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.     

Das charakteristische Anfangswertproblem füru_{x_1 x_2 x_3 } = f unter Carathéodory-Voraussetzungen

Ohne Zusammenfassung     

On the rational difference equation x_{n}=1+\frac{(1-x_{n-k})(1-x_{n-l})(1-x_{n-m})}{x_{n-k}+x_{n-l}+x_{n-m}}

In this note we consider the following higher order rational difference equations $$x_{n}=1+\frac{(1-x_{n-k})(1-x_{n-l})(1-x_{n-m})}{x_{n-k}+x_{n-l}+x_{n-m}},\quad n=0,1,\ldots,$$ where 1≤k<l<m, and the initial values x _?m ,x _?m+1,…,x _?1 are positive numbers. We give some sufficient conditions for the persistence of positive solutions for the above equation, and prove that the positive equilibrium point of this equation is globally asymptotically stable.     

On boundedness of solutions of the difference equation x_{n+1}=p+\frac{x_{n-1}}{x_{n}} for p<1

In this paper, we study the difference equation $$x_{n+1}=p+\frac{x_{n-1}}{x_n}, \quad n=0,1,\ldots, $$ where initial values x _?1,x ₀∈(0,+∞) and 0<p<1, and obtain the set of all initial values (x _?1,x ₀)∈(0,+∞)×(0,+∞) such that the positive solutions $\{x_{n}\}_{n=-1}^{\infty}$ are bounded. This answers the Open problem 4.8.11 proposed by Kulenovic and Ladas (Dynamics of Second Order Rational Difference Equations, with Open Problems and Conjectures, 2002).     

递推数列x_(n+1)=f(x_n)的单调性与收敛性讨论

利用函数单调递增对递推数列xn+1=f(xn)单调性进行讨论,在对递推数列收敛性作分析的基础上,得到使得递推数列收敛的初始迭代值的区域,讨论的方法可以用于类似问题的研究.     

On the simultaneous Diophantine equations $$ m \cdot (x_1^k+x_2^k+ \cdots + x_{t_1}^k)=n \cdot (y_1^k+y_2^k+ \cdots y_{t_2}^k)$$; $$k=1,3$$k=1,3

In this paper, we solve the simultaneous Diophantine equations \(m \cdot ( x_{1}^k+ x_{2}^k +\cdots + x_{t_1}^k)=n \cdot (y_{1}^k+ y_{2}^k +\cdots + y_{t_2}^k )\), \(k=1,3\), where \( t_1, t_2\ge 3\), and m, n are fixed arbitrary and relatively prime positive integers. This is done by choosing two appropriate trivial parametric solutions and obtaining infinitely many nontrivial parametric solutions. Also we work out some examples, in particular the Diophantine systems of \(A^k+B^k+C^k=D^k+E^k\), \(k=1,3\).     

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.     

论一阶模糊差分方程x_(n+1)=Ax_n+B

讨论一阶模糊差分方程xn+1=Axn+B(n=0,1,…)正解的存在性、有界性及正解的渐近表现.其中(xn)是正模糊数数列, A,B,x0是正模糊数.     

On the rational recursive sequence x_{n+1}=\frac{\alpha+\beta x_{n-k}}{\gamma-x_{n}}

In this paper, we investigate the global asymptotic stability, the periodicity nature and the boundedness character of the positive solutions of the difference equation x _n+1=(α+β x _n?k )/(γ?x _n ) where n=0,1,2,… and k∈{1,2,…}. The parameters α≥0, γ,β>0 and the initial conditions x _?k , x _?k+1,…,x _?1,x ₀ are real positive numbers. We show that the positive equilibrium point of this equation is a global attractor with a basin that depends on certain conditions posed on the coefficients α,β,γ.     

Dynamics of the max-type difference equation x_{n}=\max\{\frac{ 1}{ x_{n-m}} , \frac{A_{n} }{x_{n-r} }\}

In this paper, we study the periodicity, the boundedness and the convergence of the following max-type difference equation $$x_n =\max\biggl\{\frac{ 1}{ x_{n-m}} , \frac{A_n }{x_{n-r} }\biggr \},\quad n =0, 1,2,\ldots,$$ where $\{A_{n}\}^{+\infty}_{n=0}$ is a periodic sequence with period k and A _n ??(0,1) for every n??0, m??{1,2} and r??{2,3,??} with m<r, the initial values x _?r ,??,x _?1??(0,+??). The special case when $m = 1, r = 2, \{A_{n}\}^{+\infty}_{ n=0}$ is a periodic sequence with period k and A _n ??(0,1) for every n??0 has been completely investigated by Y.?Chen. Here we extend his results to the general case.     

On the Number of Solutions of the Equation a_{1}x_{1}^{m_{1}}+\cdots +a_{n}x_{n}^{m_{n}}=bx_{1}\cdots x_{n} in a Finite Field

We obtain an explicit formula for the number of solutions of a special equation in a finite field under a certain restriction on the exponents. Mathematics Subject Classifications (2000) primary: 11G25, secondary: 11T24.     

On the Recursive Sequence x_{n + 1} = {{ \mgreek{g} x_{n-1} + \mgreek{d} x_{n-2}} \over {Cx_{n-1} + Dx_{n-2}}}

We investigate the global character of solutions of the equation in the title with positive parameters and positive initial conditions. We obtain results about the global attractivity of the equilibrium, the existence and attractivity of the period-two solution and the semicycles.     

On the Dynamics of x_{\bf{n} + \bf{1}} = {{ \mgreek{a} + \mgreek{g} x_{n-1} + \mgreek{d} x_{n-2}} \over {A + x_{n-2}}}

We investigate the global stability, the periodic character and the boundedness nature of solutions of the equation in the title for all admissible nonnegative values of the parameters and the initial conditions. We show that the solutions exhibit a trichotomy character depending on how the parameter γ compares to the sum of the parameters δ and A.     

Hyperbolic reflection groups associated to the quadratic forms {-3x_0^2 + x_1^2 + \cdots + x_n^2}

We determine the maximal hyperbolic reflection groups associated to the quadratic forms ${-3x_0^2 + x_1^2 + \cdots + x_n^2, n \ge 2}$ , and present the Coxeter schemes of their fundamental polyhedra. These groups exist in dimensions up to 13, and a proof is given that in higher dimensions these quadratic forms are not reflective.