| The objective of this thesis is to investigate the global behavior of positive solutions of some nonlinear delay difference equations by using negative feedback condition, superior and inferior limit method, semicycle analysis method and convergence Theory. For the sake of precise statements, some basic definitions and some general results are presented, which is useful throughout the paper. Our results solve or partially solve some conjectures and open problems.First, we consider the global asymptotical stability, the invariant interval, semi-cycle character of the delay difference equationwhereβ,γ, A, B, C∈(0,∞) and the initial conditions x-1,x0 are nonnegative real numbers such that x-1 + x0 > 0. We show that if the equation has no period-two solution, then the unique positive equilibrium of the difference equation is a global attractor of all positive solutions. In particular, we give a condition under which the equilibrium is asymptotically stable. This result gives partially confirmation on the conjecture proposed by Kulenovic and Ladas in their monograph (Dynamics of Second Order Rational Difference Equations: with Open Problems and Conjectures, Chapman & Hall/CRC, Boca Raton, 2002)Secondly, we investigate the global asymptotical stability, boundedness, invariant interval and semicycle character of the second-order nonlinear delay difference equationwhere the parametersα,γ, A, B, C∈(0,∞) and the initial conditions x-1,x0 are nonnegative real numbers. We show that if the equation has no period-two solution, then the unique positive equilibrium of the difference equation is a global attractor of all positive solutions. Furthermore, we give a condition under which the equilibrium is asymptotically stable, which partially confirms a conjecture proposed by Kulenovic et. al.(Dynamics of Second Order Rational Difference Equations: with Open Problems and Conjectures, Chapman & Hall/CRC, Boca Raton, 2002).Thirdly, we deal with the global asymptotic stability of the second-order nonlinear delay difference equation where the parameters p, q, r∈(0,∞) and the initial conditions x0, x-1 are nonneg-ative real numbers. We show that the unique positive equilibrium of the equation is globally asymptotically stable and our results confirms the conjecture proposed by Kulenovic and Ladas in their monograph (Dynamics of Second Order Rational Difference Equations: with Open Problems and Conjectures, Chapman & Hall/CRC, Boca Raton, 2002).Finally, we study the global attractivity of the higher order nonlinear delay difference equationwhere the parametersα,αi∈(0,∞), i = 0,1, ... ,k and the initial conditions x-k, ... ,x-1 are nonnegative real numbers, x0 is a positive real number. We show that the unique positive equilibrium of the equation is a global attractor of all positive solutions under certain condition. As a corollary, our results solve the open problem proposed by Kulenovic and Ladas in their monograph (Dynamics of Second Order Rational Difference Equations: with Open Problems and Conjectures, Chapman & Hall/CRC, Boca Raton, 2002).
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