GLOBAL ASYMPTOTIC STABILITY OF x_(n 1)=F(x_n)g(x_(n-1))

1IntroductionConsiderthedifferenceequationwithpositiveinitialconditionswherethefunctionFandgsatisfythefollowinghypotheses:(HI)F,geC[(0,co),(0,co)1;(Hi)Fisincreasingandgisdecreasing;No.3LiL.T.,GLOBALASYMPTOTICSTABILITYOFxu 1=F(x.)g(x.~1)519(Hs)Theequationx…     

关于数论函数方程φ(x_1…x_(n-1)x_n)=m(φ(x_1)+…+φ(x_(n-1))+φ(x_n))

运用Euler函数的性质证明了:当n>1时,方程φ(x_1…x_(n-1)x_n)=m(φ(x_1)+…+φ(x_(n-1))+φ(x_n))仅有有限多组正整数解(x_1,…,x_(n-1),x_n),得到了这些解都满足max{x_1,…,x_(n-1),x_n}≤2m⁴(n-1)4(n-1)²n2n².     

On the Recursive Sequence x n+1 = α + (βx n−1)/(1 + g(xn ))

We investigate the boundedness character, the oscillatory and periodic nature and global attractivity of the nonnegative solutions of the difference equation where the parameters α and β are nonnegative real numbers and g(x) is a continuous function on [0, ∞), which satisfies some additional conditions.     

差分方程x_(n+1)=x_n+(x_n~α)/(n~β)解的渐近性

我们研究差分方程(1)的解的渐近性和全局吸引性.     

ALMOST PERIODIC SOLUTIONS OF THE EQUATIONx = x~3 + λg (t) x + μf (t) AND THEIR STABILITY

By using the Liapunov function and the contraction mapping principle, the authorinvestigates the existence and stability of almost periodic solutions of the first ordernonlinear equations dx/dt=-h_1(x)+h_2(x)g(t)+f(t)and dx/dt=r(t)x~n+λg(t)x+μf(t),where r(t), g(t), f(t) are given almost periodic functions, n(≥2) integer, and λ, μ realparameters.     

K周期系数差分方程x_(n+1)=p_n+(x_(n-3s-2))/(x_(n-s))的动力学性质

研究非自制差分方程x_(n+1)=p_n+(x_(n-3s-2)/x_(n-8)),其中p_n>0是一K周期序列.最后得到了方程解的有界性的一些充分条件.     

REGULARITY OF H~1 ∩L~(n((γ-1)/(2-γ))) WEAK SOLUTIONS FOR NONLINEAR ELLIPTIC SYSTEMS

Let us consider the following elliptic systems of second order-D_α(A_i~α(x, u, Du))=B_4(x, u, Du), i=1, …, N, x∈Q(?)R~n, n≥3 (1) and supposeⅰ) |A_i~α(x, u, Du)|≤L(1+|Du|);ⅱ) (1+|p|)~(-1)A_i~α(x, u, p)are H(?)lder-continuous functions with some exponent δ on (?)×R~N uniformly with respect to p, i.e.ⅲ) A_i~α(x, u, p) are differentiable function in p with bounded and continuous derivativesⅳ)ⅴ) for all u∈H_(loc)~1(Ω, R~N)∩L~(n(γ-1)/(2-γ))(Ω, R~N), B(x, u, Du)is ineasurable and |B(x, u, p)|≤a(|p|~γ+|u|~τ)+b(x), where 1+2/n<γ<2, τ≤max((n+2)/(n-2), (γ-1)/(2-γ)-ε), (?)ε>0, b(x)∈L2n/(n+2), n~2/(n+2)+e(Ω), (?)ε>0.Remarks. Only bounded open set Q will be considered in this paper; for all p≥1, λ≥0, which is clled a Morrey Space.Let assumptions ⅰ)-ⅳ) hold, Giaquinta and Modica have proved the regularity of both the H~1 weak solutions of (1) under controllable growth condition |B|≤α(|p|~γ+|u|~((n+2)/(n-2))+b, 0<γ≤1+2/n and the H~1∩L~∞ weak solutions of (1) under natural     

Stability of solutions for the recursive sequence xn+1=(α-βxn)/(γ+g(xn-k))

In this paper, we investigate the global behavior of the recursive sequence$x_{n+1}=\frac{\alpha -\beta x_n}{\gamma +g(x_{n-k})}\mathrm{for}n=0,1,\dots ,$where $\alpha ,\beta ,\gamma ⩾0$ and $g(x)$ is a continuous function defined on $(-\infty ,\infty )$, which satisfies some additional conditions.     

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 recursive sequencex n +1=α-(x n /x n −1)

We study the global asymptotic stability, global attractivity, boundedness character, and periodic nature of all positive solutions and all negative solutions of the difference equation $$x_{n + 1} = \alpha - \frac{{x_n }}{{x_{n - 1} }}, n = 0,1,...,$$ where α∈R is a real number, and the initial conditionsx?1,x ₀ are arbitrary real numbers.     

ON THE UNIQUENESS OF THE EXTREMAL FUNCTIONS OF LANDAU TYPE PROBLEM FOR THE DIFFERENTIAL OFERATOR L_(n+1)(D)=multiply from i=0 to n (d-t_i)

Introduction The wellknown landau's problem and its generalizations havebeen studied extensively (see[1-5]). Recently, A. Sharma and J.Tzimbalario presented the generalized Landau inequalities for     

2π-PERIODIC SOLUTION OF THE EQUATION x + f(x)x + bx + g(x(t - т)) = p(t)

In this paper, time delay Lienard's equations are considered, by using the theory of concidence degree, a sufficient condition of existence of at least one 2π-periodic solution is obtained.     

差分方程xn+1=xn+xαn/nβ解的渐近性

我们研究差分方程(1)的解的渐近性和全局吸引性.     

THE PLANCHEREL FORMULA FOR THE LINE BUNDLES ON SL(n + 1, R)/S(GL(1, R) × GL(n,R))

In this paper we obtain the Plancherel formula for the spaces of L~2-sections of the line bundles over the pseudo-Riemannian symmetric space G/H where G = SL(n + 1, R)and H = S(GL(1, R) × GL(n, R)). The Plancherel formula is given in an explicit form by means of spherical distributions associated with the character χ——λof the subgroup H. We follow the method of Faraut, Kosters and van Dijk.     

级数sum from n=1 to ∞(n~αsin(n~β))的收敛性

通过推导给出了使级数∑∞n=1nαsin(nβ)收敛的α与β的值.     

Iteration xn+1=αn+1f(xn) + (1 - αn+1)Tn+1xn for an Infinite Family of Nonexpansive Maps {Tn}∞n=1

Under the framework of uniformly smooth Banach spaces, Chang[1] proved in 2006 that the sequence {xn} generated by the iteration xn+1 = αn+1f(xn) + (1 - αn+1)Tn+1xn converges strongly to a common fixed point of a finite family of nonexpansive maps {Tn}, where f : C → C is a contraction. However, in this paper, the author considers the iteration in more general case that {Tn} is an infinite family of nonexpansive maps, and proves that Chang's result holds still in the setting of reflexive Banach spaces with the weakly sequentially continuous duality mapping.     

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.     

实系数的复系统■=-x,■=y-(αx+βx~(2n+1))至少有n个极限环(αβ≠0)

本文在[2]的基础上引入实系数的复微分系统的枝桠平面的概念。并证明:实系数的复微分系统(1):y=-x,x=y-(αx+βx~(2n+1)恰有2n个枝桠平面。在这2n个枝桠平面上系统(1)的极限环数的总和为n。又当β≠0(α≠0)和α(β)用变号时,n个枝桠平面上的n个极限环经过原点(无穷远)跳到另n个枝桠平面上。     

方程x~((n))+f_1(t)x~((n-1))+f_2(t)x~((n-2))+…+f_(n-1)(t)■+f_n(t)x=0解的稳定性

考虑方程z~((n))+f_1(t)x~((n-1))+f_2(t)x~((n-2))+……+f_(n-1)(t)x+f_n(t)=0 (1)在现有文献中,对方程(1)的研究几乎能集中在二阶,对于 n>2的情形,很少见到。本文应用变换技巧以及.直接方法,特别利用文[9]的推广了的方法研究了 n 为任意正整数的方程(1)的平凡解的稳定性,得到了其平凡解全局一致渐近稳定性的充分条件。特别当 n=2时所得的结果包含了文献[1-8]的有关结果。有些结果就非文[8]的条件所能得到,如本文定理4、5及其推论1。