FORMULAE AND BEHAVIOUR OF SOLUTIONS OF SOME NONLINEAR DIFFERENCE EQUATIONS

Abstract

Difference equations are often used to describe some natural phenomena. There has been recently a rapid increase in investigating such equations. The exact solutions of some equations sometimes cannot be obtained. Therefore, some researchers have studied qualitative behaviors of some equations to understand the future behavior of the relevant model. This thesis discusses the qualitative behaviors of some certain difference equations with various orders. The exact solutions of some difference equations and some systems of difference equations are also obtained. A fourth-order difference equation is discussed in Chapter II. The equilibrium points, stability, boundedness, and periodicity of this problem are intensively investigated. Moreover, some rational difference equations are introduced and investigated theoretically and numerically in Chapter III. In Chapter IV, sixth order difference equations are discussed in terms of their stability, boundedness and solutions. Chapter V investigates the solutions of third order systems of difference equations. Furthermore, fifth order systems of difference equations are solved in Chapter VI. The initial conditions of all problems are assumed to be non-zero real numbers. Two-dimensional figures for such solutions are illustrated to show the behavior of the solutions. This thesis obtains various results such as the local and global stability of equilibrium points of some rational difference equations, the boundedness of the solutions, the periodicity of the solutions, and the analytic solutions of these equations. Two-dimensional figures for the obtained results are presented to verify theoretical results. Some two-dimensional figures are also presented to confirm the theoretical work. The used method can be applied for other nonlinear difference equations.