EXISTENCE AND MULTIPLICITY OF SOLUTIONS FOR A CLASS OF p（x）-BIHARMONIC EQUATIONS

In this paper,we study a class of p(x)-biharmonic equations with Navier boundary condition.Using the mountain pass theorem,fountain theorem,local linking theorem and symmetric mountain pass theorem,we establish the existence of at least one solution and infinitely many solutions of this problem,respectively.     

ON THE WEIGHTED VARIABLE EXPONENT AMALGAM SPACE W（L^p（x）, Lm^q）

In [4], a new family W（L^p（x）, Lm^q） of Wiener amalgam spaces was defined and investigated some properties of these spaces, where local component is a variable exponent Lebesgue space L^p（x） （R） and the global component is a weighted Lebesgue space Lm^q （R）. This present paper is a sequel to our work [4]. In Section 2, we discuss necessary and sufficient conditions for the equality W （L^p（x）, Lm^q） = L^q （R）. Later we give some characterization of Wiener amalgam space W （L^p（x）, Lm^q）.In Section 3 we define the Wiener amalgam space W （FL^p（x）, Lm^q） and investigate some properties of this space, where FL^p（x） is the image of L^p（x） under the Fourier transform. In Section 4, we discuss boundedness of the Hardy- Littlewood maximal operator between some Wiener amalgam spaces.     

广义函数δ(x1,…,x2m)和δ(x1,…,x2m-1)(○) I(x2m)的乘积

用非标准分析和广义函数的调和表示,给出广义函数δ(x1,···,x2m)和δ(x1,…,x2m-1)(○)I(x2m)的乘积.

Abstract：

In this paper, we get the product of distributions δ(x1,···, x2m) and δ(x1,···, x2m-1)(○)I(x2m), defined by using nonstandard analysis and harmonic representations of distribution.     

A note about the cyclotomic polynomial Фpq（x） and some related results

Abstract. The expression of cyclotomic polynomial Фpq （x） is concerned for a long time. A simple and explicit expression of Фpq （x） in Z[x] has been showed. The form of the factors of Фpq （x） over F2 and the upper, lower bounds of their Hamming weight are provided.     

On All Fractional （a,b, k）-Critical Graphs

Let a,b,k,r be nonnegative integers with 1≤a≤b and r≥2.LetG be a graph of order n with n(a+b)(r(a+b)-2)+ak/a.In this paper,we first show a characterization for all fractional(a,b,k)-critical graphs.Then using the result,we prove that G is all fractional(a,b,k)-critical if δ(G)≥(r-1)b2/a+k and |NG(x1)∪NG(x2)∪···∪NG(xr)|≥bn+ak/a+b for any independent subset {x1,x2,...,xr} in G.Furthermore,it is shown that the lower bound on the condition|NG(x1)∪NG(x2)∪···∪NG(xr)|≥bn+ak/a+b is best possible in some sense,and it is an extension of Lu's previous result.     

迭代函数方程$G(x,f(x),\ldots,f^n(x))=F(x)$的$C^1$解

In this paper we consider the iterative equation G(x,f(x),...,f n(x)) = F(x) on R,and give the existence of C 1 solutions near the fixed point of F,which generalize some results on the leading coefficient problem from the form of the polynomial-like iterative equations to the general form.     

STABILITY OF THE SOLUTION OF THE ITERATED EQUATION sum from i=1 to n λ_if(x)=F(x)

This note is a further work of discussion on the iterated equation λ_1f(x)+λ_2f~2(x)+…λ_nf~n(x)-P(x), following the author's conclusions of the existence and uniqueness of this equation. In this note we prove that the sufficient condition of the stability of the solution is the same one as the uniqueness of the solution (see [4]).     

一类带有变指数算子的非局部问题解的存在性

The purpose of this paper is to investigate a nonlocal Dirichlet problem with(p(x),q(x))-Laplacian-like operator originated from a capillary phenomena.Using the variational methods and the critical point theory,we establish the existence of infinitely many weak solutions for this problem.     

（K1（x），K2（x））-有限伸张映射

该文首先给出(K1(x),K2(x))一有限伸张映射的定义,然后研究其自我提高的可积性性质.最后利用Caccioppoli型不等式得到可去奇异性结果.     

Inverse of a Function f（x）=y

For a function y= f（x） to have an inverse function, f must be one-to-one. Then for each x in its domian there is exactly one y in its range; furthermore, to each y in the range, there corresponds exactly one x in the domain. The correspondence from the range of f onto the domian of f is, therefore, also a function. It is this function that is the inverse of f.     

A Class Hermite Pseudospectral Approximate with ω（x） ≡1 and Application to Reaction-diffusion Equation

In this paper, we continue the discussion of [12] to establish the Hermite pseudospectral method with weight ω（x） = 1. As an application, we consider the pseudospectral approximation of the reaction-diffusion equation on the whole line, we prove the existence of the approximate attractor and give the error estimate for the approximate solution.     

TRIPLE SYMMETRIC POSITIVE SOLUTIONS TO NONLINEAR BVP WITH INTEGRAL BOUNDARY VALUE CONDITIONS

In the paper,we consider a nonlinear boundary value problem with integral boundary value conditions.By a generalization of the Leggett-Williams fixed-point theorem,we provide sufficient conditions for the existence of at least three positive solutions to the problem.An example is introduced to demonstrate our result.     

ON THE EXISTENCE OF INFINITELY MANY CRITICAL POINTS OF THE EVEN FUNCTIONAL integral from n=Q to (F(x,u,Du)) integral from n=■Q to (G(x,u))

In this paper we deal with the existence of infinitely many critical points of the even functional I(u)=integral from n=Q to (F(x,u,Du)) integral from n=(?)Q to (G(x,u)), u∈W~(1,p)(Ω),where G(x, u)=integral from n=o to u (g(x,t)dt), under the weak structure conditions on F(x, u, q) by the Mountain Pass Lemma.     

ON THE RATE OF CONVERGENCE OF BEZIER VARIANT OF SZASZ—DURRMEYER OPERATORS

In the present paper, we introduce Szasz-Durrmeyer-Bezier operators M.,.(f,x) , which generalize the Szasz-Durrmeyer operators. Here we obtain an estimate on the rate of convergence of Mn,a(f,x) for functions of bounded variation. Our result extends and improves that of Sahai and Prasad and Gupta and Pant.     

SOLUTION TO p(x)-LAPLACIAN PROBLEM WITH HARDY TERM

We study the existence of solutions to a class of p(x)-Laplacian equations involving singular Hardy terms for nonlinear terms of the type f(x, t) = ±(-λ|t|m(x)-2t +|t|q(x)-2t). First we show the existence of infinitely many weak solutions for anyλ > 0, and next prove that if λ is large enough then there exists a nontrivial weak solution. Our approach relies on direct variational methods and the theory of variable exponent Lebesgue-Sobolev spaces.     

THE REGULARITY OF QUASI-MINIMA AND ω-MINIMA OF INTEGRAL FUNCTIONALS

In this article, we have two parts. In the first part, we are concerned with the locally Hlder continuity of quasi-minima of the following integral functional ∫Ωf(x, u, Du)dx, (1) where Ω is an open subset of Euclidean N-space (N ≥ 3), u:Ω→ R,the Carath′eodory function f satisfies the critical Sobolev exponent growth condition |Du|p* |u|p*-a(x) ≤ f(x,u,Du) ≤ L(|Du|p+|u|p* + a(x)), (2) where L≥1, 1pN,p* = Np/N-p , and a(x) is a nonnegative function that lies in a suitable Lp space. In the second part, we study the locally Hlder continuity of ω-minima of (1). Our method is to compare the ω-minima of (1) with the minima of corresponding function determined by its critical Sobolev exponent growth condition. Finally, we obtain the regularity by Ekeland’s variational principal.     

COMPLEX INTERPOLATION OF Lp（C,H）SPACES WITH RESPECT TO CULLEN-REGULAR

The aim of this paper is to prove a new version of the Riesz-Thorin interpolation theorem on LP(C,H).In the sense of Cullen-regular,we show Hadamard’s three-lines theorem by means of the Maximum modulus principle on a symmetric slice domain.In addition,two applications of the Riesz-Thorin theorem are presented.Finally,we investigate two kinds of Calderón’s complex interpolation methods in LP(C,H).     

APPROXIMATION PROPERTIES OF LAGRANGE INTERPOLATION POLYNOMIAL BASED ON THE ZEROS OF （1- x^2）cosnarccosx

We study some approximation properties of Lagrange interpolation polynomial based on the zeros of （1-x^2）cosnarccosx. By using a decomposition for f（x） ∈ C^τC^τ＋1 we obtain an estimate of ‖f（x） -Ln＋2（f, x）‖ which reflects the influence of the position of the x＇s and ω（f^（r＋1）,δ）j,j = 0, 1,... , s,on the error of approximation.     

THE BRANCHING CHAIN WITH DRIFT IN SPACE-TIME RANDOM ENVIRONMENT （I）： MODEL,MARKOV PROPERTY,MOMENTS

There are three parts in this article. In Section 1, we establish the model of branching chain with drift in space-time random environment （BCDSTRE）, i.e., the coupling of branching chain and random walk. In Section 2, we prove that any BCDSTRE must be a Markov chain in time random environment when we consider the distribution of the particles in space as a random element. In Section 3, we calculate the first-order moments and the second-order moments of BCDSTRE.     

OD-CHARACTERIZATION OF ALMOST SIMPLE GROUPS RELATED TO U6（2）

Let G be a finite group and π(G) = { p 1 , p 2 , ··· , p k } be the set of the primes dividing the order of G. We define its prime graph Γ(G) as follows. The vertex set of this graph is π(G), and two distinct vertices p, q are joined by an edge if and only if pq ∈π e (G). In this case, we write p ～ q. For p ∈π(G), put deg(p) := |{ q ∈π(G) | p ～ q }| , which is called the degree of p. We also define D(G) := (deg(p 1 ), deg(p 2 ), ··· , deg(p k )), where p 1 < p 2 < ··· < p k , which is called the degree pattern of G. We say a group G is k-fold OD-characterizable if there exist exactly k non-isomorphic finite groups with the same order and degree pattern as G. Specially, a 1-fold OD-characterizable group is simply called an OD-characterizable group. Let L := U 6 (2). In this article, we classify all finite groups with the same order and degree pattern as an almost simple groups related to L. In fact, we prove that L and L.2 are OD-characterizable, L.3 is 3-fold OD-characterizable, and L.S 3 is 5-fold OD-characterizable.