| This thesis is concerned with several nonlinear delay difference equations. By using the superior and inferior limit method and semicycle analysis, we study the dynamical behavior of these equations (for example, locally asymptotically stable, boundedness, invariant interval, globally asymptotically stable and so on). Our results solve or partially solve some open problems. Concretely speaking, this paper have done the following works.Firstly, we investigate the local stability, prime period-two solutions, bound-edness, invariant intervals and global attractivity of all positive solutions of the following difference equation where the parameters p,q,r∈(0,∞),κ∈{1,2,3,…} and the initial conditions y-κ,…, y0∈(0,∞). We show that the unique positive equilibrium of this equation is a global attractor under certain conditions. The result answers the open problem proposed by Kulenovic and Ladas.Secondly, we discuss the boundedness, invariant intervals, semicycles and global attractivity of all nonnegative solutions of the equation where the parametersα,β,γ∈[0,∞),κ≥2 is an integer and the initial conditionsχ-κ,…,χ0∈[0,00). It is shown that the unique positive equilibrium of the equation is globally asymptotically stable under the conditionβ≤1. The result partially solves the open problem proposed by Kulenovic and Ladas in Dynamics of Second Order Rational Difference Equation with Open Problem and Conjecture (Chapman & Hall/CRC, Boca Raton, FL,2002).Finally, we investigate the unbounded solutions of a nonlinear difference equa-tion. Every unbounded solution exists the subsequence of even indexed (resp. odd) terms tends to∞and the subsequence of odd indexed (resp. even) terms tends to a nonnegative number is given. It also shows that two sets in the plane of initial conditions corresponding to the two cases are separated by the global stable mani-fold of the unique positive equilibrium. The result partially solves the open problem proposed by Kulenovic and Ladas.
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