 0, it is clear that there exists an integerK>0suchthat

(19)

Thenby(18),thereexistsT0> 0 such that forT>T0,

for0≤m≤K

(20)

Now by (18)-(20), forT>T0we have

withx(s)=φ(s),-1≤s≤0.

(21)

Letw(t)=x(t)+px(t-1),weclaimthatw(t)∈PAP0(R,Q).Letf=fap+fe,wherefap∈APω(R),fe∈PAP0(R,ρ),let

Wehave

Itfollowsfromdefinitionthatwap(t)isω-anti-periodic. Denote

Then {ηn}∈PAP0(R,Q).ByanargumentthesameastheproofLemma4wegetthat

Meanwhile,itfollowsfromLemma3thatthereexistssomeM> 0 such thatμs([T] + 1,Q) ≤Mμ(T,ρ) forTlarge enough. Then forTlarge enough we have

asT→∞

Thisimpliesthatwe∈PAP0(R,ρ),andhencew∈PAPω(R,ρ).

Nextweexpressxin terms ofwand then prove thatx∈PAPω(R,ρ).From

Onehasforalln∈Z+

(22)

hence,

It follows

Conversely, if we put

xis well defined andwis bounded and |p|<1,xisboundedwith|x(t)| ≤ ‖w(t)‖∞/(1-|p|),moreoveronehas

For|p|<1,rewriting(22)as

wededuceinasimilarmannerthat

If|p|<1,givenε> 0, there exists an integerK> 0 such that

(23)

Let

By a standard argument we can get thatxap∈APω(R). SincePAP0(R,ρ)withρ∈UTis translation invariant, namelyφ∈PAP0(R,ρ)ands∈Rimplythatφ(·-s)∈PAP0(R,ρ) (see[14,Lemma4.1]),wegetthatwe(· -n)∈PAP0(R,ρ)forn∈Z+. So there existsT0> 0 such that forT>T0,

for0≤n≤K

(24)

Now by (23) and (24), forT>T0,

This implies thatxe∈PAP0(R,ρ), andx∈PAPω(R,ρ). If |p|>1,let

Similarlytotheabove,wecanprovethatx∈PAPω(R,ρ).

?t∈R

Clearly,wehave

Fromtheboundednessoftheρ-pseudoω-anti-periodic function, it follows that

This means that theρ-pseudoω-anti-periodic solution of Eq. (1) is unique.

Proof of Theorem 2 (i) It is easy to seen that the spacePAPω(R,ρ)isaBanachspacewithsupremumnorm‖φ‖=supt∈R|φ(t)|.Foranyφ∈PAPω(R,ρ),usingboth(H2)andthecompositionoffunctionsinPAP0(R,ρ) (seeDiagana[19]),itfollowsthatg(t,φ(t),φ([t]))∈PAPω(R,ρ).

Weconsiderthefollowingequation:

(25)

FromTheorem1,itfollowsthatforanyφ∈PAPω(R,ρ), Eq. (25) has a unique weighted pseudo-anti-ω-periodic solution, denote byJφ. Thus, we obtain a mappingJ:φ→xφ,itfollowsthatJis a mapping fromPAPω(R,ρ)intoitself.Foranyφ,ψ∈PAPω(R,ρ),Jφ-Jψsatisfiesthefollowingequation:

where

ThisimpliesthatthereexistsK0> 0, such that

?n∈Z

Let

ThusthereexistsK1> 0 such that

We easily conclude that

We typically consider the case when |p|<1.UsingLemma5,wehave

wherea=log(1/|p|),b=1/(1-|p|). Settingt0→∞,weobtain

Hence,thereexistsη*>0, such that if 0≤η<η*,J:PAPω(R,ρ)→PAPω(R,ρ)iscontractingmapping.Thisimpliesthatthereexistsφ∈PAPω(R,ρ)suchthatJφ=φthatis,Eq. (1)hasauniqueρ-pseudoω-anti-periodic solution.

(ii) Ifω=(n0/m0)(n0,m0∈Z+)andgisρ-pseudoω-anti-periodic int, theng(t,φ(t),φ([t]))isaρ-pseudom0ω-anti-periodicfunction,foranyφ∈PAPm0 ω(R,ρ).Atthistime,itfollowsthatEq. (25)hasauniqueρ-pseudom0ω-anti-periodicsolutionJφbyusingTheorem1.Similarly,weknowthatthereexistsη*>0suchthatif0≤η<η*,Eq.(1)hasauniqueρ-pseudom0ω-anti-periodic solution. This completes the proof of Theorem 2.

[1] OKOCHI H. On the existence of periodic solutions to nonlinear abstract parabolic equations [J]. J Math Soc Japan, 1988, 40 (3): 541-553.

[2] AIZICOVICI S, MCKIBBEN M, REICH S. Anti-periodic solutions to nonmonotone evolution equations with discontinuous nonlinearities [J]. Nonlinear Anal, 2001, 43:233-251.

[3] CHEN Y, NIETO J J, O’REGAN D. Anti-periodic solutions for fully nonlinear first-order differential equations [J]. Math Comput Model, 2007, 46:1183-1190.

[4] CHEN T, LIU W, ZHANG J, et al. The existence of anti-periodic solutions for Linard equations [J]. J Math Study, 2007, 40:187-195 (in Chinese).

[5] ZHANG C Y. Pseudo almost periodic solutions of some differential equations [J]. J Math Anal Appl, 1994, 181: 62-76.

[6] ZhANG C Y. Pseudo-almost periodic solutions of some differential equations II [J]. J Math Anal Appl, 1995, 192 (2) : 543-561.

[7] DIAGANA T. Weighted pseudo-almost periodic functions and applications [J]. C R Acad Sci Paris, Ser I, 2006,343 (10): 643-646.

[8] Al-ISLAM N S, ALSULAMI S M, DIAGANA T. Existence of weighted pseudo anti-periodic solutions to some non-autonomous differential equations [J]. Applied Mathematics and Computation, 2012, 218: 6536-6548.

[9] BUSENBERG S, COOK K L. Models of vertically transmitted diseases with sequential-continuous dynamics [M]∥LAKSHMIKANTHAM V, ed. Nonlinear Phenomena in Mathematical Sciences. New York: Academic Press, 1982.

[10] COOK K L, WIENER J. A survey differential equation with piecewise continuous argument [M]∥BUSENBERG S, MARTELLI, eds. Lecture Notes in mathematics, Vol. 1475. Berlin: Springer, 1991:1-15.

[11] YUAN R. On the existence of almost periodic solutions of second order delay differential equations with piecewise constant argument [J]. Sci China, 1998, 41(3): 232-241.

[12] YUAN R. Pseudo-almost periodic solutions of second order neutral delay differential equations with piecewise constant argument [J]. Nonlinear Anal, 2000, 41: 871-890.

[13] ZHAN R L L, LI H X. Weighted pseudo-almost periodic solutions for some abstract differential equations with uniform continuity [J]. Bull Aust Math Soc, 2010, 82: 424-436.

[14] ZHANG L L, LI H X. Weighted pseudo-almost periodic solutions of second order neutral differential equations with piecewise constant argument [J]. Nonlinear Anal, 2011,74: 6770-6780.

[15] ZHUANG R K, YUAN R. The existence of pseudo-almost periodic solutions of third-order neutral differential equations with piecewise constant argument [J]. Acta Math Sin (Engl Ser), 2013,29(5): 943-958.

[16] ZHUANG R K, YUAN R. Weighted pseudo almost periodic solutions of N-th order neutral differential equations with piecewise constant arguments [J]. Acta Math Sin (Engl Ser), 2014,30(7): 1259-1272.

[17] ZHANG L, XU Y. Existence of almost periodic solutions for some nonlinear Duffing equations [J]. Acta Scientiarum Naturalium Universitatis Sunyatseni, 2010, 49(4): 6-10.

[18] ZHANG L. Almost periodic mild solutions to functional differential equations of neutral type in Banach space [J]. Acta Scientiarum Naturalium Universitatis Sunyatseni, 2016, 55(1): 30-34.

[19] DIAGANA T. Pseudo almost periodic functions in Banach spaces [M]. New York: Nova Science Publishers Inc, 2007.

2017-02-22 基金項目：國家自然科學(xué)基金(11271380，11501238) ；廣東省自然科學(xué)基金(2014A030313641，2016A030313119)；廣東省教育廳重大項目基金(2014KZDXM070)

林全文(1965年生)，男；研究方向：微分方程與動力系統(tǒng)；E-mail：linquanwen@126.com

O175

A

0529-6579(2017)03-0057-09

具有分段常變量的中立型微分方程加權(quán)偽反周期解的存在性

林全文1，莊容坤2

(1. 廣東石油化工學(xué)院數(shù)學(xué)系，廣東 茂名 525000； 2. 惠州學(xué)院數(shù)學(xué)系，廣東 惠州 516007)

通過構(gòu)造差分方程的加權(quán)偽反周期解，研究了一類含分段常變量中立型微分方程的加權(quán)偽反周期解的存在性，給出了所論方程的加權(quán)偽反周期解的存在唯一性條件。

偽反周期解；偽反周期序列；中立型時滯方程；分段常變量

10.13471/j.cnki.acta.snus.2017.03.009