共查询到20条相似文献,搜索用时 31 毫秒
1.
Marwan Aloqeili 《Journal of Applied Mathematics and Computing》2007,25(1-2):375-382
We study, firstly, the dynamics of the difference equation $x_{n + 1} = \alpha + \frac{{x_n^p }}{{x_{n - 1}^p }}$ , withp ∈ (0,1) and α∈ [0, ∞). Then, we generalize our results to the (k + 1)th order difference equation $x_{n + 1} = \alpha + \frac{{x_n^p }}{{x_{n - k}^p }}$ ,k = 2, 3,... with positive initial conditions. 相似文献
2.
In this paper we consider the difference equation $$x_{n + 1} = \frac{{a + bx_{n - k} - cx_{n - m} }}{{1 + g(x_{n - 1} )}},$$ wherea, b, c are nonegative real numbers,k, l, m are nonnegative integers andg(x) is a nonegative real function. The oscillatory and periodic character, the boundedness and the stability of positive solutions of the equation is investigated. The existence and nonexistence of two-period positive solutions are investigated in details. In the last section of the paper we consider a generalization of the equation. 相似文献
3.
Taixiang Sun Xin Wu Qiuli He Hongjian Xi 《Journal of Applied Mathematics and Computing》2014,44(1-2):61-68
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). 相似文献
4.
Taixiang Sun Hongjian Xi Caihong Han Bin Qin 《Journal of Applied Mathematics and Computing》2012,38(1-2):173-180
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. 相似文献
5.
Stevo Stević 《Journal of Applied Mathematics and Computing》2005,18(1-2):229-234
The boundedness, global attractivity, oscillatory and asymptotic periodicity of the positive solutions of the difference equation of the form $$x_{n + 1} = \alpha + \frac{{x_{n - 1}^p }}{{x_n^p }}, n = 0,1,...$$ is investigated, where all the coefficients are nonnegative real numbers. 相似文献
6.
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 \(A^k+B^k+C^k=D^k+E^k\), \(k=1,3\). 相似文献
7.
H. M. El-Owaidy A. M. Ahmed Z. Elsady 《Journal of Applied Mathematics and Computing》2004,16(1-2):243-249
Our aim in this paper is to investigate the global attractivity of the recursive sequence $$x_{n + 1} = \frac{{\alpha - \beta x_{n - k} }}{{\gamma + x_n }},$$ where α, β, γ >0 andk=1,2,… We show that the positive equilibrium point of the equation is a global attractor with a basin that depends on certain conditions posed on the coefficients. 相似文献
8.
John Mcleod 《Geometriae Dedicata》2011,152(1):1-16
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. 相似文献
9.
We investigate the boundedness nature of positive solutions of the difference equation $$ x_{n + 1} = max\left\{ {\frac{{A_n }} {{X_n }},\frac{{B_n }} {{X_{n - 2} }}} \right\},n = 0,1,..., $$ where {A n } n=0 ∞ and {B n } n=0 ∞ are periodic sequences of positive real numbers. 相似文献
10.
For a holomorphic proper map F from the ball $\mathbb{B}^{n+1}$ into $\mathbb{B}^{N+1}$ that is C 3 smooth up to the boundary, the image $M=F(\partial\mathbb{B}^{n})$ is an immersed CR submanifold in the sphere $\partial \mathbb{B}^{N+1}$ on which some second fundamental forms II M and $\mathit{II}^{CR}_{M}$ can be defined. It is shown that when 4??n+1<N+1??4n?3, F is linear fractional if and only if $\mathit{II}_{M} - \mathit{II}_{M}^{CR} \equiv 0$ . 相似文献
11.
H. M. El-Owaidy A. M. Ahmed M. S. Mousa 《Journal of Applied Mathematics and Computing》2003,12(1-2):31-37
In this paper, we investigate local stability, oscillation and boundeness character of positive solutions of the difference equation $$x_{n + 1} = \alpha + \frac{{x_{n - 1} ^p }}{{x_n ^p }},n = 0,1,...$$ under specified conditions. 相似文献
12.
V. N. Matus 《Mathematical Notes》1972,12(3):614-618
The variety \(\mathfrak{u}_{m,n} \) is defined by the system of n-ary operations ωi,..., ωm, the system of m-ary operations ?i,..., ?n, 1≤ m ≤ n, and the system of identities $$\begin{gathered} x_1 ...x_n \omega _1 ...x_1 ...x_n \omega _m \varphi _i = x_i (i = 1,...,n), \hfill \\ x_1 ...x_m \varphi _1 ...x_1 ...x_m \varphi _n \omega _j = x_j (i = 1,...,m), \hfill \\ \end{gathered} $$ It is proved in this paper that the subalgebra U of the free product \(\Pi _{i \in I}^* A_i \) of the algebras Ai (i ε I) can be expanded as the free product of nonempty intersections U ∩ Ai (i ε I) and a free algebra. 相似文献
13.
Stevo Stević 《Journal of Applied Mathematics and Computing》2006,21(1-2):223-232
The boundedness, global attractivity, oscillatory and asymptotic periodicity of the nonnegative solutions of the difference equation $$x_{n + 1} = \frac{{ax_{n - 2m + 1}^p }}{{b + cx_{n - 2k}^{p - 1} }}, n = 0, 1,...$$ wherem, k ∈ N, 2k > 2m?1,a, b, c are nonnegative real numbers andp < 1, are investigated. 相似文献
14.
ChaoHua Jia 《中国科学 数学(英文版)》2012,55(3):465-474
If n is a positive integer,let f (n) denote the number of positive integer solutions (n 1,n 2,n 3) of the Diophantine equation 4/n=1/n1 + 1/n2 + 1/n3.For the prime number p,f (p) can be split into f 1 (p) + f 2 (p),where f i (p) (i=1,2) counts those solutions with exactly i of denominators n 1,n 2,n 3 divisible by p.In this paper,we shall study the estimate for mean values ∑ p相似文献
15.
In this article, we study the global and asymptotic properties of the solutions of the difference equation $$x_{n+1}=Ax_{n}+Bx_{n-k}+(\beta x_{n}+\gamma x_{n-k})/(Cx_{n}+Dx_{n-k}),\quad n=0,1,2,\ldots,$$ where the initial conditions x ?k ,…,x ?1,x 0 are arbitrary positive real numbers and the coefficients A,B,C,D,β and γ are positive constants, while k is a positive integer number. Some numerical examples will be given to illustrate our results. 相似文献
16.
Maxim Vsemirnov 《中国科学 数学(英文版)》2018,61(11):2101-2110
Consider the sequence of algebraic integers un given by the starting values u0 = 0, u1 = 1 and the recurrence \(u_{n+1}=(4\rm{cos}^2(2\pi/7)-1)\it{u}_{n}-u_{n-\rm{1}}\). We prove that for any n ? {1, 2, 3, 5, 8, 12, 18, 28, 30} the n-th term of the sequence has a primitive divisor in \(\mathbb{Z}[2\rm{cos}(2\pi/7)]\). As a consequence we deduce that for any suffciently large n there exists a prime power q such that the group PSL2(q) can be generated by a pair x, y with \(x^2=y^3=(xy)^7=1\) and the order of the commutator [x, y] is exactly n. The latter result answers in affrmative a question of Holt and Plesken. 相似文献
17.
We study the Weinstein equation $$\Delta u - \frac{k}{{x}_{2}} \frac{\partial}{\partial{x}_{2}} + \frac{l}{x^{2}_{2}}u = 0$$ , on the upper half space ${\mathbb{R}^3_{+} = \{ (x_{0}, x_{1}, x_{2}) \in \mathbb{R}^{3} | x_2 > 0\}}$ in case ${4l \leq (k + 1)^{2}}$ . If l = 0 then the operator ${x^{2k}_{2} (\Delta - \frac{k}{x_{2}} \frac{\partial}{\partial{x}_{2}})}$ is the Laplace- Beltrami operator of the Riemannian metric ${ds^2 = x^{-2k}_{2} (\sum^{2}_{i = 0} dx^{2}_{i})}$ . The general case ${\mathbb{R}^{n}_{+}}$ has been studied earlier by the authors, but the results are improved in case ${\mathbb{R}^3_{+}}$ . If k = 1 then the Riemannian metric is the hyperbolic distance of Poincaré upper half-space. The Weinstein equation is connected to the axially symmetric potentials. We compute solutions of the Weinstein equation depending only on the hyperbolic distance and x 2. The solutions of the Weinstein equation form a socalled Brelot harmonic space and therefore it is known that they satisfy the mean value properties with respect to the harmonic measure. However, without using the theory of Brelot harmonic spaces, we present the explicit mean value properties which give a formula for a harmonic measure evaluated in the center point of the hyperbolic ball. Earlier these results were proved only for k = 1 and l = 0 or k = 1 and l = 1. We also compute the fundamental solutions. The main tools are the hyperbolic metric and its invariance properties. In the consecutive papers, these results are applied to find explicit kernels for k-hypermonogenic functions that are higher dimensional generalizations of complex holomorphic functions. 相似文献
18.
In this paper, we investigate the local and global stability and the period two solutions of all nonnegative solutions of the difference equation, where a, b, A, B are all positive real numbers, \(k \ge 1\) is a positive integer, and the initial conditions \(x_{-k},x_{-k+1},...,x_{0}\) are nonnegative real numbers. It is shown that the zero equilibrium point is globally asymptotically stable under the condition \(a+b \le A\), and the unique positive solution is also globally asymptotically stable under the condition \(a-b \le A \le a+b\). By the end, we study the global stability of such an equation through numerically solved examples.
相似文献
$$\begin{aligned} x_{n+1} = \frac{ ax_{n}+bx_{n-k}}{A+Bx_{n-k}} \end{aligned}$$
19.
A. Ivić 《Acta Mathematica Hungarica》2008,119(1-2):15-24
Some new results on power moments of the integral $$ J_k (t,G) = \frac{1} {{\sqrt {\pi G} }}\int_{ - \infty }^\infty { \left| {\varsigma \left( {\tfrac{1} {2} + it + iu} \right)} \right|^{2k} } e^{ - (u/G)^2 } du $$ (t ? T, T ? ≦ G ? T, κ ∈ N) are obtained when κ = 1. These results can be used to derive bounds for moments of $ \left| {\varsigma \left( {\tfrac{1} {2} + it} \right)} \right| $ . 相似文献
20.
V. A. Abilov F. V. Abilova M. K. Kerimov 《Computational Mathematics and Mathematical Physics》2013,53(10):1440-1446
The Fourier-Bessel integral transform $$g\left( x \right) = F\left[ f \right]\left( x \right) = \frac{1} {{2^p \Gamma \left( {p + 1} \right)}}\int\limits_0^{ + \infty } {t^{2p + 1} f\left( x \right)j_p \left( {xt} \right)dt}$$ is considered in the space $\mathbb{L}_2 \left( {\mathbb{R}_ + } \right)$ . Here, j p (u) = ((2 p Γ(p+1))/(u p ))J p (u) and J p (u) is a Bessel function of the first kind. New estimates are proved for the integral $$\delta _N^2 \left( f \right) = \int\limits_N^{ + \infty } {x^{2p + 1} g^2 \left( x \right)dx, N > 0,}$$ in $\mathbb{L}_2 \left( {\mathbb{R}_ + } \right)$ for some classes of functions characterized by a generalized modulus of continuity. 相似文献