Periodic solutions of nonlinear equations based on extended stability theorems of nonlinear systems

In this paper, we first introduce the concepts of an equilibrium region and a stable equilibrium region. Then, we state and prove conditions under which a nonlinear autonomous system has a stable equilibrium region. It will be shown that these concepts and stability conditions are generalizations of the corresponding concepts of equilibrium points and asymptotical stable equilibrium points and their stability conditions. Using these conditions and Poincare-Bendixson theorem, we discuss conditions for the existence of periodic solutions in a nonlinear system of ordinary differential equations.