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Zhouetal.
AdvancesinDierenceEquations(2019)2019:335https://doi.
org/10.
1186/s13662-019-2242-xRESEARCHOpenAccessAgraph-theoreticmethodtostudytheexistenceofperiodicsolutionsforacoupledRayleighsystemviainequalitytechniquesZhengZhou1*,HuayingLiao2andZhengqiuZhang3*Correspondence:zhouzhengslx@163.
com1SchoolofAppliedMathematicalScience,XiamenUniversityofTechnology,Xiamen,ChinaFulllistofauthorinformationisavailableattheendofthearticleAbstractInthepaper,weareconcernedwiththeexistenceofperiodicsolutionsforacoupledRayleighsystem.
BycombininggraphtheorywithcoincidencedegreetheoryaswellasLyapunovfunctionmethod,twonewsucientconditionsontheexistenceofperiodicsolutionsforthecoupledRayleighsystemareestablished.
OurresultsontheexistenceofperiodicsolutionsforthecoupledRayleighsystemimprovethoseobtainedintheexistingliteratureforcoupledRayleighsystem.
Hence,ourresultsarenewandcomplementarytotheexistingpapers.
Firstparttitle:IntroductionSecondparttitle:PreliminariesThirdparttitle:TheexistenceofperiodicsolutionsFourthparttitle:NumericaltestFiveparttitle:ConclusionKeywords:Periodicsolutions;CoupledRayleighsystem;Graphtheory;Continuationtheoremofcoincidencedegreetheory;Lyapunovfunctionmethod1IntroductionAnimportantclassofRayleighsystemsisdescribedbythefollowingform:x(t)+ft,x(t)+gt,x(t)=e(t),(1.
1)wheref,g:R*R→Rande:R→Rarecontinuousfunctions.
Thedynamicbehaviorsofsystem(1.
1)havebeenanactiveresearchtopicduetoitsextensiveapplicationsinphysics,mechanics,engineeringtechnique,andotherareas(see[1–4]andthereferencestherein).
Suchsuccessfulapplicationsaregreatlydependentontheexistenceofperiodicsolutionsforsystem(1.
1).
Hence,theperiodicityanalysisofsystem(1.
1)hasbeenasubjectofintenseactivities,andmanyresultshavebeenobtained,forexample,see[5–8]andthereferencestherein.
In[7],theauthorsinvestigatedthefollowingRayleightypeequation:x(t)+fx(t)+gt,x(t)=e(t),(1.
2)TheAuthor(s)2019.
ThisarticleisdistributedunderthetermsoftheCreativeCommonsAttribution4.
0InternationalLicense(http://creativecommons.
org/licenses/by/4.
0/),whichpermitsunrestricteduse,distribution,andreproductioninanymedium,pro-videdyougiveappropriatecredittotheoriginalauthor(s)andthesource,providealinktotheCreativeCommonslicense,andindicateifchangesweremade.
Zhouetal.
AdvancesinDierenceEquations(2019)2019:335Page2of21wheref:R→Riscontinuous,g:R2→RiscontinuousandT-periodicwithrespecttotherstvariable.
Somecriteriatoguaranteetheexistenceofperiodicsolutionsofthisequationwerepresentedin[7]byusingMawhin'scontinuationtheorem,Floquettheory,Lyapunovstabilitytheory,andsomeanalysistechniques.
In[6],theauthorsstudiedtheexistenceofperiodicsolutionsofRayleighequations:x(t)+ft,x+g(x)=e(t),(1.
3)wheref:R2→RiscontinuousandT-periodicwithrespecttotherstvariable,g,e:R→Rarecontinuous,andeisT-periodic.
TheyprovedthatthegivenequationpossessesatleastoneT-periodicsolutionundersomeconditions.
In[5],byemployingthecontinua-tiontheoremofcoincidencedegreetheory,theauthorsstudiedakindofRayleighequationwithadeviatingargumentasfollows:x(t)+fx(t)+gxt–τ(t)=p(t),(1.
4)whereg,f:R→Raretwocontinuousfunctions,τ(t)andp(t)arecontinuousandT-periodicfunctions,andestablishedsomenewresultsontheexistenceofperiodicsolutionsforsystem(1.
4).
Withthepopularityofcoupledsystems,sofar,theexistenceandglobalstabilityofpe-riodicsolutionsofcoupledsystemsonneuralnetworkshavegainedincreasingresearch[9–13],theexistenceofperiodicsolutionsofcoupledsystemsonthepredator-preysys-tems[14,15]hasbeenwidelystudied,theexistenceofperiodicsolutionsandstabilityofequilibriumpointforcoupledsystemsonnetworkshavebeenwidelyinvestigated,forexample,see[16–23]andthereferencestherein.
In[21],theauthorswereconcernedwiththefollowingcoupledRayleighsystem:xk(t)+fkt,xk(t)+gkt,xk(t)=ek(t),(1.
5)wherek=1,2,.
.
.
,n,nisapositiveinteger,fk,gk:R→Randek:R→Rarecontinuousω-periodicfunctionsintherstargumentwithperiodω>0,fk(t,xk)iscontinuouslydif-ferentiableinxk.
In[21],bytakingyk(t)=xk(t)+ηxk(t),η>0,system(1.
5)wasrewrittenasxk(t)=yk(t)–ηxk(t),yk(t)=–η2xk(t)+ηyk(t)–fk(t,yk(t)–ηxk(t))–gk(t,xk(t))+e(t).
(1.
6)Byadding–lh=1akh(yk(t)–yh(t))intothesecondequationofsystem(1.
6),in[21],theauthorsestablishedthefollowinglinearcoupledRayleighsystem:xk(t)=yk(t)–ηxk(t),yk(t)=–η2xk(t)+ηyk(t)–fk(t,yk(t)–ηxk(t))–gk(t,xk(t))+e(t)–lh=1akh(yk(t)–yh(t)),k∈K,(1.
7)whereakh(yh–yk)representstheinuenceofvertexhonvertexk,akh>0,andakh=0ifandonlyifthereexistsnoarcfromvertexhtovertexking,K,garedenedinDenition2.
1.
Zhouetal.
AdvancesinDierenceEquations(2019)2019:335Page3of21In[21,24–26],bycombininggraphtheorywithcoincidencedegreetheoryaswellasLyapunovmethod,asucientcriterionfortheexistenceofperiodicsolutionsforsystem(1.
7)wasprovidedundertheseconditions(A1)–(A5).
However,theconditionsintheresultsobtainedin[21]ontheexistenceofperiodicso-lutionsforthecoupledsystem(1.
7)aretoocomplicatedandtherearetoomanyofthem.
Thismotivatesustoobtainmoreconciseandeasilyveriednewsucientconditionsforsystem(1.
7).
Uptonow,theglobalexistenceofperiodicsolutionsfordierentialsystemshasbeeninvestigatedmainlybyemployingthefollowingvemethods:(1)Fixedpointtheoremmethods[26];(2)Combiningcontinuationtheoremofcoincidencedegreetheorywiththeaprioriestimateofperiodicsolutions[9,11,13–15,27–30];(3)Combiningcontinu-ationtheoremofcoincidencedegreetheorywithLMI[12];(4)CombiningcontinuationtheoremofcoincidencedegreetheorywithLyapunovfunctionmethod[16,18–21,31];(5)Themethodofupperandlowerfunctions.
But,intheabove-mentionedmethods,(3)and(4)areusedinrecentyearstostudytheexistenceofperiodicsolutionsfordierentsystems.
Inthispaper,weapply(4)tostudytheexistenceofperiodicsolutionforsystem(1.
7),buttheconcreteanalysistechniquesinourpaperaredierentfromthoseusedin[16,18–21].
Inthispaper,ourpurposeis,bycombininggraphtheorywithMawhin'scon-tinuationtheoremofcoincidencedegreetheoryaswellasLyapunovfunctionalmethod,toimprovetheresultsontheexistenceofperiodicsolutionsobtainedin[21]forsystem(1.
7)byremovingconditions(A4)and(A5)in[21].
Consequently,thecontributionofthispaperliesinthefollowingtwoaspects:(1)Novelinequalitytechniquesarecitedtostudytheexistenceofperiodicsolutionsfordierentequations;(2)Novelsucientconditionsaregainedforsystem(1.
7)byimprovingtheresultsobtainedintheexistingpapers.
Thispaperisorganizedasfollows.
SomepreliminariesandlemmasaregiveninSect.
2.
InSect.
3,twosucientconditionsarederivedfortheexistenceofperiodicsolutionsforsystem(1.
7).
InSect.
4,twoillustrativeexamplesaregiventoshowtheeectivenessoftheproposedtheory.
InSect.
5,aconclusionisgiven.
2PreliminariesLetRandRnbethesetofrealnumbersandann-dimensionalEuclideanspace,respectively.
Let|·|and·respectivelybenormsofRandRn.
Wecitethenotationasfollows:f=maxt∈[0,ω]f(t),wheref(t)isacontinuousω-periodicfunction.
Wemaketheassumptionsasfollows:(H1)Thereexistconstantsb>0,d>0suchthat,fork∈K,gk(t,xk)≤bxk(t)+d.
(H2)Thereexistconstantsδ0,e>0,andawithA=–η–δη2+0.
5b2+0.
5bd+η2+0.
5ηr+0.
5η|a|+0.
5ηe0suchthat,fork∈K,mk(xk)=1ωω0gk(t,xk)dtxk≥ε,xk=0,wheremk(xk)∈C1(R,R).
(A5)Fork∈K,ω0ek(t)dt=0.
Forthesakeofconvenience,weintroduceGainesandMawhin'scontinuationtheoremaboutcoincidencedegreetheory[24]andgraphtheory[25]asfollows.
Lemma2.
1([24])AssumethatXandZaretwoBanachspaces,L:D(L)X→ZisaFredholmoperatorwithindexzero.
LetΩ∈XbeanopenboundedsetandN:Ω→ZbeL-compactonΩ.
Assumethat(1)foreachλ∈(0,1),u∈Ω∩DomL,Lu=λNu;(2)foreachu∈Ω∩KerL,QNu=0;(3)deg{JQNu,Ω∩KerL,0}=0,wheredegdenotestheBrouwerdegree.
ThentheoperatorequationLu=NuhasatleastonesolutioninΩ∩DomL.
Denition2.
1([23])Adirectedgraphg=(U,K)containsasetU={1,2,.
.
.
,n}ofverticesandasetKofarcs(i,j)leadingfrominitialvertexitoterminalvertexj.
AsubgraphΓofgissaidtobespanningifΓandghavethesamevertexset.
Adirectedgraphgisweighedifeacharc(j,i)isassignedapositiveweightbij.
TheweightW(Γ)ofasubgraphHistheproductoftheweightsonallitsarc.
Adirectedpathδingissubgraphwithdistinctvertices{i1,i2,.
.
.
,im}suchthatitssetofarcsis{(ik,ik+1):k=1,2,.
.
.
,m–1}.
Foraweighteddigraphgwithlvertices,wedenetheweightmatrixB=(bij)n*nwhoseentrybij>0isequaltotheweightofarc(j,i)ifitexists,and0otherwise.
Adigraphgisstronglyconnectedif,foranypairofdistinctvertices,thereexistsadirectedpathfromonetotheother.
TheLaplacianmatrixof(g,B)isdenedasL=(pij)l*l,wherepij=–bijfori=jandpij=k=ibikfori=j.
Lemma2.
2([24])Supposethatl≥2andckdenotesthecofactorofthekthdiagonalelementoftheLaplacianmatrixof(g,B).
Thenlk,h=1ckakhGkh(xk,xh)=Q∈ΩW(Q)*(k,h)∈K(CQ)Ghk(xh,xk),whereGkh(xk,xh)isanarbitraryfunction,Qisthesetofallspan-ningunicyclicgraphsof(g,B),W(Q)istheweightofQ,CQdenotesthedirectedcycleofQ,andK(CQ)isthesetofarcsinCQ.
Inparticular,if(g,B)isstronglyconnected,thenck>0for1≤k≤l.
Zhouetal.
AdvancesinDierenceEquations(2019)2019:335Page5of21Lemma2.
3Foranyλ∈(0,1),considerthefollowingsystem:xk(t)=λ[yk(t)–ηxk(t)],yk(t)=λ[–η2xk(t)+ηyk(t)–fk(t,yk(t)–ηxk(t))–gk(t,xk(t))+ek(t)–lh=1akh(yk(t)–yh(t))],k∈K.
(2.
1)Iftheperiodicsolutionsofsystem(2.
1)exist,thentheyareboundedandtheboundaryisindependentofthechoiceofλunderassumptions(H1),(H2),and(A3).
Namely,thereexistsapositiveconstantHsuchthatx(t),y(t)T=x1(t),x2(t),.
.
.
,xl(t),y1(t),y2(t),.
.
.
,yl(t)T≤H,thenorm·isdenedintheproofofTheorem3.
1.
ProofSupposethat(x(t),y(t))T=(x1(t),x2(t),.
.
.
,xl(t),y1(t),y2(t),.
.
.
,yl(t))Tisaperiodicsolutionofsystem(2.
1)forsomeλ∈(0,1).
LettingV(x,y)=0.
5lk=1ck(x2k+y2k),whereckdenotesthecofactorofthekthdiagonalelementofLaplacianmatrixof(g,(bkh)l*l).
Accordingtoassumption(A3)andLemma2.
2,onehasck>0,k∈K.
Makinguseofas-sumptions(H1)and(H2),wehavedV(x,y)dt=λlk=1ck–ηx2k(t)–η2xk(t)yk(t)+ηy2k(t)–yk(t)fkt,yk(t)–ηxk(t)+yk(t)xk(t)–gkt,xk(t)+yk(t)ek(t)–lh=1akhyk(t)yk(t)–yh(t)≤λlk=1ck–ηx2k(t)–η2xk(t)yk(t)+ηy2k(t)–δyk(t)–ηxk(t)2–ayk(t)–ηxk(t)+yk(t)*ek(t)+xk(t)yk(t)+0.
5y2k(t)+0.
5g2k(t,xk)–ηxk(t)fkt,yk(t)–ηxk(t)–lh=1akhy2k(t)+12lh=1akhy2k(t)+y2h(t)≤λlk=1ck–ηx2k(t)–η2xk(t)yk(t)+ηy2k(t)–δyk(t)–ηxk(t)2–ayk(t)–ηxk(t)+yk(t)ek+0.
5y2k(t)+yk(t)xk(t)+0.
5b2x2k(t)+d2+2bdxk(t)+ηxk(t)ryk(t)+rηxk(t)+e+12lh=1akhy2h(t)–y2k(t)≤λlk=1ck–η–δη2+0.
5b2+0.
5bd+η2r+0.
5ηl+0.
5η|a|+0.
5ηex2k(t)Zhouetal.
AdvancesinDierenceEquations(2019)2019:335Page6of21+η–δ+0.
5ηr+0.
5|a|+1y2k(t)+–η2+2ηδ+xk(t)yk(t)+0.
5(ek)2+bd+d2+ηe+|a|η+|a|+λ2lh=1,hckakhFhk(yk,yh),(2.
2)whereFhk(yk,yh)=y2h–y2k.
ByemployingLemma2.
2,weobtainlk,h=1ckakhFhk(yk,yh)=0,fromwhich,togetherwith(2.
2),itfollowsthatdV(x,y)dt≤λlk=1ck–η+–δη2+0.
5b2+0.
5bd+η2l+0.
5ηl+0.
5η|a|+0.
5ηex2k(t)+η–δ+0.
5ηl+0.
5|a|+1y2k(t)+–η2+2ηδ+1xk(t)yk(t)+0.
5(ek)2+bd+d2+ηe+|a|η+|a|.
(2.
3)SinceA0,yk(t)=0,xk(t)=0.
Sotheequationinxk(t):Ax2k(t)+(η–δ+0.
5ηl+0.
5|a|+1)y2k(t)+(–η2+2ηδ+1)xk(t)yk(t)=0hastworealrootsx1,x2(x1x2,orxkmax{|x1|,|x2|}=r|yk|,Ax2k(t)+(η–δ+0.
5ηl+0.
5|a|+1)y2k(t)+(–η2+2ηδ+1)xk(t)yk(t)nr+1|yk|,Ax2k(t)+η–δ+0.
5ηr+0.
5|a|+1y2k(t)+–η2+2ηδ+1xk(t)yk(t)r1,Ax2k(t)+η–δ+0.
5ηr+0.
5|a|+1y2k(t)+–η2+2ηδ+1xk(t)yk(t)r|yk|,andisincreasinginxkwhenxkH,Ax2k(t)+qkη–δ+0.
5ηr+0.
5|a|+1y2k(t)+–η2+2ηδ+1xk(t)yk(t)+0.
5(ek)2+bd+d2+ηe+|a|η+|a|H,thenAx2k(t)+qkη–δ+0.
5ηr+0.
5|a|+1y2k(t)+–η2+2ηδ+1xk(t)yk(t)+0.
5(ek)2+bd+d2+ηe+|a|η+|a|0,whichisindependentofthechoiceofλ,suchthat(x(t),y(t))T0ischosensothattheboundislarger.
Hence,foranyλ∈(0,1),z∈Ω∩DomL,Lz=λNz.
Whenz∈Ω∩KerP,wewillshowQNz=0.
Whenz∈Ω∩KerL,z∈R2l(namelyzisaconstantvector)withZhouetal.
AdvancesinDierenceEquations(2019)2019:335Page9of21z=(x,y)T=H+r.
Ifzisaconstantvectorwithz=H+r,QNz=0,thenitfollowsthattheconstantvectorzwithz=H+rsatises,fork=1,2,.
.
.
,l,1ωω0Gk(t)dt=0,1ωω0Fk(t)dt=0.
Hence,thereexisttk(i=1,2),ξi∈[0,ω](k=1,2,.
.
.
,l)suchthatGk(tk)=0,Fk(ξk)=0.
(3.
1)From(3.
1),wehave0=lk=1ckxkGk(tk)+ykFk(ξk).
(3.
2)Byusingthesameproofasthoseof(2.
4)inLemma2.
3,from(3.
2),itfollowsthat0=lk=1ckxkGk(tk)+ykFk(ξk)≤lk=1ck–η+–δη2+0.
5b2+0.
5bd+η2r+0.
5ηl+0.
5η|a|+0.
5ηex2k+η–δ+0.
5ηr+0.
5|a|+1y2k+–η2+2ηδ+1xkyk+0.
5(ek)2+bd+d2+ηe+|a|η+|a|.
(3.
3)ItfollowsfromRemark1that,since(x,y)T>H,lk=1–η+–δη2+0.
5b2+0.
5bd+η2r+0.
5ηl+0.
5η|a|+0.
5ηex2k+η–δ+0.
5ηr+0.
5|a|+1y2k+–η2+2ηδ+1xkyk+0.
5(ek)2+bd+d2+η+|a|η+|a|H,wehavefromRemark1lk=1–η+–δη2+0.
5b2+0.
5bd+η2r+0.
5ηl+0.
5η|a|+0.
5ηe+x2k–η2+2ηδ+1xkyk+η–δ+0.
5ηr+0.
5|a|+1y2k+0.
5(ek)2+bd+d2+ηe+|a|η+|a|0,whichisindependentofthechoiceofλsuchthat(x(t),y(t))T0isachosenpositiveconstantsuchthattheboundofΩislarger.
Hence,foranyλ∈(0,1),z∈Ω∩DomL,Lz=λNz.
Whenz∈Ω∩KerP,wewillshowQNz=0.
Whenz∈Ω∩KerL,z∈R2l(namelyzisaconstantvector)withz=(x,y)T=H+r.
Ifzisaconstantvectorwithz=H+r,QNz=0,thenitfollowsthattheconstantvectorzwithz=H+rsatises,fork=1,2,.
.
.
,l,1ωω0Gk(t)dt=0,1ωω0Fk(t)dt=0.
Hence,thereexisttk(i=1,2),ξi∈[0,ω](k=1,2,.
.
.
,l)suchthatGk(tk)=0,Fk(ξk)=0.
(3.
16)From(3.
16),wehave0=lk=1ckxkGk(tk)+ykFk(ξk).
(3.
17)Fromtheproofofpage4inLemma3of[21],itfollowsfrom(3.
17)that0=lk=1xkGk(tk)+ykFk(ξk)=lk=1–ηx2k(t)–η2xk(t)yk(t)+ηy2k(t)–yk(t)fkt,yk(t)–ηxk(t)+yk(t)xk(t)–gkt,xk(t)+yk(t)ek(t)–lh=1akhyk(t)yk(t)–yh(t)0,ηβk(ξk,yk–ηxk)–ηxkyk≤0.
5ηβk(ξk,yk–ηxk)–ηx2k+y2kZhouetal.
AdvancesinDierenceEquations(2019)2019:335Page16of21andμnk+uk+ηη–βk(ξk,yk–ηxk)xkyk≤0.
5μnk+uk+ηβk(ξk,yk–ηxk)–ηx2k+y2k,itfollowsfrom(3.
22)that0≤lk=1ck0.
5–δ+0.
5μ1–η+0.
5ηβk(ξk,yk–ηxk)–η+μmk+η–0.
5+δ–0.
5μ1+0.
5nk+uk+0.
5ηβk(ξk,yk–ηxk)–ηx2k+0.
5ηβk(ξk,yk–ηxk)–η+η+1–βk(ξk,yk–ηxk)+μ–ηβk(ξk,yk–ηxk)–1+vk+0.
5nk+uk+η–ηβk(ξk,yk–ηxk)y2k+0.
5(ek)2=lk=1ck–122δ–μ1–1+η2+η2–βk(ξk,yk–ηxk)–12μ–2mk–4η+1–2δ+μ1+η2–nk–uk+η2–βk(ξk,yk–ηxk)x2k+–12η2+(2–η)βk(ξk,yk–ηxk)–2–2η+μ–ηβk(ξk,yk–ηxk)–1+vk+0.
5nk+|uk+ηβk(ξk,yk–ηxk)–η]y2k+(ek)2(3.
23)≤lk=1ck–η24x2k–η22y2k+(ek)2–12μ–2mk–4η+1–2δ+μ1+η2–nk–|uk|x2k–12μ2–2vk–nk–uk+2η2y2k.
(3.
24)Choosevk,mk,uk,nksuchthat2vk=2–nk–uk+2η2(3.
25)and2mk=1–4η–2δ+μ1+η2–nk–uk.
(3.
26)Substituting(3.
25)and(3.
26)into(3.
24)gives0≤lk=1ck–η24x2k–η22y2k+(ek)2.
(3.
27)FromtheproofofLemma3in[21],wehavelk=1ck–η24x2k–η22y2k+(ek)2<0.
(3.
28)Zhouetal.
AdvancesinDierenceEquations(2019)2019:335Page17of21TherestoftheproofissimilartothatofthecorrespondingpartinTheorem3.
1,anditisomitted.
Remark4InourTheorem3.
2,conditions(A4)and(A5)inTheorem1in[21]areremoved,theremainingconditions(A1)–(A3)arethesame.
Hence,ourresultimprovesTheorem1in[21].
Remark5Byapplyingnewinequalitytechniques,weestablishnewsucientconditionsfortheexistenceofperiodicsolutionsofacoupledRayleighsystem.
Ourmethodcanbeappliedtostudyingtheexistenceofperiodicsolutionsforanysecond-orderdierentialsystem.
4NumericaltestExample1ConsiderthefollowingRayleighsystem:xk(t)=yk(t)–ηxk(t),yk(t)=–η2xk(t)+ηyk(t)–fk(t,yk(t)–ηxk(t))–gk(t,xk(t))+ek(t)–lh=1akh[yk(t)–yh(t)],(4.
1)wherek=1,2,3,4,η=0.
4andgk(t,xk(t))=0.
05|xk(t)|+0.
05cosxk(t)+0.
05sinxk,fk(t,xk(t))=(0.
5+0.
6sinxk(t))xk(t)+0.
003,ek(t)=1+cost.
Wecancheckthat|gk(t,xk)|≤0.
05|xk(t)|+0.
06,andwetakeb=0.
005,d=0.
06.
xkfk(t,xk)≥–0.
1x2k+0.
03xk,andδ=–0.
1,a=0.
03.
Takingη=0.
4,wegetA=–η–δη2+0.
5b2+0.
5bd+η2+0.
5ηr+0.
5η|a|+0.
5ηe<0,thusconditions(H1),(H2)aresatised.
Sincegk(t,x)isnotdierentialinxk,thuscondition(A4)in[21]cannotbesatised;since10(1+cost)dt=0,hencecondition(A5)in[21]cannotbesatised,hencetheexistenceofperiodicsolutionsofsystem(4.
1)cannotbeveriedbytheseresultsin[21].
AssumingthatB=(akh)4*4=020.
610.
3030.
430.
50220.
620,Figure1Thephraseplan(x1(t),y1(t))Zhouetal.
AdvancesinDierenceEquations(2019)2019:335Page18of21Figure2Thephraseplan(x2(t),y2(t))Figure3Thephraseplan(x3(t),y3(t))Figure4Thephraseplan(x4(t),y4(t))wecancheckthatcondition(A3)holds.
Now,alltheconditionsinTheorem3.
1inourpaperaresatised.
Thesolutionofsystem(4.
1)isshowninFigs.
1–4,fromwhichwecanclearlyseethatsystem(4.
1)hasatleastoneperiodicsolution.
Example2Insystem(4.
1),wesetgk(t,xk(t))=(1+0.
001|sinxk(t)|+0.
001sint)xk(t),fk(t,xk(t))=0.
2xk(t)sin2t,ek(t)=sint+1.
Itiseasytoverifythat(A1),(A2),and(A3)areZhouetal.
AdvancesinDierenceEquations(2019)2019:335Page19of21Figure5Thephraseplan(x1(t),y1(t))Figure6Thephraseplan(x2(t),y2(t))satisedassumingthatB=(akh)4*4=02610.
3010.
430.
50220.
620.
But(A4)isnotsatisedsincemk(xk)contains|sinxk(t)|,whichisnotdierential.
Hence,theexistenceofperiodicsolutionsofsystem(4.
1)cannotbeveriedbytheresultsin[21].
Ontheotherhand,byourTheorem3.
1,system(4.
1)hasatleastoneω-periodicsolution.
Thesolutionofsystem(4.
1)isshowninFigs.
5–8,fromwhichwecanclearlyseethatsystem(4.
1)hasatleastoneperiodicsolution.
5ConclusionInthepaper,wediscusstheexistenceofperiodicsolutionsforaclassofcoupledRayleighsystemsbycombininggraphtheorywithcontinuationtheoremaswellasLyapunovfunc-tions.
Bytheabovestudymethodsandbyusingnovelinequalitytechniques,weobtainnewsucientconditionstoensuretheexistenceofperiodicsolutionsforsystem(1.
7).
Ourresultsandmethodarecompletelynew.
Zhouetal.
AdvancesinDierenceEquations(2019)2019:335Page20of21Figure7Thephraseplan(x3(t),y3(t))Figure8Thephraseplan(x4(t),y4(t))AcknowledgementsTheauthorswouldliketothankthehandlingeditorsandtheanonymousreviewers.
FundingNone.
CompetinginterestsTheauthorsdeclarethattheyhavenocompetinginterests.
Authors'contributionsAllauthorsreadandapprovedthenalmanuscript.
Authordetails1SchoolofAppliedMathematicalScience,XiamenUniversityofTechnology,Xiamen,China.
2DepartmentofMathematicsandComputerScience,NanchangNormalUniversity,Nanchang,China.
3CollegeofMathematicsandEconometrics,HunanUniversity,Changsha,China.
Publisher'sNoteSpringerNatureremainsneutralwithregardtojurisdictionalclaimsinpublishedmapsandinstitutionalaliations.
Received:5May2019Accepted:15July2019References1.
Li,X.
,Ma,Q.
:Boundednessofsolutionsforsecondorderdierentialequationswithasymmetricnonlinearity.
J.
Math.
Anal.
314,233–253(2006)2.
Radhakrishnan,S.
:ExactsolutionsofRayleigh'sequationandsucientconditionsforinviscidinstabilityofparallel,boundednessshearows.
Z.
Angew.
Math.
Phys.
45,615–637(1994)3.
Habets,P.
,Torres,P.
J.
:SomemultiplicityresultsforperiodicsolutionsofaRayleighdierentialequation.
Dyn.
Contin.
DiscreteImpuls.
Syst.
,Ser.
AMath.
Anal.
8,335–351(2001)4.
Cao,H.
,Liu,B.
:ExistenceanduniquenessofperiodicsolutionsforRayleigh-typeequations.
Appl.
Math.
Comput.
211,148–154(2009)5.
Lu,S.
,Ge,W.
:SomenewresultsontheexistenceofperiodicsolutionstoakindofRayleighequationwithadeviatingargument.
NonlinearAnal.
,TheoryMethodsAppl.
56,501–514(2004)Zhouetal.
AdvancesinDierenceEquations(2019)2019:335Page21of216.
Ma,T.
:PeriodicsolutionsofRayleighequationsviatime-maps.
NonlinearAnal.
,TheoryMethodsAppl.
75,4137–4144(2012)7.
Wang,Y.
,Zhang,L.
:ExistenceofasymptoticallystableperiodicsolutionsofaRayleightypeequation.
NonlinearAnal.
,TheoryMethodsAppl.
71,1728–1735(2009)8.
Lord,J.
W.
,Strutt,R.
:TheoryofSound,vol.
1.
Dover,NewYork(1877,re-issued1945)9.
Zhang,Z.
Q.
,Wang,L.
P.
:Existenceandglobalexponentialstabilityofaperiodicsolutiontodiscrete-timeCohen–GrossbergBAMneuralnetworkswithdelays.
J.
KoreanMath.
Soc.
48(4),727–747(2011)10.
Zhang,Z.
Q.
,Cao,J.
D.
:Periodicsolutionsforcomplex-valuedneuralnetworksofneutraltypebycombininggraphtheorywithcoincidencedegreetheory.
Adv.
Dier.
Equ.
2018,261(2018)11.
Zhang,Z.
Q.
,Liu,K.
Y.
:Existenceandglobalexponentialstabilityofaperiodicsolutiontointervalgeneralbidirectionalassociativememory(BAM)neuralnetworkswithmultipledelaysontimescales.
NeuralNetw.
24,427–439(2011)12.
Zhang,Z.
Q.
,Zheng,T.
:Globalasymptoticstabilityofperiodicsolutionsfordelayedcomplex-valuedCohen–GrossbergneuralnetworksbycombiningcoincidencedegreetheorywithLMImethod.
Neurocomputing289,220–230(2018)13.
Liu,K.
Y.
,Zhang,Z.
Q.
,Wang,L.
P.
:ExistenceandglobalexponentialstabilityofperiodicsolutiontoCohen–GrossbergBAMneuralnetworkswithtime-varyingdelays.
Abstr.
Appl.
Anal.
2012,ArticleID805846(2012)14.
Hu,D.
W.
,Zhang,Z.
Q.
:FourpositiveperiodicsolutionstoaLotka–Volterracooperativesystemwithharvestingterms.
NonlinearAnal.
,RealWorldAppl.
11,1115–1121(2010)15.
Zhang,Z.
Q.
,Hou,Z.
T.
:Existenceoffourpositiveperiodicsolutionsforaratio-dependentpredator–preysystemwithmultipleexploited(orharvesting)terms.
NonlinearAnal.
,RealWorldAppl.
11,1560–1571(2010)16.
Gao,S.
,Li,S.
S.
,Wu,B.
Y.
:Periodicsolutionsofdiscretetimeperiodictime-varyingcoupledsystemonnetworks.
ChaosSolitonsFractals103,246–255(2017)17.
Suo,J.
H.
,Sun,J.
T.
,Zhang,Y.
:Stabilityanalysisforimpulsivecoupledsystemonnetworks.
Neurocomputing99,172–177(2013)18.
Zhang,X.
H.
,Li,W.
X.
,Wang,K.
:Theexistenceofperiodicsolutionsforcoupledsystemonnetworkswithtimedelays.
Neurocomputing152,287–293(2015)19.
Zhang,X.
H.
,Li,W.
X.
,Wang,K.
:Periodicsolutionsofcoupledsystemsonnetworkswithbothtime-delayandlinearcoupling.
IMAJ.
Appl.
Math.
80,1871–1889(2015)20.
Zhang,X.
H.
,Li,W.
X.
,Wang,K.
:Theexistenceandglobalexponentialstabilityofperiodicsolutionforaneutralcoupledsystemonnetworkswithdelays.
Appl.
Math.
Comput.
264,208–217(2015)21.
Guo,Y.
,Liu,S.
,Ding,X.
H.
:TheexistenceofperiodicsolutionsforcoupledRayleigh.
Neurocomputing191,398–408(2016)22.
Yang,X.
,Lu,J.
:Finite-timesynchronizationofcouplednetworkswithMarkoviantopologyandimpulsiveeects.
IEEETrans.
Autom.
Control61,2256–2261(2016)23.
Li,M.
Y.
,Shuai,Z.
:Global-stabilityproblemforcoupledsystemofdierentialequationsonnetworks.
J.
Dier.
Equ.
248,1–20(2010)24.
Gaines,R.
E.
,Mawhin,J.
L.
:CoincidenceDegree,andNonlinearDierentialEquations.
LectureNotesinMathematics,vol.
568.
Springer,Berlin(1977)25.
West,D.
:IntroductiontoGraphTheory.
PrenticeHall,UpperSaddleRiver(1996)26.
Hu,J.
,Zeng,C.
N.
,Tan,J.
:Boundednessandperiodicityforlinearthresholddiscrete-timequaternion-valuedneuralnetworks.
Neurocomputing267,417–425(2017)27.
Lu,S.
P.
,Guo,Y.
Z.
,Chen,L.
J.
:PeriodicsolutionsforLienardequationwithanindenitesingularity.
NonlinearAnal.
,RealWorldAppl.
45,542–556(2019)28.
Lu,S.
P.
,Jia,X.
W.
:Homoclinicsolutionsforasecond-ordersingulardierentialequation.
J.
FixedPointTheoryAppl.
20,101–115(2018)29.
Yu,Y.
C.
,Lu,S.
P.
:AmultiplicityresultforperiodicsolutionsofLienardequationswithanattractivesingularity.
Appl.
Math.
Comput.
346,183–192(2019)30.
Du,B.
:Stabilityanalysisofperiodicsolutionforacomplex-valuedneualnetworkswithboundedandunboundeddelays.
AsianJ.
Control20,881–892(2018)31.
Du,B.
,Lian,X.
,Cheng,X.
:PartialdierentialequationmodelingwithDirichletboundaryconditionsonsocialnetworks.
Bound.
ValueProbl.
2018,ArticleID50(2018)