ON THE L_∞ CONVERGENCE AND THE EXTRAPOLATION METHOD OF A DIFFERENCE SCHEME FOR NONLOCAL PARABOLIC EQUATION WITH NATURAL BOUNDARY CONDITIONS

1. IntroductionNoulocal parabolic equaions have many aPplicatiOns. FOr exaInPle, in COn8W fiuld fiowin a saturated porous mallum, the eqwtion gOvwi the pore presure p(nt) in an odarcylindrical roch sWle is given in [1] astopether withwher c is a material constat with driensiom of velOcity, the COeffilen of conSOlidatbo, ut tyare shear coefficieats and n a material constat, f dentes radial distance and t thae. If t isrePaced by ct and be by q in (1.1), we getVarious ndtial and boundny co…     

Triebel-Lizorkin space boundedness of Marcinkiewicz integrals associated to surfaces

In the present paper, we consider the boundedness of Marcinkiewicz integral operator μΩ,h,φalong a surface Γ = {x = φ(|y|)y/|y|)} on the Triebel-Lizorkin space Fαp,q(Rn) for Ωbelonging to H1(Sn-1) and some class W Fα(Sn-1), which relates to Grafakos-Stefanov class.Some previous results are extended and improved.     

On a Proof of Roth''''s Theorem

We denote by M_(n,m)(F) the set of all n×m matrices over the field F and by M_n(F) the set of all n×n matrices over the field F. W. E. Roth has shown the following theorem in 1952, [1]. Theorem Let A∈M_n(F),B∈M_m(F) and C∈M_(n,m)(F), then the matrix equation AX-YB=C (1) has a solution X, Y∈M_(n,m)(F) if and only if the matrices     

ON THE EXISTENCE OF INFINITELY MANY CRITICAL POINTS OF THE EVEN FUNCTIONAL integral from n=Ω to (F(x, u, Du)dx) IN W_0~(1,p)

§1. Introduction In the paper [1], Pacella and Tricarico proved that the existence of infinitely many critical points of even functional integral from Ω to (F(x,u,Du)dx) in space W_0~(1,2). As far as our knowledge goes, the paper [1] is the first investigation of general functional integral from Ω to (F(x,u,Du)dx) in Hilbert space W_0~(1,2), but the techniques used in the paper[1], was restricted to Hilbert space. In the same year as when [1] had published, we independently proved [2] the     

EXISTENCE THEOREM AND FINITE ELEMENT METHOD FOR STATIC PROBLEMS OF A CLASS OF NONLINEAR HYPERELASTIC SHELLS

In this paper, various boundary value problems of hyperelastic shells are considered. It is assumed that the storede-nergy function W(x, F) of the material,of which the shell is made, satisfies polyconvex conditions proposed by Ball~([2]).Existence of minimum points of the total energy of the shell in suitably chosen function spaces, and in suitably chosen finite element spaces is proved. Convergence of the finite element solutions is proved under certain regular conditions on the minimum points and some additional assumptions on W(x, F). A Gradient type computing scheme for solving the finite element solutions is given, and global convergent result is obtained.     

Maximal Dimension of Invariant Subspaces to Systems of Nonlinear Evolution Equations

In this paper,the dimension of invariant subspaces admitted by nonlinear systems is estimated under certain conditions.It is shown that if the two-component nonlinear vector differential operator F=(F 1,F 2) with orders {k 1,k 2 } (k 1 ≥ k 2) preserves the invariant subspace W 1 n 1 × W 2 n 2 (n 1 ≥ n 2),then n 1 n 2 ≤ k 2,n 1 ≤ 2(k 1 + k 2) + 1,where W q n q is the space generated by solutions of a linear ordinary differential equation of order n q (q=1,2).Several examples including the (1+1)-dimensional diffusion system and Ito 's type,Drinfel'd-Sokolov-Wilson's type and Whitham-Broer-Kaup's type equations are presented to illustrate the result.Furthermore,the estimate of dimension for m-component nonlinear systems is also given.     

关于Ahuja的一个结果

In [1], O. P. Ahuja established the following result: If F is an element of R_n(α)for n≥0 and 0≤α<1, F(z)=(c+1)/(z~c)integral from 0 to z f(t)t~(c-1)dt with |z|<1,Rec≥-α, and 0≤β<1. then the function f is an element of R_n(β) for |z|相似文献   

L~0-CONVEX COMPACTNESS AND RANDOM NORMAL STRUCTURE IN L~0(F,B)

Let(B,||·||) be a Banach space,(?, F, P) a probability space, and L~0(F, B) the set of equivalence classes of strong random elements(or strongly measurable functions) from(?, F, P) to(B,||·||). It is well known that L~0(F, B) becomes a complete random normed module, which has played an important role in the process of applications of random normed modules to the theory of Lebesgue-Bochner function spaces and random operator theory. Let V be a closed convex subset of B and L~0(F, V) the set of equivalence classes of strong random elements from(?, F, P) to V. The central purpose of this article is to prove the following two results:(1) L~0(F, V) is L~0-convexly compact if and only if V is weakly compact;(2)L~0(F, V) has random normal structure if V is weakly compact and has normal structure.As an application, a general random fixed point theorem for a strong random nonexpansive operator is given, which generalizes and improves several well known results. We hope that our new method, namely skillfully combining measurable selection theorems, the theory of random normed modules, and Banach space techniques, can be applied in the other related aspects.     

Matrix Algebra Motivated by Essentially Stochastic Matrices

A matrix of order n whose row sums are all equal to 1 is called an essentially stochastic matrix (see Johnsen [4]). We extend this notion as the following. Let F be a field of characteristic 0 or a prime greater than n. M_n(F) denotes the set of all n×n matrices over F. Let t be an elernent of F. A matrix A=(a_(ij)) in M_n(F) is called essentially t-stochastic' provided its row sums are each equal to t. We denote by R_n(t) the set of all essentially t-stochastic matrices over F. We shall mainly study R_n(0) and. Our main references are Johnson [2,4] and Kim [5].     

THE UNIQUENESS OF THE SOLUTION OF LIENARD SYSTEM TO THE INITIAL VALUE PROBLEM

1 IntroductionThe Li6nard systemis a very important nonlinear system. Many authors have studied the asymptotic behavior of its solutions. But for this purpose we require first the uniquenessof its solutions to the initial value problem. Most references did not discuss itin detail, they generally assume that F(x) and g(x) are continuous and satisfythe conditions for the uniqueness of the solution. Only few authors gave someconcrete conditions.FOr example Dragilev [1] required that F(x) and g…     

PICARD ITERATION FOR NONSMOOTH EQUATIONS

1. IntroductionConsider the following nonsmooth equationsF(x) = 0 (l)where F: R" - R" is LipsChitz continuous. A lot of work has been done and is bellg doneto deal with (1). It is basicly a genera1ization of the cIassic Newton method [8,10,11,14],Newton-lthe methods[1,18] and quasiNewton methods [6,7]. As it is discussed in [7], the latter,quasiNewton methods, seem to be lindted when aPplied to nonsmooth caJse in that a boundof the deterioration of uPdating matrir can not be maintained w…     

Asymptotic behavior of multistep runge-kutta methods for systems of delay differential equations

1. IntroductionIn order to assess the asymptotic behavior of numerical methods for DDEs, much attention has been given in the literature to the scalar case (cL [1-6]). UP to now) only partialresults (of. [7-10]) have dealt with the delay systemswhere y(t) = (yi(t), so(t),' ) yp(t))" E Cd, which is unknown for t > 0, L and M areconstat complex p x Hmatrices, T > 0 is a constat delay and W(t) 6 CP is a specifiedinitial function.In [111, C.J. Zhang and S.Z. Zhou made an investigation on…     

L~P SOLUTION OF GENERAL MEAN-FIELD BSDES WITH CONTINUOUS COEFFICIENTS

In this paper we consider one dimensional mean-field backward stochastic differential equations (BSDEs) under weak assumptions on the coefficient.Unlike[3],the generator of our mean-field BSDEs depends not only on the solution (Y,Z) but also on the law P_(Y )of Y.The first part of the paper is devoted to the existence and uniqueness of solutions in L~p,1p≤2,where the monotonicity conditions are satisfied.Next,we show that if the generator f is uniformly continuous in (μ,y,z),uniformly with respect to (t,ω),and if the terminal valueξbelongs to L~p(?,F,P) with 1p≤2,the mean-field BSDE has a unique L~psolution.     

Linear Isometries of S^D Spaces

This paper generalizes the result about linear isometries of S^pspaces given by W.P.Novinger and D.M.Oberlin[2] for the unite dise of C to the bounded symmetrie domains of C^n。     

SOME RESULTS ON PRODUCT COMPLEX FINSLER MANIFOLDS

Let (M, F ) be the product complex Finsler manifold of two strongly pseudoconvex complex Finsler manifolds (M 1 , F 1 ) and (M 2 , F 2 ). In this paper, we obtain the relationship between the Chern Finsler connection coefficients Γ i ; k associated to F and the Chern Finsler connection coefficients Γ a ; c , Γα ; γ associated to F 1 , F 2 , respectively. As applications we prove that, if both (M 1 , F 1 ) and (M 2 , F 2 ) are strongly Ka¨hler Finsler (complex Berwald, or locally complex Minkowski, respectively) manifolds, so does (M, F ). Furthermore, we prove that the holomorphic curvature K F = 0 if and only if K F1 = 0 and K F2 = 0.     

ON UNIFORM ASYMPTOTIC STABILITY OF INFINITE DELAY DIFFERENCE EQUATIONS

51. IntroductionThe aim of thi8 paPer is to establish the stability criteria for the indnite de1ay differenceequations of the formx(n 1) = F(n,x.) for n E Z, (1'l)where F: Z x CH -- Rk, Z denotes the integer set, Rh is the n-dimensional Euclidean space,CH = {T E C: llWII < H} fOr some constan H > 0, whileC = {yt: {... l --2, --1,0} - Rk l W is bounded}withllytIl = sup IW(8)I for W E C,8<0and x.(8) = x(n 8) for 8 5 0. Here, and in the sequel, l' I is a norm in Rk, and we atwa…     

The relations among the three kinds of conditional risk measures

Let(Ω,E,P)be a probability space,F a sub-σ-algebra of E,Lp(E)(1 p+∞)the classical function space and Lp F(E)the L0(F)-module generated by Lp(E),which can be made into a random normed module in a natural way.Up to the present time,there are three kinds of conditional risk measures,whose model spaces are L∞(E),Lp(E)(1 p+∞)and Lp F(E)(1 p+∞)respectively,and a conditional convex dual representation theorem has been established for each kind.The purpose of this paper is to study the relations among the three kinds of conditional risk measures together with their representation theorems.We first establish the relation between Lp(E)and Lp F(E),namely Lp F(E)=Hcc(Lp(E)),which shows that Lp F(E)is exactly the countable concatenation hull of Lp(E).Based on the precise relation,we then prove that every L0(F)-convex Lp(E)-conditional risk measure(1 p+∞)can be uniquely extended to an L0(F)-convex Lp F(E)-conditional risk measure and that the dual representation theorem of the former can also be regarded as a special case of that of the latter,which shows that the study of Lp-conditional risk measures can be incorporated into that of Lp F(E)-conditional risk measures.In particular,in the process we find that combining the countable concatenation hull of a set and the local property of conditional risk measures is a very useful analytic skill that may considerably simplify and improve the study of L0-convex conditional risk measures.∞     

New Identities for Weak KAM Theory

This paper records for the Hamiltonian H =1/2│P│2 + W(x) some old and new identities relevant for the PDE/variational approach to weak KAM theory.     

A Note on Restricted Representations of the Witt Superalgebras

Let F be an algebraically closed field of prime characteristic p 〉 3, and W（n） the Witt superalgebra over F, which is the Lie superalgebra of superderivations of the Grassmann algebra in n indeterminates. The dimensions of simple atypical modules in the restricted supermodule category for W（n） are precisely calculated in this paper, and thereby the dimensions of all simple modules can be precisely given. Moreover, the restricted supermodule category for W（n） is proved to have one block.     

OSCILLATION CRITERIA FOR SECOND ORDER QUASILINEAR DIFFERENTIAL EQUATIONS

1 IntroductionThis paper is concerned with the problem of oscillation of second-orderquasilinear differential equationsFOr equation (1), the following conditions, which are referred to (FO), arealways assumed to be valid(a) a > 0 is a constant;(b) F: [to, co) x R x R x R x R - R is a continuous function;(c) sgn F(t, x) u? v3 w) = sgn x for each t 2 to and x, u? v? w E R,(d) r(t) E C([to, co); (0, co)) and jco ddt = co.' (e) T: [to, co) - R is continuous, T(t) 5 t for t 2 to and ill7T(t)…