On the existence of periodic solutions for nonlinear system with multiple delays

IntroductionInthispaper,westudyT_periodicsolutionsofthefollowingnonlinearsystemwithmultipledelays x(t) =f(t,x(t) ,x(t-τ1(t) ) ,… ,x(t -τm(t) ) ) ,(1 )wherex(t) ∈C(R ,R) ,fiscontinuous,f(t+T ,·) =f(t,·) ,τi(t) (i=1 ,2 ,… ,m)arecontinuousperiodicfunctionsofperiodT .AlemmaisintroducedfordiscussingtheexistenceofT_periodicsolutionofsystem (1 ) .LetXbeaBanachSpace ,considerthefollowingoperatorequation :Lx =λNx　　 (λ∈ [0 ,1 ] ) ,whereL :DomL∩X→Xisalinearoperator,λ∈ [0 ,1 ]isapa…     

Asymptotic theory of initial value problems for nonlinear perturbed Klein-Gordon equations

Introduction InthispaperasymptotictheoryofthefollowinginitialvalueproblemforanonlinearKlein Gordonequationisconsidered.tt-Δ =εF(t,x,,ε),t>0,x∈R2,(0,x,ε)=0(x,ε),t(0,x,ε)=1(x,ε),x∈R2,(1)where(t,x)isarealvaluedunknownfunction,Δ=2i     

THE ASYMPTOTIC THEORY OF INITIAL VALUE PROBLEMS FOR SEMILINEAR PERTURBED WAVE EQUATIONS IN TWO SPACE DIMENSIONS

IntroductionInthispaper,anasymptotictheoryisestablishedforthefollowinginitialvalueproblemforasemilinearperturbedwaveequation :utt-Δu=εf(u ,ε)　　 (t>0 ,x∈R2 ) ,(1 )u(0 ,x,ε) =u0 (x ,ε)　　 (x∈R2 ) ,(2 )ut(0 ,x,ε) =u1(x ,ε)　　 (x∈R2 ) ,(3 )where     

CONTROLLABILITY OF A CLASS OF HYBRID DYNAMIC SYSTEMS (Ⅱ)—SINGLE TIME_DELAY CASE

IntroductionConsideraswitchedlinearsystemwithsingletime_delayinthecontrolfunctiongivenby x(t) =Ar(t) x(t) +Br(t) u(t) +Dr(t) u(t -τ) ,(1)wherex(t) ∈Rnisthestate,u(t) ∈R mtheinput,thepiecewiseconstantscalarfunctionr(t) :R+ → 1,2 ,… ,Nistheswitchingpath ,(Ai,Bi,Di) ｜i =1,… ,Nisafinitefamilyofsystemrealizations.Moreover,r(t) =iimpliesthat(Ai,Bi,Di)ischosenasthesystemrealizationattimet.τ >0isthefixedtimedelayincontrol.Inthispaper ,wewillstudythecontrollabilityofsystem (1) .Theone_per…     

Difference scheme and numerical simulation based on mixed finite element method for natural convection problem

ntroductionLetΩ R2 beaboundeddomain .Weconsiderthefollowingnon_stationarynaturalconvectionproblem :Problem (Ⅰ )　Findu =(u1,u2 ) ,p ,andTsuchthat,foranyt1>0 ,ut- μΔu +(u· )u + p=λjT　　 ((x ,y ,t) ∈Ω× (0 ,t1) ) ,divu =0　　　　　　　　　 ((x ,y,t) ∈Ω× (0 ,t1) ) ,Tt-ΔT +λu· T =0　　 ((x,y,t) ∈Ω× (0 ,t1) ) ,u =0 ,T =0　　　　　　 ((x,y,t)∈ Ω× (0 ,t1) ) ,u(x ,y ,0 ) =0 ,　T(x,y,0 ) =f(x,y)　　 ((x,y) ∈Ω) ,whereuisthefluidvelocityvectorfield ,pthepressurefield ,Tthet…     

On the problem of preventing blowing-up and quenching for semilinear heat equation

In this paper,the global existence of solutions to the IVP=Δu+g(t)f(u) (t>0),u|_(t=0)=u_0(x)and the (?)PVPu_t=Δu-g(t,x)f(u)(t>0,x∈Ω),u|_(t-0)=u|_(?)(?)is investigated. As has been done in [6]the (?)duction of factor g(t) or g(t.x) innonlinear term is to prevent(?) occurrance of blowing-up or quenching of solutions.It isshown in this paper that most of the restrictions on f,g and u_0 in the theorems of[6] maybe cancelled or relaxed,that the smallness of g is required only for t large,and thatunder certain conditions controlling initial state can avoid blowing-up.     

NEWTON METHOD FOR SOLVING A CLASS OF SMOOTH CONVEX PROGRAMMING

1ConvexProgrammingandExactPenaltyFunction Weconsiderthefollowingconvexprogramming:(P)minf(x)s.t.x∈S={x∈Rn:gi(x)≤0,i=1,…,m}.SupposethatSisacompactset.ThusthereexistsalargeboundedboxX,suchthatS={x∈Rn:gi(x)≤0,i=1,…,m}intX.Assumethatf(x),gi(x),i=1,…,m     

NEW SIMPLE SMOOTH MERIT FUNCTION FOR BOX CONSTRAINED VARIATIONAL INEQUALITIES AND DAMPED NEWTON TYPE METHOD

IntroductionLetX Rn,F:X→Rnbe given, the variational inequality, denoted byVI(X,F),is tofind a vectorx∈ Xsuch thatF(x)T(y-x)≥0, y∈ X. (1)DenoteN∶={1,2,…,n},whenX =[a,b]∶={x∈Rn|ai≤xi≤bi,i∈N},theVI(X,F)is called the box constrained variational ine     

DYNAMICAL CHARACTER FOR A PERTURBED COUPLED NONLINEAR SCHRDINGER SYSTEM

Introduction WeconsidertheperturbedcouplednonlinearSchr dingerequations(CNLS)iq1t=q1xx 2[|q1|2 |q2|2-ω2]q1 iε(^D1q1-r1),iq2t=q2xx 2[|q1|2 |q2|2-ω2]q2 iε(^D2q2-r2),(1)whereqjis2πperiodicandeveninx.rjisconstant,^Djisboundeddissipativeoperatorand assumedtotaketheform^Djqj=-αjqj βj^Bqj,(2)αjandβjarepositiveconstants,j=1,2.^BisaFouriertruncationofLaplaceoperator xx,i.e.,^Bcos(kx)=-k2cos(kx),k0isasmallperturbation parameter.Th…     

Interative approximation of fixed points for almost asymptotically nonexpansive type mappings in Banach spaces

IntroductionThroughoutthispaperweassumethatEisarealBanachspace ,E isthedualspaceofE ,DisanonemptysubsetofEandJ:E →2 E isthenormalizeddualitymappingdefinedbyJ(x) =f∈E :〈x ,f〉=‖x‖·‖f‖,‖f‖=‖x‖,　　 x∈E .　　Definition 1　LetT :D →Dbeamapping .1 )Tissaidtobeasymptoticallynonexpansive[1],ifthereexistsasequence kn [1 ,∞)withlimn→∞kn =1suchthat　　　　　‖Tnx-Tny‖≤kn‖x-y‖forall　　x ,y∈D ,n≥0 ;(1 )　　2 )Tissaidtobeofasymptoticallynonexpansivetype[2 ],if　　　　　lims…     

Oscillation criteria for second order nonlinear differential equation with damping

1IntroductionandProblem Thefollowingisthesecondordernonlineardifferentialequationwithdamping: [p(t)ψ(y)u(y′)]′ r(t)y′(t) q(t)f(y)g(y′)=0(t≥t0),(1) inwhich,p(t)∈C′([t0,∞),(0,∞)),r(t)∈C([t0,∞),(-∞,∞)),q(t)∈C([t0, ∞),[0,∞))withtheexistenceofT≥t0,q(t)≠0,t∈[T,∞),f(y),g(y),ψ(y),u(y) ∈C((-∞,∞),(-∞,∞)),andyf(y)>0,yu(y)>0,y≠0. Thesolutiony(t)ofEq.(1)iscallednormalsolutionify(t)isthenon_constantsolutionof Eq.(1)andsupt≥t0｜y(t)｜>0(refertoRef.[1]).Anormalsolutionisoscill…     

On the boundedness and the stability properties of solution of certain fourth order differential equations

This paper investigates equation(1)in two cases:(i)P≡0,(ii)P(≠O)satisfies|P(t,x,y,z,ω)|≤(A |y| |z| |ω|)q(t),where q(t)is a nonnegative function of t.For case(i)the asymptotic stability in the large of the trivial solution x=0 is investigatedand for case(ii)the boundedness result is obtained for solutions of equation(1).Theseresults improve and include several well-known results.     

SINGULARLY PERTURBED BOUNDARY VALUE PROBLEMS FOR SEMI-LINEAR RETARDED DIFFERENTIAL EQUATIONS WITH NONLINEAR BOUNDARY CONDITIONS

IntroductionInthispaper,westudiedakindofboundaryvalueproblems (BVPs)forsemi_linearretardeddifferentialequationwithnonlinearboundarycondition :　　　　εx″(t) =f(t,x(t) ,x(t-ε) ,ε) ,　　t∈(0 ,1 ) ,(1 )　　　　x(t) =φ(t,ε) ,　t∈[-ε0 ,0 ] ,h(x(1 ) ,x′(1 ) ,ε) =A(ε) ,(2 )whereε>0isasmallparameterandε0 isasufficientlysmallpositiveconstant.ThereweremanyresultsofstudyingonsingularlyperturbedboundaryvalueproblemforretardeddifferentialequationinRefs.[1～5] .Butthosestudiespossessedanesse…     

ON REICH''''S OPEN QUESTION

1　IntroductionandPreliminariesThroughoutthispaperweassumethatEisarealBanachspace ,E isthedualspaceofE ,DisanonemptysubsetofEandJ:E → 2 E isthenormalizeddualitymappingdefinedbyJ(x) =f∈E ,　〈x ,f〉 =‖x‖‖f‖ ,　‖x‖ =‖f‖ 　 (x∈E) . (1 )　　Definition 1　LetT :D →Dbeamapping .1 .Tissaidtobeasymptoticallynonexpansive[1],ifthereexistsasequence kn [1 ,∞ )withlimn→∞kn =1suchthat‖Tnx-Tny‖ ≤kn‖x-y‖　forall　x ,y∈D ,n≥ 1 ;　　 2 .Tissaidtobenonexpansive,ifthesequence kn app…     

THE HOLDER CONTINUITY OF GENERALIZED SOLUTIONS OF A CLASS QUASILINEAR PARABOLIC EQUATIONS

I.IntroductionLetFbethen-dimensionalEuclideanspaceandGaboundeddomaininM,WI(C)andM(C)beusualSobolevspaces.Let0相似文献   

GLOBAL DYNAMICS OF DELAYED BIDIRECTIONAL ASSOCIATIVE MEMORY （BAM） NEURAL NETWORKS

IntroductionConsiderthebidirectionalassociativememory (BAM )neuralnetworkswithconstanttransmissiondelaysdescribedbyasystemofdelaydifferentialequationsoftheform[1,2 ]:dxi(t)dt =-aixi(t) nj=1bijfj(yj(t-σij) ) Ii,　　i=1 ,2 ,… ,m ,dyj(t)dt =-cjyj(t) mi=1djigi(xi(t-τji) ) Jj,　　j=1 ,2 ,… ,n ,fort >0 .Thesystem ( 1 )consistsoftwosetsofneurons (orunits)arrangedontwolayers,namely ,I_layerandJ_layer.Inthesystem ( 1 ) ,xi( ·)andyj( ·)denotemembranepotentialoftheithneuronsfromtheI_laye…     

What are the separatrix values named by Leontovich on homoclinic bifurcation

Introduction LetVλ(x),x∈R2,λ∈RkbeafamilyofplanarC∞(oranalytic)vectorfields.Suppose thatforλ=λ0,Vλ0hasahyperbolicsaddlepointattheorigin(0,0)inthephaseplaneandthere isahomoclinicorbit(oraseparatrixloop)Γ0totheorigin.Thehyperbolicityratioattheorigin isr(λ)=-λ1/λ2,whereλ1<0<λ2arethetwoeigenvaluesofthelinearizedsystematthe origin.Generally,whenλ≠λ0and｜λ-λ0｜issmallenough,nearΓ0limitcycleswillbe created.Thisso_calledhomoclinicbifurcationhasbeenstudiedbyalotofauthors[1-3].Deno…     

Existence of solutions for nonlinear elliptic boundary value problems

IntroductionInRef.[1 ]KannanandLockergivetheexistenceofatleastonesolutionofTy-h(t,y ,… ,y(n- 1) )y=f(t,y ,… ,y(n- 1) )　　 (a相似文献   

Global linear and quadratic one-step smoothing newton method for vertical linear complementarity problems

IntroductionLetN ∈Rm0 ×n(m0 ≥n)beaverticalblockmatrixoftype(m1,… ,mn) ,q∈Rm0 beaconstantvectorpartitionedconformablywithN ,thatareN =N1Nn,　q =q1qn,　m0 = ni=1mi,whereNi ∈Rmi×n,qi ∈Rmi,i ∈I =1 ,… ,n .WeconsidertheverticallinearomplementarityproblemVLCP(N ,q)associatedwith (N ,q)offindingavectorx∈Rnsuchthatx≥ 0 ,　si(x) :=Nix+qi ≥ 0 ,　xi∏mij=1sij(x) =0　　 (i∈I) ,wherexi,sij(x)denotetheithcomponentofxandthejthcomponentofsi(x) ,respectively ;∏ mj=1ajdenotesa1…am.Iti…     

ITERATIVE APPROXIMATION WITH ERRORS OF FIXED POINT FOR A CLASS OF NONLINEAR OPERATOR WITH A BOUNDED RANGE

1IntroductionandPreliminariesLetXbearealBanachspacewithnorm||·||andadualX*.ThenormalizeddualitymappingJ:X→2Xisdefinedbywhere<·,·>denotesthegeneralizeddualitypairing.Itiswell-knownthatifX*isstrictlyconvex,thenJissingle-valuedandsuchthatJ(tx)=tJxforallt≥0,x∈X.IfX*isunifomilyconvex,thenJisuniformlycontinuousonanyboundedsubsetsofX(of.Browder[1]andBarbu[2])OperatorTwithdomainD(T)andrangeR(T)inXissaidtobeaccretiveifforeveryx,y6D(T),thereexistsajeJ(x--y)suchthatTheconceptofaccretive…