ABSTRACT
In this chapter, we present a brief exposition of the qualitative theory of mathematical models involving various types of differential equations. Of the many approaches used to develop qualitative theory, we give the study in terms of the concepts of differential inequalities, comparison principle, iterative techniques using lower and upper solutions, and stability and perturbation theory. In addition to ordinary differential equations, the mathematical models included are integrodifferential equations followed by delay, stochastic, and random; impulsive, fuzzy, set, fractional, matrix, and graph differential equations; and differential equations involving retardation and anticipation. The notions of epsilon approximate solutions and Euler solutions are introduced while dealing with existence results. Various types of stability concepts, such as practical and nonuniform stability, have been included, along with standard stability concepts. The variation Lyapunov method, a combination of nonlinear variation of parameters formula and the Lyapunov method, is also given.
