This chapter provides methods and suitable criterion that describe the nature and behavior of solutions of difference systems, without actually constructing or approximating them. Since in contrast with differential equations, the existence and uniqueness of solutions of discrete initial value problems is already guaranteed we shall begin with the continuous dependence on the initial conditions and parameters. This is followed by the asymptotic behavior of solutions of linear as well a..o.; nonlinear difference systems. In particular, easily verifiable sufficient conditions are obtained so that the solutions of perturbed systems remain bounded or eventually tend to zero, provided the solutions of the unperturbed systems have the same property. Next we introduce various types of stability and give several examples to illustrate these notions. Then, for the stability of linear systems we provide necessary and sufficient conditions in terms of their fundamental matrices. This includes certain concepts which are of computational importance. This is followed by the comparison between the stability and boundedness of the solutions of linear systems with those of perturbed nonlinear systems. Next we develop a nonlinear variation of constants formula and give its application which establishes its importance. Then, for the linear difference systems we define ordinary and exponential dichotomies, provide necessary and sufficient conditions so that these systems have dichotomies, and use these dichotomies to study the behavior of the solutions of perturbed nonlinear difference systems. Then, we introduce Lyapunov fum:tions and emphasize their importance in the study of stability properties of solutions of autonomous as well as non autonomous difference systems. This is followed by the stability of solutions of several discrete models appearing in population dynamics. Next, assuming certain stability properties of the given difference systems, we shall provide the construction of the Lyapunov functions. These results known as converse theorems arc then used to study the total stability of the solutions
234 Chapter 5
of difference systems. Then, we define the concept of practical stability of the solutions, which goes beyond the classical Lyapunov stability theory and finds some applications in numerical analysis. Finally, we shall introduce the concept of mutual stability of the solutions of two given difference systems, which provides bounds on the solutions in tube like domains.