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In this paper we study the periodicity of higher order nonlinear equations. They are defined by a recursion which is generated by a mapping, whereXis a state set. Our main objective is to prove sharp conditions for the global periodicity of our equations assuming the weakest possible assumptions on the state setX. As an application of our general algebraic-like conditions we prove a new linearized global periodicity theorem assuming thatXis a normed space. We needed a new proof-technique since in the infinite dimensional case the Jacobian does not exist. We give new necessary and/or sufficient conditions as well as new examples for global periodicity, for instance whenever the state setXis a group. [ABSTRACT FROM AUTHOR]

In this paper, we study the existence of analytic invariant curves of an iterative equationwhich is from Pielou's equation. By reducing the equation with the Schröder transformation to an auxiliary equation, the author discusses not only that the constantat resonance, i.e. at a root of the unity, but also thosenear resonance under the Brjuno condition. [ABSTRACT FROM AUTHOR]

In this paper, we investigate the periodicity character of some classes of difference equations. Among other results, we give a simpler proof of an Open Problem addressed in Csörnyei and Laczkovich, 2001, “Some periodic and non-periodic recursions”, Monath. Math. , 132 , 215–236. We also consider Open Problem 2.8 in Grove and Ladas, 2005, Periodicities in Nonlinear Difference Equations , (Chapman and Hall/CRC), and the case when a difference equation has periodic solutions depending on arbitrary parameters. [ABSTRACT FROM AUTHOR]

This paper studies the behavior of positive solutions of the recursive equationwith We prove that every positive solution { y i } converges to a period two solution or to the equilibrium . This result answers Open Problem 11.4.8 (a) in Kulenovic and Ladas, 2002 Dynamics of Second Order Rational Difference Equations. With Open Problems and Conjectures (Boca Raton:Chapman and Hall/CRC). [ABSTRACT FROM AUTHOR]

In this paper, we consider a nonlinear difference equation of the form xn+1 =f(pn,xn-2s,xn-2t-1), n = 0,1,…, under some certain assumptions, where s,t ∈ {0,1,2,…} with s ⩽ t and the initial values x-2t-1 ,x-2t,ߪ, x0 ∈ (0, + ∞) and pn is the period-two sequence. We give sufficient conditions under which every positive solution of this equation converges to the period-two solution. [ABSTRACT FROM AUTHOR]

In this paper, we study the asymptotic behavior of positive solutions of nonautonomous Beverton-Holt equationwhere { K n} is a positive persistent and bounded sequence. In our main result, we proved that all positive solutions { x n} satisfy the inequalityThe obtained result represents the extension of the Cushing-Hensons's conjecture. Applications to the case when { K n} is a positive periodic sequence are given and the alternative proof of the Cushing-Henson's conjecture is also presented. [ABSTRACT FROM AUTHOR]

In this paper, we study the global attractivity, the invariant intervals, the periodic and oscillatory character of the difference equation where a , b , A , B are positive real numbers, k ≥1 is a positive integer, and the initial conditions x - k ,..., x -1 , x 0 are nonnegative real numbers such that x -k or x 0 or both are positive real numbers. We show that the positive equilibrium of the difference equation is a global attractor. As a corollary, our main result confirms a conjecture proposed by Kulenović et al. (2003) [The dynamics of x n +1 =(α+ βx n )/( A + Bx n + Cx n -1 ) facts and conjectures, Computational Mathematics Applications , 45 , 1087-1099]. § Supported by the NNSF of China (10171040), the NSF of Gansu Province of China (ZS011-A25-007-Z) and the Teaching and Research Award Program for Outstanding Young Teachers in Higher Education Institutions of Ministry of Education of China. [ABSTRACT FROM AUTHOR]

In this note, we study positive solutions of the difference equation [This symbol cannot be presented in ASCII format] where p ∈ [1.∞) and s, l ∈ N. We prove that if p > 1, then every positive solution converges to the positive equilibrium x* = p + 1 and if p = 1, then every positive solution converges to a 2s-periodic solution. The second result generalizes the main result in the paper: W.T. Patula and H.D. Voulov. On the oscillation and periodic character of a third order rational difference equation. [ABSTRACT FROM AUTHOR]

The boundedness, global attractivity, oscillatory and asymptotic periodicity of the nonnegative solutions of the difference equation in the title is investigated, where all the coefficients are nonnegative real numbers The paper is motivated by an open problem proposed by Ladas [Open problems and conjectures, J. Differ, Equations Appl., 5 (1999), 211-215]. [ABSTRACT FROM AUTHOR]