THE CONTINUITY OF THE YPbUCOH OPERATOR

1. Let X and Y be two sets of points on each of which a completely additive measure is given. Let K(x,y,u) be a real-valued function which is defined for every pair of points (x,y) ∈ (X,Y) and for every real number u, such that, for almost every point x∈X it satisfies the Caratheodory condition with respect to (y,u): K(x,y,u) is measurable in y for every u and continuous in u for almost every y. For every measurable function f(y), the functionKxf(y) = K(x,y,f(y))is measurable in y for almost every x. If this funetion is integrable with respect to y for almost every x, the value of the integral yields a function Kf:defined for almost every a∈ X. We call the functional operator K the operator generated by the function K(x,y,u).     

EXTENDING POMPEIU OPERATOR AND THE COMPACTNESS OF SOLUTION SET FOR A CLASS OF HIGHER ORDER PARTIAL DIFFERENTIAL EQUATIONS

This paper discusses chiefly the compactness of solution set of following equationswhere △ is the Laplacian in Sobolev's sense, a;(x),i= 0,1,…n, n≥ 3, are real square matrices of dimen-sion N×V , bounded and measurable in a bounded multiply connected domain Ω, the boundary S is as-sumed to be sufficiently smooth, u(x) is unknown vector, z = (x_1,x_2,…,x_n) Ω R~z,m≥1,|S_1|issuperficial measure of the unit sphere of R~Z, |i|=i_1+ i_2 +… + i., △~m=△(△~(m-1)). ,(Ω), ,(Ω), … denote the classes of vectors or matrices whose elements belong to L,(Ω),W,(Ω),…. A vector or a matrix is said to be continuous differentiable, bounded and measurable if so are its ele-     

一类非线性Schrdinger-Kirchhoff系统非平凡解的存在性

文章主要讨论带有位势V(x)的非线性Schrdinger-Kirchhoff型方程﹛(a+b∫[|▽u|~2+V(x)u~2])[-Δu+V(x)u]+λh(x)φu=g(x,u),x∈R3,-Δφ=λh(x)u~2,x∈R~3.(1)(λ≥0)非平凡解的存在性,利用山路定理得到其至少存在一个非平凡解.     

BOUNDARY VALUE PROBLEMS FOR FULLY NONLINEAR UNIFORMLY ELLIPTIC EQUATIONS OF SECOND IN UNBOUNDED DOMAINS

We consider fully nonlinear equations of the formF(z,u,Du,D~2u) = F(x,y,u,u_z,u_y,u_zz,u_zy,u_yy) = 0 (1)in unbounded open subset G = R~2\Ωof the plane R~2,where F is a real continuous function on U = G×R×R~2×R~3 and Ω= Ω_i,Ω_i is a simply connected region (i=1,2,",N) . We assume the function F hascontinuous partial der ivatives F_(u_zz), F_(u_zy), F_(u_yy). on U.For a real function r C( G) a real function u(x,y) is called a solution of (1) satisfyingu = r on G,(2)if there exists a constant P0>2 such that u C~1 ( ) W~(2.p)_(Loc)0 (G) satisfies (1) almost everywhere and (2)in the common sense.The method for treating the above exterior Dirichlet problem in a given unbounded region is as fol-     

全空间R~N上基尔霍夫方程的多解存在性(英文)

文章主要研究全空间RN上基尔霍夫方程.(a+b∫RN|▽u|2dx)Δu+V(x)u=f(x;u)。运用喷泉定理,在位势V(x)满足某些假设条件时,我们得到了该方程的多解存在性结果。     

The Existence of Positive Bounded Entire Solutions of Second Order Quasilinear Elliptic Equations

In this paper we are concerned with the existence of positive entire solutions of second order quasilinear elliptic equations of the type div(|Du|p-2Du)+f(x,u)=0, x∈RN, (1) where f(x, u) is a continuous function on RN×(0,∞). This problem appears in the study of non-Newtonian fluids and non-Newtonian filtration. The quantity p is a characteristic of the madium. Media with p>2 are called dilatant fluids and those with p<2 are called pseudoplastics. If p=2, they are Newtonian fluid. In the present paper we give new sufficient conditions which ensure the existence of positive entire solutions of (1).When p=2, the related results have been obtained by [1,2]. Our theorem for existence complement and extent to the results by [1,2].     

变号位势哈密顿椭圆系统解的存在性

考虑以下具有变号位势的非周期超二次哈密顿椭圆系统{-△u+b(x)·▽u+V(x)u=H_v(x,u,v),-△v-b(x)·▽u+V(x)v=H_u(x,u,v),x∈R~N u(x),v(x)→0,|x|→∞其中z=(u,v):R~N→R×R,当|z|→∞时,H(x,z)关于z是超二次的,在位势V变号的假设下,利用强不定泛函的临界点理论证明此系统存在一个非平凡解。     

DIFFERENCE-BOUNDARY INTEGRO DIFFERENTIAL EQUATION METHOD FOR TWO DIMENSIONAL PARABOLIC EQUATION

Let Ω be a bounded domain in R~2 with smooth boundary and Ω' be the complementary set of Ω∪ .We consider the Neumann's problem of parabolic equation as follow;Let τ be time step, t_k=k_τ and u~k(x)=u(x,t,). We discrete u/ t of (1) by use of difference and getwhereQ~K=1/t(u~K-u~(k-1)-t u~k/ t+f(u~k)-f(u~(k-1)).(2) is a family of Neumann's boundary value problems of Helmholtz equation. Let E(x,y) be the funda-mental solution to Helmholtz equation, i. e.     

具容许状态反馈Pritchard-Salamon系统的小时滞鲁棒稳定性

It has been observed that for many stable feedback control systems, the introduction of arbitrarily small delays into the loop causes instability. Therefore, robustness of stablility with respect to small delays is of great importance. The authors study the robustness with respect to small delays for exponential stability of Pritchard-Salamon systems with admissible state feedback,i.e. the exponential stability of the following systems are equivalent:{x(t)=S(t)x1 ∫o^tS(t-s)BFx(s)ds u(t)=Fx(t),x0∈V,t≥0and{x(t)=S(t)x0 ∫o^tS(t-s)BFx(s-r)ds u(t)=Fx(t-r),x0∈V,t≥0and obtain a number of necessary and sufficient conditions, particularly, frequency domain characterization for robustness with respect to small delays for exponential stability.     

具有变指数的退化抛物方程解的唯一性

考虑具有变指数的退化抛物方程ut=div(ρα丨▽a(u)|p(x)-2▽a(u))+g(x)div(b(u))弱解的存在唯一性问题,其中ρ(x)=dist(x,(e)Ω)是其到边界的距离函数,a(s)是一个严格单调上升的函数.通过选取合适的检验函数证明在无边界值条件情形下该方程弱解的唯一性成立.     

Asymptotic Behavior of the Finite Difference and the Finite Element Methods for Parabolic Equations

0IntroductionLeft(xΩ),b eg(a x)boaunnddeud0(d xo)maairne ibnouRndne,d a anndd a csosnutimneuo tuhsa t.We consider the followinginitial-boundary value problem u t(x,t)-Δu(x,t)=f(x),inΩ×R u|Ω=g(x),t>0u|t=0=u0(x),x∈Ω(1)whereΔis Laplace’s operator.Th     

一类发展的p(x)-Laplace方程解的存在唯一性

讨论一类发展的p(x)-Laplace方程ut=div(a(x,t)|▽u|p(x)-2▽u)+f(u,x,t)解的存在唯一性.不同于此前的研究,文中假设a(x,t)≥0,且当x∈Ω时,a(x,t)>0,解的稳定性是建立在一个合理的部分边界条件u(x,t)=0,(x,t)∈Σ1上,其中Σ1? ?Ω ×(0,T)仅仅是一...     

Vertex disjoint cycles in intersection graphs of bases of matroids

The intersection graph of bases of a matroid M=(E, B) is a graph G=G~I(M) with vertex set V(G) and edge set E(G) such that V(G)=B(M) and E(G)={BB′:|B∩B′|≠0, B, B′∈B(M), where the same notation is used for the vertices of G and the bases of M. Suppose that|V(G~I(M))| =n and k_1+k_2+…+k_p=n, where k_i is an integer, i=1, 2,…, p. In this paper, we prove that there is a partition of V(G~I(M)) into p parts V_1 , V_2,…, V_p such that |V_i| =k_i and the subgraph H_i induced by V_i contains a k_i-cycle when k_i ≥3, H_i is isomorphic to K_2 when k_i =2 and H_i is a single point when k_i =1.     

关于RN上非齐次Kirchhoff方程的注记

运用临界点理论中的Ekeland变分原理研究了非齐次Kirchhoff方程-(1+b∫RN|▽u|2dx)Δu+V(x)u=f(u)+h(x)x∈RN解的存在性,其中V∈C(RN,R)满足infNV(x)≥a1>0,这里a1>0是一个常数,更进一步,对每个M>0,meas({x∈RN:V(x)≤M})<∞,这里meas表示RN中的Lebesgue测度;f∈C(R,R+),f(0)=0,并且f(z)≡0当z<0;limz→0f(z)/z=0,limz→+∞f(z)/z=l<+∞.     

Generalized Galerkin approximations for pseudoinverses and operator equations of the first kind

0 IntroductionLeqtuiliHnebaer afo rHimlbaenrtd sap baoceun.dLeedtlianaenardfufnbceti oan baolu onndeHd ,serse--spectively. Our goal is to seekv∈Hsuchthata(u,v) =f(u) , u∈ H (1)As a rule, we call the expression (1) a variational equationandva solutiontothis equation.By virtue of the Riesz repre-sentationtheorem,there exists a uniquely determined boundedlinear operatorA:H→Hsuchthata(u,v) =〈u,Av〉, u,v∈ H,where〈·,·〉is the inner product onH.If the sesquilinearformais also with posit…     

一类非线性退化抛物方程的解

在带权常指数索伯列夫空间中讨论一类非线性方程的解的适定性问题。方程u_t=div(a(x)|▽u|~(p-2)▽u)+f(u,x,t)中,a(x)在边界退化。在一定条件下,方程解的稳定性可不依赖于边界条件而完全由初值控制。     

EXISTENCE AND REGULARITY OF POSITIVE SOLUTIONS OF THE ELLIPTIC EQUATIONS INVOLVING CRITICAL NONLINEAR AND SUBLINEAR

0 IntroductionLet Ωbe a smooth bounded domain in R~n with n>3, consider the problem of finding a function u(x)which satisfieswhere 0<α<1, 1 1 there is asolution of (0. 1) for any λ>0.The principal tecncihique of argument in [1] is using positive radial solution (0. 3) of equation -△u=u~2~n-1 in R~*.     

Super-Edge-Connectivity of G(k,d,s)(s≥k/2)

A graph G is super-edge-connected,for short super-λ,if every minimum edge-cut consists of edges adjacent to a vertex of minimum degree.Alphabet overlap graph G(k,d,s)is undirected,simple graph with vertex set V={v|v=1()kv…v;vi∈{1,2,…,d},i=1,…,k}.Two vertices u=(u1…uk)and v=(v1…vk)are adjacent if and only if us+i=vi or vs+i=ui(i=1,…,k-s).In particular G(k,d,1)is just an undirected de Bruijn graph.In this paper,we show that the diameter of G(k,d,s)is k s,the girth is 3.Finally,we prove that G(k,d,s)(s≥k/2)is super-λ.     

一类变系数非线性波方程弱解的存在性

文章主要用迦辽金逼近和能量估计法,证明带有变系数项div(a(x)u)的非线性波方程{utt-div(a(x)▽u)=f in UT u=g,ut=h on U×{T=0} u=0 on U×[0,T]在uT=ux[0,T]上弱解的存在性.     

BLOW-UP OF A CLASS OF SEMILINEAR PARABOLIC VARIATIONAL INEQUALITIES

In this paper we study the blow-up behavior for a class of semilinear parabolic variational inequalities;whereK = {u ∈L~2(0,T;H_0~1(Ω))|u(x,t)≥ψ(x) a. e. (x,t) ∈Ω×(0,T), u(x,0) = (x)},andis a uniformly elliptic operator.We prove the following main theorem.Theorem Let u(x,t) be a local solution of problem (I),u∈C(0,T;H~2(Ω)∩H_0~1(Q)),u_i∈L~2(0,T;L~2(Ω)), and following conditions are satisfied.(1) There exists a continuously differentiable function G(x,s) and a positive number α,such that