scholarly journals Note on Limit-Periodic Solutions of the Difference Equation xt1-[h(xt)λ]xt=rt, λ>1++

Axioms ◽  
2019 ◽  
Vol 8 (1) ◽  
pp. 19 ◽  
Author(s):  
Jan Andres ◽  
Denis Pennequin
Keyword(s):  
Difference Equation ◽  
Periodic Solutions ◽  
Differential Systems ◽  
Abstract Theorem ◽  
The Difference

As a nontrivial application of the abstract theorem developed in our recent paper titled “Limit-periodic solutions of difference and differential systems without global Lipschitzianity restricitions”, the existence of limit-periodic solutions of the difference equation from the title is proved, both in the scalar as well as vector cases. The nonlinearity h is not necessarily globally Lipschitzian. Several simple illustrative examples are supplied.

scholarly journals On the Dynamics of the Recursive Sequence

2012 ◽  
Vol 2012 ◽  
pp. 1-11 ◽  
Author(s):  
Mehmet Gümüş ◽  
Özkan Öcalan ◽  
Nilüfer B. Felah
Keyword(s):  
Periodic Solution ◽  
Difference Equation ◽  
Periodic Solutions ◽  
Positive Solutions ◽  
Initial Conditions ◽  
Recursive Sequence ◽  
Boundedness Character ◽  
The Difference

We investigate the boundedness character, the oscillatory, and the periodic character of positive solutions of the difference equation , where , , and the initial conditions are arbitrary positive numbers. We investigate the boundedness character for . Also, we investigate the existence of a prime two periodic solution for is odd. Moreover, when is even, we prove that there are no prime two periodic solutions of the equation above.

scholarly journals Equations for periodic solutions of a logistic difference equation

1981 ◽  
Vol 23 (1) ◽  
pp. 78-94 ◽  
Author(s):  
A. Brown
Keyword(s):  
Difference Equation ◽  
Periodic Solutions ◽  
New Method ◽  
Polynomial Equations ◽  
Restricted Range ◽  
Simple Transformation ◽  
The Difference ◽  
Logistic Difference Equation ◽  
Range Of Values

AbstractThe paper is concerned with periodic solutions of the difference equation un + 1 = 2aun, where a and b are constants, with and b > 0. A new method is developed for dealing with this problem and, for period lengths up to 6, polynomial equations are given which allow the periodic solutions to be determined in a precise and practical manner. These equations apply whether the periodic solutions are stable or unstable and the elements of the cycle can be determined with an accuracy which is not affected by instability of the cycle.A simple transformation puts the equation into the form , where A = a2 − a, and the detailed discussion is based on this simpler form. The discussion includes details such as the number of cyclic solutions for a given value of A, the pattern of the cycles and their stability. For practical purposes, it is enough to consider a restricted range of values of A, namely , although the equations obtained are valid for A > 2.

scholarly journals Critical values for a nonlinear difference equation

1987 ◽  
Vol 28 (3) ◽  
pp. 340-361
Author(s):  
A. Brown
Keyword(s):  
Difference Equation ◽  
Periodic Solutions ◽  
Finite Length ◽  
Critical Values ◽  
Nonlinear Difference Equation ◽  
Equilibrium Solutions ◽  
Critical Curves ◽  
Parameter Equation ◽  
The Difference ◽  
Image Position

AbstractThe paper discusses equilibrium solutions and solutions with period two and period three for the difference equationwhere Q and A are real, positive parameters. The equation was used by Bier and Bountis [1] as an example of a difference equation whose iteration diagram can show bubbles of finite length rather than the successive bifurcations usually expected. The paper examines in more detail what kind of solution can occur for given values of Q and A and establishes a series of critical curves which demarcate the regions in the (Q, A) plane where solutions of period two or period three occur and the subregions where these periodic solutions are stable. This makes it easy to see how Q and A can be combined into a one-parameter equation which gives a bubble, or a series of bubbles, in the iteration diagram.

scholarly journals On the equivalence of three-dimensional differential systems

2020 ◽  
Vol 18 (1) ◽  
pp. 1164-1172
Author(s):  
Jian Zhou ◽  
Shiyin Zhao
Keyword(s):  
Periodic Solutions ◽  
Differential System ◽  
Necessary Conditions ◽  
Three Dimensional ◽  
Sufficient Conditions ◽  
Periodic Systems ◽  
Structural Form ◽  
Differential Systems

Abstract In this paper, firstly, we study the structural form of reflective integral for a given system. Then the sufficient conditions are obtained to ensure there exists the reflective integral with these structured form for such system. Secondly, we discuss the necessary conditions for the equivalence of such systems and a general three-dimensional differential system. And then, we apply the obtained results to the study of the behavior of their periodic solutions when such systems are periodic systems in t.

scholarly journals On Boundedness of Solutions of the Difference Equation xn+1=(pxn+qxn−1)/(1+xn) for q>1+p>1

2009 ◽  
Vol 2009 ◽  
pp. 1-11 ◽  
Author(s):  
Hongjian Xi ◽  
Taixiang Sun ◽  
Weiyong Yu ◽  
Jinfeng Zhao
Keyword(s):  
Difference Equation ◽  
Boundedness Of Solutions ◽  
The Difference

scholarly journals Existence for Singular Periodic Problems: A Survey of Recent Results

2013 ◽  
Vol 2013 ◽  
pp. 1-17 ◽  
Author(s):  
Jifeng Chu ◽  
Juntao Sun ◽  
Patricia J. Y. Wong
Keyword(s):  
Differential Equations ◽  
Periodic Solutions ◽  
Impulsive Differential Equations ◽  
Differential Systems ◽  
Singular Differential Equations ◽  
Periodic Problems ◽  
Existence Of Periodic Solutions

We present a survey on the existence of periodic solutions of singular differential equations. In particular, we pay our attention to singular scalar differential equations, singular damped differential equations, singular impulsive differential equations, and singular differential systems.

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