带脉冲的p-Laplacian系统的周期解研究(英文).docx

带脉冲的p-Laplacian系统的周期解研究(英文)
Title:StudyonPeriodicSolutionsofp-LaplacianSystemwithImpulsiveEffects
Introduction:Inrecentyears,thestudyofimpulsivedifferentialequationshasreceivedincreasingattentionfromscholarsandresearchersduetoitsbroadapplicationsinphysics,biology,andengineering.Inthispaper,wefocusontheperiodicsolutionsofp-Laplaciansystemwithimpulsiveeffects.Thep-Laplacianoperatorisanonlinearoperatorwidelyusedinmanymathematicalmodels,andimpulsiveeffectscanrepresentsuddenchangesinthesystem.
Body:
1.Preliminaries
Inthissection,weintroducesomebasicconceptsandtheoryrelatedtothep-Laplacianoperatorandimpulsivedifferentialequations.Thep-Laplacianoperatorisdefinedbytheformula
∆pu=div(|∇u|p−2∇u)
wherep>1isaconstantanduisafunctionfromadomainD⊂RntoR.Thep-Laplacianoperatorisoftenusedtodescribenonlinearphenomenainvariousfieldssuchasfluidmechanics,populationdynamics,andimageprocessing.
Impulsivedifferentialequationsareatypeofdifferentialequationthatinvolvesuddenchangesordiscontinuitiesinthesystem.Theyarewidelyusedinmodelingvariousreal-lifephenomenasuchaspopulationdynamics,biologicalsystems,andcontrolsystems.Thestudyofimpulsivedifferentialequationshasbeenofgreatinteresttoresearchersinrecentyearsduetotheirpotentialapplications.
2.Periodicsolutionsofp-Laplaciansystem
Weconsiderthefollowingp-Laplaciansystemwithperiodicboundaryconditions:
{−∆pu+λv=f(u),
−∆pv+μu=g(v),
whereλ,μareconstants,f(u),g(v)arenonlinearfunctions,andu,vareperiodicfunctionswithperiodT.
Weshowthatundercertainconditionsontheparametersandfunctions,thissystemhasatleastoneperiodicsolution.Toprovethisresult,weusetheLeray-Schauderdegreetheoryandthefixedpointtheory.
3.Periodicsolutionsofp-Laplaciansystemwithimpulseeffects
Considerthefollowingimpulsivep-Laplaciansystem:
{−∆pu(t)+λv(t)=f(u(t)),
−∆pv(t)+μu(t)=g(v(t)),
whereu(t),v(t)arefunctionsdefinedon[0,∞),andfort∈Ti=(ti−1,ti](i=1,2,…),wehave
u(ti)=u(i),
v(ti)=v(i),
where(u(i),v(i))areimpulsivevalues.
Undercertainassumptionsontheparametersandfunctions,weshowtheexistenceandu