scholarly journals Growth of meromorphic solutions of some difference equations

2010 ◽  
Vol 4 (2) ◽  
pp. 309-321 ◽  
Author(s):  
Xiu-Min Zheng ◽  
Zong-Xuan Chen ◽  
Tu Jin
Keyword(s):  
Difference Equation ◽  
Difference Equations ◽  
Higher Order ◽  
Meromorphic Solutions ◽  
Entire Solutions ◽  
Order Difference ◽  
Difference Analogue ◽  
Algebraic Difference ◽  
The Difference

We investigate higher order difference equations and obtain some results on the growth of transcendental meromorphic solutions, which are complementary to the previous results. Examples are also given to show the sharpness of these results. We also investigate the growth of transcendental entire solutions of a homogeneous algebraic difference equation by using the difference analogue of Wiman-Valiron Theory.

scholarly journals Growth of Meromorphic Solutions of Some -Difference Equations

2013 ◽  
Vol 2013 ◽  
pp. 1-6
Author(s):  
Guowei Zhang
Keyword(s):  
Difference Equation ◽  
Fixed Points ◽  
Difference Equations ◽  
Meromorphic Solutions ◽  
Order Difference ◽  
Complex Difference ◽  
Complex Difference Equations ◽  
Second Order Difference Equation ◽  
Order Difference Equation ◽  
The Difference

We estimate the growth of the meromorphic solutions of some complex -difference equations and investigate the convergence exponents of fixed points and zeros of the transcendental solutions of the second order -difference equation. We also obtain a theorem about the -difference equation mixing with difference.

Dynamics of a Higher Order Rational Difference Equation

2011 ◽  
Vol 216 ◽  
pp. 50-55 ◽  
Author(s):  
Yi Yang ◽  
Fei Bao Lv
Keyword(s):  
Difference Equation ◽  
Difference Equations ◽  
Initial Conditions ◽  
Higher Order ◽  
Rational Difference Equation ◽  
The Difference

In this paper, we address the difference equation xn=pxn-s+xn-t/q+xn-t n=0,1,... with positive initial conditions where s, t are distinct nonnegative integers, p, q > 0. Our results not only include some previously known results, but apply to some difference equations that have not been investigated so far.

scholarly journals Meromorphic Solutions of Linear Difference Equations with Polynomial Coefficients

2017 ◽  
Vol 59 (1) ◽  
pp. 159-168
Author(s):  
Y. Zhang ◽  
Z. Gao ◽  
H. Zhang
Keyword(s):  
Difference Equation ◽  
Difference Equations ◽  
Necessary Conditions ◽  
Meromorphic Solution ◽  
Linear Difference Equation ◽  
Meromorphic Solutions ◽  
Linear Difference Equations ◽  
Polynomial Coefficients ◽  
Difference Analogue ◽  
Borel Exceptional Values

AbstractWe study the growth of the transcendental meromorphic solution f(z) of the linear difference equation:where q(z), p0(z), ..., pn-(z) (n ≥ 1) are polynomials such that p0(z)pn(z) ≢ 0, and obtain some necessary conditions guaranteeing that the order of f(z) satisfies σ(f) ≥ 1 using a difference analogue of the Wiman-Valiron theory. Moreover, we give the form of f(z) with two Borel exceptional values when two of p0(z), ..., pn(z) have the maximal degrees.

On the growth analysis of meromorphic solutions of finite ϕ-order of linear difference equations

Analysis ◽  
2020 ◽  
Vol 40 (4) ◽  
pp. 193-202
Author(s):  
Sanjib Kumar Datta ◽  
Nityagopal Biswas
Keyword(s):  
Difference Equation ◽  
Complex Plane ◽  
Difference Equations ◽  
Growth Analysis ◽  
Linear Difference Equation ◽  
Higher Order ◽  
Meromorphic Solutions ◽  
Linear Difference Equations ◽  
Unbounded Function ◽  
Growth Properties

AbstractIn this paper, we investigate some growth properties of meromorphic solutions of higher-order linear difference equationA_{n}(z)f(z+n)+\dots+A_{1}(z)f(z+1)+A_{0}(z)f(z)=0,where {A_{n}(z),\dots,A_{0}(z)} are meromorphic coefficients of finite φ-order in the complex plane where φ is a non-decreasing unbounded function. We extend some earlier results of Latreuch and Belaidi [Z. Latreuch and B. Belaïdi, Growth and oscillation of meromorphic solutions of linear difference equations, Mat. Vesnik 66 2014, 2, 213–222].

scholarly journals Comparison results and linearized oscillations for higher-order difference equations

1992 ◽  
Vol 15 (1) ◽  
pp. 129-142 ◽  
Author(s):  
G. Ladas ◽  
C. Qian
Keyword(s):  
Linear Equation ◽  
Difference Equations ◽  
Bounded Solution ◽  
Higher Order ◽  
Comparison Result ◽  
Oscillation Theorem ◽  
Order Difference ◽  
Comparison Results ◽  
The Difference

Consider the difference equationsΔmxn+(−1)m+1pnf(xn−k)=0,   n=0,1,…        (1)andΔmyn+(−1)m+1qng(yn−ℓ)=0,   n=0,1,….       (2)We establish a comparison result according to which, whenmis odd, every solution of Eq.(1) oscillates provided that every solution of Eq.(2) oscillates and, whenmis even, every bounded solution of Eq.(1) oscillates provided that every bounded solution of Eq.(2) oscillates. We also establish a linearized oscillation theorem according to which, whenmis odd, every solution of Eq.(1) oscillates if and only if every solution of an associated linear equationΔmzn+(−1)m+1pzn−k=0,   n=0,1,…         (*)oscillates and, when m is even, every bounded solution of Eq.(1) oscillates if and only if every bounded solution of (*) oscillates.

scholarly journals On a class of solvable higher-order difference equations

Filomat ◽  
2017 ◽  
Vol 31 (2) ◽  
pp. 461-477 ◽  
Author(s):  
Stevo Stevic ◽  
Mohammed Alghamdi ◽  
Abdullah Alotaibi ◽  
Elsayed Elsayed
Keyword(s):  
Difference Equation ◽  
Closed Form ◽  
Difference Equations ◽  
Higher Order ◽  
Real Numbers ◽  
Order Difference ◽  
Initial Values ◽  
Long Term Behavior

Closed form formulas for well-defined solutions of the next difference equation xn = xn-2xn-k-2/xn-k(an + bnxn-2xn-k-2), n ? N0, where k ? N, (an)n?N0, (bn)n?N0, and initial values x-i, i = 1,k+2 are real numbers, are given. Long-term behavior of well-defined solutions of the equation when (an)n?N0 and (bn)n?N0 are constant sequences is described in detail by using the formulas. We also describe the domain of undefinable solutions of the equation. Our results explain and considerably improve some recent results in the literature.

scholarly journals On zeros and growth of solutions of complex difference equations

2021 ◽  
Vol 2021 (1) ◽  
Author(s):  
Min-Feng Chen ◽  
Ning Cui
Keyword(s):  
Entire Function ◽  
Difference Equation ◽  
Finite Order ◽  
Difference Equations ◽  
Meromorphic Solutions ◽  
Complex Difference ◽  
Difference Polynomial ◽  
Complex Difference Equations ◽  
The Difference ◽  
Growth Of Solutions

AbstractLet f be an entire function of finite order, let $n\geq 1$ n ≥ 1 , $m\geq 1$ m ≥ 1 , $L(z,f)\not \equiv 0$ L ( z , f ) ≢ 0 be a linear difference polynomial of f with small meromorphic coefficients, and $P_{d}(z,f)\not \equiv 0$ P d ( z , f ) ≢ 0 be a difference polynomial in f of degree $d\leq n-1$ d ≤ n − 1 with small meromorphic coefficients. We consider the growth and zeros of $f^{n}(z)L^{m}(z,f)+P_{d}(z,f)$ f n ( z ) L m ( z , f ) + P d ( z , f ) . And some counterexamples are given to show that Theorem 3.1 proved by I. Laine (J. Math. Anal. Appl. 469:808–826, 2019) is not valid. In addition, we study meromorphic solutions to the difference equation of type $f^{n}(z)+P_{d}(z,f)=p_{1}e^{\alpha _{1}z}+p_{2}e^{\alpha _{2}z}$ f n ( z ) + P d ( z , f ) = p 1 e α 1 z + p 2 e α 2 z , where $n\geq 2$ n ≥ 2 , $P_{d}(z,f)\not \equiv 0$ P d ( z , f ) ≢ 0 is a difference polynomial in f of degree $d\leq n-2$ d ≤ n − 2 with small mromorphic coefficients, $p_{i}$ p i , $\alpha _{i}$ α i ($i=1,2$ i = 1 , 2 ) are nonzero constants such that $\alpha _{1}\neq \alpha _{2}$ α 1 ≠ α 2 . Our results are improvements and complements of Laine 2019, Latreuch 2017, Liu and Mao 2018.

scholarly journals Distribution of iterates of first order difference equations

1980 ◽  
Vol 22 (1) ◽  
pp. 133-143 ◽  
Author(s):  
James B. McGuire ◽  
Colin J. Thompson
Keyword(s):  
Invariant Measure ◽  
Difference Equation ◽  
Lebesgue Measure ◽  
Difference Equations ◽  
Absolutely Continuous ◽  
Order Difference ◽  
First Order ◽  
Order Difference Equation ◽  
The Difference ◽  
Biological Context

An invariant measure which is absolutely continuous with respect to Lebesgue measure is constructed for a particular first order difference equation that has an extensive biological pedigree. In a biological context this invariant measure gives the density of the population whose growth is governed by the difference equation. Further asymptotically universal results are obtained for a class of difference equations.

scholarly journals Global Dynamics of a Higher Order Difference Equation with a Quadratic Term

2020 ◽  
Author(s):  
Erkan Taşdemir
Keyword(s):  
Difference Equation ◽  
Rate Of Convergence ◽  
Difference Equations ◽  
Initial Conditions ◽  
Higher Order ◽  
Quadratic Term ◽  
Global Dynamics ◽  
Order Difference ◽  
Convergence Of Solutions ◽  
Order Difference Equation

In this paper, we investigate the dynamics of following higher order difference equation x_{n+1}=A+B((x_{n})/(x_{n-m}²)) with A,B and initial conditions are positive numbers, and m∈{2,3,⋯}. Especially we study the boundedness, periodicity, semi-cycles, global asymptotically stability and rate of convergence of solutions of related higher order difference equations.

scholarly journals A classification scheme for nonoscilatory solutions of a higher order neutral nonlinear difference equation

1999 ◽  
Vol 67 (1) ◽  
pp. 122-142 ◽  
Author(s):  
Li Wan-Tong ◽  
Sui Sun Cheng ◽  
Guang Zhang
Keyword(s):  
Difference Equation ◽  
Difference Equations ◽  
Classification Scheme ◽  
Neutral Type ◽  
Higher Order ◽  
Nonlinear Difference Equation ◽  
Nonoscillatory Solutions ◽  
Order Difference ◽  
Asymptotic Behaviors ◽  
Existence Criteria

AbstractNonoscillatory solutions of a nonlinear neutral type higher order difference equations are classified by means of their asymptotic behaviors. Existence criteria are then provided for justification of such classficiation.

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