Dynamics Of Some Nonlinear Delay Difference Equations

In this thesis, we investigate the dynamical behavior of some nonlinear delay difference equations. We solve three conjectures and one related problem by the means of fixed point theory, convergence theory, superior and inferior limit method,semicycle analysis and so on.At first, our aim is to consider the boundedness, the invariant interval, theperiodic character and the global attractivity of the difference equation: (?),where p,q,r∈(0,∞),and the initial conditions y_-k,..., y_-1, y₀are nonnegative real numbers such that they aren't all zero. We show that the positive equilibrium of the difference equation is a global attractor. As a corollary, this result can confirm a conjecture raised by Kulenovi(?) et.al: the positive equilibrium of the difference equation is globally asymptotically stable.Especially, when k = 1, the difference equation reduces to (?) Kulenovi(?) and Ladas (Dynamics of Second Order Rational Difference Equations: with Open Problems and Conjectures, Chapman & Hall/CRC, Boca Raton, 2001) have proved the existence of period-two solution. This part mainly confirmsits locally asymptotically stability.Secondly, we mainly consider the difference equation: (?), where y_-1,y₀∈R, and consider its dynamic behavior about stability andsemicycle theory and so on.Finally, our aim is to consider the asymptotic behavior of the unique positiveequilibrium and the periodic behavior of positive solution about this nonlinear delaydifference equation: (?),where b_i∈(0,∞),i= 0,1,..., k, the initial conditions y_-k,..., y_-1, y₀∈(0,∞). In particular, if k = 1, b₀=b₁=1, then our conclusion solve the open problem posed by Kulenovi(?) and Ladas in their monograph.