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.
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
3.
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
4.
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
5.
The main objective of this paper is to study the boundedness character, the periodic character and the global stability of the positive solutions of the following difference equation $x_{n + 1} = \frac{{\alpha x_n + \beta x_{n - 1} + \gamma x_{n - 2} + \delta x_{n - 3} }}{{Ax_n + Bx_{n - 1} + Cx_{n - 2} + Dx_{n - 3} }},n = 0,1,2.....$ where the coefficientsA, B, C, D, α, β, γ, δ, and the initial conditionsx -3,x -2,x -1,x 0 are arbitrary positive real numbers. 相似文献
6.
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
7.
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
8.
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
9.
Harald Riede 《The Ramanujan Journal》2011,26(1):35-43
The article on hand deals with the continued fraction $$\frac{1 |}{| z } +\frac{1 |}{| 1 } + \frac{2 |}{| z } +\frac{3 |}{| 1 } + \frac{4 |}{| z} + \cdots.$$ The famous Indian mathematician Srinivasa Ramanujan has given a pre-presentation by a power series, but he however concealed a proof. Subsequently a proof has been established, but a direct verification is intricate. Here we give a quick and direct approach with comparitively little effort. 相似文献
10.
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
11.
Antonio C. Asperti Alexandre Lymberopoulos Bárbara Corominas Valério 《Results in Mathematics》2014,65(1-2):9-25
In this work we give a local classification of connected ruled Weingarten hypersurfaces M n , n ≥ 3 in the hyperbolic space ${{\mathbb{H}}^{n+1} \subset {\mathbb{L}}^{n+2}}$ . 相似文献
12.
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
13.
The volume of the unit ball in ${\mathbb{R}^{n}}$ is defined by $$\Omega_{n} = \frac{\pi^{n/2}}{\Gamma(n/2+1)},\qquad n = 1,2,3,\ldots,$$ where Γ denotes the classical gamma function of Euler. In several recently published papers numerous authors studied properties of Ω n . In particular, various inequalities involving Ω n are given in the literature. In this paper, we continue the work on this subject and offer new inequalities. More precisely, we offer sharp upper and lower bounds for $$\frac{\Omega_{n}^{2}}{\Omega_{n-1} \Omega_{n+1}},\quad\frac{\Omega_{n}}{\Omega_{n-1}+\Omega_{n+1}} \quad {\rm and} \quad\Omega_{n}.$$ 相似文献
14.
The integral $$\int_0^{{\pi \mathord{\left/ {\vphantom {\pi 4}} \right. \kern-\nulldelimiterspace} 4}} {\ln \left( {\cos ^{{m \mathord{\left/ {\vphantom {m n}} \right. \kern-\nulldelimiterspace} n}} \theta \pm \sin ^{{m \mathord{\left/ {\vphantom {m n}} \right. \kern-\nulldelimiterspace} n}} \theta } \right)d\theta } $$ where m and n are relatively prime positive integers, is evaluated exactly in terms of elementary functions and the Catalan constant G. 相似文献
15.
In this paper a sufficient condition is obtained for the global asymptotic stability of the following system of difference equations $$x_{n + 1} = \frac{{x_n y_{n - 1}^b + 1}} {{x_n + y_{n - 1}^b }}, y_{n + 1} = \frac{{y_n x_{n - 1}^b + 1}} {{y_n + x_{n - 1}^b }}n = 0,1,2 \ldots$$ where the parameter b ∈ [0, ∞) and the initial values (x k , y k ) ∈ (0, ∞) (for k = ?1, 0). 相似文献
16.
V. Totik 《Analysis Mathematica》1980,6(2):165-184
стАтьь ьВльЕтсь пРОД ОлжЕНИЕМ пРЕДыДУЩЕИ ОДНОИМЕННОИ РАБОты АВтОРА, гДЕ ИжУ ЧАлсь пОРьДОк ВЕлИЧИН пРИ УслОВИьх, ЧтО α>-1/2, Рα >- 1 И ЧтО МАтРИцАt nk УДОВлЕтВОРьЕт НЕкОт ОРОМУ УслОВИУ РЕгУльРНОстИ. жДЕсь ДОкАжыВАЕтсь, Ч тО ЕслИf∈H Ω, тО ВыпОлНь Етсь ОцЕНкА $$\left\{ {\frac{1}{{\lambda _n }}\mathop \Sigma \limits_{k = n - \lambda _n + 1}^n \left| {\sigma _k^\alpha \left( x \right) - f\left( x \right)} \right|^p } \right\}^{{1 \mathord{\left/ {\vphantom {1 p}} \right. \kern-\nulldelimiterspace} p}} = O\left( {\left\{ {\frac{1}{{\lambda _n }}\mathop \Sigma \limits_{k = n - \lambda _n + 1}^n \left( {\frac{1}{k}\mathop \smallint \limits_{{1 \mathord{\left/ {\vphantom {1 k}} \right. \kern-\nulldelimiterspace} k}}^{2\pi } \frac{{\omega \left( t \right)}}{{t^2 }}dt} \right)^p } \right\}^{{1 \mathord{\left/ {\vphantom {1 p}} \right. \kern-\nulldelimiterspace} p}} + \left( {\frac{{\lambda _n }}{n}} \right)^\alpha \omega \left( {\frac{1}{n}} \right)} \right)$$ (λ1=1, λn+1-λn≦1), А тАкжЕ ЧтО Ёт А ОцЕНкА ОкОНЧАтЕльН А В сВОИх тЕРМИНАх; пОДОБ НыИ РЕжУль-тАт спРАВЕДлИВ тАкжЕ И Дль сОпРьжЕННОИ ФУНкцИИ . ДОкАжыВАЕтсь, ЧтО Усл ОВИьα>?1/2 Иpα>?1, кОтОРыЕ Б ылИ НАлОжЕНы В УпОМьНУтО И ВышЕ ЧАстИ I, сУЩЕстВЕН Ны. 相似文献
17.
M. V. Tuvaev 《Mathematical Notes》1992,52(3):983-989
The following uniformly elliptic equation is considered: $$\sum {\tfrac{\partial }{{\partial x_i }}a_{ij} (x)\tfrac{{\partial u}}{{\partial x_j }} = f(x,u,\nabla u)} , x \in \Omega \subset R^n ,$$ with measurable coefficients. The function f satisfies the condition $$f(x, u, \nabla u) u \geqslant C|u|^{\beta _1 + 1} |\nabla u|^{\beta _1 } , \beta _1 > 0, 0 \leqslant \beta _2 \leqslant 2, \beta _1 + \beta _2 > 1$$ . It is proved that if u(x) is a generalized (in the sense of integral identity) solution in the domain ΩK, where the compactum K has Hausdorff dimension α, and if \(\frac{{2\beta _1 + \beta _2 }}{{\beta _1 + \beta _2 - 1}}< n - \alpha \) , u(x) will be a generalized solution in the domain ω. Moreover, the sufficient removability conditions for the singular set are, in some sense, close to the necessary conditions. 相似文献
18.
J. Nahas 《Calculus of Variations and Partial Differential Equations》2013,46(1-2):427-437
We show that smooth, radially symmetric wave maps U from ${\mathbb {R}^{2+1}}$ to a compact target manifold (N, where ? r U and ? t U have compact support for any fixed time, scatter. The result will follow from the work of Christodoulou and Tahvildar-Zadeh, and Struwe, upon proving that for ${(\lambda^{\prime} \in (0,1),}$ energy does not concentrate in the set $$K_{\frac{5}{8}T,\frac{7}{8}T}^{\lambda^{\prime}} = \{(x,t) \in \mathbb{R}^{2+1} \vert \quad|x| \leq \lambda^{\prime} t, t \in [(5/8)T,(7/8)T] \}.$$ 相似文献
19.
We obtain all solutions of the equation $\frac{ax^{n+2l}+c}{abt^{2}x^{n}+c} = by^{2}$ with c??{??1,??2,??4}. 相似文献
20.
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