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:2003211206:(1948-),,.:025822724(2004)0320394203Gronwall(,402168):Gronwall,.,Gronwall.Gronwall.:;;Gronwall;:O122.3;O175.1:ARemarksonGronwallInequalityLIANGShao2jun(Dept.ofMathematics,WesternChongqingUniversity,Yongchuan402168,China)Abstract:TheclassicalGronwallinequalityanditsproofbyusinggeneralmethodswereintroduced.Theestimateoflinearcontrolwasextendedtothatofnonlinearcontrolbythecomparisontheorem.TheapplicationofGronwallinequalitytothestudyofthecharacteristicsofsolutionsofdifferentialequationswasalsodiscussed.Keywords:differentialequations;qualitativetheory;Gronwallinequality;comparisontheoremGronwall,Gronwall[1].1Gronwall1[1]x(t),(t)(t)[a,b],Pt0,(t)0,[a,b]x(t)(t)+ta(s)x(s)ds,Pt[a,b],x(t)(t)+ta(s)(s)exp[ts(u)du]ds.y(t)=ta(u)x(u)du,t[a,b],y(0)=0,y(t)=(t)x(t)(t)(t)+(t)ta(s)ds=(t)(t)+(t)y(t),t[a,b].exp(-ta(s)ds)0,ddt(y(t)exp(-ta(s)ds))(t)(t)exp(-ta(s)ds).tat39320046JOURNALOFSOUTHWESTJIAOTONGUNIVERSITYVol.39No.3Jun.2004y(t)exp(-ta(s)ds)ta(u)(tu)exp(-ta(s)ds)du,y(t)ta(u)(u)exp(1-tu(s)ds)du.x(t)(t)+y(t)(t)+y(t)(t)+ta(u)(u)exp(1-tu(s)ds)du.1(t)[a,b],1:11,(t)[a,b],Pt[a,b],x(t)(t)ta(s)ds+taexp(ts(u)du)(s)ds.2(t),1:21,(t)C,Pt[a,b],x(t)Cexp(ta(u)du).2Gronwall1,x(t),x(s),,.,x(t).2x(t)[a,b]R+,t[a,b],x(t)M+ta(s)f(x(s))ds.(1):M0,,[a,b]R+,fR+R+.t[a,b],x(t)-1((M)+ta(s)ds).RR:(u)=uu0dsu(s),uRy(t)=taf(x(s))(s)ds,t[a,b].y(a)=0,(1)y(t)f(M+y(t))(y),t[a,b].(2)(2)tat,y(t)0dsf(M+s)ta(s)ds+(M),t[a,b],(y(t)+M)ta(s)ds+(M),t[a,b].2x(s),.3x(t)[a,b]R,12x2(t)12x20+ta(s)ds,t[a,b],:x0,(s)[a,b].:x(t)x0+ta(s)ds,t[a,b].(3)5933:Gronwally(t)=12(x20+2)+ta(s)x(s)ds,t[a,b].0.x2(t)y(t).(4)y(t)(t)x(t),Pt[a,b].y(t)(t)y(t).tat:2y(t)2y(a)+ta(s)dst[a,b].(4)x(t)x0++ta(s)ds,t[a,b].(3).3Banach.x(t)=g(t)+taV(t,s,x(s))ds,t[,].(5):V[,]2XXg[,)X;XBanach.(5)C([,),X),44(5)V(1)V(t,s,x)B(t)L(s,x),t,s[,],xX,(6)L0L(t,u)-L(t,V)M(t-V)u-V,t[,),uV0,(7)B[,).(2)xC([,),X),(5),x(t)g(t)+B(t)tL(u,g(u)Xexp(tuM(s,g(s))B(s)ds)du,Pt[,).X[,)X,x(t)g(t)+tV(t,s,x)ds,t[,).(6)(7)x(t)g(t)+B(t)tL(s,x(s))ds,t[,).y(t)[,)R+y(t)=tL(s,x(s))ds,yy(t)=L(t,x(t)),y()=0.(7)y(t)L(t,g(t)+B(t)y(t))L(t,g(t)+M(t)g(t)B(t)y(t).s=y(t)exp(-tM(s,g(s))B(s)ds),1.:[1]BirkhoffG,RotaGC.Ordinarydifferentialequations[M].NewYork:JohnWiley,1978.72290.[2]BelInanR.Stabibitytheoryofdifferentialequations[M].NewYork:McGrawHill,1983.15218.[3],.[J].,1993,(3):2552261.[4],.[M].:,1985.1172130.69339