ON THE EXISTENCE OF ZERO POINTS OF A CONTINUOUS FUNCTION

IPreliminarlesTheee。etOOk totot比OO。stsllCCdCSdO‘lOll‘。山OOqll时1OO/(。)＝。,SS山 SFSSdPoint Theorems[‘’,”],Degree Theory【9,15]．In general these are used when f aPPlies a set ora space on itself．Cm眈inn眈ion Methods can be used to test the existence ofa solution whenf pplies a set or a sp。e on it咖‘－’,”－“l,or aPPlies a set or sP。m another sP。,s。M see here．wrthermore,contlnuulon methods pve us the exlstence ofcurves wb讪 can beused to CalCttlue the Sdutionsl6…     

THE BLOCH CONSTANT OF LOCALLY BIHOLOMORPHIC MAPPINGS ON BOUNDED SYMMETRIC DOMAINS

51.IntroductionIn[11,westudiedtheboundsofBlochconstantofholomorphicmappingsonirreducibleboundedsymmetricdomains.ThispaperisasuccessivepaPerof[1].WestudytheboundsoftheBlochconstantoflocallybiholomorphicmappingsonirreducibleboundedsymmetricdomains.AllthenotationsofthispaPerarethesameasthoseweusedin[1].LetDbeadomain,whichcontainstheorigin.AholomorphicmappingonDisdefinedasaBlochmappingifthefamilyofmappingsFj={g:g(z)=f(rk(z))-j(gh(o)),forallrkEAut(D)},isanormalfamily.TheBlochnormofaBlochmapp…     

ON THE RATE OF CONVERGENCE OF CONJUGATE LEGENDRE-FOURIER SERIES OF FUNCTION OF BOUNDED VARIATION

This paper gives the degree of approximation of function of bounded variation by the partial sums of con-jngate Legendre-Fourier Series.     

THE PARAMETRIC REPRESENTATION FOR SPIRAL-LIKE MAPPINGS OF TYPE α ON BOUNDED BALANCED PSEUDOCONVEX DOMAINS

LetΩbe a bounded balanced pseudoconvex domain in Cn that admits a plurisubhamonic defining function of class C2. The authors prove that if f is a spiral-like mapping of typeαonΩ, then f can be expressed as a limiting form concerning a Schwarz mapping. Moreover, the characteristics for star-like mappings and spiral-like mappings of typeαare obtained. As a direct application, the growth theorem for spiral-like mappings of typeαis set up.     

THE GROWTH THEOREM FOR STARLIKE MAPPINGS ON BOUNDED STARLIKE CIRCULAR DOMAINS

1.IntroductionOnthegeometricfunctiontheoryofonecomplexvariable,thefollowinggrowthandi-coveringtheoremiswellknown(see[2]).TheoremA.Foreachno~alizedunivalentjunctionfontheunitdiscDCC,ESPecially,theleft-handsideOftheaboveinequalityimpliesf(D)2ID.ForeachzED,z/0,equalityoccursintheaboveinequalityifandonlyiffisKoe6efunctionK(z)=theoritsrotatione--"K(e"z).Itisnaturaltoextendthisandotherresultsonthegeometricfunctiontheoryofonevariablestoseveralvariables.Butasearlyasfiftyyearsago,H.Cartanpointe…     

ON THE ERROR FUNCTION OF THE SQUARE-FULL INTEGERS

Let L(x) denote the number of square-full integers not exceeding x. It is proved in [1] thatL(x)～(ζ(3/2)/ζ(3))x~(1/2) (ζ(2/3)/ζ(2))x~(1/3) as x→∞,where ζ(s) denotes the Riemann zeta function. Let △(x) denote the error function in the asymptotic formula for L(x). It was shown by D. Suryanaryana~([2]) on the Riemann hypothesis (RH) that1/x integral from n=1 to x |△(t)|dt=O(x~(1/10 s))for every ε>0. In this paper the author proves the following asymptotic formula for the mean-value of △(x) under the assumption of R. H.integral from n=1 to T (△~2(t/t~(6/5))) dt～c log T,where c>0 is a constant.     

ON THE EXISTENCE OF PRODUCT STOCHASTIC MEASURES

In this paper, we prove that the existence of product stochastic measures depends on the axiomsystem of set theory: If one aecepts the axiom of choice, the answer is negative, and we give acounter-example where the product stochastic measure doesn't exist; but in the Solovay model (onekind of set theory which refuses the axiom of choice),the answer is positive, and we give a proof.     

A REMARK ON THE COEFFICIENTS OF BOUNDED POLYNOMIALS

Let p(z)=akzk be such that |p(eip)|≤1 for R and |p(1)|=a[0,1].An inequality of Dewan and Cavil for the sum |av|+|au|,0≤u相似文献   

EXISTENCE OF THE UNIFORMLY ULTIMATE BOUNDED SOLUTIONS AND PERIODIC SOLUTIONS

１ＩｎｔｒｏｄｕｃｔｉｏｎＣｏｎｓｉｄｅｒａｔｈｉｒｄｏｒｄｅｒｎｏｎｌｉｎｅａｒｃｏｎｔｒｏｌｓｙｓｔｅｍｉｎｔｈｅｆｏｒｍｘ…＋Ｆ（ｒ）ｘ¨＋Ｇ（ｒ）ｘ·＋Ｈ（ｒ）ｘ＝ｅ（ｔ），（１）ｗｈｅｒｅＦ（ｒ）＝１－ε［１＋ｋｎ（ε）ｒ２］，Ｇ（ｒ）＝１...     

GLOBAL EXISTENCE OF THE SOLUTIONS OF NONLINEAR PARABOLIC EQUATIONS IN EXTERIOR DOMAINS

This paper deals with the following IBV problem of nonlinear parabolic equation: where Ω is the exterior domain of a compact set in R~n with smooth boundary and F satisfies |F(λ)|=o(|λ|~2), near λ=0. It is proved that when n≥3, under the suitable smoothness and compatibility conditions, the above problem has a unique global smooth solution for small initial data. Moreover, It is also proved that the solution has the decay property ‖u(t)‖_(L∞(Ω)=o(t~(-n/2)), as t→ ∞.     

ON THE RATE OF CONVERGENCE OF THE GAMMA OPERATOR FOR FUNCTIONS OF BOUNDED VARIATION

Let f be a function defined on the infinite interval[0,∞).The Gamma operator G_n applied to f isG_n(f(t),x)=∫_0~f(nt)g_n(x/t)(dt)/t,whereg_n(u)=u~ne~(-u)/(n-1)!(see[1],P.207).There are some results of convergence and saturation of G_n     

THE SUM OF DEFICIENCIES OF ENTIRE FUNCTION ON C~n

This paper proves that: Let / be an entire function of finite order λon Cn. Then(1) , where k(X) is a nonnegative constant depending only on A;(2) If (a, f) = 1, then A is a positive integer and equals the lower order of /.     

ON THE STRUCTURE OF THEATTRACTORS OF HYPERBOLICITERATED FUNCTION SYSTEM

1.IntroductionThefractalsgeneratedbytheattractorsofiteratedfunctionsystems(i.f.s.)havebeenresearchedbymanyauthorsfl--3'6'7'9].ByaniteratedfunctionsystemwemeanacompactmetricspaceXtogetherwithacollectionofcontinuousmapsTI,T2,'tTNonit,denotedby(X,TI,',TN).IfalltheTi'sarecontractionswecall(X;TI,',TN)ahyperboliciteratedfunctionsystem(h.i.f.s.).NForanh.i.f.s.thereexistsacompactsubsetAofX,suchthatA=.UTi(A).Aiscalledtheattractoroftheh.i.f.s.DenoteZ=(1,2,',N)N,anddefineametricdonZby…     

EXISTENCE AND UNIQUENESS OF STRONG PERIODIC SOLUTION OF THE EVOLUTION SYSTEM DESCRIBING GEOPHYSICAL FLOW—PART I： IN BOUNDED DOMAINS

61. IntroductionDiscovered by R. HideI3], the following evolution system has been used to describe thegeophysical flow:where u = u(xl,',XN,t) = (ul,',UN) and B = B(xl,',XN,t) = (BI,',BN) arethe velocity vectors Of Eulerian flow and magnetic fields respectively. p(x, t) and q(x, t) arepressures. j(x, t) and g(x,t) are volUme forces. p and u are the constants of density andviscosity of the flow respeCtively, p is the constant of magnetic permeability and A = f withelectrical resis…     

α-CARLESON MEASURE AND BLOCH FUNCTIONS ON BOUNDED SYMMETRIC DOMAINS OF C~n

α－ＣＡＲＬＥＳＯＮＭＥＡＳＵＲＥＡＮＤＢＬＯＣＨＦＵＮＣＴＩＯＮＳＯＮＢＯＵＮＤＥＤＳＹＭＭＥＴＲＩＣＤＯＭＡＩＮＳＯＦＣ~ｎＯｕｙａｎｇＣａｉｈｅｎｇ（欧阳才衡）（ＷｕｈａｎＩｕｓｔ．ｏｆＭａｔｈ．Ｓｃｉ．，ＣｈｉｎｅｓｅＡｃａｄ．ｏｆＳｃｉ．，...     

A NOTE ON THE EXISTENCE AND NONEXISTENCE OF GLOBALLY BOUNDED CLASSICAL SOLUTIONS FOR NONISENTROPIC GAS DYNAMICS

Existence of globally bounded classical solution for nonisentropic gas dynamics system has long been studied, especially in the case of polytropic gas. In [4], Liu claimed that sufficient condition has been established. However, the authors find that the argument he used is not true in general. In this article, the authors give a counter example of his argument. Hence, his claim is not valid. The authors believe that it is difficult to impose general conditions on the initial data to obtain globally bounded classical solution.     

ON THE EXISTENCE OF PERIODIC SOLUTIONS OF NONLINEAR OSCILLATION EQUATIONS

This paper deals with the existence of periodic solutions of the nonlinear oscillationequation f(x)(x) ψ(x)η(x)=0.The author offers a method which can reduce(3)into the system=h(y)-e(y)F(x),=-g(x).(9)Some sufficient conditions for the existence of the limit cycles of(9)are obtained.These results generalize the results in [1,2,3,4,5,6](3)(9)obfained.     

ON THE EXISTENCE OF GLOBAL GENERAL SOLUTIONS OF POLYNOMIAL SYSTEMS

ＯＮＴＨＥＥＸＩＳＴＥＮＣＥＯＦＧＬＯＢＡＬＧＥＮＥＲＡＬＳＯＬＵＴＩＯＮＳＯＦ　ＰＯＬＹＮＯＭＩＡＬＳＹＳＴＥＭＳＺＨＡＯＸＩＡＯＱＩＡＮＧ（赵晓强）（ＩｎｓｔｉｔｕｔｅｏｆＡｐｐｌｉｅｄＭａｔｈｅｍａｔｉｃｓ，ｔｈｅＣｈｉｎｅｓｅＡｃａｄｅｍｙｏ...     

GLOBAL EXISTENCE OF THE SOLUTIONS TO NONLINEAR HYPERBOLIC EQUATIONS IN EXTERIOR DOMAINS

This paper deals with the following IBV problem of nonlinear hyperbolic equations u_(tt)- sum from i, j=1 to n a_(jj)(u, Du)u_(x_ix_j)=b(u, Du), t>0, x∈Ω, u(O, x) =u~0(x), u_t(O, x) =u~1(v), x∈Ω, u(t, x)=O t>O, x∈()Ω,where Ωis the exterior domain of a compact set in R~n, and |a_(ij)(y)-δ_(ij)|= O(|y|~k), |b(y)|=O(|y|~(k+1)), near y=O. It is proved that under suitable assumptions on the smoothness,compatibility conditions and the shape of Ω, the above problem has a unique global smoothsolution for small initial data, in the case that k=1 add n≥7 or that k=2 and n≥4.Moreover, the solution ham some decay properties as t→ + ∞.