scholarly journals On Parametric Gevrey Asymptotics for Some Cauchy Problems in Quasiperiodic Function Spaces

2014 ◽  
Vol 2014 ◽  
pp. 1-29 ◽  
Author(s):  
A. Lastra ◽  
S. Malek
Keyword(s):  
Perturbation Parameter ◽  
Formal Series ◽  
Cauchy Problems ◽  
Small Divisor ◽  
Real Numbers ◽  
Positive Real ◽  
Quasiperiodic Function ◽  
Quasiperiodic Functions ◽  
The Difference ◽  
Formal Power

We investigate Gevrey asymptotics for solutions to nonlinear parameter depending Cauchy problems with 2π-periodic coefficients, for initial data living in a space of quasiperiodic functions. By means of the Borel-Laplace summation procedure, we construct sectorial holomorphic solutions which are shown to share the same formal power series as asymptotic expansion in the perturbation parameter. We observe a small divisor phenomenon which emerges from the quasiperiodic nature of the solutions space and which is the origin of the Gevrey type divergence of this formal series. Our result rests on the classical Ramis-Sibuya theorem which asks to prove that the difference of any two neighboring constructed solutions satisfies some exponential decay. This is done by an asymptotic study of a Dirichlet-like series whose exponents are positive real numbers which accumulate to the origin.

Analogue of a theorem of Khintchine in fields of formal power series

1966 ◽  
Vol 62 (4) ◽  
pp. 637-642 ◽  
Author(s):  
T. W. Cusick
Keyword(s):  
Power Series ◽  
Positive Constant ◽  
Classical Result ◽  
Real Numbers ◽  
Linear Forms ◽  
Absolute Value ◽  
The Difference ◽  
Image Position ◽  
Formal Power ◽  
Positive Integers

For a real number λ, ‖λ‖ is the absolute value of the difference between λ and the nearest integer. Let X represent the m-tuple (x1, x2, … xm) and letbe any n linear forms in m variables, where the Θij are real numbers. The following is a classical result of Khintchine (1):For all pairs of positive integers m, n there is a positive constant Г(m, n) with the property that for any forms Lj(X) there exist real numbers α1, α2, …, αn such thatfor all integers x1, x2, …, xm not all zero.

On the global attractivity and the solution of recursive sequence

2010 ◽  
Vol 47 (3) ◽  
pp. 401-418 ◽  
Author(s):  
Elsayed Elsayed
Keyword(s):  
Difference Equation ◽  
Global Attractivity ◽  
Initial Conditions ◽  
Real Numbers ◽  
Positive Real ◽  
Special Cases ◽  
Recursive Sequence ◽  
The Difference

In this paper we study the behavior of the difference equation \documentclass{aastex} \usepackage{amsbsy} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{bm} \usepackage{mathrsfs} \usepackage{pifont} \usepackage{stmaryrd} \usepackage{textcomp} \usepackage{upgreek} \usepackage{portland,xspace} \usepackage{amsmath,amsxtra} \usepackage{bbm} \pagestyle{empty} \DeclareMathSizes{10}{9}{7}{6} \begin{document} $$x_{n + 1} = ax_{n - 2} + \frac{{bx_n x_{n - 2} }}{{cx_n + dx_{n - 3} }},n = 0,1,...$$ \end{document} where the initial conditions x−3 , x−2 , x−1 , x0 are arbitrary positive real numbers and a, b, c, d are positive constants. Also, we give the solution of some special cases of this equation.

On the solutions of a higher order difference equation

2020 ◽  
Vol 27 (2) ◽  
pp. 165-175 ◽  
Author(s):  
Raafat Abo-Zeid
Keyword(s):  
Difference Equation ◽  
Explicit Formula ◽  
Initial Conditions ◽  
Global Behavior ◽  
Real Numbers ◽  
Positive Real ◽  
Order Difference ◽  
Forbidden Set ◽  
Order Difference Equation ◽  
The Difference

AbstractIn this paper, we determine the forbidden set, introduce an explicit formula for the solutions and discuss the global behavior of solutions of the difference equationx_{n+1}=\frac{ax_{n}x_{n-k}}{bx_{n}-cx_{n-k-1}},\quad n=0,1,\ldots,where{a,b,c}are positive real numbers and the initial conditions{x_{-k-1},x_{-k},\ldots,x_{-1},x_{0}}are real numbers. We show that when{a=b=c}, the behavior of the solutions depends on whetherkis even or odd.

scholarly journals Global behavior of the difference equation $x_{n+1}=\frac{Ax_{n-1}} {B-Cx_{n}x_{n-2}}$

2011 ◽  
Vol 31 (1) ◽  
pp. 43 ◽  
Author(s):  
R. Abo-Zeid ◽  
Cengiz Cinar
Keyword(s):  
Difference Equation ◽  
Global Stability ◽  
Global Behavior ◽  
Real Numbers ◽  
Positive Real ◽  
Admissible Solutions ◽  
The Difference

The aim of this work is to investigate the global stability, periodic nature, oscillation and the boundedness of all admissible solutions of the difference equation $x_{n+1}=\frac{Ax_{n-1}} {B-Cx_{n}x_{n-2}}$, n=0,1,2,... where A, B, C are positive real numbers.

scholarly journals On the Recursive Sequencexn=A+xn−kp/xn−1r

2009 ◽  
Vol 2009 ◽  
pp. 1-8 ◽  
Author(s):  
Fangkuan Sun ◽  
Xiaofan Yang ◽  
Chunming Zhang
Keyword(s):  
Difference Equation ◽  
Dynamic Behavior ◽  
Positive Solutions ◽  
Initial Conditions ◽  
Real Numbers ◽  
Positive Real ◽  
The Difference ◽  
The Stability

This paper studies the dynamic behavior of the positive solutions to the difference equationxn=A+xn−kp/xn−1r,n=1,2,…, whereA,p, andrare positive real numbers, and the initial conditions are arbitrary positive numbers. We establish some results regarding the stability and oscillation character of this equation forp∈(0,1).

On the difference equation $y_{n + 1} = \frac{{\alpha + y_n^p }} {{\beta y_{n - 1}^p }} - \frac{{\gamma + y_{n - 1}^p }} {{\beta y_n^p }} $

2011 ◽  
Vol 61 (6) ◽  
Author(s):  
İlhan Öztürk ◽  
Saime Zengin
Keyword(s):  
Difference Equation ◽  
Global Stability ◽  
Equilibrium Point ◽  
Global Attractor ◽  
Initial Conditions ◽  
Real Numbers ◽  
Positive Real ◽  
The Difference

AbstractIn this paper, we investigate the global stability and the periodic nature of solutions of the difference equation $y_{n + 1} = \frac{{\alpha + y_n^p }} {{\beta y_{n - 1}^p }} - \frac{{\gamma + y_{n - 1}^p }} {{\beta y_n^p }},n = 0,1,2,... $ where α, β, γ ∈ (0,∞), α(1 − p) − γ > 0, 0 < p < 1, every y n ≠ 0 for n = −1, 0, 1, 2, … and the initial conditions y−1, y0 are arbitrary positive real numbers. We show that the equilibrium point of the difference equation is a global attractor with a basin that depends on the conditions of the coefficients.

Global behavior of two third order rational difference equations with quadratic terms

2019 ◽  
Vol 69 (1) ◽  
pp. 147-158 ◽  
Author(s):  
R. Abo-Zeid
Keyword(s):  
Explicit Formula ◽  
Difference Equations ◽  
Initial Conditions ◽  
Global Behavior ◽  
Real Numbers ◽  
Positive Real ◽  
Third Order ◽  
Rational Difference Equations ◽  
The Difference

Abstract In this paper, we determine the forbidden sets, introduce an explicit formula for the solutions and discuss the global behaviors of solutions of the difference equations $$\begin{array}{} \displaystyle x_{n+1}=\frac{ax_{n}x_{n-1}}{bx_{n-1}+ cx_{n-2}},\quad n=0,1,\ldots \end{array} $$ where a,b,c are positive real numbers and the initial conditions x−2,x−1,x0 are real numbers.

scholarly journals On the Recursive Sequencexn+1=A+xnp/xn−1r

2007 ◽  
Vol 2007 ◽  
pp. 1-9 ◽  
Author(s):  
Stevo Stevic
Keyword(s):  
Difference Equation ◽  
Positive Solutions ◽  
Analogous Result ◽  
Real Numbers ◽  
Positive Real ◽  
Unbounded Solutions ◽  
Boundedness Character ◽  
The Difference

The paper considers the boundedness character of positive solutions of the difference equationxn+1=A+xnp/xn−1r,n∈ℕ0, whereA,p, andrare positive real numbers. It is shown that (a) Ifp2≥4r>4, orp≥1+r,r≤1, then this equation has positive unbounded solutions; (b) ifp2<4r, or2r≤p<1+r,r∈(0,1), then all positive solutions of the equation are bounded. Also, an analogous result is proved regarding positive solutions of the max type difference equationxn+1=max{A,xnp/xn−1r}, whereA,p,q∈(0,∞).

scholarly journals On the Difference Equation

2008 ◽  
Vol 2008 ◽  
pp. 1-8 ◽  
Author(s):  
İbrahim Yalçinkaya
Keyword(s):  
Difference Equation ◽  
Higher Order ◽  
Real Numbers ◽  
Positive Real ◽  
Global Behaviour ◽  
Initial Values ◽  
The Difference

We investigate the global behaviour of the difference equation of higher order , where the parameters and the initial values and are arbitrary positive real numbers.

scholarly journals Closed-Form Solution of a Rational Difference Equation

2021 ◽  
Vol 2021 ◽  
pp. 1-12
Author(s):  
Tarek F. Ibrahim ◽  
Abdul Qadeer Khan ◽  
Burak Oğul ◽  
Dağistan Şimşek
Keyword(s):  
Difference Equation ◽  
Closed Form ◽  
Closed Form Solution ◽  
Form Solution ◽  
Real Numbers ◽  
Positive Real ◽  
Rational Difference Equation ◽  
The Difference

In this paper, we study the solution of the difference equation Ω m + 1 = Ω m − 7 q + 6 / 1 + ∏ t = 0 5 Ω m − q + 1 t − q , where the initials are positive real numbers.

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