C1 Solutions of the Iterative Equation G(x, f(x), ..., fn(x)) =F(x)

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.     

(0,1, …, m - 2, m) INTERPOLATION FOR THE LAGUERREABSCISSAS

A necessary and sufficient condition of regularity of (0,1,…, m - 2, m) interpo-lation on the zeros of the Laguerre polynomials Ln(α) (x) (α≥ -1) in a manageable form is established. Meanwhile, the explicit representation of the fundamental polynomials, when they exist, is given. Moreover, it is shown that, if the problem of (0,1,…, m - 2, TO) interpolation has an infinity of solutions, then the general form of the solutions is f0(x) Cf1(x) with an arbitrary constant C.     

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]).     

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.     

Continuously decreasing solutions for polynomial-like iterative equations

Most known results on existence,uniqueness and stability for solutions of the polynomial-like iterative equation Σni=1λifi(x)=F(x) were obtained in the case of λ 1 = 0.In this paper,we construct C 0 decreasing solutions of the iterative equation in the case that λ 1 can vanish to answer the Leading Coefficient Problem.Moreover,we also give the necessary and sufficiently condition for uniqueness of solutions.     

ARE RATIONAL COMBINATIONS OF {x~λ_n},λ≥0 ALWAYS DENSE IN C[0,∞]?

The present paper shows that far any sequence of nomegatwe numbers with infinitely many distinct elements, the rational combinations of form a dense set (in the uniform norm on the positive axis) in the space of continuous factions f on [0) with lim f(x)=f(0). The last condition can be removed whenhas an infinity cluster point.     

Isochronous Centers and Isochronous Functions

Abstract We study isochronous centers of two classes of planar systems of ordinary differential equations.Forthe first class which is the Linard systems of the form =y-F(x),=-g(x) with a center at the origin, we provethat if g is isochronous(see Definiton 1.1),then the center is isochronous if and only if F≡0.For the secondclass which is the Hamiltonian systems of the form =-g(y),=f(x) with a center at the origin,we prove thatif f or g is isochronous,then the center is isochronous if and only if the other is also isochronous.     

ON BERNSTEIN POLYNOMIALS OF FUNCTIONS OF BOUNDED VARIATION OF ORDER p

1.IntroductionLet f be a function defined on[0,1],then the Bernstein poly-nomial B_n(f,x)of f is T.Popoviciu(see[1])and G.G.Lorentz studied the operator(1.1)and estimated the rate of convergence of B_n(f,x).F.Herzogand J.D. Hill,and others studied the convergence of(1.1)fordiscontinuous functions,they proved that if f is bounded on[0,1]and x is a discontinuity point of the first kind,then     

REGULARITY AND EXPLICIT REPRESENTATION OF (0,1,…,m - 2,m) INTERPOLATION ON THE ZEROS OF (1-x)P_(n-1)~(α,β)(x)

A necessary and sufficient condition of regularity of (0,1,…,m - 2,m) interpolation on the zeros of (1-x)P_(n-1)~(α,β)(x) (α> -1,β≥- 1) in a manageable form is established, where P_(n-1)~(α,β)(x) stands for the (n-1)th Jacobi polynomial. Meanwhile, the explicit representation of the fundamental polynomials when they exist, is given.     

A Tauberian theorem for l-adic sheaves on A~1

Let K ∈ L1(R) and let f ∈ L∞(R) be two functions on R.The convolution(Kf)(x) =∫R K(x-y)f(y)dy can be considered as an average of f with weight defined by K.Wiener's Tauberian theorem says that under suitable conditions,if lim x→∞(K f)(x) = lim x→∞(K A)(x) for some constant A,then lim x→∞ f(x) = A.We prove the following-adic analogue of this theorem:Suppose K,F,G are perverse-adic sheaves on the affine line A over an algebraically closed field of characteristic p(p=l).Under suitable conditions,if(K F)|η∞≌(K G)|η∞,then F|η∞≌ G|η∞,where η∞ is the spectrum of the local field of A at ∞.     

$(0,1,\cdots,m-2,m)$ Interpolation for the Laguerre Abscissas

A necessary and sufficient condition of regularity of $(0,1,\cdots,m-2,m)$ interpolation on the zeros of Laguerre polynomials $L_n^{(α)}(x) (α≥-1)$ in a manageable form is established. Meanwhile, the explicit representation of the fundamental polynomials, when they exist, is given. Moreover, it is shown that, if the problem of $(0,1,\cdots,m-2,m)$ interpolation has an infinity of solutions, then the general form of the solutions is $f_0(x)+Cf_1(x)$ with an arbitrary constant $C$.     

$(0,1,\cdots,m-2,m)$ Interpolation for the Laguerre Abscissas

A necessary and sufficient condition of regularity of $(0,1,\cdots,m-2,m)$ interpolation on the zeros of Laguerre polynomials $L^{(α)}_n(x) (α≥-1)$ in a manageable form is established. Meanwhile, the explicit representation of the fundamental polynomials, when they exist, is given. Moreover, it is shown that, if the problem of $(0,1,\cdots,m-2,m)$ interpolation has an infinity of solutions, then the general form of the solutions is $f_0(x)+Cf_1(x)$ with an arbitrary constant $C$.     

$\mathbb{R}^n$ 上 $p(x)$-Laplace型椭圆问题的无穷多解

讨论了涉及一般散度型椭圆算子($p(x)$-Laplace算子为其特例) 非线性偏微分方程的弱解存在性和多解性问题, 假定非线性项 $f_1,\, f_2$ 其中之一是超线性的, 且满足 Ambrosetti-Rabinowitz 条件, 另一项是次线性的. 所采用的方法依赖于变指数 Sobolev 空间 $W^{1,p(x)}(\mathbb{R}^n)$ 理论. 主要结果的证明基于喷泉定理和对偶喷泉定理.     

Singularity and Quadrature Regularity of $(0,1,\cdots,m-2,m)$-Interpolation on the Zeros of Jacobi Polynomials

In this paper we show that, if a problem of $(0,1,\cdots,m-2,m)$-interpolation on the zeros of the Jacobi polynomials $P^{\alpha,β}_n(x) (\alpha,β\geq -1)$ has infinite solutions, then the general form of the solutions is $f_0(x)+Cf(x)$ with an arbitrary constant $C$, where $f_0(x)$ and $f(x)$ are fixed polynomials of degree $\leq mn-1$. Moreover, the explicit form of $f(x)$ is given. A necessary and sufficient condition of quadrature regularity of the interpolation in a manageable form is also established.     

Existence and Uniqueness of Limiting Cycle of Equations $\[\dot x = \varphi (y) - F(x),\dot y = - g(x)\]$ and its Existence of Two and only Two Limiting Cycle

In this paper, we consider the existence of limiting cycle of the system of equations $\[\dot x = \varphi (y) - F(x),\dot y = - g(x)\]$(E) its existence and uniqueness, and its existence of two and only two limiting cycle. The theorems of existence and of existence and uniquence include following conditions: 1° All orbits of system (E) rotate round origin and not all orbits rotate round origin; 2° All or some of integral $\[\int_0^{ \pm \infty } {g(x)}dx \]$ and $\[\int_0^{ \pm \infty } {F'(x)} dx\]$ diverge or converge; 3° System (E) has one or two (one of them is saddle point) singular points. These theorems include following results: 1° All orbits of system (E), if not zero, tend to the unique cycle as $\[t \to + \infty \]$. 2° The result allow us to decide the place and the number of cycle etc. In the theorem of existence of two and only two limiting cycle, F(x) and g (x) needn't odd functions; number of zero point of F(jx) may be five or over five; F (x) may ascend or descend repeatedly in certain finite interval. Combining § 2 with § 3, in fact, we can give a result of the existence of n and only n limiting cycle of system (E).     

Almost Periodic Solutions of Equation $[\dot x = {x^3} + \lambda g(t)x + \mu f(t)\]$ and Their Stability

By using the Liapunov function and the contraction mapping principle, the author investigates the existence and stability of almost periodic solutions of the first order nonlinear equations $\frac{dx}{dt}=-h_1(x)+h_2(x)g(t)+f(t)$ and $\frac{dx}{dt}=r(t)x^n+\lambdag(t)x+\muf(t)$, where r(t), g(t), f(t) are given almost periodic functions, n(\geq 2) integer, and \lambda,\mu real parameters.     

${\mbox{\boldmath $R$}}^N$上奇异非线性多调和方程的正整体解

本文研究形如△((△nu)(p-1) )=f(|x|,u,|(?)u|)u-β,x∈RN的奇异非线性多调和方程在RN上的正整体解,此处P>1,β≥0是常数,n是自然数,f:R × R ×R →R 是一个连续函数, ξδ*:=sign(ξ)·|ξ|δ,,ξ∈R,δ>0,给出了该类方程具有无穷多个其渐进阶刚好为|x|2n的正整体解的充分条件与必要条件.这些结论可以推广到更一般的方程.     

关于数论函数$\sigma(n)$的一个注记

对于两个不相同的正整数$m$和$n$, 如果满足$\sigma(m)=\sigma(n)=m+n$, 则称之为一对亲和数, 这里$\sigma(n)=\sum_{d|n}d$.本文给出了$f(x,y)=x^{2^{x}}+y^{2^{x}}(x>y\geq{1},(x,y)=1)$不与任何正整数构成亲和数对的结论, 这里$x$,$y$具有不同的奇偶性, 即, 关于$z$的方程$\sigma(f(x,y))=\sigma(z)=f(x,y)+z$不存在正整数解.     

Multiplicity of Weak Solutions for a $(p(x), q(x))$-Kirchhoff Equation with Neumann Boundary Conditions

The aim of this study is to investigate the existence of infinitely many weak solutions for the $(p(x), q(x))$-Kirchhoff Neumann problem described by the following equation : \begin{equation*} \left\{\begin{array}{ll} -\left(a_{1}+a_{2}\int_{\Omega}\frac{1}{p(x)}|\nabla u|^{p(x)}dx\right)\Delta_{p(\cdot)}u-\left(b_{1}+b_{2}\int_{\Omega}\frac{1}{q(x)}|\nabla u|^{q(x)}dx\right)\Delta_{q(\cdot)}u\+\lambda(x)\Big(|u|^{p(x)-2} u+|u|^{q(x)-2} u\Big)= f_1(x,u)+f_2(x,u) &\mbox{ in } \Omega, \\frac{\partial u}{\partial \nu} =0 \quad &\mbox{on} \quad \partial\Omega.\end{array}\right. \end{equation*} By employing a critical point theorem proposed by B. Ricceri, which stems from a more comprehensive variational principle, we have successfully established the existence of infinitely many weak solutions for the aforementioned problem.