Basic Theory

Abstract

This chapter is devoted to the basic theory of causal differential equations (CDE) and therefore forms a basis for the remaining chapters. We begin Section 2.2 with causal differential inequalities and prove fundamental results involving strict and nonstrict functional and causal differential inequalities, which are useful to prove existence of extremal solutions and important comparison results. Section 2.3 deals with local existence results, including the existence of maximal and minimal solution of CDEs, which are required for later discussion. While proving the important comparison results in this section, we realize that when imposing various conditions on the causal functions involved, it is enough to suppose such conditions on a suitable subset of the function space, which is a definite advantage in investigating the qualitative properties of solutions of CDEs.