Global Behavior Of Some Higher Order Nonlinear Difference Equations

In this thesis, we investigate the global dynamical behavior of some higher order nonlinear difference equations. By using qualitative and stability theory, superior and inferior limit method, semicycle analysis, convergence theory together with some inequality techniques, we solve or partially solve a open problem and a conjecture proposed by Ladas et al.At first, in Chapter 2, we consider the nonlinear difference equation where p, q, r and the initial conditions x-k,…,x0 are nonnegative real numbers, k∈{2,3,…}. we investigate the boundedness, the periodic character, the invari-ant interval and the global attractivity of positive solutions of the above equation. we show that the positive equilibrium of the difference equation is globally asymp-totically stable. Our results solve the open problem proposed by Kulenovic and Ladas.Secondly, we consider the difference equation where p, q are positive real numbers and the initial conditions x_k…, x0 are non-negative real numbers. We investigate the periodic character, the invariant interval and the global attractivity of positive solutions of the difference equation. We show that the unique positive equilibrium of the equation is a global attractor with a basin that depends on certain conditions posed on the coefficients.Finally, we investigate the boundedness, the periodic character, the invariant intervals and the global attractivity of positive solutions of the higher order nonlinear difference equation where a,A,B∈[0,∞), the initial conditions x-k,…,x0 are nonnegative, k∈{1,2,…}. We show that the positive equilibrium is globally asymptotically stable, which partially confirms a conjecture proposed by Amleh, Camouzis and Ladas.