Existence and uniqueness

This chapter is devoted to the existence and uniqueness and the continuation of solutions of scalar differential equations. We begin with motivating examples and then introduce the notion of Lipschitz continuity. We precisely define what it means for a function to be a solution of an initial value problem and prove that this solution satisfies an integral equation. We begin our discussion by stating and proving the two main theorems regarding Picard's local existence and uniqueness and Cauchy–Peano's existence theorems. We introduce Gronwall's inequality and apply it to prove uniqueness of solutions. We transition into the fixed point theory and prove the existence of solutions on Banach spaces. This allows us to prove two versions of Picard's theorem, global and local versions. Toward the end of the chapter, we prove an existence theorem for linear differential equations. The chapter is ended by proving results regarding continuation of solutions, their maximal intervals of existence, and the dependence of solutions on the initial data.