L_[0,1]~p(1

梅雪峰  周颂平 《数学进展》2005,(6)

综合题新编选登

题153设函数f(x)=ax-(a 1)ln(x 1),其中a>0.1)求f(x)的单调区间;2)当x>0时,证明不等式:1 xx0),由f′(x)=0,解得x=1a.当x变化时,f′(x),f(x)的变化情况如下表:x(-1,1a)1a(1a, ∞)f′(x)-0 f(x)极小值由上表可知,当x∈(-1,1a)时,f′(x)<0,函数f(x)在(-1,1a)内单调递减;当x∈(1a, ∞)时,f′(x)>0,函数f(x)在(1a, ∞)内单调递增.所以,函数f(x)的单调减区…     

带有单侧共振的Duffing方程解的存在性

对于α,β∈L(0,1),如果α(t)≤β(t)对a.e.t∈(0,1)成立,则记α≤β;如果α≤β,而且α(t)<β(t)在(0,1)的一个正测度集上成立,则记α<β.任意x∈C~1(0,1)记x满足下述的(2)式,KerL_p=.考虑周     

以Dirac测度为源的拟线性退化抛物方程

本文讨论拟线性退化抛物方程带有初值条件的Cauchy问题,其中δ(x)是Dirac测度,m>1,Q≡Rm×(0, ∞),u0(x)≥0,uo(x)∈Cβ(Rn),β∈(0,1)且0∈suppu0=Ω,Ω是Rn中的一个有界开集,证明了弱解的存在性.此外,还讨论了自由边界的Hlder连续性.     

参数函数$\epsilon$取边际值时的$GCF_\epsilon$集

Let ∈ :N → R be a parameter function satisfying the condition ∈(k) + k + 1 > 0and let T∈ :(0,1] →(0,1] be a transformation defined by T∈(x) =-1 +(k + 1)x1 + k-k∈x for x ∈(1k + 1,1k].Under the algorithm T∈,every x ∈(0,1] is attached an expansion,called generalized continued fraction(GCF∈) expansion with parameters by Schweiger.Define the sequence {kn(x)}n≥1of the partial quotients of x by k1(x) = ∈1/x∈ and kn(x) = k1(Tn-1∈(x)) for every n ≥ 2.Under the restriction-k-1 < ∈(k) <-k,define the set of non-recurring GCF∈expansions as F∈= {x ∈(0,1] :kn+1(x) > kn(x) for infinitely many n}.It has been proved by Schweiger that F∈has Lebesgue measure 0.In the present paper,we strengthen this result by showing that{dim H F∈≥12,when ∈(k) =-k-1 + ρ for a constant 0 < ρ < 1;1s+2≤ dimHF∈≤1s,when ∈(k) =-k-1 +1ksfor any s ≥ 1where dim H denotes the Hausdorff dimension.     

Some Sets of GCF_∈ Expansions Whose Parameter ∈ Fetch the Marginal Value

Let ∈ :N → R be a parameter function satisfying the condition ∈(k) + k + 1 0and let T∈ :(0,1] →(0,1] be a transformation defined by T∈(x) =-1 +(k + 1)x1 + k-k∈x for x ∈(1k + 1,1k].Under the algorithm T∈,every x ∈(0,1] is attached an expansion,called generalized continued fraction(GCF∈) expansion with parameters by Schweiger.Define the sequence {kn(x)}n≥1of the partial quotients of x by k1(x) = ∈1/x∈ and kn(x) = k1(Tn-1∈(x)) for every n ≥ 2.Under the restriction-k-1 ∈(k) -k,define the set of non-recurring GCF∈expansions as F∈= {x ∈(0,1] :kn+1(x) kn(x) for infinitely many n}.It has been proved by Schweiger that F∈has Lebesgue measure 0.In the present paper,we strengthen this result by showing that{dim H F∈≥12,when ∈(k) =-k-1 + ρ for a constant 0 ρ 1;1s+2≤ dimHF∈≤1s,when ∈(k) =-k-1 +1ksfor any s ≥ 1where dim H denotes the Hausdorff dimension.     

Travelling Wave Solutions for Some Degenerate Parabolic Equations ( I )

The existence and regularity of travelling wave front solutions are studied forsome degenerate parabolic equationswith m,n>0 and f satisfies(H):f(u)∈C~1[0,1],f(0)<0, f(1)<0 and f'(1)<0.There exists a∈(0,1),s.t.f(u)<0 for u∈(0,a) and f(u)>0 for u∈(a,1). A function u=q(z)with z=x+ct is said to be a travelling wave front solution     

具有无限多个奇性点的一维p-Laplacian方程的正解

主要讨论奇异边值问题{(фp(x′))′ a(t)f(x(t))=0,t∈(0,1),αx(0)-βx′(0)=0,γx(1) δx′(1)=0.在奇性条件下无穷多个解的存在性问题,其中:фp(s)=│s│p-2s,p>1;a(t)在0,1/2上有可数个奇性点.     

对一道世锦赛试题的拓广

题目 设a,6,c∈R+,a+b+c＝1,则M＝√3a+1+√3b+1+√3c+1 的整数部分 ∈是( ). 参考答案是这样求M的下界值的: 因为x∈(0,1)时,有x＞xn(n∈N且n≥2),所以√3x+1＞√x2+2x+1＝x+1. 即√3a+1+√3b+1+√3c+1＞a+b+c+3＝4.     

综合题新编选登

题61 已知函数f(x)(0,1)上是增函数.1)求实数a的取值范围;2)若数列{an}满足a1=c∈(0,1)且an+1=ln(2-an)+an(n∈N*),证明0相似文献   

Sturm-LiouVille算子部分谱信息的逆问题

研究了Sturm-Liouville算子Aq,h,Hj,j=1,2,…中势函数q(x)的确定性问题,即已知部分区间[a,1](a∈(0,1))上的势函数q(x),则无限组部分谱信息可唯一确定整个区间[0,1]上的势函数.推广了Simon的方法,且将选择条件的范围从一组谱扩展到了无限组.     

WEAK TRAVELLING WAVE FRONT SOLUTIONS OF GENERALIZED DIFFUSION EQUATIONS WITH REACTION

The author demonstrate that the two-point boundary value problem {p′(s)=f′(s)-λp^β(s)for s∈(0,1);β∈(0,1),p(0)=p(1)=0,p(s)&gt;0 if s∈(0,1),has a solution(λ^-，p^-（s）),where |λ^-| is the smallest parameter,under the minimal stringent restrictions on f(s), by applying the shooting and regularization methods. In a classic paper, Kohmogorov et.al.studied in 1937 a problem which can be converted into a special case of the above problem. The author also use the solution(λ^-,p^-(s)) to construct a weak travelling wave front solution u(x,t)=y(ξ),ξ=x-Ct,C=λ^-N/(N+1),of the generalized diffusion equation with reaction δ/δx（k（u）|δu/δx|^n-1 δu/δx)-δu/δt=g（u），where N&gt;0,k(s)&gt;0 a.e.on(0,1),and f(a):=n+1/N∫0ag(t)k^1/N(t)dt is absolutely continuous ou[0,1],while y(ξ) is increasing and absolutely continuous on (-∞,+∞) and (k(y(ξ))|y′(ξ)|^N)′=g(y(ξ))-Cy′(ξ)a.e.on(-∞，+∞）,y(-∞)=0,y(+∞)=1.     

一道不等式恒成立典型问题的思路与解法

<正>题目已知函数f(x)=2sinx-xcosx-ax(a∈R).当a≤1时,证明:对任意x∈(0,π),f(x)>0.思考1:变换主元法不等式2sinxxcosx-ax>0理解为二元不等式,将a视作主元,记作m(a)=-xa+2sinx-xcosx,是递减的一元一次函数,则当a=1时取最小值为2sinx-xcosx-x,于是问题转化为求证:对任意x∈(0,π),2sinx-xcosx-x>0.     

LP[0,1](1＜p＜∞)空间正系数多项式倒数逼近的一个推广

本文推广了LP[0,1](1＜p＜∞)空间函数的正系数多项式的倒数逼近的结论,即证明了:设f(x)∈LP[0,1],1＜p＜∞,且在(0,1)内严格1次变号,则存在一点x0∈(0,1)及一个n次多项式Pn(x)∈∏n(+)使得‖f(x)-x-x0/Pn(x)‖LP[0,1]≤Cpω(f,n-1/2)LP[0,1],其中∏n(+)为次数不超过n的正系数多项式的全体.     

试题研讨(21)

题目　( 2 0 0 3年南昌市高三第二次调研测试题)设函数f ( x)是定义在[- 1 ,0 )∪( 0 ,1 ]上的奇函数,当x∈[- 1 ,0 )时,f ( x) =2 ax 1x2 　( a为实数) .1求当x∈( 0 ,1 ]时,f ( x)的解析式;2若f ( x)在区间( 0 ,1 ]上为增函数,求a的取值范围;3求f( x)在x∈( 0 ,1 ]上的最大值.命题溯源　本题研究了函数y =2 ax -1x2 的单调性及最值,2 0 0 2年天津市高中质量调查理科第1 9题与2 0 0 3年合肥市高三抽样测试第2 2题都涉及此类问题.原解思路　1设x∈( 0 ,1 ],则- x∈[- 1 ,0 ) .又f ( x)为奇函数,则f ( x) =- f ( - x) =- [2 a( - x) 1( -…     

分数阶Schrodinger—Kirchhoff方程无穷多高能量解的存在性

本文研究如下带有变号势函数的分数阶Schrodinger Kirchhoff方程(a+b∫∫R^N|u(x)-u(y)|^p/|x-y|^N+p^sdxdy)^p-1(-△)p^su+λV(x)|u|^p-2u=f(x,u)-μg(x)|u|^q-2u,x∈R^N.其中s∈(0,1),p∈[2,∞),q∈(l,p),a,b>0,λ,μ>0均为正常数,在V,f,g等函数合适的条件下,运用喷泉定理获得该系统无穷多高能量解的存在性.     

台劳中值定理的一个证明

按同济书,台劳中值定理可叙述为:若f(x)在(a,b)内有n+1阶导数,x_0∈(a,b),则当x∈(a,b)时,有证明设λ满足下式,使下式成为恒等式     

自适应迎风格式求解奇异摄动两点边值问题的高精度算法(二)

0引言 考虑与文[1]相同的奇异摄动两点边值问题的数值解法: Tu(x):=-εu″(x)-p(x)u′(x)=f(x),x∈(0,1); (1) u(0)=0,u(1)=1. (2) 其中ε是一个常数,0<ε≤1,f∈C2[0,1].假定P∈C3[0,1]且存在常数β和-β使得0<β≤p(x)≤-β,|p′(x)|≤-β,(V)x∈[0,1] (3) 成立.     

半线性抛物型方程的双移动边界问题

本文运用Fourier方法和压缩映像不动点原理,证明了半线性抛物型方程的双移动边界问题 u_t=a~2u_(xx) F(x,t,u,u_x),(x,t)∈D_∞, u(l_1(t),t)=0,l_1(0)=0,t∈(0, ∞), u(l_2(t),t)=0,l_2(0)=l_0,t∈(0, ∞), u(x,0)=φ(x),0≤x≤l_0,φ(0)=φ(l_0)=0.解的存在唯一性,其中D_∞={(x,t)|l_1(t)相似文献   

具共振条件下的一类三阶非局部边值问题的可解性

本文考虑一类三阶非局部边值问题x”’(t)=f(t,x(t),x'(t),x”(t)),t∈(0,1), x(0)=0,x'(0)=0,x'(1)=(?) x'(s)dg(s),其中f:[0,1]×R3→R是一个连续函数, g:[0,1]→[0,∞)是一个非减的函数,且满足g(0)=0．在g满足共振条件g(1)=1 的情况下,通过应用重合度理论,得到了该问题解的存在性结果．